Design of Extrusion Forming Tools

Shawbury, Shrewsbury, Shropshire, SY4 4NR, UK Telephone: +44 (0)1939 250383 Fax: +44 (0)1939 251118 Web: www.polymer-books.com

Published by Smithers Rapra Technology Ltd, 2012

The design of extrusion forming tools (dies and calibrators) is a difficult task usually performed by the employment of experimental trial-and-error procedures, which can hinder the performance and cost of the tools, may increase the time to market of new extruded products and limit their complexity.

The main objective of this book is to provide detailed information on the design of extrusion forming tools. It describes the main problems to be faced when designing dies and calibrators, the most relevant polymer properties to be considered in the design process, the specific problems related to several types of conventional extrusion dies, and recent developments on the design of special dies and process modelling. It will be an updated and uncommon book on the subject, where each chapter is prepared by internationally recognised experts. Having in mind its nature, it is expected to become a useful reference book for higher education students (both undergraduate and graduate ones), teachers, researchers and engineers active in the extrusion industry.

Design of ExtrusionForming Tools

Edited by

Olga S. Carneiro

and

J. Miguel Nbrega

Design of Extrusion Form

ing ToolsO

lga S. Carneiro and J. M

iguel Nbrega

Design of Extrusion Forming Tools

Editors:

Olga S. Carneiro

and

J. Miguel Nbrega

A Smithers Group Company

Shawbury, Shrewsbury, Shropshire, SY4 4NR, United Kingdom Telephone: +44 (0)1939 250383 Fax: +44 (0)1939 251118

http://www.polymer-books.com

First Published in 2012 by

Typeset by Argil Services

ISBN: 978-1-84735-517-1 (hardback)

978-1-84735-518-8 (softback)

978-1-84735-519-5 (ebook)

Every effort has been made to contact copyright holders of any material reproduced within the text and the authors and publishers apologise if

any have been overlooked.

A catalogue record for this book is available from the British Library.

All rights reserved. Except as permitted under current legislation no part of this publication may be photocopied, reproduced or distributed in any form or by any means or stored in a database or retrieval system, without

the prior permission from the copyright holder.

2012, Smithers Rapra Technology Ltd

Smithers Rapra Technology LtdShawbury, Shrewsbury, Shropshire, SY4 4NR, UK

iii

Preface

Olga S. Carneiro and J. Miguel Nbrega

The extrusion of thermoplastics encompasses a huge number of different techniques (extrusion lines) and products, and is the most significant processing technology in terms of global thermoplastics consumption.

The key components of all extrusion lines are the forming tools, i.e., the extrusion die and the calibration system used. In fact, these tools have a crucial role in the establishment of the final geometry and dimensions of the extrudate, on its morphology and, consequently, on its properties. Additionally, these tools generally limit the maximum rate at which the extrusion line can be operated.

Despite their importance, there is a clear lack of literature devoted to their design and integrating the different phenomena involved, which motivated the authors to write the current book. This book intends to fill this gap, by addressing the phenomena and design issues associated with the tools used in the main types of conventional extrusion lines and also some special extrusion dies. It is expected that it will be a useful reference for higher education students, teachers, researchers and engineers active in the extrusion industry, since it addresses the main scientific problems associated with the design of extrusion tools; it is also intended to serve as a real practical guide for those who are involved in their design. In some cases, simple design methodologies are presented, which can help to solve a specific problem; other times, sophisticated numerical codes (developed in-house or commercially available), are used to illustrate relevant phenomena or the importance of some processing conditions or material properties in the performance of the extrusion tools. In these cases, the idea is not to provide the means to perform a specific design task, but to enable the readers to learn through examples. To guarantee the quality of the book, each chapter is written by researchers, both from the academic and the industrial communities, whose contribution in the specific field addressed is internationally recognised.

The organisation of the book follows a logical sequence, starting in Chapter 1 with the definition of the objectives and the most relevant problems associated with the design of extrusion tools, followed, in the next chapter, by the relevant polymer

iv


properties required for the design process. After these two introductory chapters, the design of tools for the production of specific geometries is addressed, namely: pipes (Chapter 3), flat film and sheet (Chapter 4), blown film (Chapter 5) and profiles (Chapter 6). Finally, there are two chapters devoted to special dies, namely, flexible dies (Chapter 7) and rotating mandrel dies (Chapter 8).

vContributors

Olga S. Carneiro

Department of Polymer Engineering, University of Minho, Campus de Azurem, Guimaraes, 4800-058, Portugal

Rafael Castillo

Dual Spiral Systems Inc., 1760 Main Street West, Hamilton, Ontario, L8S 1H2, Canada

Jos A. Covas


Heinz Gross

Kunststoffe-Verfahrenstechnik, Ringstrasse 137, Rossdorf, D64380, Germany

Jean-Marc Haudin

Ecole de Mines de Paris, Centre de Mise en Forme de Materiaux, BP207, 06904 Sophia Antipolis Cedex, France

Joo Miguel Nbrega


vi


J. Peter Mller

EDS GmbH, Garnisonstrasse 7, 4560 Kirchdorf, Austria

Nickolas Polychronopoulos

Polydynamics Inc., Dundas, Ontario, Canada

Shinichiro Tanifuji

Hyper Advanced Simulation Laboratory, Tokyo, Japan

Bruno Vergnes


Michel Vincent


John Vlachopoulos

Department of Chemical Engineering, McMaster University, 1280 Main Street West, Hamilton, L8S 4L7, Canada

vii

Contents

1 Main Issues in the Design of Extrusion Tools ............................................. 1

1.1 Introduction .................................................................................... 1

1.2 Extrusion Dies ................................................................................ 3

1.2.1 Rheological Defects ........................................................... 3

1.2.2 Postextrusion Phenomena ................................................ 14

1.2.2.1 Extrudate-swell ............................................... 14

1.2.2.2 Draw-down ..................................................... 17

1.2.2.3 Shrinkage ........................................................ 18

1.2.3 Flow Balance ................................................................... 18

1.3 Calibration/Cooling Systems ......................................................... 22

1.3.1 Types of Calibration/Cooling ........................................... 22

1.3.2 Main Parameters .............................................................. 24

1.4 Conclusion .................................................................................... 30

2 Properties of Polymers .............................................................................. 37

2.1 Introduction .................................................................................. 37

2.2 Rheological Properties in the Molten State.................................... 39

2.2.1 Viscous Behaviour ............................................................ 39

2.2.2 Viscoelastic Behaviour ..................................................... 41

2.2.3 Dependence on Temperature and Pressure ....................... 44

2.2.4 Wall Slip .......................................................................... 46

2.2.5 Flow Instabilities .............................................................. 47

2.3 Thermal Properties ....................................................................... 49

2.3.1 Conductivity and Diffusivity ............................................ 49

viii


2.3.2 Interfacial Temperature and Heat Penetration .................. 49

2.3.3 Temperature Evolution in Extrusion Flows ...................... 51

2.4 Crystallisation and Solid Properties .............................................. 52

2.4.1 Generalities of Polymer Crystallisation ............................ 52

2.4.2 Processing Effects ............................................................. 55

2.4.3 Orientation ...................................................................... 55

2.4.4 Viscoelastic Properties in the Solid State and in the Liquid-solid Transition Zone ........................................... 56

2.5 Conclusion .................................................................................... 59

3 Pipe Forming Tools .................................................................................. 63

3.1 Introduction .................................................................................. 63

3.2 Flow Through Pipe Dies ............................................................... 64

3.2.1 The Different Approaches from One-dimensional to Three-dimensional ........................................................... 64

3.2.2 One-dimensional Calculation ........................................... 65

3.2.3 Temperature Computations ............................................. 67

3.2.4 An Example of Nonaxisymmetric Flow ........................... 69

3.3 Pipe Calibration Experimental ................................................... 74

3.3.1 Technological Review ...................................................... 74

3.3.2 Objectives and Motivations ............................................. 76

3.3.3 Velocity ............................................................................ 76

3.3.4 Friction between the Pipe and the Calibrator ................... 81

3.3.5 Temperature Evolution .................................................... 83

3.3.6 Residual Stresses .............................................................. 84

3.4 Process-induced Microstructure and Properties ............................. 88

3.4.1 Orientation ...................................................................... 88

3.4.2 Crystallinity ..................................................................... 91

3.4.3 Surface State .................................................................... 93

3.4.4 Mechanical Properties ...................................................... 94

3.5 Modelling of Calibration .............................................................. 95

Contents

ix

3.5.1 Calculation of the Temperature Field ............................... 95

3.5.1.1 General Presentation ....................................... 95

3.5.1.2 Boundary Conditions ...................................... 95

3.5.1.3 Crystallisation ................................................. 96

3.5.1.4 Validation of the Model and Typical Results ... 97

3.5.2 Stress Development Model ............................................. 100

3.5.2.1 General Presentation ..................................... 100

3.5.2.2 Boundary Conditions .................................... 101

3.5.2.3 Thermoelastic Model .................................... 102

3.5.2.4 Viscoelastic Model ........................................ 103

3.5.2.5 Coupling with the Thermal Model ................ 104

3.5.3 Orientation Development .............................................. 105

3.5.4 Results of Residual Stress Calculations .......................... 107

3.6 Conclusion .................................................................................. 109

4 Flat Film and Sheet Dies ......................................................................... 113

4.1 Film Casting and Sheet Extrusion ............................................... 113

4.2 Flow Distribution and Channel Design ....................................... 114

4.3 Mathematical Modelling ............................................................. 120

4.4 Computer-assisted Flat Die Design .............................................. 125

4.5 Flat Die Coextrusion ................................................................... 130

4.6 Rheological Considerations ........................................................ 133

4.7 Mechanical and Other Construction Considerations................... 134

4.8 Concluding Remarks .................................................................. 136

5 Blown Film Dies ..................................................................................... 141

5.1 Introduction to Blown Film Extrusion ........................................ 141

5.2 Flow Distribution Considerations ............................................... 145

5.3 Mathematical Modelling ............................................................. 150

5.4 Computer-assisted Spiral Die Design .......................................... 154

xDesign of Extrusion Forming Tools

5.5 Multilayer Blown Film Extrusion ................................................ 158

5.6 Mechanical, Thermal, Gauge Control and Rheological Considerations ............................................................................ 163

5.7 Concluding Remarks .................................................................. 165

6 Profile Forming Tools ............................................................................. 169

6.1 Introduction ................................................................................ 169

6.2 Profile Extrusion Dies ................................................................. 170

6.2.1 Introduction ................................................................... 170

6.2.2 Profile Extrusion Die Constructive Solutions ................. 172

6.2.2.1 Plate Dies ...................................................... 172

6.2.2.2 Stepped Dies ................................................. 173

6.2.2.3 Streamlined Dies ........................................... 174

6.2.3 Die Design Tasks ............................................................ 175

6.2.4 Flow Balance in Extrusion Dies ..................................... 176

6.2.5 Automatic Optimisation ................................................ 180

6.2.6 Outline of the Numerical Procedure .............................. 183

6.2.7 Case Study ..................................................................... 185

6.2.7.1 Flow Channel Optimisation .......................... 186

6.2.7.2 Sensitivity Analysis ........................................ 193

6.2.8 Conclusion ..................................................................... 196

6.3 Calibration/Cooling Systems ....................................................... 198

6.3.1 Introduction ................................................................... 198

6.3.2 Design of Calibration/Cooling Systems .......................... 200

6.3.3 Outline of the Numerical Procedure .............................. 203

6.3.4 Case Study ..................................................................... 205

6.3.5 Conclusion ..................................................................... 211

7 Flexible Dies ........................................................................................... 221

7.1 Introduction ................................................................................ 221

7.1.1 Annular Dies .................................................................. 222

Contents

xi

7.1.2 Flex Ring Dies ............................................................... 226

7.1.2.1 Flex Ring Pipe Dies ....................................... 228

7.1.2.2 Flex Ring Throttle ......................................... 233

7.1.2.3 Flex Ring Blown Film Dies ............................ 234

7.1.2.4 Flex Ring Dies for the Production of Foamed Sheets and Films .............................. 237

7.1.2.5 Flex Ring Dies for Extrusion Blow Moulding 237

7.1.2.6 Flex Ring Profile Dies .................................... 242

7.1.3 Membrane Sheet Dies .................................................... 244

7.1.4 Flat Film Dies with Super Flexible Lips .......................... 246

7.1.5 Universal Slit Dies for Sheets and Films ......................... 247

7.1.6 Membrane Dies for Coextrusion .................................... 248

7.1.7 Membrane Feedblocks for Coextrusion ......................... 250

8 Rotating Mandrel Dies ........................................................................... 253

8.1 Introduction ................................................................................ 253

8.2 Rotation for Extrudate Homogenisation ..................................... 253

8.3 Rotation for Molecular Orientation/Morphology ....................... 256

8.4 Rotation for Producing Specific Product Shapes .......................... 264

8.4.1 Flexible Pipes ................................................................. 264

8.4.2 Nets ............................................................................... 265

8.4.3 Coaxial Helical Cables ................................................... 267

8.4.4 Encapsulation ................................................................ 267

8.4.5 Other Products .............................................................. 269

Abbreviations .................................................................................................... 275

Index .............................................................................................................. 279

xii


1Olga S. Carneiro and J. Miguel Nbrega

1.1 Introduction

The major objective of an extrusion line is to produce, at a high rate and quality, the required product [1]. These two goals are generally conflicting, since producing at higher rates generally results in lower quality products or, in other words, the extrusion rate required for a high quality product will limit the maximum production rate. Therefore, the optimisation of the processing conditions and of the design of extrusion tools (extrusion die and calibration/cooling system) demands a deep knowledge and careful study of all the phenomena involved during the extrusion process. The die and the calibration/cooling system are the extrusion line components or stages that play a central role in establishing the dimensions, morphology and properties of the final product, and these also generally limit the maximum allowable production rate [2].

The viscoelastic nature of polymer melts, in particular their elasticity, is responsible for some of the most important problems and/or defects that affect the extrusion process and product quality. Elasticity can lead to die drool (also called die build-up, plate out, die drip or die peel), sharkskin, stick-spurt and melt fracture. The minimisation or elimination of these defects demands a proper die design, together with the adoption of adequate processing conditions. In this chapter, these phenomena will be described and analysed with a view to defining and understanding a set of die design principles and to discussing the relevance of the main processing conditions which influence them.

Other relevant phenomena that must be taken into consideration during die design, not necessarily all related to the viscoelastic nature of the polymer melts, are the postextrusion phenomena such as extrudate-swell, stretching/draw-down promoted by the pulling system and shrinkage upon cooling. These will all have a significant influence on the cross-section dimensions and/or shape of the extrudate and, therefore, their effect must be anticipated at the design stage. Die flow balance, which is especially relevant in the case of nonaxisymmetric geometries, will also be addressed here.

1 Main Issues in the Design of Extrusion Tools

2Design of Extrusion Forming Tools

The complex behaviour of the polymer melt during flow through the die that has already been referred to, together with the expected slight variations of the operating conditions and/or polymer rheological properties, make it very difficult to produce the required melt extrudate cross-section with precise and stable dimensions. For this reason, in some cases (sheet, pipe, profiles) the calibration/cooling system is used to establish the final most relevant dimensions of the extrudate while cooling it down to a temperature that guarantees its shape along the downstream stages [2, 3]. Moreover, as the extrudate progresses along the line, it is subjected to a variety of external forces (e.g., friction, buoyancy and compression in the case of pipes and profiles). It is therefore necessary to guarantee that the extrudate is strong enough to withstand these forces without deforming [2, 4].

From a thermal point of view, the calibration/cooling system must also ensure fast rate uniform cooling of the extrudate in order to induce the adequate morphology and a reduced level of thermal residual stresses [5-8]. In practical terms, the temperature gradient along both the extrudate contour and its thickness must be minimised [7, 9-12] and its average temperature at the calibration/cooling system outlet must fall below the solidification temperature (TS), in order to avoid subsequent melting [3, 9]. This problem has particular relevance when relatively high and dissimilar thicknesses are present, as can be the case for profiles. Furthermore, to ensure that the extrudate will be properly handled, before the saw or the winding points, all cross section temperatures must fall below TS, as illustrated in Figure 1.1 for the case of pipes or profiles.

Calibration/Cooling Haul-off Extruder Saw

mTT sTT sTT sTT

Die

Figure 1.1 Evolution of the temperature of an extruded pipe or profile along the extrusion line, for the case where two calibrators are used in series (d thickness

of the solidified layer; T temperature at any point of the cross-section; Tm melting temperature; TS solidification temperature; T

average temperature

of the cross-section)

3Main Issues in the Design of Extrusion Tools

There are multiple parameters which are related to the design of cooling systems and calibrators. For each type of extruded product, these will be addressed in the corresponding chapter. For the case of extruded profiles, the main parameters and operating conditions that affect the efficiency (cooling rate) and uniformity (homogeneity of temperature) of the cooling stage are systematised in Figure 1.2. These will be addressed in this chapter and the effect of some of them will be illustrated in Chapter 6.

Coolingconditions Profilegeometry

System

geome

try

High cooling rate andtemperature homogeneity

Metalproperties

Polymerproperties

Vacuum

conditionsEx

trusio

n

cond

ition

s

Figure 1.2 Parameters affecting the design of calibration/cooling systems

1.2 Extrusion Dies

1.2.1 Rheological Defects

Melt flow instabilities may occur during the flow through the extrusion die, negatively affecting the quality of the extrudate and, eventually, giving rise to an unacceptable product. Some of the most common and limiting defects which occur in extrusion are sharkskin, stick-spurt, melt fracture and die drool. The three first are instantaneous defects, i.e., they occur if the critical value of one extrusion parameter (e.g., shear rate, shear stress or extensional stress) is attained, while die drool is a cumulative defect that develops over time. Defects related to polymer degradation can also be considered as cumulative. These problems are more difficult to handle than the instantaneous ones


since they develop over time and, therefore, depend on the duration of the extrusion run, and may possibly stay hidden during several hours of production.

The first part of this section will address the first set of defects, which are illustrated in Chapter 2 (Figure 2.7).

These defects have been classified and described in different ways by different authors, so it is difficult to systematise the results and conclusions that have been published in the countless scientific papers devoted to them. Due to this, some authors use the term sharkskin-melt fracture in order to avoid distinguishing between the two types of defect, which does not help to clarify the situation. However, in recent years, the use of more sophisticated characterisation techniques such as LDV (laser-Doppler velocimetry) and on-line measurements of flow-induced birefringence (FIB), together with numerical modelling of the flow, have provided a deep insight into the characterisation of the defects, helping to improve the systematisation of past and present studies.

Sharkskin is thought to occur at the die parallel zone or at the die exit. It is a surface defect that increases the extrudate roughness or, in a less severe case, inhibits its surface gloss [13, 14]. The main causes pointed out as the origin of this defect are: (a) the slip-stick phenomenon at the flow channel wall [14-16]; (b) the high normal stresses induced at the die exit, caused by the sudden acceleration of the melt outer layers [17-22]; and (c) coalescence of small voids promoted by negative pressures on the metal/polymer boundary or in the bulk, depending on the cohesion of both media [23].

According to recent work by Agassant and co-workers [24], the second hypothesis, which was already the most consensual, was strengthened. In fact, the studies of Merten [25], which involved the use of theoretical (flow modelling) and experimental (LDV and FIB) tools, clearly showed that the cause of sharkskin is the sudden extensional acceleration of the outer layer of the polymer melt upon leaving the die, promoted by the rearrangement of the velocity profile, in which the velocity passes from zero (if no slip at the wall is assumed) to the extrusion average velocity, as schematised in Figure 1.3.


Figure 1.3 Schematic representation of the rearrangement of the velocity profile at the die exit

Having in mind this mechanism for the onset of sharkskin, the lower the velocity difference between that of the outer layer and the average one, the lower the acceleration will be and, therefore, some slip at the wall of the die channel will minimise the severity of the defect. Slip at the wall will reduce the stress singularity but does not guarantee the elimination of the defect, as demonstrated in [26]. Despite being a defect promoted by a normal stress that exceeds the critical one that originates the rupture of the melt (which is a property of the polymer), since sharkskin is originated by the shear flow developed at the die parallel zone it is usual to employ a critical value of the shear stress at the die exit wall as a criterion for the onset of the defect [24, 27]. In practical terms, this critical value can be determined in capillary rheometry tests, and is taken as the value of the shear stress corresponding to the appearance of a rough surface/loss of gloss of the extrudate. For the majority of polymer melts, the value of the critical shear stress is approximately 0.1-0.2 MPa [28], and is independent of the melt temperature. This value may be considered as an indicative one, since the onset of sharkskin depends also on the aspect ratio of the die and on the shape of the exit region [28]. This indicative value was also confirmed for two biodegradable polymers in a recent study [29] that presented values of critical shear stresses of 0.14 and 0.18 MPa. When the value of the critical shear stress of a specific polymer is not known, one can, therefore, take the value of 0.1 MPa as the default. In simple analytical calculations, this value is commonly substituted by the corresponding shear rate which, obviously, depends on temperature. For design purposes, sharkskin has to be considered in the parallel (final) zone of the extrusion


die and will limit the maximum throughput. There are several methods to minimise, or to eliminate, this defect, namely:

Promotingmeltslipbytheadditionofanexternallubricant/processingaid[14]or by coating the die channel walls, prior to the die exit, with a low surface energy material, such as polytetrafluoroethylene [24]. Since coatings wear in a few extrusion hours, only the use of slip agents seems to be a plausible solution;

Increasingthemelttemperatureinordertodecreasetheshearstressor,equivalently,to increase the value of the critical shear rate;

Decreasingthethroughput;and

Modifyingthedieexitgeometrybyincludingacurvatureinordertoreducethestress concentration [26].

Melt fracture is a severe defect affecting the bulk extrudate [13], as can be seen in Figure 2.7 (Chapter 2). In a capillary rheometry test this defect can be detected by simple observation of the extrudate or by a decrease in pressure, which gives rise to an apparent lowering of the shear viscosity, as shown in Figure 2.6 (Chapter 2). As already mentioned, there is no clear agreement concerning the mechanism causing this defect and it is also believed that it can depend on the material and/or flow channel geometry [14]. The mechanisms usually referred to as being the cause of melt fracture are the following: (a) the slip-stick phenomena at the flow channel wall [15, 30-34]; and (b) fracture of the melt at a convergent flow channel region [17, 24, 35-40] due to the high extensional stresses experienced by the melt during the extensional flow developed.

According to Agassant and co-workers [24] it seems well-established that the second cause is the more plausible, since all the observations made using different techniques such as flow birefringence [41, 42], particle tracers [43-45], LDV [45, 46] and particle image velocimetry [47] showed that the onset of this defect happened upstream of the die, in the convergent flow zone, i.e., the defect can appear when the melt is subjected to high extensional deformations. Therefore, and for die design purposes, one should expect the criterion for the onset of melt fracture to be based on a critical normal stress or a critical extensional deformation rate value [27, 48, 49]. The critical normal stress is a property of the material, independent of temperature. As in the case of sharkskin, if a critical deformation rate is used, then the effect of temperature has to be considered. In practical terms, the critical value of the normal stress can be determined in capillary rheometry tests, using one of the available analyses of the convergent flow [50-52]. The critical normal stress will be that corresponding to the appearance of a gross melt distortion in the extrudate.


Melt fracture has to be considered in the design of the convergent zones of the extrusion die and will limit their maximum convergence angle. There are several analytical equations [27] that can be used to define the limits for the nonoccurrence of this defect (maximum extensional deformation rate, , or maximum convergence angle, a) in simple convergent channel geometries. These were deduced considering the following assumptions [27]: isothermal flow, incompressible fluid and, in each section of the channel, constant flow velocity (taken as the average velocity over the whole section):

(a) channel geometry: circular convergent (see Figure 1.4)

(1.1)

or

(1.2)

with

(1.3)

where:

critical extensional deformation rate

maximum shear rate (which occurs at the chanel outlet)

R1, R2 inlet and outlet channel radii, respectively

Q flow rate

n Power Law index


Figure 1.4 Convergent circular channel

(b) channel geometry: rectangular convergent with constant height H, inlet width W0 and outlet width W1 (see Figure 1.5).

(1.4)

or

(1.5)

with

(1.6)

These equations apply also, for example, to the channels defined between two adjacent spider legs.


Figure 1.5 Convergent rectangular channel

(c) channel geometry: annular convergent (see Figure 1.6).

; E (1.7)In this case, the convergent annular channel was substituted by a rectangular equivalent one, with outlet height H1 = R4-R3, and outlet width W1 = p (R3+R4), where R3 and R4 are the outlet inner and outer radii, respectively.

Figure 1.6 Schematic representation of a convergent annular channel

In a well-designed die, melt fracture should not occur. However, if present, there are several methods to minimise, or to eliminate, this defect, namely: (a) increasing the melt temperature, in order to increase the value of the critical extensional deformation rate [13, 14, 40, 53]; (b) decreasing the throughput; and (c) decreasing the convergence angle [13, 14, 40, 53].

10


As a rule of thumb, the convergence angle should be limited to a maximum of 30. The stick-spurt defect, when present, occurs for intermediate shear rates (see Figure 2.6 in Chapter 2) between those corresponding to the onset of sharkskin and melt fracture. During a capillary rheometry test, this defect promotes an oscillation of the pressure. In this case, the aspect of the extrudate alternates periodically between smooth and rough, as can be seen in Figure 2.7 (Chapter 2). The origin of this defect seems to be related to wall slip in the land zone of the die [24], which is favoured by high L (length)/D (diameter) values of the land zone. For die design purposes, this defect does not seem so relevant as the others since it occurs for shear rates higher than that corresponding to the onset of sharkskin which, in practice, should limit the extrusion throughput.

Another important issue is the die drool phenomenon, also called die build up, plate out, die drip or die peel. This defect can be present in the main polymer processing technologies, including calendering, injection moulding and extrusion [54]. In extrusion, the defect results from the accumulation of material in the surface of the die, at its exit, as illustrated in Figure 1.7.

Front view Side view Extrusion

Extrudate

Extrudate

Extrudate

time

Die

Die

Die

Figure 1.7 Schematic illustration of die drool formation, accumulation and eventual removal by the extrudate

11

Main Issues in the Design of Extrusion Tools

Die drool can affect not only the aesthetics of the extrudate but also its properties, thereby reducing its performance. For example, it can negatively influence the dielectric properties of coated wires or may be responsible for the stoppage of the weaving process if present in a fibre [54]. When present, the only way to remove the excess material from the die surface is to stop the extrusion line and clean the die. This is obviously, an expensive solution that should be avoided. The objective is, therefore, to eliminate or minimise the defect via a proper die design and/or by the adoption of the most favourable processing conditions.

Several factors have been suggested in the open literature as sources of this defect, namely [55]: (a) low molecular weight polymer species; (b) volatiles, including moisture; (c) the presence of a filler; (d) poor dispersion of pigments; (e) draw down rates; (f) the amount and rate of extrudate-swell; (g) die exit angles, land length and land entrance size; (h) dissimilar component viscosities; (i) die condition (including cleanliness, presence of damage, defects, etc.); (j) pressure fluctuations in screw channel; and (k) inadequate melt temperature. Some of these sources are still under debate and are controversial. For example, some authors refer to the swelling of the melt after leaving the die as a possible cause for die drool [56] but it is well-known that the incorporation of solid fillers reduces melt elasticity [57] and, on the other hand, highly filled formulations are known to be more prone to induce this defect.

Little has been published on this subject in the open literature. Presently, the most active group in this area is that of Zatloukal and co-workers, whose experimental and modelling work helped to improve the characterisation and understanding of the die drool phenomenon [58-60]. The main conclusions of the research on die drool are the following [58]:

There are two types of die drool: external and internal. The external type occurs due to the negative pressure developed at the die exit. For die design purposes, it is therefore crucial to understand the conditions that lead to the development of negative pressures in order to avoid their occurrence or to minimise their absolute value. According to Zatloukal [59], based on the results of numerical modelling of the flow, the negative pressure is a consequence of the elasticity of the melt and streamline curvature (promoted by the velocity profile rearrangement that occurs at the die exit). This leads to the generation of normal stress that causes a local pressure decrease. Since this decrease in pressure occurs in the vicinity of the nil pressure zone (at the die exit), it can result in a negative value. The resultant suction effect, together with the extrudate free surface rupture (sharkskin), may promote the adhesion and accumulation of material at the external surface of the die. Also, the negative pressure may promote the migration of low molecular weight components or fillers, when present, and their accumulation at the die exit surface. According to the same authors, and based on results of numerical modelling of the flow, the suppression of

12


this type of die drool can be achieved by modifying the die exit angle, a solution that was also proposed in previous patents [61, 62]. This hypothesis was also supported by Ding and co-workers [63], who showed, through numerical modelling that the suppression of the die drool promoted by flared dies was related to the maximisation of the undershoot of the first normal stress difference, N1, occurring at the die exit region. They argued that the important parameter is the magnitude of the undershoot of N1, and not its absolute value, at the die exit. In practice, one needs to select the upstream gap (h1) that maximises the N1 undershoot at the die wall, and to select a flared length (L2) which is long enough to stabilise the value of N1 near the die exit (see Figure 1.8, where h is the gap, L the length and subscripts 1 and 2 refer to upstream and flared zones, respectively).

h1 h2

L2L1

Flow

Figure 1.8 Flared die. Adapted from F. Ding, L. Zhao, A.J. Giacomin and J.S. Gander, Polymer Engineering and Science, 2000, 40, 10, 2113 [63]

The internal type of die drool can be caused by the molecular weight fractionation, induced by the flow before the die exit, which causes the accumulation of the lower molecular weight fractions on the die wall surface. According to Ohhata and co-workers [61] the ratios D2/D1 = 1.1 - 2.0 and L2/D2 = 0.6 - 1.6 (see Figure 1.9, where D and L are diameter and length dimensions of different zones of the channel) should be adopted to supress the die drool, whereas Rakestraw and Waggoner [62] recommend values of D3/D1 = 1.15 - 1.2, L1/L3 = 2 - 10 and q = 45 to 90 (see Figure 1.10, where D and L are diameter and length dimensions of different zones of the channel and q is the divergent angle).

13


Flow D1

L1

D2

L2

Figure 1.9 Flared die and associated geometrical parameters. Adapted from T. Ohhata, H. Tasaki, T. Yamagushi, M. Shiina, M. Fukuda and H. Ikeshita,

inventors; General Electric Company, assignee; US 5,417,907, 1995 [61]

Flow

L1

T

D1 D3

L3

Figure 1.10 Flared die and associated geometrical parameters. Adapted from J. Rakestraw and M. Waggoner, inventors; E.I. Du Pont de Nemours and

Company, assignee; US 5,458,836, 1995 [62]

There are several other recommendations and findings, published in the literature, which are intended to diminish or supress the die drool defect, and which become more or less obvious after the explanations given. In summary, the objective is to minimise the elasticity of the melt, or to minimise the tendency for the migration of low molecular weight fractions or solid particles to the surface of the flow channel. This can be attained through the adoption of one, or several, of the following actions:

Avoidhighextrusiontemperatures.

In the extrusion of high density polyethylene (HDPE), the degradation of which causes an increase in the average molecular weight, high extrusion temperatures can lead to an increased tendency for the onset of die drool [58].

14


Limitthevalueoftheflowrate.

The higher the mass flow rate, the higher the die drool weight, for a constant extrusion time [58].

Promotewallslipusinglubricants[64]orbycoatingthediesurfacewithalowsurface energy material [65, 66].

1.2.2 Postextrusion Phenomena

In addition to the eventual shape changes occurring along the die flow channel and at the calibration/cooling system, other changes taking place along the extrusion line must also be considered [67-69], namely:

Extrudate-swell:Duetothevelocityprofilerearrangementandtotheelasticityof the polymer melt, the extrudate cross-section dimensions increase and, for the case of nonaxisymmetric cross-sections, it can also be heavily distorted [2, 70]. This effect decreases with increasing parallel zone length due to the relaxation of upstream deformations, until a minimum limit value is reached [71-75].

Draw-down:Sincetheprofileispulledbythehaul-offunitwithavelocityhigherthan the average melt velocity at the die flow channel, its cross-section is stretched in the vicinity of the die exit [67], where the material temperature is higher.

Shrinkage:This effect is promotedby the decrease of the specific volumeofthe material which occurs during the cooling down process from the extrusion temperature to the storage one; therefore, semicrystalline polymers will shrink to a greater extent than amorphous ones.

The geometry and dimensions of the cross-section of the parallel zone of the extrusion die should anticipate these postextrusion effects through the adequate corresponding corrections.

1.2.2.1 Extrudate-swell

As already mentioned, extrudate-swell is promoted both by the velocity profile rearrangement at the die exit and by the elastic character of the melt. The value of the first component is, therefore, independent of the viscoelastic characteristics of the melt, being approximately 13% for circular channels and approximately 19% for rectangular ones [76]. This component has to be added to the elastic one which, in a simple analysis, can be predicted by one of several semi-empirical equations available to describe the dependence of the extrudate-swell on the properties of the melt at the

15


extrusion temperature. One of the most popular models used in the simulation of the global extrusion process is that of Tanner [77]. This is recognised as being based on adequate physical assumptions [53, 76-79], namely, isothermal flow, incompressible melt and high ratio between the length and the gap (L/D) of the die parallel zone (i.e., assumption of fully developed flow). The main limitations concern its inadequacy to deal with short dies and high deformations. The corresponding equations for circular (Equations 1.8 and 1.9) and rectangular channels/flow between parallel plates (Equations 1.10 and 1.11) are the following:

BDD N1

81

w0

12 1 6

x= = + ` j8 B (1.8)

or B2 1r2 6c = -^ h (1.9)

and

BHH N1

121

w0

12 1 4

x= = + ` j8 B (1.10)

or B3 1r2 4c = -^ h (1.11)

where:

B extrudate-swell

D0, H0 diameter or thickness of the circular or rectangular die, respectively

D, H diameter or thickness of the circular or rectangular extrudate, respectively

N1 first normal stress difference

16


t - shear stress

gr recoverable shear deformation

w indicating that the value is taken at the wall of the channel

The use of this model requires the characterisation of the shear elasticity of the melt through N1 or gr. Besides the existing direct experimental methods available to characterise these properties (through, for example, the use of rotational rheometers), some indirect methods have been proposed to estimate the elastic properties from data collected in capillary rheometry experiments, namely those based on the pressure at the exit of the capillary [80], on the hole pressure [81], on the entrance effects [36] and on the shear flow curve. Concerning the last approach, several analyses have been developed by Gleissle [82], Bird [83, 84] and Wagner [85]. Comparing values for direct measurements of extrudate-swell with the predictions resulting from these different analyses, it was shown [27] that the most accurate indirect method, based on the shear flow viscosity, to predict N1 was that based on the Wagner analysis [85], which has the form:

N

n d

d11 2

1} c

c

c

c

ch= =-:

:

:

:

:^ ^ ^h h h (1.12)

where:

Y1 first normal stress difference coefficient

n material parameter (typically, 0.13 n 0.20 for polymer melts)

h shear viscosity

c: - shear rate

To anticipate the effect of the extrudate-swell in the design of an extrusion die, the die gap has to be corrected with the value of the swelling at the corresponding shear

rate DBD

0 =` j. If the flow rate is maintained the resulting decrease in the dimension of the gap will promote an increase in the shear rate and in the value of the extrudate-swell. Consequently, this corrective process must be iterative.

17


1.2.2.2 Draw-down

The stretching promoted by the haul-off system originates a decrease in the dimensions of the cross-section of the extrudate, contributing also to increase the level of molecular orientation in the extrusion direction. The draw-down ratio (DDR), or take-up ratio, is defined as the ratio of the linear velocity of the haul-off system (or linear extrusion velocity, v) and the velocity at which the extrudate emerges from the die (v0):

DDRvv0

= (1.13)

In the extrusion of pipes and profiles, this ratio should be maintained as low as possible, in order to reduce the level of residual stresses frozen in the extrudate. Typical values are shown in Table 1.1.

Table 1.1 Typical draw-down ratios used in the extrusion of pipes and profiles

Material Draw-down ratio

HDPE 1.15

Low density polyethylene (LDPE) 1.60

Polyurethane 1.3-2.0

Plasticised polyvinyl chloride (P-PVC) 1.15-2.0

Unplasticised-PVC (U-PVC) 1.15

In other extrusion processes (such as in the production of fibres, filaments or films) the DDR value can be much higher in order to enable the production of thin products, such as films, at high rates (which would be impossible with very small gap dies), to deliberately induce a high degree of molecular orientation of the polymer to improve its performance (mechanical, optical and/or barrier) in the orientation direction(s), or to get shrink products (as in the case of films, for example).

18


Using a simple mass balance approach, the effect of draw-down ratio on the diameter of the die (D0) will be given by:

.D D DDR0 2= (1.14)

where:

D is the diameter required for the extrudate.

1.2.2.3 Shrinkage

Shrinkage will also cause a decrease of the cross-section dimensions of the extrudate. Its extent depends on the difference in the density of the material at the extrusion and at room temperatures. Supposing that during cooling the material can shrink freely (i.e., the process is not constrained by the solidified material), the correction of any characteristic dimension of the cross-section of the die (D0) can be done through:

(1.15)

where:

rm, r density at melt temperature or room temperature, respectively

Vm, V specific volume at melt temperature or room temperature, respectively

1.2.3 Flow Balance

Using the simplest definition, it can be said that in a balanced die the flow will be distributed in such a way that will originate a uniform melt velocity of the polymer melt all over the die exit contour. This feature can be easily guaranted for axisymmetric dies, i.e., for the dies where all the possible paths of the melt are similar and, therefore, have similar flow restriction. Unfortunately, this only happens for rod dies and pipe mandrel dies with spider legs, if they are in-line with the extruder. In any other case (for example, annular dies for blow moulding, spiral mandrel pipe dies, wire coating

19


dies, sheet dies and profile dies), there are different paths defined for the melt along the extrusion die and, as a consequence, the issue of flow balance has to be taken into account during the design stage. The solutions to the problems associated with this issue have been systematised for some types of geometries, namely, sheet and spiral mandrel dies, and will be addressed in the appropriate chapters.

The problem of flow balance is particularly difficult to solve, or to generalise, for the case of profile dies since they can present an infinite variety of geometries and may comprise walls of different thicknesses. Extruded profiles will be addressed in more detail in Chapter 6 and illustrated with a case study using a numerical flow modelling code. Here, the objective is to highlight the relevance of flow balance and to provide a simple analytical process, useful for those that do not have access to sophisticated numerical tools, which can help in the design of a new die.

The simplest and most intuitive technique to balance the flow is that based on the control of the parallel zone length. In this case, a balanced die should promote a similar velocity of the melt along the entire die exit contour. For this purpose, the methodology to be adopted is that described in Figure 1.11:

Step1:Divisionofthecross-sectionoftheparallelzoneofthedieinelementalsections (ES). A different ES shall be considered whenever there is a change in geometry and/or in thickness.

Step2:Determinationoftherelativeflowrate(Qi) required in each ES. Bearing in mind the requirement of similar average melt velocity in all the ES, Qi = Q(Ai/A), where Ai is the cross-section area of each ES, Q is the total flow rate, which is still unknown, and A is the total cross-section area of the parallel zone.

Step3:IdentificationofthecriticalES.Thissectionwillbethatwherethehighershear rate will occur, when the required relative flow rates are used.

Step4:DeterminationofthemaximumflowrateinthecriticalES.Thiswillbeafunction of its geometry and of the critical shear rate of the melt at the extrusion temperature.

Step5:Determinationoftheflowrates(Qi) required in the remaining ES. Bearing in mind the relationships between the total flow rate (Q) and that required in each ES (Qi), as determined in Step 2, the flow rate in each ES can then be computed.

Step6:Determinationoftherelativelengths(Li) of the parallel zone of each ES. The only way to force the required flow rates to happen is by imposing similar pressure drops for each of the paths of the melt at the corresponding flow rates. Therefore, using the cross-section geometry of each ES and its flow rate (Qi), a relationship for their relative lengths will be obtained. If flow separators (walls)

20


between the ES are not used, this will be a rough solution since the equations available to compute the pressure drops are adequate for isolated channels (closed geometries) but not for those sharing one or various faces/segments with neighbouring ES.

Step7:DeterminationofthelengthsofeachES.ThelengthoftheshortestESmust be fixed. In the previous step, the relative lengths were determined. Now, having established one of them, the others can be computed.

1

2

3

4

5

6

7

Figure 1.11 Methodology for the analytical determination of the lengths of each elemental section (ES) leading to flow balance

Table 1.2 shows the equations needed to compute the shear rate and pressure drop for some simple geometry channels.

21


Tab

le 1

.2 E

quat

ions

to

com

pute

she

ar r

ate

and

pres

sure

dro

p in

sim

ple

geom

etry

cha

nnel

s, u

sing

the

pow

er la

w

to d

escr

ibe

the

shea

r vi

scos

ity

of t

he p

olym

er

Geo

met

rySh

ear

rate

Pres

sure

dro

p

Cir

cula

r

nnRQ

41

34

3

+=

pg

(1

.16)

(1.1

7)

Para

llel

plat

es

(1

.18)

(1.1

9)

Ann

ular

(1

.20)

(1.2

1)

Cir

cula

r C

onve

rgen

t

(

1.22

)

Not

es:

subs

crip

t o

sta

nds

for

oute

r or

out

let;

sub

scri

pt i

sta

nds

for

inne

r or

inle

t.G

eom

etry

: R

rad

ius,

L

leng

th; H

h

eigh

t; W

w

idth

; Fp

co

rrec

tion

fac

tor.

Rhe

olog

ical

pro

pert

ies:

n

pow

er la

w in

dex;

K

con

sist

ency

inde

x.Pr

oces

sing

par

amet

ers:

Q

flow

rat

e;

- s

hear

rat

e.

22


The problem with this simple analytical methodology is that it will most probably not provide an adequate solution after the first trial, due to the development of lateral flow between adjacent ES. To illustrate this issue, consider the plastic profile and the corresponding three ES defined, illustrated in Figure 1.12a, and suppose that after applying the methodology described in Figure 1.11, the lengths that resulted for the three ES are L1, L2 and L3, with L1 > L2 > L3. The pressure evolution along the length of each ES of the parallel zone of the extrusion die is schematised in Figure 1.12b.

(a) (b)

ES1 ES2 ES3 Pres

sure

L1 L2

L3

Extrusion direction

Lateral flow ES2-ES3

Lateral flow ES1-ES2

Length

ES3ES2ES1

Figure 1.12 Flow balance: (a) cross-section of the plastic profile (or of the die land) and ES considered; and (b) evolution of pressure in each ES of the parallel zone

As referred to in Step 6 of the methodology, if no separators are being used, lateral flow (cross flow) will develop from areas of higher pressure to areas of lower pressure at any axial location of the parallel zone, as illustrated in Figure 1.12b, and so it will alter the desired balance. In practice, this will involve a trial-and-error process of adjustment of the lengths that can be very expensive (in terms of human and equipment resources and raw material). The use of flow separators would simplify the solution (in this case, the first trial solution will be a good guess) since lateral flow is absent, but it can have a high negative impact on the mechanical strength of the plastic profile.

1.3 Calibration/Cooling Systems

1.3.1 Types of Calibration/Cooling

The type of cooling used to cool down the extrudate depends on the type of product

23


or extrusion line. For blown films, forced air convection air rings are used and for flat film or sheet, the heat is removed by conduction (through contact of the extrudate with cooled rolls or calenders). For the majority of the remaining products, cooling is promoted by direct contact with water or a refrigerated calibrator. In any case, however, the problem to be solved is a steady heat conduction problem, associated with different boundary conditions (related to the type of cooling process employed: conduction and/or convection). Calibration is only used for sheets (via the gap defined between a pair, or a series, of rolls) and hollow extrudates such as pipes or other complex profiles.

For hollow geometries, calibration may be performed by internal pressure or external vacuum and cooling can be wet and/or dry [86, 87], as illustrated in Figures 1.13 and 1.14. In dry cooling there is no contact between the hot profile and the cooling medium, the heat being removed through the contact between the calibrator surface and the material. However, in wet cooling at least part of the heat is removed directly by the cooling fluid, the remainder being removed through the contact with the calibrator [2].

Extrusion die

Extrusion die

Vacuum

Water inlet

Water outlet

Compressed air

Perforated sleeve

Water

Plug

Figure 1.13 Pressure and vacuum calibration in pipe extrusion

24


Figure 1.14 Dry calibration in profile extrusion

In order to assure integrity of the extrudate, the layer of cooled material must be thick enough along the entire cooling line to withstand the forces required to pull it [3]. Since higher production rates involve higher pulling forces, due to increased friction, this requirement may limit the maximum production rate, especially for dry cooling systems [88], where higher friction forces are generated (see also Chapter 3).

1.3.2 Main Parameters

The parameters which are expected to have some influence on the thermal performance of the calibration/cooling system may be grouped as follows:

Systemgeometrynumberofcalibration/coolingunits(watertanks,airrings,rolls, calibrators), their effective cooling length and distance separating them. It is advantageous to use several cooling units in series instead of a single unit with the same total effective length. This will be illustrated in Chapter 6, for the case of profile cooling with calibrators.

Coolingconditionstemperatureandflowrateofthecoolingfluid.Lowcoolingfluid temperatures increase the cooling efficiency but also the thermal gradients within the extrudate.

Extrusionconditionsmassflowrate,extrusiontemperatureandextrudatecross-section temperature field at the die exit. The first will be one of the most relevant

25


factors determining the cooling length required for the system. The cross-section extrudate temperature should be as uniform as possible if thermal residual stresses are to be minimised [7, 8-12].

Polymerthermophysicalpropertiesthermaldiffusioncoefficientandshrinkage.In order to increase the cooling efficiency, the progressive decrease in extrudate dimensions should be matched by those of the calibration/cooling system, in the case of flat film/sheet and profiles [2, 89, 90].

In practice, the objective is to determine the cooling time (or residence time of the extrudate in the cooling system) in order to define the required effective cooling length (for a given extrusion rate), or to define the maximum extrusion rate (for a given cooling length). Calculation of the cooling time can be done using different types of approaches of increasing complexity, namely graphical, analytical or numerical, respectively. For the analytical case, this problem can be considered a one-dimensional transient heat transfer one, assuming constant material thermophysical properties, perfect thermal contact at the interface between the material and the cooling medium and an initial uniform polymer temperature. Considering the model illustrated in Figure 1.15, where D is the location of the cooling surface (and the thickness of the polymer slab which must be cooled down) and the simplifications above, the solution of the energy equation results in one of the equations solved with respect to the dimensionless temperature (Equation 1.23), q, or dimensionless time (Equation 1.24), F0 [2], respectively, at any location x (see Figure 1.15):

(1.23)

or

(1.24)

where:

(degree of cooling) (1.25)

26


(to be determined iteratively) (1.26)

(Biot number) (1.27)

(Fourier number) (1.28)

t time

D thickness of the slab (and, also, location of the cooling surface)

(polymer thermal diffusivity) (1.29)

with

FT - cooling fluid temperature

MT - initial extrudate temperature

h - heat transfer coefficient

k , r , - polymer thermal conductivity, specific mass and specific heat capacity, respectively.

27


!

x

D

T

Figure 1.15 Model for the cooling of a plastic slab that is in contact with the cooling medium at x = D: d thickness of the solidified layer; D total thickness of the plastic layer. Adapted from W. Michaeli, Extrusion Dies for Plastics and Rubber: Design and Engineering Computations, 2nd Edition, Hanser Publishers,

Munich, Germany, 1992 [2]

Having in mind the model described in Figure 1.15, Equation 1.24 can be used to determine, for example:

Theresidencetimeinthefirstcooling/calibrationunitrequiredtoavoidfurtherdistortions of the extrudate in the subsequent stages. In this case the solidification temperature of the polymer (Ts) will be imposed at dDx = where d is the thickness of the solidified layer which is capable of promoting adequate strength (see Figure 1.1, for the case of pipe or profile extrusion).

The residence time required in the global cooling/calibration system. Thisrequirement, together with the linear extrusion velocity, will define the effective cooling length required (which can be, depending the type of product being extruded, the total length of the calibrator(s), the total length of the water bath(s), the height of the blown film extrusion line or the contact length with the cooling rolls). In this case, Ts can be imposed at x = 0, i.e., the entire thickness will be solidified. It should be noted that this requirement is overstated since it will be enough to reach an average temperature lower than Ts. To avoid this overestimate of the cooling time, one can determine the temperature profile along the extrudate thickness through Equation 1.23, for different residence times, until the condition required for the average temperature is satisfied.

28


In Tables 1.3 and 1.4 typical values of the heat transfer coefficient and thermal properties of polymers commonly used in extrusion, respectively, are listed.

Table 1.3 Typical values of the heat transfer coefficient corresponding to different cooling conditions

Condition Heat transfer coefficient, h (W/m2K)

Natural convection (air) 5

Forced convection (air) 50

Water bath 300

Water spray 1500

Calibrator (dry cooling) 300-1000

Table 1.4 Typical thermal properties of some common polymers

Polymer Thermal conductivity, k (W/mK)

Density, r(kg/m3)

Specific heat capacity,

cp (kJ/kgK)

Acrylonitrile-butadiene-styrene

0.19-0.34 1010-1040 1.25-1.67

HDPE 0.46-0.52 940-960 2.3

LDPE 0.33-0.35 910-930 2.3

Polypropylene 0.12 900-910 1.4

U-PVC 0.13-0.29 1300-1450 0.84-1.25

The use of Equations 1.23 and 1.24 is illustrated in Figures 1.16 and 1.17, respectively. A HDPE plate of 2 mm thickness, cooled at x = 2 mm (i.e., D = 2 mm), was studied as an example. The data used in the calculations are listed in Table 1.5.

As it can be seen in Figure 1.16, after a short cooling time (i.e., 3 s), the temperature of the surface exposed to the cooling media decreases significantly (from 200 to approximately 128 C) while the opposed surface is almost unaffected (here the

29


temperature only decreases by approximately 2 C). On the other hand, after a cooling time of 100 s, the temperature of the plate is almost uniform, varying between 21 C (exposed surface) and 22 C (opposed surface).

Table 1.5 Data considered to compute the results shown in Figures 1.16 and 1.17

Heat transfer coefficient (water), h 300 W/m2K

HDPE thermal diffusivity, a 2.20 x 10-7 m2/s

HDPE thermal conductivity, k 0.5 W/mK

Initial plate temperature, TM 200 C

Cooling fluid temperature, TF 20 C

b (computed with Equation 1.26) 0.9178 rad

Figure 1.16 Temperature profiles for a HDPE plate cooled at x = 2 mm (D = 2 mm), at different cooling times

30


In Figure 1.17 the same type of information is shown in a different way. If the solidification temperature of the polymer, Ts, is known, it can be used to determine the time required to have a specific thickness of the plate sufficiently cooled (solid). In this case, it will be enough to compute the time required to attain the solidification temperature (70 C in the case of HDPE, for example) at x = D - d.

Figure 1.17 Time required to attain different specific temperatures in a HDPE plate cooled at x = 2 mm (D = 2 mm)

1.4 Conclusion

Despite the complexity of the phenomena involved in the design of extrusion tools, which can be addressed through the use of sophisticated numerical modelling codes, when available, there are some simple analytical approaches that enable the designer to obtain a first trial solution. In the subsequent experimental trial-and-error process needed to fine-tune the design, the better the knowledge of the designer the lower will be the number of required iterations until a final solution is reached. A more efficient methodology will consist of performing the trial-and-error process with simulations done by numerical modelling codes, which should save human and material resources.

31


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37

Bruno Vergnes, Michel Vincent, Jean-Marc Haudin

2.1 Introduction

In order to design extrusion forming tools, it is necessary to understand and to be able to control the polymer properties not only in the molten state, but also during the post-extrusion stages, i.e., cooling, crystallisation and, finally, in the solid state. In the present chapter, we will first present the rheological and thermal properties in the molten state, then the crystallisation kinetics and viscoelastic behaviour in the solid state.

First of all, we have to recall that polymers have specific properties, which have important consequences on their processing conditions. For example, molten polymers are highly viscous fluids, with a viscosity usually in the range 100-100,000 Pa.s. As a consequence:

(a) the Reynolds number Re, ratio of inertia to viscous forces, is low. For a flow in a pipe of diameter D, Re is defined as:

VDRe rh

= (2.1)

where r is the melt density, V the average velocity and h the viscosity. By considering classical values in polymer processing (r = 103 kg/m3, V = 10 cm/s, D = 1 cm, h = 103 Pa.s), we obtain a value of 10-3. It means that molten polymer flows are almost always laminar and that inertia forces are negligible;

(b) the gravity forces can also be neglected, at least in processes where the difference in height is less than one metre, typically horizontal processes like extrusion or injection; and

2 Properties of Polymers

38


(c) viscous dissipation in a die flow may be important. The dissipated power by unit volume W can be expressed as:

(2.2)

where is the shear rate. For a shear rate between 10 and 100 s-1, it leads to values between 105 and 107 W/m3. Consequently, the temperature may be difficult to control and temperature heterogeneities may occur across the flow.

As we will see more in more detail in Section 2.2, molten polymers are non-Newtonian fluids. The viscosity is a function of the rate of deformation and the elasticity may also play a significant role, especially in elongational situations and in free surface flows. Moreover, flow instabilities may appear above critical flow conditions, limiting the processing rates (see Section 2.2.5).

Another key point is that viscosity is highly dependent on temperature. Typically, an increase of 20 C may divide the viscosity by a factor 2. This induces a strong coupling between mechanical and thermal flow parameters: when the polymer is flowing, its temperature changes, thus its viscosity changes, and this modifies the flow conditions.

Finally, polymers are also good thermal insulators, with low values of thermal diffusivity and conductivity. For example, thermal conductivity is around 0.2 W.m-1.C-1 for a polymer, while it is around 200 W.m-1.C-1 for copper. This will induce long heating and cooling times. Typically, a sheet 2 mm thick can be cooled in 10 s, when it would take 17 min for a thickness of 2 cm. It is why most produced parts are usually thin (less than 5 mm). Moreover, as the thermal diffusivity is low, the viscous dissipation occurring in the sheared zones cannot diffuse rapidly through the material, so this leads to temperature heterogeneities across the flow.

This short introduction shows that molten polymers are very specific fluids, which require appropriate processing conditions and optimised forming tools, as will be explained in detail in the following chapters.

39

Properties of Polymers

2.2 Rheological Properties in the Molten State

2.2.1 Viscous Behaviour

The viscosity is defined as the ratio between the stress and the rate of deformation. It is expressed in Pa.s. In simple shear, it is the ratio between the shear stress t and the shear rate :

(2.3)

but we can also define the elongational viscosity, Eh , for example in uniaxial extension:

(2.4)

where s is a tensile stress and an extension rate.

In the present chapter, we will focus on the shear viscosity, which is the most relevant for the flow in extrusion dies. The simplest behaviour is the Newtonian one, in which the viscosity is constant. Then, the shear stress is proportional to the shear rate:

(2.5)

However, as said before, molten polymers are non-Newtonian. As shown in Figure 2.1, a polymer is generally Newtonian at low shear rate (Newtonian plateau) but above a certain value, the viscosity decreases with the shear rate. This is called shear thinning behaviour.

40


106

105

104

Vis

cosi

ty (

Pa.s

)Newtonian plateau

n - 1

1/

Power law

103

102

104 103 102 101

Shear rate (s1)

100 101 102 103

Figure 2.1 Viscosity curve of a molten polymer

Different laws can be used to describe this particular behaviour. The simplest one is the power law (also called the Ostwaldde Waele relationship [1, 2]), written here in simple shear:

(2.6)

where K is the consistency (expressed in Pa.sn) and n the power law index. As shown in Figure 2.1, this law only describes the shear thinning part of the viscosity curve, n 1 being the slope of this curve in log-log scale. The main interest of the power law is to allow the flow conditions for simple geometries to be calculated analytically. For example, the flow in a pipe of diameter D can be described by the following equations:

Velocity profile:

1 11/1 2( ) 11 2 2

n nnn nn p D rw r

n K L D

+ + D = + (2.7)

41


Flow rate:

1 31/13 1 2 2

nnnn p DQ

n K L

+

D = + (2.8)

where r is the radial coordinate, Dp the pressure drop and L the pipe length.

However, a much better description of the viscous behaviour on the whole range of shear rates is obtained by using a Carreau (or Carreau-Yasuda) law [3, 4]:

(2.9)

where h0 is the zero-shear viscosity (Newtonian plateau), l a characteristic time and a a parameter describing the transition between the Newtonian plateau and the shear thinning region; n has the same meaning as in the power law. As shown in Figure 2.1, 1/l is approximately the shear rate corresponding to the intercept between the Newtonian plateau and the power law. From Equation 2.9, it can be seen that, at low

shear rate, the viscosity tends to h0 while, at high shear rate, it tends to 1

0 ( )nh h lg = ,

i.e., a power law. Consequently, the same equation can be used to account for the whole viscosity curve and is thus much more accurate than a simple power law. Unfortunately, it is not possible to derive analytical solutions for the Carreau law, even in a simple case such as flow through a pipe. Consequently, this type of law can only be used when a numerical approach is envisaged.

Other rheological laws, such as the Cross law, can be found in the literature. We refer the reader to specialised books for more details [5-7].

2.2.2 Viscoelastic Behaviour

Besides its viscosity, a molten polymer is also characterised by a certain level of elasticity. Some experimental facts demonstrate this elastic character, for example, the Weissenberg effect shown in Figure 2.2 or the extrudate swell at the die exit, shown in Figure 2.3.

42


Figure 2.2 Weissenberg effect: the polymer solution climbs along the rotating drill. Reproduced with permission from J. Bico, R. Welsh and G. McKinley, Non-

Newtonian Fluids Lab, MIT

Figure 2.3 Extrudate swell: the polymer solution exiting the die shows a larger diameter which increases with flow rate. Reproduced with permission from C.

MacMinn and G. McKinley, Non-Newtonian Fluids Lab, MIT

43


It is obviously much more difficult to take into account this viscoelastic behaviour, and the rheological models existing in the literature are numerous and frequently difficult to handle. Once again, we refer to specialised books for more information [8, 9] and we will limit our description to the simplest Maxwell model. As shown in Figure 2.4, in a one-dimensional (1D) approach, the Maxwell model can be seen as the association of a spring (elastic part) and a dashpot (viscous part) in series; s is the applied stress, e1 and e2 the deformations of the spring and the dashpot, respectively, G is the elastic modulus.

Figure 2.4 Maxwell model describing a viscoelastic behaviour

By considering that the total strain e is the sum of e1 and e2, we can derive the expression for the Maxwell law:

d ddt dts e

s l h+ = (2.10)

where l = h/G is a relaxation time.

Despite its simplicity, Equation 2.10 describes the stress development and relaxation in shear flows, and provides a qualitative explanation of phenomena such as extrudate swell, the Weissenberg effect and the existence of normal stresses. However, it is insufficient to provide realistic results. Indeed, a polydisperse polymer has a distribution of relaxation times, but not a unique one. Moreover, physical phenomena at the chain level, such as chain extension and orientation, must be accounted for in order to be closer to reality. Unfortunately, calculations with such constitutive equations remain a challenge, except for very simple geometries. By chance, elastic effects are mainly dominant in elongational flows and in free surface flows, which is not the case for the flows in extrusion dies, where the polymer is usually confined in a geometry with only smooth changes in successive cross-sections.

44


2.2.3 Dependence on Temperature and Pressure

The viscosity of a molten polymer is highly dependent on temperature. For example, Figure 2.5a shows the viscosity curves for a polystyrene (PS) at different temperatures between 160 and 220 C.

(a)

Vis

cosi

ty (

Pa.s

)

Shear rate (s1)

(b)

Red

uced

vis

cosi

ty (

Pa.s

)

Reduced shear rate (s1)

Figure 2.5 a) Viscosity curves of a polystyrene at different temperatures: ( ) 160 C, (n) 180 C, () 200 C, (l) 220 C; and (b) Mastercurve obtained by

time-temperature superposition of the data of Figure 2.5a at 180 C

45


It can be shown that usual polymers obey a time-temperature superposition principle, in which the viscosity at any temperature T can be deduced from the viscosity at a reference temperature T0 by using a

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Design of Extrusion Forming Tools