Barriers to Progress in the Simulation of Viscoelastic Flows of Molten Plastics

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Bell & Howell Information and Leaming 300 North Zeeb Road, Ann Arbori MI 48106-1346 USA 800-521-0600 • • • BARRIERS TO PROGRESS IN THE SIMULATION OF VISCOELASTIC FLOWS OF MOLTEN PLASTICS by Marie-Claude Heuzey Department of Chemical Engineering McGill.University l\'lontreal, Canada September 1999 A Thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment ofthe requirements of the degree of Doctor ofPhilosophy © Marie-Claude Heuzey 1999 1+1 National Ubrary of Canada Acquisitions and Bibliographie Services 395 Wellington Street OttawaON K1A ON4 Canada Bibliothèque nationale du Canada Acquisitions et services bibliographiques 395. rue Wellington OttawaON K1A0N4 C.anada Your 61e Votte rëfërence Our file Norre référence The author has granted a non- exclusive licence allowing the National Library ofCanada to reproduce, loan, distribute or sell copies of this thesis in microfonn, paper or electronic formats. The author retains ownership ofthe copyright in this thesis. Neither the thesis nor substantial extracts from it may be printed or othetwise reproduced without the author' s perrmSSIon. L'auteur a accordé une licence non exclusive permettant à la Bibliothèque nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la forme de microfiche/film, de reproduction sur papier ou sur format électronique. L'auteur conserve la propriété du droit d'auteur qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation. 0-612-55341-8 Canada • • • ABSTRACT Polymer melts exhibit sorne degree of viscoelasticity in rnost industrial fonning operations, and elasticity is particularly important in flows involving an abrupt contraction or expansion in the flow direction. However, the incorporation of a viscoelastic constitutive equation into computer models for polymer processing poses many problems, and for this reason inelastic models have been used aImost exclusively to represent rheological behavior for flow simulation in the plastics industry. In order to explore the limits.. of v:ïscoelastic flow simulations, we used two nonlinear viscoelastic models (Leonov and Phan-ThienfTanner) to simulate axisymmetric and planar contraction tlows and extrudate swell. Their predictions were compared with those obtained using a strictly viscous model (Carreau-Yasuda) and with experimental results. The models are implemented in a modified Elastic Viscous Split Stress (EVSS) mixed finite element formulation. The viscoelastic constitutive equations are calculated using the Lesaint-Raviart method,. and the divergence-free Stokes problem is solved applying Uzawa's algorithme The decoupled iterative scheme is used as a preconditioner for the Generalized Minimal Residual (GMRES) method. Numerical instability was observed starting at quite low elasticity levels. For the converging flows, the predicted flow patterns were in fair agreement with experimental results, but there was a large discrepancy in the entrance p r e s s u r ~ drop. In the case of extrudate swell, the agreement with observation was poor, and c<>nvergence was impossible except at the Lowest flow rate. After exploring the limits of simaLations using viscoelastic models, we conclude that there are serious barriers to progress in the simulàtion of viscoelastic flows of industrial importance. The ultimate source C)f the problem is the melt elasticity, and traditional numerical methods and rheologjcal modeIs do not provide a suitable basis for simulating practical flows. A new approach is required, and we propose that a rule-based expert system be used. • e· À la mémoire de mon père. • • • RÉsUMÉ Les polymères fondus présentent généralement un caractère viscoélastique lors des procédés de transformation industriels. L'élasticité devient particulièrement importante dans les géométries comportant une contraction ou une expansion abrupte en direction de l'écoulement. Toutefois, Pincorporation de lois de comportement viscoélastique pour la modélisation des procédés de transformation des polymères s'avère difficile, et de ce fait des modèles inélastiques ont presque exclusivement été utilisés lors des simulations . d' éc?ulements d'importance industrielle. Afin d'explorer les limites des simulations d'écoulements viscoélastiques, nous avons utilisé deux modèles viscoélastiques non linéaires (Leonov et Phan-Thienffanner) afin de simuler les écoulements dans des contractions plane et axisymmétrique ainsi que le gonflement d'extrudat. Les prédictions des deux modèles ont été comparées à celles obtenues à l'aide d'un modèle strictement visqueux (Carreau-Yasuda) et à des mesures expérimentales. Les lois de comportement sont incorporées à une fonnulation mixte d'éléments finis de type "Elastic Viscous Split Stress" (EVSS) modifiée. La méthode de Lesaint-Raviart sert à la résolution des modèles viscoélastique, et le calcul du problème de Stokes est effectué à l'aide de l'algorithme d'Uzawa. Une approche découplée basée sur l'algorithme "Generalized Minimal Residual" (GMRES) est utilisée pour traiter la non linéarité du problème. Lors des simulations, des instabilités numériques ont été observées à faible niveaux d'élasticité. Pour les géométries convergentes, les écoulements prédits ont montré un accord satisfaisant avec les données expérimentales, sauf les pertes de pression en entrée qui sont largement sous-estimées. Dans le cas du gonflement d'extrudat, la convergence numérique n'a été possible qu'au plus faible débit et le gonflement prédit est largement sous-estimé. • • • Suite à l'exploration des limites des simulations à l'aide de lois de comportement viscoélastiques, nous concluons qu'il existe toujours de sérieux obstacles au progrès de la simulation d'écoulements d'importance industrielle. La source ultime du problème réside dans l'élasticité du polymère fondu, et les méthodes numériques traditionnelles de même que les modèles rhéologiques existants ne comportent pas la solution au succès des simulations d'écoulements d'intérêt pratique. Une nouvelle approche est requise, et nous suggérons l'utilisation d'un système expert afin de guider l'évolution des calculs numériques. • • REMERCIEMENTS En premier lieu j'aimerais remercier mes directeurs de thèse, les professeurs John M. Dealy et André Fortin, qui ont toujours su m'encourager et faire preuve d'une patience infinie tout au long de ce projet. Je leur suis aussi fort reconnaissante pour leur générosité et l'ambiance agréable qu'ils ont su établir dans leur groupe de recherche respectif: tant à l'Université McGill qu'à l'École Polytechnique, ce qui a passablement allégé les trois années qu'ont duré mes études doctorales. l'inclus aussi dans mes remerciements le personnel technique des deux universités qui a rendu possible la mise à terme de ce projet: spécialement fvflvI. Alain Gagnon, Walter Greenland, Charles Dolan et Luciano Cusmish de l'Université McGill, ainsi que Benoît Forest de l'École Polytechnique. Je suis fort reconnaissante envers Mazen Samara et le Prof. Kama! pour avoir permis l'utilisation de l'appareillage de mesure du gonflement d'extrudat. Aussi un immense merci au personnel du CE:MEF à Sophia-Antipolis en France, particulièrement Lucia Gavrus, Erik Brotons, Bruno Vergnes et le Prof. Agassant qui m'ont accueillie au centre de recherche et assistée dans les mesures de l'écoulement plan. Je garde un souvenir impérissable de la Côte d'Azur. Merci aussi à tous mes collègues de travail pour leur camaderie et entraide, particulièrement Seung Oh Kim, François Koran, Stewart McGlashan et ma grande amie Paula Wood-Adams de l'Université McGill, ainsi que Alain Béliveau, Alain Lioret et Steven Dufour de l'École Polytechnique. Finalement, je remercie spécialement mes proches dont ma famille et mon compagnon Luc pour leur patience, leur amour et encouragement qui m'ont permis d'aller au bout de cette étape. • • l TABLE OF CONTENTS List of Tables. •••••••••.•••••••••.••••••••••••••••••••••••••••••••••••••••••••••.••.•••..•••••••••••••.•••••••.••••••••••••••••••• V List ofFigu.res•••••••••••••.••••••••••••••••••••••••••••••••••••••••••••••••••.•••..•.•.••••••••.••••.••••••••••••••••••••••••••'VI 1. Introduction 1 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 3 2.1 CONSTITUTIVE EQUATIONS FORPOLYMERMELTS 3 2.1.1 Viscoelastic FlowPhenomena 3 2.1.2 Definitions ofTensors and Materia! Functions 5 2.1.3 Geli.eralized Newtonian Models 7 2.1.4 Viscoelastic Models 8 2.2 MATHEMATICAL TREATMENT OF VISCOELASTIC FLOW 12 2.2.1 Set ofEquations to Solve and Inherent Difficulties 12 2.2.2 Numerica! Approaches to Viscoelastic Simulations 14 2.2.3 Boundary Conditions 15 2.3 ALGORITfIM DEVELOPMENT .•..••••.•.••.•••....••...•..••..•...••...•..•...•..........•......•.....•..••..•. 16 2.3.1 Failure ofTraditional Numerical Methods 16 2.3.2 First Successful Approach 17 2.3.3 Economical Storage Techniques 19 2.3.4 Projection Technique 20 2.3.5 Influence ofTheoretical Wark 22 2.4 COMPARISONS WITH EXPERIMENTS 23 2.4.1 First Comparisons: Boger Fluids 23 2.4.2 Rheological Models Frequently Used in Simulations 24 2.4.3 Determination ofModel Parameters 25 2.4.4 Planar Abrupt Contraction Flow 26 2.4.5 Axisymmetric Abrupt Contraction Flow 28 2.4.6 Axisymmetric Converging Flow 29 2.4.7 Extrudate Swell 30 2.4.8 Other Complex Flows 34 2.5 CONCLUSIONS 35 • II 2.5.1 Limitations ofNumerical Simulations ofViscoelastic Flows 35 2.5.2 Kinetic Theory Models: A Promising Approach? 36 3. PIa.n of the Resea..reh 43 3.1 OBJECflVES •••••••••••.••••.•........•..••..•.•....•..........•.•...••...•...........•.............•.•...........•.... 43 32 MonELS SELEcrED FORTInS RESEARCH 43 3.2.1 Carreau-y asuda Model 43 3.2.2 Leonov Model 44 3.2.3 Phan-Thienffanner Model 46 4. Rh.eological Characterization 50 4.1 MA1"ERIALS USEO •••••••.••••.••.•••••••.••••••••.•.•••.••••••••••••••••.••••••...••..••...••.•••.••••...•.•.•...•• 50 4.2 EXPERIMENTAL METHons 50 4.2.1 Small Amplitude Oscillatory Shear _ 50 4.2.2 Steady Shear 51 4.2.3 Extensional Flow 52 4.3 LINEAR-Low DENSITY POLYETIIYLENE .•.....••.•....•.•...........................•...•........•..... 53 4.3.1 Storage an.d Loss Moduli 53 4.3.2 Viscosity' 55 4.3.3 Tensile Stress Growth Coefficient. 57 4.4 HIGH-DENSITY POLYETHYLENE••..............•.•...........•.•........................ _ 59 4.4.1 Storage an.d Loss Moduli 59 4.4.2 Viscosity 60 4.4.3 Tensile Stress Growth Coefficient. 63 5. DeterDIination ofModel Parameters 65 5.1 LINEAR-Low DENSITY POLYETHYLENE 65 5.1.1 Viscous Model 65 5.1.2 Leonov Model Parameters 66 5.1.3 Phan-ThienITanner Param.eters 68 5.1.4 Viscosity 68 5.1.5 Tensile Stress Growth Coefficient 71 5.1.6 Prediction ofNormal Stress Differences 72 • 5.1.7 Prediction ofElongational Viscosity 74 • • ID 5.2 HrGH-DENSIITPOLYETHYLENE•.•••.••••..•.•.•••.•.••.••..•.••..•..•..••...••.......•...•••••••••••••••••. 75 5.2.1 Viscous Model 75 5.2.2 Leonov Model Param.eters 76 5.2.3 Phan.-Thienffanner Parameters 77 52.4 Viscosity' 77 5.2.5 Tensile Stress Growth Coefficient 79 5.2.6 Prediction ofNormal Stress Differences 80 5.2.7 Prediction ofElongational Viscosity 82 6. Numerica.l Method .•.•..•..•...•.•.•..•••...••...•.....•.....••.•...............••...••.•.......•...........••...... 84 6.1 DESCRIPTION OF THE PROBLEM ••••••.•••••••••.••••.•..•••••.••..............•.•....•......••...•...•.••••• 84 6.1.1 Four-Field Mixed Formulation 84 6.1.2 Free Surface 86 6.1.3 Weak Form.ulation 87 6.2 DISCRETIZATION 88 6.3 N ~ R I C A L SOLUTION.•.•.•.••.•••••••••.••.•••.••....•••..••.....•........•..•............•.•••.•.•••.•••..... 90 6.3.1 Stokes Problem 90 6.3.2 Discontinuous Galerkin Problem 91 6.3.3 Gradient Tensor ofVelocity 94 6.3.4 Resolution ofthe Global System 95 7. Planar Abrupt Contraction Flowof High-Density Polyethylene ............•......••.• 99 7.1 EXPERIMENTAL METRons 99 7.1.1 Die Geometry 99 7.1.2 Laser-Doppler Velocimetry 100 7.1.3 Flow-Induced Birefringence 102 7.1.4 FlowRate-Pressure Curve 104 7.2 NUMERICAL PROCEDURE•••.•••••••••••.•••.•••••••...•••••••....•............•....•...................•..... 105 7.2.1 Meshes 105 7.2.2 Viscoelastic Converged Solutions 106 7.3 TOTAL PRESSURE DROP 107 7.4 VELOCITY 110 7.5 BIREFRIN'GENCE 112 • • • IV 7.6 DISCUSSION •...•...•....•...•..•..•...................••.•••...........•...••.•..•........•.....•....••.....•.•.•..• 118 8. Axisymmetric Entrance Flow and Extrudate Swell of Linear Low-Density Polyethylene 122 8.1 E:x:PERThŒNTAI.MEïHOD•••.•.••••••••••:••••••••.•••.•.•...•....••••••H •••••••••••••••••••••••••••••••••••• 122 8.1.1 Capillary Extrusion 122 8.1.2 Extrudate Swell Measurements 122 8.2 NU1ŒRICAL PROCEDURE.••........•••.•••••..•.•••.•••.•••••.••.•.••••...•..•••••••••••••.•••.•.••••••••...• 125 8.2.1 Meshes 125 8.2.2 Viscoelastic Converged Solutions 126 8.3 TOTAL PRESSURE DROP•..............••.•...•...•...•...•.•................•........................•...•.••. 127 8.4 EN1RANCEPRESSURE DROP 130 8.5 EXIRUDATE SWELL .•••.••••.•••••.•••••.•••.••.•.•••••••••.•••••.•.•••.•••••.............••.•.•...........•.•.. 133 8.5.1 Results 133 8.5.2 Comparison with. Simulations 135 8.6 DISCUSSION 137 9.. Alternative Approaches 140 9.1 VrSCOrvIETRIC FLOW..... ..•...•......•.••••.•••.•..•. .••.....•........... 140 9.2 POWER-LAW MODELS FOR ELONGATIONAL VISCOSITY 142 9.3 APPROXIMATE PREDICTIONOF EXTRUDATE SWELL 143 9.4 RULE-BASEO FLowSThfULATION ••.•••.•..•••••••••...•••••.••••..•••.•••.•................•.....•...... 143 9.4.1 Fundamentals of Knowledge-Based Expert Systems 145 9.4.2 Application to Flows 146 9.4.3 Incorporation ofüther Phenomena : 148 10. Conclusions and Recommendations 152 10.1 CONCLUSIONS 152 10.2 ORIGINAL CONTRIBUTIONS TO KNOWLEDGE .•..•...•.•.••••.•.•..•••.•••.•..•...............•. 153 10.3 RECOMMENDATIONS FORFU1URE WORK•............•.••........•...•......................••.. 154 Bibliography 155 Appendix 167 • Table 2-1 Table 2-2 Table 2-3 Table 2-4 Table 2-5 Table 2-6 Table 2-7 Table 2-8 Table 4-1 Table 4-2 Table 4-3 Table 4-4 Table 5-1 Table 5-2 Table 5-3 Table 5-4 Table 5-5 Table 5-6 Table 7-1 Table 7-2 Table 7-3 Table 7-4 Table 7-5 Table 8-1 Table 8-2 Table 8-3 • v LIST OF TABLES Numerical Simulations ofPlanar Abrupt Contraction Flow 27 Numerical Simulations ofAxisymmetric Abrupt Contraction Flow 29 Numerical Simulations ofAxisymmetric Converging Flow 30 Numerical Simulations ofAnnular Extrudate Swell 31 Numerical Simulations ofAxisymmetric Extrudate Swell. 32 Comparisons between Predicted and Measured Extrudate Swell (long die) 33 Comparisons between Predicted and Measured Extrudate Swell (orifice) 33 Numerical Simulations ofPlanar FlowAround a Cylinder 34 Experimental Materials 50 Compression Molding Procedure at ISO°C 51 Discrete Relaxation Spectra for the LLDPE at 150°C 54 Discrete Relaxation Spectra for the HDPE at Isaoc 60 Carreau-Yasuda Model Parameters for the LLDPE at 150°C 65 Leonov Model Parameters for the LLDPE at ISaoC 65 Carreau-Yasuda Model Parameters for the LLDPE at 150°C 65 PIT Model Parameters for the HDPE at 2aOoe 65 Leonov Model Parameters for the HDPE at 20aoC 65 PIT Model Parameters for the HDPE at 2aOoe 65 Geometrical Characteristics ofthe Die 100 Velocimetry Measurements Locations 101 Mesh Characteristics for 2D Flow Simulations 105 Converged Solutions with the PIT Model (max=ï2) 107 Converged Solutions with the Simple Leonov Model (max=12) 107 Mesh Characteristics for Axisymmetric Flow Simulations 125 Converged Solutions with the PIT Model (max=7) 127 Converged Solutions with the Simple Leonov Model (max=7) 127 • • VI LIST OF FIGURES Figure 4-1 Dynamic Moduli ofthe LLDPE at 150°C 54 Figure 4..2 Fits ofthe Discrete Spectra for the LLDPE at 150°C 55 Figure 4..3 Entrance Pressure Drop by Two Methods for the LLDPE at 150°C 56 Figure 4..4 FlowCurve by Two Rheometers for the LLDPE at 150°C 56 Figure 4..5 Complex Viscosity and Viscosity ofthe LLDPE at 150°C 57 Figure 4-6 Transient Elongational Stress for the LLDPE at 150°C 58 Figure 4-7 Dynamic Moduli ofthe HDPE at 200°C 59 Figure 4-8 Fits ofthe Discrete Spectra for the HDPE at 200°C 60 Figure 4-9 FlowCurve by Three Rheometers for the HDPE at 200°C 61 Figure 4-10 Complex Viscosity and Viscosity ofthe HDPE at 200°C 62 Figure 4-11 FirstNormal Stress Difference ofthe HDPE Shifted from 190° to 200°C .. 62 Figure 4-12 Transient Elongational Stress for the HDPE at 200°C 63 Figure 5-1 Viscosity Fit of Carreau-Yasucla Equation for the LLDPE at 150°C 66 Figure 5-2 Fit ofViscosity with Leonov and PIT Models for the LLDPE at ISOaC 69 Figure 5-3 Viscosity Curve ofPIT Model with a.=1/8 for the LLDPE at I50 a C 70 Figure 5-4 Flow Curve Predicted by the PIT Model with a.=0 71 Figure 5-5 Fit ofTensile Stress Growth Coefficient for the LLDPE at IS0 a C 72 Figure 5-6 Prediction ofFirst Normal Stress Difference for the LLDPE at ISOaC 73 Figure 5-7 Ratio of Second to First Normal Stress Difference for the LLDPE at 150°C 74 Figure 5-8 Prediction ofElongational Viscosity for the LLDPE at 150 a C 75 Figure 5-9 Viscosity Fit of Carreau-Yasuda Equation for the HDPE at 200°C 76 Figure 5-10 Fit ofViscosity with Leonov and PIT Models for the HDPE at 200 0 e 78 Figure 5-11 FlowCurve for the HDPE at 200°C Predicted by the PTT Model without the Viscous Contribution 78 Figure 5:-12 Viscosity Curve ofPTI Model with a.=1/8 for the HDPE at 200 a C 79 Figure 5-13 Fit of Tensile Stress Growth. Coefficient for the HDPE at 200°C 80 • • VII Figure 5-14 Prediction ofFirstNormal Stress Difference for the HDPE at 200°C 81 Figure 5-15 Ratio of Second ta First Normal Stress Difference for the HDPE at 200°C 81 Figure 5-16 Prediction ofElangational Viscosity for the HOPE at 200°C 82 Figure 6-1 The P2+ - Pl - Pl - Pl elements for variables 1:!h,[1h, ~ and@t (and Fh). •....... 89 Figure 6-2 Discontinuity Jump at the Interface ofTwo Elements 93 Figure 7-1 Geometry ofthe Planar 8: 1 Contraction 99 Figure 7-2 Experimental Set-Up ofLDVMeasurements 101 Figure 7-3 Positions ofthe Velocity Profile Measurements 102 Figure 7-4 Experimental Set-Up for the Measurement ofFlow-Induced Birefringence 103 Figure 7-5 Characteristic Curve ofDie-Extruder at 200°C 104 Figure 7-6 Comparison ofExtruder Data with Viscosity Curve ofHDPE at 200°C 105 Figure 7-7 Mesh Planar_1. 106 Figure 7-8 Mesh Planar 2. 106 Figure 7-9 Comparison ofPredicted Total Pressure Drop with Experimental Values. 108 Figure 7-10 Influence ofDiscrete Spectrum. on Prediction ofTotal Pressure Drop 109 Figure 7-11 Mesh Independence for the Prediction of Total Pressure Drop 109 Figure 7-12 Velocity Profile (Reservoir, position 1) 111 Figure 7-13 Velocity Profile (Reservoir, position 2) III Figure 7-14 Velocity Profile (Reservoir, position 3) 111 Figure 7-15 Velocity Profile (Channel, position 1) III Figure 7-16 Velocity Profile (Channel, position 2) 111 Figure 7-17 Velocity Profile (Channel, position 3) 111 Figure 7-18 Mesh Independence (Reservoir, position 3) 112 Figure 7-19 Influence ofSpectrum (Reservoir, position 3) 112 Figure 7-20 Evaluation ofâ max on the Axis ofSymmetry 113 Figure 7-21 Calculation ofthe Stress Optical Coefficient.............................................. 114 Figure 7-22 Comparison ofMeasured and Calculated Birefringence (Leonov Model, ra = 14.4 5- 1 ) .••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••.•••••••.••.••••••.••••••.••••••••••••••••• 115 Figure 7-23 Comparison ofMeasured and Calculated Birefringence (pTT Model, ra =14.4s- 1 ) •••••••••••••••••••••••••••.••••••••••••••••••••••••••••••••••••.•••••••.••••••.••.•••••••••••••••••••••••• 115 • • vm Figure 7-24 Comparison ofMeasured and Calculated Birefringence (pTT Model, i a =56.4S-I)••...•....•.•.•...•.....••..••••••••••••••.••••.•.••.•.•..•.•.•..•.•.•.••••••••.•••••.•••••.••••••.•••••••• 116 Figure 7-25 Predicted and Measured Birefringence on the Axis of Symmetry (Ya =4.6 S-I)••.••••..•.••••..••••.•.•.••••••••••••••.•••••••••••..•.....•.•..•••.•.•..•.•••••........••••......•••••••• 117 Figure 7-26 Predicted and Measured Birefringence on the Axis of Symmetry (fa =9.4 S-I)••.•.••.....•.••.••••.•••••..•••••••••••...••.•.•..•••......•••..•.•.••••••••..•.•......•••.•.•.••.•.••.•.• 117 Figure 7-27 Predicted and Measured Birefringence on the Axis ofSymmetry (fa =14.4 5- 1 ) •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••.••••••••••••••• 117 Figure 7-28 Predicted and Measured Birefringence on the Axis ofSymmetry (ra =56.4 5- 1 ) 117 Figure 7-29 Influence of Spectrum on Birefringence (ia = 28.7 S-1)•.••..••••..•••••••••••••••• 118 Figure 7-30 Influence of Mesh on Birefringence Cia = 47.7 S-I).•...............•........•......• 118 Figure 7-31 Influence of tlx on Velocity Profile (Reservoir, position 2) 119 Figure 7-32 Influence oftlx on Velocity Profile (Reservoir, position 3) 119 Figure 7-33 Extension Rate on the Axis ofSymmetry (ia = 56.4 S·l) 120 Figure 8-1 Capillary Rheometer 123 Figure 8-2 Experimental Set-up for Extrudate Swell Measurements 124 Figure 8-3 Mesh Axi_1 125 Figure 8-4 Mesh Axi_2 126 Figure 8-5 Mesh Swell 1..... 126 Figure 8-6 Ultimate Swell Predicted by the SL.'"D.ple Leonov Madel 127 Figure 8-7 Comparison ofPredicted Total Pressure Drop with Experimental Values. 128 Figure 8-8 Influence ofDiscrete Spectra on Prediction ofTotal Pressure Drop 129 Figure 8-9 Mesh Independence on Prediction ofTotal Pressure Drop 129 Figure 8-10 Comparison ofPredicted Entrance Pressure Drop with Experiments 131 Figure 8-11 Extension Rate on the Axis ofSymmetry (ra =88.8s- 1 ) ••••••••••••••.•••••••••••• 131 Figure 8-12 Influence ofDiscrete Spectra on Prediction ofEntrance Pressure Drop 132 Figure 8-13 Mesh Independence on Prediction ofEntrance Pressure Drop 132 Figure 8-14 Total Swell and Oil Swell as a Function ofTime 134 • • IX Figure 8-15 Net Swell as a Function ofTime after Oil Swell Substraction 135 Figure 8-16 Calculated Interface with PIT Model (truncated spectrum, mesh Swell_l) 136 Figure 8-17 Measured and Predicted Ultimate Swell. 136 Figure 9-1 Expert Systems vs. Conventional Programs 145 Figure 9-2 Essential Elements ofa Possible Rule-Based Expert System for Viscoelastic Flow Simulation 149 • 1. Introduction • The broadening of the range of application of plastic parts requires improved final product quality, in terms of both mechanical properties and exterior appearance. Standards for product quality have been raised, while increased competitiveness requires reductions in produet development time and unit cost. However, plastic processing continues to involve long product development cycles, high tooling costs, low process yields and product inconsistencies. Numerical simulation can reduce the time required to develop newprocesses and machines and can aid process optimization by: 1. extrapolation or scale-up ofdesigns, 2. exploration ofthe effects ofindividual variables, 3. sensitivity and stability analysis, aIl this being achieved at a lower cost than ifcarried out experimentally. An important issue in polymer processing is melt rheology, which deals with the relationship between stress and strain in defonnable Polymeric liquids are rheologically complex, since they exhibit both viscous resistance to deformation and elasticity. An understanding oftheir rheological behavior is essential in the development, production and processing of polymerie materials. However, flow phenomena arising from viscoelasticity are very complex, and these cause severa! problems in the mathematical modeling of certain forming processes. Inelastic models May be adequate to simulate flows in which shear stresses are predominant, but if elongational effects are aIso important, the viscoelastic nature ofpolymerie materials can not be ignored. Plastics can be molded, extruded, formed, machined and joined into various shapes. In severa! processing operations such as fiber spinning, thermoforming and injection and blow molding, the polymer undergoes bath shear and elongation. Viscoelasticity not _..._- ,.) ooly affects melt flow but aIso plays a major role in the development of the microstructure and physicaI properties of the final plastic part. However, since the incorporation of a viscoelastic constitutive equation into a computer model for polymer processing poses many problems, inelastic modeIs have been used aImost exclusively to represent rheological behavior for flow simulation in the plastics industry. • Chapter 1. Introduction 2 • Moreover, aIl previous attempts to find a general viscoelastic fluid model applicable to aIl classes of flow and capable of realistic predictions have failed. The motivation for this research was first to the models for the numerical simulation ofthe flow of second objective was to establish the cause of these limitations and sJlggest methods for overcoming them. To / achieve these objectives, two complex were subjected to both numerical simulation / and experimental observation. // //// / In the next chapter, the state-of-the-art ofnumerical simulation ofthe f10w of viscoelastic materials is In"Coopter 3, we present the plan of the research and the ----- • ----..:.. Chapter 4 describes the materials used for the -_-=-:.::..=.:::==:::=:::------- experimental study and presents their rheological properties, while Chapter 5 gives the method used to determine the various model parameters. In Chapter 6 we describe the numerical procedure used in the simulations. Experimental methods and results are presented in Coopter 7 for the planar abrupt contraction flow, and in Chapter 8 for the axisymmetric entrance tlow and extrudate swell. Chapter 9 presents alternatives to full viscoelastic simulations and proposes a rule-based system for flow simulation. FinaIly, conclusions, original contributions to knowledge and recommendations for future work are given in Coopter 10. • •• 2. Numerical Simulation of Viscoelastic Flow: State-of-the-Art 2.1 Constitutive Equations for Polymer Melts Polymer melts behave very differently from purely viscous materials, because in addition to viscous resistance to deformation, they exhibit elasticity. The reason why polymerie liquids are non-Newtonian is related to their molecular structurel. Polymer molecules can he represented by long chains with many joints allowing relative rotation of adjacent links, and -the presence of this large number of joints allows many different configurations and makes the molecule flexible. At rest, there will he a unique average value ofthe end-to-end distance, R, for the molecules of a given polymer. When the melt is deformed, this average length is altered, and when the deformation is stopped, Brownian motion will tend to retum R ta its equilibrium value. This is the molecular origin ofthe elastic and relaxation phenomena that occur in polymeric liquids, which are said to have a "fading memoryfl . In this section, we will describe briefly sorne flow phenomena associated with viscoelasticity and present several constitutive equations that have heen developed for polymerie fluids. 2.1.1 Viscoelastic FlowPhenomena A number of phenomena encountered in the flow ofpolymers cannot be explained on the basis of purely viscous behavior. We describe a few of these phenomena, and a more thorough discussion on the subject can be found in references 2, 3 and 4. Weissenberg Effect This effect, aIso called rod climbing, was reported by Weissenberg in 1947 s • It is the tendency of an elastic liquid to rise up around a rotating shaft partially immersed in il. This is very different from the behavior of a Newtonian fluid, which will flow towards the walls due to inertia forces. In the absence of inertial effects, the free surface of a Newtonian fluid would remain tlat. Weissenberg deduced that the phenomenon was caused by an elastic normal stress acting along the circular streamlines. This normal stress exists in aIl shear flows of polymerie liquids except at very low shear rates. Dealy and Vu studied rod climbing in molten polymers 6 • • Chapter 2. Numericai Simulation ofViscoelastic Flow: State-of-the-Art 4 • Extrudate Swell Extrudate swell is defined as the increase inthe diameter ofa stream as it exits adie. For a Newtonian fluid at lowReynolds number (Re), axisymmetric extrudate swell at the exit of a capillary is of the order of 12-13%, and it decreases as Re increases, even becoming negative. For a viscoelastic liquid, this extrudate swell ean be as high as 200-300%. This phenomenon is related to normal stress differences and inereases with inereasing flow rate. Due te the fluid' s memory, the swell depends on the flow at the entrance to the capillary, but for a sufficiently long capillary, this effect beeomes negligible because it is a "fading memory". Tubeless Siphon In 1908, Fano 7 reported a remarkable manifestation offluid elasticity. In an experiment where he immersed the tip of a capillary tube in a beaker filled with a biological macromolecular fluid, he began withdrawing liquid from the beaker through the tube. However, even after the fluid level in the beaker dropped below the tip of the capillary t tube, the tluid continued to flow upwards to the tip ofthe no-Ionger-submerged capillary. This phenomenon is known as a tubeless siphon and results from the large tensile stress that the fluid can support due to its elasticit)? Other flow phenomena such as vortex formation in contraction flows are aIso related to the elasticity ofpolymerie fluids. Simple Flows For a Newtonian fluid, the viscosity " is independent of shear rate, but for a polymerie fluid " decreases with increasing shear rate; Le. these fluids are "shear tbinning". But shear tbinning alone does not imply that a fluid is viscoelastic. A stronger indication of viscoelasticity is the existence of normal stress differences in shearing deformations, which give rise, for example, ta the Weissenberg effect • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 5 Another manifestation of viscoelasticity is stress relaxation. Stresses will persist in these materials after deformation has ceased, and the duration ofthe time during which stresses persist is called the relaxation rime of the material. Recoil, the reverse defonnation that occurs when the tluid is suddenly relieved of an extemally imposed stress, is anotber manifestation of viscoelasticity_ A realistic viscoelastic constitutive equation must be able to predict all the phenomena mentioned above. 2.1.2 Definitions of Tensors and Material Functions We present here definitions ofthe tensors and rheological material functions that are used throughout this thesis. The reader is referred to Dealy and Wissbrun 1 for a thorough discussion ofthese quantities. Tensors The strain rate or rate ofdefonnation tensor D is given by: Eqn.2-1 where Mis the velocity vector. The extra stress or viscous stress tensor ~ is defined as follows: Eqn.2-2 • wherep is the pressure, cr is the Cauchy stress tensor and [ is the unit tensor. Useful measures of strain in polymer rheology are the Cauchy tensor C if Ctl,t 2 ) and the Finger tensor B if (t 1 ,t 2 ). The Finger tensor is the inverse of the Cauchy tensor and can aIso be written ci;t (t 1 '(2). The first, second and third invariants of the Finger tensor are defined as follows: • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 6 Eqn.2-3 Eqn.2-4 Eqn. 2-5 Material Functions In steady shear, the following materiaI functions (the ''viscometric'' functions) are defined (D 12 == y): Eqn.2-6 Eqn.2-7 Eqn.2-8 (,,;) _ ~ 1 2 7lv =-.- r (viscosity) (fust nonnal stress difference) (second normal stress difference) Sometimes the normal stress coefficients are used: Eqn.2-9 Eqn.2-10 \fi = Ni 1 - .2 r (fust nonnal stress coefficient) (second normal stress coefficient) Start-up and cessation of shear flow is denoted by a plus or minus sign, respectively: • Eqn.2-11 (start-up flow) Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-.the-Art • Eqn.2-12 (cessation oftlow) 7 Insimple uniaxial elongation, we defined a principal stretching stress crE: Eqn.2-13 The elongational viscosity is defined as follows for steady uniaxial elongational tlow: Eqn.2-14 ( ') (JE TJE & =-.- e For start-up of elongational f1ow, we have the tensile stress growth coefficient: Eqn.2-15 (j(t,ë) ë • 2.1.3 Generalized Newtonian Fluid Models The fust requirement of a constitutive equation to be used for flow simulation is that it describes adequately the rheological behavior that govemthe flow of interest. For certain restricted flows, there are general constitutive equations that relate the stress tensor to the deformation history in terms of a few material parameters or functions. For example, viscometric flows are flows that, from the point of view of a material element, cannot be distinguished from simple shea1. Although the shear rate may vary from particle to particle, the defonnation history of each tluid element is constant as it flows along a streamline. The extra stress tensor for viscometric flows is described by the Criminale-Ericksen- Filbey (CEF) equation 4 • 8 . If it is fonnulated using the upper-convected derivative of the v rate ofdeformation tensor D, it can be written as follows: Eqn.2-16 where the viscosity and first and second normal stress coefficients are functions of IID, the second invariant ofthe rate ofdeformation tensor D: • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 8 Eqn.2..17 Ifone is primarily interested in the shearing component of the stress tensor, assuming that normal stresses do not affect the pressure gradient, then Eqn. 2-16 can he simplified to: Eqn.2-18 A material that behaves in this manner is called a "generalized Newtonian fluid". It is appropriate to use this equation in the conservation of momentum equation to compute the pressure drop and viscous dissipation in a tube or channel with constant cross-section. Eqn. 2-18 has the tensorial character of a Newtonian fluid but can exhibit shear tbinning. It is regarded as a purely viscous constitutive equation, since it does not predict normal stress differences, which are manifestations of elasticity. A popular form for the viscosity as a function ofIID is the Carreau-y asuda equation 9 : Eqn.2-19 • which bas Newtonian behavior with viscosity 110 at low shear rates and power-law behavior at high shear rates. Eqn. 2-18 is often used in complex polymeric flows for which it is not valid for polymerie liquids, but more realistie constitutive equations almost invariably lead ta difficult numerieal problems 2 • This will be discussed in more detail in section 2.2.1. 2.1.4 Viscoelastic Models Sïnee the viseometrie tlow simplification is not always approprlate, a truly viscoelastic model must often he used. Viscoelastie behavior may be linear or nonlinear. Linear viscoelastic behavior is observed ooly when the deformation is very small or slow, and is therefore not directly relevant ta the behavior of molten polymers in processîng operations where deformations are always large and rapide Linear viscoelastic behavior is independent of the kinematics of the deformation and the magnitude of past strains. These simplifications make possible the addition of the effects of successive deformations. The Boltzmann superposition principle (Eqn. 2-20) describes weIl the linear viscoelastic response ofa material to an arbitrary strain history: • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-ot:the-Art 9 Eqn.2-20 t ~ ( t ) = 2 fG(t-t')D(t')dt' • where ~ is the extra stress tensor, G(t-t') is the relaxation modulus, and D(t') is the rate of deformation tensor at time t'. Conversely, in the nonlinear regime the response to an imposed defonnation depends on the sïze, the rate and the kinematics of the deformation. Therefore, it is not possible to measure a response in one type of deformation and use the result to predict the response in another type of deformation, unless the rate, magnitude and kinematics of the deformation are all the same in both cases 1 • Two approaches have been used to fonnulate nonlinear viscoelastic constitutive equations. The:first one is based on the derivation of molecular theories for melts and dilute solutions together with the use of statistical mechanics, but this approach does not always yield closed fOIm constitutive models. Because of the mathematical complexity involved, many simplifying assm.'nptions must be made, and this limits the practical value ofthe molecular approach. Examples of such theories include the Rouse model for dilute solutions10, and the Doi_Edwards ll • 1 2.13 without the independent alignment assumption and Curtiss-Bird models l4 ,15 for entangled melts, the last two being based on the reptation concept. The second method is an empirical approach based on continuum mechanics concept 1 • Complications in building the model arise from the involvement of tensor-valued quantities and the fact that the response of the material depends on strains imposed at previous times. These complications make it difficult to establish an acceptable form. of nonlinear constitutive equation. Larson 2 has presented an exhaustive review of closed form continuummodels. • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 10 At the present time, there is no universal theory that describes nonlinear rheological phenomena. Nevertheless, severa! models have been used in numerical simulations, and these are either differential or integral constitutive equations. These are reviewed in the following sub-sections. Recent numerical approaches called micro-macro formulations are based on kinetic theories rather than closed form constitutive models16.17.18.19,20,21. These are dèscribed in section 2.5.2. 2.1.4.1 Differentiai Viscoelastic Models Many differential modeIs can be written in the following general form: Eqn.2-21 Here, Â. is a relaxation time and Tl=ÂG is a viscosity. The symboll:: represents a model- dependent tensor function, and the operator E9 stands for the convected Gordon- Schowalter derivative 22 , which is a combination of the upper V and lower Il convected derivatives: Eqn.2-22 Eqn.2-23 Eqn.2-24 v Dr T = Vu Vu = Dt. - = = - â Dr.. T Dt=- -= • where D IDt is the material time derivative and T indicates the transpose. In order to obtain a more realistic representation of the behavior of polymer liquids, Eqn. 2-21 can be modified to include a discrete spectrum of relaxation times Â.b where Nis the number ofmodes: • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art Il Eqo.2-25 Eqo.2-26 N = 1 i=l= The simplest differential model is the upper-convected Maxwell model, with Li;-1, ; 2 and constant Tl. The addition of a Newtonian viscosity Tls yields the Oldroyd-B mode!. However, these constitutive equations do not provide a realistic description of polymerie fluids. More complex models, such as those of Giesekus 23 , LeonoV24 and Phan- (PT!) have been proposed to give better agreement with data, but identifying the correct model and the suitable parameters remains a difficult task. 2.1.4.2 Integral Viscoelastic Models For the purpose of expressing integraj constitutive equations, deformations are measured relative to the fluid configuration at the present time t. Single integral constitutive equations give the extra stress at a fluid particle as a time integral of the defonnation history, as shown by Eqn. 2-27: Eqo.2-27 Eqo.2-28 l = ( ') dG(t-t') (-Ct-t')] m t - t = = LJ- exp ---.,;;---.;... dt' i=1 ;Li  i • where we consider a time integral taken along the particle path parameterized by the time t '. The factor m(t-t? is the time dependent memory function that incorporates the concept of fading memory. This means that the defonnation experienced by a fluid element in the recent past contributes more to the current stress than earlier deformations 26 • Again, the extra stress can he expressed as a SUffi of individual contributions (Eqn. 2-25). Fis a model-dependent tensor that describes the defonnation ofthe fluide • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 12 One of the simplest nonlinear integral constitutive equations is the Lodge rubberlike liquid modet2 7 , in which F=lb where B is the Finger tensor. This madel is equivalent to the upper-convected Maxwell differential model with N=-l. Other madels of the integral type that have been found to give a better fit ta data are of the K-BKZ type 28 ,29. In these models, the memory function depends on the strain as weIl as on time. The strain dependence ofthe memory function is called the damping function, h(1},12). In this case, the memory function is said ta he "separable" or factorable: Eqn.2-29 M[(t -t'),1 1 , [2] = met - t')h(/l '/2) • One of the K-BKZ model with a particu1ar form of damping function was proposed by Papanastasiou et al. 30 and fitted weil experimental data for simple shear and simple elongational flows. 2.2 Mathematical Treatment of Viscoelastic Flow 2.2.1 Set of Equations to Solve and Inherent Difficulties The simulation of the flow of viscoelastic materials involves the solution of a set of coupled partial differential (or integral-differential) equatians: conservation of mass, conservation of momentum and a rheological equation. If the flow is compressible or non-isothermal, conservation of energy and an equation of state for density are aIso required. For sïmplicity, we assume that the f10w is isothermal and incompressible. We present the goveming equations using a one-mode differential viscoelastic model, since the majority of the numerical methods have been developed for differential models. For incompressible, isothermal flows the conservation equations for mass and momentum are: • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 13 Eqn.2-30 Eqn.2-31 Du p-==-Vp+V·r+f Dt = - where p denotes a constant density, p is the pressure, ~ is the extra stress tensor andfis a body force. For a Newtonian fluid and in the absence of inertia forces, this set of equations is referred to as the " Stokes problem" (Eqn. 2-30 and Eqn. 2-31). The extra stress tensor is sometimes split into viscous and viscoelastic contributions, for example inthe Oldroyd-B fluid: Eqn.2-32 N ~ == 2TJsD + L'r've.i i=l= where ve represents the viscoelastic character, and 115 is a constant viscosity that can he a solvent viscosity for a polymer solution or a fraction of the zero-shear viscosity for a polymer melt: Eqn.2-33 N 7]0 =TJs + LTJi i=1 This splitting is often arbitrary but it bas a strong stabilizing effect on most numerical schemes. The last equation is the rheological model, an example of which is the upper- convected Maxwell model: Eqn.2-34 'il '!::+Â'!::-271DVD =0 • The mathematical type of a system of equations can be described in terms of characteristics 31 • Ellipticity is represented by complex characteristics, while hyperbolicity is represented by real characteristics. Ellipticity and hyperbolicity are the mathematical concepts associated with diffusion and propagation. Ellipticity bas a regularizing effect on a singularity while hyperbolicity propagates it. For example, long distance effects controlled by an elliptic set of equations depend only on averaged quantities, whereas phenomena controlled by a hyperbolic system of equations, like shocks or singularities, are transported. The set of equations for a purely viscous fluid is elliptic, but a viscoelastic constitutive equation like Eqn. 2-34 is hyperbolic. The combination of ellipticity and hyperbolicity complicates the simulation of viscoelastic flows. For example, the introduction of a geometrical singularity, such as an abrupt contraction, induces a stress singularity and results in both mathematical and numerical difficulties, since the rheological equation transports the effect of the singularity and pollutes the velocity field 31 • In the next section, we will look at severa! numerical approaches that have been proposed to overcome these difficulties. • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 14 2.2.2 Numerical AJ2Proaches to Viscoelastic Simulations The solution of viscoelastic flow problems presents severa! numerical challenges in the case of both differentia! and integral constitutive models. However, certain similarities exist between the two cases, for example the nonlinear character of the goveming equations brought about by the constitutive equation for the viscoelastic stresses. Many discretization techniques have been used ta solve the conservation and constitutive equations, and these include finite element, boundary element, finite difference and spectral methods. The majority of published simulations have been carried out using finite element methods for both differential and integral constitutive equations. This is due to the advantages of the finite element methods in discretizing arbitrary geometries and imposing simply and accurately a variety ofcomplex boundary conditions 32 • If we assume an Eulerian framework, the upper-convected Maxwell model (Eqn. 2-34) can he expressed as follows: • Eqn.2-35 The convective term. brings the complication of having to solve a partial differential equation rather than an ordinary differential equation, as would be the case in a Lagrangian framework. Two fundamentally different approaches are used to solve the set of partial differential equations 33 • In the first approach, a mixed fonnulation is adopted, where the velocity, pressure and extra stress are treated as unknowns, and each of the equations are multiplied independently by a weighting function and transformed into a weighted residual fOlm 34 • In the second approach, the constitutive equation is transfonned into an ordinary differential equation either by using a Lagrangian formulation for unsteady flows 35 or by integration along streamlines. Many integral ·models cannot be expressed in differential form, so either a Lagrangian or streamIine formulation must be adopted. ·-Details of these computations have been described by Luo and HuIsen and van der Zanden 37 , Goublomme et al. 38 and Rajagopalan et al. 39 • Luo40 introduced a control volume approach to solve integral viscoelastic models. Recently, Rasmussen 41 presented a new technique to solve time-dependent three- dimensional viscoelastic flow with integral models, based on a Lagrangian kinematic description of the fluid flow. In this work, we will discuss more thoroughly mi.xed finite element formulations designed for differential constitutive equations in an Eulerian framework. • • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of:the-Art 15 • 2.2.3 Boundary Conditions In order ta complete the mathematical description of a viscoelastic f1ow, appropriate boundary conditions must be specified. The nature of these boundary conditions is intimately related to the mathematical nature of the governing equations. In the case of elliptic systems with complex characteristics, the problem is well-posed for boundary conditions ofthe Dirichlet type and for conditions of the Neumann type when the net flux (Green's theorem) is zero, while for hyperbolic systems with real characteristics, it is well-posed for conditions of the Cauchytype on the entrance boundary. Coupled equations of a mixed elliptic-hyperbolic type have both real and complex characteristics. To date, there is no complete mathematical theory that guides the selection of boundary conditions for the fiow of viscoelastic materials 4 2. 4 3. The general approach is to use the same boundary conditions as for purely viscous fluids: no-slip at solid walls and specification of the velocity components at the entrance and exit of the tlowboundaries. Because ofthe memory effect, the extra stresses must aIso be specified at the inflow boundary to reflect the fiow history. In practice, conditions that result from a fully developed velocity profile are usually applied at the inlet boundary. However, for extra stress this boundary condition is sometimes as complex as the solution of the flow itseIt: because ofthe laek of anaIytical solutions for the more complex models. It is then eommon practiee to specify inlet extra stresses and known kinematics for the upper- convected Maxwell fluid (Eqn. 2-34). These approximate boundary conditions result in a rearrangement of the velocity and stresses at the inflow and outfiow, and this rearrangement is sometimes important enough to restrain numericaI convergence in simulations. It is then highly desirable to improve the quality ofthe boundary conditions. • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 16 • In summary, the issue of appropriate boundary conditions for viscoelastie computations is still an open question 42 • The approach adopted may make it possible to proceed numerically but not be completely supported by either mathematical or experimental results. For example, it is known that the no-slip boundary condition does not apply in all situations for polymerie fluids. AIso, the imposition of proper boundary conditions at geometric singularities, such as capillary exit, is still a subject ofdiscussion. 2.3 A1gorithm Development 2.3.1 Fanure ofTraditional Numerical Methods NumericaI analysts have been interested in simulating viscoelastic fiows for many years. As mentioned above, the numerical computation of viscoelastic flows involves strongly nonlinear, coupled equations of a mixed elliptic-hyperbolic type. The use of conventional numerical techniques has proven unsuccessful due to a loss of convergence, even at low levels ofmelt elasticity. It took aImost ten years to identify what is ealled the " high Deborah number problem ". The Deborah number (De) is a dimensionless group that indicates the degree of elasticity of the flow. Classical finite element techniques based on the application of the Galerkin method to the mixed formulation failed at relatively lowDe, and it was unknown at the time ifthese limits were intrinsically related to the continuous problem or to inadequate numerical schemes. It is now understood and accepted that the computational difficulties are numerical in nature and are due ta the hyperbolic nature of the constitutive equations, the existence of stress singularities, and the method of resolution of the coupled equations. Debbaut and Crochet 44 were able to increase the De limit for an abrupt 4: 1 contraction by imposing more strongly the incompressibility constraint. For the same problem Keunings 45 concluded, after a critical exarnjnation ofthe numerical solutions and an intensive mesh refinement study, that the limiting values of De were numerical :utifacts caused by excessive approximation errors. From then on, it was suggested 45 !hat efforts be focused on complex flows with singularities or boundary layers in order to test algorithms robustness, since strong gradients have a tendency to increase numerical imprecision. • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 17 2.3.2 First Successful Approach Most of the early work on viscoelastic flow analysis was based on the three-field mixed finite element formulation 46 • 47 ,48. The momentum, continuity and constitutive equations are expressed in a weighted residual form, and this leads to an approximation of the Stokes problem (Eqn. 2-30 and Eqn. 2-31) when applied to a Newtonian flow. For an Oldroyd-B fluid with one-mode, and neglecting body forces, the equations in the weak formulation are: Eqn.2..36 Find eu, p, ~ e V x Q x S such that: (Vy)T,2TJ s D + ~ ) - (V . ~ p) =0 'ifv eV where è ~ and q are weighting functions, and (...,...) is the appropriate inner product. • Eqn.2..37 Eqn.2..38 (q, V· ~ ) = O 'ifqeQ 'ifseS In 1987, based on this formulation, Marchal and Crocheé 9 proposed a new method that showed good numerical behavior. First, they introduced a new computational element for the stress components composed of 16 sub-elements (4 x 4), in order to have a discrete representation of the extra stresses that would satisfy the equivalence compatibility condition. The polynomial approximations ofthe three variables need to be carefully selected with respect to each other to satisfy the so-called generalized inf-sup (Brezzi-Babuska) compatibility condition 50 • When this condition is satisfied, the mixed fonnulation including extra stresses provides a convergent approximation for the different variables. The velocity is approximated by biquadratic polynomials, while the · p r e s ~ e is bilinear and continuous. Since the approximation for the stress is also continuous, the inf-sup condition js satisfied by using a sufficient number of interior nodes in each element, such as the 4 x 4 sub-elements used by Marchal and Crochet 49 , as proven later by Fortin and Pierre 51 • • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-ot:the-Art 18 • Once the compatibility of spatial discretization was settled, Marchal and Crochet 49 used two methods to account for the convective term in the constitutive equation: the Streamline Upwind Petrov Galerkin (SUPG) method, and the Streamline Upwind (SU) method: Eqn.2-39 Eqn. '2-40 SUPG: SU: • Those are examples of finite element methods in which the weighting functions differ from the basis functions. In both methods, artificial diffusion is introduced in the tlow direction by the upwind term a.!f' \7~ in order to stabilize the convection-dominated transport. Over the years, several variations of the parameter a. have been proposed, but are all ofthe forro 33 : Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art • Eqn.2-41 h a=- U 19 where h is a characteristic length of the geometry and U a characteristic velocity of the flow. The SU m e t h o ~ where the upwind term is applied only to the convective term of the constitutive equation, showed an increased robustness compared to SUPG, which produced oscillatory stress fields at steep stress boundary layers or near singularities. However, the SU formulation is an inconsistent substitution of the exact solution, since the second term of Eqn. 240 remains as a residual, and even if it converges weil it is not always toward the correct solution S 2., S3. The relative success of this approach showed that a combination of compatible discretization spaces with a discretization technique adapted to the rheological equation could eliminate the 10ss of convergence at high De. Unfortunately, the new element was rapidly shawn ta be too costly in memory space for practical use. 2.3.3 Economical Storage Techniques Marchal and Crochet' s element could only he used for single mode simulations, considering the large memory space needed to solve viscoelastic flow problems, but it became possible to do multimode calculations when Fortin and Fortin 54 intI'oduced their economical storage technique. Fortin and Fortin 54 presented a discontinuous Galerkin (DG) method to handle the constitutive equation, based on ideas of Lesaint and Raviart 55 • Upwinding is introduced in this method by the jump of discontinuous variables at the interfaces ofelements: • Eqn.2-42 where !! is the outward unit normal on the boundary of element e, r in e is the part of element e boundary where!!· n < 0, and J,.nu p the extra stress tensor in the neighboring upwind e l e m e n ~ 3 . • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 20 • • The equivalence compatibility condition was also satisfied in their method by the respective discretizations, but fuis time the solution was decoupled; velocity, pressure and extra stresses were not solved simultaneously. By use of the DG technique, the solution for the stresses did not imply the' inversion of large linear systems but only local calculations on elementary systems. However, the method was found to be insufficiently robust because of the fixed-point algorithm used ta couple the Stokes problem and the viscoelastic equation. Later, "Fortin and Fortin 56 used a quasi-Newton iterative solver (GMRES, for .... Generalized Minimal Residual ") for the solution of the linear system. This increased the algoritbm. robustness but did not -achieve the performance of Marchal and Crochet's method 49 due to oscillatory behavior produced by the DG formulation near singularities. However, the economical storage technique was a major contribution to the simulation of viscoelastic flo\vs; the only large system to store is the one related to the Stokes problem, for which a LU factorization is done. The DG method allows the solution of the constitutive equation on an element by element basis, resulting in very efficient solvers and greatly reducing the memory space needed to solve flow problems using differential multimode models. Later, the robustness of GivIRES was tested with more success by Guénette and Fortin 57 • 2.3.4 Projection Technique The major contnbution of Rajagopalan et al. 58 was the introduction of a fourth field in the miXed finite element formulation of viscoelastic fiow, opening the door to all the projection techniques that are now used to introduce compatibility of discretized spaces. Rajagopalan et al. implemented the EVSS method (Elastic Viscous Split Stress) which is a projection technique based on the splitting of the extra-stress tensor into viscous and elastic parts as shown in Eqn. 2-43: • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art Eqn.2-43 21 'd'v eV Eqn.2-44 A change of variable is performed in the momentum and the constitutive equations yielding a set of equations involving the velocity lb the pressure p and the newvariable E. Moreover, the rate of deformation tensor D is introduced as an additional. unknown, g, leading to a four-field UbP,E,J!) problem. In summary, the EVSS method based on the four-field formulation can be formulated as follows: Find Û!,p,E,.Jl) e V x Q x S x C such that: ;2(77s .y,p)=0 Eqn.2-45 (q, V 'd'qeQ Eqn.2-46 'd'seS Eqn.2-47 'd'cee where g is a suitable weighting function. This method is remarkably stable. However, it requires the convected derivative of the rate of deformation tensor Cin Eqn. 2-46), and the change of variable of Eqn. 2-43 does not yield a closed expression for all constitutive equations. These disadvantages were corrected by Guénette and Fortin 57 who recently proposed a modification of the EVSS formulation, in which the main difference lies in the introduction of a stabilizing elliptic operator in the discrete version of the momentum equation: • Eqn.2-48 (CVy)T,2C77:; .y,p)=0 The parameter a. is positive and a priori arbitrary, but ifproperly chosen it improves the overall convergence of the algorithme In the discrete approximation (Eqn. 2-48), the difference in the discretization of the a. terms combined with the inf-sup condition has a stabilization effect on the discretization. Their method is not restricted to a particular class ofconstitutive equations and is easier to implement than the original EVSS method. It was tested in conjunction with the non-consistent (SU) streamline upwind method on the 4:1 contraction and stick-slip problems. The algorithm seems robust, and no limiting De number was reached when using the one-mode PIT model for the stick-slip problem s1 • • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of..the-Art 22 • • Recently, Fan et al. 59 introduced another stabilized formulation, based on the incompressibility residual of the finite element discretization and the SUPG technique. They claim that the new method has ~ e same level of stability and robustness of the modified EVSS method s1 and is superioF to the EVSS technique 58 , since it does not require the solution ofthe convected derivative ofthe rate ofdeformation tensor. 2.3.5 Influence of Theoretical Work Fortin and Pierres 1 made a mathematical analysis of the Stokes problem for the three- field formulation used by Marchal and Crochet 49 • They confinned that the success of the method was dependent, among others, on the compatibility between discretized spaces. They showed that in the a b ~ e n c e of a purely viscous contribution (115=0) and using a regular Lagrangian interpolation, three conditions must hold 33 • 1. The velocity-pressure interpolation in the Stokes problem must satisfy the Brezzi- Babuska condition to prevent spurious oscillation phenomena. 2. .If a discontinuous interpolation of the viscoelastic stress ~ is used (as in the DG technique), the space ofthe strain rate tensor D.(y) obtained after differentiation ofthe velocity field!! must he in the same discretization space as 't. 3. If a continuous interpolation of the extra stress J; is used (as in the SUPG or SU techniques), the number ofintemal nodes must be larger than the number ofnades on the side ofthe element used for the velocity interpolation. The compatibility between discretized spaces also contributes to the success of the four- fielcÏ modified EVSS technique 57 , as Iater proven by Fortin et aL 60 based on a generalized theory of mixed problems. However, the influence of the discretization of the constitutive equation near a singularity remains unexplained. • . Chapter 2. Numerical Simulation ofVlScoelastic Flow: State-of:the-Art 23 • • 2.4 Comparisons with Experiments With the improved performance of numerical methods for viscoelastic flow simulations, direct comparison of the numerical results with the experimental data became . increasingly feasihle and necessary. Actual comparison may be based on global flow features such as die swell or·· pressure drop measurements or on local tlow kinematics measured by streakline photography, laser Doppler velocimetry (LDV) or birefringence. 2.4.1 First Comparisons: Boger Fluids The first comparisons between numerical simulations and experimental data were made using the upper-convected Maxwell (UCM) and Oldroyd-B models, which can not capture the full nonlinear behavior of polymer fluids. To narrow the gap between experimental observation and prediction, the so-called Boger fluids were developed 61 • These elastic fluids are specially prepared solutions ofhigh molecular weight polymers in viscous solvents and are thought to be nearly free of shear tbinni n g 2 • They are aIso characterized by. a first nOfID.al stress difference proportional to the square of the shear rate, a zero second normal stress difference and an elongational viscosity that increases with elongation rate. Boger tluids are reasonably weIl described up to high frequency in dynamic shear experiments by a two-mode version of the UCM equation 62 • At low and moderate frequencies, the faster of these modes can be collapsed into a retardation terro, and a single-mode Oldroyd-B equation is then a sufficient rheological description.!. Experimental elongational data for Boger fluids have compared favorably with numerical simulations using the Oldroyd-B Debbaut et al. M , Luo and Tanner 65 and Coates et ai. 66 aIso tried to reproduce numericaIly the tlow kinematics measured by streakline photography of a Boger tluid in abrupt 4: 1 and 8: 1 contractions 67 • Most of the results were in qualitative but not quantitative agreement with. the measurements. Recent sources of disagreement between experiments and simulations appear to indicate that Boger fluids are not accurately represented by the Oldroyd-B model and that improved agreement is obtained by using a single-mode finitely extensible nonlinear elastic (FENE-P) dumbbell modeI15.68.69. • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of:.the-Art 24 • • However, Boger fluids do nc;>t retlect the viscoelastic behavior of polymer melts. With the development of new and more realistic constitutive models, further investigation of Boger fluids is now of less interest. In the following sections, we present a short review . on studies that compare experimental data for polymer melts to of complex flows with more realistic rheological models. 2.4.2 Rheological Models Frequently Used in Simulations Simulations of flows of molten polymers under typical processing conditions require the use of suitable rheological constitutive equations. Before looking at complex viscoelastic flow phenomena like rod-climbing, extrudate swell and vortex formation in contraction flows, the constitutive equation must be able to describe the simplest flows, i.e. flows used to establish rheological parameters. The model should predict shear tb;nn;ng, normal stress differences in shear, stress relaxation, recoil and sensitivity to kinematics. Sorne of the models that have been proposed for polymers have been mentioned earlier: these inclu4e the Giesekus, Leonov, Phan-Thien/Tanner (PTT) and K- BKZ rnodels. Their superiority has also been shown in providing numerical solutions for viscoelastic tlows with geometric singularities, such as occur at sharp corners in contraction or expansion flows. These flows are notoriously difficult to simulate, in particular for the UCM and Oldroyd-B models. A significant portion of the work on· the numerical analysis of viscoelastic tlow of polymer melts is based on streamline methods, employing K-BKZ type constitutive models and the damping functions proposed by Papanastasiou et al. 30 (pSM) or Wagner 7 . O • The popularity of the PSM damping function arises from the possibility of fitting shear and elongational data independently. Of the clifferential type models, PTT and Giesekus are the ones most often used, which is surprising considering that the simple Leonov model gives more realistic values of limiting extensional viscosities than Giesekus 2 and contains no nonlinear parameters. The Giesekus model has one parameter that controls nonlinearity, whereas the PTT model bas two. The latter has the ability to fit shear and extensional properties independently. However it gives spurious oscillations during start-up of shear flow71. Therefore, none of the existing constitutive models are entirely suitable for the simulation ofcomplex flows. • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 25 • 2.4.3 Determination of Model Parameters It was pointed above that the use of a viscoelastic constitutive equation in simulations requires the detennination of severa! parameters. Linear viscoelastic data, usually the storage and 10ss moduli (G', G"), which are easy to measure, are used to determine a discrete relaxation spectrum [Gi, Âj]. Very long, entangled polymer chains have many modes of motion and thus of relaxation, and these can not be described in terms of a single relaxation time. Usually, between five and ten modes are necessary ta provide a good fit ofthe relaxation modulus l . However, such a fit may not be adequate for properties that depend in detail on molecular weight distribution (MWD). For example, the compliance and other manifestations of melt elasticity, such as normal stress difference and extrudate swell are very sensitive to MWD 1 • While the rheological behavior depends strongly on molecular structure, rheological data have been used to provide information about molecular structure: this bas been called the "inverse problem". A number of researchers have had significant success in the calculation of the viscosity MWD oflinear polymers72.73.74 by using linear viscoelastic data. Obviously, important information about the MWD is lost when using a discrete spectrum. of relaxation times, and this might affect the predictions in tlow simulations ofphenomena tbat depend strongly on MWD, i.e. on the spectrum. The experimental determination of the nonlinear parameters in the various models is a more diificult task. Some parameters are related to shear behavior and can be easily obtained by measuring viscosity, but for parameters related to extensional behavior or normal stresses, the available experimental data are quite limited. In particular, we lack reliable data for normal stresses at high shear rates and the response to large, rapid extensional flows. As a result, important parameters in a viscoelastic constitutive equation are often determined by fitting the viscosity curve and the few available data for fust normal stress difference in shear or extensional data at low strain rates. Inadequate parameter evaluation can result in poor predictions if we try to simulate typical processing conditions, where deformations tend to be complex, large and rapide In any event, it appears that no existing model is able to fit the available data for typical commercial thermoplastics. • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of:the-Art 26 • In addition, there is no straightforward way to evaluate model parameters for complex flows where both shear and extensional modes of deformation occur simultaneously, for example in converging and diverging flows. Nouatin et al. 75 presented preliminary work using a numerical procedure to identify rheological parameters of Oldroyd-B and PIT models by finite element simulation and inverse problem solving. They reported encouraging results and will carry out future work to identify parameters by use of actual experimental data. 2.4.4 Planar Abrupt Contraction Flow Contraction flow bas received more attention than any other complex flow. This can be explained by the simplicity of the geometry and the fact that it has features that arise in polymer processing. Accelerating flows from a large cross-section via an abrupt or angular entry into a smaller cross-section, Le. entry flows, arise in polymer processing applications such as extrusion and injection molding. Planar contraction flow bas also been chosen as a benchmark problem to eValuate numerical methods and constitutive equations by comparing numerical predictions with experimental results76. A detailed review of comparisons of experiment and simulation for planar abrupt contraction flow has been presented by Schoonen 77 • In Table 2-1 we summarize numerical simulations of planar abrupt contraction flows that have been compared with experiments, including the number ofmodes used in the caIcuIations. • Chapter 2. Numerical Simulation ofViscoelastic Flow: 27 Table 2-1 Numerical Simulations ofPlanar Abrupt Contraction Flow • • Author(s) Type of Model Numberof Numerical Material Model Modes Method White et ai. DifferentiaI PTT 1 PenaltyFEM PS, (1988)78.79 LOPE Park et al. Integral K-BKZ 8 Streamline HDPE . (1992)80 FEM 36 Maders Differentiai White- 1 Decoupled LLOPE (1992)8i Metzner 82 FEM Kiriakidis et Integral K-BKZ 8 Streamline LLOPE al. (1993)83 FEM 36 Ahmed et al. Integral K-BKZ 8 FEM HDPE, (1995)84 (polyflow)85 LOPE Fiegl and Integral Rivlin- 14 FEM LOPE 6ttinger Sawyers 87 (1996)86 Beraudo et al. DifferentiaI PTT 7,8 DGFEM LLDPE, (1998)88 LOPE Schoonen Differential PTT, Giesek:us 4,8 FEM LDPE (1998f7 EVSSDG FEM: Finite Element Method OG: Discontinuous Galerkin EVSS: Elastic Viscous Split Stress It is obvious frOID Table 2-1 tllat the use of integral models makes it possible ta use more relaxation times. This is because in integral models, the contribution from each mode gets summed up in a single integral, while for differential models the number of equations ta solve for the extra stress is multiplied by the number ofmodes considered. Authors of papers frequently fail to numerical convergence problems near the singularities, although White et aI. 78 • 79 and Schoonen 77 did explain how they circumvented the problem. The loss of 'convergence was delayed by White et al. Newtonian behavior in the elements that were in contact with the re-entrant corner, while Schoonen had to use a parameter set for the PIT model that was inconsistent with uniaxial extensional flow behavior. • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of:the-Art 28 • Interestingly, two simulations using different constitutive equations for the same material and same flow gave conflicting results. Comparing their results with the data of Beaufils 89 , Maders et al. 81 used a one-mode White-Metzner model and reported that the preclicted stresses along the centerline showed a slower decay than the experiments. Comparing with the same experimental the simulations of Kiriakidis et al. 83 underestimated the stresses inthe downstream channel using a multimode K-BKZ mode!. Finally, Ahmed et al. 84 and Fiegl and Ôttinger 86 explained some of the discrepancies between experimental results and simulations in terms of the 2D nature of the simulation and a 3D component ofthe experimental flow. Most authors refer to the study of Wales 90 who empirically found the effects of confining walls to be negligible for an aspect ratio exceeding 10 for fully developed shear flow. However, this ratio is mostly used in the channel (downstream section), whereas quantitative comparisons are also made in the upstream. section where the ratio is rarely larger than 10 77 . 2.4.5 Axisymmetric Abrupt Contraction Flow Most of the simulations ofaxisymmetric abrupt contraction flow have been done using integral models. In Table 2-2, we present a summary of numerical simulations of this flow that were compared with experimental results. When comparison was made with experimental entrance pressure drop in these studies, poor agreement with the simulations was obtained. Fiegl and reported reasonable quantitative agreement in terms of vortex growth, but computed values of the entrance pressure 10ss were underestimated. Barakos and Mitsoulis 92 also reported the underestimation of entrance pressure loss, measured by Meissner 93 , by their finite element simulations. 29 Numerical Simulations ofAxisymmetric Abrupt Contraction Flow Table 2-2 Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art FEM: FlnIte Element Method SIMPLE: Semi-Implicit Method for Pressure Linked Equations Author(s) Type of Model Numberof Numerical Material Model Modes Method Dupont and Integral K-BKZ 8 Streamline LDPE Crochet FEM (1988)94 Luo and Integral K-BKZ 8 Streamline LDPE Mitsoulis FEM 36 (1990)95 Hulsenand Differential Giesekus 8 Streamline LDPE van der FEM Zanden (1991i 7 Fiegl and Integral Rivlin- 9 FEM LDPE Ôttinger Sawyers (1994)91 Barakos and Integral K-BKZ 8 Streamline LDPE Mitsoulis FEM 36 (1995)92 Luo (1996r U Integral K-BKZ 8 Control LDPE volume SIMPLE . . • 2.4.6 Axisymmetric Converging Flow The prediction of entrance pressure.loss in converging axisymmetric f10w bas been the subject of severa! recent studies using both integral and differential models. These are Sllrnmarized in Table 2-3. Most studies report that numerical predictions of entrance pressure drop were substantially underestimated when compared to experimental data. Mitsoulis et al. 96 claimed that the entrance pressure 10ss was insensitive to extensional rheology. • Beraud0 97 and Fiegl and Ôttinger H tried to explain these discrepancies. According to the authors, the experimental values include die exit effects, and the degree to which these effects contribute to the experimentally determined entrance pressure 10ss is unknown. However, exit pressure drop is known to be very sma1l 98 and would not compensate for the large difference between experimental and predicted entrance pressure drop. • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of:.the-Art 30 Table 2-3 Numerical Simulations ofAxisymmetric Converging Flow • Author(s) Type of Model Numberof Numerical Material Madel Modes Method Beraudo Differential PIT 7 DGFEM LLDPE (1995)97 Guillet et al. Integral Wagner 1UU 7,8 Stream-tube LLDPE, (1996)99 mapping LDPE Hatzikiriakos Integral K-BKZ 6 Streamline LLDPE and· Mitsoulis FEM (l996)1Ol ." Mitsoulis et al. Integral, K-BKZ, 6,8 Streamline LLDPE (1998)96 differential, PIT, FEM, viscous Carreau l02 EVSS-SUPG FEM: Finite Element Method DG: Discontinuous Galerkin EVSS: Elastic Viscous Split Stress SUPG: Streaniline Upwind Petrov Galerkin 2.4.7 Extrudate Swell 2.4.7.1 Annular Swell • Many attempts have been made ta simulate extrudate swell. In Table 2-4, we present a summary of simulations ofannular extrudate swell that were compared with experimental data. Two independent swell ratios are required to describe annular swell: the diameter swell and the thickness swelL Two studies comparing numerical simulations with annular extrudate swell measurements (Luo and Mitsoulisl03 and Tanoue et al. l04) used the experimental data of Orbey and DealylOS for three HDPE melts. Convergence problems were reported for the tapered geometries by Luo and Mitsoulis103 and Garcia-Rejon et al. L06 • Sorne of t h ~ results presented were not fully converged solutions (Luo and MitsoulisL03), and computations failed to converge for diverging angles beyond +30 0 and for converging angles beyond -30 0 (Garcia-Rejon et al. l06 ). 31 Numerical Simulations ofAnnular Extrudate Swell Table 2-4 Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art FEM: FlDIte Element Method Author(s) Type of Model Numberof Numerical Material Model Modes Method Luo and Integral K-BKZ 8 Streamline HDPE Mitsoulis FEM 36 (1989)103 Garcia-Rejon Integral K-BKZ 6 FEM HDPE et al. (1995)106 (pOlyflow)85 Otsuki et al. Integral K-BKZ 6 Streamline HDPE (1997)107 FEM Tanoue etai. DifferentiaI Giesekus 1 Under HDPE (1998)104 relaxation mixedFEM .. • For the straight annular die, results from Luo and Mitsoulis l03 showed reasonable agreement of prediction and experiment for the diameter swell, but the thickness swell was no1: close to the experimental data. Garcia-Rejon 106 et al. reported opposite results: predictions ofthickness swell were in reasonably good agreement with experimental data, but the agreement was poor for diameter swell. Tanoue et aI. 104 reported deviations as high as 34 % between simulation and experiment. According to the authors, the use of a discrete spectrum of relaxation times would improve the predictions. 2.4.7.2 Capillary Swell Axisymmetric swell was also the subject of numerous studies. Table 2-S presents numerical simulations ofaxisymmetric extrudate swell that were compared with experimental results. • Again, most of the studies were done using integral models. Many studies used the data of M e i s s n e ~ 3 to compare with their simulations. The temperature was kept constant in these swell experiments by extruding into silicone oil at ISOaC. The comparisons between simulations and measured values for a long and a short die are presented in Tables 2-6 and 2-7. Model parameters were fitted to shear and elongationai rheologicai data. The number ofmodes used in the simulations is given in Table 2-5. • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 32 Table2-S Numerical Simulations ofAxisymmetric Extrudate Swell FEM: FlDlte Element Method BEM: Boundary Element Method DG: Discontinuous Galerkin Author(s) Type of Model Numberof Numerical Material Model Modes Method Luo and Integral K-BKZ 8 Streamline LDPE Tanner FEM 36 (1986)108 Bush Differentiai Leonov 7 BEM LDPE (1989) 109 Goublomme Integral K-BKZ 6 FEM HDPE and Crochet (1992i 8 Goublomme Integral Wagner 6 FEM HDPE and Crochet (1993)110 Barakos and Integral K-BKZ 8 Streamline LDPE Mitsoulis FEM 36 (1995)92 Sun et al. Integral K-BKZ 8 Streamline LDPE (1996)111 FEM Beraudo et al. Differentiai PIT 7 DGFEM LLDPE (1998)88 . . In generaI, moderate agreement is obtained for the low and medium apparent shear rates, while swell is largely overestimated at the highest shear rate. We aIso see that predictions differ considerably from one study to another. • Comparing with their own measurements, Beraudo et aI. 88 reported good agreement for a long die but poor a g r e e m ~ n t for a short die. The computation largely underpredicted the experimental value for the short die, and the underestimation was larger at high shear rates. It is worth mentioning that in their swell experiments, the polymer was extruded directly into air n , thus involving sagging due to gravity and non-isothermai effects that couid result in an underestimation ofthe ultimate swell. Finally, Goublomme and simulated the extrudate swell of an HOPE melt thathad been measured by Koopmans 1 12. Extrusion was into a silicone oil bath at 190°C. Inclusion of a converging upstream conical section in the simulation ta match the experimental set-up resulted in a very high calculated swelling ratio (- 9 calculated vs. 2.38-2.61 measuredi 8 • With a difIerent Wagner damping function modified to give a non-zero second normal stress difference, quantitative agreement with experimental results was obtained. the required ratio of second ta fust nonnal stress difference (N21N I =_0.3)110 was larger than typical reported values. • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 33 In conclusion, the simulation of extrudate swell has praduced contradictory results, underestimating or overestimating swell, usually differing considerably from measured values. Table 2-6 Comparisons between Predicted and Measured Extrudate Swell Oong die) Reference Model Swell (0.1 s-a) Swell (1.0 S-L) Swell (10.0 S-i) Luo and K-BKZ 31 % 51 % 82% Tanner lo8 Barakas and K-BKZ 30% 69% 84% Mitsoulis 92 * Sun LU ** K-BKZ 30% 62% 94% Bush 1 \r.? *** Leonov 37% 53 % 77% Meissnei"'" **** Experiments 34% 52% 56% • MeshM2 •• MeshM3 ... Special form ofLeonov model, results taken from figure •••• Extrapolated values for an inf'mite long die Table 2-7 Comparisons between Predicted and Measured Extrudate Swell (orifice) • Reference Model SweU (0.1 S-I) Swell (1.0 S-I) Swell (10.0 s-I.) Barakos and K-BKZ 54% 133 % 195% Mitsoulis 92 * Sun ll1 ** K-BKZ 49% 163 % 223% Meissner'" Experiments 58% 125% 195% • MeshM2 •• MeshM3 We provide here a brief summary of other complex polymer flows such as planar flow . around a cylinder and cross-siot flow. The planar flow around a cylinder has been proposed as a benchmark problem for numerical tecbniques 1l3 , but it has received orny limited attention up to now. This flow is of interest because along the centerline, a material element undergoes various types of deformation: it will be compressed when approaching the cylinder, then sheared along the cylinder's surface and finally stretched in the wake of the cylinder. In cross-slot flow, or planar stagnation, two liquids impinge to create a steady extensionai deformation. At the stagnation point a material element experiences a very large extensional strain rate, which will produce a high Ievei of orientation. • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 2.4.8 üilier Complex Flows 34 Table 2-8 presents recent numerical simulations of planar flow around a cylinder where predictions were compared with experiments. • Table 2-8 Numerical Simulations of Planar Flow around a Cylinder • Author(s) Type of Modei Numberof Numerical Material Model Modes Method Hartt and DifferentiaI, PIT, 1 (pIT), FEM LLDPE, Baird integral Rivlin- 7 (RS) (POlyflow)85 LDPE (1996)1I4 Sawyers Baaijens et aI. Differential PIT, 4,8 FEM LDPE (1997)115 Giesekus EVSSDG Schoonen Differential PIT, 4,8 FEM LDPE (1998)77 Giesekus EVSSDG FEM: Finite Element Method EVSS: Elastic Viscous Split Stress DG: Discontinuous Galerkin Baaijens et 'al. Ils tested two differential constitutive equations, PTI and Giesekus, with model parameters fitted to shear data including viscosity and first normal stress difference. For the PIT model, no unique set ofparameters could be identified without the use of elongational data. Three parameter sets were tried, all giving equally good fits of the shear data. However, comparison with experimentally obtained birefringence patterns revealed that neither of the models could predict quantitatively the observed stress patterns. These results were consistent with those reported by Hartt and Baird114 for the same flow ofpolyethylene melts. • Chapter2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 35 • Schoonen 77 did an experimental/numerical study of an LDPE melt in both planar tlow around a cylinder and cross-slot flow. He compared the predictions of the multimode Giesekus and PTT models with velocities measured using particle tracking velocimetry and stresses measured by fieldwise flow induced birefringence. Where the PTT model gave better agreement than the Giesekus model witb. both velocity and stress data in planar abrupt contraction flow (section 2.4.4), the Giesekus model performed better for the cylinder and cross-siot flow. The PTT model, which described rheological properties best, gave convergence problems. Schoonen had to use a parameter set for the PTT model that was consistent with shear data but not uniaxial extensional behavior in order to obtain numerical convergence. 2.5 Conclusions Major difficulties persist in the simulation of polymer fiows, both on the experimental and the numerical side, and these are surnmarized below. 2.5.1 Limitations ofNumerical Simulations of Viscoelastic Flows Over the past decade, significant progress has been made in the numerical and experimental analysis of viscoelastic flows. Within the category of numerical mixed methods, the modified EVSS technique fust introduced by Guénette and Fortin 57 , appears to provide the most robust formulation currently available. To achieve accurate results, the modified EVSS method should be combined with an upwind scheme like the SUPG formulation or the discontinuous Galerkin (Lesaint-Raviart) Methode However, despite the improved performance of numerical methods for viscoelastic flow simulations, serious convergence problems remain. For flow geometries with a singularity, such as sharp corners for enclosed flows and discontinuities in boundary conditions for free-surface flows, a numerical breakdown occurs above a modest elasticity level or De number. Simulations of these flows are still limited, in general, to low Deborah numbers, i.e., either ta low flow rates or materials with short relaxation times. • Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 36 • Apart from purely numerical issues, such as convergence problems, the predictive capability ofany numerical analysis is only as good as the input data; i.e., the constitutive model, material parameters and boundary conditions. Attempts to find a general viscoelastic fluid model that would be applicable to all classes of flow problems and be able to give reliable results have 50 far failed. If good success is 0 btained in comparison with experimental data for a particular class of flow problem, poor agreement is usually observed with another class of flows. Even when convergence is possible, confrontation with. experimental results Încreasingly reveals the inability of existing constitutive equations to predict the complicated stress fields of industrial forming operations. This holds in particular for flow regions with strong elongational components. At the present time, it is not clear if the poor agreement between simulations and experiments is due to the inadequacy of the models or ta the poor evaluation of the model parameters. We lack, in particular, reliable data for normal stresses at high shear rates and response ta large, rapid extensional flows ta fit the model parameters. And as was mentioned previously, there is no straightforward way ta evaluate model parameters for complex flows where both shear and extensional modes of deformation are occurring simultaneously, for example in converging and diverging flo\vs. 2.5.2 Kinetic Theory Models: A Promising Approach? A relatively recent approach that does not require closed form constitutive models is the so-called micro-macro fonnulation based on kinetic theories l 6-21. Hua and Schieber 16 recently presented calculations of viscoelastic flow through fibrous media using kinetic theory models and a combined finite element and Brownian dynamics technique. Comparisons were made using the same numerical technique with analogous models that lead to closed-fonn constitutive equations, specifically the FENE-P dumbbell, which is an approximate PENE model, and the Doi-Edwards reptation model with and without the independent alignment assumption. They reported significant quantitative differences between the predictions of the approximate and the more realistic models. For example, the FENE-P dumbbell underestimated the magnitude of the normal stress by as much as 25% compared to FENE model, and reptation with independent alignment underestimated it by 22% compared to the more realistic reptation modeL The use of kinetic theory models improves considerably the quality of the prediction, but although no constitutive equation is solved explicitly, problems still arise that prevent convergence of the flow field at high De number. According to Hua and Schieber 16 , a way ta avoid the numerical instability problem might be the use of more accurate finite element methods, such as the modified EVSS formulation or a higher order technique 1 16. Using their current method, they had to modify the momentum equation by adding a relaxation parameter ta make the equation more elliptic in order ta obtain numerical convergence. • Chapter 2. 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In arder to achieve these objectives, we canied out viscoelastic flow simulations using a numerical procedure described in Chapter 6. For these simulations, we used two differential viscoelastic constitutive equations and compared their predictions with. those ofa strictly viscous model. The models selected are presented in the next section. We aIso conducted an experimental study of two complex flows: a planar abrupt 8:1 contraction tlow and an axisymmetric entrance flow and extrudate swell. The materials used for these experiments are described in Chapter 4. 3.2 ModeIs Selected for this Research 3.2.1 Carreau-y asuda Model In order to appreciate the added value of using a viscoelastic constitutive equation in tlow simulations, we carried out calculations using a strictly viscous model. We chose the Carreau-Yasuda 1 model, a generalized Newtonian model that describes fairly weIl the shear dependence of the viscosity of polymer melts. This model was previously introduced in Chapter 2: • Eqn.3-1a where !Ct) = 77[IID Ct) ~ C t ) • Chapter 3. Plan ofthe Research Eqn.3-1b m-I 71[ilD (t)] =710 (1 + CU?ilD 44 • The scalar viscosity is a function of IID, the second invariant ofthe rate of deformation tensor. As stated before, normal stress differences in shear and transient stresses in start- up flows are not predicted by this purely viscous mode!. Four parameters must he specified in order to use the model: the zero-shear viscosity (110), which can be estimated using the discrete relaxation spectrum, a characteristic time (À.), a power-Iaw index (m), which is related to the slope of the viscosity curve at high shear rates, and a second power-Iaw index (s), which controls the transition between the plateau and the power-Iaw regions. The Carreau-Yasuda model parameters for the two materials studied are presented in sections 5.1.1 and 5.2.1. 3.2.2 Leonov Model The fust viscoelastic constitutive equation we selected is a modified Leonov model 2 • This model was initially chosen because it is the ooly one that satisfies basic stability criteria 3 , Hadamard (thermodynamic) and dissipative stability while describing weIl the material functions usually measured 2 • The original model proposed by Leonov 4 .5, which is derived from irreversible thermodynamic principles, is an extension of the theory of rubber elasticity ta viscoelastic liquids. The simple Leonov model, which uses ooly the parameters of the discretized linear viscoelastic spectrum., has been found to provide a good representation of data for shear flows and to be easy to use 6 • However, it provided a poor fit to extensional flow data. Recently, a new fonnulation of the general class of differential constitutive equations proposed by Leonov was presented 2 • This latest version eHminates sorne of the recognized deficiencies of the original "simple" model by adding ooly one or two nonlinear parameters. This modification makes possible a fairly accurate description of simple flows for polymers such as low and high density polyethylenes (LDPE and HOPE), polystyrene (PS) and polyisobutylene (pmi. For the sake of simplicity, we present here the modified Leonov model for a single relaxation time and an incompressible fluide However, the model can easily he extended to coyer the multimode case and compressibility. Derivation of the compressible case for the simple model is presented in references 7, 8 and 9. A multimode incompressible version was used in our simulations. • Chapter 3. Plan ofthe Research 45 The evolution equations for the Finger tensor Bij and the associated dissipation function D are 4 : Eqn.3-2 Eqn.3-3 v Bij+2B ik • ekjp =0 (tr(e-- ) =0) Yp v Here, B ij is the upper-convected time derivative of the Finger tensor, eijp =eijp (Bij) is the irreversible strain rate tensor, and 't jj is the extra stress tensor. For incompressible liquids, the invariants of the Finger tensor are as fol1ows: Eqn.3-4 1 3 =detBij =1. The irreversible rate ofdeformation tensor, eij , has the following general form: p Eqn.3-5 e-- = .!!-[B-- - B _ . ~ l + (1 2 - Il )8.. ] lJ p 42 y y 3 lJ • Here Ôij is the unit tensor, A is the relaxation time in the linear Maxwell limit, and b=b(IhI0 can be thought of as a deformation-history-dependent scaling factor for the linear relaxation times and is an adjustable parameter. The simplest choice is to let b=I, which is known as the standard Leonov mode!. It is sufficient that b{Il,IJ be positive for the dissipation function ta he positive definite, which is required by the Second Law of therm.odynamics. The faIm ofthe evolution equation (Eqn. 3-2) becomes: • Chapter 3. Plan ofthe Research Eqn.3-6 46 In order to relate the extra stress tensor to the elastic Finger tensor during the defonnation history, a functional form. for the elastic potential W(I., 1 2 , n=pof must be provided. Here, Po is the density, andfis the specific Helmholtz free energy. The following general elastic potential bas been suggested 2 : Eqn.3-7 where G is the linear Hookean elastic modulus, and ~ and n are numerical parameters. Eqn. 3-7 yields the Mooney potential for n=O, and the neo-Hookean potential for n=J3=O, which is the case for HDPE and pS2. A constitutive equation with b=1 and the neo- Hookean potential (n=J3=O) gÏves back the simple Leonov model in its original forme Finally, the extra stress tensor can be written in the Finger form as: Eqn.3-8 T .. =2(Boo üVV - B..- r 8VVJ - G5.. y Y;r lJ a lJ (/.1.} 2 • The functional form. of the b function is determined from extensional flow data, and we present in sections 5.1.2 and 5.2.2 the results for our LLDPE and HDPE. We aIso compare the predictions of the modified Leonov model with those obtained using the simple mode!. 3.2.3 Phan-Thien Tanner Model Calculations using the simple Leonov model in numerical flow simulations indicated serious convergence problems. While a major effort was directed at achieving satisfactory simulations, it was found that the use of this equation exacerbated the convergence problems of the numerical analysis and made it impossible to obtain solutions for mast cases. In order to achieve the objectives of the research, therefore, a second viscoelastic constitutive equation was selected for use. A differential model was chosen so that the existing numerical code could be used. The Phan-ThïenfTanner model1o,II (Eqn. 3-9) had shown itself to behave weIl in numerical simulations, although it did not give as good a fit of measurable material functions. Fortin l2 reported having successful results in terms of numerical convergence with that model. However, even this equation gave convergence problems, as will be seen in Chapter 8. • Chapter 3. Plan ofthe Research 47 Like the Leonov model, the PIT model does not predict a separable relaxation modulus, although it is derived from network theory. Separability refers ta the possibility of separating time and strain effects in the nonlinear relaxation modulus, as discussed in section 2.1.4 (Eqn. 2-29). The PIT model is expressed as follows: Eqn.3-9 where E9 stands for the convected Gordon-Schowalter derivative 13 , which is a combination ofthe upper 'V and lower tl convected derivatives: Eqn.3-10 where ;-2 represents the upper-convected derivative ;=0, the lower-convected derivative ç=l, it is the corotional derivative. Phan-Thien and Tanner suggested two possibilities for the function Y ( t r ~ ) : • Eqn.3-11 • Chapter 3. Plan ofthe Research Eqn.3-12 48 • With either choice, the PIT model has two nonlinear parameters (ç and g) to be determined for each materia! in addition to the discrete spectrum. Nonlinear shear and extensional data are needed to determine the nonlinear parameters of the mode!. The parameter ; contraIs the level of shear tbinning, while E bas little influence on shear properties and serves mainly to biunt the extensional singularity that would otherwise be present 6 ,lO,ll. Larson 6 pointed out that there are disadvantages to using the Gordon-Schowalter derivative for melts. These disadvantages are inherited by the PTT model if ; * 2. Primary among these are unphysical oscillations in the shear stress and first normal stress during start-up of steady shear at large shear rates. Other problems have aIso been reported when ç =:1= 2; for example a maximum in the fiow curve, which is physically unrealistic 14 , and the violation ofLodge-Meissner relationshipls. Using ç= 2 avoids these problems, but then the model predicts N 2 =O, which is also unrealistic. Despite these problems, the PTT model is known to give a moderate to good fit 6 of shear and extensional rheological properties of polymer melts using Eqn. 3-12 for Y(tr;;) and of polymer solutions with Eqn. 3-11. Eqn. 3-12 is suitable for polymer melts, since the exponentiaI term results in a maximum in the steady-state elongational viscosity and is thus in better agreement with experimental data. The linear equation for Y (Eqn. 3-11) predicts a plateau value of the elongational viscosity at high strain rates, which is more suitable for polymer solutions. The PTT model parameters we used for our materials are presented in sections 5.1.3 and 5.2.3. • Chapter 3. Plan ofthe Research REFERENCES 49 • 1 Yasuda K.Y., Armstrong R.C., Cohen R.E., Rheo/. p. 163 (1981) 2 Simhambhatla M., Leonov A.I., "On the Rheological Modeling of Viscoelastic Polymer Liquids with Stable Constitutive Equations", Rheol. Acta, p. 259-273 (1995) 3 Leonov Al., Constitutive Equations and Rheology for High Speed Polymer Processing", Polym. Intem., 36(2), p. 187-193 (1995) 4 Leonov A.l., ''Nonequilibrium Thennodynamics and Rheology of Viscoelastic Polymer Media", Rheol. .li, p. 85-98 (1976) 5 Leonov A.I., Lipkina E.R., Paskhin E.D., Prokunin AN., "Theoretical and Experimental Investigation ofShearing in Elastic Polymer Liquids", Rheo/. Acta, li, p. 41 1-426 (1976) 6 Larson R.G., "Constitutive Equations for Polymer Melts and Solutions", Butterworths Series in Chem. Eng., Boston (1988) 7 Baaijens F.P.T., "Calculation of Residual Stresses in Injection Molded Products", Rheo/. Acta, dQ, p. 284 (1991) 8 Flaman AA.M., "Buildup and Relaxation of Molecular Orientation in Injection Molding. Part 1: Formulation", PoZym. Eng. Sei., 33(4), p. 193 (1993) 9 Flaman A.AM., "Buildup and Relaxation of Molecular Orientation in Injection Molding. Part fi: Experimental Verification", Po/ym. Eng. Sei., 33(4), p. 202 (1993) 10 Phan Thien N., Tanner R.I., "ANew Constitutive Equation Derived from Network Theory", J. Non-Newt. FI. Meck, b p. 353-365 (1977) Il Phan Thien N., "ANonlinear Network Viscoelastic Madel", J. Rheo!., 2b p. 259-283 (1978) 12 Fortin A., Beliveau A., "Injection Molding for Charged Viscoelastic Fluid Flow Problems", PPS-14, p. 327-328, Yokohama, Japan (1998) 13 Gordon R.J., Schowalter W.R., Trans. Soc. Rheol.• lQ, p. 79 (1972) 14 Beraudo C., "Méthodes éléments finis pour la simulation numérique d'écoulements de fluides viscoelastiques de type différentiel dans des géométries convergentes", Ph.D. thesis, École Nationale Supérieure des Mines de Paris, Sophia Antipolis, France (1995) 15 Carrot C., Guillet J., Revenu P., Arsac A., "Experimental Validation of Non Linear Network Models", in "Rheology of Polymer Melt Processing ll , J.M. Piau and J.-f. Agassant (eds.), Elsevier Science, Amsterdam (1996) • 4. Rheological Characterization 4.1 Materials Used ln the original research plan, a single polymer was to be studied in !Wo types of complex flow. This was intended to make possible the performance evaluation of the constitutive equations of interest in different tlow situations. However, the linear low-density polyethylene (LLDPE) that was studied in the axisymmetric extrudate swell experiments proved to be unstable inthe extruder used for the planar abrupt contraction flow. It was not possible ta obtain a steady flow rate, and it was necessary ta use another material for these experiments, a high-density polyethylene (HDPE). The LLDPE was Dowlex™ (Dow Chemical Company) 2049A, which is a commercial high stability copolymer made using a Ziegler-Natta catalyst. It is used primarily in packaging and is processed by extrusion and film blowing. The HDPE XI010 was an experimental DSM material polymerized using a catalyst different from the one used to make DSM commercial blow molding resins and sold as Stamylan Th ! lID. Some characteristics of the two resins are presented in Table 4-1. Table 4-1 Experimental Materials • Resin Density MeltIndex Molecular Weight Polydispersity (g/cm 3 ) (g/lO min.) <Mw) (g/mol) (M,jMJ LLDPE 0.926 1.0 * 119600 3.82 HDPE 0.957 1.15 ** 195000 35.0 • 2.16 l c ~ 190°C (ASTM D1238, Method E) •• Test 15 4.2 Experimental Methods 4.2.1 Smalt Amplitude Oscillatory Shear The experiment most widely used to determine the linear viscoelastic properties of polymer melts is small amplitude oscillatory shear. The storage and 10ss moduli were measured using a Rheometrics Dynamic Analyzer II (RDA II), a controlled strain rheometer. The instrument was operated in a parallel plate configuration (plate diameter=25 mm), under a nitrogen atmosphere to prevent thermal degradation. Thermal stability tests were performed to ensure that the resins did not degrade under test conditions, and strain sweeps were performed to verify that the measurements were within the linear viscoelastic regime at aIl frequencies. The nominal frequency range of the RDA II is from 0.001 to 500 radis, but the useful frequency range is resin dependent and must be determined experimentally. • Chapter 4. Rheological Charaeterization 51 Samples for dynamic measurements were compression molded. The molding temperature and pressure cycle was optimized to produce homogeneous samples without residual stresses or voids and to rnjnjmjze thermal degradation. The samples were prepared at 180°C according to the sequence oftimes and applied pressures presented in Table 4.2. Table 4-2 Compression Molding Procedure at 180°C • Cycle melting compression compression compression Cooling * Pressure (l\1Pa) 0 1.2 2.4 3.6 3.6 Time (min.) 5 5 5 5 -10 • pressure released when T < 40°C 4.2.2 Steady Shear Capillary Rheometer The viscosity of the LLDPE was measured on an Instron, piston-driven, constant speed capillary rheometer. In this instrument, the apparent wall shear rate is calculated from the piston speed, and the wall shear stress is calculated from the force measured by the load. cell. Accurate control of the temperature in the barrel (±lOC) is achieved through three independent heating zones with PID controllers and J-type thermocouples 1 • Circular dies of 1.4 mm diameter and LlO ratios of 4.75, 9.5 and 19.0 were used to obtain accurate values of the Bagley corrections which agreed with values ob1ained using an orifice die 2 (LID <0.5). • Cbapter 4. Rheological Characterization 52 Siiding Plate Rheometer The sliding plate rheometer shears a melt between two stainless steel plates. The moving plate, which can produce shear rates up to 500 S-I, is driven by a servo-hydraulic linear actuator. The shear stress is measured at the center of the sample by a shear stress transducer developed at McGill University. AlI measurements take place in a temperature-controlled oven. A detailed description can be found in reference 3. Measurements were made at 200°C for the HDPE resin. 4.2.3 Extensional Flow Transient elongational experiments were performed using a Meissner 4 type apparatus, the Rheometrics Melt Elongational rheometer (RNfE), by Plastech Engineering AG laboratory in Zurich. The sample, clamps and leaf springs are installed in a temperature- controlled oven, which is heated by electrical heater wires em.bedded in the walls. The sample is supported by a cushion of inert gas (nitrogen) and stretched homogeneously between two sets of conveyor belt clamps at a constant Hen.cky strain rate defined as follows: Eqn.4-1 è = _d..:.,.(ln.....;:(:--L:...:...)) dt • where L is the actual sample length. The strain rate range of the instrument is 0.001 to 3 S·I, but the attainable strain rate range is resin dependent and must be determined experimentally. Measurements were performed at 150°C for the LLDPE and at 200°C for the HDPE_ • Chapter 4. Rheological Charaeterization 4.3 Linear-Low Density Polyethylene 53 • AIl rheological measurements for the LLDPE were performed at which is the temperature at which the extrudate swell measurements were conducted. In order to make sure that the polymer was fully molten at that we consulted data previously collected for this material s . Reat capacity measurements carried out in a differential scanning calorimeter showed a melting peak temperature of 126°C. Pressure- volume-temperature (pVn data obtained in cooling gave the following crystaJUzation as a function ofpressure: Eqn. 4-2 Tc = 113.52 +CO.281)P with P the pressure in MPa The highest pressure reached in the capillary rheometer was 42 which gives a crystallization temperature of 125°C. We are therefore confident that all measurements were perfonned inthe molten state. 4.3.1 Storage and Loss Moduli The storage and loss CG") moduli were measured for tbree with a maximum between-sample variation of 3 %. Measurements were made over a frequency range of 0.02 to 500 rad/s. Results are presented in Fig. 4-1. The discrete spectrum ofrelaxation times was determined from the oscillatory shear data by linear regression as described in reference 6. In that technique we specify N values of  i distributed around a center time and with a specified distance between times, within the range of experimental frequencies. The values of G i are then determined by a least- squares procedure, using M sets ofdata: Eqn.4-3 MINf[(f 1 G. (aJj'Â,i)2 G i j=l i=1 G' Cmj) 1 1+ (aJp.t i )2 A minimum of six modes was necessary to obtain a good fit of the experimental moduli. We determined two discrete spectra: one for all the experimental data available and one for a truncated set ofdata where we removed the measurements at the lowest frequencies. This was done to obtain a smaller longest relaxation time in order to improve the numerical convergence of the simulations. Both fits are shown in Fig. 4-2. The parameter sets [G j , Â.J are presented in Table 4-3. • Chapter 4. Rheological Charaeterization 54 1.0E+06 __-------.------------------.. 1.0E+OS a • • • G' ••• • 1!J1'DDO D 8 1 G" D D • • - - • D D a _ a ri ri • a • ri • .. - 1.0E+04 ~ ~ - ë) .. ë) 1.0E+03 1.0E+02 Il • 1000 100 1 10 Frequency (radIs) 0.1 1.0E+01 .L.. ~ - - - - - - - - - - - - - - _ _ . . . 0.01 Figure 4-1 Dynamic Moduli ofthe LLDPE at ISO°C. Table 4-3 Discrete Relaxation Spectra for the LLDPE at 150°C •• AllData Truncated Data Mode G(pa) À (s) G(pa) Â. (s) 1 2.51 x lOS 1.43 X 10- 3 2.38 x 105 2.00 X 10- 3 2 2.09 x IDs 1.00 x 10- 2 1.77 x 105 1.00 X 10- 2 3 6.38 X 10 4 7.00 X 10- 2 6.25 X 10 4 5.00 X 10- 2 4 1.17 X 10 4 4.90 X 10- 1 2.16 X 10 4 2.50 X 10- 1 5 1.11 X 10 3 3.43 x 10 u 2.00 x 10-" 1.25 x 10 0 6 1.10 X 10 2 24.0 x 100 8.36 X 10 2 6.25 x 10 0 • Chapter 4. Rheoiogical Charaeterization 55 1.0E+06 .....------------------------.... 1000 100 -Based on Full Data • • Based on Truncated Data 1 10 Frequency (radis) 0.1 1.0E+02 1.0E+01 .....- - - - ~ - - - - - - - - - - - - - - - - - - - - ~ 0.01 1.0E+OS G" - 1.0E+04 ta Q. - (, ... (, 1.0E+03 Figure 4-2 Fits ofthe Discrete Spectra for the LLDPE at 150°C. 4.3.2 Viscosity Measurements were made at 150°C, and the results were compared with others 5 obtained at 160 0 , 200° and 240°C using a master curve at 2aaoe and an activation energy (EJ of 26.1 kI/mol. The Bagley corrections were in reasonable agreement with values obtained by Kim? using an orifice die (L/D < 0.5) (Figure 4-3). With ail corrections applied (Bagley and Rabinowitch), the flow curve compared weil with data obtained from the sliding plate rheometer at the same temperature by Koran? The flow curve is presented in Fig. 4-4. Fina11y, in Fig. 4-5 the viscosity is plotted along with the complex viscosity, showing that the empirical Cox-Merz rule (Eqn. 4-4) does not apply for this material at high shear rates, which is unusual for a linear material. • Eqn.4-4 (0) = y) 2.5 .....-------------------------.... • Chapter 4. Rheological Charaeterization 56 2 - ta a. :i 1.5 - • Determined By Orifice Die o Determined by Bagley Plot CD U c: ! .. c: CD Q. <1 1 0 • 0.5 ,i 0 0 20 40 o 1 60 80 100 120 140 Shear Rate (1/s) Figure 4-3 Entrance Pressure Drop by Two Methods for the LLDPE at 150°C. 1.,..--------------------------... -rD ri 0.1 - en en ! o ta o 0.01 o Capillary Rheometer à Siiding Plate Rheometer 1 10 100 1000 Shear Rate (1/s) FlowCurve by Two Rheometers for the LLDPE at 150°C. Figure 4-4 0.001 ..... 0.1 • • Chapter 4. Rheological Charaeterization 1.0E+OS ......------------------------... 57 0 0 1.0E+04 - fi) ni tL - ~ .. .- ~ 1.0E+03 o RDA Il o Capillary rheometer 6 Sliding Plate Rheometer 0.01 0.1 1 10 100 1000 Frequency (radis), Shear Rate (1/s) Complex Viscosity and Viscosity of the LLDPE at ISO°C. Figure 4-5 1.0E+02 +.- - - - ~ - - - ~ - - - ~ - - - - - - - - - - - - ....... 0.001 4.3.3 Tensile Stress Growth Coefficient Transient elongational experiments were performed at ISO°C for nominal strain rates of 0.01, 0.1 and 1.0 S-I. Using particle tracking and video images of three zones in the sample, it was possible ta obtain an accurate value of the strain rate. The true strain rates were found ta be 0.01024, 0.09460 and 1.004 S-l. Even if we elimjnate the uncertainty in the strain rate, an important source of errer that remains is the uncertainty in the force measurement. According to the instrument manufacturer, this uncertainty is 0.001 N. The error on the stress can be determined by performing an errer propagation analysis 8 • The stress is calculated by the following equatien: • Eqn.4-S where FCt) is the force as a function of time and A(t) is the cross sectional area as a function of time. Ac is the cross sectional area ofthe sample just before the deformation begins: • Chapter 4. Rheological Charaeterization 58 ( J 2I3 Eqn. 4-(j A o = HRTWRT ;: The subscripts RT and TI refer to the room and test temperatures, and H and W are the initial height and width of the sample. The LLDPE melt density at test temperature (150°C) is 0.785 glcm 3 • Neglecting the uncertainty in the cross-sectional the error propagation analysis gives: Eqn.4-7 !::.(jE Mi' 0.001 --=-=-- (jE F F In Fig. 4-6, we have included the error bars calculated with Eqn. 4-7. We can see that at low strain rates, the uncertainty in the force leads to a very large uncertainty in the rheological properties ofinterest. 1.0E+OS ....------------------------... Figure 4-6 Transient Elongational Stress for the LLDPE at 150°C. 1000 100 1 10 lime Cs) 0.1 o rate=0.01 5-1 o rate=0.1 s-1 6. rate=1.0 s-1 -Linear Viscoelasticity 1.0E+Q4 - .... ...... 0.01 .c 1- o fi! .. cg t'a.. 11)- cne ! CD 1.0E+OS rn..2 CD= = CD II) 0 1:0 CD 1- • • Chapter 4. Rheological Charaeterization 4.4 High Density Polyethylene 59 4.4.1 Storage and Loss Moduli The storage (G') and 1055 (G'') moduli were measured on five samp1es, with a maximum between-sample variation of 4 %. Measurements were made over a frequency range of 0.02 to 500 rad/s. Average values of G' and G" for the five samples are presented in Figure 4-7. We determined aIso for this material two discrete spectra: one for all the experimental data available, and one for a truncated set of data, where we removed the data at the lowest frequencies. Again, a minimum of six modes was necessary to obtain a good fit of the experimental moduli. Both fits are presented in Fig. 4-8. The parameter sets [G j , ÂJ are presented in Table 4-4. 1000 100 0.1 1 10 frequency (rad/sec) Figure 4-7 Dynamic Moduli ofthe HDPE at 200°C. o G' (average) DG" (average) 000 00 0 0 800000000 ~ B 0 e i ~ oS9ij 0 00 0 0 0 0 00 0 DO 00 DO 00 00 0 o 0 0 0 0 0 1.0E+03 1.0E+02 0.01 1.0E+OS 1.0E+06 - ca a. - (, 1.0E+Q4 è> • 60 1000 100 1 10 frequency (rad/sec) 0.1 1.0E+02 ....- ~ ...--..01!-----!__-------------__. 0.01 1.0E+O& - r - - - - - - - - - - - - - - - - - - - - - - - - - ~ 1.0E+03 1.0E+OS -Based on Ali Data - - Based on Truncated Data -ca a. - (, 1.0E+04 ë Chapter 4. Rheological Characterization • Figure 4-8 Fits of the Discrete Spectra for the HDPE at 200 o e. Table 4-4 Discrete Relaxation Spectra for the HDPE at 200 0 e AllData Truncated Data Mode G(pa) À (s) G(pa) Â. (s) 1 1.31 X 10 5 1.43 X 10- 3 1.23 X 10 5 2.00 X 10- 3 2 1.09 X 10 5 1.00 X 10- 2 9.29 X 10 4 1.00 X 10-:l 3 4.81 X 10 4 7.00 X 10- 2 4.21 X 10 4 5.00 X 10- 2 4 1.66 X 10 4 4.90 X 10- 1 2.51 X 10 4 2.50 X 10- 1 5 3.08 X 10 3 3.43 x 10 0 3.54 X 10 3 1.25 x 10° 6 5.98 x 10 2 24.0 x 10 0 2.95 X 10 3 6.25 x 10° 4.4.2 Viscosity and FirstNormai Stress Difference • The viscosity of the HDPE was measured using the sliding plate rheometer at 200°C. Our results were compared with data obtained at DSM using capillary and cone and plate rheometers at 190°C and shifted to 200°C using an activation energy E a of 27.6 kJ/mol which was determined by DSM. Good agreement was found between the three sets of data (Fig. 4-9). In Fig. 4-10, the viscosity is plotted along with the complex viscosity. The cone and plate data do not follow the same trend as the measurements in small amplitude oscillatory f1ow, which is incorrect since both the complex viscosity and the viscosity should reach the same plateau-value, Tto. If we disregard the cone and plate data, since it is not possible to evaluate the error involved in these measurements, we conclude that the empirical Cox-Merz nùe is valid for this material. • Chapter 4. Rheological Characterization 61 The first normal stress difference was aIso measured using the cone and plate rheometer at 190°C. We present these data, shifted to 200°C, in Fig. 4-11, using the same shift factor as previously. These data were not used to fit the model parameters. 1.0E+03 .....--------------------------. 6. Sliding Plate Data o Capillary Data (shifted at 200°C) x Cane and Plate Data (shifted al 200°C) 1.0E+02 - ca Q. ~ - en en ! 1.0E+01 - U) ... ca CD .c U) 1.0E+00 o 0 o 10000 100000 1000 1 10 100 Shear Rate (1/s) 0.1 0.01 1.0E-D1 ....- - ~ - - - - - - ~ - - ~ - - - - - ~ - - - - - ...... 0.001 • Figure 4-9 Flow Curve by Three Rheometers for the HDPE at 200°C. • Chapter 4. Rheological Characterization 1.0E+OS X 62 1.0E+04 - CI) ai 1.0E+03 o 1.0E+02 o RDA Il (average data) o Capillary Data (shifted at 200°C) ô Sliding PJate Data ::te Cane and Plate Data (shitted at 200°C) o o o 0.01 0.1 1 10 100 1000 10000 100000 Frequency (radis), Shear Rate (1/s) Figure 4-10 Complex Viscosity and Viscosity ofthe HDPE at 200 0 e. 1.0E+01 0.001 i.OE+OS ..,...------------------------........ x Cane and Plate Data (shifted at 200°C) - ca e:.. 1.0E+04 .... Z 1 0.1 1.0E+03 .....----------------------------' 0.01 Shear Rate (1/s) Figure 4-11 First Normal Stress Difference of the HDPE Shifted from 190° to 200 o e. • • Chapter 4. Rheological Characterization 4.4.3 Tensile Stress Growth Coefficient 63 We saw in section 4.3.3 (Fig. 4.6) that a large error is present in the stress at low extensional rates due to the low forces. On the other hand, artificial strain hardening is often observed at high rates with the RME rheometer for non-strain hardening materials 9 • For the HDPE, we chose a medium. rate and had two replicates done in arder to verify reproducibility. Three measurements were done at a rate of 0.5 S-I at 200°C. Using particle tracking and video images of three zones, it was possible ta obtain an accurate value of the strain rate. The true strain rates were 0.4845, 0.4959 and 0.4946 5- 1 • (The HDPE melt density at test temperature is 0.742g1cm 3 ). Excellent agreement was found between the three sets of data (Fig. 4.12). The error bars are smaller than the size of the symbols. 1.0E+O& ..- --. .c i - 1.0E+OS o fi! .. ca C)A- ",- "'1: ! eu 0 13 e u ~ ~ (J 1.0E+Q4 ~ o rate=O.5 s-1 (1) o rate=O.5 s-1 (2) 6. rate=O.5 s-1 (3) - Linear Viscoelastïcity 100 10 0.1 1 Tirne (s) Figure 4-12 Transient Elongational Stress for the HOPE at 200°C. 1.0E+03 L ~ - - - - - - = = = = = = = = = = = ~ 0.01 • • Chapter 4. Rheological Charaeterization REFERENCES 64 • 1 Sentmanat M.L., Dealy JM., "Sorne Factors Affecting Capillary Rheometer Performance", ANTEC '95, Society ofPlostie Engineers, p. 1068-1072 (1995) 2 Kim S.O., unpublished work, McGill University, Montreal, Canada (1999) 3 Giacomin AJ., Samurkas T., Dealy J.M., " A Novel Sliding Plate Rheometer for Molten Plastics", Polym. Eng. Sei, 29(8), p. 499 (1989) 4 Meissner, J., Hostettler J., ~ ~ A New ElongationaI Rheometer for Polymer Melts and Other Highly Viscoelastic Liquids ", Rheal. Acta, li, p. 1-21 (1994) 5 Heuzey M.C., C ; ~ The Occurrence of Flow Marks during Injection Molding of Linear Polyethylene", Master's Thesis, McGill University, Montreal, Canada (1996) 6 Orbey N., Dealy J.M., "Detennination of the Relaxation Spectrum from Oscillatory Shear Data", J. Rheal., 35(6), p. 1035-1049 (1991) 7 Koran F., "The effect of Pressure on the Rheology and Wall Slip of Polymer Melts", Ph.D. Thesis, MeGill University, Montreal, Canada (1998) 8 Wood-Adams P., "The Effeet of Long Chain Branching on the Rheological Behavior of Polyethylenes Synthesized using Single Site Catalysts", Ph.D. Thesis, McGill University, Montreal, Canada (1998) 9 Hepperle, J., Saito T., unpublished work, Institute of Polymer Materials, University of Erlangen-Nuremberg, Erlangen, Germany (1998) • s. Determination of ModeI Parameters 5.1 Linear Low-Density Polyethylene 5.1.1 Viscous Model The Carreau-Yasuda generalized Newtonian model (Eqn. 5-1) contains four parameters: the zero-shear viscosity 110' a characteristic time A., and two power-Iaw indexes, m and s. Eqo.5-1 The zero-shear viscosity was calculated from the discrete spectrum based on aIl the data presented in Table 4-3, as follows: Eqo.5-2 N 770 = LGi i =19079 Pa.s i=1 The other three parameters were determined by a least-squares procedure, using M sets of data: Ego. 5-3 where T'lmeas represents the experimental viscosity, and 11calc the viscosity calculated using Eqn. 5-1. The parameters determined for the LLDPE are presented in Table 5-1, and the corresponding fit ofthe viscosity is shown in Fig. 5-1. Table 5-1 Carreau-YasudaModel Parameters for the LLDPE at I50 a C • Parameter Value 110 [pa.s] 19079 A. [s] 0.39 m 0.41 s 0.76 • Chapter S. Determination ofModel Parameters 1.0E+OS -r-------------------------... 66 en 1.0E+Q4 câ a. - b a u fi) :> 1.0E+03 o Capillary rheometer 6. Sliding Plate Rheometer -Carreau-Yasuda 1000 100 0.01 1.0E+02 ..... 0.001 0.1 1 10 Shear Rate (1/5) Figure 5-1 Viscosity Fit ofCarreau-Yasuda Equation for the LLDPE at 150°C. 5.1.2 Leonov Madel Parameters Specification of both in the evolution equation (Eqn. 5-4) and the elastic potential (Eqn. 5-5) are required in order to use the modified Leonov model 1 , which we write as follows: Eqn.5-4 Eqn.5-5 • For polyethylene, Simbambhatla and Leonov 1 suggested using a Neo-Hookean elastic potential (n=JFO in Eqn. 5-5). The procedure recommended by the authors 1 for the choice of b(Il,I,) is as follows. First, perform preliminary calculations for various tlows using the simple model (b=l). Then, ifthere is disagreement with experimental data, a functional form of b that brings the calculations into qualitative agreement with the data can be systematically developed. To model the viscoelastic behavior of various polymers (LDPE, HDPE, PS, and pm), simple power-Iaw or exponential functions of the invariants of the Finger tensor with one or two adjustable parameters are usually sufficient. • Chapter 5. Determination ofModel Parameters 67 We found that the best result for non-strain hardening materials was obtained with an exponential function ofIv, the first invariant ofthe Finger tensor: Eqn.5-6 To fit the m parameter, we used the transient elongational data, since this form of the b function bas very little influence on the prediction of viscosity. The best value of the parameter was determined using an IMSLlFortran™ subroutine (DUVMIF)2. This routine is based on a quadratic interpolation method to find the least value of a function specified by the user. The value obtained was m=0.02. The complete set ofparameters is presented in Table 5-2. Table 5-2 Leonov Model Parameters for the LLDPE at 150°C • Parameter Value n 0 J3 0 m 0.02 • Chapter 5. Determination ofModel Parameters 5.1.3 Phan-ThienITanner Model Parameters The PTT mode1 3 • 4 can be expressed as follows for polymer melts. 68 Eqn.5-7 ;L(Ç (1- +!: exp( À.& trrJ = 211D 2- 2 - - Tl - - It bas two nonlinear parameters (ç and E) to be determined in addition to the discrete spectrum. The parameter ç controis the level of shear tbinning in shear flows and E the extensional behavior. We used the IMSLIFortranTM subroutine described earlier to determine the parameters. The value of ç was determined by fitting the viscosity curve and that of E by fitting the transient elongational data. The resulting parameters are presented in Table 5-3. Table 5-3 PTT Model Parameters for the LLDPE at ISOCC 5.1.4 Viscosity Parameter Value 1.79 0.16 • We already mentioned that the simple and modified Leonov models do not differ in their shear predictions ifEqn. 5-6 is used, so we show only the results for the mode!. In Fig. 5-2, we show the fit of the Leonov and PTT models for the viscosity. With one parameter fitted exclusively to shear data (ç), the PTT model reproduces more realistically the observed behavior. The Leonov model predicts the Cox-Merz rule, which is not valid for this material at high shear rates (Chapter 4). • Chapter 5. Determination ofModel Parameters 1.0E+OS ~ - - - - - - - - - - - - - - - - - - - - - - - - .... 69 .. 1.0E+04 ca a. - b 8 u fi) :> 1.0E+03 o Capillary rheometer ~ Sliding Plate Rheometer -Modified Leonov Madel PTT Madel (alpha=O) " " 1000 100 10 0.1 0.01 1 Shear Rate (ils) Fit ofViscosity with Leonov and PIT modeIs for the LLDPE at 150°C. 1.0E+02 ....- - - - ~ - - - ~ - - - ~ - - - - - - - - - - - _ . . . 0.001 Figure 5-2 It was mentioned in Chapter 3 that one ofthe problems with the PTT model when .; =#= 2 is the presence of a maximum. in the flow curve, which is physically unrealistic s . Crochet et al. 6 showed that it is possible to avoid this problem by using a non-zero viscous component (11 s =#=O)' This way, the shear stress becomes a strictly increasing function of the shear rate ifthe viscosity ratio fulfilIs the following condition: Eqn.5-8 with Eqn.5-9 lls+1]=1]o • However, when a spectrum of relaxation times is used instead of a single mode, this condition can. be relaxed. The use of a multimode spectrum for polymer melts is always strongly recommended since a single mode can not fit data properly and leads to numerical problems. Beraudo S used a viscosity ratio CTl/TJo) of 1/8 in her simulations, although she used a spectrum of relaxation times. We verified. the effect ofthat ratio on the prediction of the viscosity for our material, and the result is presented in Fig. 5-3. We found that this criterion could not be applied blindly in simulations. Instead, we fitted an additional parameter, cx., that govems the viscosity ratio based on shear data. • Chapter 5. Determination ofModel Parameters 70 Eqn.5-10 a =t'Js t'Jo We obtained cx.-lO- 5 for the LLDPE, which shows that the condition given by Eqn. 5-8 can be ignored when using a spectrum ofrelaxation times. Fig. 5-4 shows the flow curve with cx.=O, and we can see that there is no maximum in the curve over the range of experimental data, even without a viscous contribution. 1.0E+OS -p---------------------------. en 1.0E+04 ca a. - ~ ëii o u en :; 1.0E+03 o Capillary rheometer 6. Sliding Plate Rheometer -PTTMadel. (alpha=118) 0.01 0.1 1 10 100 1000 Shear Rate (115) Viscosity Curve ofPTT Model with cx.=1/8 for the LLDPE at ISO°C. 1.0E+02 . . . a - - - - ~ - - - . . 0 1 1 ! " - - - - - - - - - - - - - - - - ~ 0.001 Figure 5-3 • Chapter 5. Determination ofModel Parameters 71 • 1..,...----------------. Cà ~ 0.1 - o Capillary Rheometer ~ Sliding Plate Rheometer -PITModel (alpha=O) 1000 1 10 100 Shear Rate (1/5) Flow Curve Predicted by the PIT Model with a=O. Figure 5-4 0.001 .....- - - - - - ~ - - - - - - - - - - - - - - - - - - ..... 0.1 5.1.5 Tensile Stress Growth Coefficient In Fig. 5-5, we show the fit of the tensile stress growth coefficient provided by the Leonov and PTT models. The r e ~ t i n g curves are very similar for the two models. Good agreement is obtained for the highest strain rate (1.0 S·I), but at the lowest rates less strain hardening is predicted than is seen experimentally. However, considering the large error involved in the measurements at these rates, we suspect that these data are not reliable. • Chapter 5. Determination ofModel Parameters 1.0E+O& ..,------------------------... 72 o rate=0.01 s-1 o rate=0.1 s-1 A rate=1.0 s-1 -Modified Leonov Model • • PTT Model 0.1 1 10 100 1000 Time (5) Fit ofTensile Stress Growth Coefficient for the LLDPE at 150°C. 1.0E+04 ........- - - ~ - - - - ~ - - - _ . . . o i ! - - - - - - - - - ~ 0.01 FigureS-5 .c 1- o CI! .. as C)Q. cn- cn'E ! CD 1.0E+OS ...- r:Ë = CD fi) 0 cU ~ • 5.1.6 Prediction ofNonnal Stress Differences We had no data for the LLDPE to compare with the predicted fust nonnal stress difference, so we used Laon' s 7 empirical relation for 'Pl: Eqn.5-11 [ 2] 0.7 Gr "G' '1'1ct) =2(J)2 l+l Gu ) • This relationship was verified by Laun 7 for low and high-density polyethylenes, polypropylene and polystyrene. The comparison for NI is presented in Fig. 5-6 for the Leonov and PIT models. The agreement is good with both models at low and moderate shear rates, but the Leonov model follows more closely Laun' s empirical relation at high shear rates. 1.0E+07 ....------------------------.... 73 1000 100 10 1 0.1 /j. Laun's Empirical Relation -Modified Leonov Model - - PTT Model 1.0E+06 1.0E+OS 1.0E+03 1.0E+01 : . - - - - - ~ - - - - - - - - - - - - - - - - - - - - - - 4 0.01 1.0E+02 êi ~ 1.0E+04 op Z Chapter 5. Determination ofModel Parameters • Shear Rate (1/5) Figure 5-6 Prediction ofFirst Normal Stress Difference for the LLDPE at IS0 a C. For NzlN h the ratio of the second to first normal stress difference, the Leonov model predicts a value that varies with shear rate. At low shear rates, N/N l = -0.25 and approaches zero at high shear rates (Fig. 5-7). Experimental data for steady simple shear of a number of materials indicate that N 2 is negative and bas a magnitude about lOto 30% that ofN l s . ForNJNI' PTT gives a constant ratio of-o.l05 accordingto Eqn. 5-12. Eqn.5-12 N 2 2 - ~ -=-- NI 2 • 74 - - - - - - . -. - - - - - - - -- - - - - -- - - - - . - - -0.2 0.-==============---------=====1 -0.1 -Modified Leonov Model - • ·PTT Model -G.05 -G.25 .....- ....- ~ z è'i -0.15 Z Chapter 5. Determination ofModel Parameters • 0.1 10 1000 100000 1000000 Shear Rate (1/s) Ratio of Second to First Normal Stress Difference for the LLDPE at ISOaC. -0.3 .....- - - - ~ - - - - - - - - - _ - - - - - - - - - - .... 0.001 Figure 5-7 5.1.7 Prediction ofElongational Viscosity • We present in Fig. 5-8 the elongational viscosity predicted by both versions of the Leonov model and by the PTT model. The modified Leonov and PTT models both predict a maximum in the elongational viscosity curve, which is more realistic for polymer melts. We observe that both. models agree over the range of strain rates where the respective parameters were fitted. However, the predictions differ considerably at higher elongational rates. The simple Leonov model predicts a plateau value of the elongational viscosity of six times the zero-shear viscosity at high strain rates. The inaccurate elongational flow behavior is the main deficiency of the simple model since linear polymers usually exhibit little strain hardening. • Chapter S. Determination of Madel Parameters 1.0E+06 -r--------------------------.. 75 , , - • Linear Viscoelasticity -Modified Leonov Model x Simple Leonov Model .PTT Model en :. - 1.0E+OS X X XX>OOI8K X X ·Ci o ...... _ :JI - - - • - u -_ en :> 'ii c o ëij 1.0E+04 c S Ji( w 0.1 1 10 100 1000 10000 Extensional Rate (1/5) Prediction of Elongational Viscosity for the LLDPE at 150°C. Figure 5-8 1.0E+03 ... , ........ 0.01 5.2 High-Density Polyethylene 5.2.1 Viscous Model Using the procedure described in section 5.1.1, we determined the parameters for the Carreau-Yasuda equation for the HDPE. The zero-shear viscosity is calculated from the discrete spectrum. based on ail the data given in Table 4-4. The resulting parameters are presented inTable 5-4, and the corresponding fit ofthe viscosity is shown in Fig. 5-9. As discussed previously in Chapter 4, we see a disagreement with the cone and plate data. Table 5-4 Carreau-Yasuda Model Parameters for the HDPE at 200°C • Parameter Value t'lo [pa.s] 37694 Â. [s] 0.48 m 0.20 s 0.45 • Chapter 5. Determination ofModel Parameters 1.0E+OS ---------------------.... le 76 1.0E+04 - fi) :. - 1.0E+03 ëi 0 u en :> 1.0E+02 o Capillary Data (shifted at 200°C) 6. Sliding Plate Data le Cone and Plate Data (shifted al 200°C) -Carreau-Yasuda 1000 10000 100000 1 10 100 Shear Rate (115) 0.1 0.01 1.0E+01 .... .....__ 0.001 Figure 5-9 Viscosity Fit by Carreau-Yasuela Equation for the HDPE at 200°C. 5.2.2 Leonov Mode! Parameters The functional form. of b(Il' 1,) given by Eqn. 5-6 was used for the HDPE. The parameters m, n and J3 (Eqn. 5-5), determined using the IMSL/Fortran™ subroutine, are shown in Table 5-5. Table 5-5 Leonov Model Parameters for the HDPE at 200°C n Value o o m 0.3 • The parameters of the PTT model (Eqn. 5-7) for the HDPE are presented in Table 5-6. Again, çwas fitted ta the viscosity curve and s ta the transient elongational data. • Chapter S. Determination ofModel Parameters 5.2.3 Phan-Thien/Tanner Madel Parameters 77 Table 5-6 PTT Model Parameters for the HDPE at 200°C 5.2.4 Viscosity Parameter Value 1.82 0.32 • In Fig. 5-10, we show the fit of the modified Leonov and PIT models for the viscosity. We see that none of the models reproduce weIl the viscosity at high shear rates. However, at these high rates, flow instabilities can become important, sa the reliability of these data points can be questioned. With the PTT model, we had to use a small viscous contribution to avoid a maximum in the flow curve (Fig. 5-11). We used a.=3.6xl0"4, which is still much smaller than the recommended value of "/110=1/8. In Fig. 5-12, we show what the viscosity curve looks like if 0.=1/8 is used. This emphasizes that one should be careful in using the rule suggested by Crochet et al. 6 (Eqn. 5-8) along with. a spectrumofrelaxation times. • Chapter 5. Determination ofModel Parameters 1.0E+OS ~ - - - - - - - - - - - - - - - - - - - - - - - - . ~ 78 1.0E+04 - en ca Q. - b 1.0E+03 ëi 0 Co) en :> 1.0E+02 o Capillary Data (shifted at 200 G C)  Sliding Plate Data ~ Cone and Plate Data (shifted at 200 G C) -Modified Leonov Madel - - PTT Model (alpha=3.6e-4) 1000 10000 100000 0.1 0.01 1.0E+01 ....= = = = = = = = : : ; : : : : = = = = = = ~ - - - - ~ ~ 0.001 1 10 100 Shear Rate (1/s) Figure 5-10 Fit ofViscosity by Leonov and PTT Models for the HDPE at 200°C. 1.0E+03 ....--------------------------. 1.0E+02 êG Q. ~ - en en ! 1.0E+01 û5 ~ ca CP oC en 1.0E+00  Sliding Plate Data o Capillary Data (shifted at 200°C) ~ Cane and Plate Data (shifted at 200°C) -PTTModel (alpha=O) 1000 10000 100000 100 10 1 0.1 0.01 1.0E-G1 ........- ~ - - ~ - - - - - - - - - - - - - - - - - - - - . . 0.001 Shear Rate (1/s) Figure 5-11 FlowCurve for the HOPE at 200°C Predicted by the PIT Model without the Viscous Contribution. • • Chapter 5. Determination of Model Parameters 1.0E+OS ...,------------------------... X 79 1.0E+04 - en ai Q. - ~ 1.0E+03 ëii o u en :> 1.0E+02 o o Capillary Data (shifted at 200°C) 6 Sliding Plate Data X Cone and Plate Cata (shifted at 200°C) -PTTModel (alpha =1/8) o o o o o 1000 10000 100000 1 10 100 Shear Rate (1/5) 0.1 0.01 1.0E+01 ....=====::;::=====;:::====--------..L:l...J 0.001 Figure 5-12 Viscosity Curve ofPTI Madel with a=1/8 for the HDPE at 200°C. 5.2.5 Tensile Stress Growth Coefficient In Fig. 5-13, we show the fit of the tensile stress growtb. coefficient. Bath models give equally good fits, except for data at rate 1, which shows a higher level than the predicted steady state. • 1.0E+0& ... • Cbapter 5. Detennination ofModel Parameters 80 s:- I fi! 1.0E+OS .. ca en- en'" ! i ....- U) CD= := CD en 0 1.0E+04 cO <> rate=O.S s-1 (1) o rate=O.S 5-1 (2) rate=O.S 5-1 (3) -Modified Leonov Madel PTT Model 10 1 0.1 1.0E+03 .... 0.01 Time (s) Figure 5-13 Fit ofTensile Stress Grovvth Coefficient for the HDPE at 200°C. 5.2.6 Prediction ofNormal Stress Differences We had only a few cone and plate data for the :first normal stress difference, 50 we again used Laun' s empirical relation ta verify the predictions. The comparison for NI is presented in Fig. 5-14. The agreement is good with both models at low and moderate shear rates, and again for this material, the Leonov model is closer to Laun's empirical relation at high shear rates. The cane and plate data are in fair agreement with the predictions, except at the lowest shear rate. This is similar to what is seen in the viscosity corvee • For the ratio of the second to tirst normal stress difference, the Leonov model predicts N 2 /N 1 = -0.25 at low values of the shear rate. This ratio decreases toward zero with increasing shear rate (Fig. 5-15). The PTT model gives a constant ratio of-0.09. • Chapter 5. Determination ofModel Parameters 1.0E+07..,-------------------------... 81 1000 100 D.. Prediction trom Laun PTT Model X Cone and Plate Data (shifted at 200°C) 10 -Modified Leonov Model 1 0.1 1.0E+OS 1.0E+03 1.0E+02 0.01 1.0E+05 -ca Q. - Z 1.0E+04 Shear Rate (1/s) Figure 5..14 Prediction ofFirst Normal Stress Difference for the HDPE at 200°C. 0-.----------------------------. -G.05 . - . - . - - . - - . - - - - - - - _. - - . - - - - - . -- -0.1 ... z N -0.15 Z -0.2 -0.25 .....---- - Modified Leonov Model PTT Model 10000000 100000 0.1 -0.3 .... 0.001 10 1000 Shear Rate (1/5) Figure 5-15 Ratio of Second ta Fbst Normal Stress Difference for the HDPE at 200°C. • • Chapter 5. Detennination ofModel Parameters 5.2.7 Prediction ofElongational Viscosity 82 We present in Fig. 5-16 the elongational viscosity predicted by bath versions of the Leonov model and by the PIT modeL Agaïn, the modified Leonov and PIT models predict a maximum in the elongational viscosity curve. We see that both models agree over a range around the strain rate at which the parameters were fitted (0.5 S-I) and at higher rates. The simple Leonov model always predicts a plateau value of the elongational viscosity that is six times the zero-shear viscosity at high strain rates. 1.0E+OS . . - - - - - - - - - - - - - - - - - - - - - - - - - - ~ x ~ X X ~ x x ~ xx x x ~ x - 1 Linear Viscoelasticity -Modified Leonov Madel x Simple Leonov Madel PTT Madel en tG Q. - ~ 1.0E+OS ~ u fi) :> 'ii c o in 1.0E+04 c S >< W 1.0E+03 ....- - - - - - - - ~ - - - ~ - - - - - - - - - - ~ .... 0.01 0.1 1 10 100 1000 10000 Extensional Rate (5-1) Figure 5-16 Prediction of Elongational Viscosity for the HDPE at 200°C. • In conclusion, we can say that both viscoelastic models give similar fits and predictions of the rheological properties. Also, our results showed that for the PTT model the addition of a viscous contribution to avoid the maximum in the flow curve should be done with caution. • Cbapter 5. Determination ofModel Parameters REFERENCES 83 • 1 Simbambhatla Leonov A.L, "On the Rheological Modeling of Viscoelastic Polymer Liquids with Stable Constitutive Equations", Rhea/. Acta. p. 259-273 (1995) 2 IMSL MathlLibrary, Version 2.0, p. 972 (1991) 3 Phan Thien N., Tanner "A New Constitutive Equation Derived from Network Theory", J. Non-Newt. FI. Mech., 2, p. 353-365 (1977) 4 Phan Thien N., "ANonlinear Network Viscoelastic J. Rhea!., 2b p. 259-283 (1978) 5 Beraudo C., "Méthodes éléments finis pour la simulation numérique d'écoulements de fluides viscoelastiques de type différentiel dans des géométries convergentes", Ph.D. thesis, École Nationale Supérieure des Mines de Paris, Sophia Antipolis, France (1995) 6 Crochet Davies A.R., Walters K., Simulation ofNon-Newtonian Flows", Ed. Elsevier. Am.sterdam (1984) 7 Laun H.M., Rheo/. Acta. p. 459 (1986) 8 Dealy J.M., Wissbrun K.F., "Melt Rheology and its RaIe in Plastics Processing", Van Nostrand Reinhold. New York (1990) • 6. Numerical Method 6.1 Description of the Problem 6.1.1 Four-Field Mixed Formulation In Chapter 2, we presented the system of equations that must be solved to simulate the flow of viscoelastic materials. We recall here that for isothermal flows, neglecting compressibility and body forces, the conservation equations for mass and momentum. can be written as follows: Eqn.6-1 Eqn.6-2 Du p-=+Vp-V·t"=O Dt = where p denotes a constant density,p is the pressure, and ~ is the extra stress tensor. For a Newtonian fluid and neglecting inertia forces, this set of equations constitutes the "Stokes problem". In the absence of a viscous contribution, and for a multimode representation, the extra stress tensor is expressed as follows: Eqn.6-3 L N 1:= 1:. = ;=1 1 where N represents the number of modes. The last equation required to close the above system is the rheological model, which in our work is the Leonov or the Phan-ThienITanner model: Eqn.6-4 • where t'lz-Â,-Gi• For the Leonov model: Chapter 6. Numerical Method • Eqn.6...5 Eqn.6-6 For the PIT model: Eqn.6-7 Eqn.6-8 A(".)=;'.;. = ~ 1 1 1 85 A modified elastic-viscous stress splitting (EVSS) method i is introduced to facilitate the choice of the discretization spaces. This requires the explicit introduction and discretization ofa tensor variable, ~ which is the gradient ofthe velocity M: d=Vu Thus, the rate ofdeformation tensor"DM can he expressed as: Eqn.6-10 The conservation equation for momentum (Eqn. 6-2) becomes: • The parameter a. can take any value, but a.=t'lo has been reported to be an optimal choice 1 • • Cbapter 6. Numerical Method 6.1.2 Free Surface 86 In the presence of a free surface, as in extrudate swell, a pseudo-concentration method is used 2 , requiring the solution ofthe additional equation: Eqn.6-12 aF -+u·VF=O ôt - where the function F represents the pseudo-concentration that describes the interface between the polymer and another fluid, in this case air: Eqn.6-13 { o inair F= 1 inthe polymermelt (fluidO) (fluid 1) This function specifies the material properties appropriate for each region of the computational domaine For example, the relaxation time Â. as a function of the pseudo- concentration is defined as follows: Eqn.6-14 Since air is not viscoelastic, Ào=O, and Eqn. 6-14 simplifies to: Eqn.6-15 Â(F) =Â.tF • AlI the material properties are evaluated using equations similar to Eqn. 6-14. Preliminary calculations were done using a fictitious tluid instead of air with a viscosity equal to 10- 3 times the polymer viscosity. Later, the viscosity of this fluid was set equal to zero, using the same value of ex. (Eqn. 6-11) in the entire flow domain, based on melt properties, without changing the results ofthe simulations. • In snmmary, the global systemof equations to solve is shown below. -vo(2aQ)+Vp- V o;;=-v{ d=V!f 87 Eqn.6-16 aF -+u·VF=O ar - 6.1.3 WeakFormulation In order ta obtain the weak fonnulation corresponding to this system, each equation is multiplied by an associated weighting function and integrated- over a domain Q. After applying the divergence theorem to the Stokes problem, we have the following weighted residual problem: Find fup.'t, dl e Vx 0 x k x L such that: "iIqeO Eqn.6-17 "iIye V • where q, !, Il and Ji{ are suitable weighting functions. The additional equation associated with the pseudo-concentration bas the following weak formulation: • Chapter 6. Numerical Method 88 Eqn.6-18 r[aF +u.VF~ d x = O -bat - j- 'tIGeç • 6.2 Discretization ûnly triangular elements are considered in this work. In arder ta solve the weighted residual problem by the finite element method, we need to discretize the variables. This is a difficult task, since compatibility conditions exist in the four-field Stokes problem for the discretized variables !!h, Ph' ~ and gp. Their respective discretization must therefore satisfy a generalization of Brezzi's condition 3 (the inf-sup condition) valid for the usual velocity-pressure formulation of the Stokes problem, presented in Chapter 2. The ch<>sen discrete subspaces are V h , Qh' ~ and ~ and the chosen discretization is illustrated in Fig. 6-1. Pt is the quadratic polynomial set deflned on element K, to which a cubic "bubble" function associated with the centroid node is added, and Plis the discontinuous linear polynomial set defined on element K. We chose identical discretization spaces for the variables ~ and g, but it is possible to select another discretization space for g. The discretized variable F h is approximated by piecewise linear polynomials, as for ~ and ~ . By replacing the spaces V, Q and ~ by their respective subspaces ~ , Qh and ~ in the continuous weighted residual problem (Eqn. 6-17), we obtain the following discrete weak formulation: • Chapter 6. Numerical Method such that: 89 Eqn.6-19 where tA and J/4 are suitable discretized weighting functions. For the pseudo- we have the folIowing additional equation: Eqn.6-20 We can see that the variable is a projection of DŒJ in the discrete subspace and as was mentioned in Chapter 2, there is no equality between and Df16t) in the discrete problem. • Figure 6-1 The P2+ - Pl - Pl - Pl elements for variables llh, Ph' and (and F.J 4. We aIso note the similarity between the constitutive equation (Eqn. 6-4) and the pseudo- concentration equation (Eqn. 6-12). Indeed, the upper and lower convected derivatives in the constitutive equation are defined as follows: • Chapter 6. Numerical Method Eqn.6-21 Eqn.6-22 v a! T 'Z"=-=+u· V,-Vu "-"(V0 = at - = -== a a'Z" ·,+'Z"·\7u = at - = = = - 90 • Both the constitutive equation and the pseudo-concentration equation are suitable for the discontinuous Galerkin method, which is described in section 6.3.2. 6.3 Numerical Solution The global system presented in Eqn. 6-19 and Eqn. 6-20 can be quite large, especially when a multimode representation of the extra stress tensor is employed. A coupled approach to solve this problem would require too much memory space, 50 we use instead a decoupled approach. We solve separately the following sub-systems: the Stokes problemfor fixed and the constitutive equation for fixed lib the projection = D(y,J for fixed lib. The coupling of these sub-systems, along with the treatment of nonlinearity, is done through GNmES (for .... Generalized Minimal Residual "l a Newton-Krylov iterative solver. When a free surface is present, the discretized pseudo-concentration F b is updated for fixed M& outside GMRES, by a fixed-point algorithme In the following sections, we describe the solution of each decoupled sub-system and the solution of the global four- field system by GMRES. 6.3.1 Stokes Problem For fixed the discretized Stokes problem is as follows: • Chapter 6. Numerical Method Find (UtJh) E V ~ such that: 91 Eqn.6-23 1bis discrete problem can be represented by the following matrix problem: Eqn.6-24 We use Uzawa's algorithm 6 to solve this system, allowing a reduction in the size of the global matrix. Furthermore, four degrees of freedom per element, two in velocity and two in pressure, are removed according to the condensation technique of Fortin and Fortin 7 • The system is finally solved througb. a direct method by aLU factorization. 6.3.2 Discontinuous Galerkin Method We will only describe the method for the constitutive equation (Eqn. 6-4), but a similar treatment is applied to the pseudo-concentration (Eqn. 6-12). In the more general time- dependent case, a fully implicit second order Gear scheme is used for the rime derivative. For stationary problems, like those studied in this work, the time derivative is simply dropped. The discrete weighted residual formulation of the constitutive equation is defined as follows in the traditional Galerkin method: • Eqn.6-25 If we decompose this formulation inta elementary sub-systems, we obtain the following weak formulation on each element K: • Cbapter 6. Numerical Method 92 Eqo.6-26 We use the upper-convected derivative for A ~ . J to simplify the last formulation, but a similar treatment can be applied to the Gordon-Schowalter derivative ofEqn. 6-7. A discontinuous polynomial approximation cP1) for each variable !ï of mode i is used on each element K. The velocity field Hh is known from the previous iteration. The discontinuous approximation gives rise to a discontinuity jump at the element interface in the derivative o f ~ , as illustrated in Fig. 6-2. The convective terro. in Eqn. 6-26 becomes: where l1K is the unit normal vector to the boundary 8K pointing outside K (Fig. 6-2). The inflow boundary 8K: with respect to the velocity field is defined as: Eqo.6-28 1Jr.J is the jump in the variable !i,h and is expressed as: • Eqo.6-29 • Chapter 6. Numerical Method u Figure 6-2 Discontinuity lump at the Interface of Two Elements 8 • 93 When we replace Eqn. 6-27 in the weighted residual formulation of Eqn. 6-26, and assume that ~ . h + =~ . h ' we obtain a new formulation: Eqn.6-30 This is the discrete fonnulation proposed by Lesaint and Raviart 9 • It results in a small nonlinear system on each element K, which is linearized by a Newton method: Eqn.6-31 • However, the resolution of this system requires knowledge of the quantity ~ - on elements adjacent ta BK: so that a particular numbering of the elements is necessary. A perfect numbering is not always possible. In the presence of recirculation zones, for example, the best possible numbering is provided and the elements are swept many times so that the resolution can be seen as a block relaxation (Gauss-Seidel type) method. ln conclusion, we have to solve for each i mode the nonlinear system given by Eqn. 6-30 of size 9 x 9 for 2D problems and 12 x 12 for axisymmetric problems, on each element. For the pseudo-eoncentration, we have to solve a 3 x 3 linear system given by the following equation: • Chapter 6. Numerical Method 94 Eqn.6-32 or in matrix forro: Eqn.6-33 2.3.3 Gradient Tensor ofVelocity The projection of D(yJ in the discrete subspace ~ can be calculated by considering the following problem for a fixed !lb: Eqn.6-34 The projection is achieved on an elementary basis since no continuity is required at the elements interface for variables ~ and !I!b. Therefore, we have to consider only these elementary systems: Eqn.6-35 The last equation can be expressed in matrix form: Eqn.6-36 • The small elementary systems are solved by a direct method with aLU factorization. Following the presentation of the various sub-systems, the global system, composed of Eqn. 6-19 and Eqn. 6-20 if a free surface is can now be expressed in the following way. • Chapter 6. Numerica1 Method 6.3.4 Resolution orthe Global System 95 Eqn.6-37 Eqn.6-38 =0 +B T Ph M K('!:!.K' 'ri J'ri = + H KTi - Ci = 1, N equations) =K =K =K OKdK Eqn. 6-37 is a nonlinear system, so we need ta use an iterative method to solve it. A fixed-point method is not applicable, since it results in a 10ss of numerical convergence as viscoelasticity becomes more important. The Newton method is more suitable for this problem. It generally applies to problems ofthe form: Find x such that RCx)=O. The algorithm. ofthe Newton method is as follows: 1. Given&, an initial guess; 2. Eqn.6-39 • where J n is the Jacobian matrix ofR(K,z) evaluated in&; 3. &+l=Xn+BX; 4. If IlôXll < &and IIR(Kn)11 < &, stop. Otherwise, go back to step 2. Our specifie problem can he expressed as follows: • Chapter 6. Numericai Method snch !hat d0=O where R(utvptv't bJ dJ is given by: +B T Ph ( ) E ( ) H - K M.K' t"i "i - K - K"i K =K =K =K K (i =1, N equations) 96 The difficuity for this large system is to solve the problem without calculating explicitly the Jacobian matrix of Eqn. 6-39, which would require too much memory space. To avoid building this large global matrix at every we use an approximation based on a finite central difference: Eqn.6-40 Jd Rex. +h4)-R(K-h4J - 2h • whereX=(utvptv't hJ dJTand h is small (usuaIly ofthe order of 10-6). The system to solve is highly nonlinear, and this causes convergence problems. In order to obtain a better conditioning of the system, we use instead a preconditioned residual 1.10, defined as (8!bôp,t5r,,5dr, where Ôlb ôp, and ôg are determined as follows: 1. Solve the following Stokes problem: =-B1l.h 8p =-A1l.h _B T Ph and obtain a solution corrected in velocity, y =1k + ôy. 2. Solve the constitutive equation by a Newton method for fixed MK, for all K elements: J K (!!K' "i )5,,; =-MK(!!K't"i )"i +EK&K)+HK"i - +HK5"i- =K =K =K =K =K =K (i =1, N equations) where JI<. is the Jacobian matrix obtained by the linearization due to the Newton method, ~ being fixed. The residual ô ~ is made up of all the elementary residuals ô!K. Note that it is possible ta ignore the last term of the previous equation and to perform. only one sweep, which accelerates the calculations. 3. Compute the local projections oftensor DC1lJ on all K elements: 0K t5d K =-OK d K +WU!K) The residual ô ~ is aIso made ofall the elementary residuals ôgI<.. • Chapter 6. Numerical Method 97 • The definition of this newresidual R is simply the preconditioning of the residual R with the inverse of a matrix J, which is a block-diagonal matrix bullt with the respective Jacobian matrix of the decoupled sub-systems. For the resolution of the coupled system ofEqn. 6-39, we used the G:MR.ES iterative method. This method has been designed to solve non-symmetricallinear systems. In the GMRES algorithm, the Jacobian matrix is only present in matrix-vector products. It is then not necessary to explicitly build the large global Jacobian matrix. Instead we use the approximation given by Eqn. 6-40, which reduces considerably the memory space needed. We tried ta incorporate the calculation of the pseudo-concentration (Eqn. 6-38) in the GMRES method for the calculation of the free surface, but it caused convergence problems. Instead, the free surface was updated outside GIvffiES, using a fixed-point algorithm. for fixed 1J.K. The other variables, solved in G ~ S , were calculated for a fixed position of the free surface. More details on the discontinuous Galerkin method applied to the pseudo-concentration can be found in reference 2. Finally, for sorne simulations at low Deborah number it was found useful ta do one to tbree fixed point iterations as a preconditioner to GMRES. However, this caused convergence problems at high levels of elasticity and had to be avoided for these flow conditions. • Chapter 6. Numerical Method REFERENCES 98 • 1 Guénette R., Fortin M., nA New Mixed Finite Element Method for Computing Viscoelastic Flows", J. Non-Newt. FI. Mech., ~ p.27-52 (1995) 2 Béliveau A., « Méthodes de Lesaint-Raviart Modifiées : Applications au Calcul des Surfaces Libres », Ph.D. Thesis, École Polytechnique de Montréal, Montréal, Canada (1997) 3 Brezzi F., Fortin M., "Mixed and Hybrid Finite Element Methods: Series in Computational Mechanics", Springer, NewYork (1991) _ 4 Fortin A., Béliveau A., Heuzey M.C., Lioret A., «Ten Years Using Discontinuous Galerkin Methods for Polymer Processing », to he published in the Proceedings of Int. Symp. on Discontinuous Galerkin Methods: Theory, Computations and Applications, Saive Regina College, Newport, RI, USA (1999) 5 Fortin A., Fortin M., "A Preconditioned GeneraIized Minimal ResiduaI Aigorithm for the NumericaI Solution ofViscoelastic Fluid Flows", J. Non-Newt. FI. Mech, ~ p. 277-288 (1990) 6 Fortin M., Fortin A., "A Generalization of Uzawa's Algorithm for the Solution of Navier- Stokes Equations", Comm. App/. Num.. Metk, 1, p. 205-208 (1985) 7 Fortin M., Fortin A., ''Experiments with Several Elements for Viscous Incompressible Flows", Int. J. Num.. Meth. F/., ~ p. 911-928 (1985) 8 Lioret A., "Méthodes d'Éléments Finis Adaptative pour les Écoulements de Fluides Viscoélastiques », Master's Thesis, École Polytechnique de Montréal, Montréal, Canada (1998) 9 Lesaint P. Raviart P.•<\., "On a Finite Element Method for Solving the Neutron Transport Equations", in "Mathematical Aspects of Finite Elements in Partial Differentiai Equations", C. de Boer (ed.),Academic Press, NewYork, p. 89-123 (1974) 10 Fortin A., Zine A., Agassant J.F., "Computing Viscoelastic Fluid Flowat Low Cost", J. Non- New!. FI. Meck, ~ p. 209-229 (1992) • 7. Planar Abrupt Contraction Flow offfigh-Density Polyethylene 7.1 Experimental Methods 7.1.1 Die Geometry Extrusion experiments were carried out using a slit die with transparent walls made of PyrexTM. The die is continuously fed by a Kaufinann PK 25 single screw extruder. It consists of a large reservoir, in which the pressure and the temperature are m e a s u r e ~ and a die land, which can be adjusted to vary the local geometry. Details of the die design are given in reference 1. We used only one configuration consisting of an abrupt 8: 1 contraction. The geometry of the slit die is shown in Fig. 7-1, and the dimensions are given in Table 7-1. With regard to the aspect ratios W/H of the reservoir and channel we note that these are smaller than the mjnimum value 2 of 10 recommended to ensure 2D flow, especially inthe case ofthe reservoir. This issue will be discussed when comparing data witb. the results of numerical simulations. AIl the experiments were camed out at 200 ± 2°C and at mass flow rates varying between 0.21 and 2.6 kg/h, which correspond to apparent shear rates in the channel in the range of 4.6 to 56 S-I. The transparent glass walls permit the use of laser-Doppler velocimetry and the measurement of flow-induced birefringence. FlowDirection w " • H Channel Reservoir Figure 7-1 Geometry of the Planar 8:1 Abrupt Contraction. H H L W WIH WIH (reservoir) (channel) (channel) (reservoir) (channel) 20 mm 2.5 mm. 20 mm 16mm 0.8 6.4 Chapter 7. Planar Abrupt Contraction Flow ofHDPE • Table 7-1 Geometrical Characteristics ofthe Die 100 • 7.1.2 Laser-Doppler Velocimetry Due to its high spatial and temporal resolution, laser-Doppler velocimetry (LDV) is a useful technique for local velocity measurement in translucent fluids, as it does not disturb the flow. LDV is based on the frequency shift of Iight that is scattered by small moving particles. The technique is as follows: a measuring volume is created by the intersection of two light beams of equal intensity, and the frequency of one beam is shifted with respect to the other. The frequency shift (Doppler shift) ofthe scattered light with respect to the reference dual beams is directIy proportional to the velocity of a moving particle in the fluid passing through the measuring volume. Details of the LDV method can be found in reference 3. We used the Flowlite™ dual-beam system of Dantec, controlled by the Flow Velocity Analyzer™ and operated from a persona! computer with the Burstware™ software (Fig. 7-2). A 10 mW He-Ne laser generates the incident light beam (red light: Â=632.8 nm). The beam is divided in two by a prism, and one beam has its frequency shifted (40 MHz) with respect to the other by a Bragg cell. The two beams are transrnitted by optical fibers to a probe or optical head, where a lens if = 160 mm) focuses the beams into the measuring volume. The dimensions of this ellipsoidal measuring volume are ôx = ôy = 108 J.1m and ôz = 913 JllD. 4 • The scattered light is received by the same probe and transmitted to the Doppler analyzer, which can perform fast-Fourier transform (FFT) analysis. The probe is placed on a XY translation table (Dantec Lightweight Traverse) that is controlled by Burstware™. Displacement along the z-axis is done manual.ly with a micrometric positioning screw. The x component ofthe velocity eux) was measured at six fixed x positions (three in the reservoir and three in the channel) as a function ofy. In Table 7-2 and Fig. 7-3 we present the fixed x positions where the velocity profiles were measured. Only one half of the geometry is shown, since it is symmetric with respect to the y axis. • Chapter 7. Planar Abrupt ConttaetioD Flow ofHDPE 101 Doppler signal analyzer He-Ne laser beam splitter -.-- .- ~ ~ ~ ~ ~ . : ~ ~ ~ =- ," '. :., ~ ; ; . . 4 photomultiplier Optical Head Flow ::::: ..... ::::: ..... ..... ..... ..... ..... .... .:: :: ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... • Figure 7-2 Experimental Set-Up ofLOV Measurements. Table 7-2 Velocimetry Measurements Locations Position Reservoir Channel 1 -14mm +2.0 mm 2 - 6.5 mm +10mm 3 - 3.0 mm +17mm Chapter 7. Planar Abrupt Contraction FlowofHDPE • 1 Reservoir 1 "'"--'---..-.1 o 00 102 Channel FlowDirection .. .. .. . . .. .. .. .. .. : : : : ·- ··- .. - ·- ··~ .. -.,. er-=- ·_. ·t--. · - · 1- . . . . . . . . . . : : : : : : : : : : (0,0) Figure 7-3 Positions ofthe Velocity Profile Measurements. 7.1.3 Flow-Induced Birefringence Birefringence occurs in an optically anisotropic material in which there are different velocities of light for different directions of propagation. Anisotropy is a result of the material structure and can appear in sorne materials at rest, such as calcite crystals. It can aIso be induced in materials that are isotropic at rest, for example by flow in polymer melts or elastic deformation ofsolid polymers. Inthe last two examples, birefringence is a result of the difference in polarizability of a polymer chain along its backbone and perpendicular to it. Polarizability is a measure of the strength of the response of the electrons in a bond between the atoms ofthe polymer molecule. • The use of birefringence to infer the components of the stress tensor in a flowing or defonned polymer depends upon the existence and validity of a "stress-optical relatïon"s. This relation is given by the semi-empirical stress optical law as a proportionality between the components ofthe refractive index and stress tensors: Eqn.7-1 where C is the stress optical coefficient. For a flexible polymer, this coefficient is proportional to the polarizabilities parallel and transverse to the polymer backbone. For a given polymer, C is essentially independent of the molecular weight and its distribution and is relatively insensitive to temperature s . • Chapter 7. Planar Abrupt Contraction Flow ofHDPE 103 The experimental set-up to measure flow-induced birefringence in polymer melts is shown in Fig. 7-4. A beam of polarized monochromatic light is directed at the melt flowing through the transparent slit die and then passes through an analyzer. We used a diffuse monochromatic light source of sodium (yellow light: Â.=589.8 nm). The optical anisotropy of the flowing polymer produces an interference pattern consisting of isochromatic and isoclinic fringes. The measurement of the isoclinic fringes is generally inaccurate and tedious 6 , 50 we added two quarter-waves plates on each side of the die to obtain only the isochromatic fringe pattern. analyzer transparent slit die polarizer diffuse monochromatic light source (sodium) 1 t quarter waves plates Experimental Set-Up for the Measurement ofFlow-Induced Birefringence. camera Figure 7-4 ....~ . • • Chapter 7. Planar Abrupt Contraction Flow ofHDPE 7.1.4 FlowRate-Pressure Curve 104 Extrusion experiments were carried out at twelve flow rates. In Fig. 7-5, we present the characteristic curve of flow rate vs. pressure for the extruder and die used, using pressure transducer data. If we neglect the correction for the entrance pressure drop, we can convert these data to shear rate vs. viscosity by applying the Rabinowitch correction. These results are presented in Fig. 7-6, along with rheological data, and they are in agreement with. the viscosity curve measured in the laboratory. 25 • • • 20 • • • • - • - ns 15 J:a • - ! • ~ en en 10 e • Q. • 5 3 2.5 0.5 1 1.5 2 Mass Flow Rate {kg/hl Figure 7-5 Characteristic Curve ofDie-Extruder at 200°C. o...- - - - ~ - - - - ~ - - - - - - - - - - - - - - - - - - ... o • Chapter 7. Planar Abrupt Contraction Flow ofHDPE 105 10000 100000 1000 0.1 0.01 x X X â1 âX âl1 ~ 6 i l ~ , 0 0 0 o Capillary Data (shifted at 200°C) 6. Sliding Plate Data 0 x Cone and Plate Data (shifted at 200°C) 0 • Extruder 0 n 1 10 100 Shear Rate (1/5) Comparison ofExtruder Data with Viscosity Curve ofHDPE at 200°C. 1.0E+OS 1.0E+01 0.001 1.0E+04 - et) m a. - ~ 1.0E+03 -Ci 0 u fi) :> 1.0E+02 Figure 7-6 • 7.2 Numerical Procedure . 7.2.1 Meshes Two meshes were used for the 2D simulations in order to verify the mesh independence of the solutions. Both meshes are composed of triangular elements. One is an unstructured mesh, while the other is structured. The number of elements for each mesh is presented in Table 7-3, and the meshes are shawn in Figs. 7-7 and 7-8 in the region near the contraction. The numerical procedure used was described in Chapter 6. Table 7-3 Mesh Characteristics for 2D Flow Simulations • Mesh Planar 1 Planar 2 Unstructured Structured Nomber of Elements 1277 2976 • Chapter 7. Planar Abrupt Contraction Flow ofHDPE Figure 7-7 Mesh Planar_l. 106 • Figure 7-8 Mesh Planar_2. 7.2.2 Viscoelastic Converged Solutions Numerical convergence is always limited in viscoelastic simulations of fields that contain flow singularities. The maximum tlow rate for convergence depends on the mesh, the longest relaxation time of the discrete spectrum, and the constitutive equation. As the mesh is more refined near the flow singularity or reentrant corner, convergence becomes more difficult as in the case of mesh Planar_2. Convergence was aIso more difficult when using the discrete spectrum based on all data as compared ta the spectrum based on truncated data (Table 4-4), since the longest relaxation time is longer in the former case (24.0 sec. vs. 6.25 sec.). We will refer to these spectra as the "full" and "truncated" spectra. We aIso found that the simple Leonov model gave more convergence problems than the PIT model, possibly because of stranger nonlinearity in the extra stress. It was not possible to determine exactly why this occurred. We did not try simulations with the modified Leonov model, considering the difficulties encountered using the simple mode!. We present a map of the converged solutions for the various cases (Tables 7-4 and 7-5). Convergence is expressed in terms of the number of flow rates converged, where the maximum is twelve (12). Note that for the PIT model, the same nonlinear parameters (ç, E) were used with the full and the truncated spectra, these parameters giving equal1y good fits ofthe rheological data. • Chapter 7. Planar Abrupt Contraction Flow ofHDPE 107 Table 7-4 Converged Solutions with the PTT Model (max=12) Spectrum Mesh Planar 1 Planar 2 Full 6 0 Truncated 12 12 Table 7-5 Converged Solutions with the Simple Leonov Madel (max=12) * third solution not fuIly converged Spectrum Mesh Planar 1 Planar 2 Full 1 0 Truncated 3* 1 . . 7.3 Total Pressure Drop For each flow rate we compared the predicted total pressure drop with the one indicated by the pressure transducer in the reservoir. The predicted pressure drop was evaluated as follows: Eqn.7-2 t:J'ca!c =Preservoir - PL=20mm • where Preservoir is the pressure in the reservoir at the position of the pressure transducer, and PL=20mm is the pressure in the slit die at L=20mm which corresponds ta the die exit In Fig. 7-9 the results of both viscoelastic models and the viscous model are compared ta the measured values. Note that only tbree points were obtained with the simple Leonov mode!. The calculated pressures are slightly overestimated by the viscoelastic models, while they are underestimated at high shear rates by the viscous model, but in general the agreement is good. Since the melt is predominantly in shear flow, even the viscous model can give a good approximation of the total pressure drop. In Fig. 7-10, we compare the predictions ofthe PIT model with the full and truncated spectra and we see that they agree very weIl. This shows that there is no advantage in using the full spectrum, which usually gives more convergence problems. Finally, mesh independence is demonstrated in Fig. 7-1 L • Chapter 7. Planar Abrupt Contraction Flow ofHDPE 108 25 ....----------------------------. 20 - ~ ca 15 ..Q - l! = CI) CI) 10 l! ~ 5 • • Measured o PTT, truncated spectrum, Planar_1 /j" Leonov, truncated spectrum, Planar_1 x Carreau-Yasuda, Planar_1 0.5 1 1.5 2 2.5 3 Mass Flow Rate (kglh) Comparison ofPredicted Total Pressure Drop with Experimental Values. O ~ - - - - . . . o i ! ~ - - - ~ - - - - - - - - - - - - - - - - - - .... o Figure 7-9 • Chapter 7. Planar Abrupt Contraction Flow ofHDPE 109 • 25 20 - .. as 15 .a - ! :::J CD CD 10 ! a. • 5 • , • • Measured o PTT, truncated spectrum, Planar_1 + PTT, full spectrum, Planar_1 3 2.5 0.5 O... o 1 1.5 2 Mass Flow Rate (kglh) Figure 7-10 Influence ofDiscrete Spectrum on Prediction of Total Pressure Drop. 25 .....---------------------------.. il 20 - .. as 15 .a - CI» .. :::J CD CD 10 ! D- i 5 • il il • Measured o PTT, truncated spectrum, Planar_1 o PTT. truncated spectrum, Planar_2 • o .... o 0.5 1 1.5 2 2.5 3 Mass Flow Rate (kglh) Figure 7-11 Mesh Independence for the Prediction of Total Pressure Drop. • Chapter 7. Planar Abrupt Contraction FlowofHDPE 7.4 Velocity 110 • The velocity profiles were measured for the three lowest flow rates, and the results are shown in Figs. 7-12 to 7-17 for the six measurement positions shown in Fig. 7-3 and Table 7-2. The predicted velocity profiles were computed on mesh Planar_l for all models using the truncated spectrum for the viscoelastic models. Note that bath viscoelastic models predict the same velocity profiles. Where the flow is dominated by shear (position 1 in the reservoir and all three positions in the channel) all models including the viscous model give good predictions of the velocity profiles. However, where the extensional component of the flow becomes important, as we get closer to the contraction (positions 2 and 3 in the reservoir), we start seeing disagreement between the predictions and the measurements. There is aIso more dispersion in the experimental results at these positions, since a small difference in tJx or 8y results in a large difference in velocity. The viscous model mas! underestimates the velocity as the extension becomes more important, but the viscoelastic models are aIso unable to capture weil the behavior ofthe flow. In Fig. 7-15 the viscous model is seen to be in better agreement with experimental data where the flow is dominated by shear. It appears that the flow becomes fully developed faster than predicted by the viscoelastic models. Again mesh independence was verified by comparing the results with mesh Planar_2. This is shown in Fig. 7-18 using the truncated spectrum for the third tlow rate of position 3 in the reservoir. We aIso verified for the same flow conditions that the use of the full spectrum did not improve the velocity profile predictions (Fig. 7-19). 10 1 4 • Position y (mm) % .- o.....ii! ..i!!.g:::::::::!..- .J o 111 ur;:;=:;::;:;:;:::===::::::;-----------:;:-""'l 0"'.' a 'A.., 14..1 .., -Laonovand PTT Model$ - - Canu.. ModIr 1.5 10 a 4 • Position y (mm) % ---' o 1.% a tA.1 à ,4..1., -t.--UIdPTTlIbIIIa Chapter 1. Planar Abrupt Contraction Flow ofHDPE • Figure 7-12 Velocity Profile (Reservoir, position 1). Figure 7-13 Velocity Profile (Reservoir, Position 2). 1.2.5 0.5 0.15 Position y (mm) 0.2.5 % OF;;....----------------' o 10 l'F o ::=:' ..::;;:.'=".==,= ........=:;----------, o ,.A., a PTTModils - - canuu Madel 10 1 4 • Posftlon y (mm' % 4r;::::=:::;::;;::::;====:::;'------7""1 o U.1 a 1..1., à 14..1., -1Aonovand PTTliIocIeJa - - can.u lIodeI 3.5 Figure 7-14 Velocity Profile (Reservoir, Position 3). Figure 7-15 Velocity Profile (Channel, Position 1). 1.%5 0.5 0.75 Position y (mm) 0.2.5 OF;;....----------------I o 2 10 .. .....----------, a 1.A.1 a - - Camau Modal 1.%5 0.5 0.75 Position Y (lIUII) 0.%5 % OF;;....- ------_--""" o 10 ... 01..1., a and PTTModel$ - - can.u Mode! Figure 7-16 Velocity Profile (Channel, Position 2). Figure 7-17 Velocity Profile (Channel, Position 3). • 10 a " 6 Position y (mm) 2 oAr---""l!I!!--==s::==-----L- ...J o 3 112 rr==::6=::17: U :=:="=:='=====::::;------:'""l 3.5 -PIT(II'Ul'lc:a1H spe:ll'Um. - - PIT(long spectrum. Planar_1) 0.5 e - 2 € 1.5 > 10 5 PosltJon y (mm) 2 o!-_-..__=s:::::::::.--Al.....- --J o Q.S Chapter 1. Planar Abrupt Contraction Flow ofHDPE • Figure 7-18 Mesh Independence (Reservoir, Position 3). Figure 7-19 Influence of Spectrum (Reservoir, Position 3). 7.5 Birefringence It was mentioned that the use of birefringence data to infer the components of the stress tensor in a flowing polymer depends upon the validity ofthe stress-optical relation: • where C is the stress optical coefficient. In order to compare the experimental birefringence with the calculated one, we need to know the value of this coefficient. An approximate value can be obtained by comparing the maximum. values of the fust principal stress difference ((u i -(2)calc ) calculated by the finite element simulations with the corresponding maximum birefringence (6 max ) measured on the axis of symmetry for all flow rates. The maximum. birefringence (6 max ) is obtained by fitting a cubic spline to the data at each flow rate and finding the maximum from the spline (Fig. 7-20). We obtained. a linear relation between (u} -(2)calcand Ll max (Fig. 7-21), which implies the validity of the stress-optical relation. By measuring the slope, we can evaluate the stress optical coefficient. We obtained a value of 1.47 x 10- 9 m 2 /N, which is lower than the value reported by Beraudo' for a LLDPE (2.1 x 10- 9 m 2 /N) but equal to that reported by Schoonen 8 (C= 1.47 X 10- 9 m 2 /N) for a low-density polyethylene. 45r------------------------------. Chapter 7. Planar Abrupt Contraction FlowofHDPE • 40 35 iô ~ 30 - )( ';"25 u c CD ~ 2 0 '5 = 15 m 10 5 ·4.65-1 09As-1 â14.4s-1 -19.1 5-1 %24.0 s-1 -28.75-1 -33.55-1 038.75-1 -42.85..1 <>47.75..1 Â52.2s-1 XS6.45-1 113 20 15 10 -505 Position (mm) -10 -15 0 ...--......--------------....---------..... -20 Figure 7..20 Evaluation of Ô max on the Axis ofSymmetry. We can make two types of comparison between the measured and the calculated birefringence, one qualitative and the other more quantitative. In the qualitative case, we look at the entire tlow domain and compare the shapes of the calculated birefringence fringes and their relative positions with experimental data. In the quantitative case, we compare the birefringence predicted by the various models with the measured birefringence on the axis of symmetry. In a 2D planar flow domain, the :first principal stress difference, which is used to calculate birefringence, is given by the following equation: Eqn.7-3 • Chapter 7. Planar Abrupt Contraction Flow ofHDPE 114 • 45 40 0 PTT pO 0 Leonov - - C =1.47 X 10- 9 m 2 /N 09 35 0/ 30 0/ 0/ ...... ." -y 0 25 op Y )( -K 20 ca E ~ O <j 15 /60 10 cD 5 / / 0.35 0.30 0.25 0.20 0.15 0.05 0.10 o ~ - - - - - - - - - - - - - - - - - - - - - - - - - _ ......__... 0.00 Figure 7-21 Calculation ofthe Stress Optical Coefficient. • In Figs. 7-22 and 7-23 we present the qualitative comparison on the entire flow domain for the third apparent shear rate (14.4 S-1), which is the highest rate for which we obtained a converged solution with the Leonov modeL Fig. 7-22 shows the solution obtained for the Leonov model, while Fig. 7-23 was obtained for the PTT mode!. Both models represent well the measured birefringence in terms of the shape and position of the fringes in the reservoir, but the Leonov model is closer to the data in the channeL In Fig. 7-24 we show the calculated birefringence for the highest apparent shear rate (56.4 S-1) for the PTT model. Agam, good qualitative agreement is obtained. • Chapter 7. Planar Abrupt Contraction Flow ofHDPE 115 • Figure 7-22 Comparison ofMeasured and Calculated Birefringence (Leonov Model, ra =14.45- 1 ). Figure 7-23 Comparison ofMeasured and Calculated Birefringence (pTT Model, ra =14.45- 1 ). 116 • • Chapter 7. Planar Abrupt Contraction Flow ofHDPE Figure 7-24 Comparison ofMeasured and Calculated Birefringence (pIT Madel, ra = 56.4s- 1 ). To evaluate more quantitatively the performance of the models, we compare the calculated andmeasured birefringence on the axis ofsymmetry. In Figs. 7-25 to 7-27, we show the results of the three models for the three lowest apparent shear rates. The Carreau-Yasuda model gives clearly the poorest prediction; the normal stresses relax immediately, going to zero as soon as the streamlines become paraIlel in the channel. While the predictions of the velocity profiles were similar for the two viscoelastic models, they now differ considerably; the Leonov model gives a much better representation of the experimental birefringence, with. the PIT model relaxing too fast in the channel, as was seen in the quantitative comparison on the entire flow domaine This fast relaxation is shown again for the highest apparent shear rate where we compare the predictions ofthe PTT and Carreau-Yasuda models. • Chapter 7. Planar Abrupt Contraction Flow ofHDPE 10 r.=.=U;:;::: ..1::=::=:;------------, -PTT ... - -c.n.u-V.... a •u i !' 4 ii z o __ __ __ ......" .aD .15 -10 -4 0 5 10 15 20 PaItIon (nwn) 117 1. ==;-----------, 1$ -PTT • LAanov :; -canuu Yuuca. a 1Z • t •a. c , ii oJ-__ __ _ -20 ·15 -10 -4 0 5 10 15 ZO Position (rrwn) Figure 7-25 Predicted and Measured Birefringence on the Axis ofSymmetry (fa =4.65- 1 ). Figure 7-26 Predicted and Measured Birefringence on the Axis of Symmetry Cra = 9.4 S-I). • 25 T'i=;::.=:::1 .... ..1:===;----------, -PTT 2D •• L..- -·can-Vauâ . & c:: 10 ! ii oL-__=====-__ __...;;:;;:;;ii........, ·15 .10 .s 0 5 10 15 2D Pœi!ian Elna) .zo -15 -la ..s a 5 10 15 2G Position (rrwn) Figure 7-27 Predicted and Measured Birefringence on the Axis of Symmetry (ra = 14.45- 1 ). Figure 7-28 Predicted and Measured Birefringence on the Axis of Symmetry (ra =56.4 S-I). • We saw previously that there was no difference in the predictions of the pressure drop ()or the velocity when using the full spectrum vs. the truncated one. For the birefringence on the axis of symmetry, there is a small difference in the maximum. birefringence if we the full spectrum. (Fig. 7-29). This difference amounts ta 1.2 % at an apparent shear rate of28.7 S-I. Similarly, there is a small difference in the maximum birefringence between the predictions with the two meshes (Fig. 7-30) of 2.4 %. What is remarkable with structured mesh is the smoothness of the curve and the absence of oscillations in the stresses. r;=:::;:::::?F::;===,---------, 1 · 47.7 .., 1 -PTT(UI1SU'UCIUftd) 118 oZ -15 -10 ..s 0 5 10 15 20 Position (mm) :> . .. c: •CIl ;E 15 e iD o.L-__:::::::;:::=-- ..... oZ .15 .10 ..s 0 5 10 15 20 PosItIon (mm) __""'------"""" 5 ='25 o - oc ";"20 f 15 i 10 iD Chapter 7. Planar Abrupt Contraction Flow ofHDPE • Figure 7-29 Influence ofSpectrum on Birefringence (fa =28.7s- 1 ). Figure 7-30 Influence ofMesh on Birefringence (ra =47.7 S-I). 7.6 Discussion In general, the results of simulations of planar abrupt contraction flow are disappointing. The model giving the most realistic representation of the experimental stresses, the simple Leonov model, is aIso the one posing the most serious numerical convergence problems. The predictions of total pressure drop by the viscoelastic models are good, but so are those of the strictly viscous Carreau-Yasuda modeL The viscous model is aIso able to give an. excellent prediction of the velocity field when there is no strong elongational component. However, when this component becomes important, even the viscoelastic models have difficulty in capturing all aspects ofthe behavior ofthe melt. • Noting the rapid change in the flow field near the contraction, we did a sensibility study in this region to evaluate the efIect of a small error in on the velocity profile. We looked at the predicted velocity fields assuming the position of the measurement was in errar by ± 0.5 mm in x at positions 2 and 3 in the reservoir. The results of the study are shown in Figs. 7-31 and 7-32. Surprisingly, the predicted velocity profiles are DOW in perfect agreement with the measurements. This shows that results of the simulations in terms of velocity might he hetter than originally thought, and that a small error in the measurement position where the f10w is changing rapidly can result in a large error in velocity. 10 4 1 Position y (mm) z 119 __---J o o.s '0 • , 5 PosIIIon Y(mm) 0.-=---..-------------1 o ,.1 r;::::=====::;--------;:;::::="'.... 'U.., O ........ fIII .. _. ,.5 -.0.5_ -- -u_ Chapter 7. PIanar Abrupt Contraction Flow ofHDPE • Figure 7-31 Influence of LU on Velocity Profile (reservoir, position 2). Figure 7-32 Influence of on Velocity Profile (reservoir, position 3). It was aIso conc1uded that there is little advantage in working with the full spectrum vs. the truncated. spectrum.. Of course, there is a limitation on the amount of truncation that can he done before starting to loose information about the material. Our truneation was sufficient to facilitate numerical convergence without affecting the predictions of the variables studied. The faster relaxation of the stresses in the channel predicted by the PTT model in comparison with the experiments was aIso observed by Beraud0 7 and Schoonen 8 . It is bard to say why the model behaves in this way. For one thing, the parameter that controis elongation, E, was determined from uniaxial elongation experiments, while we applied it to a flow where the elongation is planar. Furthermore, this parameter was determined at low extension rates, while this rate can become quite large in the actual fiow situations. For example, at the highest apparent shear rate (56.4 S-I) the extension rate on the axis ofsymmetry reaches a value of 8 S-1 near the contraction (x=O), as seen in Fig. 7-33. • Finally, the fact that we had a ratio of WIR < 10 neither in the reservoir nor in the channel may invalidate the assumption of 2D fiow. This could explain the difference between the shape of the predicted fringes in the reservoir ("butterfly" shape) and the experimental ones, which are Chapter 7. Planar Abrupt Contraction Flow ofHDPE 120 • 8 7 6 - ~ • (1) 5 - CD .. as 4 ~ c 0 3 -;; c G) 2 .. Je W 1 0 -1 -20 -15 -10 -5 0 5 10 15 20 X Figure 7-33 Extension Rate on the Axis of Symmetry Cfa =56.4 5- 1 ) . • • Chapter 7. Planar Abrupt Contraction Flow ofHDPE REFERENCES 121 • 1 Beaufils P., Vergnes B., Agassant J.F., "Characterization of the Sharkskin Defeet and its Development with the Flow Conditions ", lnt. Polym. Process.• IV(2), p. 78-84 (1989) 2 Wales IL.S., "The Application of Flow Birefringence to Rheological Studies of Polymer Melts ", PhD. Thesis, Delft University ofTechnology, The NetherIands (1976) 3 Dorst F., Melling A., Whitelaw J.H., " Principles and Practice of laser-Doppler Anemometry", Academie Press, 2 nd edition (1981) 4 Gavrus L., "Étude du comportement des polyéthylenes: Biréfringence d'écoulement et anémometrie laser", rapport d'étape, École Nationale Supérieure des Mines de Paris, Sophia 5 Dealy J.M., Wissbrun K.F., "Melt Rheology and its Role in Plastics Processing", Van Nostrand ReinholtL NewYork (1990) 6 McHugh A.J., Mackay M.E., Khomani B., " Measurements of Birefringence by the Method of Isoclinics '\ J. Rheal., ll(7), p. 619-634 (1987) 7 Beraudo, C., "Méthodes éléments finis pour la simulation numérique d'écoulements de fluides viscoelastiques de type différentiel dans des géométries convergentes", Ph.D. thesis, École Nationale Supérieure des Mines de Paris, Sophia Antipolis, France (1995) 8 Schoonen J.F.M., "Determination of Rheological Constitutive Equations using Complex. Flows", Ph.D. thesis, Technische Universiteit Eindhoven, Eindhoven, The Netherlands (1998) 9 Baaijens F.P.T., private communication (1999) • 8. Axisymmetric Entrance Flow and Extrudate Swell of Linear Low-Density Polyethylene 8.1 Experimental Methods 8.1.1 CapiIIaty Extrusion Extrusion experiments were carried out in a constant speed, piston-driven capillary rheometer, as described in section 4.2.2. Circular dies with a 90° tapered entrance angle of 1.4 mm diameter and three LJD ratios were used (4.75,9.5, 19.0). We aIso included in the study a longer die (L/D=38.0) for the extrudate swell experiments, in arder to ensure fully developed flow at the exit. A schematic viewofthe rheometer is shawn in Fig. 8-1. In this instrument, the apparent wall shear rate is related ta the flow rate, and the wall shear stress to the force measured by the load cell. AIl the experiments were carried out at 150 ± 1°C and at apparent shear rates between 2.22 and 88.8 5- 1 • The quantities measured are the total pressure drop given by the load cell, the entrance pressure drop, inferred from a Bagley plot or measured directly for an orifice die (see Fig. 4-3), and the extrudate swell. The experimental set-up for the extrudate swell measurement is described inthe next section. 8.1.2 Extrudate Swell Measurement The die swell experiments were aIso performed at ISO°C and apparent shear rates from 2.22 ta 88.8 S-I. The ratio measured is the time-dependent extrudate swell B(!}, defined as: Eqn.8-1 • where De is the capillary diameter and De(f) is the diameter of the extrudate, which is a funetion of time. In addition to the capillary rheometer, the experimental apparatus to measure die swell consists of a thermostating chamber, an opticai detection system and a data acquisition system. For the accurate measurement of the diameter of a soft, delicate, hot, and moving abject, a non-contacting sensor must be used. In this work an optical system was used, consisting of a photodiode array and a 2 mW He-Ne laser beam. This systemwas designed by Samara1. • Chapter 8. Axisymmetric Flow and Extrudate Swell ofLLDPE 123 1 A P total _t_ A P entrance Swell load ceIl Reservoir 1 Piston 1 Capillary Extrudate • Figure 8-1 Capillary Rheometer. A schematie view of the experimental apparatus is shown in Fig. 8-2. To measure die swell, the resin is extruded from the die into a thermostating chamber. An expanded and collimated laser beam is used as a back-light source to cast a shadow of the polymerie extrudate onto a linear photodiode array. An achromatic lens magnifies this shadow. The electrical signal from the photodiode array is then processed using a specially designed circuit. The data logger we used was designed to enable real rime acquisition of data using a microcomputer. Two problems can arise during extrusion of polymer directly into air. First, extrusion into ambient air leads to premature freezing and the development of frozen-in stresses. Second, if the polymer is extruded into a heated oven, it will sag under its own weight. The use of an oil-filled thermostating chamber elirninates both of these problems. The thermostating chamber is composed ofa stainless steel container, an outer and two rectangular stainless steel inner compartments. The inner compartments were equipped with two quartz windows each to allow the laser beam. to pass through. The chamber was equipped with two plug-screw300 watt immersion heaters. A tem..]>erature controller, operating in conjunction with a thermocouple, was used to control and monitor the temperature in the inner compartments. • Chapter 8. Axisymmetric Flow and Extrudate Swell ofLLDPE 124 signal conversion and data acquisition unit magnifying lens comFuter oscilloscope photodiode array l , . l thennostating chamber D············..··.....· He-Ne laser Figure 8-2 Experimental Setup for Extrudate Swell Measurements. • The outer compartment was filled with 200 centistokes (cS) silicone oil Fluid from Dow Corning). The inner compartments contained a mixture of silicone oils,. mainly 2 cS and 5 cS Dow Coming 200® Fluid. The oils were mixed in proportions found by trial-and-error to achieve a density slightly lower than that of the meIt at Üe test temperature. We found that a 52 % - 48 % by volume mixture of 2 cS - 5 cS gave the best results. Vle aIso determined the small amount of swell due to the oil and sul:>traeted it from the measured swell. This procedure is discussed in section 8.5.1. Note that the maximum. temperature to which the ails could be heated was 150°C, which expLains the selection ofthe operating temperature for the experiments. A complete descripti<>n ofthe experimental apparatus and procedure can be found inreference 1. • Cbapter 8. Axisymmetric Flow and Extrudate Swell ofLLDPE 8.2 Numerical Procedure 8.2.1 Meshes 125 Simulations of entrance pressure drop and extrudate swell were performed separately. The boundary conditions of fully developed flow were used for the extrudate swell simulation so that it could he decoupled from the flow in the reservoir. Two meshes were used for the axisymmetric entrance flow simulations in order to verify mesh i n d e ~ n d e n c e of the solutions. Both meshes are composed of triangular elements and are unstructured. Several meshes were tested for extrudate swell, but it was possible to obtain numerical convergence at our test conditions on only one ofthem. This mesh is aIso unstructured and made of triangular elements. The number of elements for each mesh is presented in Table 8-1, and the meshes are shown in Figs. 8-3, 8-4 and 8-5. Table 8-1 Mesh Characteristics for Axisymmetric Flow Simulations • Mesh An 1 AD 2 SweU 1 Unstructured Unstructured Unstructured Number of Elements 1611 3402 1822 Figure 8-3 Mesh Axi 1. • Chapter 8. Axisymmetric Flow and Extrudate Swell ofLLDPE Figure 8-4 Mesh Axi 2. 126 • Figure 8-5 Mesh Swell_l. 8.2.2 Viscoelastic Converged Solutions Numerical convergence can be easily summarized for the axisymmetric entrance flow simulations. The PIT model gave converged solutions for all seven experimental flow rates, on both meshes and with both the full and truncated spectra shown in Table 4-3. On the other band, it was impossible to obtain a converged solution with the simple Leonov model for any ofthese conditions at one ofthe experimental flow rates. For extrudate swell simulations, a pecuIiar behavior was observed with the simple Leonov model. As we slowly increased the flow rate from Newtonian behavior to viscoelastic, the ultimate swell ( t ~ C ( » ) started to decrease from its Newtonian value of 13 %. This behavior is shown in Fig. 8-6 (mesh Swell_l, truncated spectrum). A sunHar observation bas been reported by Rekers 2 . However, we lost convergence at a very low shear rate, and it was not possible to verify if this trend was continuing at higher shear rates. We present below a map of the converged solutions for the various cases of entrance flow and extrudate swell. 1t is expressed in tenns of the number of flow rates converged, where the maximum is seven. Spectrum Mesh Axi 1 Axi 2 Swell 1 Full 7 7 0 Truncated 7 7 1 Chapter 8. Axïsymmetric Flow and Extrudate Swell ofLLDPE • Table 8-2 Converged Solutions with the PIT Model (max=7) 127 Table 8-3 Converged Solutions with the Simple Leonov Model (max=7) • Spectrum Mesh Axi 1 Axi 2 Swell 1 Full 0 0 0 Truncated 0 0 0 13.5 ....---------------------------. - 13 0 :::..e 0 ~ 'Ci) ~ fi) S 0 ftI E .- 5 12.5 0 12 ......------------.-------------...... 0.0001 0.001 0.01 0.1 1 ApparentShear Rate (1/5) Figure 8-6 Ultimate Swell Predicted by the Simple Leonov Madel. 8.3 Total Pressure Drop The numerical results presented here were obtained with the PTT and Carreau-Yasuda models. We compare the calculated and experimental total pressure drop for an LID of 19.0, since the viscosity was thought to be affected by the pressure with the LID of38.0, and pressure effects are not included in the finite element modeL The predicted total pressure drop was evaluated as follows: • Chapter 8. Axisymmetric Flow and Extrudate SwelllClfLLDPE 128 Eqn.8-2 In Fig. 8-7, the results of both models are e=ompared to measured values. The calculated pressures are slightly underestimated at high shear rates, especially by the Carreau- Yasuda model, but in general the agreement is good. Since the melt is predominantly in shear tlow, even the strictly viscous moder.. can give a good approximation of the total pressure drop. In Fig. 8-8, we compare the predictions of the PTT model with the full and the truncated spectra, and we see that they agree very weil. Finally, mesh independence is demonstrated in Fig. 8-9. 25 o Measured 6. PTT {full spectrum, axi_1l 20 o Carreau-Yasuda (axi_1) ~ ~ 0 -ca 15 D- ~ :i 6. - Ci 0 - 0 - 10 ~ D- A <1 5 o 1 10 Apparent Shear Rate (1/5) 100 • Figure 8-7 Comparison ofPredicted Total Pressure Drop with Experimental VaIues. 25 p- .... Chapter 8. Axisymmetric Flow and Extrudate Swell ofLLDPE 10 129 o Measured ~ PTT (full spectrum, axi_1) o PTT (truncated spectrum, axi_1) 5 20 - :. 15 :lE - 1 a. <1 • o 1 10 Apparent Shear Rate (1/5) 100 Figure 8-8 Influence of Discrete Spectra on Prediction ofTotal Pressure Drop. 25 o Measured ~ PTT (full spectrum, axL1) o PTT (full spectrum, axi_2) 5 20 - ca 15 a. :e ~ -=- Q 'Ci - 0 - 10 Q. <1 100 10 Apparent Shear Rate (1/s) OL.. ..... ... 1 Figure 8-9 Mesh Independence on Prediction of Total Pressure Drop. • • Chapter 8. Axisymmetric Flow and Extrudate Swell of LLDPE 8.4 Entrance Pressure Drop 130 ln the entrance region the melt undergoes both shear and elongation, 50 we expect to see a difference between the predictions ofthe viscoelastic and the viscous models. Fig. 8-10 shows the measured values of the entrance pressure drop along with the predictions of both. models. The predicted entrance pressure drop was evaluated as follows: Eqn.8-3 M'cale =Presen;oir - PLI R=O-S • The Carreau-Yasuda model underestimates the entrance pressure drop, as expected because of the strong elongational component of the flow. For example, at the highest apparent shear rate (88.8 S-l) the extension rate on the axis of symmetry reaches a value of22 S-1 near the contraction (x=O), as seen in Fig. 8-11. The oscillations in Ë after the contraction are typical of results obtained on unstructured meshes when using the discontinuous Galerkin method 3 • However, the PIT model also underestimates the entrance pressure drop by about 50%. Earller simt!lations of entrance flow have also produced entrance pressure drops weil below those observed experimentally4.5.6.7.8. In Fig. 8-12, we compare the predictions of the PIT model with the full and truncated s p e c ~ and we see again that they agree weIl. Finally, mesh independence is demonstrated in Fig. 8-13. • Chapter 8. Axisymmetric Flow and Extrudate Swell of LLDPE 3 o Measured (orifice die) 6 PTT (full spectrum, axL1) o Carreau-Yasuda (axi_1) 131 êâ 2 a.. :& - 100 o 6 o 6- o f ê t o O .... ..... ~ 1 10 Apparent Shear Rate (ils) Figure 8-10 Comparison ofPredicted Entrance Pressure Drop with Experiments. CD U c ! ... c CD a.. 1 <1 25 ..... ..... 5 10 -5 0 ....... -- - 15 ~ • en - S ca a: c o ëii c ~ W 20 40 30 20 10 o -10 -20 -10 .....- - . - 0 1 ! ' - - - " " " ' ! - - - - ~ - - - - - - - - - - - - - ..... ..30 x Figure 8-11 Extension Rate on the Axis ofSymmetry (ra =88.8s- 1 ). • • Chapter 8. Axisymmetric Flow and Extrudate SweII ofLLDPE 3 <> Measured (orifice die) 6 PTT (full spectrum, axi_1) o PTT (truncated spectrum, axi_1) 132 li 2 D. :e - ~ u c ! .- c CD D. 1 <] o i il 100 o 1 10 Apparent Shear Rate (1/s) Figure 8-12 Influence ofDiscrete Spectrum on Prediction of Entrance Pressure Drop. 3 100 <> Measured (orifice die) 6 PTT (full spectrum, axi_1) o PTT (full spectrum, axi_2) . l t 0 . m 0 t f Q . i Q ~ Q il . 10 Apparent Shear Rate (1/5) Figure 8-13 Mesh Independence on Prediction ofEntrance Pressure Drop. o 1 :: 2 :e - CD U c ! .- c CD D. 1 <] • • Chapter 8. Axisymmetric Flow and Extrudate Swell ofLLDPE 8.5 Extrudate Swell 8.5.1 Experimental Results 133 • The measurements were made with an L/D of38.0 to ensure fully developed flow at the die exit. Comparisons with swell data for L/D = 19.0 showed that fully developed flow was not reached in the shorter die since swell was larger. Fig. 8-14 shows the measured swell as a function oftime for the seven apparent shear rates. Utracki et al.9,1 0 have found that Dow Coming 200® fluid silicone oils do not swell polyethylene if medium viscosity grades (50 cS and 100 cS) are used. However, sorne interaction between the oil and polymer occurs when low viscosity grades are used with polyethylene. In the present study, a mixture of low viscosity grades of silicone oil was used (2 cS and 5 cS) in arder to pravide an ail density slightly lower than the polymer melt density. The swell due to the oil was measured so that it could be subtracted from the measured swell using a method similar to that ofreference 1. The resin was extruded directly into air at an. apparent wall shear rate of 13.3 S-I. Cut extrudate strips were annealed far 30 minutes in an oyen at 150°C in a bath of500 cS silicone ail. We verified that the high viscosity oil was not absorbed by the polymer by weighing the sample before and after annealing. The annealed samples were placed in the thermostated chamber, and extrudate swell was measured at 150°C using the apparatus. It seems reasonable to assume that after annealing, all the stresses are relaxed and the equilibrium swell is reached. Consequently, any additional swell would be due ta interaction with the ail. The oil swell curve is shown at the bottom of Fig. 8-14. This swell was subtracted from all the extrudate swell measured values. This i<; shawn in Fig. 8-15 for the apparent shear rate of88.8 5- 1 • In this way the net swell without the effect of oil was obtained. We assumed that the oil absorption effect was the same at all shear rates. No data are available at short times for the lowest apparent shear rates in Fig. 8-14. The method used does not allow an accurate capture of the instantaneous swell that occurs in the first seconds of extrusion for these rates, because a significant time is required for the sample to reach the viewing chamber at these shear rates. Finally, the ultimate net swell, obtained at long times, is shown in Fig. 8-17 as a function of apparent shear rate. • Chapter 8. Axisymmetric Flow and ExtnIdate Swell ofLLDPE 134 1.8 1.6 - -; tn 1.4 1.2 • 1.0 o 200 40Q 600 800 1000 1200 1400 Time (5) 0 Oïl Swell 0 2.221/5 6- 4.441/5 "V 8.881/5 0 13.3 1/5 0 22.21/5 0 44.41/5 0 88.81/5 Figure 8-14 Total Swell and Oil Swell as a Function ofTime. • Chapter 8. Axisymmetric Flow and Extrudate Swell ofLLDPE 135 1.8 1.6 a 1.4 o Oil Swell o Total Swell (88.8 1/s) A Net Swell (88.8 1I/s) 1.2 600 800 1000 12()0 1400 Time (5) 400 200 1.0 o Figure 8-15 Net Swell as a Function ofTime after Oil Swell Subtraction. 8.5.2 Comparison with Simulations • As was mentioned in section 8.2.2, it was not possible to calculate the viscoelastic swell for most conditions, although the Newtonian swell (13 %) was calculated on severa! meshes. The Leonov mode! was discarded early, fust because of its pathological prediction of decreasing swell, and second because of numerical conv-ergence problems. With the PIT mode!, which showed aImost no convergence problems in other situations, we were able to obtain a converged solution only for the lowest apparent shear rate (2.22 S·I) and only on one mesh. In Fig. 8-16, we show the predicted interface calculated on mesh Swell_l with the truncated spectrum. The ultimate swell (20 %), shawn in Fig. 8- 17 along with experimental values, is aImost half of the measured swell for that flow condition (37 %). The experimental ultimate swell data is presented in the appendix. Since we could not obtain a converged solution using the full spectrum, it is not possible to know at this time if the low calculated swell is related to the use of the truncated spectrum. As mentioned previously in section 2.4.3, extrudate swell depends stronglyon molecular weight distribution, Le. on the detailed speetrum. • Chapter 8. Axisymmetric Flow and Extrudate Swell ofLLDPE 136 - - [ - - - ~ Figure 8-16 Calculated Interface with PIT Model (truncated spectrum, mesh Swell_l). 1.8 r--------------------------..... 1.6 - ~ ; ~ fi) ~ S 1.4 eu t E ;; Measured 5 <> t::.. PTT (truncated spectrum, swell_1) 1.2 t::,. .Newtonian Value .. -- - - - .. - . - - -- . - - . - - - - - - - - - - - - - - • 1 o 20 40 &0 80 Apparent Shear Rate (1/s) Figure 8-17 Measured and Predicted illtimate Swell. 100 • Chapter 8. Axïsymmetric Flow and Extrudate Swell ofLLDPE 8.6 Discussion 137 • In general, the results of simulations ofaxisymmetric entrance flow and extrudate swell are disappointing. The performance of the simple Leonov model is much worse than in the planar case, in terms of numerical convergence problems and the prediction of decreasing swell. The Leonov model gave realistic predictions for planar contraction flow (see Chapter 7) but was not useful for axisymmetric extrudate sweli. The PIT model was much easier ta work with in terms of numerical convergence for the prediction of entrance flow but not for the free surface flow. The predictions of total pressure drop are good, but so are those of the strictly viscous Carreau-Yasuda model. Predictions of entrance pressure drop are largely underestimated, even with the viscoelastic PIT model. We also tried using E=Ü in the simulations, which gives the strongest strain-hardening behavior in elongation, but this ooly increased the prediction ofthe entrance pressure drop by 6 %. This poor performance is similar to what is seen in planar flow for velocity fields with a strong elongational component (section 7.4). At this time, it is difficult to say why this is 50. There are two possibilities: either the models are inappropriate, or the parameters determined in simple flows can not reproduce the features of (;omplex flows. However, the models must aIso be correct for the simple flows. We also note that the elongation rate in the flow simulated can be much larger than the range of extension rates for which it was possible to fit the viscoelastic model parameters. Beraudo 7 suggested that the most appropriate quantity to be compared with the measured pressure is the rr component of the Cauchy stress tensor, evaIuated at the wall. Using this suggestion for the orifice die, we have: Eqn.8-4 M ca/e = -cr,.,. reser1loir + (j,.,. LI R=O.S = Preser1loir - PL1R=O.S + Il,,rr where â ~ rr = ~ ,.,. LI R=O.s - ~ rr l'eser1loir The contribution of the extra stres:s difference, ~ ' t r r , is negative and decreases the computed entrance pressure drop bF 3-4 %, thus increasing the discrepancy with the measured values. • Chapter 8. Axisymmetric Flow and Extrudate Swell of LLDPE 138 Finally, for the free surface predictio:n, the difficulty was not with the calculation of the free surface position itself but in t:ryiI::J.g to converge the extra stresses for a fixed position of the interface. We also tried an alternative to the use of the pseudo-concentration, where the interface is modeled by a ::function h (for height) calculated from the velocity profile. Eqn.8-5 • But this did not facilitate the convergence of the extra stresses and failed to give a converged solution for any ofour exp<rimental conditions. • Chapter 8. Axisymmetric Flow and Extrudate Swell ofLLDPE REFERENCES 139 • 1 Samara M.K., 1;1; Design and Construction of Computerized Die Swell Apparatus", Master's Thesis, McGill University, Montreal, Canada (1985) 2 Rekers G., 1;' Numerical Simulation ofUnsteady Viscoelastic Flow ofPolymers", Ph.D. Thesis, University ofTwente, Twente, The Netheriands (1995) 3 Lioret A., "Méthodes d'Éléments Finis Adaptative pour les Écoulements de Fluides Viscoélastiques », Master's Thesis, École Polytechnique de Montréal, Montréal, Canada (1998) 4 GuilletJ., Revenu P., Béreaux Y., Clermont J.-R., and Numerical Study ofEntry FlowofLow-Density Polyethylene Melts", Rheol. Acta, & p. 494-507 (1996) S Fiegl K., ôttinger H.C., 1;' The Flow of a LDPE Melt through an Axisymmetric Contraction: A Numerical Study and Comparison to Experimental Results ", J. Rheo!., 38(4), p. 847-874 (1994) 6 Mitsoulis E.; Hatzikiriakos S.G., Christodoulou K., Vlassopoulos D., "Sensitivity Analysis of the Bagley Correction to Shear and Extensional Rheology", Rheo!. Acta, TI, p. 438-448 (1998) 7 Beraudo C., "Méthodes éléments finis pour la simulation numérique d'écoulements de fluides viscoelastiques de type différentiel dans des géométries convergentes", Ph.O. thesis, École Nationale Supérieure des Mines de Paris, Sophia Antipolis, France (1995) 8 Hatzikiriakos S.G., Mitsoulis E., "Excess Pressure Losses in the Capillary Flow of Molten Polymers", Rheal. Acta, J1, p. 545-555 (1996) 9 Utracki L.A., Catani A., Lara J., Kamal M.R., "Polymer Processing and Properties'" G. Astarita and L. Nicolais, eds., Capri, Italy, p. 79 (1983) 10 Utracki L.A., Lara J., Intern. Workshop on Extensional Flows, Mulhouse-La Bress, France (1983) • 9. Alternative Approaches • Inorder to optimize the design. of plastics processing equipment and the processability of plastics resins other than by trial and error, it is necessary to have a reasonably accurate model of the process of interest. However, the numerous difficulties involved in viscoelastic flow simulation have led to the development of alternative approaches that avoid the explicit modeling of fluid elasticity. These make use of specifie material functions, particularly viscosity, normal stress differences and elongational viscosity. Highly simplified constitutive models are used, which are strictly valid only in steady simple shear or elongational flows and employ empirical equations fitted to experimental data. Inthis chapter, we describe severa! of these simplified approaches and propose the use of a rule-based expert system as an alternative to full viscoelastic flow simulations. 9.1 Viscometric Flow The viscometric flow approximation bas been used successfully by Vlachopoulos et al. 1.2.3. 4 • This is based on the observation that most polymer processing operations involve flow in channels where there is a main flow direction such that the streamlines are almost parallel, therefore justifying the treatment of the flow as locally viscometric. This approach is based on the CriminaJe-Ericksen-Filbey (CEF) equation (Eqn. 2-16) and requires a minimum amount of experimental information, i.e. the viscosity and normal stress differences. It takes into account the effect of normal stresses, which is a manifestation ofviscoelasticity, but avoids explicit reference to the fluid memory. In the field ofnumerical simulation, one often talks about the "high Weissenberg number problem". However, this revea1s an error in terminology. The reference should be to the "high Deborah number problem", the term introduced in Chapter 2. The Weissenberg number (We) indicates the degree of nonlinearity or anisotropy that is manifested in a given flow, while the Deborah number (De) indicates the relative importance of elasticity. • Chapter 9. Alternative Approaches Eqo.9-1 Eqo.9-2 We=Â.Y 141 • The Weissenberg number is the product of a charaeteristic relaxation time ofthe material and a typical strain rate, while the Deborah number is the characteristic relaxation time of the fluid Âjluid divided by the characteristic time of the flow Â-jlow. The characteristic time ofthe flow Âflow is a time reflecting the variation ofthe strain history. The confusion between the two arises from the fact that in many flows of practical importance they are directly related to each other. The distinction is particularly important in flow stability analyses, and those working in this field have been very careful not to confuse the two expressions. For example, Larson s has properly distinguished between De and We. M o r e o ~ e r , McKinley6 has explained in detail how the !Wo are related in converging flows and has introduced a "scaling factor" that expresses this relationship quantitatively. Fully developed flow in a channel of constant cross section, which is a viscometric flow, is a flow in which De is zero because of constant-stretch history but in which We increases with the tlow rate. Vlachopoulos et al., among others, have had considerable success in the modeling of melt flows in which De is small but We can he quite large. Examples of forming processes simulated using the viscometric flow approach include extrusion 7 , film blowing 8 , thermoformini and calendering lO for polymer melts and elastomers ll . However, we have to keep in mind that the viscometric approach remains an approximation, which is similar to the lubrication approximation used to simplify the momentumequation in certain polymer processes, such as injection molding. GUpta 12 bas also adopted a simplified approach, but this time to simulate the axisymmetric entrance flow of polymers. Generalized Newtonian models, which have been successfully used in shear-dominated flows, do not provide an accurate simulation of applications involving an elongational flow, such as flow at the entrance of an extrusion die. On the other hand, viscoelastic constitutive equations have not produced simulations that are in good quantitative agreement with experimental data and are subject to convergence problems at high Deborah number. Gupta has developed software for the simulation ofaxisymmetric polymerie flow that requires only knowledge of the viscosity and the strain-rate dependence of the elongational viscosity of the polymer (which is not easy to measure). According to the author, bis software can accurately preclict the velocity and pressure field in applications involving a significant elongational flow component. However, the constitutive equations employed in the software do not predict nonnaI stresses in shear flow. Apparently, the incorporation of a first normal stress difference prediction in the software gives convergence problems13, and the incorporation of a second normal stress difference bas not been tried yet. Viscosity and elongational viscosity are modeled using the following equations: • Chapter 9. Alternative Approaches 9.2 Power-Law Models for Elongational Viscosity 142 Eqn.9-3 Eqn.9-4 T'fe = 3"10 for IID ~ IIDt • The strain rate for transition from Newtonian to power-Iaw behavior can be different for the shear and elongational viscosities. A more elaborate four-parameter model for the elongational viscosity has aIso been used, but the author bas not revealed its form13. The parameters are determined by inverse problem solving; for a given shear rate, the viscosity and the entrance pressure drop are measured, and the parameters ruling the elongational viscosity are found by trial-and- error fitting. GUpta 12 reported that the (elongational viscosity)/(shear viscosity) ratio was important in deterrninjng the recirculating vortex and the extra pressure 10ss in axisymmetric entrance flow. Obviously, the predictive capabilities and the versatility of this approach are limited. Two more references, Gotsis and Odriozola l4 and Schunk: and Scrîven15, explain how elongational viscosity can be incorporated in simulations without using a viscoelastic constitutive equation. • Cbapter 9. Alternative Approaches 143 • • 9.3 Approximate Prediction of Swell Seriai et al. 16 derived a semi empirical equation to predict the extrudate swell of linear polymers. The model is based on the rubher-like elasticity theory and on the calculation ·of the elongational strain recovery of a Lodge fluid17,18. The theoretical extrudate swell ratio mainly depends on the relaxation modulus, the extension ratio, and the recoverable shear strain. The main advantage ofthis model is that it provides good accuracy for short dies over a wide range of shear rates, where other available semi-empirical theories assume fuIly developed tlow in a long die1.9• In extrusion processes, relatively short dies are generally used, and transient flow dominates. The strain history of the melt is thus of central importance and can not be neglected. Bush 20 has also a simplified simulation of extrudate swelI. He models extrudate swell as the stratified flow oftwo, Newtonian, isothermal fluids with different viscosities. By suitable selection of the viscosity ratio, the model can take into account nonisothermal, shear-thinning and elastic effects. The model provides a means of simulating complex geometriçal effects in profile extrusion without the burden of a full viscoelastic model and provides a practical aid for die design. The application to three- dimensional flow problems is also feasible using this method. 9.4 Rule-Based Flow Simulations The simplified approaches described above have been helpful in certain applications, such as the design of certain types of die, extruders and injection molds. However, the capabilities and versatility of these approaches remain limited. On the other hand, no practical method has been developed for dealing with viscoe1astic flow situations in which there is a strong convergence or divergence of streamlines or a singularity in the boundary conditions. • Chapter 9. Alternative Approaches 144 • • While there is no question that the rheological mode!s used to date in the numerical simulation of the flow of polymerie liquids are quite inadequate, the computational problems encountered in viscoelastic flow simulations occur even when the inaccuracy of the mode! is unlikely to be the cause. _These problems arise under conditions where the models are thought to be reasonably reliable, i.e., at low De and when slip or fracture is not anticipated. Therefore, it is concluded that J:he of the problem is the viscoelasti.city ofthe fluid, i.e.,' the dependence of its state of stress on the past history of the deformation. In particular, the. notorious convergence problems mentioned above do not occur in the simulation of flow at small De. This is strengthened by the observation that it has not been possible by elaboration or variation of the numerical method used in the simulation to eliminate the convergence problem. Neither has it been possible ta eliminate the problem by the use ofmolecular dynamic models ofrheological behavior. It is very unlikely that any single, closed fonn constitutive equation will ever be developed that can describe accurately all aspects of the rheological behavior of highly entangled molten polymers. Furthermore there are phenomena, such as slip and fracture, that can not be predicted by a continuum mechanics mode! but that must be taken into account in the modeling ofmany industrial forming operations. Obviously, a new approach is needed to deal with such tlows. The most likely approach appears to be a rule-based expert system that is able to select, from a repertoire of possible models of rheological and slip behavior, the one that is most appropriate in each region of the tlow field. A major challenge in the development of snch a system is a method for matching the solutions at the boundaries of neighboring regions. In this last section, we provide background information on rule-based expert systems and an overview ofhow such a system might be used in the simulation ofviscoelastic tlows. Expert systems, based on techniques developed from artificia1 intelligence research, are currently used in a number of engineering applications, especially in the manufacturing and process control domains 21 • An excellent review of knowledge-based expert systems for materia1s processing has heen presented by Lu 22 . • Chapter 9. Alternative Approaches 9.4.1 Fundamentals ofKnowledge-Based Expert Systems 145 • An expert system can he distmguished from a conventiona1 computer programin terros of its three basic components 22 , which are shawn in Fig. 9-1 and listed be1ow. 1. A knowledge base - which contains domain knowledge such as rules, facts, and other mformation that may be useful in formulating a solution. 2. An inference engine - whiêh applies proper domain knowledge and controIs the order and strategies ofproblem solving. 3. A blackboard - which records the interm.ediate hypotheses and resuIts that the expert system manipulates. Knowledge-Based Expert Systems 1Blackboard (data) I ~ " - - - - - , Control Inference 1 Knowledge Base • Conventional Programs 1... D_a_t_a II.- ..-----, P_r_o_g_ra_m 1 Figure 9-1 Expert Systems vs. Conventiona1 Programs 22 • A conventional program contains ooly two parts, the program and the data, as shown in Fig. 9-1, and all the required control and domain knowledge is coded into the program. that manipulates data in a sequential manner with many conditional branches. During execution, the status of the problem is represented by the current values of the variables in the program. • Chapter 9. Alternative Approaches 146 '. • In an expert system, the knowledge base is treated as a separate entity rather than appearing ooly implicitly as part of the program. It contains facts and mIes that are used as the basis for decision-making. The knowiedge related to the control of decision- 'making is collected in the inference engine, which is physically separate from the knowledge base. The inference engine has an interpreter that decides how to select and apply the mIes from the knowledge base to infer new information, and a scheduler that determines t h ~ order in which the selected rules should be applied. The blackboard is a separate entity that serves as a working memory for the system. It can be viewed as a global database for keeping track of the current problem status and other relevant information during problem. solving. The separation of the task-level knowledge (stored in the knowledge base) from the control knowledge (stored in the inference engine), and the addition of a separate working mem.ory (the blackboard), gives expert systems greater tleJC'ibility in both the implementation and execution stages, Many applications can benefit from the use of such a system. 9.4.2 Application ta Viscoelastic Flow Simulations One way to approach effectively viscoelastic flow simulations is through computational domain decomposition23. For example, the Schur complement method is based on a domain decomposition that leads to a decoupling of the system on a sub-domain basis rather than on a component basis. The advantage of this technique is that all the unknowns in an element (stress, velocity and pressure) exhibit strong local coupling, which makes a decoupling on a component basis less attractive 23 • Another strong advantage of this technique is that it enables the use of the most appropriate rheological model in each region of the flow field or sub-domain. However, the size and boundaries· of these sub-domains are not known a priori, and they should be allowed to established themselves according to flow conditions through an automated process. This is where knowledge-based expert system based on m.1e representation become attractive. An expert system would be able to select from a bank of rheological equations the most appropriate model according ta the flow conditions calculated in a "start-up element" ta get the cycle going 22 • The mIes for model selection could be based, for example, on the examination of the velocity gradient. Two main difficulties . would have ta be overcome: fust, the matching of the solutions at the boundaries of neighboring regions, and second the achievement of convergence towards a unique model in each sub-domain, 50 it would not change from one iteration to the nexr 4 . • Chapter 9. Alternative Approaches 147 • • The use of a rule-based expert system would enable many simplifications, such as the viscometric approach in flow regions far frOID flow singularities or in shear-dominated zones. More complete calculations would be done only in complex flow zones, allowing a solution for a small system with a full Newton method, which would facilitate numerical convergence due ta its larger radius ofconvergence. Sorne issues have to be explored systematically to develop the expert system. For example, a procedure to determine the number of modes needed, based on the material and the nature of flow. This number could also vary depending on the location of the sub-domain. AIso, a method is needed to determine the best approach for use near a singularity. Three possibilities are: 1. The use of a Newtonian model to ensure flow convergence, which is the simplest . approach, but would not be valid for extrudate swell. 2. A boundary layer simplification, similar to the boundary layer theory of cIassical fluid mechanics. An example of such equations are those derived by Renardy25 for the PIT and Giesekus models for encI.osed flows, but they have not been tested in large finite element codes. 3. A coupled macroscopic description of the conservation laws with a kinetic theory of the polymer dynamics, along with. a full Newton method to allow convergence. This approach would provide a more realistic description of the flow in these small regions, but would require more computational resources. • Chapter 9. Alternative Approaches 148 • • The use of a more realistic description would lead to an improvement of the simplified approach by adding to our understanding ofmaterial . .9 Incorporation of Oilier Phenomena Another advantage of the expert system approach for polymer processing is the possibility of incorporating other phenomena such as slip or fracture that cau not be predicted by a continuum mechanics model, but that must be taken into account in the modeling of many industrial forming operations. Since reliable models of these phenomena are not yet available, they could be incorporated into the expert system by means of fuzzy logic 27 • Fuzzy logic or possibility is especially suitable for complex, ill-defined, nonlinear phenomena, where human experience is superior to mathematical models 28 • Fuzzy logic represents a mathematical way of looking at vagueness in a form that a computer can deal with. ft might be possible in this way to model phenomena such as the wall slip and fracture of moiten polymers for which explicit models are not available. The advantage of this approach is that a fuzzy expert system is sunilar to a rule-based expert system, except that it contains mIes with ÏInprecise relationships29. Fuzzy-logic could be easily incorporated to the computer program architecture presented earlier. The essential elements of a possible rule-based expert system for viscoelastic flow simulation are presented in Fig. 9-2. The fust calculation might be carried out for a Newtonian fluid in order to insert a start-up solution in the working memory (DATA) to get the cycle of"reasoning n (CONTROL) going. DATA CONTROL -mesh information -FEMcode -boundary conditions , , -decision making: -material parameters -order of actions -storage of intermediate -establishment of sub-domains solutions boundaries -selection of rheological models KNOWLEDGE -selection of slip and fracture models (fuzzy logic reasoning) -rules on order of actions -selection of boumdary conditions . -mies on selection of: . , -matching of solutions at boundaries -sub-domains boundaries -convergence verification: -rheological models -towards single modeI in each zone -slip models -numerical convergence -fracture models -boundary conditions -roles on matching of solutions at boundaries • • Chapter 9. Alternative Approaches 149 • Figure 9-2 Essential'Elements ofa Possible Rule-Based Expert System for Viscoelastic Flow Simulation. • Chapter 9. Alternative Approaches REFERENCES 150 • • 1 Mitsoulis E., Vlachopoulos J., Mirza FA., "A Numerical Study of the Effect of Normal Stresses and Elongational Viscosity on Entry Vortex Growth and Extrudate Swell", Polym. Eng. Sei.• 25(11), p. 677-689 (1985) 2 Mitsoulis E., Vlachopoulos J., Mirza FA., cc Simulation of Vortex Growth in Planar Entry Flow ofa Viscoelastic Fluid", J. App!. Polym. Sei.. JQ, p. 1379-1391 (1985) 3 Mitsoulis E., Vlachopoulos J., Mirza F.A., cc Numerieal Simulation ofEntry and Exit Flows in Slit Dies ", Polym. Eng. Sei.• 24(9), p. 707-715 (1984) . 4 Mitsoulis E., Vlachopoulos J., Mirza F.A., cc Simulation of Extrudate Swell from Long Slit and Capillary Dies", Polym. Proc. Eng.• b p. 153-177 (1984) 5 Larson R.G. cc Instabilities in Viscoelastic Flows ", Rheo!. Acta. J.L p. 213-263 (1992) 6 McKinley G.R., cc Rheological and Geometrie Scaling of Purely Elastic Flow Instabilities '\ J. Nan-Newt. FI. Mech.• QI, p. 19-47 (1996) 7 Vlcek J., Perèlikoulis J., Vlachopoulos J., "Extrusion Die Flow Simulation and Design with FLATCAD, COEXCAD and SPIRALCAD", in Applications of CAE in Extrusion and Other Continuous Polymer pracesses. K.T. O'Brien (ed.), p. 487-504, Carl Hanser Verlag (1992) 8 Sidiropoulos V., Tian J.J., Vlachopoulos J., 'c Computer Simulation ofFilm Blowing ", J. Piast. Film Sheet.1b p. 107-129 (1996) 9 Ghafur M.O., -Kozïey B.L., Vlachopoulos J., 'c Simulation ofThermoforming and Blowmolding: Theory and Experiments", in Rheological Fundamentals in Polymer Processing. J. Covas et al. (eds.), p. 321-383, Kluwer, Netherlands (1995) 10 Mitsoulis E., Vlachopoulos J., Mirza FA., cc Calendering Analysis without the Lubrication Approximation ", Polym. Eng. Sei.. p. 6-18 (1985) 11 Nassehi V., Salemi R., "Finite Element Modeling of Non-isothermal Viscometric Flows in Rubber Mixing ", fntern. Palym. Process., !X(3), p. 199-204 (1994) 12 Gupta M., cCEffect of Elongational Viscosity on Axisymmetric Entrance Flow of Polymers", Palym. Eng. Sei. in review 13 Gupta M., 'c Entrance Flow Simulation Using Elongational Properties ofPlastics ",. presentation atANTEC·99. Society ofPlastic Engineers, New York (1999) . 14 Gotsis AD., Odriozola A., "The relevance of Entry Flow Measurements for the Estimation of Extensional Viscosity ofPolymer Melts", Rheo/. Acta. 37(5), p. 430-437 (1998) 15 Schunk P.R, Scnven L.E., Equations for Modeling Mixed Extension and Shear in Polymer Solution Processing". J. Rheol.• 34(7), p. 1085-1119 (1990) 16 Seriai M., Guillet J., Carrot C., " A Simple Mode1 to Predict Extrudate Swell of Polystyrene and Linear Polyethylenes", Rheo!. Acta. Jb p. 532-538 (1993) 17 Taylor C.R., Ferry J., "Nonlinear Stress Relaxation of PolYisobutylene in Simple Extension and RecoveIY After Partial Stress Relaxation ", J. Rheo!.. 2J., p. 533-542 (1979) 18 Lodge A.S., 'c A Network TheoIY of Flow Birefringence and Stress in Concentrated Polymer Solutions ", Trans. Faraday Sac., g p. 120-130 (1956) 19 VIachopoulos J., cc Extrudate Swell in Polymers ", in Reviews an the DefOrmation Behavior of Materials. ID(4), P. Feltham (ed.), p. 219-248, Freund Publishing Rouse, Tel-Aviv, Israel (1981) 20 Bush M.B., c, A Method for Approximate Prediction of Extrudate SweU ", Polym. Eng. Sei.• 33(15), p. 950-958 (1993) - 21 Crowe E.R., Vassiliadis C.A., "Artificial Intelligence: Starting to Realize its Practical Promise", Chem. Eng. Pracess. Jan. issue, p. 22-31 (1995) 22 Lu S. C.-Y., cc Knowledge-Based Expert Systems for Materials Processing'\ in Fundamentals ofComputer Modeling for Palymer processing. C.L. Tucker ID (ed.), p. 535-573, Carl Hanser Verlag (1989) .- 23 Debbault B., Goublomme A., Homerin O., Koopmans R., Liebman D., Meissner J., Schroeter B., Reckman B., Daponte T., Verschaeren P., Agassant J.-F., Vergnes B., Venet C., " of High Quality LLDPE and Optimized Processing for Film Blowing", Intern. Polym.process.• XIII(3), p. 262-270 (1998) 24 Guénette Robert, private communication (June 1999) 2S Hagen T., Renardy M., "Boundary Layers Analysis of the Phan-Thienffanner and Giesekus Models in High Weissenberg Number Flow", J. Non-Newt. FI. Mech.• 73(1), p. 181-190 (1997) 26 Keunings R., " Recent Developments in Computational Rheology", Seminar at the Chemical Engineering Dept. ofMcGill University (June 1999) 27 Zadeh L.A., " Fuzzy Sets as a Basis for a Theory of Possibility ", in Fuzzy Sets and Systems. North-Rolland, Amsterdam (1978) 28 Rhinehart R.R., Li R.H., Murugan P., "Improve Process Control Using Fuzzy Logic", Chem. Eng. Progress. Nov. issue, p. 60-65 (1996) 29 Crowe E.R., Vassiliadis CA., "Artificial Intelligence: Starting to Realize its Practical .Promise", Chem. Eng. Progress. Jan. issue, p. 22-31 (1995) • • • Chapter 9. Alternative Approaches 151 • • 10. Conclusions and Recommendations 10.1 Conclusions The Iack of success in the simulation of viscoelastic flows in which there is a strong convergence or divergence ofstreamIines, or a singularity in the boundary conditions, bas been thought ta arise from the following sources: - The inadequacy ofthe rheological model for the material. - The presence ofthe f10w singularity. - The elasticity, i.e., history dependence, ofthe material. It is concluded here that the principal source ofthe problem is the elasticity ofthe fluid in combination with flow singularities. Numerical difficulties are aIso encountered with Newtonian fluids in the presence offlow singularities, but since they have no memory the effect ofthe stress overshoot is not carried into the rest ofthe flow field. Viscoelasticity, as described by multiple relaxation times, sorne of which may be quite long, renders even the most aclvanced numerical methods incapable of modeling industriaI flows of melts of typical molten plastics in which elasticity plays an important raIe. One ofthe difficulties encountered in this project was the lack offlexibility ta change the numeric-al aIgorithms in arder ta improve convergence. The amount of work involved ta change the finite element Fortran code restrained us from trying more powerful (but costly) solvers like the Newton method. For faster progress in the future, new programming methods must be developed to eliminate this restriction. With regard ta the inadequacy of existing rheological models, and in particular the Leonov and PTT models, it is concluded that: 1. With the best constitutive equation from a rheological point of view (Leonov model), solutions for situations of practical importance are impossible, even when one of the most advanced numerical procedure is used. The model aIso gives conflicting results, • • Chapter 10. Conclusions and Recommendations 153 -----------------------------_-: with realistic predictions in planar flow but not in axisymmetric flow. The incorrect prediction of decreasing swell confums the findings ofRekers 1 • 2. Even a mode! that is thought ta be one of the best with regard ta convergence (pIT model) is very limited in its capability. 1t now seems quite probable that no model expressed solely in terms of continuum mechanics variables will ever be able ta provide a universal description of the flow behavior ofpolymeric liquids, and this observation bas given rise ta a relatively new area, molecular dynamics simulation. However, it bas not been possible up to now ta elirninate tlie problem of numerical convergence by the use ofmolecular dynamic models even with sorne ofthe most advanced numerical methods. The major conclusions ofthis work are as follows: 1. The barriers that have arisen in trying ta use numerical methods ta simulate complex flows of viscoelastic fluids cannot be e1irnjnated using the constitutive equations and numerical techniques that have been applied to date. 2. It is likely that even if more realistic rheological models were available, the basic problem would persist, as it arises from the history dependence of the material. This history dependence leads ta an intrinsic instability in numerical solutions. 3. An entirely new approach is needed for the modeling of complex flows. This might be based on a rule-based expert system approach to the problem. By use of such a technique, one could guide the direction of the computation in the neighborhood of singularities, thus ensuring convergence. Such an approach could also readily incorporate models for slip and fracture phenomena. 10.2 Original Contributions to Knowledge The original contributions ta knowledge resulting from this work are derived from the objectives presented in Chapter 3: 1. The determination of the limitations in present numerical methods for the simulation ofthe flow of viscoelastic materials. • • • Chapter 10. Conclusions and Recommendations 154 ---"-----------------------------------.: 2. The con:fi.rmation of the inadequaey of the Leonov model in axisymmetrie extrudate swell. 3. The establishment ofthe cause of these limitations, i.e. the history dependence of the materiaIs. 4. The proposaI that a rule-based expert system shows promise for overcoming existing limitations. 10.3 Recommendations for Future Work ·Future work should be aimed at developing the new approaeh described in Chapter 9, a rule-based expert s y s t ~ able to, select, from a repertoire of possible models of rheologieaI and slip behavior, the one that is most appropriate in each region of the flow field. The development of an expert system should be the foeus of future work in this field rather than continuing the futile effort to elaborate a single model that will be adequate over the entire field of a complex flow. The use of a multimode spectrum., truncated or not, for nonlinear viscoelastic models is strongly recommended since a single mode can not fit data properly and leads ta numericaI problems. FinaIly, we need not only more efficient numerical methods but aIso more efficient programming tools that will I?rovide more flexibility for trying various algorithms. An example of such a tool is the program MEF++ 2 (Méthode d'Eléments Finis) wrÏ.tten in the programming language C++. This code facilitates changes in the mesh, .mesh adaptativity, solvers, and upwinding methods in order to improve numerical convergence. 1 Rekers G., H Numerical Simulation ofUnsteady Viscoelastic Flow ofPolymers", Ph.D. Thesis, University of Twente, Twente, The Netherlands (l995) 2 GIREF (Groupe Interdisciplinaire de Recherche en Elements Finis), Université Laval, Québec, Canada • • Bibliography Agassant J.F., Avenas P., Sergent J.-Ph., Carreau P.J., « Polymer processing. Principles and Modeling », Hanser Publishers, NewYork (1991) Ahmed R., Liang R.F., Macldey M.R., "The Experimental Observation and Numerical Prediction ofPlanar Entry Flow and Die Swell for Molten Polyethylenes", J. Non-Newt. FI. Meek, 22, p. 129-153 (1995) Baaijens F.P.T., "Calculation of Residual Stresses in Injection Molded Products", Rheo/. Acta, J.Q, p. 284 (1991) Baaijens f.P.T., Selen S.R.A., Baaijens H.P.W., Peters G.W.M., Meijer R.E.H., " Viscoelastic Flow Past a Confined Cylinder of a LDPE Melt ", J. Non-Newt. F/. Meck, ~ p.113-203 (1997) Baaijens, F.P.T., "Mixed Finite Element Methods for Viscoelastic Flow Analysis: a Review", J. Non-Newt. F7. Meek, 19, p. 361-385 (1998) Barakos G., Mitsoulis E., ''Numerical Simulation of Extrusion through Orifice Dies and Prediction of Bagley Correction for an IUPAC-LDP Melt", J. Rheo!., 39(1), p. 193-209 (1995) Beaufils P., Vergnes B., Agassant J.F., " Characterization of the Sharkskin Defect and its Development with. the Flow Conditions", Int. Polym. 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Mech., JQ, p. 47-71 (1988) Wood-Adams P., "The Effect of Long Chain Branching on the Rheological Behavior of Polyethylenes Synthesized using Single Site Catalysts", Ph.D. Thesis, McGill University, Montreal, Canada (1998) Wood-Adams P.M., "Determination of Molecular Weight Distribution from Rheological Measurements", Master's Thesis, McGill University, Montreal, Canada (1995) Yasuda K.Y., Armstrong R.C., Cohen R.E., Rheal. Acta, ~ p. 163 (1981) Zadeh L.A., "Fuzzy Sets as a Basis for a Theory of Possibility", in Fuzzy Sets and Systems. North-Holland, Amsterdam (1978) • • APPENDIX: Ultimate Extrudate Swell ofLLDPE at 150°C Apparent Shear Rate (s-I.) UItimate Extrudate SweD 2.22 1.369 ± 0.019 4.44 1.419 ± 0.018 8.88 1.488 + 0.017 13.3 1.538 ± 0.017 22.2 1.602 ± 0.016 44.4 1.664 ± 0.016 88.8 1.736 ± 0.015 ... 167 • BARRIERS TO PROGRESS IN THE SIMULATION OF VISCOELASTIC FLOWS OF MOLTEN PLASTICS by Marie-Claude Heuzey Department of Chemical Engineering McGill.University l\'lontreal, Canada • September 1999 A Thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment ofthe requirements of the degree of Doctor ofPhilosophy • © Marie-Claude Heuzey 1999 1+1 National Ubrary of Canada Acquisitions and Bibliographie Services 395 Wellington Street OttawaON K1A ON4 Canada Bibliothèque nationale du Canada Acquisitions et services bibliographiques 395. rue Wellington OttawaON K1A0N4 C.anada Your 61e Votte rëfërence Our file Norre référence The author has granted a nonexclusive licence allowing the National Library of Canada to reproduce, loan, distribute or sell copies of this thesis in microfonn, paper or electronic formats. L'auteur a accordé une licence non exclusive permettant à la Bibliothèque nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la forme de microfiche/film, de reproduction sur papier ou sur format électronique. L'auteur conserve la propriété du droit d'auteur qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation. The author retains ownership ofthe copyright in this thesis. Neither the thesis nor substantial extracts from it may be printed or othetwise reproduced without the author's perrmSSIon. 0-612-55341-8 Canada • ABSTRACT Polymer melts exhibit sorne degree of viscoelasticity in rnost industrial fonning operations, and elasticity is particularly important in flows involving an abrupt contraction or expansion in the flow direction. However, the incorporation of a viscoelastic constitutive equation into computer models for polymer processing poses many problems, and for this reason inelastic models have been used aImost exclusively to represent rheological behavior for flow simulation in the plastics industry. In order to explore the limits.. of v:ïscoelastic flow simulations, we used two nonlinear viscoelastic models (Leonov and Phan-ThienfTanner) to simulate axisymmetric and planar contraction tlows and extrudate swell. Their predictions were compared with those obtained using a strictly viscous model (Carreau-Yasuda) and with experimental results. The models are implemented in a modified Elastic Viscous Split Stress (EVSS) • mixed finite element formulation. The viscoelastic constitutive equations are calculated using the Lesaint-Raviart method,. and the divergence-free Stokes problem is solved applying Uzawa's algorithme The decoupled iterative scheme is used as a preconditioner for the Generalized Minimal Residual (GMRES) method. Numerical instability was observed starting at quite low elasticity levels. For the converging flows, the predicted flow patterns were in fair agreement with experimental results, but there was a large discrepancy in the entrance pressur~ drop. In the case of extrudate swell, the agreement with observation was poor, and c<>nvergence was impossible except at the Lowest flow rate. After exploring the limits of simaLations using viscoelastic models, we conclude that there are serious barriers to progress in the simulàtion of viscoelastic flows of industrial importance. The ultimate source C)f the problem is the melt elasticity, and traditional numerical methods and rheologjcal modeIs do not provide a suitable basis for simulating • practical flows. A new approach is required, and we propose that a rule-based expert system be used. • À la mémoire de mon père. e· Pincorporation de lois de comportement viscoélastique pour la modélisation des procédés de transformation des polymères s'avère difficile. • . Une approche découplée basée sur l'algorithme "Generalized Minimal Residual" (GMRES) est utilisée pour traiter la non linéarité du problème. la convergence numérique n'a été possible qu'au plus faible débit et le gonflement prédit est largement sous-estimé. La méthode de Lesaint-Raviart sert à la résolution des modèles viscoélastique. Les prédictions des deux modèles ont été comparées à celles • obtenues à l'aide d'un modèle strictement visqueux (Carreau-Yasuda) et à des mesures expérimentales. d'éc?ulements d'importance industrielle. Pour les géométries convergentes. des instabilités numériques ont été observées à faible niveaux d'élasticité. Dans le cas du gonflement d'extrudat. Les lois de comportement sont incorporées à une fonnulation mixte d'éléments finis de type "Elastic Viscous Split Stress" (EVSS) modifiée. Lors des simulations. Toutefois. L'élasticité devient particulièrement importante dans les géométries comportant une contraction ou une expansion abrupte en direction de l'écoulement. nous avons utilisé deux modèles viscoélastiques non linéaires (Leonov et Phan-Thienffanner) afin de simuler les écoulements dans des contractions plane et axisymmétrique ainsi que le gonflement d'extrudat. les écoulements prédits ont montré un accord satisfaisant avec les données expérimentales.• RÉsUMÉ Les polymères fondus présentent généralement un caractère viscoélastique lors des procédés de transformation industriels. Afin d'explorer les limites des simulations d'écoulements viscoélastiques. et de ce fait des modèles inélastiques ont presque exclusivement été utilisés lors des simulations . sauf les pertes de pression en entrée qui sont largement sous-estimées. et le calcul du problème de Stokes est effectué à l'aide de l'algorithme d'Uzawa. Une nouvelle approche est requise. et les méthodes numériques traditionnelles de même que les modèles rhéologiques existants ne comportent pas la solution au succès des simulations d'écoulements d'intérêt pratique.• Suite à l'exploration des limites des simulations à l'aide de lois de comportement viscoélastiques. • • . nous concluons qu'il existe toujours de sérieux obstacles au progrès de la simulation d'écoulements d'importance industrielle. et nous suggérons l'utilisation d'un système expert afin de guider l'évolution des calculs numériques. La source ultime du problème réside dans l'élasticité du polymère fondu. Stewart McGlashan et ma grande amie Paula Wood-Adams de l'Université McGill. Finalement. ainsi que Alain Béliveau. les professeurs John M. Kama! pour avoir permis l'utilisation de l'appareillage de mesure du gonflement d'extrudat. leur amour et encouragement qui m'ont permis d'aller au bout de cette étape. l'inclus aussi dans mes remerciements le personnel technique des deux universités qui a rendu possible la mise à terme de ce projet: spécialement fvflvI. . ce qui a passablement allégé les trois années qu'ont duré mes études doctorales. Alain Lioret et Steven Dufour de l'École Polytechnique. Je suis fort reconnaissante envers Mazen Samara et le Prof. Agassant qui m'ont accueillie au centre de recherche et assistée dans les mesures de l'écoulement plan. Walter Greenland.• REMERCIEMENTS En premier lieu j'aimerais remercier mes directeurs de thèse. qui ont toujours su m'encourager et faire preuve d'une patience infinie tout au long de ce projet. Dealy et André Fortin. je remercie spécialement mes proches dont ma famille et mon compagnon • Luc pour leur patience. Erik Brotons. particulièrement Lucia Gavrus. Je garde un souvenir impérissable de la Côte d'Azur. Alain Gagnon. Charles Dolan et Luciano Cusmish de l'Université McGill. ainsi que Benoît Forest de l'École Polytechnique. Aussi un immense merci au personnel du CE:MEF à Sophia-Antipolis en France. François Koran. Bruno Vergnes et le Prof. Je leur suis aussi fort reconnaissante pour leur générosité et l'ambiance agréable qu'ils ont su établir dans leur groupe de recherche respectif: tant à l'Université McGill qu'à l'École Polytechnique. Merci aussi à tous mes collègues de travail pour leur camaderie et entraide. particulièrement Seung Oh Kim. 5 2.••••.•..2.3.••••••••.•..•••••••••••••••••••••••••• 'VI 1.•...•..••. 16 2.1 2.1 2.••.•••••••••.3.••.2 16 17 19 20 22 23 23 24 25 2.1.3.••••••••••••••••••••••••••••••••••••••••••••••••••...•..•••..••.•.•. Introduction Numerical Simulation ofViscoelastic Flow: State-of-the-Art 2.•.eralized Newtonian Models Viscoelastic Models Set ofEquations to Solve and Inherent Difficulties Numerica! Approaches to Viscoelastic Simulations Boundary Conditions Failure ofTraditional Numerical Methods First Successful Approach Economical Storage Techniques Projection Technique Influence ofTheoretical Wark First Comparisons: Boger Fluids Rheological Models Frequently Used in Simulations Determination of Model Parameters Planar Abrupt Contraction Flow Axisymmetric Abrupt Contraction Flow Axisymmetric Converging Flow Extrudate Swell Other Complex Flows 3 5 7 8 12 12 14 15 2.•••.•.. •••••••••.4 COMPARISONS WITH EXPERIMENTS 2.4.2..8 2.4.•.2 MATHEMATICAL TREATMENT OF VISCOELASTIC FLOW 2.... 2.l • TABLE OF CONTENTS List of Tables.4.4...4 1 3 3 Viscoelastic Flow Phenomena Definitions ofTensors and Materia! Functions Geli..1 CONSTITUTIVE EQUATIONS FOR POLYMERMELTS 2.1 2.3 2.1.4 2.4.6 26 28 29 2.4....3.•••••••.•••••••••••••.•.•.2 2..••••••••••••••••••••••••••••••••••••••••••••••..••.•.3 2..2 2.4 2..2..4.5 2.3 2.3 ALGORITfIM DEVELOPMENT .5 CONCLUSIONS .4.7 30 34 35 • 2..1 2.••.1..3 2.1.res•••••••••••••.....2 2..•••.3......••••••••••••••••••• V List ofFigu.••••. .....3.•....1. 43 32 MonELS SELEcrED FOR TInS RESEARCH 3. _ DeterDIination ofModel Parameters 5.•.......II • 2.•.5.•.....5 5.•.1 5..•...•.....••.d Loss Moduli Viscosity Tensile Stress Growth Coefficient..1.•.n of the Resea...4.. 43 43 44 Carreau-y asuda Model Leonov Model Phan-Thienffanner Model 46 50 Rh..••••..•.....••...2..4.3.1.4 HIGH-DENSITY POLYETHYLENE••....•.3 4....3 4.. 53 4..•..•.•...d Loss Moduli Viscosity' Tensile Stress Growth Coefficient.2...3 LINEAR-Low DENSITY POLYETIIYLENE . 4...••..•.••...•..•••....1 4.1 3.4.... Storage an......••••..2.1..2.....••..1.... 50 _ 50 51 52 53 55 57 59 59 60 63 65 Small Amplitude Oscillatory Shear Steady Shear Extensional Flow Storage an.2 5.•.......1 4.•.•..2..••••••••••••••••.••••••••..1 2..5.6 • 5....•...3 4...•••••••...•...•...2.•• 50 4..1.....••.•.....2 4..•.•..•...••••••....eters Viscosity Tensile Stress Growth Coefficient Prediction ofNormal Stress Differences Prediction ofElongational Viscosity 65 65 66 68 68 71 72 74 .1.3 5....2 EXPERIMENTAL METHons 4..3...2 4.4 5..1 LINEAR-Low DENSITY POLYETHYLENE 5...eological Characterization 4.1 4...1 OBJECflVES •••••••••••....2 3.....••.......reh 3..•...7 Viscous Model Leonov Model Parameters Phan-ThienITanner Param..2 4........3 5.1 MA1"ERIALS U SEO •••••••.....••••..2 3......•. Limitations ofNumerical Simulations ofViscoelastic Flows Kinetic Theory Models: A Promising Approach? 35 36 43 PIa.•••. ...4 5....4 VELOCITY 7.•.•.....6 5......•...1 7.5 BIREFRIN'GENCE .•••••••••. 75 5.•...•...3 6..•. 84 6.•..••..4 7...•••.1....•.•.•......•..•....•.•••••••••.......•...•...•••••••••••..1.• 99 7.•...2 6.ulation 84 86 87 88 90 90 91 94 95 99 99 100 102 104 105 106 107 110 112 6.3 Four-Field Mixed Formulation Free Surface Weak Form...1..1.•••....-Thienffanner Parameters Viscosity' Tensile Stress Growth Coefficient Prediction ofNormal Stress Differences Prediction ofElongational Viscosity 75 76 77 77 79 80 82 Numerica.3 TOTAL PRESSURE DROP • 7...•.. Stokes Problem Discontinuous Galerkin Problem Gradient Tensor ofVelocity Resolution ofthe Global System Planar Abrupt Contraction Flow of High-Density Polyethylene ..1......••••......2 HrGH-DENSIIT POLYETHYLENE•.3..... 5..3 7..•..•...•••..2.2 DISCRETIZATION N~RICAL SOLUTION ..1 7.••..2..3.ID • 6.•••••••••••••••••...••.•..••.eters Phan..•.....••.•.••...1 6..•..••.•....•.2.•••••••.. 6...•••....3.2..2.•..••...••.3 52..4 7..•....•.•••...1 5...•.2 5............ 105 7....•••••••.•••.1 DESCRIPTION OF THE PROBLEM ••••••.•..1..2 NUMERICAL PROCEDURE •••......2 6.......l Method ......•••..••.7 Viscous Model Leonov Model Param..•...•••••.2 7.••.3 6..•...•..•.•••.•.•••..••••.•...2.5 5........••.••.•...••.••••• 84 6....1 EXPERIMENTAL METRons 7..••...2..•...2 Die Geometry Laser-Doppler Velocimetry Flow-Induced Birefringence Flow Rate-Pressure Curve Meshes Viscoelastic Converged Solutions 7.3..1.2.••...••..1 6. •..•......••......•••.•...••....•••.....•••.....••.•..•... Simulations • 8... 9.2 10...•.5......•......•...4 RULE-BASEO FLow SThfULATION ••..1 8.....•....•. 143 9.1..5 EXIRUDATE SWELL .......•....••......•....•..........•.•.•.5..•..•••••.....•••••. Axisymmetric Entrance Flow and Extrudate Swell of Linear Low-Density Polyethylene 8. Alternative Approaches 9. 154 ~ Bibliography Appendix 155 167 • ....1 CONCLUSIONS ORIGINAL CONTRIBUTIONS TO KNOWLEDGE .••.•...........•....1 8....... 133 8..2 9..•.•...3 RECOMMENDATIONS FORFU1URE WORK •.•.••••.....•••••...•...1.•...••••••••••:••••••••..•.•.•••••.••••..6 DISCUSSION •.4.•...2 POWER-LAW MODELS FOR ELONGATIONAL VISCOSITY 9..2 Meshes Viscoelastic Converged Solutions 125 126 8..•..•.•.•...3 Fundamentals of Knowledge-Based Expert Systems Application to Visco~lastic Flows Incorporation of üther Phenomena : 145 146 148 10....•.....1 9...........•••.......••••....•••••...IV • 7......••••..3 APPROXIMATE PREDICTION OF EXTRUDATE SWELL 143 9.• 118 8.••..MEïHOD•••.•••••.•••••••••••••...•.........•••....•... ..•....•••.•••...•.6 DISCUSSION 9.•.1 8...•........•..••..4 EN1RANCE PRESSURE DROP 130 8...•••...•.•••.2.••...2.•••.•..•.4...2 Expe~ental Results 133 135 137 140 140 142 Comparison with.••••.....•....•••............3 TOTAL PRESSURE DROP •.1 VrSCOrvIETRIC FLOW.••••••••......•.•.•....... Conclusions and Recommendations 10.. ...•.•••••••••.••. 127 8.....4.•••.2 NU1ŒRICAL PROCEDURE .••.2 Capillary Extrusion Extrudate Swell Measurements H •••••••••••••••••••••••••••••••••••• 122 122 122 122 8.....•.•••.•.•••••••••..•••••....•....•.•••..•••..••......•..1 E:x:PERThŒNTAI....• 125 8.•... 152 152 153 10...•...•...•.•.......•..••...•••••• 8.•. v • Table 2-1 Table 2-2 Table 2-3 Table 2-4 Table 2-5 Table 2-6 Table 2-7 Table 2-8 Table 4-1 Table 4-2 Table 4-3 Table 4-4 Table 5-1 Table 5-2 Table 5-3 Table 5-4 Table 5-5 Table 5-6 Table 7-1 Table 7-2 Table 7-3 Table 7-4 Table 7-5 Table 8-1 Table 8-2 LIST OF TABLES Numerical Simulations of Planar Abrupt Contraction Flow Numerical Simulations ofAxisymmetric Abrupt Contraction Flow Numerical Simulations ofAxisymmetric Converging Flow Numerical Simulations ofAnnular Extrudate Swell Numerical Simulations ofAxisymmetric Extrudate Swell. Comparisons between Predicted and Measured Extrudate Swell (orifice) Numerical Simulations ofPlanar Flow Around a Cylinder Experimental Materials Compression Molding Procedure at ISO°C Discrete Relaxation Spectra for the LLDPE at 150°C Discrete Relaxation Spectra for the HDPE at Isaoc Carreau-Yasuda Model Parameters for the LLDPE at 150°C Leonov Model Parameters for the LLDPE at ISaoC Carreau-Yasuda Model Parameters for the LLDPE at 150°C PIT Model Parameters for the HDPE at 2aOoe Leonov Model Parameters for the HDPE at 20aoC PIT Model Parameters for the HDPE at 2aOoe Geometrical Characteristics ofthe Die Velocimetry Measurements Locations Mesh Characteristics for 2D Flow Simulations Converged Solutions with the PIT Model (max=ï2) Converged Solutions with the Simple Leonov Model (max=12) Mesh Characteristics for Axisymmetric Flow Simulations Converged Solutions with the PIT Model (max=7) Converged Solutions with the Simple Leonov Model (max=7) 27 29 30 31 32 33 34 50 51 54 60 65 65 65 65 65 65 100 101 105 107 107 125 127 127 Comparisons between Predicted and Measured Extrudate Swell (long die) 33 • Table 8-3 . =1/8 for the LLDPE at I50 aC Figure 5-4 Flow Curve Predicted by the PIT Model with a.=1/8 for the HDPE at 200 aC Figure 5-13 Fit of Tensile Stress Growth.. Coefficient for the HDPE at 200°C .3 Entrance Pressure Drop by Two Methods for the LLDPE at 150°C Figure 4..VI • LIST OF FIGURES Figure 4-1 Dynamic Moduli ofthe LLDPE at 150°C Figure 4.5 Complex Viscosity and Viscosity ofthe LLDPE at 150°C Figure 4-6 Transient Elongational Stress for the LLDPE at 150°C Figure 4-7 Dynamic Moduli ofthe HDPE at 200°C Figure 4-8 Fits ofthe Discrete Spectra for the HDPE at 200°C Figure 4-9 Flow Curve by Three Rheometers for the HDPE at 200°C Figure 4-10 Complex Viscosity and Viscosity ofthe HDPE at 200°C Figure 4-12 Transient Elongational Stress for the HDPE at 200°C Figure 5-1 Viscosity Fit of Carreau-Yasucla Equation for the LLDPE at 150°C Figure 5-2 Fit ofViscosity with Leonov and PIT Models for the LLDPE at ISOaC Figure 5-3 Viscosity Curve of PIT Model with a.4 Flow Curve by Two Rheometers for the LLDPE at 150°C Figure 4.2 Fits ofthe Discrete Spectra for the LLDPE at 150°C Figure 4.=0 Figure 5-5 Fit of Tensile Stress Growth Coefficient for the LLDPE at IS0 aC Figure 5-6 Prediction ofFirst Normal Stress Difference for the LLDPE at ISOaC Figure 5-7 Ratio of Second to First Normal Stress Difference for the LLDPE at 150°C Figure 5-8 Prediction of Elongational Viscosity for the LLDPE at 150 aC Figure 5-9 Viscosity Fit of Carreau-Yasuda Equation for the HDPE at 200°C Figure 5-10 Fit ofViscosity with Leonov and PIT Models for the HDPE at 200 0 e Viscous Contribution 54 55 56 56 57 58 59 60 61 62 63 66 69 70 71 72 73 74 75 76 78 78 79 80 Figure 4-11 FirstNormal Stress Difference ofthe HDPE Shifted from 190° to 200°C . 62 Figure 5-11 Flow Curve for the HDPE at 200°C Predicted by the PTT Model without the • Figure 5:-12 Viscosity Curve ofPTI Model with a.... . Figure 7-10 Influence ofDiscrete Spectrum.. 89 Figure 6-2 Discontinuity Jump at the Interface ofTwo Elements Figure 7-1 Geometry ofthe Planar 8:1 Contraction Figure 7-2 Experimental Set-Up ofLDV Measurements Figure 7-3 Positions ofthe Velocity Profile Measurements Figure 7-4 Experimental Set-Up for the Measurement ofFlow-Induced Birefringence Figure 7-5 Characteristic Curve ofDie-Extruder at 200°C Figure 7-6 Comparison ofExtruder Data with Viscosity Curve ofHDPE at 200°C Figure 7-7 Mesh Planar_1.. position 1) Figure 7-13 Velocity Profile (Reservoir..Pl .[1h....•••••••••••••••••••••••• 115 . ~ and@t (and Fh).••••••.. position 3) Figure 7-15 Velocity Profile (Channel..... 108 III 111 III 111 111 112 112 113 Figure 7-21 Calculation ofthe Stress Optical Coefficient.. position 3) Figure 7-20 Evaluation of â max on the Axis ofSymmetry Figure 7-22 Comparison ofMeasured and Calculated Birefringence (Leonov Model..4 1 5.•••••••.. position 3) Figure 7-18 Mesh Independence (Reservoir. 114 ra = 14........... Figure 7-8 Mesh Planar 2..••••••... position 2) Figure 7-14 Velocity Profile (Reservoir...••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••..... position 1) Figure 7-16 Velocity Profile (Channel...4s- 1 ) •••••••••••••••••••••••••••.Pl elements for variables 1:!h.Pl . ra =14..••••••••••••••••••••••••••••••••••••....••••••••••••••••• 115 • Figure 7-23 Comparison ofMeasured and Calculated Birefringence (pTT Model...••••••.) ..... position 3) Figure 7-19 Influence ofSpectrum (Reservoir... 103 104 105 106 106 109 109 111 93 99 101 102 Figure 7-9 Comparison ofPredicted Total Pressure Drop with Experimental Values.. •..••.VII • Figure 5-14 Prediction ofFirstNormal Stress Difference for the HDPE at 200°C Figure 5-16 Prediction ofElangational Viscosity for the HOPE at 200°C 81 82 Figure 5-15 Ratio of Second ta First Normal Stress Difference for the HDPE at 200°C 81 Figure 6-1 The P2+ . on Prediction of Total Pressure Drop Figure 7-11 Mesh Independence for the Prediction of Total Pressure Drop Figure 7-12 Velocity Profile (Reservoir..••. position 2) Figure 7-17 Velocity Profile (Channel.•••••••. .7 S-1)•.•.••••••••.....••••.•.••.•••..•...• 118 119 119 120 123 124 125 126 126 127 129 129 131 131 132 132 134 Figure 7-31 Influence of tlx on Velocity Profile (Reservoir...••••••••••••••• 117 1 Figure 7-28 Predicted and Measured Birefringence on the Axis ofSymmetry (ra =56.•.••..vm • Figure 7-24 Comparison ofMeasured and Calculated Birefringence (pTT Model.•••••••••••••••• 118 Figure 7-30 Influence of Mesh on Birefringence Cia = 47..•.•... i a = 56..•.•.•..•.4 S-I)••.•••••.•...•••••••• 116 Figure 7-25 Predicted and Measured Birefringence on the Axis of Symmetry (Ya =4.6 S-I)••.••..•.•...•..7 S-I)..••••••••••••••...•....•••••.4 S-I)••.••.•.•••••.•.4 5...••••••.•...•....•••••..'"D.•..• 117 Figure 7-27 Predicted and Measured Birefringence on the Axis ofSymmetry (fa =14.•...••..••.•.•..•..•••••••• 117 Figure 7-26 Predicted and Measured Birefringence on the Axis of Symmetry (fa =9.•..••••••••••••••..•.....•..•••••••••••....•..•••.•.•••••••••••• Figure 8-7 Comparison ofPredicted Total Pressure Drop with Experimental Values..8s-1) ••••••••••••••....••••.••••...••••••••.•..) •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••. position 2) Figure 7-32 Influence of tlx on Velocity Profile (Reservoir.....••••.•.ple Leonov Madel Figure 8-8 Influence ofDiscrete Spectra on Prediction ofTotal Pressure Drop Figure 8-9 Mesh Independence on Prediction of Total Pressure Drop Figure 8-10 Comparison ofPredicted Entrance Pressure Drop with Experiments Figure 8-11 Extension Rate on the Axis ofSymmetry (ra =88.4 1 5.•.••••.......••••..•..•..•••.) 117 Figure 7-29 Influence of Spectrum on Birefringence (ia = 28...•••.4 S·l) Figure 8-1 Capillary Rheometer Figure 8-2 Experimental Set-up for Extrudate Swell Measurements Figure 8-3 Mesh Axi_1 Figure 8-4 Mesh Axi_2 Figure 8-5 Mesh Swell 1.•.••••.•..••..••..•.•••••••••••....•... 128 Figure 8-12 Influence ofDiscrete Spectra on Prediction ofEntrance Pressure Drop • Figure 8-13 Mesh Independence on Prediction ofEntrance Pressure Drop Figure 8-14 Total Swell and Oil Swell as a Function ofTime ... Figure 8-6 Ultimate Swell Predicted by the SL... position 3) Figure 7-33 Extension Rate on the Axis ofSymmetry (ia = 56. mesh Swell_l) Figure 8-17 Measured and Predicted Ultimate Swell.IX • Figure 8-15 Net Swell as a Function ofTime after Oil Swell Substraction Figure 8-16 Calculated Interface with PIT Model (truncated spectrum. Figure 9-1 Expert Systems vs. Conventional Programs Flow Simulation 135 136 136 145 149 Figure 9-2 Essential Elements of a Possible Rule-Based Expert System for Viscoelastic • . while increased competitiveness requires reductions in produet development time and unit co st. machined and joined into various shapes. thermoforming and injection and blow molding. low process yields and product inconsistencies. sensitivity and stability analysis. since they exhibit both viscous resistance to deformation and elasticity. 3. but if elongational effects are aIso important. Inelastic models May be adequate to simulate flows in which shear stresses are predominant. the polymer undergoes bath shear and elongation. and these cause severa! problems in the mathematical modeling of certain forming processes. In severa! processing operations such as fiber spinning. which deals with the relationship between stress and strain in defonnable mat~rials. aIl this being achieved at a lower cost than if carried out experimentally. in terms of both mechanical properties and exterior appearance. Standards for product quality have been raised. However.. extruded..• 1. high tooling costs. Introduction The broadening of the range of application of plastic parts requires improved final product quality. . extrapolation or scale-up of designs. the viscoelastic nature ofpolymerie materials can not be ignored. Viscoelasticity not _. flow phenomena arising from viscoelasticity are very complex. plastic processing continues to involve long product development cycles. production and processing of polymerie materials. However. 2. exploration ofthe effects of individual variables. formed. since the .) _ • ooly affects melt flow but aIso plays a major role in the development of the microstructure and physicaI properties of the final plastic part. Numerical simulation can reduce the time required to develop new processes and machines and can aid process optimization by: 1.. However. Plastics can be molded. An important issue in polymer processing is melt rheology. Polymeric liquids are rheologically complex. An understanding oftheir rheological behavior is essential in the development. FinaIly. Moreover. • Chapter 4 describes the materials used for the experimental study and presents their rheological properties. two complex flo~s' were subjected to both numerical simulation / and experimental observation. aIl previous attempts to find a general viscoelastic fluid model applicable to aIl classes of flow and capable of realistic predictions have failed.=. / // //// In the next chapter. Experimental methods and results are presented in Coopter 7 for the planar abrupt contraction flow. we present the plan of the research and the materials --------.:.. conclusions. • .Chapter 1. Chapter 9 presents alternatives to full viscoelastic simulations and proposes a rule-based system for flow simulation. the state-of-the-art of numerical simulation ofthe f10w of viscoelastic co~ve eQ.::.. and in Chapter 8 for the axisymmetric entrance tlow and extrudate swell.. To achieve these objectives. In"Coopter 3. inelastic modeIs have been used aImost exclusively to represent rheological behavior for flow simulation in the plastics industry. The motivation for this research was first to numerical simulation of the flow of viscoelasti~teriaIs:--Thé-econd objective was s / det~~ the limita~o~d models for the to establish the cause of these limitations and sJlggest methods for overcoming them.:::==:::=:::------- is r~vi~w~g.~tions s~i~ -_-=-:. while Chapter 5 gives the method used to determine the various model parameters. In Chapter 6 we describe the numerical procedure used in the simulations. Introduction 2 • incorporation of a viscoelastic constitutive equation into a computer model for polymer processing poses many problems. original contributions to knowledge and recommendations for future work are given in Coopter 10. Numerical Simulation of Vis coelastic Flow: State-of-the-Art 2. this average length is altered. there will he a unique average value of the end-to-end distance. and when the deformation is stopped. This is the molecular origin of the elastic and relaxation phenomena that occur in polymeric liquids. 3 and 4.1. In this section. Polymer molecules can he represented by long chains with many joints allowing relative rotation of adjacent links. because in addition to viscous resistance to deformation. aIso called rod climbing. Weissenberg Effect This effect. This is very different from the behavior of a Newtonian fluid. we will describe briefly sorne flow phenomena associated with viscoelasticity and present several constitutive equations that have heen developed for polymerie fluids. which will flow towards . We describe a few of these phenomena.1 Constitutive Equations for Polymer Melts Polymer melts behave very differently from purely viscous materials. which are said to have a "fading memoryfl .• 2. R. they exhibit elasticity. for the molecules of a given polymer. 2. The reason why polymerie liquids are non-Newtonian is related to their molecular structurel. Brownian motion will tend to retum R ta its equilibrium value.1 Viscoelastic Flow Phenomena A number of phenomena encountered in the flow of polymers cannot be explained on the basis of purely viscous behavior. was reported by Weissenberg in 1947 s• It is the •• tendency of an elastic liquid to rise up around a rotating shaft partially immersed in il. and -the presence of this large number of joints allows many different configurations and makes the molecule flexible. At rest. When the melt is deformed. and a more thorough discussion on the subject can be found in references 2. Dealy and Vu studied rod climbing in molten polymers6 • Extrudate Swell Extrudate swell is defined as the increase in the diameter ofa stream as it exits a die. This phenomenon is known as a tubeless siphon and results from the large tensile stress that the fluid can support due to its elasticit)? Other flow phenomena such as vortex formation in contraction flows are aIso related to the elasticity of polymerie fluids. In an experiment where he immersed the tip of a capillary tube in a beaker filled with a biological macromolecular fluid. axisymmetric extrudate swell at the exit of a capillary is of the order of 12-13%. For a Newtonian fluid at low Reynolds number (Re). but for a sufficiently long capillary. and it decreases as Re increases. the tluid continued to flow upwards to the tip of the no-Ionger-submerged capillary. this extrudate swell ean be as high as 200-300%. this effect beeomes negligible because it is a "fading memory" . However. But . In the absence of inertial effects. Weissenberg deduced that the phenomenon was caused by an elastic normal stress acting along the circular streamlines. the free surface of a Newtonian fluid would remain tlat. For a viscoelastic liquid. Due te the fluid' s memory. these fluids are "shear tbinning". Numericai Simulation ofViscoelastic Flow: State-of-the-Art 4 • the walls due to inertia forces. but for a polymerie fluid " decreases with increasing shear rate. This phenomenon is related to normal stress differences and inereases with inereasing flow rate. Tubeless Siphon In 1908. the viscosity " is independent of shear rate. Fano 7 reported a remarkable manifestation offluid elasticity. even becoming negative. Le.Chapter 2. even after the fluid level in the beaker dropped below the tip of the capillary tube. the swell depends on the flow at the entrance to the capillary. This normal stress exists in aIl shear flows of polymerie liquids except at very low shear rates. he began withdrawing liquid from the beaker through the tube. t Simple Flows • For a Newtonian fluid. 2-2 where p is the pressure. 2. for example. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 5 • shear tbinning alone does not imply that a fluid is viscoelastic. Recoil.1. The Finger tensor is the inverse of the Cauchy tensor and can . which give rise. Tensors The strain rate or rate of defonnation tensor D is given by: Eqn. Stresses will persist in these materials after deformation has ceased.Chapter 2.2-1 where M is the velocity vector. and the duration ofthe time during which stresses persist is called the relaxation rime of the material.2 Definitions of Tensors and Material Functions We present here definitions ofthe tensors and rheological material functions that are used throughout this thesis. ta the Weissenberg effect Another manifestation of viscoelasticity is stress relaxation.t2 ). A stronger indication of viscoelasticity is the existence of normal stress differences in shearing deformations. is anotber manifestation of viscoelasticity_ A realistic viscoelastic constitutive equation must be able to predict all the phenomena mentioned above. the reverse defonnation that occurs when the tluid is suddenly relieved of an extemally imposed stress. The extra stress or viscous stress tensor ~ is defined as follows: Eqn. cr is the Cauchy stress tensor and [ is the unit tensor. • Useful measures of strain in polymer rheology are the Cauchy tensor C if Ctl. The reader is referred to Dealy and Wissbrun 1 for a thorough discussion ofthese quantities.t2 ) and the Finger tensor Bif (t 1. 2-11 (start-up flow) • . the following materiaI functions (the ''viscometric'' functions) are defined (D 12 == y): Eqn.2-6 7lv =-.- (. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 6 • aIso be written ci..t (t1'(2). second and third invariants of the Finger tensor are defined as follows: Eqn.2-9 \fi = Ni 1- r .2-8 (fust nonnal stress difference) (second normal stress difference) Sometimes the normal stress coefficients are used: Eqn.2-3 Eqn.2-4 Eqn.2-7 Eqn.Chapter 2. respectively: Eqn.2-10 (second normal stress coefficient) Start-up and cessation of shear flow is denoted by a plus or minus sign..2 (fust nonnal stress coefficient) Eqn. The first.) _ ~12 r (viscosity) Eqn. 2-5 Material Functions In steady shear. we defined a principal stretching stress crE: Eqn.3 Generalized Newtonian Fluid Models The fust requirement of a constitutive equation to be used for flow simulation is that it describes adequately the rheological behavior that govem the flow of interest.Chapter 2.- e For start-up of elongational f1ow.ë) ë 2. it can be written as follows: • Eqn. cannot be distinguished from simple shea1.the-Art 7 • Eqn.2-12 (cessation of tlow) In simple uniaxial elongation. there are general constitutive equations that relate the stress tensor to the deformation history in terms of a few material parameters or functions. For certain restricted flows. If it is fonnulated using the upper-convected derivative of the v rate of deformation tensor D.2-14 (JE TJE ( &') =-. from the point of view of a material element. viscometric flows are flows that. the defonnation history of each tluid element is constant as it flows along a streamline. Numerical Simulation ofViscoelastic Flow: State-of-.2-16 .1.2-13 The elongational viscosity is defined as follows for steady uniaxial elongational tlow: Eqn. we have the tensile stress growth coefficient: Eqn. For example. The extra stress tensor for viscometric flows is described by the Criminale-EricksenFilbey (CEF) equation4•8.2-15 (j(t. Although the shear rate may vary from particle to particle. then Eqn.2-19 which bas Newtonian behavior with viscosity 110 at low shear rates and power-law behavior at high shear rates.Chapter 2. a truly viscoelastic model must often he used. It is regarded as a purely viscous constitutive equation. which are manifestations of elasticity. Viscoelastie behavior may be linear or nonlinear. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 8 • where the viscosity and first and second normal stress coefficients are functions of IID. Eqn. assuming that normal stresses do not affect the pressure gradient.2-18 A material that behaves in this manner is called a "generalized Newtonian fluid".2.4 Viscoelastic Models • Sïnee the viseometrie tlow simplification is not always approprlate.1.17 If one is primarily interested in the shearing component of the stress tensor.2. Linear viscoelastic behavior is observed ooly when the deformation is very small or slow.. Eqn. and is . but more realistie constitutive equations almost invariably lead ta difficult numerieal problems2 • This will be discussed in more detail in section 2. 2-16 can he simplified to: Eqn. 2-18 is often used in complex polymeric flows for which it is not valid for polymerie liquids. A popular form for the viscosity as a function of IID is the Carreau-y asuda equation9 : Eqn. 2-18 has the tensorial character of a Newtonian fluid but can exhibit shear tbinning. the second invariant ofthe rate ofdeformation tensor D: Eqn. since it does not predict normal stress differences. 2. It is appropriate to use this equation in the conservation of momentum equation to compute the pressure drop and viscous dissipation in a tube or channel with constant cross-section.1. Because of the mathematical complexity involved. the last two being based on the reptation concept. in the nonlinear regime the response to an imposed defonnation depends on the sïze. many simplifying assm. Numerical Simulation ofViscoelastic Flow: State-ot:the-Art 9 • therefore not directly relevant ta the behavior of molten polymers in processîng operations where deformations are always large and rapide Linear viscoelastic behavior is independent of the kinematics of the deformation and the magnitude of past strains.13 without the independent alignment assumption and Curtiss-Bird models l4. it is not possible to measure a response in one type of deformation and use the result to predict the response in another type of deformation. but this approach does not always yield closed fOIm constitutive models.'nptions must be made. the rate and the kinematics of the deformation. 2-20) describes weIl the linear viscoelastic response of a material to an arbitrary strain history: Eqn. The:first one is based on the derivation of molecular theories for melts and dilute solutions together with the use of statistical mechanics. Examples of such theories include the Rouse model for dilute solutions 10.15 for entangled melts. The second method is an empirical approach based on continuum mechanics concept 1 • • Complications in building the model arise from the involvement of tensor-valued quantities and the fact that the response of the material depends on strains imposed at . These simplifications make possible the addition of the effects of successive deformations. G(t-t') is the relaxation modulus.Chapter 2. Therefore. Conversely. and this limits the practical value of the molecular approach. and D(t') is the rate of deformation tensor at time t'. and the Doi_Edwards ll •12. The Boltzmann superposition principle (Eqn.2-20 ~(t) = 2 fG(t-t')D(t')dt' t where ~ is the extra stress tensor. unless the rate. magnitude and kinematics of the deformation are all the same in both cases 1• Two approaches have been used to fonnulate nonlinear viscoelastic constitutive equations. T Dt.2. Recent numerical approaches called micro-macro formulations are based on kinetic theories rather than closed form constitutive models16.2-24 ~=~+T·VU+VU â T Dt=- -= . of nonlinear constitutive equation. .20.19.21.= = - Eqn. and the operator E9 stands for the convected GordonSchowalter derivative22. Larson2 has presented an exhaustive review of closed form continuum models.17.5. is a relaxation time and Tl=ÂG is a viscosity. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 10 • previous times. These are reviewed in the following sub-sections.2-21 Here.Chapter 2.Vu . 2. The symboll:: represents a model- dependent tensor function. which is a combination of the upper V and lower Il convected derivatives: Eqn. there is no universal theory that describes nonlinear rheological phenomena. At the present time. These are dèscribed in section 2.~-~. Nevertheless. • .. severa! models have been used in numerical simulations.~ where D IDt is the material time derivative and T indicates the transpose. Vu Dr Dr.1.1 Differentiai Viscoelastic Models Many differential modeIs can be written in the following general form: Eqn.18.2-22 Eqn.4. Â. and these are either differential or integral constitutive equations. These complications make it difficult to establish an acceptable form.2-23 ~ = v = ~. 4.. The addition of a Newtonian viscosity Tls yields the Oldroyd-B mode!.1. 2-27: l Single integral constitutive equations give the extra stress at a fluid particle as a time integral of the defonnation Eqo. 2..t ') = dG(t-t') = ~Gi exp(-Ct-t')] LJ ---.2-27 ~(t) = Jm(t-t')F(~)dt' Eqo. This means that the defonnation experienced by a fluid .---.Li Âi where we consider a time integral taken along the particle path parameterized by the time • t '. However.2 Integral Viscoelastic Models For the purpose of expressing integraj constitutive equations. such as those of Giesekus23 . 2-21 can be modified to include a discrete spectrum of relaxation times number of modes: Eqo. history. ..2-28 m ( t . The factor m(t-t? is the time dependent memory function that incorporates the concept of fading memory. Eqn.2-26 The simplest differential model is the upper-convected Maxwell model.b where Nis the = 'r"=~'r". but identifying the correct model and the suitable parameters remains a difficult task. as shown by Eqn.. N ~ 1 i=l= Eqo. 2 and constant Tl... dt' i=1 . LeonoV24 and PhanThien/Tanne~ (PT!) have been proposed to give better agreement with data. More complex models.2-25 Â. these constitutive equations do not provide a realistic description of polymerie fluids.-1.Chapter 2. with Li.. deformations are measured relative to the fluid configuration at the present time t. Numerical Simulation ofViscoelastic Flow: State-of-the-Art Il • In order to obtain a more realistic representation of the behavior of polymer liquids. In this case. Again. [2] = met . We present the goveming equations using a one-mode differential viscoelastic model.11. 2.t')h(/l '/2) One of the K-BKZ model with a particu1ar form of damping function was proposed by Papanastasiou et al.1 Set of Equations to Solve and Inherent Difficulties The simulation of the flow of viscoelastic materials involves the solution of a set of coupled partial differential (or integral-differential) equatians: conservation of mass.29. In these models. 2-25). since the majority of the numerical methods have been developed for differential models. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 12 • element in the recent past contributes more to the current stress than earlier deformations26• ofthe fluide One of the simplest nonlinear integral constitutive equations is the Lodge rubberlike liquid modet27.2 Mathematical Treatment of Viscoelastic Flow 2. h(1}.2-29 M[(t -t'). This madel is equivalent to the upper-convected Maxwell differential model with N=-l.Chapter 2. we assume that the f10w is isothermal and incompressible. the memory function is said ta he "separable" or factorable: The strain dependence ofthe memory function is called the damping function. For • . If the flow is compressible or non-isothermal. the extra stress can he expressed as a SUffi of individual contributions (Eqn. conservation of energy and an equation of state for density are aIso required. 30 and fitted weil experimental data for simple shear and simple elongational flows. For sïmplicity. in which F=lb where B is the Finger tensor. Other madels of the integral type that have been found to give a better fit ta data are of the K-BKZ type28.2. Fis a model-dependent tensor that describes the defonnation Eqn.12). the memory function depends on the strain as weIl as on time. conservation of momentum and a rheological equation. Chapter 2. body force. 2-30 and Eqn.2-31 Dt = - where p denotes a constant density.2-34 '!::+Â'!::-271DVD = 0 'il The mathematical type of a system of equations can be described in terms of • characteristics31 • Ellipticity is represented by complex characteristics. while hyperbolicity is represented by real characteristics. The last equation is the rheological model.2-33 7]0 =TJs + LTJi i=1 This splitting is often arbitrary but it bas a strong stabilizing effect on most numerical schemes.2-32 ~ == 2TJs D + L N 'r've. an example of which is the upperconvected Maxwell model: Eqn. and 115 is a constant viscosity that can he a solvent viscosity for a polymer solution or a fraction of the zero-shear viscosity for a polymer melt: N Eqn. p is the pressure. 2-31). ~ is the extra stress tensor andfis a For a Newtonian fluid and in the absence of inertia forces. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 13 • incompressible. The extra stress tensor is sometimes split into viscous and viscoelastic contributions. Ellipticity and hyperbolicity are the . for example in the Oldroyd-B fluid: Eqn.i i=l= where ve represents the viscoelastic character. this set of equations is referred to as the " Stokes problem" (Eqn. isothermal flows the conservation equations for mass and momentum are: Eqn.2-30 Du p-==-Vp+V·r+f Eqn. The combination of ellipticity and hyperbolicity complicates the simulation of viscoelastic flows. However. induces a stress singularity and results in both mathematical and numerical difficulties. 2.Chapter 2. finite difference and spectral methods. regularizing effect on a singularity while hyperbolicity propagates it. The majority of published simulations have been carried out using finite element methods for both differential and integral constitutive equations. the introduction of a geometrical singularity. This is due to the advantages of the finite element methods in discretizing arbitrary geometries and imposing simply and accurately a variety of complex boundary conditions32• If we assume an Eulerian framework. The set of equations for a purely viscous fluid is elliptic. like shocks or singularities. 2-34) can he expressed as follows: • Eqn. are transported. we will look at severa! numerical approaches that have been proposed to overcome these difficulties. Many discretization techniques have been used ta solve the conservation and constitutive equations. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 14 Ellipticity bas a • mathematical concepts associated with diffusion and propagation. such as an abrupt contraction. certain similarities exist between the two cases. boundary element. long distance effects controlled by an elliptic set of equations depend only on averaged quantities. since the rheological equation transports the effect of the singularity and pollutes the velocity field31 • In the next section. the upper-convected Maxwell model (Eqn.2 Numerical AJ2Proaches to Viscoelastic Simulations The solution of viscoelastic flow problems presents severa! numerical challenges in the case of both differentia! and integral constitutive models. For example. but a viscoelastic constitutive equation like Eqn. for example the nonlinear character of the goveming equations brought about by the constitutive equation for the viscoelastic stresses. For example.2. whereas phenomena controlled by a hyperbolic system of equations. and these include finite element.2-35 . 2-34 is hyperbolic. there is no complete mathematical theory that guides the .3 Boundary Conditions In order ta complete the mathematical description of a viscoelastic f1ow. so either a Lagrangian or streamIine formulation must be adopted. as would be the case in a Lagrangian framework. In this work. 38 and Rajagopalan et al. HuIsen and van der Zanden37. while for hyperbolic systems with real characteristics. 2. Rasmussen41 presented a new technique to solve time-dependent three- • dimensional viscoelastic flow with integral models.xed finite element formulations designed for differential constitutive equations in an Eulerian framework. based on a Lagrangian kinematic description of the fluid flow. Goublomme et al. the problem is well-posed for boundary conditions of the Dirichlet type and for conditions of the Neumann type when the net flux (Green's theorem) is zero.. Two fundamentally different approaches are used to solve the set of partial differential equations33 • In the first approach. we will discuss more thoroughly mi. The nature of these boundary conditions is intimately related to the mathematical nature of the governing equations. • Coupled equations of a mixed elliptic-hyperbolic type have both real and complex characteristics. ·-Details of these computations have been described by Luo and Tann~6. and each of the equations are multiplied independently by a weighting function and transformed into a weighted residual fOlm34• In the second approach.Chapter 2. To date. Numerical Simulation ofViscoelastic Flow: State-of:the-Art 15 • The convective term.. appropriate boundary conditions must be specified.2. 39 • Luo40 introduced a control volume approach to solve integral viscoelastic models. In the case of elliptic systems with complex characteristics. where the velocity. the constitutive equation is transfonned into an ordinary differential equation either by using a Lagrangian formulation for unsteady flows 35 or by integration along streamlines. V~.. pressure and extra stress are treated as unknowns. Many integral ·models cannot be expressed in differential form. a mixed fonnulation is adopted. brings the complication of having to solve a partial differential equation rather than an ordinary differential equation. Recently. it is well-posed for conditions of the Cauchytype on the entrance boundary. It is then highly desirable to improve the quality ofthe boundary conditions. • even at low levels of melt elasticity. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 16 • selection of boundary conditions for the fiow of viscoelastic materials42. In summary. Because of the memory effect.4 3. the numerical computation of viscoelastic flows involves strongly nonlinear. for extra stress this boundary condition is sometimes as complex as the solution of the flow itseIt: because of the laek of anaIytical solutions for the more complex models. It is then eommon practiee to specify inlet extra stresses and known kinematics for the upperconvected Maxwell fluid (Eqn.3.1 Fanure of Traditional Numerical Methods NumericaI analysts have been interested in simulating viscoelastic fiows for many years. The use of conventional numerical techniques has proven unsuccessful due to a loss of convergence. conditions that result from a fully developed velocity profile are usually applied at the inlet boundary. 2-34). coupled equations of a mixed elliptic-hyperbolic type. AIso. the issue of appropriate boundary conditions for viscoelastie computations is still an open question42• The approach adopted may make it possible to proceed numerically but not be completely supported by either mathematical or experimental results. As mentioned above. such as capillary exit. However.3 A1gorithm Development 2. In practice. These approximate boundary conditions result in a rearrangement of the velocity and stresses at the inflow and outfiow. For example. the imposition of proper boundary conditions at geometric singularities. the extra stresses must aIso be specified at the inflow boundary to reflect the fiow history.Chapter 2. and this rearrangement is sometimes important enough to restrain numericaI convergence in simulations. 2. The general approach is to use the same boundary conditions as for purely viscous fluids: no-slip at solid walls and specification of the velocity components at the entrance and exit of the tlow boundaries. is still a subject ofdiscussion. The Deborah number (De) is a dimensionless group . It took aImost ten years to identify what is ealled the " high Deborah number problem ". it is known that the no-slip boundary condition does not apply in all situations for polymerie fluids. .38 'ifseS • where è ~ and q are weighting functions. Classical finite element techniques based on the application of the Galerkin method to the mixed formulation failed at relatively low De.. the equations in the weak formulation are: Find eu.. 2-30 and Eqn. It is now understood and accepted that the computational difficulties are numerical in nature and are due ta the hyperbolic nature of the constitutive equations.2.48. For the same problem Keunings 45 concluded. and neglecting body forces. and (. continuity and constitutive equations are expressed in a weighted residual form. 2. The momentum. p. From then on. ~ e V x Q x S such that: (Vy)T . Debbaut and Crochet44 were able to increase the De limit for an abrupt 4:1 contraction by imposing more strongly the incompressibility constraint.2 First Successful Approach Most of the early work on viscoelastic flow analysis was based on the three-field mixed finite element formulation46•47. that the limiting values of De were numerical :utifacts caused by excessive approximation errors.. and it was unknown at the time if these limits were intrinsically related to the continuous problem or to inadequate numerical schemes.37 (q.3.36 Eqn. the existence of stress singularities..2. ~ p) = 0 'ifv eV Eqn.2. V· ~)=O 'ifqeQ Eqn.Chapter 2.. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 17 • that indicates the degree of elasticity of the flow. 2-31) when applied to a Newtonian flow. . it was suggested45 !hat efforts be focused on complex flows with singularities or boundary layers in order to test algorithms robustness. For an Oldroyd-B fluid with one-mode. after a critical exarnjnation of the numerical solutions and an intensive mesh refinement study. and this leads to an approximation of the Stokes problem (Eqn. since strong gradients have a tendency to increase numerical imprecision..2TJs D + ~).) is the appropriate inner product.(V . and the method of resolution of the coupled equations... In both methods. Marchal and Crocheé9 proposed a new method that showed good numerical behavior. several variations of the parameter a. have been proposed. in order to have a discrete representation of the extra stresses that would satisfy the equivalence compatibility condition. First. Over the years.2-39 SUPG: Eqn.!f' \7~ in order to stabilize the convection-dominated transport. Numerical Simulation ofViscoelastic Flow: State-ot:the-Art 18 • In 1987. the inf-sup condition js satisfied by using a sufficient number of interior nodes in each element. '2-40 SU: Those are examples of finite element methods in which the weighting functions differ from the basis functions. The velocity is approximated by biquadratic polynomials. while the ·pres~e is bilinear and continuous. based on this formulation. Since the approximation for the stress is also continuous. and the Streamline Upwind (SU) method: Eqn. they introduced a new computational element for the stress components composed of 16 sub-elements (4 x 4). such as the 4 x 4 sub-elements used by Marchal and Crochet49 . the mixed fonnulation including extra stresses provides a convergent approximation for the different variables. artificial diffusion is introduced in the tlow direction by the upwind term a. The polynomial approximations of the three variables need to be carefully selected with respect to each other to satisfy the so-called generalized inf-sup (Brezzi-Babuska) compatibility condition50 • When this condition is satisfied. as proven later by Fortin and Pierre51 • Once the compatibility of spatial discretization was settled.Chapter 2. but are all of the forro 33 : • . Marchal and Crochet49 used • two methods to account for the convective term in the constitutive equation: the Streamline Upwind Petrov Galerkin (SUPG) method. 2. the SU formulation is an inconsistent substitution of the exact solution. S3. the new element was rapidly shawn ta be too costly in memory space for practical use. which produced oscillatory stress fields at steep stress boundary layers or near singularities.. Fortin and Fortin54 presented a discontinuous Galerkin (DG) method to handle the constitutive equation.Chapter 2.2-42 • . but it became possible to do multimode calculations when Fortin and Fortin54 intI'oduced their economical storage technique. showed an increased robustness compared to SUPG. However. considering the large memory space needed to solve viscoelastic flow problems.3 Economical Storage Techniques Marchal and Crochet's element could only he used for single mode simulations.3. The relative success of this approach showed that a combination of compatible discretization spaces with a discretization technique adapted to the rheological equation could eliminate the 10ss of convergence at high De. The SU metho~ where the upwind term is applied only to the convective term of the constitutive equation. since the second term of Eqn.2-41 a=- h U where h is a characteristic length of the geometry and U a characteristic velocity of the flow. and even if it converges weil it is not always toward the correct solutionS2. 240 remains as a residual. based on ideas of Lesaint and Raviart55 • Upwinding is introduced in this method by the jump of discontinuous variables at the interfaces of elements: Eqn. Unfortunately. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 19 • Eqn. . However. and J. pressure and extra stresses were not solved simultaneously... By use of the DG technique. Later. velo city. for . the economical storage technique was a major contribution to the simulation of viscoelastic flo\vs. This increased the algoritbm. Later. the only large system to store is the one related to the Stokes problem.4 Projection Technique The major contnbution of Rajagopalan et al.3. The DG method allows the solution of the constitutive equation on an element by element basis. "Fortin and Fortin56 used a quasi-Newton iterative solver (GMRES.Chapter 2. 2-43: . implemented the EVSS method (Elastic Viscous Split Stress) which is a projection technique based on the splitting of the extra-stress tensor into viscous and • elastic parts as shown in Eqn. resulting in very efficient solvers and greatly reducing the memory space needed to solve flow problems using differential multimode models. However.nup the extra stress tensor in the neighboring upwind elemen~3. 58 was the introduction of a fourth field in the miXed finite element formulation of viscoelastic fiow. but fuis time the solution was decoupled. for which a LU factorization is done. the robustness of GivIRES was tested with more success by Guénette and Fortin57 • 2.. Generalized Minimal Residual ") for the solution of the linear system. opening the door to all the projection techniques that are now used to introduce compatibility of discretized spaces. Rajagopalan et al. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 20 • where !! is the outward unit normal on the boundary of element e. the method was found to be insufficiently robust because of the fixed-point algorithm used ta couple the Stokes problem and the viscoelastic equation. the solution for the stresses did not imply the'inversion of large linear systems but only local calculations on elementary systems. robustness but did not -achieve the performance of Marchal and Crochet's method49 due to oscillatory behavior produced by the DG formulation near • singularities. The equivalence compatibility condition was also satisfied in their method by the respective discretizations. r ine is the part of element e boundary where!!· n < 0. and the change of variable of Eqn.2-45 ((V~T.y. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 21 • Eqn. the rate of deformation tensor D is introduced as an additional.2-47 where g is a suitable weighting function.Jl) e V x Q x S x C such that: g.J!) problem. four-field formulation can be formulated as follows: Find Û!. Moreover. +a)D(ID-a~ +~J-(v .2-44 Eqn.p. unknown. However.2-43 A change of variable is performed in the momentum and the constitutive equations yielding a set of equations involving the velocity lb the pressure p and the new variable E.2C77:.y.2-48 (CVy)T . In summary.2-46 'd'seS Eqn. in which the main difference lies in the introduction of a stabilizing elliptic operator in the discrete version of the momentum equation: Eqn. 2-46).. 2-43 does not yield a closed expression for all constitutive equations. V ·~)=O 'd'v eV 'd'qeQ Eqn.E.Chapter 2.2(77s +1])DÛ!)+~-(V .p )=0 • . 'd'cee This method is remarkably stable. the EVSS method based on the Eqn.E. leading to a four-field UbP. it requires the convected derivative of the rate of deformation tensor Cin Eqn.p)= 0 (q. These disadvantages were corrected by Guénette and Fortin57 who recently proposed a modification of the EVSS formulation. on the compatibility between discretized spaces. The algorithm seems robust. It was tested in conjunction with the non-consistent (SU) streamline upwind method on the 4:1 contraction and stick-slip problems.If a discontinuous interpolation of the viscoelastic stress ~ is used (as in the DG technique). .. is used (as in the SUPG or SU • techniques).the-Art 22 • The parameter a.5 Influence of Theoretical W ork Fortin and Pierres 1 made a mathematical analysis of the Stokes problem for the threefield formulation used by Marchal and Crochet49 • They confinned that the success of the method was dependent. Fan et al.3. since it does not require the solution of the convected derivative of the rate ofdeformation tensor. 2. and no limiting De number was reached when using the one-mode PIT model for the stick-slip problems1 • Recently.(y) obtained after differentiation of the velocity field!! must he in the same discretization space as 3. based on the incompressibility residual of the finite element discretization and the SUPG technique. but if properly chosen it improves the overall convergence of the algorithme In the discrete approximation (Eqn. 't. Their method is not restricted to a particular class of constitutive equations and is easier to implement than the original EVSS method. Numerical Simulation ofViscoelastic Flow: State-of. 59 introduced another stabilized formulation. . terms combined with the inf-sup condition has a stabilization effect on the discretization. 2-48). the difference in the discretization of the a. the space of the strain rate tensor D. They showed that in the ab~ence of a purely viscous contribution (115=0) and using a regular Lagrangian interpolation. among others. is positive and a priori arbitrary.Chapter 2. If a continuous interpolation of the extra stress J. the number ofintemal nodes must be larger than the number ofnades on the side ofthe element used for the velocity interpolation. The velocity-pressure interpolation in the Stokes problem must satisfy the BrezziBabuska condition to prevent spurious oscillation phenomena. • 2. three conditions must hold33 • 1. They claim that the new method has ~e same level of stability and robustness of the modified EVSS methods1 and is superioF to the EVSS technique58. Experimental elongational data for Boger fluids have compared favorably with numerical simulations using the Oldroyd-B equation~3.. To narrow the gap between experimental observation and prediction. increasingly feasihle and necessary. which can not capture the full nonlinear behavior of polymer fluids. Luo and Tanner65 and • Coates et ai. a first nOfID. a zero second normal stress difference and an elongational viscosity that increases with elongation rate. M . and a single-mode Oldroyd-B equation is then a sufficient rheological description. the influence of the discretization of the constitutive equation near a singularity remains unexplained. as Iater proven by Fortin et aL60 based on a generalized theory of mixed problems.!. direct comparison of the numerical results with the experimental data became .4. 66 aIso tried to reproduce numericaIly the tlow kinematics measured by streakline photography of a Boger tluid in abrupt 4: 1 and 8: 1 contractions67 • Most of the . the faster of these modes can be collapsed into a retardation terro. 2. Numerical Simulation ofVlScoelastic Flow: State-of:the-Art 23 • The compatibility between discretized spaces also contributes to the success of the fourfielcÏ modified EVSS technique57. Actual comparison may be based on global flow features such as die swell or·· pressure drop measurements or on local tlow kinematics measured by streakline photography.1 First Comparisons: Boger Fluids • The first comparisons between numerical simulations and experimental data were made using the upper-convected Maxwell (UCM) and Oldroyd-B models. the so-called Boger fluids were developed61 • These elastic fluids are specially prepared solutions ofhigh molecular weight polymers in viscous solvents and are thought to be nearly free of shear tbi nni n g 2 • They are aIso characterized by. Boger tluids are reasonably weIl described up to high frequency in dynamic shear experiments by a two-mode version of the UCM equation62• At low and moderate frequencies. However.al stress difference proportional to the square of the shear rate.4 Comparisons with Experiments With the improved performance of numerical methods for viscoelastic flow simulations.Chapter 2. 2. Debbaut et al. laser Doppler velocimetry (LDV) or birefringence. Phan-Thien/Tanner (PTT) and KBKZ rnodels.the-Art 24 • results were in qualitative but not quantitative agreement with. With the development of new and more realistic constitutive models. 30 (pSM) or Wagner7. we present a short review . Boger fluids do nc. simula~ons of complex 2. The model should predict shear tb. In the following sections. However. These flows are notoriously difficult to simulate. A significant portion of the work on· the numerical analysis of viscoelastic tlow of polymer melts is based on streamline in~egration methods.>t retlect the viscoelastic behavior of polymer melts.2 Rheological Models Frequently Used in Simulations Simulations of flows of molten polymers under typical processing conditions require the • use of suitable rheological constitutive equations. the constitutive equation must be able to describe the simplest ~~rheometric" flows.nn. extrudate swell and vortex formation in contraction flows. in particular for the UCM and Oldroyd-B models.ng.e. Leonov. normal stress differences in shear.4.O• The popularity of the PSM damping function arises from the possibility of . i. stress relaxation. the measurements. further investigation of Boger fluids is now of less interest. Numerical Simulation ofViscoelastic Flow: State-of:. Before looking at complex viscoelastic flow phenomena like rod-climbing. Recent sources of disagreement between experiments and simulations appear to indicate that Boger fluids are not accurately represented by the Oldroyd-B model and that improved agreement is obtained by using a single-mode finitely extensible nonlinear elastic (FENE-P) dumbbell modeI15. such as occur at sharp corners in contraction or expansion flows. on studies that compare experimental data for polymer melts to flows with more realistic rheological models. employing K-BKZ type • constitutive models and the damping functions proposed by Papanastasiou et al. recoil and sensitivity to Sorne of the models that have been proposed for polymers have been mentioned earlier: these inclu4e the Giesekus. kinematics. flows used to establish rheological parameters.Chapter 2.68. Their superiority has also been shown in providing numerical solutions for viscoelastic tlows with geometric singularities.69. which are easy to measure. which is surprising considering that the simple Leonov model gives more realistic values of limiting extensional viscosities than Giesekus2 and contains no nonlinear parameters. PTT and Giesekus are the ones most often used. Âj]. For example. between five and ten modes are necessary ta provide a good fit ofthe relaxation modulus l . Therefore.Chapter 2. However.4. Linear viscoelastic data. • . Of the clifferential type models. entangled polymer chains have many modes of motion and thus of relaxation. rheological data have been used to provide information about molecular structure: this bas been called the "inverse problem". and these can not be described in terms of a single relaxation time. Very long.74 by using linear viscoelastic data. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 25 • fitting shear and elongational data independently. Usually. such as normal stress difference and extrudate swell are very sensitive to MWD 1• While the rheological behavior depends strongly on molecular structure.73. such a fit may not be adequate for properties that depend in detail on molecular weight distribution (MWD). none of the existing constitutive models are entirely suitable for the simulation of complex flows. A number of researchers have had significant success in the calculation of the viscosity MWD oflinear polymers72. and this might affect the predictions in tlow simulations ofphenomena tbat depend strongly on MWD.e. the compliance and other manifestations of melt elasticity.3 Determination of Model Parameters It was pointed above that the use of a viscoelastic constitutive equation in simulations requires the detennination of severa! parameters. The latter has the ability to fit shear and extensional properties independently. Obviously. whereas the PTT model bas two. The Giesekus model has one parameter that controls nonlinearity. on the spectrum. i. usually the storage and 10ss moduli (G'. of relaxation times. are used to determine a discrete relaxation spectrum [Gi. 2. However it gives spurious oscillations during start-up of shear flow71. G"). important information about the MWD is lost when using a discrete spectrum. As a result.Chapter 2. large and rapide In any event. Numerical Simulation ofViscoelastic Flow: State-of:the-Art 26 • The experimental determination of the nonlinear parameters in the various models is a more diificult task. the available experimental data are quite limited. A detailed review of comparisons of experiment and simulation for planar abrupt contraction flow . They reported encouraging results and will carry out future work to identify parameters by use of actual experimental data. In particular. Le. arise in polymer processing applications such as extrusion and injection molding. Some parameters are related to shear behavior and can be easily obtained by measuring viscosity. important parameters in a viscoelastic constitutive equation are often determined by fitting the viscosity curve and the few available data for fust normal stress difference in shear or extensional data at low strain rates. Planar contraction flow bas also been chosen as a benchmark problem to eValuate numerical methods and constitutive • equations by comparing numerical predictions with experimental results 76. 75 presented preliminary work using a numerical procedure to identify rheological parameters of Oldroyd-B and PIT models by finite element simulation and inverse problem solving.4 Planar Abrupt Contraction Flow Contraction flow bas received more attention than any other complex flow.4. there is no straightforward way to evaluate model parameters for complex flows where both shear and extensional modes of deformation occur simultaneously. it appears that no existing model is able to fit the available data for typical commercial thermoplastics. where deformations tend to be complex. but for parameters related to extensional behavior or normal stresses. we lack reliable data for normal stresses at high shear rates and the response to large. 2. for example in converging and diverging flows. Accelerating flows from a large cross-section via an abrupt or angular entry into a smaller cross-section. entry flows. In addition. rapid extensional flows. Inadequate parameter evaluation can result in poor predictions if we try to simulate typical processing conditions. This can be explained by the simplicity of the geometry and the fact that it has features that arise in polymer processing. N ouatin et al. LOPE LDPE FEM: Finite Element Method OG: Discontinuous Galerkin EVSS: Elastic Viscous Split Stress It is obvious frOID Table 2-1 tllat the use of integral models makes it possible ta use more relaxation times. (1993)83 Ahmed et al. while for differential models the number of equations ta solve for the extra stress is multiplied by the number of modes considered. Giesek:us 14 Streamline FEM36 Decoupled FEM Streamline FEM36 FEM (polyflow)85 FEM PS. This is because in integral models. LOPE LOPE Beraudo et al. PTT K-BKZ Numerical Method PenaltyFEM Material Park et al. Table 2-1 Author(s) Numerical Simulations ofPlanar Abrupt Contraction Flow Type of Model DifferentiaI Model Numberof Modes 1 8 White et ai. although White et aI. the contribution from each mode gets summed up in a single integral. (1988)78.79 . (1992)80 Integral Differentiai Integral Integral Integral Maders (1992)8i WhiteMetzner82 K-BKZ K-BKZ 1 8 8 Kiriakidis et al.8 DGFEM FEM EVSSDG Schoonen (1998f7 LLDPE. (1995)84 • Fiegl and 6ttinger (1996)86 RivlinSawyers87 PTT PTT. •79 and Schoonen77 did explain how they circumvented the problem.the-~ 27 • has been presented by Schoonen77• In Table 2-1 we summarize numerical simulations of planar abrupt contraction flows that have been compared with experiments.8 4. including the number of modes used in the caIcuIations.. The loss of 'convergence was delayed by White et al.Chapter 2. • as~uming Newtonian behavior in the elements that were in contact with the re-entrant . Numerical Simulation ofViscoelastic Flow: State-of. (1998)88 DifferentiaI Differential 7. m~ntion 78 Authors of papers frequently fail to numerical convergence problems near the singularities. LOPE HDPE LLOPE LLOPE HDPE. 84 and Fiegl and Ôttinger86 explained some of the discrepancies between experimental results and simulations in terms of the 2D nature of the simulation and a 3D component ofthe experimental flow. Most authors refer to the study of Wales90 who empirically found the effects of confining walls to be negligible for an aspect ratio exceeding 10 for fully developed shear flow. whereas quantitative comparisons are also made in the upstream. Comparing with the same experimental d~ta. the simulations of Kiriakidis et al. we present a summary of numerical simulations of this flow that were compared with experimental results. Finally. Fiegl and Ôttinge~1 reported reasonable quantitative agreement in terms of vortex growth. Comparing their results with the data of Beaufils89. However. while Schoonen had to use a parameter set for the PIT model that was inconsistent with uniaxial extensional flow behavior. but computed values of the entrance pressure 10ss were underestimated. section where the ratio is rarely larger than 1077 . Maders et al. by their finite element simulations. this ratio is mostly used in the channel (downstream section). two simulations using different constitutive equations for the same material and same flow gave conflicting results.5 Axisymmetric Abrupt Contraction Flow Most of the simulations ofaxisymmetric abrupt contraction flow have been done using integral models. measured by Meissner93 . 2.4. • . Numerical Simulation ofViscoelastic Flow: State-of:the-Art 28 • corner. 81 used a one-mode White-Metzner model and reported that the preclicted stresses along the centerline showed a slower decay than the experiments. poor agreement with the simulations was obtained. Interestingly. Ahmed et al. In Table 2-2. When comparison was made with experimental entrance pressure drop in these studies. Barakos and Mitsoulis92 also reported the underestimation of entrance pressure loss.Chapter 2. 83 underestimated the stresses in the downstream channel using a multimode K-BKZ mode!. 96 claimed that the entrance pressure 10ss was insensitive to extensional rheology. the experimental values include die exit effects.6 Axisymmetric Converging Flow The prediction of entrance pressure. Beraud097 and Fiegl and ÔttingerH tried to explain these discrepancies..4. .Chapter 2. and the degree to which these effects contribute to the experimentally determined entrance pressure 10ss is unknown.loss in converging axisymmetric f10w bas been the subject of severa! recent studies using both integral and differential models. 2. Most studies report that numerical predictions of entrance pressure drop were substantially underestimated when compared to experimental data. Mitsoulis et al. According to the • authors. These are Sllrnmarized in Table 2-3. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 29 • Table 2-2 Author(s) Dupont and Crochet (1988)94 Numerical Simulations ofAxisymmetric Abrupt Contraction Flow Type of Model Integral Integral Model K-BKZ K-BKZ Numberof Modes 8 8 Numerical Method Streamline FEM Streamline FEM36 Streamline FEM Material LDPE Luo and Mitsoulis (1990)95 LDPE Hulsenand van der Zanden (1991i 7 Differential Giesekus 8 LDPE Fiegl and Ôttinger (1994)91 Integral RivlinSawyers 9 8 FEM LDPE Barakos and Mitsoulis (1995)92 Integral K-BKZ Streamline FEM36 Control volume SIMPLE LDPE Luo (1996r U Integral K-BKZ 8 LDPE FEM: FlnIte Element Method SIMPLE: Semi-Implicit Method for Pressure Linked Equations . EVSS-SUPG 6. l06). LDPE LLDPE Integral Integral 7. Convergence problems were reported for the tapered geometries by Luo and Mitsoulis 103 and Garcia-Rejon et al. Table 2-3 Author(s) Numerical Simulations ofAxisymmetric Converging Flow Type of Madel Differential Model Numberof Modes 7 Numerical Method DGFEM Material • Integral.7 Extrudate Swell 2.8 LLDPE 2.Chapter 2. exit pressure drop is known to be very sma1l98 and would not compensate for the large difference between experimental and predicted entrance pressure drop. differential. and computations failed to converge for diverging angles beyond +30 0 and for converging angles beyond -30 0 (Garcia-Rejon et al. K-BKZ.1 Annular Swell Many attempts have been made ta simulate extrudate swell. (1996)99 Hatzikiriakos and·Mitsoulis (l996)1Ol Mitsoulis et al. we present a summary of simulations of annular extrudate swell that were compared with experimental data. viscous Carreau l02 FEM: Finite Element Method DG: Discontinuous Galerkin EVSS: Elastic Viscous Split Stress SUPG: Streaniline Upwind Petrov Galerkin Beraudo (1995)97 Guillet et al.4. L06 • Sorne of th~ results presented were not fully converged • solutions (Luo and Mitsoulis L03)." LLDPE LLDPE. l04) used the experimental data of Orbey and DealylOS for three HDPE melts. Numerical Simulation ofViscoelastic Flow: State-of:.7. In Table 2-4.4.8 6 Stream-tube mapping Streamline FEM Streamline FEM. Two independent swell ratios are required to describe annular swell: the diameter swell and the thickness swelL Two studies comparing numerical simulations with annular extrudate swell measurements (Luo and Mitsoulisl03 and Tanoue et al. (1998)96 PIT Wagner1UU K-BKZ . PIT. .the-Art 30 • However. Garcia-Rejon106 et al.Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 31 • Table 2-4 Author(s) Luo and Mitsoulis (1989)103 Numerical Simulations ofAnnular Extrudate Swell Model Numberof Modes 8 Numerical Method Streamline FEM36 FEM (pOlyflow)85 Streamline FEM Under relaxation mixedFEM Material HDPE Type of Model Integral K-BKZ K-BKZ K-BKZ Giesekus Garcia-Rejon et al. Table 2-S presents numerical simulations ofaxisymmetric extrudate swell that were compared with experimental results. The comparisons • between simulations and measured values for a long and a short die are presented in . . most of the studies were done using integral models. the use of a discrete spectrum of relaxation times would improve the predictions. but the agreement was poor for diameter swell. (1995)106 Otsuki et al. Many studies used the data of Meissne~3 to compare with their simulations. The temperature was kept constant in these swell experiments by extruding into silicone oil at ISOaC.. (1998)104 FEM: FlDIte Element Method For the straight annular die.4. Tanoue et aI.7.2 Capillary Swell Axisymmetric swell was also the subject of numerous studies. 2. but the thickness swell was no1: close to the experimental data. According to the authors. Again. reported opposite results: predictions ofthickness swell were in reasonably good agreement with experimental data. results from Luo and Mitsoulis l03 showed reasonable agreement of prediction and experiment for the diameter swell. (1997)107 Integral Integral DifferentiaI 6 6 1 HDPE HDPE HDPE Tanoue etai. 104 reported deviations as high as 34 % between simulation and experiment. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 32 • Tables 2-6 and 2-7. (1998)88 PIT FEM: FlDlte Element Method BEM: Boundary Element Method DG: Discontinuous Galerkin . and the underestimation was larger at high shear rates. . (1996)111 Integral Differentiai K-BKZ 8 7 LDPE LLDPE Beraudo et al. while swell is largely overestimated at the highest shear rate. The number ofmodes used in the simulations is given in Table 2-5. Model parameters were fitted to shear and elongationai rheologicai data. We aIso see that • . predictions differ considerably from one study to another. Beraudo et aI. In generaI. Table2-S Author(s) Numerical Simulations ofAxisymmetric Extrudate Swell Type of Model Integral Model Numberof Modes 8 Numerical Method Streamline FEM36 Material Luo and Tanner (1986)108 K-BKZ LDPE Bush (1989) 109 Differentiai Integral Leonov K-BKZ 7 BEM FEM LDPE HDPE Goublomme and Crochet 8 (1992i Goublomme and Crochet (1993)110 6 Integral Wagner 6 FEM HDPE Barakos and Mitsoulis (1995)92 Integral K-BKZ 8 Streamline FEM36 Streamline FEM DGFEM LDPE Sun et al. 88 reported good agreement for a long die but poor agreem~nt for a short die. Comparing with their own measurements. the polymer was extruded directly into airn .Chapter 2. It is worth mentioning that in their swell experiments. moderate agreement is obtained for the low and medium apparent shear rates. The computation largely underpredicted the experimental value for the short die. thus involving sagging due to gravity and non-isothermai effects that couid result in an underestimation of the ultimate swell. Special form ofLeonov model.) 195% 223% 195% Reference Barakos and Mitsoulis92 * ll1 Sun ** Meissner'" • • MeshM2 •• MeshM3 .? *** Meissnei"'" **** LU • MeshM2 •• MeshM3 . quantitative agreement with experimental results was obtained.9 calculated vs.1 S-I) 54% 49% 58% Swell (1..3)110 was larger than typical reported values.61 measuredi8 • With a difIerent Wagner damping function modified to give a non-zero second normal stress difference. Inclusion of a converging upstream conical section in the simulation ta match the experimental set-up resulted in a very high calculated swelling ratio (. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 33 • Finally. the simulation of extrudate swell has praduced contradictory results.1 s-a) 31 % 30% 30% 37% 34% Swell (1.. results taken from figure •••• Extrapolated values for an inf'mite long die Table 2-7 Comparisons between Predicted and Measured Extrudate Swell (orifice) Model K-BKZ K-BKZ Experiments SweU (0. Goublomme and Croche~8.0 s-I. Extrusion was into a silicone oil bath at 190°C. 2.1l0 simulated the extrudate swell of an HOPE melt thathad been measured by Koopmans 1 12.0 S-i) 82% 84% 94% 77% 56% Luo and Tanner lo8 Barakas and Mitsoulis92 * Sun ** Bush1 \r.Chapter 2.0 S-I) 133 % 163 % 125% Swell (10. usually differing considerably from measured values. In conclusion. Table 2-6 Reference Comparisons between Predicted and Measured Extrudate Swell Oong die) Model K-BKZ K-BKZ K-BKZ Leonov Experiments Swell (0.0 S-L) 51 % 69% 62% 53 % 52% Swell (10. However~ the required ratio of second ta fust nonnal stress difference (N21N I = _0.38-2. underestimating or overestimating swell. For the PIT model. (1997)115 Giesekus Differential Schoonen PIT. However. with model parameters fitted to shear data including viscosity and first normal stress difference. but it has received orny limited attention up to now. around a cylinder and cross-siot flow. all giving equally good fits of the shear data. This flow is of interest because along the centerline. then sheared along the cylinder's surface and finally stretched in the wake of the cylinder. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 34 • 2. or planar stagnation. 7 (RS) 4. PIT. Table 2-8 presents recent numerical simulations of planar flow around a cylinder where predictions were compared with experiments. integral Numerical Method FEM (POlyflow)85 FEM EVSSDG FEM EVSSDG Material LLDPE. PTI and Giesekus.8 4.4.8 Hartt and PIT. no unique set ofparameters could be identified without • the use of elongational data. At the stagnation point a material element experiences a very large extensional strain rate. a material element undergoes various types of deformation: it will be compressed when approaching the cylinder. (1998)77 Giesekus FEM: Finite Element Method EVSS: Elastic Viscous Split Stress DG: Discontinuous Galerkin Type of Model DifferentiaI. The planar flow around a cylinder has been proposed as a benchmark problem for numerical tecbniques 1l3.Chapter 2. which will produce a high Ievei of orientation. two liquids impinge to create a steady extensionai deformation. RivlinBaird (1996)1I4 Sawyers Differential Baaijens et aI.8 üilier Complex Flows We provide here a brief summary of other complex polymer flows such as planar flow . LDPE LDPE LDPE Baaijens et 'al. In cross-slot flow. Three parameter sets were tried. Ils tested two differential constitutive equations. comparison with experimentally obtained birefringence . • Table 2-8 Author(s) Numerical Simulations of Planar Flow around a Cylinder Modei Numberof Modes 1 (pIT). the modified EVSS technique fust introduced by Guénette and Fortin57. Within the category of numerical mixed methods.1 Limitations ofNumerical Simulations of Viscoelastic Flows Over the past decade. To achieve accurate results. serious convergence problems remain. significant progress has been made in the numerical and experimental analysis of viscoelastic flows. These results were consistent with those reported by Hartt and Baird114 for the same flow ofpolyethylene melts.5 Conclusions Major difficulties persist in the simulation of polymer fiows. The PTT model.4). Schoonen had to use a parameter set for the PTT model that was consistent with shear data but not uniaxial extensional behavior in order to obtain numerical convergence. For flow geometries with a • . 2. gave convergence problems. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 35 • patterns revealed that neither of the models could predict quantitatively the observed stress patterns.Chapter2. which described rheological properties best. appears to provide the most robust formulation currently available.4. both velocity and stress data in planar abrupt contraction flow (section 2. He compared the predictions of the multimode Giesekus and PTT models with velocities measured using particle tracking velocimetry and stresses measured by fieldwise flow induced birefringence. both on the experimental and the numerical side. and these are surnmarized below. 2. the Giesekus model performed better for the cylinder and cross-siot flow.5. despite the improved performance of numerical methods for viscoelastic flow simulations. Where the PTT model gave better agreement than the Giesekus model witb. the modified EVSS method should be combined with an upwind scheme like the SUPG formulation or the discontinuous Galerkin (Lesaint-Raviart) Methode However. Schoonen77 did an experimental/numerical study of an LDPE melt in both planar tlow around a cylinder and cross-slot flow. the predictive capability ofany numerical analysis is only as good as the input data. This holds in particular for flow regions with strong elongational components. reliable data for normal stresses at high shear rates and response ta large. rapid extensional flows ta fit the model parameters. Hua and Schieber16 recently presented calculations of viscoelastic flow through fibrous media using kinetic • theory models and a combined finite element and Brownian dynamics technique. Comparisons were made using the same numerical technique with analogous models that . either ta low flow rates or materials with short relaxation times.. a numerical breakdown occurs above a modest elasticity level or De number. 2. material parameters and boundary conditions. Even when convergence is possible. Attempts to find a general viscoelastic fluid model that would be applicable to all classes of flow problems and be able to give reliable results have 50 far failed. the constitutive model. i. there is no straightforward way ta evaluate model parameters for complex flows where both shear and extensional modes of deformation are occurring simultaneously. it is not clear if the poor agreement between simulations and experiments is due to the inadequacy of the models or ta the poor evaluation of the model parameters. If good success is 0 btained in comparison with experimental data for a particular class of flow problem.. such as sharp corners for enclosed flows and discontinuities in boundary conditions for free-surface flows. for example in converging and diverging flo\vs. At the present time. Simulations of these flows are still limited. such as convergence problems. in particular. i.5. in general. And as was mentioned previously.Chapter 2. confrontation with. to low Deborah numbers. Apart from purely numerical issues. experimental results Încreasingly reveals the inability of existing constitutive equations to predict the complicated stress fields of industrial forming operations.e. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 36 • singularity. We lack.2 Kinetic Theory Models: A Promising Approach? A relatively recent approach that does not require closed form constitutive models is the so-called micro-macro fonnulation based on kinetic theories l 6-21. poor agreement is usually observed with another class of flows.e. • . problems still arise that prevent convergence of the flow field at high De number. According to Hua and Schieber16 . For example. the FENE-P dumbbell underestimated the magnitude of the normal stress by as much as 25% compared to FENE model. a way ta avoid the numerical instability problem might be the use of more accurate finite element methods. which is an approximate PENE model. such as the modified EVSS formulation or a higher order technique 116. and reptation with independent alignment underestimated it by 22% compared to the more realistic reptation modeL The use of kinetic theory models improves considerably the quality of the prediction. Using their current method. specifically the FENE-P dumbbell.Chapter 2. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 37 • lead to closed-fonn constitutive equations. they had to modify the momentum equation by adding a relaxation parameter ta make the equation more elliptic in order ta obtain numerical convergence. 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Part 1: Aspects of Molecular Structure and Rheological Characterization Methods. ~ p. Revenu P. Peters G.. ~ p. Meck. Non-Newt. Eng. Polym... J. J. Fi.. Baaijens H. "Memory Phenomena in Extrudate Swell Simulations for Annular Dies". Part II: Time Dependency and Effects of Cooling and Sagging ". 409-419 (1998) 105 OrbeyN. W. p. 281-287 (1993) 111 Sun J..W.-L.P. 625-645 (1984) 101 Hatzikiriakos S.. Meeh. IQ.T.. Fi. Madison (1969) 103 Luo X.I.. Meck. ~ p. Clermont J.J.M.. of Wisconsin..E.L.~ "Extrudate Swell Through an Orifice Die "...H. Non-New!. FI. FI. QQ.G.-R. p. 33(8). thesis.. DiRaddo R. Béreaux Y.. Khomani B. École Nationale Supérieure des Mines de Paris.. 32(23)~ p. "Excess Pressure Losses in the Capillary Flow of Molten Polymers". Rheol. Tanner R. Crochet M.173-203 (1997) 116 Talwar K. Fi... Mitsoulis E.. 107-128 (1995) 107 Otsuki Y. Thesis. 247-268 (1996) Ils Baaijens F. 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Meek. Applications of Higher Order Finite Element Method to Viscoelastic Flow in Poraus Media ".. Funatsu K. Numerical Simulation ofViscoelastic Flow: State-of-the-Art 42 • 97 Beraudo C. Non-Newt. 37(7).J.L p.. Tanner R.Chapter 2.. ~ p.D. Baird D. J.H. "Numerical Simulations of Annular Extrudate Swell of Polymer Melts '~. 179-191 (1989) 110 Goublomme A. Polym. 2-5 (1988) 114 Hartt W.P.. 1-12 (1996) 112 Koopmans RJ.. Sophia Antipolis.E. Sei.W... Rheol. Iemoto Y. 171-185 (1983) 99 Guillet J. "Experimental and Numerical Study ofEntry Flow ofLow-Density Polyethylene Meltsn . Funatsu K.. J. For these simulations.3-1a !Ct) = 77[IID Ct) ~Ct) • where . The materials used for these experiments are described in Chapter 4.2 ModeIs Selected for this Research 3. To establish the cause ofthe limitations and suggest methods for overcoming them. The models selected are presented in the next section. 3.• 3. we canied out viscoelastic flow simulations using a numerical procedure described in Chapter 6. introduced in Chapter 2: This model was previously Eqn. we used two differential viscoelastic constitutive equations and compared their predictions with. In arder to achieve these objectives. Plan of the Research 3. We aIso conducted an experimental study of two complex flows: a planar abrupt 8:1 contraction tlow and an axisymmetric entrance flow and extrudate swell. those of a strictly viscous model. a generalized Newtonian model that describes fairly weIl the shear dependence of the viscosity of polymer melts. we carried out calculations using a strictly viscous model.1 Objectives The objectives ofthis work were as follows: 1.1 Carreau-y asuda Model In order to appreciate the added value of using a viscoelastic constitutive equation in tlow simulations. 2. To determine the limitations of present methods for the numerical simulation of the flow ofviscoelastic materiaIs. We chose the Carreau-Yasuda1 model.2. ). has been found to provide a good representation of data for shear flows and to be easy to use6 • However. which can be estimated using the discrete relaxation spectrum.2 Leonov Model The fust viscoelastic constitutive equation we selected is a modified Leonov model2• This model was initially chosen because it is the ooly one that satisfies basic stability criteria3. it provided a poor fit to extensional flow data. the second invariant of the rate of deformation tensor.5. is an extension of the theory of rubber elasticity ta viscoelastic liquids. which controls the transition between the plateau and the power-Iaw regions.1.Chapter 3.2. Recently. Four parameters must he specified in order to use the model: the zero-shear viscosity (110). a characteristic time (À. As stated before. Plan ofthe Research 44 m-I • Eqn. and a second power-Iaw index (s). a power-Iaw index (m). a new fonnulation of the general class of differential constitutive equations proposed by Leonov was presented2 • This latest version eHminates sorne of the recognized deficiencies of the This original "simple" model by adding ooly one or two nonlinear parameters. normal stress differences in shear and transient stresses in startup flows are not predicted by this purely viscous mode!.1 and 5. • modification makes possible a fairly accurate description of simple flows for polymers such as low and high density polyethylenes (LDPE and HOPE). . which uses ooly the parameters of the discretized linear viscoelastic spectrum. Hadamard (thermodynamic) and dissipative stability criteri~ while describing weIl the material functions usually measured2 • The original model proposed by Leonov4.3-1b 71[ilD (t)] = 710 (1 + CU? ilD Ct))~)-s The scalar viscosity is a function of IID. The simple Leonov model.. which is derived from irreversible thermodynamic principles. polystyrene (PS) and polyisobutylene (pmi.2. The Carreau-Yasuda model parameters for the two materials studied are presented in sections 5. 3. which is related to the slope of the viscosity curve at high shear rates.1. It is sufficient that b{Il.B_. which is required by the Second Law of • therm. has the following general form: Eqn. which is known as the standard Leonov mode!.Chapter 3.~l + (1 42 y y 2 - 3 Il )8.= lJ p .3-3 v Bij+ 2Bik • ekjp =0 (tr(e-.!!-[B-.] lJ Here Ô ij is the unit tensor. and b=b(IhI0 can be thought of as a deformation-history-dependent scaling factor for the linear relaxation times and is an adjustable parameter. For incompressible liquids. the invariants of the Finger tensor are as fol1ows: Eqn. eij 13 p ..) Yp = 0) v Here. Plan of the Research 45 • For the sake of simplicity.odynamics. the model can easily he extended to coyer the multimode case and compressibility.. 3-2) becomes: . The evolution equations for the Finger tensor Bij and the associated dissipation function D are4 : Eqn. The simplest choice is to let b=I. we present here the modified Leonov model for a single relaxation time and an incompressible fluide However. The faIm ofthe evolution equation (Eqn. eijp =eijp (Bij) is the irreversible strain rate tensor. = detBij = 1.IJ be positive for the dissipation function ta he positive definite.3-4 The irreversible rate ofdeformation tensor. and 'tjj is the extra stress tensor. 8 and 9. Derivation of the compressible case for the simple model is presented in references 7. A is the relaxation time in the linear Maxwell limit.3-2 Eqn. B ij is the upper-convected time derivative of the Finger tensor.3-5 e-. A multimode incompressible version was used in our simulations. A constitutive equation with b= 1 and the neoHookean potential (n=J3=O) gÏves back the simple Leonov model in its original forme Finally. 3-7 yields the Mooney potential for n=O. and ~ and n are numerical parameters. While a major effort was directed at achieving satisfactory simulations.3-7 where G is the linear Hookean elastic modulus.2. which is the case for HDPE and pS2.r lJ a2 lJ (/. for the elastic potential W(I.G5.2 the results for our LLDPE and HDPE.3-6 In order to relate the extra stress tensor to the elastic Finger tensor during the defonnation history. and the neo-Hookean potential for n=J3=O.} The functional form.3 Phan-Thien Tanner Model Calculations using the simple Leonov model in numerical flow simulations indicated serious convergence problems. elastic potential bas been suggested2 : n=po f must be provided. a functional form. 3. Here. . of the b function is determined from extensional flow data. the extra stress tensor can be written in the Finger form as: Eqn.1. Eqn. and we present in sections 5..3-8 T .2 and 5. We aIso compare the predictions of the modified Leonov model with those obtained using the simple mode!.B..Chapter 3. The following general Eqn. andfis the specific Helmholtz free energy.2. 12.. Y.1. Po is the density. Plan ofthe Research 46 • Eqn.8VVJ . it was found that the use of this equation exacerbated the • convergence problems of the numerical analysis and made it impossible to obtain solutions for mast cases.. y r = 2(Boo üVV .. as will be seen in Chapter 8. even this equation gave convergence problems. However.-2 represents the upper-convected derivative .3-11 . 3-9) had shown itself to behave weIl in numerical simulations. Separability refers ta the possibility of separating time and strain effects in the nonlinear relaxation modulus.=0. Phan-Thien and Tanner suggested two possibilities for the function Y(tr~): • Eqn.1. as discussed in section 2. which is a combination ofthe upper 'V and lower tl convected derivatives: Eqn. The Phan-ThïenfTanner model1o. Fortin l2 reported having successful results in terms of numerical convergence with that model. The PIT model is expressed as follows: Eqn. the lower-convected derivative ç= l. therefore.3-10 where .4 (Eqn. a second viscoelastic constitutive equation was selected for use. 2-29). although it is derived from network theory. although it did not give as good a fit of measurable material functions.Chapter 3. Plan ofthe Research 47 • In order to achieve the objectives of the research. it is the corotional derivative.II (Eqn. Like the Leonov model. the PIT model does not predict a separable relaxation modulus. A differential model was chosen so that the existing numerical code could be used.3-9 where E9 stands for the convected Gordon-Schowalter derivative 13 . * 2. which is more suitable for polymer solutions. while present6. contraIs the level of shear tbinning. the PIT model has two nonlinear parameters (ç and g) to be determined for each materia! in addition to the discrete spectrum. • .) and of polymer solutions with Eqn. These disadvantages are inherited by the PTT model if .3 and 5.1.2. Despite these problems. The linear equation for Y (Eqn. the PTT model is known to give a moderate to good fit6 of shear and extensional rheological properties of polymer melts using Eqn. which is also unrealistic. Nonlinear shear and extensional data are needed to determine the nonlinear parameters of the mode!. 3-11) predicts a plateau value of the elongational viscosity at high strain rates. The PTT model parameters we used for our materials are presented in sections 5. E bas little influence on shear properties and serves mainly to biunt the extensional singularity that would otherwise be Larson6 pointed out that there are disadvantages to using the Gordon-Schowalter derivative for melts. Plan ofthe Research 48 • Eqn. which is physically unrealistic 14 . 3-12 is suitable for polymer melts. Eqn. and the violation ofLodge-Meissner relationshipls. for example a maximum in the fiow curve. 3-11. since the exponentiaI term results in a maximum in the steady-state elongational viscosity and is thus in better agreement with experimental data. reported when Other problems have aIso been ç =:1= 2.3.Chapter 3. but then the model predicts N 2=O.3-12 With either choice. 3-12 for Y(tr. Using ç = 2 avoids these problems..ll. The parameter .lO. Primary among these are unphysical oscillations in the shear stress and first normal stress during start-up of steady shear at large shear rates. . Yokohama. • . p. Acta... "Buildup and Relaxation of Molecular Orientation in Injection Molding.. p. dQ. Eng.. Rheol. Po/ym. 202 (1993) 10 Phan Thien N.. "Buildup and Relaxation of Molecular Orientation in Injection Molding. "A New Constitutive Equation Derived from Network Theory". p. 193 (1993) 9 Flaman A.. Acta. Act~ . École Nationale Supérieure des Mines de Paris.D. in "Rheology of Polymer Melt Processing J.. Prokunin AN. 353-365 (1977) Il Phan Thien N. FI.R.. "Calculation of Residual Stresses in Injection Molded Products". "Injection Molding for Charged Viscoelastic Fluid Flow Problems".P. J..Y. 259-283 (1978) 12 Fortin A... Paskhin E.I. Elsevier Science.. Acta....... Guillet J.T. Rheo/. Eng.AM. Part 1: Formulation". "Méthodes éléments finis pour la simulation numérique d'écoulements de fluides viscoelastiques de type différentiel dans des géométries convergentes". Cohen R. Rheol. France (1995) 15 Carrot C.. 79 (1972) 14 Beraudo C. "Theoretical and Experimental Investigation ofShearing in Elastic Polymer Liquids". "On the Rheological Modeling of Viscoelastic Polymer Liquids with Stable Constitutive Equations". 259-273 (1995) 3 Leonov Al. Tanner R.li. Trans. "Constitutive Equations for Polymer Melts and Solutions". Beliveau A. Schowalter W. Sei. Non-Newt.C.J.. 36(2).I.. Meck. 327-328. p. Leonov A. 284 (1991) 8 Flaman AA. Part fi: Experimental Verification". Rheo/. Japan (1998) 13 Gordon R.. Amsterdam (1996) ll .. Polym.R. 187-193 (1995) 4 Leonov A. p.. b p.M. Sei. li. Rheo/. thesis. ~ p. 33(4). p. Intem. Piau and J.D. Soc. Ph. 41 1-426 (1976) 6 Larson R. p. 2b p. Sophia Antipolis. Arsac A. Rheol..G. '~iscoelastic Constitutive Equations and Rheology for High Speed Polymer Processing".E. Armstrong R. p. PPS-14.. 33(4). Rheo!. Lipkina E. Butterworths Series in Chem.). "A Nonlinear Network Viscoelastic Madel". Eng.Chapter 3. PoZym. "Experimental Validation of Non Linear Network Models"... Revenu P.l. ''Nonequilibrium Thennodynamics and Rheology of Viscoelastic Polymer Media"..M.I. J.. Agassant (eds. Boston (1988) 7 Baaijens F. Plan ofthe Research 49 • REFERENCES 1 Yasuda K.-f. 163 (1981) 2 Simhambhatla M. Act~ ~ p. 85-98 (1976) 5 Leonov A..• lQ. Some characteristics of the two resins are presented in Table 4-1. It was not possible ta obtain a steady flow rate.) 1. the linear low-density polyethylene (LLDPE) that was studied in the axisymmetric extrudate swell experiments proved to be unstable in the extruder used for the planar abrupt contraction flow.1 Materials Used ln the original research plan. Table 4-1 Experimental Materials Molecular Weight <Mw) (g/mol) 119600 195000 Polydispersity (M.jMJ 3. which is a commercial high stability copolymer made using a Ziegler-Natta catalyst. This was intended to make possible the performance evaluation of the constitutive equations of interest in different tlow situations.0 * 1. The storage and 10ss moduli were .16 lc~ 190°C (ASTM D1238.0 Resin LLDPE HDPE Density (g/cm3) 0.1 Smalt Amplitude Oscillatory Shear • The experiment most widely used to determine the linear viscoelastic properties of polymer melts is small amplitude oscillatory shear. and it was necessary ta use another material for these experiments. Rheological Characterization 4.957 MeltIndex (g/lO min.2 Experimental Methods 4.2.• 4.82 35. a single polymer was to be studied in !Wo types of complex flow. However.15 ** • 2. Method E) •• Test 15 4. It is used primarily in packaging and is processed by extrusion and film blowing. The HDPE XI010 was an experimental DSM material polymerized using a catalyst different from the one used to make DSM commercial blow molding resins and sold as StamylanTh! lID.926 0. The LLDPE was Dowlex™ (Dow Chemical Company) 2049A. a high-density polyethylene (HDPE). Chapter 4. Rheological Charaeterization 51 • measured using a Rheometrics Dynamic Analyzer II (RDA II), a controlled strain rheometer. The instrument was operated in a parallel plate configuration (plate diameter=25 mm), under a nitrogen atmosphere to prevent thermal degradation. Thermal stability tests were performed to ensure that the resins did not degrade under test conditions, and strain sweeps were performed to verify that the measurements were within the linear viscoelastic regime at aIl frequencies. The nominal frequency range of the RDA II is from 0.00 1 to 500 radis, but the useful frequency range is resin dependent and must be determined experimentally. Samples for dynamic measurements were compression molded. The molding temperature and pressure cycle was optimized to produce homogeneous samples without residual stresses or voids and to rnjnjmjze thermal degradation. The samples were prepared at 180°C according to the sequence of times and applied pressures presented in Table 4.2. Table 4-2 Cycle Pressure (l\1Pa) Time (min.) Compression Molding Procedure at 180°C compression compression compression 2.4 3.6 1.2 5 5 5 Cooling * 3.6 -10 melting 0 5 • pressure released when T < 40°C 4.2.2 Steady Shear Capillary Rheometer The viscosity of the LLDPE was measured on an Instron, piston-driven, constant speed capillary rheometer. In this instrument, the apparent wall shear rate is calculated from the piston speed, and the wall shear stress is calculated from the force measured by the load. cell. Accurate control of the temperature in the barrel (±lOC) is achieved through three independent heating zones with PID controllers and J-type thermocouples 1• Circular dies • of 1.4 mm diameter and LlO ratios of 4.75, 9.5 and 19.0 were used to obtain accurate Cbapter 4. Rheological Characterization 52 • values of the Bagley corrections which agreed with values ob1ained using an orifice die2 (LID <0.5). Siiding Plate Rheometer The sliding plate rheometer shears a melt between two stainless steel plates. The moving plate, which can produce shear rates up to 500 S-I, is driven by a servo-hydraulic linear actuator. The shear stress is measured at the center of the sample by a shear stress transducer developed at McGill University. temperature-controlled oven. AlI measurements take place in a A detailed description can be found in reference 3. Measurements were made at 200°C for the HDPE resin. 4.2.3 Extensional Flow Transient elongational experiments were performed using a Meissner4 type apparatus, the Rheometrics Melt Elongational rheometer (RNfE), by Plastech Engineering AG laboratory in Zurich. The sample, clamps and leaf springs are installed in a temperaturecontrolled oven, which is heated by electrical heater wires em.bedded in the walls. The sample is supported by a cushion of inert gas (nitrogen) and stretched homogeneously between two sets of conveyor belt clamps at a constant Hen.cky strain rate defined as follows: Eqn.4-1 è= _d..:.,.(ln.....;:(:--L:...:...)) dt where L is the actual sample length. The strain rate range of the instrument is 0.001 to 3 S·I, but the attainable strain rate range is resin dependent and must be determined experimentally. performed at 150°C for the LLDPE and at 200°C for the HDPE_ Measurements were • Chapter 4. Rheological Charaeterization 53 • 4.3 Linear-Low Density Polyethylene AIl rheological measurements for the LLDPE were performed at 150°C~ which is the temperature at which the extrudate swell measurements were conducted. In order to make sure that the polymer was fully molten at that temperature~ we consulted data previously collected for this materials. Reat capacity measurements carried out in a differential scanning calorimeter showed a melting peak temperature of 126°C. Pressurevolume-temperature (pVn data obtained in cooling gave the following crystaJUzation temperature~ Tc~ as a function ofpressure: Tc = 113.52 + CO.281)P Eqn. 4-2 with P the pressure in MPa The highest pressure reached in the capillary rheometer was 42 MP~ which gives a crystallization temperature of 125°C. We are therefore confident that all measurements were perfonned in the molten state. 4.3.1 Storage and Loss Moduli The storage (G~) and loss CG") moduli were measured for tbree samples~ with a maximum between-sample variation of 3 %. Measurements were made over a frequency range of 0.02 to 500 rad/s. Results are presented in Fig. 4-1. The discrete spectrum of relaxation times was determined from the oscillatory shear data by linear regression as described in reference 6. In that technique we specify N values of Âi distributed around a center time and with a specified distance between times, within the range of experimental frequencies. The values of G i are then determined by a leastsquares procedure, using M sets of data: Eqn.4-3 MIN Gi j=l • f[(f 1 G. 1 i=1 G' Cm j) (aJj'Â,i)2 1 + (aJp.ti )2 . Rheological Charaeterization 54 • A minimum of six modes was necessary to obtain a good fit of the experimental moduli.09 x 6.Chapter 4.0E+03 ..00 X 10-2 4. G' 1.00 X 10-2 5. .43 X 10-3 1..00 x 8.01 1 10 Frequency (radIs) 100 1000 Figure 4-1 Dynamic Moduli ofthe LLDPE at ISO°C..38 X 1.0E+06 __.10 X lOS IDs 104 104 103 102 Truncated Data (s) 1.50 X 10-1 1.0E+OS • 1!J1'DDO ••• ~ ~ 1..00 X 10-3 1.... parameter sets [Gj .-...38 x 1.J are presented in Table 4-3. This was done to obtain a smaller longest relaxation time in order to improve the numerical convergence of the simulations....36 X 105 105 104 104 10-" 102 (s) 2...25 X 2.11 X 1.16 X 2.0E+01 ~--------------__.... Table 4-3 Mode 1 2 3 4 5 6 G(pa) Discrete Relaxation Spectra for the LLDPE at 150°C AllData 2. 1.25 x 100 Â..1 •• • a ri ri a • •a D D _ D D D • 8 1 G" • 1.... 0..0E+04 ë) ë) 1. . ri a 1. The Â. Both fits are shown in Fig.51 x 2.00 x 10-2 7.77 x 6...90 X 10-1 3.25 x 100 6....00 X 10-2 2. We determined two discrete spectra: one for all the experimental data available and one for a truncated set of data where we removed the measurements at the lowest frequencies. • .0E+02 •• Il 0.43 x 10u 24.L.0 x 100 À G(pa) •• 2. 4-2. .17 X 1. 200° and 240°C using a master curve at 2aaoe and an activation energy (EJ of 26.0E+03 .0E+04 (.. 1....5) (Figure 4-3).. 1..... 1..-... 1. • Eqn... the flow curve compared weil with data obtained from the sliding plate rheometer at the same temperature by Koran? The flow curve is presented in Fig.0E+OS G" 1. With ail corrections applied (Bagley and Rabinowitch). which is unusual for a linear material. in Fig.1 100 Frequency (radis) Figure 4-2 Fits ofthe Discrete Spectra for the LLDPE at 150°C.. The Bagley corrections were in reasonable agreement with values obtained by Kim? using an orifice die (L/D < 0... 4....----~--------------------~ 1000 1 10 0..Chapter 4.....01 0... and the results were compared with others5 obtained at 1600 .. 4-4) does not apply for this material at high shear rates.0E+06 ..2 Viscosity Measurements were made at 150°C. (.1 kI/mol. Rheoiogical Charaeterization 55 • ta Q. Fina11y..0E+02 -Based on Full Data • • Based on Truncated Data 1... 4-5 the viscosity is plotted along with the complex viscosity. showing that the empirical Cox-Merz rule (Eqn....4-4 (0) = y) .0E+01 . 4-4..3. .1 • Shear Rate (1/s) Figure 4-4 Flow Curve by Two Rheometers for the LLDPE at 150°C..5 0 . Rheological Charaeterization 56 • :i 2. ! c: CD CD U a...oIIi!'--=========_J 1 10 100 1000 0...-.1 ! en en ta ~ o 0.5 . .... o ~ ri rD 0.--------------------------..... 2 .5 o 1 1 c: Q....... 1..... ta 1.-----~----..i 0 ~i 20 • • Determined By Orifice Die 0 o Determined by Bagley Plot 40 60 80 100 120 140 Shear Rate (1/s) Figure 4-3 Entrance Pressure Drop by Two Methods for the LLDPE at 150°C...001 ......01 o Capillary Rheometer à Siiding Plate Rheometer 0...Chapter 4...... <1 0..... 0.. Shear Rate (1/s) Figure 4-5 Complex Viscosity and Viscosity of the LLDPE at ISO°C...4-S • .... According to the instrument manufacturer.-~ ~ 1. 1000 Frequency (radis). f i) 1.1 1 10 100 .0 S-I...001 0... an important source of errer that remains is the uncertainty in the force measurement....Chapter 4...-..01.. 4.0E+03 o RDA Il o Capillary rheometer 6 Sliding Plate Rheometer 1..3 Tensile Stress Growth Coefficient Transient elongational experiments were performed at ISO°C for nominal strain rates of 0.. Even if we elimjnate the uncertainty in the strain rate. this uncertainty is 0..09460 and 1. 0..0E+02 +.004 S-l. Rheological Charaeterization 57 • -. The true strain rates were found ta be 0. it was possible ta obtain an accurate value of the strain rate.001 N..----~---~---~-----------0..3. The error on the stress can be determined by performing an errer propagation analysis8 • The stress is calculated by the following equatien: Eqn......0E+OS ...01024.1 and 1. Using particle tracking and video images of three zones in the sample..0E+04 tL ni ..01 0..... 0 0 1.. --~----. rate=1..1 s-1 6..-. .-.. o rate=0.. we have included the error bars calculated with Eqn. 4-6....(jE Mi' 0..Chapter 4.0E+OS ...01 0. cg 11)- .... and H and W are the initial height and width of the sample......01 5-1 o rate=0..2 II) ~- ! cne 1..01!'-------------100 1000 1 10 0..0 s-1 -Linear Viscoelasticity 1o fi! t'a.... • lime Cs) Figure 4-6 Transient Elongational Stress for the LLDPE at 150°C...c rn......: The subscripts RT and TI refer to the room and test temperatures.. 1.. the uncertainty in the force leads to a very large uncertainty in the rheological properties ofinterest. 4-7. . We can see that at low strain rates..1 .001 --=-=-(jE F F In Fig.. begins: 2I3 Ac is the cross sectional area of the sample just before the deformation Eqn. Rheological Charaeterization 58 • where FCt) is the force as a function of time and A(t) is the cross sectional area as a function of time..4-7 !::..0E+Q4 ~.785 glcm3 • Neglecting the uncertainty in the cross-sectional are~ the error propagation analysis gives: Eqn. The LLDPE melt density at test temperature (150°C) is 0.. 4-(j Ao = H RTWRT ( J ..0E+OS CD CD= = 0 CD CD 1- 1:0 1. Chapter 4. Average values of G' and G" for the five samples are presented in Figure 4-7. a minimum of six modes was necessary to obtain a good fit of the experimental moduli. Both fits are presented in Fig. with a maximum between-sample variation of 4 %.0E+OS ca a. ÂJ are presented in Table 4-4.02 to 500 rad/s.01 0.0E+03 o 0 1.1 Storage and Loss Moduli The storage (G') and 1055 (G'') moduli were measured on five samp1es. • . Rheological Charaeterization 59 • 4. Measurements were made over a frequency range of 0. where we removed the data at the lowest frequencies. oS9ij 00 DO 00 DO 00 00 0 0 0 0 0 0 0 00 0 0 0 0 ei~ 0 0 800000000 ~B 0 00 000 ( .0E+02 0.1 1 10 100 1000 frequency (rad/sec) Figure 4-7 Dynamic Moduli ofthe HDPE at 200°C.0E+Q4 è> 1. Again. 1.4 High Density Polyethylene 4.4. The parameter sets [Gj. and one for a truncated set of data. 1. We determined aIso for this material two discrete spectra: one for all the experimental data available.0E+06 o G' (average) DG" (average) 1. 4-8. . Table 4-4 Discrete Relaxation Spectra for the HDPE at 200 0 e AllData Truncated Data À Mode 1 2 3 4 5 6 G(pa) 1.00 X 10-2 2.2 Viscosity and FirstNormai Stress Difference The viscosity of the HDPE was measured using the sliding plate rheometer at 200°C.81 X 104 1.1 1 10 100 1000 frequency (rad/sec) Figure 4-8 Fits of the Discrete Spectra for the HDPE at 200 o e.00 X 10-3 1. 0.0E+O& -r-------------------------~ -Based on Ali Data .00 X 10-2 4.23 X 9.-~.00 X 10-:l 5.31 X 105 1.90 X 10-1 3.25 x 10° 6.43 X 101.50 X 10-1 1. (s) 5 1.98 x 102 (s) 3 G(pa) Â..0E+OS ca ( . 0..95 X 10 104 104 104 103 103 2..Chapter 4..4.25 x 10° 4.0E+03 1.51 X 3.0 x 100 1.01 --.29 X 4.21 X 2.66 X 104 3. 1..43 x 100 24.0E+04 ë 1.08 X 103 5. 1.01!-----!__-------------__.6 kJ/mol .Based on Truncated Data 1. • Our results were compared with data obtained at DSM using capillary and cone and plate rheometers at 190°C and shifted to 200°C using an activation energy Ea of 27.09 X 105 4.0E+02 . Rheological Characterization 60 • a.54 X 2..00 X 10-2 7. which is incorrect since both the complex viscosity and the viscosity should reach the same plateau-value.. The first normal stress difference was aIso measured using the cone and plate rheometer at 190°C..0E-D1 ... in Fig. 4-11.0E+00 6. Good agreement was found between the three sets of data (Fig.. Q.0E+03 .... the viscosity is plotted along with the complex viscosity.. we conclude that the empirical Cox-Merz nùe is valid for this material. Rheological Characterization 61 • which was determined by DSM....c U) 1... shifted to 200°C.. These data were not used to fit the model parameters. In Fig... since it is not possible to evaluate the error involved in these measurements..... 1000 10000 100000 1 10 100 0..0E+02 0 o ca U) . 1..--~------~--~-----~----. • . 4-10.001 Shear Rate (1/s) Figure 4-9 Flow Curve by Three Rheometers for the HDPE at 200°C. The cone and plate data do not follow the same trend as the measurements in small amplitude oscillatory f1ow..... If we disregard the cone and plate data..0E+01 en en . 4-9)....Chapter 4.1 0.... Sliding Plate Data o Capillary Data (shifted at 200°C) x Cane and Plate Data (shifted al 200°C) 1. ~ o 1. We present these data.... Tto. ca CD ! 1.. using the same shift factor as previously.....01 0. . ......0E+04 1. .-..OE+OS ........001 Frequency (radis).JJ.......0E+04 ai ~ 1...0E+OS -r-~~---------------------'" X 1..01 • Shear Rate (1/s) Figure 4-11 First Normal Stress Difference of the HDPE Shifted from 190° to 200 o e.1 1 0.0E+03 ~ o 1.1 0.....=======:::::.01 0. ...................0E+02 o RDA Il (average data) o Capillary Data (shifted at 200°C) ô Sliding o o o P Jate Data ::te Cane and Plate Data (shitted at 200°C) 1..0E+01 ~=~==:... ca Z 1.0E+03 ... Shear Rate (1/s) Figure 4-10 Complex Viscosity and Viscosity ofthe HDPE at 200 0 e. Rheological Characterization 62 • CI) 1...J 10 100 1000 10000 100000 0. .....' 1 0....=~---_.... . .......Chapter 4.. x Cane and Plate Data (shifted at 200°C) e:. i. .. . 4. it was possible ta obtain an accurate value of the strain rate.5 s-1 (2) Linear Viscoelastïcity 10 o - 6. Excellent agreement was found between the three sets of data (Fig. Using particle tracking and video images of three zones.6) that a large error is present in the stress at low extensional rates due to the low forces.1 1 Tirne (s) 100 Figure 4-12 Transient Elongational Stress for the HOPE at 200°C. artificial strain hardening is often observed at high rates with the RME rheometer for non-strain hardening materials9• For the HDPE. 1.4946 5. • .5 S-I at 200°C.742g1cm3). ca 0 "'1: ! eu ~ eu~ (J 1. The error bars are smaller than the size of the symbols.c io fi! 13 1.0E+O& . 4. rate and had two replicates done in arder to verify reproducibility.5 s-1 (3) 1.- --.- . On the other hand.• 1 (The HDPE melt density at test temperature is 0. .3 Tensile Stress Growth Coefficient We saw in section 4.4845.3 (Fig. we chose a medium.. The true strain rates were 0. 0.4959 and 0. Rheological Characterization 63 • 4. 4.01 L ~------===========~ 0..0E+Q4 ~ o rate=O.3.0E+03 0.5 s-1 (1) rate=O.Chapter 4. Three measurements were done at a rate of 0.12).0E+OS C)A". rate=O. ...C. Saito T. Canada (1999) 3 Giacomin AJ. Sei.. p. "The effect of Pressure on the Rheology and Wall Slip of Polymer Melts". Rheal. Rheological Charaeterization 64 • REFERENCES 1 Sentmanat M. p. J. Rheal. unpublished work.O.. C. "Detennination of the Relaxation Spectrum from Oscillatory Shear Data". Dealy J M. Ph.. Montreal. McGill University. MeGill University. ANTEC '95. Society ofPlostie Engineers. McGill University.. " A Novel Sliding Plate Rheometer for Molten Plastics".M. 29(8). Montreal. Germany (1998) • . Master's Thesis.. li. ~~ A New ElongationaI Rheometer for Polymer Melts and Other Highly Viscoelastic Liquids ". unpublished work.M. Canada (1998) 9 Hepperle. Montreal.~ The Occurrence of Flow Marks during Injection Molding of Linear Polyethylene". Institute of Polymer Materials. 1035-1049 (1991) 7 Koran F.Chapter 4. Thesis. J. "The Effeet of Long Chain Branching on the Rheological Behavior of Polyethylenes Synthesized using Single Site Catalysts". Ph. Dealy J. 1-21 (1994) 5 Heuzey M. Canada (1998) 8 Wood-Adams P.D. Acta... Canada (1996) 6 Orbey N..D. Polym. J. Dealy J... p.. University of Erlangen-Nuremberg.L. 35(6). 499 (1989) 4 Meissner. Samurkas T.. Thesis. Erlangen. McGill University. Eng. "Sorne Factors Affecting Capillary Rheometer Performance". 1068-1072 (1995) 2 Kim S.. p. Montreal. Hostettler J. Determination of ModeI Parameters 5. 5-3 where T'lmeas represents the experimental viscosity.5-2 770 = LGiÂi =19079 Pa. 5-1. Eqo.s] A. and the corresponding fit ofthe viscosity is shown in Fig.39 0. using M sets of data: Ego.76 . [s] m s Value 19079 0.1 Linear Low-Density Polyethylene 5.41 0. 5-1. and two power-Iaw indexes. m and s.. 5-1) contains four parameters: the zero-shear viscosity 110' a characteristic time A.5-1 The zero-shear viscosity was calculated from the discrete spectrum based on aIl the data presented in Table 4-3. Table 5-1 Carreau-YasudaModel Parameters for the LLDPE at I50 a C • Parameter 110 [pa.1 Viscous Model The Carreau-Yasuda generalized Newtonian model (Eqn. The parameters determined for the LLDPE are presented in Table 5-1. as follows: N Eqo.1.s i=1 The other three parameters were determined by a least-squares procedure. and 11calc the viscosity calculated using Eqn.• s. .Chapter S.1 1 10 100 0.0E+02 . 5.01 Shear Rate (1/5) Figure 5-1 Viscosity Fit ofCarreau-Yasuda Equation for the LLDPE at 150°C... 5-5).001 0.0E+OS -r-------------------------.----~---~---_"!"----~-------_. which we write as follows: Eqn. b fi) câ :> 1....0E+Q4 a u a.1. 5-4) and the elastic potential (Eqn.0E+03 o Capillary rheometer 6. 1. The procedure recommended by the authors 1 for the .5-5 • For polyethylene. Simbambhatla and Leonov1 suggested using a Neo-Hookean elastic potential (n=JFO in Eqn. 1000 0. Sliding Plate Rheometer -Carreau-Yasuda 1. Determination ofModel Parameters 66 • en 1.I~ in the evolution equation (Eqn. 5-5) are required in order to use the modified Leonov model 1..2 Leonov Madel Parameters Specification of both b(ll.5-4 Eqn. Then. HDPE. The value obtained was m=0.) is as follows. First.02. simple power-Iaw or exponential functions of the invariants of the Finger tensor with one or two adjustable parameters are usually sufficient. perform preliminary calculations for various tlows using the simple model (b=l).5-6 To fit the m parameter. we used the transient elongational data. Determination ofModel Parameters 67 • choice of b(Il. We found that the best result for non-strain hardening materials was obtained with an exponential function of Iv. ifthere is disagreement with experimental data. since this form of the b function bas very little influence on the prediction of viscosity. The complete set of parameters is presented in Table 5-2.I. This routine is based on a quadratic interpolation method to find the least value of a function specified by the user.Chapter 5. and pm).02 • . To model the viscoelastic behavior of various polymers (LDPE. the first invariant ofthe Finger tensor: Eqn. The best value of the parameter was determined using an IMSLlFortran™ subroutine (DUVMIF)2. PS. a functional form of b that brings the calculations into qualitative agreement with the data can be systematically developed. Table 5-2 Leonov Model Parameters for the LLDPE at 150°C Parameter n J3 m Value 0 0 0. Table 5-3 PTT Model Parameters for the LLDPE at ISOCC Parameter Value 1.16 5. 5-6 is used.4 Viscosity We already mentioned that the simple and modified Leonov models do not differ in their shear predictions if Eqn.~) -~J + !: exp( À.1.1. With one parameter fitted exclusively to shear data (ç).5-7 . the We used the IMSLIFortranTM subroutine described earlier to determine the parameters. the PTT model reproduces more realistically the observed behavior. which is not valid for this material at high shear rates (Chapter 4). so we show only the results for the modifil"~d mode!.& trrJ = 211-D 22 Tl (ç and E) to be determined in addition to the discrete E It bas two nonlinear parameters spectrum.L(Ç ~+ (1. Determination of Model Parameters 68 • 5. Eqn. The value of ç was determined by fitting the viscosity curve and that of E by fitting the transient elongational data. • . The parameter ç controis the level of shear tbinning in shear flows and extensional behavior.Chapter 5. we show the fit of the Leonov and PTT models for the viscosity. In Fig. The resulting parameters are presented in Table 5-3.79 0. 5-2. The Leonov model predicts the Cox-Merz rule.3 Phan-ThienITanner Model Parameters The PTT mode13•4 can be expressed as follows for polymer melts. 6 showed that it is possible to avoid this problem by using a non-zero viscous component (11 s=#=O)' This way. =#= 2 is the presence of a maximum. the shear stress becomes a strictly increasing function of the shear rate if the viscosity ratio fulfilIs the following condition: Eqn. The use of a multimode spectrum for polymer melts is always • strongly recommended since a single mode can not fit data properly and leads to numerical problems.Chapter 5. which is physically unrealistics.. Determination ofModel Parameters 69 • 8 u :> fi) 1. this condition can.1 1 100 1000 0.0E+03 Capillary rheometer ~ Sliding Plate Rheometer -Modified Leonov Madel PTT Madel (alpha=O) ca o "" 1.01 0..5-9 lls+1]=1]o However..0E+OS ~-----------------------. ...0E+04 a. when a spectrum of relaxation times is used instead of a single mode.. 10 0. Crochet et al.001 Shear Rate (ils) Figure 5-2 Fit ofViscosity with Leonov and PIT modeIs for the LLDPE at 150°C. in the flow curve. .5-8 with Eqn.----~---~---~-----------_. be relaxed.. b 1... . 1.0E+02 . It was mentioned in Chapter 3 that one ofthe problems with the PTT model when . =O.5-10 a = t'Js t'Jo We obtained cx. 1.0E+OS -p---------------------------. that govems the viscosity ratio based on shear data. . Fig.001 Shear Rate (115) • Figure 5-3 Viscosity Curve ofPTT Model with cx. We found that this criterion could not be applied blindly in simulations. cx.=1/8 for the LLDPE at ISO°C. even without a viscous contribution. o en ~ 1.0E+02 .Chapter 5. Determination ofModel Parameters 70 • BeraudoS used a viscosity ratio CTl/TJo) of 1/8 in her simulations. 1.011!"----------------~ 0. Eqn.. 5-3.0E+03 o Capillary rheometer 6. Sliding Plate Rheometer en -PTT Madel.1 1 10 100 1000 0. 5-8 can be ignored when using a spectrum of relaxation times. and the result is presented in Fig. (alpha=118) 1. we fitted an additional parameter. Instead.0E+04 :..01 0. although she used a spectrum of relaxation times..a----~---.-lO-5 for the LLDPE. and we can see that there is no maximum in the curve over the range of experimental data. We verified. ëii u ca a. the effect ofthat ratio on the prediction of the viscosity for our material. which shows that the condition given by Eqn.. 5-4 shows the flow curve with cx. ..1..1 o ~ Capillary Rheometer Sliding Plate Rheometer -PIT Model (alpha=O) 0. 5-5...------~-----------------.----------------.1 1000 Shear Rate (1/5) Figure 5-4 Flow Curve Predicted by the PIT Model with a=O. but at the lowest rates less strain hardening is predicted than is seen experimentally. we show the fit of the tensile stress growth coefficient provided by the Leonov and PTT models... Good agreement is obtained for the highest strain rate (1. 5. 1 10 100 0. we suspect that these data are not reliable.... - Cà ~ 0. The re~ting curves are very similar for the two models.Chapter 5. Determination ofModel Parameters 71 • 1..5 Tensile Stress Growth Coefficient In Fig. considering the large error involved in the measurements at these rates.001 .. However. • .0 S·I).. ..01 Time (5) FigureS-5 Fit ofTensile Stress Growth Coefficient for the LLDPE at 150°C...1 s-1 ...c A rate=1. 5.. ..Chapter 5. . ..r:Ë fi) cncn'E 1.. but the Leonov model follows more closely Laun's empirical relation at high • shear rates.... 5-6 for the Leonov and PIT models...6 Prediction ofNonnal Stress Differences We had no data for the LLDPE to compare with the predicted fust nonnal stress difference... . so we used Laon' s7 empirical relation for 'Pl: Gr Eqn. . as = . ..0E+04 .1 1 10 100 1000 0. Determination ofModel Parameters 72 • 1 CI! o C)Q... polypropylene and polystyrene..---~----~---_.....0E+O& ..01 s-1 o rate=0.5-11 '1'1 ct) =2 (J)2 [ l+l "G' Gu ) 2] 0.oi!---------~ 0.. 1... The agreement is good with both models at low and moderate shear rates....7 This relationship was verified by Laun7 for low and high-density polyethylenes.0 s-1 -Modified Leonov Model • • PTT Model .. The comparison for NI is presented in Fig.0E+OS ! CD ~ CD 0 cU 1..1.. o rate=0.. . the Leonov model predicts a value that varies with shear rate. 5-12...PTT Model 1.0E+03 Z 1.. Eqn. Experimental data for steady simple shear of a number of materials indicate that N 2 is negative and bas a magnitude about lOto 30% that ofNl s .5-12 -2= .0E+04 1... /j.. At low shear rates..0E+06 .l05 accordingto Eqn. Laun's Empirical Relation -Modified Leonov Model 1.1 10 100 1000 Shear Rate (1/5) Figure 5-6 Prediction ofFirst Normal Stress Difference for the LLDPE at IS0 aC....25 and approaches zero at high shear rates (Fig.0E+OS ~ 1.. ForNJNI' PTT gives a constant ratio of-o.-----~----------------------4 1 0.0E+07 .01 0.0E+01 :.-...... For NzlNh the ratio of the second to first normal stress difference....0E+02 1.. 5-7).NI 2 N 2-~ • ...... N/Nl = -0...Chapter 5... Determination ofModel Parameters 73 • êi op 1. ...001 Shear Rate (1/s) Figure 5-7 Ratio of Second to First Normal Stress Difference for the LLDPE at ISOaC......- -0. However.... models agree over the range of strain rates where the respective parameters were fitted. the predictions differ considerably at higher elongational rates..-==============---------=====1 -G.7 Prediction ofElongational Viscosity We present in Fig.. 0..3 .. 5-8 the elongational viscosity predicted by both versions of the Leonov model and by the PTT model.-.1.25 ... ...-...Chapter 5. 5. -. The modified Leonov and PTT models both predict a maximum in the elongational viscosity curve...15 -0.... The inaccurate elongational flow behavior is the main deficiency of the simple model since linear polymers usually exhibit little strain hardening.- -0.05 -Modified Leonov Model .. We observe that both. The simple Leonov model predicts a plateau value of the elongational viscosity of six times the zero-shear viscosity at high strain rates..2 -G.----~---------_---------... ..1 10 1000 100000 1000000 0.- .• ·PTT Model -0. which is more realistic for polymer melts. • .1 .. Determination ofModel Parameters 74 • z è'i Z ~ 0.... 0E+03 . 1.0E+06 -r--------------------------..48 0..2.. As discussed previously in Chapter 4.. 5. we see a disagreement with the cone and plate data..Chapter S.1 1 10 100 1000 10000 Extensional Rate (1/5) Figure 5-8 Prediction of Elongational Viscosity for the LLDPE at 150°C. ·Ci o u en ~ 1.01 0.... . we determined the parameters for the Carreau-Yasuda equation for the HDPE.0E+04 c en S Ji( w • Linear Viscoelasticity -Modified Leonov Model x Simple Leonov Model ._---==-_ ~ X X~ X XX>OOI8K X X _ :> 'ii c o ëij 1..PTT Model .. 0. Determination of Madel Parameters 75 • 1.. :.1.20 0.2 High-Density Polyethylene 5. :JI .. [s] m • s Value 37694 0.----~-------~-----------...1.s] Â.0E+OS 1~~-_-=~~~~-:Sx~X~:::=_. and the corresponding fit ofthe viscosity is shown in Fig.• ~. The zero-shear viscosity is calculated from the discrete spectrum.45 .. 5-9. Table 5-4 Carreau-Yasuda Model Parameters for the HDPE at 200°C Parameter t'lo [pa.1 Viscous Model Using the procedure described in section 5.. based on ail the data given in Table 4-4. The resulting parameters are presented in Table 5-4.. .Chapter 5.=======~ 0..... Table 5-5 Leonov Model Parameters for the HDPE at 200°C Par~...01 0..0E+OS .0E+04 u 0 ëi ~ 1.0E+01 . determined using the IMSL/Fortran™ subroutine....-' 1000 10000 100000 Shear Rate (115) Figure 5-9 Viscosity Fit by Carreau-Yasuela Equation for the HDPE at 200°C.:::==:.3 • ..2..0E+03 :> 1.. are shown in Table 5-5.-~ .. Determination of Model Parameters 76 • :..) given by Eqn...__~_b.... 5-5)...::=::::. le 1.... 5.-.. The parameters m.0E+02 en o Capillary Data (shifted at 200°C) 6..2 Leonov Mode! Parameters The functional form. n and J3 (Eqn.001 1 10 100 0.meter Value n m o o 0. Sliding Plate Data le Cone and Plate Data (shifted al 200°C) -Carreau-Yasuda 1. of b (Il' 1..1 .-==::. f i) 1.. 5-6 was used for the HDPE. Chapter S. 5-10.3 Phan-Thien/Tanner Madel Parameters The parameters of the PTT model (Eqn. we show the fit of the modified Leonov and PIT models for the viscosity. Again.32 5. We see that none of the models reproduce weIl the viscosity at high shear rates. which is still much smaller than the recommended value of "/110=1/8. With the PTT model.4 Viscosity In Fig. at these high rates.2. 5-7) for the HDPE are presented in Table 5-6. we had to use a small viscous contribution to avoid a maximum in the flow curve (Fig. ç was fitted ta the viscosity curve and s ta the transient elongational data. sa the reliability of these data points can be questioned. Determination ofModel Parameters 77 • 5.6xl0"4. 5-12.=3. • . 5-8) along with. 6 (Eqn.82 0. we show what the viscosity curve looks like if 0. a spectrum of relaxation times. flow instabilities can become important.2. However. In Fig. 5-11).=1/8 is used. Table 5-6 PTT Model Parameters for the HDPE at 200°C Parameter Value 1. We used a. This emphasizes that one should be careful in using the rule suggested by Crochet et al. ..01 0.0E+00 Sliding Plate Data o Capillary Data (shifted at 200°C) ~ Cane and Plate Data (shifted at 200°C) -PTT Model (alpha=O)  1.0E+04 ëi 0 b 1..0E+01 ca ~ 1.. 1. 1..1 1 0. 1000 10000 100000 100 10 0...0E+02 Q.. Determination ofModel Parameters 78 • Q....0E+03 :> 1.::::======~----~~ 1000 10000 100000 10 100 1 0....6e-4) o 1....001 • Shear Rate (1/s) Figure 5-11 Flow Curve for the HOPE at 200°C Predicted by the PIT Model without the Viscous Contribution..0E+01 ......========::...Chapter 5.......0E+02 en Co) Capillary Data (shifted at 200 G C)  Sliding Plate Data G ~ Cone and Plate Data (shifted at 200 C) -Modified Leonov Madel ..01 0....PTT Model (alpha=3.. 1.0E+03 .. ~ ca en 1....1 0.001 Shear Rate (1/s) Figure 5-10 Fit ofViscosity by Leonov and PTT Models for the HDPE at 200°C.0E-G1 . ~ êG û5 en CP oC ! en en 1.-~--~--------------------. .0E+OS ~------------------------... Bath models give equally good fits...0E+01 .:::====--------.0E+OS ... 0.::=====...L:l... coefficient.0E+02 o Capillary Data (shifted at 200°C) 6 Sliding Plate Data X en o o o Cone and Plate Cata (shifted at 200°C) 1... except for data at rate 1.. .. Determination of Model Parameters 79 • ëii 1. X Q.. 5. ..01 0.5 Tensile Stress Growth Coefficient In Fig...J 0.. we show the fit of the tensile stress growtb. ..... 1...0E+04 en ai ~ 1.2.. • ..1 -PTT Model (alpha =1/8) 1 o o 100 Shear Rate (1/5) 10 1000 10000 100000 Figure 5-12 Viscosity Curve ofPTI Madel with a=1/8 for the HDPE at 200°C.... which shows a higher level than the predicted steady state. .0E+03 o o u :> 1.. 5-13..001 =====::.Chapter 5.-. Cbapter 5. Detennination ofModel Parameters 80 • s:- 1.0E+0& ~-----------------------... I.. fi! ca C)~ 1.0E+OS U) .~ ! i .... ~ enen'" := CD en 0 cO 1.0E+04 CD= <> rate=O.S s-1 (1) o rate=O.S 5-1 (2) ~ rate=O.S 5-1 (3) -Modified Leonov Madel PTT Model 1.0E+03 ....-------..01!"---~=========~ 10 0.01 0.1 1 Time (s) Figure 5-13 Fit of Tensile Stress Grovvth Coefficient for the HDPE at 200°C. 5.2.6 Prediction ofNormal Stress Differences We had only a few cone and plate data for the :first normal stress difference, used Laun's empirical relation ta verify the predictions. 50 we again The comparison for NI is presented in Fig. 5-14. The agreement is good with both models at low and moderate shear rates, and again for this material, the Leonov model is closer to Laun's empirical relation at high shear rates. The cane and plate data are in fair agreement with the predictions, except at the lowest shear rate. This is similar to what is seen in the viscosity corvee For the ratio of the second to tirst normal stress difference, the Leonov model predicts N 2/N1 = -0.25 at low values of the shear rate. This ratio decreases toward zero with • increasing shear rate (Fig. 5-15). The PTT model gives a constant ratio of-0.09. Chapter 5. Determination ofModel Parameters 81 • Q. 1 . 0 E + 0 7 . . , - - - - - - - - - - - - - - - - - - - - - - - - -... 1.0E+OS 1.0E+05 ca Z ~ 1.0E+04 D.. Prediction trom Laun -Modified Leonov Model 1.0E+03 X PTT Model Cone and Plate Data (shifted at 200°C) 1.0E+02 .w:....---~---~---!===========!J 10 0.1 1 100 1000 0.01 Shear Rate (1/s) Figure 5..14 Prediction ofFirst Normal Stress Difference for the HDPE at 200°C. 0-.----------------------------. -G.05 -0.1 . - . - . - - . - - . - - - - - - - _. - - . - - - - - . - - z -0.15 N Z ... -0.2 -0.25 .....- - - - - Modified Leonov Model PTT Model • -0.3 ~.-----,....----~----._01!"---------100000 10000000 0.1 0.001 Shear Rate (1/5) 10 1000 .. Figure 5-15 Ratio of Second ta Fbst Normal Stress Difference for the HDPE at 200°C. Chapter 5. Detennination ofModel Parameters 82 • 5.2.7 Prediction ofElongational Viscosity We present in Fig. 5-16 the elongational viscosity predicted by bath versions of the Leonov model and by the PIT modeL Agaïn, the modified Leonov and PIT models predict a maximum in the elongational viscosity curve. We see that both models agree over a range around the strain rate at which the parameters were fitted (0.5 S-I) and at higher rates. The simple Leonov model always predicts a plateau value of the elongational viscosity that is six times the zero-shear viscosity at high strain rates. 1.0E+OS ..--------------------------~ Q. ~ en tG xx~ x x~ XX~ xx~ xx 1.0E+OS u ~ :> 'ii c fi) o in c 1.0E+04 Linear Viscoelasticity -Modified Leonov Madel x Simple Leonov Madel PTT Madel 1 S W >< 1.0E+03 ....--------~---~----------~ 0.01 0.1 1 10 100 1000 Extensional Rate (5-1) .... 10000 Figure 5-16 Prediction of Elongational Viscosity for the HDPE at 200°C. In conclusion, we can say that both viscoelastic models give similar fits and predictions of the rheological properties. Also, our results showed that for the PTT model the addition of a viscous contribution to avoid the maximum in the flow curve should be • done with caution. "On the Rheological Modeling of Viscoelastic Polymer Liquids with Stable Constitutive Equations".F. Rhea/. p. p. Am. Tanner R. Rheo/. Acta. ~ p. Non-Newt.M. 353-365 (1977) 4 Phan Thien N. Ph. 459 (1986) 8 Dealy J.0. Version 2. Walters K. France (1995) 6 Crochet M. ~ p. 2. Elsevier.J.L. 259-283 (1978) 5 Beraudo C. Sophia Antipolis. thesis. Determination ofModel Parameters 83 • REFERENCES Simbambhatla M... 2b p.D... Ed. "A Nonlinear Network Viscoelastic Model"~ J. Rhea!. "Méthodes éléments finis pour la simulation numérique d'écoulements de fluides viscoelastiques de type différentiel dans des géométries convergentes".~ Leonov A.~ "A New Constitutive Equation Derived from Network Theory"..~ Davies A... École Nationale Supérieure des Mines de Paris.M. J.. Wissbrun K. 259-273 (1995) 2 IMSL MathlLibrary.sterdam (1984) 7 Laun H. ~~umerical Simulation of Non-Newtonian Flows". Mech. "Melt Rheology and its RaIe in Plastics Processing"..I.R. Van Nostrand Reinhold.. 972 (1991) 3 Phan Thien N. Acta. New York (1990) 1 • . FI.Cbapter 5. can be written as follows: Eqn.1. For a Newtonian fluid and neglecting inertia forces. the conservation equations for mass and momentum.6-4 where t'lz-Â.• 6. We recall here that for isothermal flows. and for a multimode representation. we presented the system of equations that must be solved to simulate the flow of viscoelastic materials. neglecting compressibility and body forces. and ~ is the extra stress tensor.6-1 Eqn. 1 where N represents the number of modes.6-2 p-=+Vp-V·t"=O Du Dt = where p denotes a constant density. In the absence of a viscous contribution. the extra stress tensor is expressed as follows: Eqn.=1 1:.-Gi• • .p is the pressure. Numerical Method 6. which in our work is the Leonov or the Phan-ThienITanner model: Eqn.6-3 1:= = L N . this set of equations constitutes the "Stokes problem".1 Description of the Problem 6. The last equation required to close the above system is the rheological model.1 Four-Field Mixed Formulation In Chapter 2. Numerical Method 85 • For the Leonov model: Eqn.6-8 A modified elastic-viscous stress splitting (EVSS) method i is introduced to facilitate the choice of the discretization spaces. 6-2) becomes: • The parameter a.5 =~ A(".. ~ which is the gradient ofthe velocity M: d=Vu Thus.6-10 The conservation equation for momentum (Eqn.Chapter 6. 1 1 1 Eqn.. This requires the explicit introduction and discretization of a tensor variable. but a. can take any value.'.6-6 For the PIT model: Eqn. the rate ofdeformation tensor"DM can he expressed as: Eqn.6.6-7 Eqn.=t'lo has been reported to be an optimal choice 1• ...)=. a pseudo-concentration method is used2. as in extrudate swell.1. and Eqn. Ào=O. Numerical Method 86 • 6. requiring the solution ofthe additional equation: Eqn. using the same value of ex. (Eqn.Cbapter 6.6-14 Since air is not viscoelastic. the relaxation time concentration is defined as follows: Â. the viscosity of this fluid was set equal to zero. • .2 Free Surface In the presence of a free surface.tF AlI the material properties are evaluated using equations similar to Eqn. without changing the results of the simulations. 6-11) in the entire flow domain. Later.6-12 -+u·VF=O ôt - aF where the function F represents the pseudo-concentration that describes the interface between the polymer and another fluid.6-15 Â(F) =Â. 6-14. as a function of the pseudo- Eqn. based on melt properties.6-13 F= { inair 1 in the polymermelt o (fluidO) (fluid 1) This function specifies the material properties appropriate for each region of the computational domaine For example. 6-14 simplifies to: Eqn. in this case air: Eqn. Preliminary calculations were done using a fictitious tluid instead of air with a viscosity equal to 10-3 times the polymer viscosity. After applying the divergence theorem to the Stokes problem.'t.Chapter6.=-v{~~+24TJJ d=V!f Eqn.1.V o.6-16 -+u·VF=O aF ar - 6. the global system of equations to solve is shown below. we have the following weighted residual problem: Find fup.Numerical~ethod 87 • In snmmary.over a domain Q.3 WeakFormulation In order ta obtain the weak fonnulation corresponding to this system. each equation is multiplied by an associated weighting function and integrated..6-17 "iIye V • . v·~=o -vo(2aQ)+Vp. dl e Vx 0 x k x L such that: "iIqeO Eqn. VF~dx=O j- 'tIGeç 6. This is a difficult task. Q and by their respective subspaces Qh and ~ in the continuous weighted residual problem (Eqn.6-18 -bat r[aF +u. Qh' ~ and ~ and the chosen discretization is illustrated in Fig. 6-1. 6-17). we need to discretize the variables. !. Numerical Method 88 • where q. we obtain the following discrete weak formulation: • . presented in Chapter 2. as for ~ and ~ ~. By replacing the spaces V. since compatibility conditions exist in the four-field Stokes problem for the discretized variables !!h. The ~. Ph' ~ and gp. Their respective discretization must therefore satisfy a generalization of Brezzi's condition3 (the inf-sup condition) valid for the usual velocity-pressure formulation of the Stokes problem. but it is possible to select another discretization space for g. to which a cubic "bubble" function associated with the centroid node is added. discretized variable F h is approximated by piecewise linear polynomials. The ch<>sen discrete subspaces are Vh. We chose identical discretization spaces for the variables ~ and g.Chapter 6.2 Discretization ûnly triangular elements are considered in this work. In arder ta solve the weighted residual problem by the finite element method. The additional equation associated with the pseudo-concentration bas the following weak formulation: Eqn. Il and Ji{ are suitable weighting functions. Pt is the quadratic polynomial set deflned on element K. and Plis the discontinuous linear polynomial set defined on element K. Pl . For the pseudo- concentratio~we have the folIowing additional equation: Eqn. tA and J/4 are suitable discretized weighting functions.6-20 We can see that the variable ~ is a projection of DŒJ in the discrete subspace was mentioned in Chapter 2. 6-12). Numerical Method 89 such that: • Eqn. there is no equality between problem.Chapter 6. the upper and lower convected derivatives in the constitutive equation are defined as follows: . We aIso note the similarity between the constitutive equation (Eqn. ~ ~~ and as and Df16t) in the discrete Figure 6-1 The P 2+ .Pl . Ph' ~ and ~ (and F. Indeed.J 4. 6-4) and the pseudo- • concentration equation (Eqn.6-19 where qh~ ~.Pl elements for variables llh. approach to solve this problem would require too much memory space.2. is done through GNmES (for .-Vu "-"(V0 • Eqn. which is described in section 6..J for fixed lib. by a fixed-point algorithme In the following sections.6-21 = at - = -== Eqn. When a free surface is present. 6-20 can be quite large.6-22 a = a'Z" 'Z"=--=-+U·\7T+(V~T ·.+'Z"·\7u at - = = = - Both the constitutive equation and the pseudo-concentration equation are suitable for the discontinuous Galerkin method.3 Numerical Solution A coupled The global system presented in Eqn. Generalized Minimal Residual "l a Newton-Krylov iterative solver. 6.. 50 we use instead The coupling of these sub-systems. along with the treatment of nonlinearity. Numerical Method 90 v a! T 'Z"=-=+u· V. we describe the solution of each decoupled sub-system and the solution of the global fourfield system by GMRES.3.1 Stokes Problem • For fixed ~ and~. 6-19 and Eqn.. a decoupled approach. We solve separately the following sub-systems: the Stokes problem for fixed ~ and ~ the constitutive equation for fixed lib the projection ~ = D(y. the discretized pseudo-concentration F b is updated for fixed M& outside GMRES. especially when a multimode representation of the extra stress tensor is employed. the discretized Stokes problem is as follows: .3. 6.Chapter 6. The discrete weighted residual formulation of the constitutive equation is defined as follows in the traditional Galerkin method: • Eqn. the time derivative is simply dropped. but a similar treatment is applied to the pseudo-concentration (Eqn. two in velocity and two in pressure. Furthermore.3. a fully implicit second order Gear scheme is used for the rime derivative. Numerical Method 91 • Find (UtJh) E V~ such that: Eqn. 6-12). allowing a reduction in the size of the global matrix.6-24 We use Uzawa's algorithm6 to solve this system. In the more general timedependent case. For stationary problems. a direct method by aLU factorization. like those studied in this work.Chapter 6. 6. 6-4).2 Discontinuous Galerkin Method We will only describe the method for the constitutive equation (Eqn.6-23 1bis discrete problem can be represented by the following matrix problem: Eqn.6-25 . four degrees of freedom per element. are removed according to the condensation technique of Fortin and Fortin7 • The system is finally solved througb. as illustrated in Fig. A discontinuous polynomial approximation each element K.J is the jump in the variable !i. Numerical Method 92 • If we decompose this formulation inta elementary sub-systems. but a similar treatment can be applied to the Gordon-Schowalter derivative of Eqn.6-29 • . 6-2).6-28 1Jr. 6-26 where l1K is the unit normal vector to the boundary 8K pointing outside K (Fig.Cbapter 6.6-26 We use the upper-convected derivative for A~. 6-7. The inflow boundary 8K: with respect to the velocity field is defined as: Eqo.h and is expressed as: Eqo.J to simplify the last formulation. we obtain the following weak formulation on each element K: Eqo. The discontinuous approximation gives rise to a discontinuity jump at the element interface in the derivative becomes: of~. The convective terro. cP 1) for each variable !ï of mode i is used on The velocity field Hh is known from the previous iteration. in Eqn. 6-2. In the presence of recirculation zones. • . 6-26. the best possible numbering is provided and the elements are swept many times so that the resolution can be seen as a block relaxation (Gauss-Seidel type) method. and assume that ~. Numerical Method 93 • u Figure 6-2 Discontinuity lump at the Interface of Two Elements8 • When we replace Eqn. A perfect numbering is not always possible. 6-27 in the weighted residual formulation of Eqn. the resolution of this system requires knowledge of the quantity ~- on elements adjacent ta BK: so that a particular numbering of the elements is necessary. for example. which is linearized by a Newton method: Eqn.6-31 However.h+ = ~.h' we obtain a new formulation: Eqn.Chapter 6.6-30 This is the discrete fonnulation proposed by Lesaint and Raviart9 • It results in a small nonlinear system on each element K. 6-35 The last equation can be expressed in matrix form: Eqn.6-32 or in matrix forro: Eqn.6-34 The projection is achieved on an elementary basis since no continuity is required at the elements interface for variables elementary systems: ~ and !I!b.6-33 2. we have to solve for each i mode the nonlinear system given by Eqn. we have to solve a 3 x 3 linear system given by the following equation: Eqn. Therefore. 6-30 of size 9 x 9 for 2D problems and 12 x 12 for axisymmetric problems.3 Gradient Tensor ofVelocity The projection of D(yJ in the discrete subspace following problem for a fixed !lb: ~ can be calculated by considering the Eqn.6-36 The small elementary systems are solved by a direct method with aL U factorization.3.Chapter 6. on each element. we have to consider only these Eqn. Numerical Method 94 • ln conclusion. For the pseudo-eoncentration. • . B~h =0 Eqn.Ci = 1. stop. Forn~O: Eqn. The algorithm. 6-19 and Eqn. the global system. &+l=Xn+BX. It generally applies to problems ofthe form: Find x such that RCx)=O. Numerica1 Method 95 • 6. • Our specifie problem can he expressed as follows: . A fixed-point method is not applicable. an initial guess.Chapter 6. since it results in a 10ss of numerical convergence as viscoelasticity becomes more important. ofthe Newton method is as follows: 1. 6-37 is a nonlinear system.6-37 A~h +B T Ph =C~h'~h) M K('!:!. 6-20 if a free surface is presen~ can now be expressed in the following way. Otherwise. composed of Eqn.K' 'ri J'ri = EK~K) + H K Ti . 2.6-38 Eqn. go back to step 2.z) evaluated in &. N equations) =K =K =K OKdK =W~K) Eqn.6-39 where J n is the Jacobian matrix of R(K. so we need ta use an iterative method to solve it. The Newton method is more suitable for this problem. 4. If IlôXll < & and IIR(Kn )11 < &. 3.3.4 Resolution orthe Global System Following the presentation of the various sub-systems. Given &. To avoid building this large global matrix at every iteratio~ we use an approximation based on a finite central difference: Eqn.Ph'~h'~h) 1.dh)= ~MK( M. 2. N equations) =K =K =K LOKdK-W~K) The difficuity for this large system is to solve the problem without calculating explicitly the Jacobian matrix of Eqn. Solve the constitutive equation by a Newton method for fixed MK. d0=O where R(utvptv 't bJ dJ is given by: B~h R~h. y T = 1k + ôy. The system to solve is highly nonlinear.E K (~K ) .5dr.dh) ) "i . Solve the following Stokes problem: Bt5~ = -B1l. which would require too much memory space. and this causes convergence problems. N equations) =K . where Ôlb ôp.6-40 Jd ~ Rex.(i = 1.K' t"i ~ K K A~h +B T Ph -C~h.+HK5"i=K =K =K =K (i =1. we use instead a preconditioned residual R&.t5r. +h4)-R(K -h4J 2h whereX=(utvptv 't hJ dJT and h is small (usuaIly of the order of 10-6).. Numericai Method 96 • ~~) snch !hat R(u~..10.h. for all K elements: • J K (!!K' "i =K )5.!:.h At5~+BT 8p =-A1l.h _B Ph +CC!:h'~h) and obtain a solution corrected in velocity..Chapter 6. =-MK(!!K't"i )"i +EK&K)+HK"i . In order to obtain a better conditioning of the system. defined as îi&.H K"i .Ph'!:h'~h)=(8!bôp.h.Ph''f. 6-39. ô~ and ôg are determined as follows: 1.._h. Instead we use the approximation given by Eqn. this caused convergence problems at high levels of elasticity and had to be avoided for these flow • conditions.. for sorne simulations at low Deborah number it was found useful ta do one to tbree fixed point iterations as a preconditioner to GMRES. which is a block-diagonal matrix bullt with the respective Jacobian matrix of the decoupled sub-systems. 3. which accelerates the calculations. The other variables. The residual ô~ is made up of all the elementary residuals ô!K. More details on the discontinuous Galerkin method applied to the pseudo-concentration can be found in reference 2. were calculated for a fixed position of the free surface. ~ being fixed. 6-39. solved in G~S. using a fixed-point algorithm. Instead. Finally. Numerical Method 97 • where JI<. .ES iterative method. For the resolution of the coupled system ofEqn. we used the G:MR. the Jacobian matrix is only present in matrix-vector products. for fixed 1J. the free surface was updated outside GIvffiES. The definition of this new residual R is simply the preconditioning of the residual R with the inverse of a matrix J. We tried ta incorporate the calculation of the pseudo-concentration (Eqn. Note that it is possible ta ignore the last term of the previous equation and to perform.Chapter 6. Compute the local projections oftensor DC1lJ on all K elements: 0K t5d K =-OK d K +WU!K) The residual ô~ is aIso made of all the elementary residuals ôgI<. 6-38) in the GMRES method for the calculation of the free surface. This method has been designed to solve non-symmetricallinear systems. 6-40. but it caused convergence problems. which reduces considerably the memory space needed. is the Jacobian matrix obtained by the linearization due to the Newton method.K. only one sweep. It is then not necessary to explicitly build the large global Jacobian matrix. In the GMRES algorithm. However. ). "Computing Viscoelastic Fluid Flowat Low Cost". Fortin M. Canada (1998) 9 Lesaint P. J... "Méthodes d'Éléments Finis Adaptative pour les Écoulements de Fluides Viscoélastiques ». FI. Agassant J.Chapter 6.Academic Press.. C.. RI. 205-208 (1985) 7 Fortin M.. FI. Béliveau A. NonNew!. Thesis.. F/. École Polytechnique de Montréal. Fortin A. Fortin M..D. Fortin M. Num. Non-Newt.. de Boer (ed. USA (1999) 5 Fortin A. 277-288 (1990) 6 Fortin M.. New York (1991) _ 4 Fortin A. Newport. «Ten Years Using Discontinuous Galerkin Methods for Polymer Processing ». Canada (1997) 3 Brezzi F. Meck. FI. Springer. p. ~ p. ''Experiments with Several Elements for Viscous Incompressible Flows". J. in "Mathematical Aspects of Finite Elements in Partial Differentiai Equations". Fortin A. Comm.. Computations and Applications.F.. p. to he published in the Proceedings of Int. "On a Finite Element Method for Solving the Neutron Transport Equations".. Int. Heuzey M. 89-123 (1974) 10 Fortin A.. nA New Mixed Finite Element Method for Computing Viscoelastic Flows"... J. Saive Regina College. « Méthodes de Lesaint-Raviart Modifiées : Applications au Calcul des Surfaces Libres ».. Numerical Method 98 • REFERENCES 1 Guénette R. Non-Newt. ~ p..C. Montréal.. Lioret A. Mech. Raviart P. App/.... Symp.•<\. Mech. New York. J. École Polytechnique de Montréal. 911-928 (1985) 8 Lioret A. Ph. ~ p. 1. Zine A. "A Generalization of Uzawa's Algorithm for the Solution of NavierStokes Equations". 209-229 (1992) • . "Mixed and Hybrid Finite Element Methods: Series in Computational Mechanics".. "A Preconditioned GeneraIized Minimal ResiduaI Aigorithm for the NumericaI Solution ofViscoelastic Fluid Flows". Montréal.. on Discontinuous Galerkin Methods: Theory. ~ p. Master's Thesis.. Metk. Num.27-52 (1995) 2 Béliveau A. Meth. Flow Direction S-I. and the dimensions are given in Table 7-1. 7-1. in which the pressure and the temperature are given in reference 1. The geometry of the slit die is shown in Fig.21 and 2. the results of numerical simulations. measure~ and a die land. especially in the case ofthe reservoir. Details of the die design are We used only one configuration consisting of an abrupt 8: 1 contraction. With regard to the aspect ratios W/H of the reservoir and channel we note that these are smaller than the mjnimum value2 of 10 recommended to ensure 2D flow. The transparent glass walls permit the use of laser-Doppler velocimetry and the measurement of flow-induced w H " Channel Reservoir • Figure 7-1 Geometry of the Planar 8:1 Abrupt Contraction. The die is continuously fed by a Kaufinann PK 25 single screw extruder. . which can be adjusted to vary the local geometry.1 Experimental Methods 7. AIl the experiments were camed out at 200 ± 2°C and at mass flow rates varying between 0.6 to 56 birefringence.1. This issue will be discussed when comparing data witb. which correspond to apparent shear rates in the channel in the range of 4.• 7. Planar Abrupt Contraction Flow offfigh-Density Polyethylene 7.1 Die Geometry Extrusion experiments were carried out using a slit die with transparent walls made of PyrexTM. It consists of a large reservoir.6 kg/h. The probe is placed on a XY translation table (Dantec Lightweight Traverse) that is controlled by Burstware™.1.8 WIH (channel) 6. laser-Doppler velocimetry (LDV) is a useful technique for local velocity measurement in translucent fluids.ly with a micrometric positioning screw. In Table 7-2 and Fig. The dimensions of this ellipsoidal measuring volume are ôx = ôy = 108 J. A 10 mW He-Ne laser generates the incident light beam (red light: Â=632. The two beams are transrnitted by optical fibers to a probe or optical head. The frequency shift (Doppler shift) of the scattered light with respect to the reference dual beams is directIy proportional to the velocity of a moving particle in the fluid passing through the measuring volume. Displacement along the z-axis is done manual. and one beam has its frequency shifted (40 MHz) with respect to the other by a Bragg cell. where a lens if = 160 mm) focuses the beams into the measuring volume. Planar Abrupt Contraction Flow ofHDPE 100 • Table 7-1 Geometrical Characteristics of the Die H (reservoir) 20 mm H (channel) L (channel) 20 mm W 2.5 mm. which can perform fast-Fourier transform (FFT) analysis. The x component of the velocity eux) was measured at six • fixed x positions (three in the reservoir and three in the channel) as a function of y.8 nm).4 • The scattered light is received by the same probe and transmitted to the Doppler analyzer.Chapter 7. 7-3 we present the fixed x positions where the velocity profiles were . The technique is as follows: a measuring volume is created by the intersection of two light beams of equal intensity. 7-2). The beam is divided in two by a prism. and the frequency of one beam is shifted with respect to the other. LDV is based on the frequency shift of Iight that is scattered by small moving particles.2 Laser-Doppler Velocimetry Due to its high spatial and temporal resolution. 16mm WIH (reservoir) 0. We used the Flowlite™ dual-beam system of Dantec. controlled by the Flow Velocity Analyzer™ and operated from a persona! computer with the Burstware™ software (Fig.1m and ôz = 913 JllD. Details of the LDV method can be found in reference 3. as it does not disturb the flow.4 7. . Doppler signal analyzer Optical Head Figure 7-2 Experimental Set-Up ofLOV Measurements..:~ ~ =.0 mm +10mm +17mm • .Chapter 7... .. ... Table 7-2 Position 1 Velocimetry Measurements Locations Reservoir -14mm 2 3 ... . ......... .......... Planar Abrupt ConttaetioD Flow ofHDPE 101 • measured.0 mm Channel +2.. .5 mm . . ...-~ .. .3........ ...... ~..6. .... ::::: ..:: :: .... He-Ne laser beam splitter -. :....... ... ... ... since it is symmetric with respect to the y axis.... ..- ~~~~~.." '. Only one half of the geometry is shown..4 photomultiplier Flow ::::: .... . ·_.. : Flow Direction ... . ·. · - · 1- : Figure 7-3 : : (0.·· . It can aIso be induced in materials that are isotropic at rest.. such as calcite crystals..·..1 1 o 00 .. Anisotropy is a result of the material structure and can appear in sorne materials at rest.0) : : : : : : .. This relation is given by the semi-empirical stress optical law as a proportionality between the components ofthe refractive index and stress tensors: • Eqn....3 Flow-Induced Birefringence Birefringence occurs in an optically anisotropic material in which there are different velocities of light for different directions of propagation. -.. . : : Positions ofthe Velocity Profile Measurements. .. ~ : : t--.1. birefringence is a result of the difference in polarizability of a polymer chain along its backbone and perpendicular to it. er-=-.. 7. for example by flow in polymer melts or elastic deformation of solid polymers.··. The use of birefringence to infer the components of the stress tensor in a flowing or defonned polymer depends upon the existence and validity of a "stress-optical relatïon"s. ·. ...Chapter 7. In the last two examples.-.7-1 . Polarizability is a measure of the strength of the response of the electrons in a bond between the atoms of the polymer molecule.. Planar Abrupt Contraction Flow ofHDPE 102 Channel • 1 Reservoir "'"--'---.. . . For a given polymer. C is essentially independent of the molecular weight and its distribution and is relatively insensitive to temperatures. analyzer transparent slit die polarizer .. A beam of polarized monochromatic light is directed at the melt flowing through the transparent slit die and then passes through an analyzer.Chapter 7. We used a diffuse monochromatic light source of sodium (yellow light: Â. Planar Abrupt Contraction Flow ofHDPE 103 For a flexible polymer. 50 we added two quarter-waves plates on each side of the die to obtain only the isochromatic fringe pattern.8 nm).. The optical anisotropy of the flowing polymer produces an interference pattern consisting of isochromatic and isoclinic fringes. this coefficient is • where C is the stress optical coefficient. proportional to the polarizabilities parallel and transverse to the polymer backbone.. camera ~ . 7-4. The measurement of the isoclinic fringes is generally inaccurate and tedious 6. The experimental set-up to measure flow-induced birefringence in polymer melts is shown in Fig. Figure 7-4 Experimental Set-Up for the Measurement of Flow-Induced Birefringence.=589. t quarter waves plates 1 diffuse monochromatic light source (sodium) • . • . we can convert these data to shear rate vs.5 o Mass Flow Rate {kg/hl Figure 7-5 Characteristic Curve ofDie-Extruder at 200°C. viscosity by applying the Rabinowitch correction. along with rheological data. In Fig. and they are in agreement with. 7-6. 25 20 J:a ns 15 ~ ! en en • 10 • • • • • • • • • Q.. 2.Chapter 7. using pressure transducer data. Planar Abrupt Contraction Flow ofHDPE 104 • 7.----~----~-----------------..1..4 Flow Rate-Pressure Curve Extrusion experiments were carried out at twelve flow rates. These results are presented in Fig. pressure for the extruder and die used.5 1 1. we present the characteristic curve of flow rate vs. If we neglect the correction for the entrance pressure drop. 7-5. the viscosity curve measured in the laboratory.5 2 3 0.. e • 5 • o . Chapter 7.0E+03 . 7. -Ci u 0 fi) 1.01 0. Planar Abrupt Contraction Flow ofHDPE 105 • a. 7-7 and 7-8 in the region near the contraction. 0 0 :> 1. The numerical procedure used was described in Chapter 6.001 0. Table 7-3 Mesh Mesh Characteristics for 2D Flow Simulations Planar 1 Unstructured Planar 2 Structured 2976 • Nomber of Elements 1277 .0E+02 o Capillary Data (shifted at 200°C) 6. One is an unstructured mesh.2 Numerical Procedure .0E+OS x X X â1 âX âl1 m e t) 1.0E+01 0. Both meshes are composed of triangular elements.1 1 10 100 1000 10000 100000 Shear Rate (1/5) Figure 7-6 Comparison ofExtruder Data with Viscosity Curve ofHDPE at 200°C. Sliding Plate Data 0 0 0 x Cone and Plate Data (shifted at 200°C) • Extruder 0 n 1.2.1 Meshes Two meshes were used for the 2D simulations in order to verify the mesh independence of the solutions.0E+04 ~6 il~ ~ 1. The number of elements for each mesh is presented in Table 7-3. and the meshes are shawn in Figs. while the other is structured. 7. 2 Viscoelastic Converged Solutions Numerical convergence is always limited in viscoelastic simulations of fields that contain flow singularities. and the constitutive equation.2. The maximum tlow rate for convergence depends on the mesh. It was . We aIso found that the simple Leonov model gave more convergence problems than the PIT model. 6. the longest relaxation time of the discrete spectrum.0 sec.25 sec. Planar Abrupt Contraction Flow ofHDPE 106 • Figure 7-7 Mesh Planar_l. convergence becomes more difficult as in the case of mesh Planar_2. As the mesh is more refined near the flow singularity or reentrant corner. possibly because of stranger nonlinearity in the extra stress. Figure 7-8 Mesh Planar_2.). Convergence was aIso more difficult when using the discrete spectrum based on all data as compared ta the spectrum based on truncated data (Table 4-4). 7. vs. We will refer to these spectra as the "full" and "truncated" • spectra. since the longest relaxation time is longer in the former case (24.Chapter 7. Chapter 7. Planar Abrupt Contraction Flow ofHDPE 107 • not possible to determine exactly why this occurred. We did not try simulations with the modified Leonov model, considering the difficulties encountered using the simple mode!. We present a map of the converged solutions for the various cases (Tables 7-4 and 7-5). Convergence is expressed in terms of the number of flow rates converged, where the maximum is twelve (12). Note that for the PIT model, the same nonlinear parameters (ç, E) were used with the full and the truncated spectra, these parameters giving equal1y good fits of the rheological data. Table 7-4 Converged Solutions with the PTT Model (max=12) Spectrum Full Truncated Mesh Planar 1 6 12 Planar 2 0 12 Table 7-5 Converged Solutions with the Simple Leonov Madel (max=12) Spectrum Planar 1 . * third solution not fuIly converged Mesh 1 3* Full Truncated . Planar 2 0 1 7.3 Total Pressure Drop For each flow rate we compared the predicted total pressure drop with the one indicated by the pressure transducer in the reservoir. The predicted pressure drop was evaluated as follows: Eqn.7-2 t:J'ca!c = Preservoir - PL=20mm where P reservoir is the pressure in the reservoir at the position of the pressure transducer, and PL=20mm is the pressure in the slit die at L=20mm which corresponds ta the die exit In Fig. 7-9 the results of both viscoelastic models and the viscous model are compared ta • the measured values. Note that only tbree points were obtained with the simple Leonov mode!. The calculated pressures are slightly overestimated by the viscoelastic models, Chapter 7. Planar Abrupt Contraction Flow ofHDPE 108 • while they are underestimated at high shear rates by the viscous model, but in general the agreement is good. Since the melt is predominantly in shear flow, even the viscous model can give a good approximation of the total pressure drop. that they agree very weIl. is demonstrated in Fig. 7-1 L In Fig. 7-10, we compare the predictions of the PIT model with the full and truncated spectra and we see This shows that there is no advantage in using the full spectrum, which usually gives more convergence problems. Finally, mesh independence 25 ....- - - - - - - - - - - - - - - - - - - - - - - - - - - - . 20 ~ ..Q ca 15 l! CI) CI) = 5 ~ l! 10 • • Measured o PTT, truncated spectrum, Planar_1 /j" Leonov, truncated spectrum, Planar_1 x Carreau-Yasuda, Planar_1 O~----...oi!~---~-----------------0.5 1 1.5 2 2.5 3 o Mass Flow Rate (kglh) .. Figure 7-9 Comparison ofPredicted Total Pressure Drop with Experimental Values. • Chapter 7. Planar Abrupt Contraction Flow ofHDPE 109 • ! :::J 25 20 . -. .a CD CD • as 15 ! a. 10 , • • • Measured o PTT, truncated spectrum, Planar_1 + PTT, full spectrum, Planar_1 5 O ...---~----~--------.....,.--------.....t! 3 o 2.5 1 1.5 2 0.5 Mass Flow Rate (kglh) Figure 7-10 Influence ofDiscrete Spectrum on Prediction of Total Pressure Drop. 25 .....- - - - - - - - - - - - - - - - - - - - - - - - - - -.. il 20 . -. .. CI» il • as .a 15 :::J CD CD il D- ! 10 i 5 • Measured o PTT, truncated spectrum, Planar_1 o PTT. truncated spectrum, Planar_2 o ....----..oIi!"----~---~------------_... o 0.5 1 1.5 2 2.5 3 • Mass Flow Rate (kglh) Figure 7-11 Mesh Independence for the Prediction of Total Pressure Drop. viscoelastic models predict the same velocity profiles. Planar Abrupt Contraction Flow ofHDPE 110 • 7. 7-19). 7-12 to 7-17 for the six measurement positions shown in Fig.4 Velocity The velocity profiles were measured for the three lowest flow rates. 7-3 and Table 7-2. • . Again mesh independence was verified by comparing the results with mesh Planar_2. We aIso verified for the same flow conditions that the use of the full spectrum did not improve the velocity profile predictions (Fig. where the extensional component of the flow becomes important. 7-18 using the truncated spectrum for the third tlow rate of position 3 in the reservoir. Note that bath Where the flow is dominated by shear (position 1 in the reservoir and all three positions in the channel) all models including the viscous model give good predictions of the velocity profiles. In Fig. we start seeing disagreement between the predictions and the measurements. The viscous model mas! underestimates the velocity as the extension becomes more important. 7-15 the viscous model is seen to be in better agreement with experimental data where the flow is dominated by shear. It appears that the flow becomes fully developed faster than predicted by the viscoelastic models. but the viscoelastic models are aIso unable to capture weil the behavior of the flow. as we get closer to the contraction (positions 2 and 3 in the reservoir). This is shown in Fig. since a small difference in tJx or 8y results in a large difference in velocity. However. There is aIso more dispersion in the experimental results at these positions. and the results are shown in Figs.Chapter 7. The predicted velocity profiles were computed on mesh Planar_l for all models using the truncated spectrum for the viscoelastic models. .:..g:::::::::!. • .' a 'A .A...====:::..75 Position Y (lIUII) 1."!=:!'=~~.'=".====:::..==. Planar Abrupt Contraction Flow ofHDPE 111 • 1... -1Aonov and PTT liIocIeJa a o a ~~and PTTModils ...5 ~ 14.Camau Modal % 2 OF..1 .t .- ------_--""" 0..------------..:.&~ . .can.. . ...2. ModIr ---' % o...5 0. Figure 7-17 Velocity Profile (Channel...5 10 Figure 7-13 Velocity Profile (Reservoir. Position 3)......1.A...II~ ur...'------7""1 o U.1 3.:.::::... 1"P"'O~4.1...Chapter 1.%5 o 0.5 1. position 1).. Position 3).. ::=:'::.canuu Madel .::::=:::.::.. . 10 1'P""o~"l!"..75 o 0. 1. -Laonov and PTT Model$ a tA.J 1 o ~ ..4.----------------I 0.:::o=..i!i!!....=:..Canu. 4r.2...u lIodeI % % 4 • Posftlon y (mm' 1 10 OF..- .. = .. l'Fo .=:..i ...1.. .----------....5 0..1... Position 2)... 01..15 Position y (mm) 1.1 a ~~ and PTTModel$ .1 0"'.----------------' o 0.. o % 4 • .5 0..% rr=::.-----------:..:.So'~1~--. Position 1). à 14. " " o 4 • Position y (mm) a 10 10 Position y (mm) Figure 7-12 Velocity Profile (Reservoir. 10 Figure 7-15 Velocity Profile (Channel.~----.---------""" ..:::===::::::...::.%5 OF.can..u Mode! a ~~andPTTModels .2.. ..5 0.%5 Position y (mm) Figure 7-16 Velocity Profile (Channel.5 Figure 7-14 Velocity Profile (Reservoir.. .:-""'l 1. Position 2).U I d PTT lIbIIIa à --can. a 1. _ . The maximum.. We obtained. By measuring the slope. we can evaluate the stress optical coefficient.._ _=s:::::::::. we need to know the value of this coefficient. P~nar_11 . Planar_1) • Q. An approximate value can be obtained by comparing the maximum. which implies the validity of the stress-optical relation.5 € - 2 0.S ~2.47 x 10-9 m 2/N. . which is lower than the value reported by Beraudo' for a LLDPE (2.J a 10 Figure 7-18 Mesh Independence (Reservoir. values of the fust principal stress difference ((ui -(2)calc ) calculated by the finite element simulations with the corresponding maximum birefringence (6 max) measured on the axis of symmetry for all flow rates. Planar Abrupt Contraction Flow ofHDPE 112 ~ rr==::6=::17::=:="=:='= ====::::.5 3 -PIT(II'Ul'lc:a1H spe:ll'Um. 7.. • .5 Birefringence It was mentioned that the use of birefringence data to infer the components of the stress tensor in a flowing polymer depends upon the validity ofthe stress-optical relation: where C is the stress optical coefficient..5 e > ~ 1.Chapter 1. 7-20).5 o! . In order to compare the experimental birefringence with the calculated one.. 7-21). Figure 7-19 Influence of Spectrum (Reservoir.47 X 10-9 m2/N) for a low-density polyethylene... Position 3).- --J 10 o 2 ~ 5 PosltJon y (mm) o Ar---""l!I!!--==s::==-----Lo 2 " 6 Position y (mm) . birefringence (6max) is obtained by fitting a cubic spline to the data at each flow rate and finding the maximum from the spline (Fig.. Position 3).--Al.PIT (long spectrum.------:'""l U 3. a linear relation between (u} -(2)calcand Llmax (Fig.1 x 10-9 m2/N) but equal to that reported by Schoonen8 (C = 1. We obtained a value of 1. 0 s-1 -28.1 Â52....."25 c u CD ~20 '5 = 15 m 10 ·4.1 5-1 %24.... which is used to calculate birefringence....85..75.75-1 -33.65-1 09As-1 â14.55-1 038. -20 -15 -10 -505 10 15 20 Position (mm) Figure 7. we compare the birefringence predicted by the various models with the measured birefringence on the axis of symmetry.. In the qualitative case...7-3 • . we look at the entire tlow domain and compare the shapes of the calculated birefringence fringes and their relative positions with experimental data.45-1 5 0 .... one qualitative and the other more quantitative.1 <>47.4s-1 -19.-. We can make two types of comparison between the measured and the calculated birefringence.............. is given by the following equation: Eqn...20 Evaluation of Ô max on the Axis ofSymmetry..-..75-1 -42. In a 2D planar flow domain. Planar Abrupt Contraction Flow ofHDPE 113 • ~ )( 45r------------------------------.. 40 35 iô 30 '...2s-1 XS6.-..Chapter 7. In the quantitative case. the :first principal stress difference.. 7-23 was obtained for the PTT mode!. 0 op 0/ 0/ )( K -y <j E ca ~O /60 / / 0. which is the highest rate for which we obtained a converged solution with the Leonov modeL Fig. • .. good qualitative agreement is obtained. cD 5 o~-------------------------_ 0.. 7-22 and 7-23 we present the qualitative comparison on the entire flow domain for the third apparent shear rate (14.Chapter 7.. .05 0. but the Leonov model is closer to the data in the channeL In Fig.15 0. Planar Abrupt Contraction Flow ofHDPE 114 • .30 0. Agam. 7-22 shows the solution obtained for the Leonov model.. . __.. In Figs. Both models represent well the measured birefringence in terms of the shape and position of the fringes in the reservoir." 45 40 35 30 25 20 15 10 0 0 - PTT Leonov C 1.10 Y .25 0.20 0.47 X 10-9 m 2/N pO 09 0/ = . 7-24 we show the calculated birefringence for the highest apparent shear rate (56.4 S-1).4 S-1) for the PTT model..00 0.35 Figure 7-21 Calculation ofthe Stress Optical Coefficient. while Fig. Planar Abrupt Contraction Flow ofHDPE 115 • Figure 7-22 Comparison ofMeasured and Calculated Birefringence (Leonov Model.45- 1 ).Chapter 7. .45- 1 ). ra =14. ra =14. • Figure 7-23 Comparison ofMeasured and Calculated Birefringence (pTT Model. The Carreau-Yasuda model gives clearly the poorest prediction. as was seen in the quantitative comparison on the entire flow domaine This • fast relaxation is shown again for the highest apparent shear rate where we compare the predictions ofthe PTT and Carreau-Yasuda models. While the predictions of the velocity profiles were similar for the two viscoelastic models. we compare the calculated andmeasured birefringence on the axis ofsymmetry. ra To evaluate more quantitatively the performance of the models. we show the results of the three models for the three lowest apparent shear rates. Planar Abrupt Contraction Flow ofHDPE 116 • Figure 7-24 Comparison ofMeasured and Calculated Birefringence (pIT Madel. they now differ considerably. In Figs. going to zero as soon as the streamlines become paraIlel in the channel.Chapter 7. the Leonov model gives a much better representation of the experimental birefringence. with. . 7-25 to 7-27.4s. the normal stresses relax immediately.1). the PIT model relaxing too fast in the channel. = 56. .1:===.=U..10 . 1$ ~. Figure 7-28 Predicted and Measured Birefringence on the Axis of Symmetry (ra = 56...I=:_--"".. We saw previously that there was no difference in the predictions of the pressure drop ()or the velocity when using the full spectrum vs..::..65-1). :.. - r... the truncated one. ~ i !' 4 z o ~ ~ ii • ~ •...=::. -c. Figure 7-26 Predicted and Measured Birefringence on the Axis of Symmetry Cra = 9. (Fig. a c t .. Planar Abrupt Contraction Flow ofHDPE 117 • 10 a • u ii ~ .. 7-29)... ...-·can-Vauâ ~15 .1.... . o J-_ _ -20 ===~ __ . ._::=__I .:.----------.4 S-I).:. ~ 2D •• ~ ~ ~ -PTT L.:.. 7-30) of 2.......... ~ o~-~~---.~t:=:A~"::=l -PTT • LAanov ==.2 % at an apparent shear rate S-I.ii.15 -===~ __ __ ~ ~ . there is a small difference in the maximum birefringence between th~ the predictions with the two meshes (Fig..4 S-I)..4 %.45-1)... Similarly..aD -10 -4 0 5 10 ·15 -10 -4 PaItIon (nwn) 0 5 Position (rrwn) 10 15 ZO Figure 7-25 Predicted and Measured Birefringence on the Axis ofSymmetry (fa =4...u-V. . .n.s Pœi!ian Elna) a 5 Position (rrwn) 10 15 2G Figure 7-27 Predicted and Measured Birefringence on the Axis of Symmetry (ra = 14. birefringence if we of28..:=:::=. . & c:: ! ii 10 • o L-_ _ ~ =====-__ __.~.Chapter 7... 25 T'i=..:::1 : : = : : = : .. What is remarkable with stresses..=.I ·15 .zo -15 -la ...s 0 5 10 15 2D . ~ ~. For the birefringence on the axis of symmetry.-----------.7 us~ the full spectrum.. This difference amounts ta 1. structured mesh is the smoothness of the curve and the absence of oscillations in the • . -PTT ~ 1." 15 20 __ _ ~ _. a 1Z -canuu Yuuca. there is a small difference in the maximum... but so are those of the strictly viscous Carreau-Yasuda modeL The viscous model is aIso able to give an. Noting the rapid change in the flow field near the contraction. Figure 7-30 Influence ofMesh on Birefringence (ra =47.......• C Il ~ iD .. .E e iD . 20 o ~ _ . the predicted velocity profiles are DOW in perfect agreement with the measurements. 7. excellent prediction of the velocity field when there is no strong elongational component. we did a sensibility study in this region to evaluate the efIect of a small error in ~ on the velocity profile.. The results of the study are shown in Figs. The model giving the most realistic representation of the experimental stresses. . However.7 S-I). 7-31 and 7-32.Chapter 7.s 0 5 Position (mm) 10 15 20 Figure 7-29 Influence ofSpectrum on Birefringence (fa =28.s 0 5 PosItIon (mm) 10 15 oZ -15 -10 ." " " " o oc ='25 :> ~ r. .10 .15 ... · 47.=:::.---------. and that a small error in the measurement position where the f10w is changing rapidly can result in a large error in • velocity.::~==__ 15 o . even the viscoelastic models have difficulty in capturing all aspects of the behavior of the melt...7s-1). is aIso the one posing the most serious numerical convergence problems.7 .:::::?F::."20 f i ~ 15 10 5 .. .:::=-oZ . This shows that results of the simulations in terms of velocity might he hetter than originally thought... -PTT (UI1SU'UCIUftd) 1 1 ~ ·-PTT(~ ~30 c: ". We looked at the predicted velocity fields assuming the position of the measurement was in errar by ± 0.===. Planar Abrupt Contraction Flow ofHDPE 118 ~ • 35T'P""~~~ _ _" " ' . Surprisingly..5 mm in x at positions 2 and 3 in the reservoir...6 Discussion In general. the simple Leonov model.. the results of simulations of planar abrupt contraction flow are disappointing..L-_ _:::::::. - ~ . c : : : : : : : . when this component becomes important. The predictions of total pressure drop by the viscoelastic models are good. which are rounde~.1 . 'U . It was aIso conc1uded that there is little advantage in working with the full spectrum vs. E.-=---. as seen in • between the shape of the predicted fringes in the reservoir ("butterfly" shape) and the experimental ones.... This could explain the difference S-1 near the contraction (x=O). there is a limitation on the amount of truncation that can he done before starting to loose information about the material. this parameter was determined at low extension rates. . The faster relaxation of the stresses in the channel predicted by the PTT model in comparison with the experiments was aIso observed by Beraud07 and Schoonen8 . position 3). Our truneation was sufficient to facilitate numerical convergence without affecting the predictions of the variables studied... the truncated..-u_ ~ O . For example.. -.s 0. spectrum...0.::::="'. the fact that we had a ratio of WIR < 10 neither in the reservoir nor in the channel may invalidate the assumption of 2D fiow. Of course... Figure 7-32 Influence of ~ on Velocity Profile (reservoir.:. PosIIIon 5 (mm) oL-_-_I!!!!!!!!~:L. o..::::=====::. while this rate can become quite large in the actual fiow situations.Chapter 7.--------.. while we applied it to a flow where the elongation is planar. For one thing. . _ . position 2). 7-33.5_ -.. at the highest apparent shear rate (56. 4 1 o z Position y (mm) __---J 10 Figure 7-31 Influence of LU on Velocity Profile (reservoir. was determined from uniaxial elongation experiments.. Furthermore. the parameter that controis elongation. Finally. fIII . It is bard to say why the model behaves in this way.4 S-I) the extension rate on the axis of symmetry reaches a value of 8 Fig.5 r...-------------1 o '0 • Y . PIanar Abrupt Contraction Flow ofHDPE 119 • . Planar Abrupt Contraction Flow ofHDPE 120 • -• .Chapter 7. c G) Je W ~ as 3 .. • . ~ 8 7 6 5 4 (1) CD c 0 -. 2 1 0 -1 -20 -15 -10 -5 0 5 10 15 20 X Figure 7-33 Extension Rate on the Axis of Symmetry Cfa = 56..4 5-1) .. F.T..Fr.. Wissbrun K... C. PhD.F. Delft University ofTechnology. rapport d'étape. "The Application of Flow Birefringence to Rheological Studies of Polymer Melts ". École Nationale Supérieure des Mines de Paris. 2nd edition (1981) 4 Gavrus L. Flows". 78-84 (1989) 2 Wales IL. École Nationale Supérieure des Mines de Paris. p.. thesis. Van Nostrand ReinholtL New York (1990) 6 McHugh A. Eindhoven. Technische Universiteit Eindhoven. thesis. Thesis.H. "Melt Rheology and its Role in Plastics Processing". "Méthodes éléments finis pour la simulation numérique d'écoulements de fluides viscoelastiques de type différentiel dans des géométries convergentes". France (1995) 8 Schoonen J. private communication (1999) • .. Whitelaw J. Process. Agassant J. Rheal. "Étude du comportement des polyéthylenes: Biréfringence d'écoulement et anémometrie laser".Chapter 7. The Netherlands (1998) 9 Baaijens F...P.D. Ph. Vergnes B.. ll(7). Academie Press.S. Ph..M... The NetherIands (1976) 3 Dorst F. "Determination of Rheological Constitutive Equations using Complex. 619-634 (1987) 7 Beraudo. lnt.M.ance(1995) 5 Dealy J.J... Sophia 1 ~tipolis.E.F.. Melling A. " Principles and Practice of laser-Doppler Anemometry". "Characterization of the Sharkskin Defeet and its Development with the Flow Conditions ". " Measurements of Birefringence by the Method of Isoclinics '\ J. Mackay M.D.• IV(2). Polym.. Khomani B. Planar Abrupt Contraction Flow ofHDPE 121 • REFERENCES Beaufils P.. Sophia Antipolis. p. AIl the experiments were carried out at 150 ± 1°C and at apparent shear rates between 2.0).8 S-I. For the accurate measurement of the diameter of a soft. The ratio measured is the time-dependent extrudate swell B(!}.5.2 Extrudate Swell Measurement The die swell experiments were aIso performed at ISO°C and apparent shear rates from 2. and the wall shear stress to the force measured by the load cell. 8-1.2.1 CapiIIaty Extrusion Extrusion experiments were carried out in a constant speed. In this work an optical . as described in section 4. Circular dies with a 90° tapered entrance angle of 1. We aIso included in the study a longer die (L/D=38. In this instrument.1. an opticai detection system and a • data acquisition system. In addition to the capillary rheometer. 8.• The quantities measured are the total pressure drop given by the load cell. and moving abject. hot. delicate. the apparent wall shear rate is related ta the flow rate. the entrance pressure drop. inferred from a Bagley plot or measured directly for an orifice die (see Fig.8 1 5.1 Experimental Methods 8. Axisymmetric Entrance Flow and Extrudate Swell of Linear Low-Density Polyethylene 8.75. which is a funetion of time.0) for the extrudate swell experiments. a non-contacting sensor must be used. 4-3). and the extrudate swell. 19. A schematic view of the rheometer is shawn in Fig. defined as: Eqn.4 mm diameter and three LJD ratios were used (4. piston-driven capillary rheometer.1.22 ta 88.9.2.• 8. in arder to ensure fully developed flow at the exit.22 and 88. the experimental apparatus to measure die swell consists of a thermostating chamber.8-1 where De is the capillary diameter and De(f) is the diameter of the extrudate. The experimental set-up for the extrudate swell measurement is described in the next section. extrusion into ambient air leads to premature freezing and the development of frozen-in stresses. First. Second. To measure die swell.Chapter 8. Axisymmetric Flow and Extrudate Swell ofLLDPE 123 • system was used. if the polymer is extruded into a heated oven. the resin is extruded from the die into a thermostating chamber. An expanded and collimated laser beam is used as a back-light source to cast a shadow of the polymerie extrudate onto a linear photodiode array. Two problems can arise during extrusion of polymer directly into air. consisting of a photodiode array and a 2 mW He-Ne laser beam. it will sag under its own weight. An achromatic lens magnifies this shadow. The data logger we used was designed to enable real rime acquisition of data using a microcomputer. A schematie view of the experimental apparatus is shown in Fig. This system was designed by Samara1. load ceIl Reservoir Piston 1 A P total 1 _t_ A P entrance 1 Capillary Extrudate Swell Figure 8-1 Capillary Rheometer. The • . The electrical signal from the photodiode array is then processed using a specially designed circuit. 8-2. The use of an oil-filled thermostating chamber elirninates both of these problems. . The inner compartments contained a mixture of silicone oils..48 % by volume mixture of 2 cS . The inner compartments were equipped with two quartz windows each to allow the laser beam. The oils were mixed in proportions found by trial-and-error to achieve a density slightly lower than that of the meIt at Üe test temperature. temperature to which the ails could be heated was 150°C. The chamber was equipped with two plug-screw 300 watt immersion heaters. Axisymmetric Flow and Extrudate Swell ofLLDPE 124 com~ent. (200~ The outer compartment was filled with 200 centistokes (cS) silicone oil Fluid from Dow Corning). A complete descripti<>n of the experimental apparatus and procedure can be found in reference 1. • thermostating chamber is composed of a stainless steel container. mainly 2 cS and 5 cS Dow Coming 200® Fluid. was used to control and monitor the temperature in the inner compartments. magnifying lens ~ He-Ne laser D············..5 cS gave the best results.. photodiode array oscilloscope thennostating chamber l signal conversion and data acquisition unit comFuter Figure 8-2 Experimental Setup for Extrudate Swell Measurements. an outer and two rectangular stainless steel inner compartments. operating in conjunction with a thermocouple. to pass through.Chapter 8.5.]>erature controller.. . A tem.. which expLains the • selection of the operating temperature for the experiments. Vle aIso determined the small amount of swell due to the oil and sul:>traeted it from the measured swell. This procedure is discussed in section 8.··.. ~ l .1. Note that the maximum.· . We found that a 52 % . The number of elements for each mesh is presented in Table 8-1. The boundary conditions of fully developed flow were used for the extrudate swell simulation so that it could he decoupled from the flow in the reservoir. 8-4 and 8-5. Several meshes were tested for extrudate swell. • . Table 8-1 Mesh Mesh Characteristics for Axisymmetric Flow Simulations An 1 Unstructured Number of Elements 1611 AD 2 Unstructured 3402 SweU 1 Unstructured 1822 Figure 8-3 Mesh Axi 1. 8-3. Axisymmetric Flow and Extrudate Swell ofLLDPE 125 • 8. Both meshes are composed of triangular elements and are unstructured. and the meshes are shown in Figs. This mesh is aIso unstructured and made of triangular elements.2 Numerical Procedure 8. Two meshes were used for the axisymmetric entrance flow simulations in order to verify mesh inde~ndence of the solutions.2.Cbapter 8.1 Meshes Simulations of entrance pressure drop and extrudate swell were performed separately. but it was possible to obtain numerical convergence at our test conditions on only one of them. on both meshes and with both the full and truncated spectra shown in Table 4-3. where the maximum is seven. . and it was not possible to verify if this trend was continuing at higher shear rates.2. a pecuIiar behavior was observed with the simple Leonov model. As we slowly increased the flow rate from Newtonian behavior to (t~C(») started viscoelastic. The PIT model gave converged solutions for all seven experimental flow rates.2 Viscoelastic Converged Solutions Numerical convergence can be easily summarized for the axisymmetric entrance flow simulations. However. 8-6 (mesh Swell_l. A sunHar observation bas been reported by Rekers2 . We present below a map of the converged solutions for the various cases of • entrance flow and extrudate swell. Figure 8-5 Mesh Swell_l. truncated spectrum).Chapter 8. On the other band. 8. For extrudate swell simulations. Axisymmetric Flow and Extrudate Swell ofLLDPE 126 • Figure 8-4 Mesh Axi 2. 1t is expressed in tenns of the number of flow rates converged. we lost convergence at a very low shear rate. the ultimate swell to decrease from its Newtonian value of 13 %. it was impossible to obtain a converged solution with the simple Leonov model for any of these conditions at one ofthe experimental flow rates. This behavior is shown in Fig. .....001 0. 8......-........5 S ftI E 0 12.......Chapter 8. Axïsymmetric Flow and Extrudate Swell ofLLDPE 127 • Table 8-2 Converged Solutions with the PIT Model (max=7) Axi 1 7 7 Spectrum Full Truncated Mesh Axi 2 7 7 Swell 1 0 1 Table 8-3 Converged Solutions with the Simple Leonov Model (max=7) Axi 1 0 0 Spectrum Full Truncated Mesh Axi 2 0 0 Swell 1 0 0 13..01 0.0001 0. .1 1 ApparentShear Rate (1/5) Figure 8-6 Ultimate Swell Predicted by the Simple Leonov Madel..... 0.... 'Ci) ~ fi) :::.... We compare the calculated and experimental total pressure drop for an LID of ..5 ................e ~ 13 0 0 ........3 Total Pressure Drop • The numerical results presented here were obtained with the PTT and Carreau-Yasuda models.....5 0 12 ...... 0 ~ 10 A ~ 5 o 1 10 100 Apparent Shear Rate (1/5) • Figure 8-7 Comparison ofPredicted Total Pressure Drop with Experimental VaIues. The calculated pressures are slightly underestimated at high shear rates. especially by the CarreauYasuda model. Axisymmetric Flow and Extrudate SwelllClfLLDPE 128 • 19. Since the melt is predominantly in shear tlow. even the strictly viscous moder. independence is demonstrated in Fig. . Finally.0. 8-8. In Fig.0. 8-7. but in general the agreement is good. and pressure effects are not included in the finite element modeL The predicted total pressure drop was evaluated as follows: Eqn. 8-9.8-2 In Fig. PTT {full spectrum. since the viscosity was thought to be affected by the pressure with the LID of38. axi_1l o Carreau-Yasuda (axi_1) 20 ~ ~ Ci 0 <1 ca :i D- D- 0 15 6. can give a good approximation of the total pressure drop.Chapter 8. and we see that they agree very weil. we compare the predictions of the PTT model with the full and the truncated spectra. the results of both models are e=ompared to measured values. mesh 25 o Measured 6.. . 10 .... 10 5 o 1 10 Apparent Shear Rate (1/5) 100 Figure 8-8 25 Influence of Discrete Spectra on Prediction of Total Pressure Drop.Chapter 8. axi_2) ~ :e ca a. 15 -='Ci 0 Q.. axi_1) o PTT (truncated spectrum.. 1 . axL1) o PTT (full spectrum.... axi_1) :lE <1 15 1 a.. Q ~ 10 <1 5 OL. 100 • Apparent Shear Rate (1/s) Figure 8-9 Mesh Independence on Prediction of Total Pressure Drop. o Measured 20 PTT (full spectrum. 25 p- 20 o Measured ~ PTT (full spectrum. • :. . Axisymmetric Flow and Extrudate Swell ofLLDPE 129 . as seen in Fig.8 S-l) the extension rate on the axis of symmetry reaches a value of22 S-1 near the contraction (x=O). 8-11.Chapter 8. models. The predicted entrance pressure drop was evaluated as follows: Eqn. at the highest apparent shear rate (88. • .6. mesh independence is demonstrated in Fig. For example. The oscillations in Ë after the contraction are typical of results obtained on unstructured meshes when using the discontinuous Galerkin method3 • However. as expected because of the strong elongational component of the flow. In Fig. 8-12.8. 8-13. 8-10 shows the measured values of the entrance pressure drop along with the predictions of both. Axisymmetric Flow and Extrudate Swell of LLDPE 130 • 8. the PIT model also underestimates the entrance pressure drop by about 50%.5. 50 we expect to see a difference between the predictions ofthe viscoelastic and the viscous models. Earller simt!lations of entrance flow have also produced entrance pressure drops weil below those observed experimentally 4. Finally.8-3 M'cale = Presen.oir - PLI R=O-S The Carreau-Yasuda model underestimates the entrance pressure drop. Fig.4 Entrance Pressure Drop ln the entrance region the melt undergoes both shear and elongation.7. we compare the predictions of the PIT model with the full and truncated spec~ and we see again that they agree weIl. .... -- -5 -10 . • x Figure 8-11 Extension Rate on the Axis ofSymmetry (ra =88...... CD êâ 2 ! . axL1) o Carreau-Yasuda (axi_1) a. 20 • en ~ 15 10 a: c ëii c S ca o 5 W ~ 0 .8s.. Axisymmetric Flow and Extrudate Swell of LLDPE 131 • :& U 3 o Measured (orifice die) 6 PTT (full spectrum.1). c <1 c a. . Comparison ofPredicted Entrance Pressure Drop with Experiments...... ê 10 o 6- o ~ 100 Apparent Shear Rate (ils) Figure 8-10 25 ....--.30 10 -20 20 30 -10 40 o .Chapter 8... CD 1 6 o f O ..-01!'---"""'!----~------------........ 1 t o ... ... axi_1) o PTT (full spectrum.c CD o 1 D. <> Measured (orifice die) 6 PTT (full spectrum. axi_2) :: :e U 2 .Chapter 8. . 10 100 • Apparent Shear Rate (1/5) Figure 8-13 Mesh Independence on Prediction ofEntrance Pressure Drop. 2 ! . <] 1 . <] o 1 il i 10 100 Apparent Shear Rate (1/s) Figure 8-12 3 Influence ofDiscrete Spectrum on Prediction of Entrance Pressure Drop. axi_1) o PTT (truncated spectrum. ~ 1 o il i t Q f Q Q . Axisymmetric Flow and Extrudate SweII ofLLDPE 132 • li 3 <> Measured (orifice die) 6 PTT (full spectrum.c CD c D. . axi_1) :e u c ~ D. t m 0 l 0 CD ! . 8-14. This swell was subtracted from all the extrudate swell measured values. No data are available at short times for the lowest apparent shear rates in Fig. a mixture of low viscosity grades of silicone oil was used (2 cS and 5 cS) in arder to pravide an ail density slightly lower than the polymer melt density. We verified that the high viscosity oil was not absorbed by the polymer by weighing the sample before and after annealing. Cut extrudate strips were annealed far 30 minutes in an oyen at 150°C in a bath of500 cS silicone ail.• In this way the net swell without the effect of oil was obtained. This i<. sorne interaction between the oil and polymer occurs when low viscosity grades are used with polyethylene. 8-14. 8-14 shows the measured swell as a function oftime for the seven apparent shear rates.1 0 have found that Dow Coming 200® fluid silicone oils do not swell polyethylene if medium viscosity grades (50 cS and 100 cS) are used.5.3 S-I.Chapter 8. The • method used does not allow an accurate capture of the instantaneous swell that occurs in the first seconds of extrusion for these rates. apparent wall shear rate of 13. shawn in Fig. In the present study.5 Extrudate Swell 8. 8-15 for the apparent shear rate of88. reasonable to assume that after annealing. Fig. and extrudate swell was measured at 150°C using the apparatus.1 Experimental Results The measurements were made with an L/D of38. Consequently. all the stresses are relaxed and the equilibrium swell is reached. The annealed samples were placed in the thermostated It seems chamber. Axisymmetric Flow and Extrudate Swell ofLLDPE 133 • 8.9. We assumed that the oil absorption effect was the same at all shear rates. Utracki et al.0 showed that fully developed flow was not reached in the shorter die since swell was larger. Comparisons with swell data for L/D = 19. any additional swell would be due ta interaction with the ail. However.8 1 5. The swell due to the oil was measured so that it could be subtracted from the measured swell using a method similar to that of reference 1. because a significant time is required for the . The oil swell curve is shown at the bottom of Fig. The resin was extruded directly into air at an.0 to ensure fully developed flow at the die exit. tn 1.221/5 4.81/5 Figure 8-14 Total Swell and Oil Swell as a Function ofTime.. the ultimate net swell..Chapter 8. is shown in Fig.441/5 8.41/5 88.2 1.21/5 44. obtained at long times.0 ~1oIIQiiii-"""""'''''''''''''''''''''''. Axisymmetric Flow and ExtnIdate Swell ofLLDPE 134 • sample to reach the viewing chamber at these shear rates.3 1/5 22.8 -..4 1. • .881/5 13.I''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' o 200 40Q 600 800 1000 1200 1400 Time (5) 0 0 6- "V 0 0 0 0 Oïl Swell 2.-. Finally. 8-17 as a function of apparent shear rate.. 1.6 1. ..... In Fig..2..... With the PIT mode!..6 a o 1.4 o A Oil Swell Total Swell (88....2... although the Newtonian swell (13 %) was e~ily calculated on severa! meshes. The Leonov mode! was discarded early...8 1/s) Net Swell (88..2 Comparison with Simulations As was mentioned in section 8....... we were able to obtain a converged solution only for the lowest apparent shear rate (2.. shawn in Fig..... it was not possible to calculate the viscoelastic swell for most conditions.0~"""'"""""""""... The ultimate swell (20 %)..5... Axisymmetric Flow and Extrudate Swell ofLLDPE 135 • 2.. 8-16. is aImost half of the measured swell for that flow . we show the predicted interface calculated on • mesh Swell_l with the truncated spectrum.......0 ~l.... 8....~~ 1.... which showed aImost no convergence problems in other situations..8 1I/s) 1.22 S·I) and only on one mesh.. and second because of numerical conv-ergence problems.. fust because of its pathological prediction of decreasing swell.....Chapter 8..2 1......... 817 along with experimental values......lo"""""""""""'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' o 200 400 600 800 1000 12()0 1400 Time (5) Figure 8-15 Net Swell as a Function ofTime after Oil Swell Subtraction..8 1.."....IIQiiiiiiliiiO.... on the detailed speetrum. Axisymmetric Flow and Extrudate Swell ofLLDPE 136 • condition (37 %).8 r--------------------------. 1. it is not possible to know at this time if the low calculated swell is related to the use of the truncated spectrum.. The experimental ultimate swell data is presented in the appendix. As mentioned previously in section 2. 5 ~ fi) S 1.20 40 &0 Apparent Shear Rate (1/s) 80 ---100 o • Figure 8-17 Measured and Predicted illtimate Swell.. Since we could not obtain a converged solution using the full spectrum.2 t t::... PTT (truncated spectrum... Le..4 eu E .. ~ <> Measured t::.4. 1 -- .Newtonian Value .-. .Chapter 8. 1...... --[---~ Figure 8-16 Calculated Interface with PIT Model (truncated spectrum.. extrudate swell depends stronglyon molecular weight distribution. mesh Swell_l). swell_1) . ~ 1.... . .. ...3..6 . . .. the models must aIso be correct for the simple flows. reser1loir + ( j . This poor performance is similar to what is seen in planar flow for velocity fields with a strong elongational component (section 7. we have: Eqn.rr • where â ~rr = ~. We also tried using E=Ü in the simulations. even with the viscoelastic PIT model.4).S = Preser1loir - - PL 1R=O. Predictions of entrance pressure drop are largely underestimated. The PIT model was much easier ta work with in terms of numerical convergence for the prediction of entrance flow but not for the free surface flow. The Leonov model gave realistic predictions for planar contraction flow (see Chapter 7) but was not useful for axisymmetric extrudate sweli.. or the parameters determined in simple flows can not reproduce the features of (. We also note that the elongation rate in the flow simulated can be much larger than the range of extension rates for which it was possible to fit the viscoelastic model parameters.8-4 M ca/e = -cr.. but this ooly increased the prediction ofthe entrance pressure drop by 6 %. LI R=O.S + Il.Chapter 8. Using this suggestion for the orifice die.... Beraudo7 suggested that the most appropriate quantity to be compared with the measured pressure is the rr component of the Cauchy stress tensor. However. which gives the strongest strain-hardening behavior in elongation... The predictions of total pressure drop are good. At this time.6 Discussion In general.omplex flows. evaIuated at the wall. Axïsymmetric Flow and Extrudate Swell ofLLDPE 137 • 8. LI R=O. in terms of numerical convergence problems and the prediction of decreasing swell. the results of simulations ofaxisymmetric entrance flow and extrudate swell are disappointing. There are two possibilities: either the models are inappropriate.s ~rr l'eser1loir . but so are those of the strictly viscous Carreau-Yasuda model... The performance of the simple Leonov model is much worse than in the planar case. it is difficult to say why this is 50.. for the free surface predictio:n. thus increasing the discrepancy with the Finally. Eqn. ~'trr.8-5 But this did not facilitate the convergence of the extra stresses and failed to give a converged solution for any of our exp<rimental conditions. where the interface is modeled by a ::function h (for height) calculated from the velocity profile. measured values. is negative and decreases the computed entrance pressure drop bF 3-4 %. We also tried an alternative to the use of the pseudo-concentration.Chapter 8. the difficulty was not with the calculation of the free surface position itself but in t:ryiI::J.g to converge the extra stresses for a fixed position of the interface. Axisymmetric Flow and Extrudate Swell of LLDPE 138 • The contribution of the extra stres:s difference. • . Ph. École Polytechnique de Montréal. p. Rheal. & p.O.G. "Excess Pressure Losses in the Capillary Flow of Molten Polymers". Montréal..' 1. Rheo!. Béreaux Y. Ph. "Polymer Processing and Properties'" G. École Nationale Supérieure des Mines de Paris. J. Lara J.' • . Catani A. Mulhouse-La Bress.K.. Capri.. Canada (1985) 2 Rekers G.A.1. McGill University. p. Twente. ôttinger H.. France (1995) 8 Hatzikiriakos S. Canada (1998) 4 GuilletJ. Christodoulou K. Master's Thesis. Acta.. p. 494-507 (1996) S Fiegl K.. 438-448 (1998) 7 Beraudo C...R. University ofTwente.. Nicolais. 545-555 (1996) 9 Utracki L..~xperimental and Numerical Study ofEntry Flow ofLow-Density Polyethylene Melts".. Kamal M. Lara J. The Flow of a LDPE Melt through an Axisymmetric Contraction: A Numerical Study and Comparison to Experimental Results ".... Italy. Intern.G.. Astarita and L.A. 847-874 (1994) 6 Mitsoulis E. Clermont J.. France (1983) 1.-R.. 1. Thesis. Master's Thesis. The Netheriands (1995) 3 Lioret A. J1. "Méthodes d'Éléments Finis Adaptative pour les Écoulements de Fluides Viscoélastiques ».C. Workshop on Extensional Flows. "Sensitivity Analysis of the Bagley Correction to Shear and Extensional Rheology". Montreal. Acta. Rheo!. Axisymmetric Flow and Extrudate Swell ofLLDPE 139 • REFERENCES 1 Samara M..Chapter 8. Rheol. I.. 79 (1983) 10 Utracki L.. p.. "Méthodes éléments finis pour la simulation numérique d'écoulements de fluides viscoelastiques de type différentiel dans des géométries convergentes". Hatzikiriakos S. Design and Construction of Computerized Die Swell Apparatus". 38(4). Vlassopoulos D. eds.. TI. Acta. Numerical Simulation ofUnsteady Viscoelastic Flow ofPolymers". Sophia Antipolis. thesis.. Revenu P.D.. Mitsoulis E. i. Highly simplified constitutive models are used.3. this revea1s an error in terminology.e. 1. 2-16) and requires a minimum amount of experimental information. the term introduced in Chapter 2.• 9. which is a manifestation ofviscoelasticity. we describe severa! of these simplified approaches and propose the use of a rule-based expert system as an alternative to full viscoelastic flow simulations. particularly viscosity. but avoids explicit reference to the fluid memory. However. the numerous difficulties involved in viscoelastic flow simulation have led to the development of alternative approaches that avoid the explicit modeling of fluid elasticity. In this chapter. However. it is necessary to have a reasonably accurate model of the process of interest. one often talks about the "high Weissenberg number problem". 9. normal stress differences and elongational viscosity. therefore justifying the treatment of the flow as locally viscometric. The reference should be to the • "high Deborah number problem". which are strictly valid only in steady simple shear or elongational flows and employ empirical equations fitted to experimental data. The Weissenberg number (We) indicates the degree of nonlinearity or anisotropy that is manifested in a . It takes into account the effect of normal stresses.4 • involve flow in channels where there is a main flow direction such that the streamlines are almost parallel. the viscosity and normal stress differences. This approach is based on the CriminaJe-Ericksen-Filbey (CEF) equation (Eqn. In the field of numerical simulation. These make use of specifie material functions.1 Viscometric Flow This is based on the observation that most polymer processing operations The viscometric flow approximation bas been used successfully by Vlachopoulos et al. of plastics processing equipment and the processability of plastics resins other than by trial and error.2. Alternative Approaches In order to optimize the design. have had considerable success in the modeling of melt flows in which De is small but We can he quite large.Chapter 9. McKinley6 has explained in detail how the !Wo are related in converging flows and has introduced a "scaling factor" that expresses this relationship quantitatively. The distinction is particularly important in flow stability analyses.Y Eqo. Moreo~er. For example. However. among others.. we have to keep in mind that the viscometric approach remains an approximation. while the Deborah number is the characteristic relaxation time of the fluid Âjluid divided by the characteristic time of the flow Â-jlow. The confusion between the two arises from the fact that in many flows of practical importance they are directly related to each other. Alternative Approaches 141 • given flow. Examples of forming processes simulated using the viscometric flow approach include extrusion7 . is a flow in which De is zero because of constant-stretch history but in which We increases with the tlow rate. • . Fully developed flow in a channel of constant cross section. film blowing8 .9-2 The Weissenberg number is the product of a charaeteristic relaxation time ofthe material and a typical strain rate. such as injection molding. The characteristic time ofthe flow Âflow is a time reflecting the variation ofthe strain history. Vlachopoulos et al. and those working in this field have been very careful not to confuse the two expressions. which is similar to the lubrication approximation used to simplify the momentum equation in certain polymer processes. which is a viscometric flow.9-1 We=Â. while the Deborah number (De) indicates the relative importance of elasticity. thermoformini and calendering lO for polymer melts and elastomers ll . Larsons has properly distinguished between De and We. Eqo. but this time to simulate the axisymmetric entrance flow of polymers. On the other hand. GUpta12 reported that the (elongational viscosity)/(shear viscosity) ratio was important in deterrninjng the recirculating vortex and the extra pressure 10ss in . Apparently. The parameters are determined by inverse problem solving. Alternative Approaches 142 • 9.2 Power-Law Models for Elongational Viscosity GUpta12 bas also adopted a simplified approach. which have been successfully used in shear-dominated flows. the incorporation of a first normal stress difference prediction in the software gives convergence problems 13. Viscosity and elongational viscosity are modeled using the following equations: Eqn. and the incorporation of a second normal stress difference bas not been tried yet. the viscosity and the entrance pressure drop are • measured. viscoelastic constitutive equations have not produced simulations that are in good quantitative agreement with experimental data and are subject to convergence problems at high Deborah number. and the parameters ruling the elongational viscosity are found by trial-anderror fitting. do not provide an accurate simulation of applications involving an elongational flow. for a given shear rate.Chapter 9. such as flow at the entrance of an extrusion die. A more elaborate four-parameter model for the elongational viscosity has aIso been used.9-4 T'fe = 3"10 for II D ~ II Dt The strain rate for transition from Newtonian to power-Iaw behavior can be different for the shear and elongational viscosities. Generalized Newtonian models. the constitutive equations employed in the software do not predict nonnaI stresses in shear flow. Gupta has developed software for the simulation ofaxisymmetric polymerie flow that requires only knowledge of the viscosity and the strain-rate dependence of the elongational viscosity of the polymer (which is not easy to measure). However.9-3 Eqn. According to the author. but the author bas not revealed its form 13. bis software can accurately preclict the velocity and pressure field in applications involving a significant elongational flow component. Obviously. Two more references. The strain history of the melt is thus of • central importance and can not be neglected. no practical method has been developed for dealing with viscoe1astic flow .3 Approximate Prediction of Extr~date Swell Seriai et al.Cbapter 9. Gotsis and Odriozolal4 and Schunk: and Scrîven15. the extension ratio. explain how elongational viscosity can be incorporated in simulations without using a viscoelastic constitutive equation. The application to threedimensional flow problems is also feasible using this method. However.9• In extrusion processes. On the other hand. 9. Alternative Approaches 143 • axisymmetric entrance flow. The model is based on the rubher-like elasticity theory and on the calculation ·of the elongational strain recovery of a Lodge fluid 17. and transient flow dominates. He models extrudate swell as the stratified flow oftwo. the predic~ve • capabilities and versatility of these approaches remain limited. The model provides a means of simulating complex geometriçal effects in profile extrusion without the burden of a full viscoelastic model and provides a practical aid for die design. extruders and injection molds. 9. The main advantage ofthis model is that it provides good accuracy for short dies over a wide range of shear rates. relatively short dies are generally used.4 Rule-Based Flow Simulations The simplified approaches described above have been helpful in certain applications. 16 derived a semi empirical equation to predict the extrudate swell of linear polymers. where other available semi-empirical theories assume fuIly developed tlow in a long die 1. such as the design of certain types of die. the predictive capabilities and the versatility of this approach are limited. Bush20 has also ma~e a simplified simulation of extrudate swelI. Newtonian. the model can take into account nonisothermal.18. and the recoverable shear strain. shear-thinning and elastic effects. The theoretical extrudate swell ratio mainly depends on the relaxation modulus. isothermal fluids with different viscosities. By suitable selection of the viscosity ratio. . s~urce Therefore. In this last • section. _These problems arise under conditions where the models are thought to be reasonably reliable. The most likely approach appears to be a rule-based expert system that is able to select.e.. . at low De and when slip or fracture is not anticipated. the one that is most appropriate in each region of the tlow field. we provide background information on rule-based expert systems and an overview ofhow such a system might be used in the simulation ofviscoelastic tlows. Neither has it been possible ta eliminate the problem by the use of molecular dynamic models of rheological behavior.Chapter 9. i.e. such as slip and fracture.' the dependence of its state of stress on the past history of the deformation. the. particular. Obviously. Alternative Approaches 144 • situations in which there is a strong convergence or divergence of streamlines or a singularity in the boundary conditions.city of the fluid. While there is no question that the rheological mode!s used to date in the numerical simulation of the flow of polymerie liquids are quite inadequate. it is concluded that J:he of the problem is the viscoelasti. the computational problems encountered in viscoelastic flow simulations occur even when the inaccuracy of the mode! is unlikely to be the cause. from a repertoire of possible models of rheological and slip behavior. A major challenge in the development of snch a system is a method for matching the solutions at the boundaries of neighboring regions. Furthermore there are phenomena. This concl~sion is strengthened by the observation that it has not been possible by elaboration or variation of the numerical method used in the • simulation to eliminate the convergence problem. notorious convergence problems mentioned above do not occur in the simulation of flow at small De. closed fonn constitutive equation will ever be developed that can describe accurately all aspects of the rheological behavior of highly entangled molten polymers. that can not be predicted by a continuum mechanics mode! but that must be taken into account in the modeling of many industrial forming operations. In i. It is very unlikely that any single. a new approach is needed to deal with such tlows. 4. and other mformation that may be useful in formulating a solution. which are shawn in Fig..... especially in the manufacturing and process control domains21 • An excellent review of knowledge-based expert systems for materia1s processing has heen presented by Lu22. Conventiona1 Programs 22• • .1 Fundamentals ofKnowledge-Based Expert Systems Expert systems. D_a_t_a Figure 9-1 II. A blackboard . A knowledge base . 2.- P_r_o_g_ra_m 1 Expert Systems vs.Chapter 9..ediate hypotheses and resuIts that the expert system manipulates..which records the interm. are currently used in a number of engineering applications. • Knowledge-Based Expert Systems 1 Blackboard (data) I~"-----.which contains domain knowledge such as rules. Alternative Approaches 145 • 9. facts. Control Inference 1 Knowledge Base Conventional Programs 1 . 3. An expert system can he distmguished from a conventiona1 computer program in terros of its three basic components22. ..whiêh applies proper domain knowledge and controIs the order and strategies ofproblem solving. based on techniques developed from artificia1 intelligence research. 9-1 and listed be1ow.. 1. An inference engine .. which is physically separate from the knowledge base. gives expert systems greater tleJC'ibility in both the implementation and execution stages. solving. Many applications can 9. The separation of the task-level knowledge (stored in the knowledge base) from the control knowledge (stored in the inference engine). In an expert system.4. that manipulates data in a sequential manner with many conditional branches. benefit from the use of such a system. and all the required control and domain knowledge is coded into the program. the knowledge base is treated as a separate entity rather than appearing ooly implicitly as part of the program. which makes a decoupling on a component basis less attractive23 • • . During execution. It can be viewed as a '. It contains facts and mIes that are used as the basis for decision-making.2 Application ta Viscoelastic Flow Simulations One way to approach effectively viscoelastic flow simulations is through computational domain decomposition23. velocity and pressure) exhibit strong local coupling. the program and the data. the Schur complement method is based on a domain decomposition that leads to a decoupling of the system on a sub-domain basis rather than on a component basis. 9-1.Chapter 9. The inference engine has an interpreter that decides how to select and apply the mIes from the knowledge base to infer new information. the status of the problem is represented by the current values of the variables in the program. global database for keeping track of the current problem status and other relevant information during problem. Alternative Approaches 146 • A conventional program contains ooly two parts. The blackboard is a separate entity that serves as a working memory for the system.ory (the blackboard). and the addition of a separate working mem. as shown in Fig. and a scheduler that determines th~ order in which the selected rules should be applied. The advantage of this technique is that all the unknowns in an element (stress. For example. The knowiedge related to the control of decision- 'making is collected in the inference engine. Two main difficulties . which would facilitate numerical convergence due ta its larger radius of convergence. would have ta be overcome: fust. allowing a solution for a small system with a full Newton method. on the examination of the velocity gradient. but would not be valid for extrudate swell. approach. similar to the boundary layer theory of cIassical fluid mechanics. . An example of such equations are those derived by Renardy 25 for the PIT and Giesekus models for encI.1e representation become attractive. such as the viscometric approach in flow regions far frOID flow singularities or in shear-dominated • zones. AIso.Chapter 9. Sorne issues have to be explored systematically to develop the expert system. based on the material and the nature of flow.osed flows. This is where knowledge-based expert system based on m. The use of a rule-based expert system would enable many simplifications. for example. a procedure to determine the number of modes needed. This number could also vary depending on the location of the sub-domain. For example. Alternative Approaches 147 • Another strong advantage of this technique is that it enables the use of the most appropriate rheological model in each region of the flow field or sub-domain. The use of a Newtonian model to ensure flow convergence. More complete calculations would be done only in complex flow zones. However. A boundary layer simplification. which is the simplest . Three possibilities are: 1. the size and boundaries· of these sub-domains are not known a priori. and they should be allowed to established themselves according to flow conditions through an automated process. 2. 50 it would not change from one iteration to the nexr4 . An expert system would be able to select from a bank of rheological equations the most appropriate model according ta the flow conditions calculated in a "start-up element" ta get the cycle going22 • The mIes for model selection could be based. the matching of the solutions at the boundaries of neighboring regions. but they have not been tested in • large finite element codes. and second the achievement of convergence towards a unique model in each sub-domain. a method is needed to determine the best approach for use near a singularity. they could be incorporated into the expert system by • means of fuzzy logic27 • Fuzzy logic or possibility theo~ is especially suitable for complex. 9-2. where human experience is superior to mathematical models28 • Fuzzy logic represents a mathematical way of looking at vagueness in a form that a computer can deal with. but that must be taken into account in the modeling of many industrial forming operations. nonlinear phenomena.Chapter 9. The fust calculation might be carried out for a Newtonian fluid in order to insert a start-up solution in the working memory (DATA) to get the cycle of"reasoningn (CONTROL) going. The use of a more realistic description would lead to an improvement of the simplified approach by adding to our understanding ofmaterial behavio~6. Fuzzy-logic could be easily incorporated to the computer program architecture presented earlier. . Since reliable models of these phenomena are not yet available. A coupled macroscopic description of the conservation laws with a kinetic theory of the polymer dynamics.9 . a full Newton method to allow convergence. The essential elements of a possible rule-based expert system for viscoelastic flow simulation are presented in Fig. ill-defined. The advantage of this approach is that a fuzzy expert system is sunilar to a conven~onal rule-based expert system. except that it contains mIes with ÏInprecise relationships29. but would require more computational resources. along with. ft might be possible in this way to model phenomena such as the wall slip and fracture of moiten polymers for which explicit models are not available. Alternative Approaches 148 • 3. • .~ Incorporation of Oilier Phenomena Another advantage of the expert system approach for polymer processing is the possibility of incorporating other phenomena such as slip or fracture that cau not be predicted by a continuum mechanics model.4. This approach would provide a more realistic description of the flow in these small regions. • . -mies on selection of: . . CONTROL -FEMcode -decision making: -order of actions -establishment of sub-domains boundaries -selection of rheological models KNOWLEDGE -rules on order of actions -selection of slip and fracture models (fuzzy logic reasoning) -selection of boumdary conditions . . Alternative Approaches 149 • DATA -mesh information -boundary conditions -material parameters -storage of intermediate solutions .Chapter 9. -sub-domains boundaries -rheological models -slip models -fracture models -boundary conditions -matching of solutions at boundaries -convergence verification: -towards single modeI in each zone -numerical convergence • -roles on matching of solutions at boundaries Figure 9-2 Essential'Elements ofa Possible Rule-Based Expert System for Viscoelastic Flow Simulation. 950-958 (1993) 21 Crowe E.. ID(4).).. C. Liebman D. Jb p.• QI. Vlachopoulos J.. Acta. J.. p. Feltham (ed... 321-383. 533-542 (1979) 18 Lodge A. Polym. Eng. cc Extrudate Swell in Polymers ".• 33(15). cc Simulation of Vortex Growth in Planar Entry Flow ofa Viscoelastic Fluid".. in Applications of CAE in Extrusion and Other Continuous Polymer pracesses. Koopmans R.. Film Sheet. p. COEXCAD and SPIRALCAD". Tel-Aviv..R. in Rheological Fundamentals in Polymer Processing.R.).T.. 213-263 (1992) 6 McKinley G. Tian J. Rheol. Sei. Sei. "Artificial Intelligence: Starting to Realize its Practical Promise". p.. Acta. Polym... P.. "Extrusion Die Flow Simulation and Design with FLATCAD. Jan. in review 13 Gupta M. cc Rheological and Geometrie Scaling of Purely Elastic Flow Instabilities '\ J. Ferry J. c.A. Schroeter • • .. 14 Gotsis AD.. Palym. fntern. 535-573. g p.R.L p.. Mirza F. Perèlikoulis J. Rheo!. 487-504. Salemi R. Proc. 707-715 (1984) .L. Sei.. J.. Vassiliadis C. cc Simulation of Extrudate Swell from Long Slit and Capillary Dies". Mech. cCEffect of Elongational Viscosity on Axisymmetric Entrance Flow of Polymers".. issue. C. New York (1999) . Eng. Piast...-Y.. Sei. Scnven L. 1085-1119 (1990) 16 Seriai M.. p. Sei. Pracess.B. Vlachopoulos J.. Eng. Odriozola A. K. 37(5).. cc Numerieal Simulation ofEntry and Exit Flows in Slit Dies ".. 219-248. Meissner J. p. J. 430-437 (1998) 15 Schunk P .. Israel (1981) 20 Bush M. Goublomme A. Vlachopoulos J.. Society ofPlastic Engineers. Freund Publishing Rouse.. Carl Hanser Verlag (1992) 8 Sidiropoulos V.A. p. Carl Hanser Verlag (1989) .• 25(11). p. "The relevance of Entry Flow Measurements for the Estimation of Extensional Viscosity ofPolymer Melts". Rheo!. Kluwer. p. Eng. " A Simple Mode1 to Predict Extrudate Swell of Polystyrene and Linear Polyethylenes".. 2J.. (eds. Faraday Sac. p. J. Polym. 'c Computer Simulation of Film Blowing ". J. p.).. Tucker ID (ed.S. 199-204 (1994) 12 Gupta M.O. Carrot C.• 24(9). Rheo!. Vlachopoulos J. 'c Entrance Flow Simulation Using Elongational Properties ofPlastics "..J. Mirza FA. "Nonlinear Stress Relaxation of PolYisobutylene in Simple Extension and RecoveIY After Partial Stress Relaxation "..G. Eng.. 'c A Network TheoIY of Flow Birefringence and Stress in Concentrated Polymer Solutions ".. "A Numerical Study of the Effect of Normal Stresses and Elongational Viscosity on Entry Vortex Growth and Extrudate Swell".. Sei. Nan-Newt. 107-129 (1996) 9 Ghafur M. J... Vlachopoulos J.). 'c Simulation ofThermoforming and Blowmolding: Theory and Experiments".E.. 120-130 (1956) 19 VIachopoulos J. Polym. A Method for Approximate Prediction of Extrudate SweU ". Eng. Mirza F. 6-18 (1985) 11 Nassehi V. 153-177 (1984) 5 Larson R. Rheo/. 532-538 (1993) 17 Taylor C. Netherlands (1995) 10 Mitsoulis E. Process.. p. Acta. Covas et al. '~Constitutive Equations for Modeling Mixed Extension and Shear in Polymer Solution Processing". -Kozïey B. ~ p. p. 19-47 (1996) 7 Vlcek J.. "Finite Element Modeling of Non-isothermal Viscometric Flows in Rubber Mixing ".R.. cc Calendering Analysis without the Lubrication Approximation ". 23 Debbault B. cc Instabilities in Viscoelastic Flows ".• 34(7).Chapter 9.. FI. 1379-1391 (1985) 3 Mitsoulis E.L. in Reviews an the DefOrmation Behavior of Materials.. Mirza FA.. Polym. App!. cc Knowledge-Based Expert Systems for Materials Processing'\ in Fundamentals ofComputer Modeling for Palymer processing. O'Brien (ed.. Trans.. 22-31 (1995) 22 Lu S.. Homerin O. JQ. !X(3). Vlachopoulos J. Palym. Eng.• b p. 4 Mitsoulis E. Vlachopoulos J. 677-689 (1985) 2 Mitsoulis E. p...A.1b p. presentation atANTEC·99. Mirza FA... Vlachopoulos J. Polym. Chem.. Alternative Approaches 150 • REFERENCES 1 Mitsoulis E. p.. Guillet J. .• XIII(3). Progress... "Artificial Intelligence: Starting to Realize its Practical .R.Promise". 262-270 (1998) 24 Guénette Robert. 60-65 (1996) 29 Crowe E. " Develop~ent of High Quality LLDPE and Optimized Processing for Film Blowing".• 73(1).. Vergnes B. ofMcGill University (June 1999) 27 Zadeh L. p.R. Eng.. Jan.. Eng. "Boundary Layers Analysis of the Phan-Thienffanner and Giesekus Models in High Weissenberg Number Flow".. FI.. Polym. Intern. in Fuzzy Sets and Systems.H. Venet C. issue.. Mech. Daponte T. p. Alternative Approaches 151 • B. 181-190 (1997) 26 Keunings R... Verschaeren P. J. private communication (June 1999) 2S Hagen T. " Recent Developments in Computational Rheology". p. Chem. Renardy M. Seminar at the Chemical Engineering Dept.-F. Chem.. North-Rolland. Reckman B. Progress. p. Non-Newt. issue. " Fuzzy Sets as a Basis for a Theory of Possibility ". Agassant J.. "Improve Process Control Using Fuzzy Logic". Nov.process.. 22-31 (1995) • • .A. Li R.Chapter 9. Vassiliadis CA.. Amsterdam (1978) 28 Rhinehart R.. Murugan P. The amount of work involved ta change the finite element Fortran code restrained us from trying more powerful (but costly) solvers like the Newton method. bas been thought ta arise from the following sources: .The elasticity. sorne of which may be quite long. . renders even the most aclvanced numerical methods incapable of modeling industriaI flows of melts of typical molten plastics in which elasticity plays an important raIe. For faster progress in the future. but since they have no memory the effect ofthe stress overshoot is not carried into the rest of the flow field.The presence ofthe f10w singularity. history dependence.1 Conclusions The Iack of success in the simulation of viscoelastic flows in which there is a strong convergence or divergence of streamIines. It is concluded here that the principal source ofthe problem is the elasticity ofthe fluid in combination with flow singularities. • solutions for situations of practical importance are impossible. and in particular the Leonov and PTT models. even when one of the most advanced numerical procedure is used. new programming methods must be developed to eliminate this restriction. One ofthe difficulties encountered in this project was the lack of flexibility ta change the numeric-al aIgorithms in arder ta improve convergence.e. .• 10. With the best constitutive equation from a rheological point of view (Leonov model). it is concluded that: 1. . ofthe material. Viscoelasticity. i. as described by multiple relaxation times. With regard ta the inadequacy of existing rheological models. or a singularity in the boundary conditions. Numerical difficulties are aIso encountered with Newtonian fluids in the presence of flow singularities.The inadequacy ofthe rheological model for the material.. Conclusions and Recommendations 10. The model aIso gives conflicting results. It is likely that even if more realistic rheological models were available. Even a mode! that is thought ta be one of the best with regard ta convergence (pIT model) is very limited in its capability. An entirely new approach is needed for the modeling of complex flows. thus ensuring convergence. The determination of the limitations in present numerical methods for the simulation of the flow of viscoelastic materials. as it arises from the history dependence of the material. This history dependence leads ta an intrinsic instability in numerical solutions. 1t now seems quite probable that no model expressed solely in terms of continuum mechanics variables will ever be able ta provide a universal description of the flow behavior of polymeric liquids. one could guide the direction of the computation in the neighborhood of singularities. By use of such a technique. The incorrect prediction of decreasing swell confums the findings ofRekers 1• 2.Chapter 10. Conclusions and Recommendations -----------------------------_-:153 • with realistic predictions in planar flow but not in axisymmetric flow. . However. it bas not been possible up to now ta elirninate tlie problem of numerical convergence by the use of molecular dynamic models even with sorne ofthe most advanced numerical methods. Such an approach could also readily incorporate models for slip and fracture phenomena. molecular dynamics simulation. 3.2 Original Contributions to Knowledge The original contributions ta knowledge resulting from this work are derived from the objectives presented in Chapter 3: • 1. The barriers that have arisen in trying ta use numerical methods ta simulate complex flows of viscoelastic fluids cannot be e1irnjnated using the constitutive equations and numerical techniques that have been applied to date. 10. This might be based on a rule-based expert system approach to the problem. 2. the basic problem would persist. The major conclusions ofthis work are as follows: 1. and this observation bas given rise ta a relatively new area. The proposaI that a rule-based expert system shows promise for overcoming existing limitations. • 1 Rekers G. the one that is most appropriate in each region of the flow field. Thesis. and upwinding methods in order to improve numerical convergence.e. a rule-based expert syst~ able to. Conclusions and Recommendations ---"-----------------------------------. for nonlinear viscoelastic models is strongly recommended since a single mode can not fit data properly and leads ta numericaI problems.:154 • 2. solvers. 3. This code facilitates changes in the mesh.mesh adaptativity.tten in the programming language C++. select. . • The use of a multimode spectrum.rmation of the inadequaey of the Leonov model in axisymmetrie extrudate swell. 10. University of Twente. The establishment of the cause of these limitations. Québec. The Netherlands (l995) 2 GIREF (Groupe Interdisciplinaire de Recherche en Elements Finis).. we need not only more efficient numerical methods but aIso more efficient programming tools that will I?rovide more flexibility for trying various algorithms.. Twente. 4. truncated or not. The development of an expert system should be the foeus of future work in this field rather than continuing the futile effort to elaborate a single model that will be adequate over the entire field of a complex flow. Canada H . An example of such a tool is the program MEF++2 (Méthode d'Eléments Finis) wrÏ. The con:fi. the history dependence of the materiaIs. from a repertoire of possible models of rheologieaI and slip behavior.D.Chapter 10. Numerical Simulation ofUnsteady Viscoelastic Flow ofPolymers".3 Recommendations for Future Work ·Future work should be aimed at developing the new approaeh described in Chapter 9. i. Université Laval. Ph. 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O'Brien (ed.. 017 1.016 1.736 ± 0.017 1. • .167 • APPENDIX: Ultimate Extrudate Swell ofLLDPE at 150°C Apparent Shear Rate (s-I.3 22.419 ± 0.488 + 0.018 1.88 13..015 .369 ± 0.019 1.602 ± 0.664 ± 0.4 88.016 1.) 2.538 ± 0..44 8.2 44.22 4.8 UItimate Extrudate SweD 1. 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