Design of Extrusion Forming Tools

Design of Extrusion Forming Tools

Edited by Olga S. Carneiro and J. Miguel Nóbrega

Design of Extrusion Forming Tools Editors: Olga S. Carneiro and J. Miguel Nóbrega

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

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©2012, Smithers Rapra Technology Ltd

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ISBN: 978-1-84735-517-1 (hardback) 978-1-84735-518-8 (softback) 978-1-84735-519-5 (ebook)

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P

reface

Olga S. Carneiro and J. Miguel Nóbrega 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

iii

Design of Extrusion Forming Tools 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).

iv

C

ontributors

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

José A. Covas Department of Polymer Engineering, University of Minho, Campus de Azurem, Guimaraes, 4800-058, Portugal

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

João Miguel Nóbrega Department of Polymer Engineering, University of Minho, Campus de Azurem, Guimaraes, 4800-058, Portugal v

Design of Extrusion Forming Tools

J. Peter Müller EDS GmbH, Garnisonstrasse 7, 4560 Kirchdorf, Austria

Nickolas Polychronopoulos Polydynamics Inc., Dundas, Ontario, Canada

Shinichiro Tanifuji Hyper Advanced Simulation Laboratory, Tokyo, Japan

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

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

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

vi

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ontents

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

1.2.3 1.3

1.4 2

1.2.2.1

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

1.2.2.2

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

1.2.2.3

Shrinkage......................................................... 18

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

Calibration/Cooling Systems.......................................................... 22 1.3.1

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

1.3.2

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

Conclusion..................................................................................... 30

Properties of Polymers............................................................................... 37 2.1

Introduction................................................................................... 37

2.2

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

2.3

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

Thermal Properties........................................................................ 49 2.3.1

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

vii

Design of Extrusion Forming Tools

2.4

2.5 3

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

2.3.3

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

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

Conclusion..................................................................................... 59

Pipe Forming Tools................................................................................... 63 3.1

Introduction................................................................................... 63

3.2

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

3.3

3.4

3.5 viii

2.3.2

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

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

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

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

Contents 3.5.1

3.5.2

3.6 4

5

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

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

Conclusion................................................................................... 109

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

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

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Design of Extrusion Forming Tools

6

5.5

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

5.6

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

5.7

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

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

7

6.2.2.2

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

6.2.2.3

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

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

Conclusion...................................................................... 196

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

Flexible Dies............................................................................................ 221 7.1

Introduction................................................................................. 221 7.1.1

x

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

6.2.3

6.2.8 6.3

6.2.2.1

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

Contents 7.1.2

8

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

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

xi

Design of Extrusion Forming Tools

xii

1

Main Issues in the Design of Extrusion Tools

Olga S. Carneiro and J. Miguel Nóbrega 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 buildup, 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

Design 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.

δ

δ

T

Extruder

Die

Tm

Tδ

Ts

T

Ts

Calibration/Cooling

T Ts

Haul-off

Saw

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)

2

Main 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.

tem Sys etry m geo

Cooling conditions

P geo rofile me try

um s cu ion Va dit n co

E co xtr nd usi iti on on s

High cooling rate and temperature homogeneity

Metal properties

Polymer properties

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

3

Design of Extrusion Forming Tools 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 (laserDoppler 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.

4

Main Issues in the Design of Extrusion Tools

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

5

Design of Extrusion Forming Tools die and will limit the maximum throughput. There are several methods to minimise, or to eliminate, this defect, namely: • Promoting melt slip by the addition of an external lubricant/processing aid [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; • Increasing the melt temperature in order to decrease the shear stress or, equivalently, to increase the value of the critical shear rate; • Decreasing the throughput; and • Modifying the die exit geometry by including a curvature in order to reduce the stress 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.

6

Main Issues in the Design of Extrusion Tools 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)



(1.3)

with



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

7

Design of Extrusion Forming Tools

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.

8

Main Issues in the Design of Extrusion Tools

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]. 9

Design of Extrusion Forming Tools As a rule of thumb, the convergence angle should be limited to a maximum of 30º. 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

Die

Extrudate

Die

Extrudate

Die

time

Extrudate

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

10

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

11

Design of Extrusion Forming Tools 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).

Flow

h1

h2

L1

L2

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 coworkers [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 90º (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).

12

Main Issues in the Design of Extrusion Tools

Flow

D1

D2

L1

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]

T Flow

D1

L1

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: • Avoid high extrusion temperatures.

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].

13

Design of Extrusion Forming Tools • Limit the value of the flow rate.

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

• Promote wall slip using lubricants [64] or by coating the die surface with a low surface 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: Due to the velocity profile rearrangement and to the elasticity of 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: Since the profile is pulled by the haul-off unit with a velocity higher than 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 promoted by the decrease of the specific volume of the 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

14

Main Issues in the Design of Extrusion Tools 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:

2 B = D = 81 + 1 ` N 1 j B D0 8 x w

(1.8)

c2r = 2 ^ B 6 - 1h

(1.9)

1



or

6

and

2 B = H = 81 + 1 ` N 1 j B H0 12 x w

(1.10)

c2r = 3 ^ B 4 - 1h

(1.11)

1



or

4

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

15

Design of Extrusion Forming Tools

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:



: : dh ^ c h N 1 ^ch 1 }1 ^ch = : 2 =n d c: c :

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

16

Main Issues in the Design of Extrusion Tools

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):

DDR = v v0



(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).

17

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



D 0 = D 2 .DDR

(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

18

Main Issues in the Design of Extrusion Tools 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: • Step 1: Division of the cross-section of the parallel zone of the die in elemental sections (ES). A different ES shall be considered whenever there is a change in geometry and/or in thickness. • Step 2: Determination of the relative flow rate (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. • Step 3: Identification of the critical ES. This section will be that where the higher shear rate will occur, when the required relative flow rates are used. • Step 4: Determination of the maximum flow rate in the critical ES. This will be a function of its geometry and of the critical shear rate of the melt at the extrusion temperature. • Step 5: Determination of the flow rates (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. • Step 6: Determination of the relative lengths (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)

19

Design of Extrusion Forming Tools 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. • Step 7: Determination of the lengths of each ES. The length of the shortest ES must 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.

20

g =

4Q 3n + 1 pR 3 4n

Shear rate



(1.20)

(1.18)

(1.16)



Notes: subscript ‘o’ stands for outer or outlet; subscript ‘i’ stands for inner or inlet. Geometry: R – radius, L – length; H – height; W – width; Fp – correction factor. Rheological properties: n – power law index; K – consistency index. Processing parameters: Q – flow rate; - shear rate.

Circular Convergent

Annular

Parallel plates

Circular

Geometry

Pressure drop



(1.22)

(1.21)

(1.19)

(1.17)

Table 1.2 Equations to compute shear rate and pressure drop in simple geometry channels, using the power law to describe the shear viscosity of the polymer

Main Issues in the Design of Extrusion Tools

21

Design of Extrusion Forming Tools

Pressure

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.

ES1

ES2

Extrusion direction ES3

ES2

ES1

ES3

Lateral flow ES1-ES2 Lateral flow ES2-ES3

L3 (a)

L2

L1

Length

(b)

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 22

Main Issues in the Design of Extrusion Tools 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].

Water outlet

Extrusion die

Plug

Water inlet Compressed air

Vacuum

Extrusion die

Water

Perforated sleeve

Figure 1.13 Pressure and vacuum calibration in pipe extrusion

23

Design of Extrusion Forming Tools

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: • System geometry – number of calibration/cooling units (water tanks, air rings, 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. • Cooling conditions – temperature and flow rate of the cooling fluid. Low cooling fluid temperatures increase the cooling efficiency but also the thermal gradients within the extrudate. • Extrusion conditions – mass flow rate, extrusion temperature and extrudate crosssection temperature field at the die exit. The first will be one of the most relevant

24

Main Issues in the Design of Extrusion Tools 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]. • Polymer thermophysical properties – thermal diffusion coefficient and shrinkage. 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 onedimensional 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)

25

Design of Extrusion Forming Tools

(to be determined iteratively)

(Biot number)

(1.27)

(Fourier number)

(1.28)





(1.26)

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

(polymer thermal diffusivity)

(1.29)

with

TF - cooling fluid temperature TM - initial extrudate temperature h - heat transfer coefficient k, r,

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

26

Main Issues in the Design of Extrusion Tools

T !

x

δ D 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: • The residence time in the first cooling/calibration unit required to avoid further distortions of the extrudate in the subsequent stages. In this case the solidification temperature of the polymer (Ts) will be imposed at x = D − d 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. This requirement, 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.

27

Design of Extrusion Forming Tools 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 Heat transfer coefficient, h (W/m2K)

Condition 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 Thermal conductivity, k (W/mK)

Density, r (kg/m3)

Specific heat capacity, cp (kJ/kgK)

Acrylonitrilebutadiene-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

0.12

900-910

1.4

0.13-0.29

1300-1450

0.84-1.25

Polymer

Polypropylene U-PVC

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 ºC) while the opposed surface is almost unaffected (here the

28

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

Table 1.5 Data considered to compute the results shown in Figures 1.16 and 1.17 Heat transfer coefficient (water), h HDPE thermal diffusivity, a HDPE thermal conductivity, k

300 W/m2K 2.20 x 10-7 m2/s 0.5 W/mK

Initial plate temperature, TM

200 ºC

Cooling fluid temperature, TF

20 ºC

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

29

Design of Extrusion Forming Tools 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 ºC 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.

30

Main Issues in the Design of Extrusion Tools

References 1.

B. Endrass, Kunststoffe/Plast Europe, 1999, 89, 8, 48.

2.

W. Michaeli, Extrusion Dies for Plastics and Rubber: Design and Engineering Computations, 2nd Edition, Hanser Publishers, Munich, Germany, 1992.

3.

V. Kleindienst, Kunststoffe, 1973, 63, 1, 7.

4.

S. Levy, Advances in Plastics Technology, 1981, 1, 1, 8.

5.

G. Menges, M. Kalwa and J. Schmidt, Kunststoffe-German Plastics, 1987, 77, 8, 797.

6.

F. Thibault, L. Fradette and P.A. Tanguy in Proceedings of the 10th Annual Meeting of the Polymer Processing Society, Akron, OH, USA, 1994.

7.

L. Fradette, P.A. Tanguy, F. Thibault, P. Sheehy, D. Blouin and P. Hurez, Journal of Polymer Engineering, 1995, 14, 4, 295.

8.

R.J. Brown in Proceedings of ANTEC 2000, Orlando, FL, USA, 2000.

9.

J.M. Nóbrega, O.S. Carneiro, J.A. Covas, F.T. Pinho and P.J. Oliveira, Polymer Engineering and Science, 2004, 44, 2216.

10. J.M. Nóbrega and O.S. Carneiro, Plastics, Rubber and Composites: Macromolecular Engineering, 2006, 35, 387. 11. J.M. Nóbrega, O.S. Carneiro, A. Gaspar-Cunha and N.D. Gonçalves, International Polymer Processing, 2008, 23, 331. 12. J.M. Nóbrega and O.S. Carneiro in Optimization in Polymer Processing, Eds., A. Gaspar-Cunha and J.A. Covas, Nova Science Publishers Inc., New York, NY, USA, 2011, p.145. 13. P. L. Clegg, The Plastics Institute Transactions, 1959, 26, 151. 14. C. Rauwendaal, Polymer Extrusion, 4th Edition, Hanser Publishers, Munich, Germany, 2001, p.682. 15. M.M. Denn in Proceedings of the 11th International Congress on Rheology, Brussels, Belgium, 1992. 16. S.Q. Wang, P.A. Drda and Y.W. Inn, Journal of Rheology, 1996, 40, 875.

31

Design of Extrusion Forming Tools 17. E.R. Howells and J.J. Benbow, Transactions of the Plastics Institute, 1962, 30, 240. 18. F.N. Cogswell, Journal of Non-Newtonian Fluid Mechanics, 1977, 2, 37. 19. J.M. Piau, N. El Kissi and B. Tremblay, Journal of Non-Newtonian Fluid Mechanics, 1990, 34, 145 20. J.M. Piau and N. El Kissi in Proceedings of the 11th International Congress on Rheology, Brussels, Belgium, 1992. 21. R. Rutgers and M. Mackley, Journal of Rheology, 2000, 44, 1319. 22. K.B. Migler, Y. Son, F. Qiao and K. Flynn, Journal of Rheology, 2002, 46, 2, 383. 23. B. Tremblay, Journal of Rheology, 1991, 35, 6, 985. 24. J-F. Agassant, D.R. Arda, C. Combeau, A. Merten, H. Munstedt, M.R. Mackley, L. Robert and B. Vergnes, International Polymer Processing, 2006, 21, 3, 239. 25. A. Merten, M. Schwetz and H. Munstedt, International Journal of Applied Mechanics and Engineering, 2003, 8, 283. 26. R.D. Arda and M.R. Mackley, Journal of Non-Newtonian Fluid Mechanics, 2005, 126, 47. 27. O.S. Carneiro, Design of Pipe Extrusion Dies, University of Minho, Portugal, 1994. [Ph.D. Thesis]. [In Portuguese] 28. K. Migler in Rheological Measurement: Control and Understanding, Eds., S. Hatzikiriakos and K. Migler, Marcel Dekker, Monticello, NY, USA, 2005, p.121. 29. D. Kanev, E. Takacs and J. Vlachopoulos, International Polymer Processing, 2007, 22, 5, 395. 30. Y-H. Lin, Journal of Rheology, 1985, 29, 6, 605. 31. D.S. Kalika and M.M. Denn, Journal of Rheology, 1987, 31, 8, 815. 32. S.T. Kurtz in Proceedings of the 11th International Congress on Rheology, Brussels, Belgium, 1992.

32

Main Issues in the Design of Extrusion Tools 33. J. Perez-Gonzalez, L. de Vargas, V. Pavlinek, B. Hausnerova and P. Saha, Journal of Rheology, 2000, 44, 3, 441. 34. L. Robert, B. Vergnes and Y. Demay, Journal of Rheology, 2000, 44, 5, 1183. 35. D.L.T. Beynon and B.S. Glyde, British Plastics, 1960, 33, 414. 36. E.B. Bagley and H.P. Schreiber, Transactions of the Society of Rheology, 1961, 5, 341. 37. A.E. Everage and R.L. Ballman, Journal of Applied Polymer Science, 1974, 18, 3, 933. 38. M.T. Shaw, Journal of Applied Polymer Science, 1975, 19, 10, 2811. 39. J.M. Piau, N. El Kissi and B. Tremblay, Journal of Non-Newtonian Fluid Mechanics, 1990, 34, 2, 145 40. C. Rauwendaal, Plastics Technology, 2001, 47, 10. 41. Y. Goutille, J.C. Majesté, J.F. Tassin and J. Guillet, Jounal of Non-Newtonian Fluid Mechanics, 2003, 111, 175. 42. R. Muller and B. Vergnes in Rheology for Polymer Processing, Eds., J.M. Piau and J.F. Agassant, Elsevier, New York, NY, USA, 1996, p.257. 43. J.P. Tordella, Transactions of the Society of Rheology, 1957, 1, 203. 44. Y. Oyanagi, Applied Polymer Symposium, 1973, 20, 123. 45. K. Nakamura, S. Ituaki, T. Nishimura and A. Horikawa, Journal of Textile Engineering, 1987, 36, 49. 46. J.R. Rothstein and G.H. McKinley, Journal of Non-Newtonian Fluid Mechanics, 2001, 98, 33. 47. S. Nigen, N. El Kissi, J.M. Piau and S. Sadun, Journal of Non-Newtonian Fluid Mechanics, 2003, 112, 177. 48. S. Kim and J.M. Dealey, Polymer Engineering and Science, 2002, 42, 482. 49. S. Kim and J.M. Dealey, Polymer Engineering and Science, 2002, 42, 495. 50. F.N. Cogswell, Journal of Non-Newtonian Fluid Mechanics, 1978, 4, 23.

33

Design of Extrusion Forming Tools 51. A.G. Gibson and G.A. Williamson, Polymer Engineering and Science, 1985, 25, 980. 52. D.M. Binding, Journal of Non-Newtonian Fluid Mechanics, 1988, 27, 173. 53. Z. Tadmor and C.G. Gogos in Principles of Polymer Processing, 2nd Edition, John Wiley & Sons Inc., New York, NY, USA, 1979, p.677. 54. J.D. Gander and A.J. Giacomin, Polymer Engineering and Science, 1997, 37, 1113. 55. S.W. Horvatt and G.A. Hattrich, Equistar, Lyondell Company. http://www. lyondellbasell.com 56. I. Klein, Plastics World, 1981, May, 112. 57. M. Xanthos in Functional Fillers for Plastics, Ed., M. Xanthos, Wiley-VHC Verlag GmbH & Co., Weinheim, Germany, 2005, p.32 58. J. Musil and M. Zatloukal, Chemical Engineering Science, 2010, 65, 6128. 59. K. Chaloupková and M. Zatloukal, Polymer Engineering and Science, 2007, 47, 871. 60. K. Chaloupková and M. Zatloukal, Journal of Applied Polymer Science, 2009, 111, 1728. 61. T. Ohhata, H. Tasaki, T. Yamagushi, M. Shiina, M. Fukuda and H. Ikeshita, inventors; General Electric Company, assignee; US 5,417,907, 1995. 62. J. Rakestraw and M. Waggoner, inventors; E.I. Du Pont de Nemours and Company, assignee; US 5,458,836, 1995. 63. F. Ding, L. Zhao, A.J. Giacomin and J.S. Gander, Polymer Engineering and Science, 2000, 40, 10, 2113. 64. C-M. Chan, International Polymer Processing, 1995, 10, 200. 65. M. Prober and J.E. Vostovich, inventors; General Electric Company, assignee; US 3,942,937, 1976. 66. D.E. Priester and R.E. Tarney, inventors; E.I. Du Pont de Nemours and Company, assignee; US 5,064,594, 1991.

34

Main Issues in the Design of Extrusion Tools 67. R.M. Griffith and J.T. Tsai, Polymer Engineering and Science, 1980, 20, 18, 1181. 68. J.F. Stevenson, Plastics and Rubber Processing and Applications, 1985, 5, 4, 325. 69. J.F. Stevenson, L.J. Lee and R.M. Griffith, Polymer Engineering and Science, 1986, 26, 3, 233. 70. F. Rothemeyer, Kunststoffe, 1969, 59, 333. 71. F. Rothemeyer, Kunststoffe, 1970, 60, 7, 235. 72. M.A. Huneault, P.G. Lafleur and P.J. Carreau, Polymer Engineering and Science, 1990, 30, 23, 1544. 73. E.B. Rabinovitch, J.W. Summers and P.C. Booth in Proceedings of ANTEC ‘91, Montreal, Canada, 1991. 74. M. Huneault, Extrusion of PVC Profiles: Rheology and Die Design, University of Montreal, Canada, 1992. [PhD Thesis]. [ In French] 75. E.B. Rabinovitch, J.W. Summers and P.C. Booth, Journal of Vinyl Technology, 1992, 14, 1, 20. 76. R.I. Tanner in Engineering Rheology, Oxford University Press, Oxford, UK, 1985, p.323. 77. R.I. Tanner, Journal of Polymer Science, Part A-2: Polymer Physics, 1970, 8, 2067. 78. S. Middleman, Fundamentals of Polymer Processing, McGraw-Hill Book Co., New York, NY, USA, 1977, p.464. 79. J.L. White, Principles of Polymer Engineering Rheology, John Wiley & Sons, Hoboken, NJ, USA, 1990, p.289. 80. C.D. Han, Transactions of the Society of Rheology, 1974, 18, 163. 81. D.G. Baird, Journal of Rheology, 1975, 19, 147. 82. W. Gleissle, Rheologica Acta, 1982, 21, 484. 83. S.I. Abdel-Khalik, O. Hassager and R.B. Bird, Polymer Engineering and Science, 1974, 14, 859.

35

Design of Extrusion Forming Tools 84. R.B. Bird, O. Hassager and S.I. Abdel-Khalik, AIChE Journal, 1974, 20, 1041. 85. M.H. Wagner, Rheologica Acta, 1977, 16, 43. 86. H.O. Schiedrum, Kunststoffe-German Plastics, 1983, 73, 1, 2. 87. B. Endrass, Kunststoffe-German Plastics, 1993, 83, 8, 584. 88. D.H. Neudell, Plastics Technology, 1982, 28, 2, 64. 89. U. Conrad and J.F.T. Pittman in Proceedings of the 3rd ESAFORM Conference on Material Forming, Stuttgart, Germany, 2000. 90. L. Placek, J. Svabik and J. Vlcek in Proceedings of ANTEC2000, Orlando, FL, USA, 2000.

36

2

Properties of Polymers

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:



Re =

rVD h

(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

37

Design of Extrusion Forming Tools

(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.

38

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,h E , for example in uniaxial extension:

where s is a tensile stress and



(2.4)

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.

39

Design of Extrusion Forming Tools 106 Power law Newtonian plateau

Viscosity (Pa.s)

105

104 n-1 103

1/λ 102 10–4

10–3

10–2

10–1

100

101

102

103

Shear rate (s–1)

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 Ostwald–de 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/ n

40

n  1 Dp   D  w(r ) =   n + 1  2 K L   2 

1+ n n

1+ n   n 2 r   1 −     D   

(2.7)

Properties of Polymers Flow rate:

1/ n



n  1 Dp   D  Q =π   3n + 1  2 K L   2 

1+ 3 n n



(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 h = h0 (lg ) n −1 , 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.

41

Design of Extrusion Forming Tools

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

42

Properties of Polymers 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:



s +l

ds de =h dt dt

(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.

43

Design of Extrusion Forming Tools

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.

Viscosity (Pa.s)

(a)

Shear rate (s–1)

Reduced viscosity (Pa.s)

(b)

Reduced shear rate (s–1)

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

44

Properties of Polymers 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 shift factor aT defined by:



aT =

h (T ) T0 r0 h (T0 ) T r

(2.11)

where r and r0 are the densities at temperature T and T0, respectively. If we want to apply this principle to the data of Figure 2.5a, we have to plot the reduced viscosity (h(T)/aT) as a function of the reduced shear rate (g aT ). This is done in Figure 2.5b, where we obtain a mastercurve on an extended shear rate domain. The dependence on temperature is expressed by the shift factor aT. Depending on the temperature range considered, it may be defined using different forms. Between the glass transition temperature Tg and Tg + 100 °C, it is defined by the WilliamsLandel-Ferry equation [10]:



 −C1g (T − Tg )  aT = exp  g   T − Tg + C2 

(2.12)

where C1g and C2g are two constants depending on the polymer. Above Tg + 100 °C, the Arrhenius law is preferred:



 E  1 1  aT = exp  a  −    R  T T0  

(2.13)

where Ea is the activation energy, R the gas constant and T0 the reference temperature. Ea is usually in the range 20-100 kJ/mol for many polymers. The application of the time-temperature superposition principle implies that, for a power law behaviour (Equation 2.6), only K is a function of temperature, with an activation energy equal to nEa, where n is the power law index. For a Carreau-Yasuda fluid (Equation 2.9), the zero-shear viscosity h0 and the characteristic time l depend on temperature, with the same activation energy Ea.

45

Design of Extrusion Forming Tools Molten polymers are generally considered as incompressible, but the viscosity may be also influenced by the pressure (p), generally according to an expression of the form:



h ( p ) = h0 exp( c p)

(2.14)

where h0 is the viscosity at atmospheric pressure and c a compressibility coefficient. c is usually of the order of 10-8-10-9 Pa-1 [11], which implies that pressure effects are only visible above 30-40 MPa. They are thus very often negligible in extrusion.

2.2.4 Wall Slip In usual conditions, molten polymers stick to the die walls and the appropriate boundary condition is that the velocity at the wall is equal to zero. However, for some formulations or above critical conditions, slip at the wall may occur. It is thus important to detect wall slip, to evaluate slip laws and, if necessary, to introduce them in numerical simulations. To detect wall slip in rheological measurements, two main methods can be used. The first one is to use both smooth and rough tools of the same geometry. The rough surface is assumed to suppress wall slip, and thus the comparison between flow curves obtained with the two types of surface will provide the answer. If the curves are superimposed, there is no slip; if they differ, wall slip is present and the difference between the curves at constant stress gives the value of the slip velocity Vs. For example, for a capillary of diameter D:





(2.15)

where a,s and a,r are the apparent shear rates for smooth and rough surfaces, respectively. The second method, proposed by Mooney in 1931 [12], consists of using dies of different diameters (or thicknesses). As the rheological properties do not depend on geometry, these experiments should provide the same flow curve. If it is not the case, wall slip is present and the method allows then to evaluate the slip velocity. As said before, wall slip is usually observed in the flow of highly filled systems or with formulations including lubricants, for example for polyvinyl chloride compounds. 46

Properties of Polymers However, slip may occur even with simple polymers when critical conditions are reached. For example, a linear polyethylene (PE) exhibits strong slip above a critical stress, which is explained by a disentanglement between bulk chains and chains adsorbed on to the die wall [13]. In these conditions, the flow may become unstable, leading to extrusion defects, as explained in Section 2.2.5.

2.2.5 Flow Instabilities Above critical conditions, all molten polymers exhibit flow instabilities, leading to a perturbed shape of the extrudate. According to their molecular structure and viscoelastic properties, these defects are usually classified into two families. The first one concerns the linear polymers, such as high density polyethylene (HDPE) or linear low density polyethylene. For these polymers, the flow curve is discontinuous and presents two stable branches, separated by an unstable zone (Figure 2.6) [14, 15].

28 26 P1 24 Pressure (MPa)

22 P2 20 18 16 stable flow oscillating flow osciliating flow

14 12 10 0.0

Q1 0.1

0.2

0.3

0.4

Q2 0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Mass flow rate (g/s)

Figure 2.6 Flow curve of a HDPE, with the two stable branches and the unstable zone. Reproduced with permission from D. Kay, P.J. Carreau, P.G. Lafleur, L. Robert and B. Vergnes, Polymer Engineering and Science, 2003, 43, 78. © 2003, Society of Plastics Engineers [15]

47

Design of Extrusion Forming Tools Along the first branch, when the flow rate (or the apparent shear rate) is increased, the extrudate is at first smooth. Then, the surface becomes matte and presents surface defects, which are more important and organised, and usually called sharkskin [16] (see Figure 2.7a). Above a critical value Q1, the flow becomes unstable and, at imposed flow rate, presents regular oscillations of both pressure and instantaneous flow rate (called stick-slip). The extrudate then presents alternate portions of smooth and sharkskinned aspect (Figure 2.7b). It has been shown that these oscillations are related to an hysteresis cycle followed along the flow curve between the two stable branches: pressure and flow rate increase along the first branch, leading to a sharkskin defect. At the critical point P1, Q1 (see Figure 2.6), a jump to the second branch occurs, with a sharp increase in flow rate. Then, this branch is followed towards the second critical point P2, Q2, where a jump takes place, back to the first branch, and the cycle goes on. Along the second branch, strong slip conditions prevail [13, 17], which leads to a smooth extrudate surface. When the imposed flow rate is above Q2, the flow conditions are stable again and the extrudate is smooth, but it rapidly becomes highly distorted when the flow rate is increased. This region is usually called gross melt fracture (Figure 2.7d).

(a)

(b)

(c)

(d)

Figure 2.7 Different examples of extrusion defects: a) sharkskin; b) stick-slip; c) helical defect; and d) gross melt fracture

48

Properties of Polymers The second family of flow instabilities is encountered for highly branched polymers which have a high level of elasticity, such as low density polyethylene (LDPE) or PS. For these materials, the flow curve is continuous. However, above a critical flow rate, volume defects appear, which are either organised (helical defect, see Figure 2.7c) or not organised (e.g., gross melt fracture). These instabilities have their origin in the converging part of the die, before the final land, and can be postponed by modifying the geometry and using small converging angles [18].

2.3 Thermal Properties 2.3.1 Conductivity and Diffusivity As said before, molten polymers are good thermal insulators. They have a low conductivity, typically in the range 0.1-0.3 W. °C-1.m-1. The heat capacity rcp is generally between 1 and 3x106 J. °C-1.m-3, leading to values of diffusivity (ratio of conductivity to heat capacity) from 1 to 7x10-7 m2.s-1. The consequences of these properties on the flow conditions have already been evoked in Section 2.1.

2.3.2 Interfacial Temperature and Heat Penetration When two materials initially at two different temperatures T1 and T2 are put in contact (assumed perfect), an interfacial temperature appears, which remains constant with time during the temperature evolution of each material. This temperature can be expressed as:



Ti =

b1T1 + b2T2 b1 + b2

(2.16)

where b1 and b2 are the thermal effusivities, defined as:



b1 = k1 r1c p1 , b2 = k2 r 2 c p 2

(2.17)

where k is the heat conductivity and c p the specific heat capacity.

49

Design of Extrusion Forming Tools From Equation 2.17, it can be seen that the interfacial temperature is imposed by the material having the largest effusivity. For a PE, for example, the value of effusivity is around 920 J. °C-1.m-2.s-0.5, when it is around 6 for air, 1630 for water and 10,000 for steel. As a consequence, when a polymer extrudate exits from a die at 200 °C into ambient air at 20 °C, its surface will stay at the temperature it had in the die. However, if it is extruded in a water bath at the same temperature (20 °C), the surface will reach an intermediate value of 85 °C, leading to the solidification of the skin. If we assume now that the polymer is in contact with a steel calibrator at 20 °C, the interfacial temperature will be 40 °C, inducing a rapid cooling of the external part of the extrudate. It is worth noting that, when the polymer is in contact with a fluid such as air or water, heat exchange by convection is usually larger than by conduction. This will be the case for pipes in water, for example. Another important problem in polymer processing is the heat penetration. Let us consider a semi-infinite medium at an initial temperature T0. At time t = 0, we impose at the surface a temperature Ts. The temperature in the semi-infinite medium will then change with time, reaching a homogeneous value of Ts after a very long time t ∞ (Figure 2.8).

Figure 2.8 Temperature profiles in a semi-infinite medium; t1 and t 2 are successive times between 0 and t ∞

The evolution of temperature with time is given by [19]:



T − Ts  y = erf  T0 − Ts  2 αt

  

where α is the thermal diffusivity and erf a function defined as:

50

(2.18)

Properties of Polymers

erf ( x) =

π ∫

2

x

exp(−u 2 )du

(2.19)

0

Equation 2.18 provides a relationship between the time t and the distance y where the change in temperature is felt by the medium. If we consider arbitrarily the distance where half of the initial difference in temperature is felt ( T = T0 −

T0 − Ts , 2

see Figure  2.8), we obtain a very simple expression, which allows the rapid determination of the orders of magnitude of heating or cooling times:



t≈

y2

α



(2.20)

For example, using Equation 2.20, we can estimate, as explained in Section 2.1, that a sheet which is 1 mm thick can be cooled in 10 s, whereas it would take 17 min for a thickness of 1 cm.

2.3.3 Temperature Evolution in Extrusion Flows When a polymer is flowing through a die, its temperature changes according to the heat dissipated by deformation (Equation 2.2) and the heat transfer towards the die walls. The respective weight of these terms is given by the Brinkman number (Br):



(2.21)

where Tw is the wall temperature, T the average melt temperature and V the average velocity. If we consider a viscosity of 103 Pa.s, a conductivity of 0.2 W.m-1. °C-1 and an average velocity of 10 cm/s, we can see that the dissipated power is more important than heat transfer as soon as T − Tw is less than 50 °C. If we consider as an example the 1D flow of a molten polymer in a pipe, assuming an imposed wall temperature, we can propose the following evolution for the temperature

51

Design of Extrusion Forming Tools (Figure 2.9): at the beginning, the temperature is assumed to be homogeneous and equal to T0. According to the velocity profile, the shear rate is at its maximum at the wall. Consequently, the temperature increases by viscous dissipation in this region and, due to the low conductivity, this local increase progresses only slowly towards the centre. After a long time, viscous dissipation and heat transfer towards the wall will equilibrate, leading to an equilibrium profile which will no longer change along the flow.

   Velocity   Temperature Figure 2.9 Velocity and temperature profiles during the flow in a pipe

The temperature profile T(r, z) in the coordinate system defined in Figure 2.9 can be calculated by solving the thermal balance equation:



(2.22)

where w is the velocity component in the flow direction and (Equation 2.2), with appropriate boundary conditions.

the dissipated power



2.4 Crystallisation and Solid Properties 2.4.1 Generalities of Polymer Crystallisation If the macromolecular chain is regular, and if enough time is left for its organisation, a polymer can crystallise, but never completely, hence the concept of a semicrystalline

52

Properties of Polymers polymer. The crystallinity ratio (or crystallinity) is the volume or mass fraction of the polymer in the crystalline state. A popular method for the measurement of crystallinity is differential scanning calorimetry (DSC): the mass crystallinity is deduced from the area of the melting peak recorded by DSC. Crystals of synthetic polymers have lamellar shapes, with a thickness usually of about 10 nm. They are organised into more complicated aggregates, such as spherulites, in which they are separated by amorphous layers. A spherulite is a polycrystalline aggregate, consisting of radiating crystallites separated by an amorphous phase, which grow from a centre to fill the available space (Figure 2.10). Its size may vary between 1 mm and several millimetres. In three dimensions, the spherulite is spherically symmetric, whereas in two dimensions (e.g., in laboratory experiments between glass slides), it has a cylindrical symmetry. To ensure space filling, the lamellae are branched. The spherical (three-dimensional [3D]) or circular (two-dimensional [2D]) outlines are maintained until the impingement with other spherulites, and the spherulites ultimately acquire polyhedral (3D) or polygonal (2D) outlines.

40 µm Figure 2.10 Observation of growing polypropylene spherulites by light microscopy between crossed polarisers

53

Design of Extrusion Forming Tools As for any material, the crystallisation of a polymer consists of two stages [20]: (a) a nucleation stage, i.e., the formation, within the liquid phase, of entities called active nuclei, from which crystals can appear and (b) a growth stage, i.e., the development of crystals or more complicated morphologies (e.g., spherulites) from these nuclei. Crystallisation can be also considered in an overall manner [21], by describing the evolution of the volume (or surface) fraction c(t) transformed into semicrystalline entities, e. g., spherulites, as a function of time. According to the Avrami-Evans theory [22, 23], c(t) is represented by the general expression:





(2.23)

where E (t ) depends on nucleation and growth modes as well as on growth geometry. For the case of isothermal crystallisation, Equation 2.23 is used in the form:





(2.24)

where n is the Avrami exponent and k a temperature function. Ozawa [24] has adapted Avrami-Evans theory to crystallisation during cooling at a constant cooling rate . Then, c becomes a function of temperature T and cooling rate :





(2.25)

where q(T) is a temperature function. Experimentally, c(t) is generally determined by measuring by DSC the heat released by crystallisation as a function of time. The phenomena described above (nucleation and growth of lamellar crystals, spherulitic organisation, etc.) are involved in the so-called ‘primary crystallisation’. After their occurrence, the structures may perfect themselves, and crystallinity may still increase. This is generally called ‘secondary crystallisation’. Two possible mechanisms are discussed in the literature [25]: (a) the formation of additional crystallites or (b) the thickening of existing lamellae. For the latter mechanism, the thickness of the lamellae increases linearly with the logarithm of time.

54

Properties of Polymers

2.4.2 Processing Effects During processing, flow and cooling rate play major roles [26]. Thermoplastics are composed of flexible macromolecules, which adopt a random-coil conformation at rest, and can be easily deformed by flow. Under flow, they also tend to be oriented along the flow direction. Stretching and orientation of macromolecules in the melt affect crystallisation thermodynamics and kinetics, as well as the subsequent morphologies. Flow increases the equilibrium melting temperature of the polymer and enhances all the kinetics (nucleation, growth, overall kinetics). It means that, during a cooling, flow-induced crystallisation occurs at higher temperatures than quiescent crystallisation, and this is experimentally observed. Increasing the cooling rate shifts the crystallisation temperature towards lower values and lowers the crystallinity. As a result, ‘slow crystallising’ polymers such as polyethylene terephthalate can be quenched into the amorphous state. From a morphological point of view, an increase in cooling rate generally involves a decrease of the spherulite size, because of an increase of the nucleation density. The perfection of the crystalline arrangements can also be affected, the crystalline lamellae being poorly organised in a nonspherulitic manner. The effects of processing conditions are generally combined, which makes it difficult to interpret the development of the structure. Numerical simulation is a useful tool to understand and predict these combined phenomena. It requires the introduction of a crystallisation model into a computer code dedicated to a thermomechanical description of the process.

2.4.3 Orientation As mentioned above, flow induces an orientation of macromolecules in the melt. After solidification, this orientation is partly frozen in the amorphous phase. On the other hand, molecular orientation has an influence on nucleation and growth of crystals, if the polymer is semicrystalline. Therefore, a major effect of processing is to create orientation in both the crystalline and amorphous phases [26]. The global orientation (amorphous and crystalline phases) can be characterised by birefringence. Orientation of the crystalline phase has been mainly characterised by wide angle X-ray diffraction. The Debye-Scherrer method with a flat-film camera makes it possible to obtain rapidly a first appraisal of the degree of orientation. A monochromatic X-ray beam is incident upon the specimen, perpendicular to its surface. A photographic film records the diffracted beams. They form circular rings, each ring corresponding to a specific hkl crystalline reflection. If the specimen is

55

Design of Extrusion Forming Tools isotropic, the rings are continuous with a uniform intensity. If the specimen is oriented (or textured), the intensity distribution is generally not uniform. Continuous rings with intensity reinforcements, arcs, and finally isolated spots may be observed for increasing orientation. In spite of its rapidity and its simplicity, the Debye-Scherrer method gives only a partial view of the orientation. For a complete characterisation, the use of pole figures is necessary. Pole figures are stereographic projections giving the space distribution of the normals Nhkl to the hkl crystallographic planes. To construct a pole figure, the intensity diffracted by the hkl planes is recorded as a function of specimen position. This intensity represents the density of Nhkl normals, which is plotted on the stereographic projection. Isodensity lines are then drawn, and the representative points are called poles. From X-ray results, the average state of orientation can be described by orientation factors f j ,i , which are simple functions of the mean-square cosines of the angles ϕ j ,i between a crystallographic axis j of the unit cell (j = a, b, c) and a macroscopic axis i of a reference frame of the specimen (i = OX, OY or OZ axis of a Cartesian coordinate system):



f j ,i =

3 < cos 2 ϕ j ,i > −1 2



(2.26)

2.4.4 Viscoelastic Properties in the Solid State and in the Liquid-solid Transition Zone Viscoelastic properties of polymers in the solid state are treated within the same framework as for the molten state (see Section 2.2.2). They can be measured in creep (constant stress or load) or in relaxation (constant strain) experiments. Very often, they are determined through dynamic mechanical analysis. The material is subjected to a small sinusoidal strain, e.g., a shear strain g applied in torsion:



g = g 0 exp(iw t )

(2.27)

where t is the time, g0 the maximum strain and w the angular frequency. The corresponding stress is recorded:

56

Properties of Polymers

t = t 0 exp(iw t + d )



(2.28)

where t0 is the maximum stress and d the phase angle. The ratio complex modulus G*:

t / g defines the

G * = G ′ + i G ′′



(2.29)

where G′ is the elastic (or storage) modulus and G′′ is the viscous (or loss) modulus. The ratio G′′/ G′ is the tangent of the phase angle, tan d. From experimental results, master curves can be established. In the case of semicrystalline polymers, the situation is more complicated compared to the molten state, since the material is not thermorheologically simple, because of its crystallinity. A usual way to solve the problem is to introduce an additional vertical shift to the classical horizontal one used in the time-temperature superposition (see Section 2.2.3). Figure 2.11 shows the tan d curves obtained for a LDPE and a HPDE used in pipe extrusion. The superposition can be considered as satisfactory for HDPE. This is not the case for LDPE in spite of the double shift.

0.3 tan δ

0.25

1

2

0.2 0.15 0.1

1 2

0.05 0.0 – 4.0

HDPE LDPE Log10 (ω(rad/s))

– 3.0

– 2.0

– 1.0

– 0.0

1.0

2.0

Figure 2.11 tan d master curves (symbols) for a HDPE and a LDPE used in tube extrusion. Reference temperature: 35 °C. The solid lines have been calculated with a multimode Maxwell model

57

Design of Extrusion Forming Tools The Maxwell model with one relaxation time presented in Section 2.2.2 is not able to describe the behaviour of solid polymers, but this can be done by using a multimode Maxwell model, consisting of N simple Maxwell models (see Figure 2.4) in parallel. Each element is characterised by its modulus G(n) and its relaxation time λ(n). The G′ and G′′ components of the complex modulus are given by:

N



G ′(w ) = ∑ 1

N



G ( n ) l( n ) 2 w 2 1 + w 2 l( n ) 2

(2.30)

G ( n ) l( n ) w 1 + w 2 l( n ) 2

(2.31)

G ′′(w ) = ∑ 1

where w is the angular frequency introduced in Equation 2.27. The values of G′, G′′ and tan d calculated with the model can be compared to the experimental ones. This is done for tan d in Figure 2.11. Fifteen relaxation modes have been used for both polymers. Multimode Maxwell models are able to describe, with appropriate coefficients, the polymer behaviour in both the liquid and the solid state. In spite of interesting works in the field, (e.g., [27, 28]), the treatment of the rheological behaviour in the liquidsolid transition zone remains an open question. An empirical solution for numerical simulation is to use a mixing rule combining quantities for the fluid (f) and the solid (s) states. For instance, in the case of PE [29], G′ has been written:









(2.32)

(2.33)

where xc denotes the crystallinity; the value xc = 0.25 corresponds to the end of primary crystallisation, which can be described by Ozawa’s model (Equation 2.25). This means that the polymer is considered as solid at the end of primary crystallisation. The same approach has been applied to polyamide 12 [30].

58

Properties of Polymers

2.5 Conclusion Polymers are complex materials whose properties, both in the liquid and solid states, control the relationships between processing conditions, structure and final properties. We will see in the following chapters that the understanding of these specific properties is ofv vital importance for the control and the optimisation of the extrusion processes and the manufacturing of a variety of industrial products such as pipes, sheets, films and profiles.

References 1.

W. Ostwald, Kolloid Zeitung, 1925, 36, 99.

2.

A. de Waele, Journal of the Oil and Colour Chemists Association, 1923, 6, 33.

3.

P.J. Carreau, Transactions of the Society of Rheology, 1972, 16, 99.

4.

K.Y. Yasuda, R.C. Amstrong and R.E. Cohen, Rheologica Acta, 1981, 20, 163.

5.

C.W. Macosko, Rheology, Principles, Measurements and Applications, WileyVCH, New York, NY, USA, 1994.

6.

P.J. Carreau, D.C.R. De Kee and R.J. Chhabra, Rheology of Polymeric Systems. Principles and Applications, Hanser, Munich, Germany, 1997.

7.

J.M. Dealy and K.F. Wissbrun, Melt Rheology and its Role in Plastics Processing, Kluwer Academic Publishers, Dordrecht, Netherlands, 1999.

8.

R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston, MA, USA, 1988.

9.

J.M. Dealy and R.G. Larson, Structure and Rheology of Molten Polymers. From Structure to Flow Behavior and Back Again, Hanser, Munich, 2006.

10. M.L. Williams, R.F. Landel and J.D. Ferry, Journal of the American Chemical Society, 1955, 77, 3701. 11. A. Goubert, J. Vermant, P. Moldenaers, A. Göttfert and B. Ernst, Applied Rheology, 2001, 11, 26. 12. M. Mooney, Transactions of the Society of Rheology, 1931, 2, 210.

59

Design of Extrusion Forming Tools 13. A. Allal and B. Vergnes, Journal of Non-Newtonian Fluid Mechanics, 2009, 164, 1. 14. S.G. Hatzikiriakos and K.B. Migler, Polymer Processing Instabilities: Understanding and Control, Marcel Dekker, New York, NY, USA, 2005. 15. D. Kay, P.J. Carreau, P.G. Lafleur, L. Robert and B. Vergnes, Polymer Engineering and Science, 2003, 43, 78. 16. C. Venet and B. Vergnes, Journal of Rheology, 1997, 41, 873. 17. L. Robert, Y. Demay and B. Vergnes, Rheologica Acta, 2004, 43, 89. 18. C. Combeaud, Y. Demay and B. Vergnes, Journal of Non-Newtonian Fluid Mechanics, 2004, 121, 175. 19. R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, New York, NY, USA, 1960. 20. J.M. Haudin and B. Monasse in Structure Development During Polymer Processing, NATO Science Series, Series: Applied Sciences, Volume 370, Eds., A.M. Cunha and S. Fakirov, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000, p.93. 21. E. Piorkowska, A. Galeski and J.M. Haudin, Progress in Polymer Science, 2006, 31, 549. 22. M. Avrami, Journal of Chemical Physics, 1940, 8, 212. 23. U.R. Evans, Transactions of the Faraday Society, 1945, 41, 365. 24. T. Ozawa, Polymer, 1971, 12, 150. 25. G. Strobl, The Physics of Polymers. Concepts for Understanding Their Structures and Behavior, 3rd Edition, Springer, Berlin, Heidelberg, Germany, 2007, p.544. 26. J.M. Haudin in Handbook of Polymer Crystallization, John Wiley, Hoboken, NJ, USA, 2012, Chapter 16. 27. K. Boutahar, C. Carrot and J. Guillet, Macromolecules, 1998, 31, 1921. 28. V. Janssens, C. Block, G. Van Assche, B. Van Mele and P. Van Puyvelde, International Polymer Processing, 2010, 25, 304.

60

Properties of Polymers 29. O. Parant, Etude Expérimentale et Calcul des Contraintes Résiduelles dans des Tubes Extrudés en Polyéthylène, Ecole Nationale Supérieure des Mines de Paris, 2002. [Ph.D Thesis]. [In French] 30. J.M. Haudin, A. Carin, O. Parant, A. Guyomard, M. Vincent, C. Peiti and F. Montezin, International Polymer Processing, 2008, 23, 55.

61

Design of Extrusion Forming Tools

62

3

Pipe Forming Tools

Jean-Marc Haudin, Michel Vincent, Bruno Vergnes 3.1 Introduction The production of pipes is usually performed according to the scheme presented in Figure 3.1:

Circular saw

Haull-off system

Die

Cooling bath

Extruder

Pipe

Calibrator

Figure 3.1 Extrusion line for pipe production

An extruder continuously feeds an axisymmetric die, by means of which the geometry of the pipe is obtained. At the die exit, the pipe passes through a calibrator, in order to cool down its surface and to freeze the outside diameter. Then, the pipe is pulled through different cooling baths, before being eventually cut and stored at the end of the extrusion line. In the present chapter, we will focus only on the die and the calibrator. Different types of die may be used [1, 2], but the most common is the spider leg die, shown in Figure 3.2.

63

Design of Extrusion Forming Tools Spider leg Mandrel

θ r z

Figure 3.2 Example of geometry of a spider leg die

The axisymmetric die consists of a mandrel connected to the carter by streamlined parts, called spider legs, which are regularly distributed around the periphery. The molten polymer flows around these spider legs and joins again downstream, forming welding lines. The final part of the die, called the land, has a constant cross section and defines the final dimensions of the pipe. This type of die allows one to produce a wide variety of pipes, from medical catheters of millimetre size to water pipes up to 2 m in diameter.

3.2 Flow Through Pipe Dies 3.2.1 The Different Approaches from One-dimensional to Threedimensional Even though the geometry of a pipe die is three-dimensional (3D), different assumptions may be used to simplify the calculation of the flow. First of all, inside the die, the flows are confined, without free surface and without abrupt changes of cross-section (see Figure 3.2). They are primarily controlled by shear, with elongation playing a limited role. Accordingly, as explained in Section 2.2.2, elastic effects can be neglected and a viscous behaviour (power or Carreau type laws) is generally sufficient to obtain a good approximation of the flow conditions. However, thermal effects can be important and a coupling between mechanical and thermal equations is necessary.

64

Pipe Forming Tools As pipes are usually extruded horizontally (see Figure 3.1), mass and inertia forces can be neglected (see Section 2.1). Moreover, due to the progressive change of the crosssection area in the flow direction, lubrication approximations can be systematically applied in order to simplify the flow equations [3]. As a consequence, three approaches can be developed to characterise the flow conditions in a pipe die: (a) a very simple one-dimensional (1D) approach, by means of which the orders of magnitude of pressure and temperature along the die may be obtained (see Section 3.2.2); (b) a two-dimensional (2D) approach, using lubrication approximations and based on finite volume or finite element calculations (see section 3.2.4); and (c) a full 3D approach, for example for calculating the flow around the spider legs.

3.2.2 One-dimensional Calculation Let us consider the die presented in Figure 3.2, used to produce polyvinyl chloride (PVC) pipes at a flow rate of 300 kg/h. The pipe has an internal diameter of 10 mm and a thickness of 5 mm. If we neglect the zone around the spider legs, the flow can be considered as axisymmetric. Applying the lubrication approximations (the velocity in the radial direction is negligible compared to the velocity in the flow direction) and considering a power law behaviour, Stokes equations reduce to the following expression, relating the pressure gradient dp/dz to the volume flow rate Q:

dp −2 K (Q / 2π ) n = n 1/ n 1/ n dz  R∗ r  ∗2  R2 R2   R R ∗2   r − u  dudr + ∫ ∗ r ∫  u −  dudr  R r  ∫R1 ∫R1  u u     

(3.1)

where K is the polymer consistency (see Section 2.2.1), n the power law index, R1 and R2 the radius of the mandrel and carter, respectively, and R* the location where the velocity is at its maximum (Figure 3.3). This expression is written locally, meaning that R1, R2 and R* vary with the axial coordinate z.

65

Design of Extrusion Forming Tools

R2(z) r

R*(z) R1(z)

R1(z)

R2(z) z

Figure 3.3 Local geometry of a pipe die

The average temperature T across the flow thickness can be calculated by solving:





(3.2)

where α is the thermal diffusivity, Tw the temperature of the carter, assumed to be constant, and rcp the heat capacity. The mandrel is supposed to be adiabatic and Nu is a Nusselt number, qualifying the heat transfer between polymer and carter [4]. Equations 3.1 and 3.2 are coupled through the value of the consistency K, which depends on temperature, for example according to the Arrhenius law (see Section 2.2.3):



E K = K 0 exp  a R

 1 1   −   T T0  

(3.3)

where T is the temperature, K0 the consistency at the reference temperature T0, Ea the activation energy and R the gas constant.

66

Pipe Forming Tools The resolution of Equations 3.1 and 3.2 step by step all along the pipe die enables the evolution of the pressure and temperature to be calculated, as shown in Figure 3.4. We observe that the pressure drop is 21 MPa, mainly concentrated along the final land. The average temperature increases also principally along the die land, from 186 to 192 °C (the fixed carter temperature Tw is 185 °C).

PRESSURE p (bar)

Spider legs 200

T

P 100

191 189 187 185

MEAN TEMPERATURE

193

10 30 50 DIE LENGTH z (cm)

Figure 3.4 Pressure and average temperature evolution along the pipe die. Adapted from B. Vergnes and J.F. Agassant, Advances in Polymer Technology, 1986, 6, 441. ©1986, Wiley [3]

These first results show that the die land is the ‘sensitive’ zone of the die where the thickness is the weakest and, consequently, the shear rates the highest. To better understand the flow conditions in this section, we can now develop a 2D approach for the temperature.

3.2.3 Temperature Computations In this section, we keep the same mechanical 1D approach, but we consider a temperature profile across the flow thickness. For that, we have to solve the following equation using, for example, a finite difference method:

67

Design of Extrusion Forming Tools



∂T k ∂  ∂T  dw = r +K rcp w   ∂z r ∂r  ∂r  dr

n +1



(3.4)

where w is the velocity component in z direction and k the thermal conductivity. It is possible to show that the adiabatic condition on the mandrel, previously used in the 1D model, is no more realistic [5]. We assume now that the mandrel is in thermal equilibrium: it receives a heat flux from the flowing polymer and releases calories to the external carter through the spider legs. After a certain time, it will reach equilibrium, with a temperature resulting from the flow conditions.

Dimensionless radius

The calculations have been performed in the conditions mentioned previously (300 kg/h, T0 = 183 °C, Tw = 185 °C). Figure 3.5 shows the temperature profiles at different locations between the spider legs and the land end (from points A to C, see Figure 3.4).

1

A

B

C

0,6 0,2 0 180

190

200 Temperature (°C)

Figure 3.5 Temperature profiles at different locations along the pipe die. The dimensionless radius is defined as (r-R1)/(R2-R1)

We can see that the temperature of the mandrel is 198 °C, i.e., much higher than the imposed temperature on the carter (185 °C). It induces a large temperature increase close to the mandrel wall, which could possibly lead to polymer degradation (especially in the case of PVC extrusion). The temperature profile is heterogeneous and presents two hot spots in the sheared zones, close to the walls. Overheating can reach 17 °C at the land exit, which could lead to possible troubles in the post-extrusion operations (calibration, cooling, residual stresses, etc). A possible solution would be to implement

68

Pipe Forming Tools a thermal regulation of the mandrel, by circulation of a cooling fluid, in order to impose a controlled temperature.

3.2.4 An Example of Nonaxisymmetric Flow In the previous sections, the flow in the pipe die was considered as axisymmetric. It can sometimes be different. To illustrate this point, we refer to the problem of thickness homogeneity encountered in the production of pipes, in the conditions previously presented [6]. Instead of having a constant thickness of 5 mm, the pipe presented a thinner value in the vertical plane (4.7 mm) and a thicker one in the horizontal plane (5.3 mm) (Figure 3.6).

emin = 4.7 mm emax = 5.3 mm e = 5 mm

(a)

(b)

Figure 3.6 Example of correct pipe (a); and defect of symmetric thickness heterogeneities (b); e is the pipe thickness

Because of the symmetry, this defect could not be due to a geometrical problem (mandrel off-centring would lead to an asymmetric defect). In fact, it was related to temperature heterogeneities created by the flow in the counter-rotating twin screw extruder used to plasticise the PVC and to feed the die. Measurements in the channel between extruder and die have effectively shown that, in the cross-section, two hot zones were present in a horizontal plane (Figure 3.7).

69

Design of Extrusion Forming Tools

Mandrel

a

a

a 10 °C

a

Figure 3.7 Temperature field in a cross-section of the channel between the twin screw extruder and the pipe die. The grey zones are the hottest. The temperature profile along the a-a line is also shown

These hot zones were around 10 °C hotter than the polymer in the periphery. We can then imagine that, when arriving on the mandrel, these zones were separated and flowed on each side of the mandrel, leading to a local increase of flow rate due to the lower viscosity. To validate this assumption and to calculate the flow conditions, it is no longer possible to consider an axisymmetric situation. We have to use a 2D model, applying lubrication approximations. Moreover, to simplify the problem, we focus on the final die land (we have seen previously that it was the ‘sensitive’ part) and we ‘unroll’ the annular space between mandrel and carter as shown in Figure 3.8.

70

Pipe Forming Tools

Figure 3.8 Unrolling of the die land zone

71

Design of Extrusion Forming Tools The calculation of the flow and temperature fields can be achieved by solving the following set of equations [3]:

hDp +

2n + 1  ∂p ∂h ∂p ∂h  h  ∂p ∂K ∂p ∂K  + − + n  ∂x ∂x ∂y ∂y  Kn  ∂x ∂x ∂y ∂y  2

2



2 ∂p ∂p ∂ 2 p  ∂p  ∂ 2 p  ∂p  ∂ p + +  2   2 2 ∂x ∂y ∂x∂y  ∂y  ∂y 1 − n  ∂x  ∂x +h =0 2 2 n  ∂p   ∂p    +   ∂x   ∂y 

1/ n

(2 n +1) / n

 ∂p  2  ∂p  2    +     ∂x   ∂y  

1/ n

(2 n +1) / n

 ∂p   ∂p    +    ∂x   ∂y 

1 2n  1   h  u ( x, y ) = −     h 2n + 1  K   2  v ( x, y ) = −



1 2n  1   h      h 2n + 1  K   2 

2

2

(1− n ) / 2 n

  

 ∂p     ∂x 

(1− n )/2 n

(3.5)



(3.6)

 ∂p     ∂y 

(3.7)

where p(x, y) is the pressure, h(x, y) the local thickness, u and v the two components of the average velocity, and T (x, y) the local average temperature. The consistency K is a function of the average temperature through an Arrhenius law (Equation 3.3). Equations 3.5 to 3.7 are solved using a finite difference iterative model. The results are shown in Figure 3.9. We have selected initial conditions corresponding to the experimental measurements, i.e., initial temperatures varying from 180 °C (cold zones) to 192 °C (hot zones).

72

Pipe Forming Tools Tr(°C)

Tr(°C)

200 185 180

200 180 y 192

180

y 192

y

192

181

181 190

192 y

180

190

193

193

184

184 187 194 x

186 194

191 x

191

188

190 191

191

191

qx/Q qx/Q

1,1

1,08 1,04

1 Position along y

1 Position along y

0,96

0,9 (a)

(b)

Figure 3.9 Regulation temperature Tr , temperature field in the die land and final flow rate distribution (from top to bottom); a) homogeneous thermal regulation, and b) heterogeneous thermal regulation; q x is the local flow rate and Q the average one

73

Design of Extrusion Forming Tools If we consider a homogeneous wall temperature Tw of 185 °C (Figure 3.9a), we can see that, during the flow along the die land, the temperature increases slightly, due to viscous dissipation and heat transfer. Even though the temperature difference between cold and hot zones is reduced (from 12 to 7 °C), we observe that the exit flow rate is heterogeneous. We have reported the ratio of the local flow rate to the average flow rate in Figure 3.9. A homogeneous distribution would provide a constant value of 1. In the present case, we observe higher values (+ 10%) at positions corresponding to the hot zones and lower values (– 10%) at positions corresponding to the cold zones. This confirms that the initial heterogeneity in temperature is at the origin of the thickness defect. Moreover, modelling may also provide a solution to overcome this problem. Let us assume now that we impose around the die a heterogeneous thermal regulation, with higher values of Tw (200 °C) in the cold zones and lower values (180 °C) in the hot zones (Figure 3.9b). We can see that, in this case, the temperature difference can be suppressed before the land exit, leading to a much better homogeneity of the flow rate, and hence of the pipe thickness. It is now common practice to use such heterogeneous regulation systems in pipe extrusion.

3.3 Pipe Calibration – Experimental 3.3.1 Technological Review After the die exit, the pipes travel on a short distance in the air before entering a calibration device, which is placed in a tank where a cold water bath or jets ensure cooling and polymer solidification (Figure 3.10a). The objective of calibration is to control and adjust the outer circular section of the pipe before solidification allows it to resist gravity. Even if internal calibration exists, where the internal radius is ensured by a calibrating device, the simpler outer calibration is more widely used. A difference of pressure between the inside and outside of the pipe is imposed, so that the outer surface sticks to the calibrating device. The difference of pressure is usually imposed by an adjustable pressure lower than the atmospheric one in the cooling tank, although a higher pressure inside the tube may also be used. The calibration device may be simply rings with decreasing diameter, but the preferred system is a tubular sizing device (Figure 3.10). The calibrator is generally made of brass, copper or bronze. Holes are drilled in the sizing device, so that the pressure difference sticks the tube against the calibrator, ensuring the correct tube dimension. The calibrator diameter is usually several per cent larger than the final required tube diameter to counterbalance the effect of thermal shrinkage. 74

Pipe Forming Tools

Water tank

Pipe

Calibrator

Die

(a)

Annular channel

∆P

Water inlet

∆P Water channels

∆P

∆P

Extrusion direction

Water outlet (b)

Figure 3.10 (a) View of the die and water tank with calibrator; and (b) shape of the calibrator used for polyamide pipes

The contact between the pipe and the calibrator must be lubricated to reduce friction and haul-off force, and improve surface properties. Lubrication is usually provided by a water flow, which arrives at the entrance of the calibrator through holes, slits or annular feeding channel, with an imposed water flow rate (Figure 3.10b). Sometimes, adhesion of the polymer on the calibrator can create pick-up marks, resulting in surface defects [7, 8]. To reduce friction, a lubricant (oil or surface active agent) can be added to the lubricating water [9, 10] or the calibrator contact surface can be coated with polytetrafluoroethylene. The design of the calibrator can be optimised [11]. A gas pressure in the space between the calibrator surface and the extrudate leads also to friction reduction [12]. Rotary calibrators improve the surface finish and reduce haul-off force [13].

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Design of Extrusion Forming Tools

3.3.2 Objectives and Motivations The objective of this study is to establish relations between processing parameters, induced microstructure and mechanical and aspect properties. The main extrusion parameters are: • Extrusion die diameter (Ddie); • Mandrel diameter (Dm); • Calibrator inner diameter (Dc) and length; • Distance between the extruder die and the entrance of the calibrator (Ddc); • Level of vacuum in the calibration tank (Dp); • Water flow rate applied upstream of the calibrator (Qw); • Cooling water temperature; and • Extrusion line speed (wl). The results shown below refer to: (a) Polyamide 12 (PA12) pipes (material provided by Arkema). The final pipe outer diameter (Dtube) varies between 7.20 and 10.50 mm depending on the extrusion conditions. All the pipes are 1 mm thick, which allows the mechanical properties to be compared. The extrusion temperature is 230 °C, the calibrator design is shown in Figure 3.10b and its length is 40 mm. These pipes are used, for instance, for fuel lines in cars. (b) Polyethylene (PE) pipes, 50 mm in diameter and 0.5 mm thick. Three materials are used: low density polyethylene (LDPE) (Lacqtène FB3010, Elf Atochem), high density polyethylene (HDPE) (Hostalen 4750, Basell), and linear low density polyethylene (LLDPE) (Escorene Y 1002, Exxon Mobil). The extrusion temperature is 220 °C. The calibrator design is similar to the one shown in Figure  3.10b, with, of course, a higher diameter which is related to the pipe diameter, and a length of 160 mm. These pipes are used for the packaging of cosmetics.

3.3.3 Velocity The evolution of the polymer velocity from the die exit towards complete solidification is important, as the material is stretched, and elongation promotes molecular orientation, affects crystallisation and induces surface defects. 76

Pipe Forming Tools There are several methods for measuring polymer velocity. They are based on the observation of markers [14, 15], so this is only possible between the extrusion die exit and the entrance to the calibrator, and downstream from the calibrator, if special windows are set up in the cooling tanks. It is not possible to measure the polymer velocity when the tube is inside the calibrator. These methods are: (a) surface marking, for measuring the outer surface velocity between the die exit and the calibrator entrance; (b) ink jet inside the pipe, for measuring inner surface velocity; and (c) aluminium particles mixed with the polymer, making it possible to measure the velocity within the extrudate. If a multilayer extrusion line is available, the introduction of markers in certain layers enables the velocity profile to be measured in the whole thickness of the pipe. Observations of these different markers lead to the conclusion that the velocity is homogeneous in the thickness of the pipe wall, upstream and downstream of the calibrator. Figure 3.11 shows the velocity evolution between the die exit and the calibrator, and downstream of the calibrator for PA12 pipes. The length of the sizing sleeve is 30 mm, and the overall calibrator length is around 40 mm. The die and mandrel diameters are 18 and 12 mm, respectively. The extrusion parameters are reported in Table 3.1.

Table 3.1 Extrusion parameters for polyamide 12 pipe extrusion Condition # 1′ 2′ 1′′ 2′′

wl m.min-1

V0 m.min-1

20

3.48

10

1.78

Dp hPa

Qw cm3.s–1

Ddc mm

VC m.min-1

Drtank

100

21.5

60

10.73

1.86

200

13.1

60

8.86

2.26

100

21.5

30

4.90

2.04

200

13.1

30

3.67

2.72

V0 is the pipe velocity at the die exit, VC is the pipe velocity at the calibrator entrance, Drtank is the draw ratio in the calibration tank.

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Design of Extrusion Forming Tools

10

1′

Extrusion line speed

Tube velocity (m/min)

Tube velocity (m/min)

20

2′

15

1′ 10

2′ 5

1′′

Extrusion line speed

8

2′′

6

1′′

4

2′′

2

0

0 0

20

40

60

80

100

120

Distance from extrusion die (mm)

(a)

140

0

20

40

60

80

100

120

Distance from extrusion die (mm)

(b)

Figure 3.11 Velocity evolution between the extrusion die and the calibrator entrance, and downstream of the calibrator for conditions of (a) type #1′ (❍) and # 2′ (▲) at 20 m.min-1; and of (b) type #1′′ (❍) and #2′′ (▲) at 10 m.min-1

The extrusion line speed is 20 m.min-1 for conditions #1′ and #2′ (Figure 3.11a), and 10 m.min-1 for conditions #1′′ and #2′′ (Figure 3.11b). The speed of the polymer at the die exit has been adjusted accordingly: V0 = 3.48 m.min-1 and V0= 1.78 m.min-1, respectively. Conditions #1′ and #2′ (and respectively, #1′′ and #2′′) differ in the level of vacuum and water flow rate: low level of vacuum (Dp) and large water flow rate (Qw) for conditions #1′ and #1′′, high Dp and low Qw for conditions #2′ and #2′′. Between the extrusion die exit and the calibrator entrance, the velocity is at first nearly constant, and then it increases regularly. For the 20 m.min-1 extrusion line velocity (Figure 3.11a), the polymer velocities at the entrance of the calibrator (VC) are different; VC is 10.73 m.min-1 for condition #1′ and 8.86 m.min-1 for condition #2′. After the calibrator exit, the velocity is around 18 m.min-1, which is close to the extrusion line velocity. This means that the pipe is submitted to elongation firstly between the die exit and the calibrator, and secondly in the calibrator itself. At the end of the calibrator, the material temperature is low enough so that the pipe resists elongational stresses. The same kinds of observations and conclusions hold for an extrusion line velocity of 10 m.min-1 (conditions #1′′ and #2′′, Figure 3.11b). For PE pipes, of 50 mm diameter and 0.5 mm thickness, the increase in velocity between the extrusion die exit and the calibrator entrance is much larger, and the

78

Pipe Forming Tools extrusion line velocity is almost reached at the calibrator entrance (Figure 3.12). This is probably related to the smaller pipe thickness, leading to a rapid solidification soon after the entrance to the calibrator.

Tube velocity (m.min–1)

6 5 4 3 2 1 0 0

0,005 0,01

0,015 0,02

0,025

0,03 0,035

0,04

Distance from die (m)

Figure 3.12 Velocity evolution between the die and the calibrator, for PE pipes, of 50 mm diameter and 0.5 mm thickness. The extrusion line speed is 6 m.min-1

For the thicker PA12 pipes, the repartition of elongation upstream of the calibrator, and in the calibrator itself, can be related to the lubrication effect of the water layer thickness ewater between the outer tube surface and the calibrator. This thickness has been evaluated in the following way. The incoming water flow rate (Qw) is divided into three parts (Figure 3.10b). Most of the water (Qexit) is rejected by a symmetrical conduit downstream of the annular section. The remaining water goes through the 45°-oriented channels. A part of this water goes into the sizing sleeve and supplies the lubricant water layer between the tube and the calibrator (Qlub). The remaining part falls down at the entrance (Qleakage). The lubricating flow rate Qlub is deduced by difference:





(3.8)

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Design of Extrusion Forming Tools Then, the water layer thickness is obtained assuming that this layer creates a ring around the tube moving at velocity VC. Numerous measurements with various extrusion parameters showed that the thickness of the water layer decreases when the level of vacuum increases or when the extrusion line velocity increases [15]. A nearly-linear decrease of the water layer between 160 µm and 10 µm is obtained when the level of vacuum increases from 50 hPa to 250 hPa, and a linear decrease between 80 and 10 µm has been obtained increasing the extrusion line speed from 13 to 20 m.min-1. For the 20 m.min-1 line velocity, the water layer ewater was around 150 µm for the lower Dp of 100 hPa (condition #1′), and 30 µm for the higher Dp of 200 hPa (condition #2′). Therefore, the respective importance of drawing in the air and drawing in the calibration tank is dependent on the level of lubrication. For conditions #1′ and #2′, the elongation parts in the calibration tank are 54% and 63%, and the air drawing parts are 46% and 37%, respectively. Therefore, drawing kinematics is dependent on lubrication conditions at the calibrator inlet. Experiments achieved with soap in the water layer confirm this fact. Indeed, elongation repartition between air and the calibration tank is modified by disturbing the level of lubrication in the calibrator: VC increases by 3 m.min-1 in the presence of soap. The draw ratio in the tank, defined as Drtank = wl / VC, correlates well with the water layer thickness (Figure 3.13).

4

Drtank

3

2

1

0 0

50

100

150

200

ewater ( m)

Figure 3.13 Draw ratio in the calibration tank versus thickness of water layer for a large variety of extrusion and calibration conditions 80

Pipe Forming Tools

3.3.4 Friction between the Pipe and the Calibrator In order to elucidate the precise effect of lubrication on the evolution of the draw ratio, it is useful to measure the friction force on the calibrator, using special devices mounted on the calibrator [14, 15]. For the PA12 pipes, the friction forces range from 3.9 to 13.4 N, with a precision of 1.5 N. The friction force is correlated to the level of vacuum Dp, if the latter exceeds 100 hPa (Figure 3.14). Tests carried out without lubrication lead to a higher friction force.

14

With lubrication

Friction force (N)

12

Without lubrication

10 8 6 4 2 0 0

50

100

150

200

250

300

Vacuum level (hPa)

Figure 3.14 Friction forces versus vacuum level for PA12 pipes

Figures 3.15 and 3.16 show that friction forces are small and independent of the level of vacuum, or depression, below 120 hPa for the LDPE pipes, and below 200 hPa for the HDPE pipes. For such low depression, the traction force leads to a decrease of the diameter which cannot be counteracted by the depression. For larger depression, the friction force increases almost linearly up to around 50 N at 170 hPa for the LDPE, and 70 N at 300 hPa for the HDPE (Figure 3.16). The friction force increases with the calibrator length and extrusion line velocity. A decrease of the water temperature from 20 to 15 °C leads to a decrease of the friction force. This can be due to thermal shrinkage. For a given friction force, the depression is smaller for LDPE than for HDPE (Figure 3.16). This can be related to the mechanical properties

81

Design of Extrusion Forming Tools of both polymers. The elastic modulus of LDPE is much lower than that of HDPE (215 and 900 MPa, respectively), and a lower depression is enough to stick the tube against the calibrator and induce friction.

60

Friction force (N)

50 40 30 20 10 0 100

120

140

160

180

200

220

240

Level of vacuum (hPa)

Figure 3.15 Friction force versus depression for LDPE pipes. (n) Calibrator length: 160 mm. Extrusion line speed: 6 m.min-1, (o) Calibrator length: 300 mm. Extrusion line speed: 6 m.min-1, and (▲) Calibrator length: 300 mm. Extrusion line speed: 9 m.min-1

The friction forces for the PE pipes (up to 70 N) are much higher than those of the PA12 pipes (11 N), because the diameters are very different (50 mm and 8 mm, respectively).

82

Pipe Forming Tools 80

Friction force (N)

70 60 50 40 30 20 10 0 100

150

200

250

300

Depression (hPa)

Figure 3.16 Friction force as a function of the depression for LDPE () and HDPE (n). The extrusion line velocity is 6 m.min-1, the calibrator length is 0.3 m, and the cooling water temperature is 15 °C

3.3.5 Temperature Evolution As expected, cooling in the air travel between the extrusion die exit and the calibrator is small. Infrared pyrometer measurements show that the outer PA12 pipe surface temperature decreases slightly by 3 to 4 °C. For the thin PE pipes, a thermocouple was fixed on a flexible rod which applies a small pressure on the inside of the pipe. Measurements show that a pipe 0.4 mm thick extruded at 6 m.min-1 is completely cooled at the end of the 60 cm long calibrator. Figure 3.17 shows for the LDPE that cooling is faster for the lower extrusion line velocity. Slope changes due to the heat of crystallisation can be observed around 90 °C for the LDPE, and similar measurements show a slope change at 110 °C for the HDPE.

83

Design of Extrusion Forming Tools 250

with cristallinity without cristallinity measured with cristallinity without cristallinity measured

Temperature (ºC)

200

150

12 m.min–1

6 m.min–1

6 m.min–1 100

50 12 m.min–1 0 0

0.2

0.4

0.6

0.8

1.0

Distance from extrusion die (m)

Figure 3.17 Temperature evolution on the inner LDPE pipe surface for extrusion line velocities of 6 (l) and 12 m.min-1 (Í) and pipe thickness of 0.6 mm. Position 0 is at the die exit, and position 80 cm is at the calibrator exit. The calibrator length is 60 cm. Solid and dashed lines represent the calculated evolution with and without including crystallisation (see Section 3.5.1)

3.3.6 Residual Stresses Residual stresses have two origins. First, stresses in the molten state which are not fully relaxed are frozen in at solidification. These stresses can be flow stresses generated in the die, or stresses exerted by the haul-off device. The second origin comes from the heterogeneous cooling and thermal volume contraction. They are referred to as thermal stresses. Frozen-in stresses induce a certain level of molecular orientation in the pipes which can be characterised as indicated in Section 2.4.3. Residual stresses affect the resistance to impact, especially if the surface is in tension. In injection moulded parts, residual stresses induce visible warpage, but this is not possible in a pipe, unless it has a finite length, such as the tubes used in cosmetics or in dental paste packaging. Then, a deformation of the tip of the tube can be observed (Figure 3.18). This defect causes problems for subsequent injection of the cap or for welding of the bottom of the packaging.

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Pipe Forming Tools

Figure 3.18 Deformation of the tip of a thin tube for packaging application

Now let us focus on the mechanisms for the build-up of thermal stresses. First we will consider free quenching of a pipe and assume that, below a certain temperature, the polymer is an elastic solid and above it, a fluid with a negligible elastic modulus. In order to understand qualitatively the build-up of residual stresses, let us imagine that cooling proceeds in two steps (Figure 3.19), and assume that heat transfer inside the pipe is negligible. In the first step, the outer layer solidifies. It shrinks freely because the inner molten layer accommodates deformation. In the second step, the inner layer solidifies, but its thermal shrinkage is partially blocked by the outer solidified layer. Therefore, the inner layer is in traction, and the outer one is in compression. If the pipe is cut along its axis (or z direction), it will close. If the tank pressure is low enough to counteract the reduction of the pipe section due to thermal shrinkage and haul-off force, the pipe diameter remains constant and equal to the calibrator diameter. This is constrained quenching (Figure 3.20). In the first step, the outer layer solidifies, its shrinkage is prevented by the low tank pressure, and this layer becomes in tension. Then, the second layer solidifies and becomes in tension as well. As the temperature variation is the same throughout the thickness, the stress is homogeneous. At the exit of the calibrator, the whole pipe will be submitted to an elastic recovery, and finally the pipe will be stress-free! In the case where the depression is not large enough to stick the tube against the calibrator until the end of cooling, the residual stress profile will be intermediate between the stress-free state of constrained cooling, and the compression/traction stress profile of free quenching.

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Design of Extrusion Forming Tools Solid

Liquid

2nd step

1st step

Figure 3.19 Build-up of residual stresses in free quenching

Solid

Liquid

1st step

2nd step

Figure 3.20 Residual stresses in constrained cooling

Another external loading comes from the pulling or haul-off force. This force is generated by the drawing deformation of the polymer between the die exit and the calibrator entrance, and by the friction on the calibrator. In the case of constrained cooling, that is if the depression sticks the tube against the calibrator, axial stresses lead to tensile hoop stresses in the first solidified layer (Figure 3.21). The tensile stresses are lower in the inner layer solidified in the second step, because the first outer layer already solidified resists the tensile stresses. When leaving the calibrator, the boundary condition against the calibrator disappears, and elastic recovery takes place, leading to a tension/compression stress profile, opposite to the one of free cooling. If the pipe is cut along the axis (or z) direction, it will open.

86

Pipe Forming Tools

Solid

Liquid

1st step

2nd step

After unloading

Figure 3.21 Residual stresses in constrained cooling and axial loading

The previous analyses are modified by the viscoelastic behaviour of both fluid and solid. Stress relaxation takes place in the solid phase if the liquid phase is still neglected, so that even in the case of constrained cooling without axial loading, a stress profile builds up. Moreover, the mechanical behaviour of the fluid regions cannot be neglected in front of those of the solid regions, but stress relaxation is faster in the liquid state than in the solid state, so that finally, most of the loadings will concentrate on the solid. In summary, the residual stresses are associated with: (a) a change of mechanical properties with cooling; (b) a change of density; (c) heterogeneous cooling; (d) circumferential and axial loadings; and (e) viscoelasticity and stress relaxation. The precise measurement of the profile of residual stresses in the pipe thickness is difficult. Of the different techniques, the layer removal method [16] has been used most often for injection moulding parts. The application to pipes is not straightforward, as an initially planar sample is needed. Nevertheless, it is very easy to see the consequences of residual stresses in a thin pipe by cutting it parallel to the pipe axis. Figure 3.22 shows that, for a low depression, the LDPE pipe closes on itself, whereas for a high depression, the pipe opens. The HDPE pipe always closes on itself, more than one revolution for the lowest tested depression.

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Design of Extrusion Forming Tools

200 α0

–400 –500 0

200 300 400 100 Level of vacuum (hPa)

500

Figure 3.22 Evolution of the angle of opening or closure (α) for thin PE pipes, as a function of the level of vacuum

3.4 Process-induced Microstructure and Properties It is important to carefully characterise the microstructure in the pipes, because it is a key point for a correct understanding of their properties, as will be shown later. The general concepts concerning polymer crystallisation and structure development in processing have been outlined in Section 2.4. The experimental results presented here concern the pipes introduced in Section 3.3.2. More detailed information can be found in [17-19] for PA12, and in [14] for PE.

3.4.1 Orientation No semicrystalline morphology can be distinguished by optical microscopy in either the PA12 or the PE pipes. On the other hand, light microscopy can be used to characterise the global orientation by birefringence measurements. The identification of the extinction lines observed on polished sections of PA12 pipes (Figure 3.23a) reveals high birefringence values at the skin, decreasing rapidly from the external surface to the inside of the pipe. At the external surface, they are always greater than 0.018, while they are lower than 0.01 beyond the 100 microns below the external surface. Observation of thin microtomed sections confirms a considerable orientation of the external skin, which appears as a bright region in an overexposed picture

88

Pipe Forming Tools (Figure 3.23b). The thickness of this skin varies from 11.6 µm to 23.3 µm and its birefringence is between 0.018 and 0.033. For given processing conditions, the level of birefringence depends on the polymer rheology, which is related to the molecular architecture. For instance, high birefringence values are observed in LLDPE (Figure 3.24), but not in LDPE and HDPE. Melting experiments have shown that birefringence is due to orientation and not to thermal residual stresses; the skin keeps its brightness up to melting. Thermal residual stresses would have relaxed at a lower temperature.

100 µm

(a)

20 µm

(b)

Figure 3.23 PA12 pipe (r, z) sections observed by transmission optical microscopy (r, radial direction and z, extrusion direction): (a) polished section (thickness about 200 mm) observed in monochromatic light (l = 546 nm); and (b) microtomed section observed in white light (thickness about 10 mm). Reproduced with permission from J.M. Haudin, A. Carin, O. Parant, A. Guyomard, M. Vincent, C. Peiti and F. Montezin, International Polymer Processing, 2008, 23, 55. ©2008, Hanser [22]

The interpretation of birefringence in terms of orientation is confirmed by wide angle X-ray diffraction (see Section 2.4.3), which clearly shows crystalline orientation in the skin. In Debye-Scherrer patterns, the 001 ring of the PA12 structure presents intensity reinforcements perpendicular to the extrusion direction (Figure 3.25a). The level of orientation has been quantified by the 001 pole figure (Figure 3.25b). As in PA12 the chain axis is the b axis of the unit cell, these results demonstrate an orientation of the crystalline chains in the extrusion direction.

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Design of Extrusion Forming Tools

Birefringence ( × 100)

2.5 0,025 2.0 0,02

20 mmin .min-1 m/min 2020 m/

1.5 0,015 1.0 0,01

0.5 0,005

8 m/min m.min Epaisseur (rayon en mm)8

-1

00 25.1 25,1

25 25

25.2 25,2

/

25.3 25,3

i

25.4 25,4

25.5 25,5

Radius (mm) Figure 3.24 Birefringence profiles in the thickness of LLDPE pipes obtained for two extrusion line speeds: 8 m.min-1 and 20 m.min-1

1.00 1.20 1.40 1.60

TD

ND

(a)

ED

(b)

Figure 3.25 Characterisation of the crystalline orientation in the external layer of a PA12 pipe: (a) Debye-Scherrer diagram; the extrusion direction is horizontal; (b) 001 pole figure; ED - extrusion direction (z); TD - transverse direction (q); and ND - normal direction (r)

90

Pipe Forming Tools The high level of orientation in the external surface layer, as well as its increase with extrusion line speed (Figure 3.24), can be attributed to the calibration step. The combination of quenching and mechanical drawing in the calibrator induces a plastic deformation of the outer layers. This statement is validated by the relation between birefringence of the pipe external skin and the parameter Drtank introduced in Section 3.3.3 (Figure 3.26). Logically, macromolecular orientation appears to be affected by the strength of the tube drawing through the calibrator: increasing Drtank leads to larger deformation at the pipe external surface, and to greater subsequent orientation.

4.0

Skin birefringence (× 100)

3.5 3.0 2.5 2.0 1.5 1.0 1.25

1.75

2.25

2.75

3.25

3.5

Drtank Figure 3.26 Skin birefringence versus draw ratio in calibration tank

3.4.2 Crystallinity Crystallinity in the pipe thickness can be measured in differential scanning calorimetry (DSC) experiments on thin slices cut out of the pipe at different depths. In PA12 pipes, three types of DSC traces are observed as a function of the location in the thickness (Figure 3.27). A single peak is observed in the skin region. For the intermediate layers (100, 200 and 300 µm from the outer surface), the main peak is preceded by

91

Design of Extrusion Forming Tools a shoulder, whose importance decreases with increasing depth. Finally, the internal region exhibits a specific behaviour: an exothermic peak is observed before the melting peak, which indicates that part of the polymer has recrystallised during heating in the calorimeter. This means that the crystallinity of these internal layers was initially lower (about 20 to 25% lower than in intermediate zones, whose crystallinity is in the 16-17% range).

Internal surface

External surface 0-25 µm

100-125 µm

200-225 µm

300-325 µm

500-1000 µm

(a)

Equivalent specific heat

1 2 3 4

4 J kg–1 ºC–1

5

140

150

160

170

180

190

200

Temperature ºC

(b) Figure 3.27 Crystallinity analysis in the thickness of a PA12 pipe: (a) location and thickness of (θ, z) cuts; (b) melting curves of the different cuts: (1) 0-25 μm; (2) 100-125 μm; (3) 200-225 μm; (4) 300-325 μm; and (5) 500-1,000 μm. Reproduced with permission from J.M. Haudin, A. Carin, M. Vincent, N. Amouroux, G. Bellet and F. Montezin, International Journal of Material Forming, 2010, 3, 225. ©2010, Springer [19]

92

Pipe Forming Tools These results can be understood by considering that crystallisation occurs under the combined effects of cooling and stress (drawing in the calibrator), according to the location in the pipe. In the skin zone, stress effects are very important, and lead to a significant orientation of the crystalline phase. In the intermediate region, orientation is weak, but stress effects seem to remain important enough to partly counterbalance the effects of cooling, which are weaker than in the skin zone. In the internal layers, there is no stress effect. Therefore, cooling predominates and quenching of the polymer leads to a lower crystallinity.

3.4.3 Surface State In some processing conditions, surface analysis reveals defects on the pipes. These defects are aligned either parallel (z direction), or perpendicular (q direction) to the extrusion direction. During tensile testing, the defects along q grow (Figure 3.28) and generally lead to rupture.

µm 30

61.9 mm

25

z

20 15 10 5

θ

0 .5

100 µm

.10 .15 .20 .25 .30

0.527 mm 0.529 mm

Figure 3.28 Three-dimensional image of the surface defects along q after an elongation of 80% along z. Reproduced with permission from A. Carin, J.M. Haudin, M. Vincent, B. Monasse, G. Bellet and N. Amouroux, International Polymer Processing, 2006, 21, 70. ©2006, Hanser [18]

The influence of the calibration parameters on the final surface state of the pipe is difficult to quantify. Indeed, the dimensions of the defects are the same for all the samples; no direct relation has been found between defect depth and calibration conditions. Only the number of defects varies from one condition to another. It seems to depend on the lubrication level in the calibrator and particularly on the lubricating water layer thickness.

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Design of Extrusion Forming Tools

3.4.4 Mechanical Properties Figure 3.29 shows the relationship between elongation at break (eB) and birefringence of the pipe external surface (DnSKIN). Elongation at break significantly decreases as the skin orientation increases: eB rises from 120 to 222% when DnSKIN is reduced from 0.033 to 0.018. A significant initial orientation limits the ability of macromolecular chains to be stretched during tensile testing, and consequently reduces the elongation at break. Since the level of chain extension is maximum at the external surface after extrusion, these are the layers that limit the elongation at break.

Elongation at break (%)

240 220 200 180 160 140 120 100 1

1.5

2

2.5

3

3.5

4

Skin birefringence (× 100) Figure 3.29 Elongation at break versus skin birefringence

The origin, number and arrangement of surface defects have also to be integrated into a general explanation of the origin of fracture. Performances of pipes are explained by the coupled influence of orientation and surface defects. Fracture is initiated by the surface defects, but a high level of orientation is necessary for the growth of initial defects. Thus, it is possible to optimise the elongation at break of the pipe by fitting the calibration parameters to reduce the calibrator draw ratio and increase the lubrication level, in order to limit the orientation and the number of defects.

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Pipe Forming Tools

3.5 Modelling of Calibration 3.5.1 Calculation of the Temperature Field

3.5.1.1 General Presentation The temperature field is calculated by solving the energy equation with appropriate boundary conditions [20-22]. The process is broken down into different cooling steps, characterised by their length and the boundary conditions on the pipe surfaces. During the first step (path in air between die exit and calibrator entrance), the polymer can be stretched. Pipe extrusion is considered as a steady-state process and temperature is assumed to depend only on r and z. Heat conduction in the flow direction (z) is neglected with respect to heat convection in that direction. Conversely, convection along r is neglected. Viscous heat dissipation is supposed to be small compared to other terms. With such assumptions, the energy equation is reduced to:

w( z )

k ∂ 2T (r , z ) ∂T (r , z ) = rc p ∂r 2 ∂z

(3.9)

This equation is solved by an explicit finite difference method, with a 2D mesh along z (i) and r (j).

3.5.1.2 Boundary Conditions In air or water, convective conditions are considered at the pipe surfaces:

h(Tsurface − T fluid ) = k

∂T ∂r



(3.10)

pipe surface

where h is the heat transfer coefficient, Tsurface is the surface temperature of the pipe and Tfluid is the temperature of the cooling fluid. When temperature measurements are available, h is usually adjusted to fit the experimental data.

95

Design of Extrusion Forming Tools For thermal exchanges in the calibrator, the heat transfer coefficient hcal can be calculated by a simple model (Figure 3.30):

e e 1 1 = + cal + film hcal hcal − wat kmetal kwater



(3.11)

where hcal-wat is the heat transfer coefficient between the calibrator and the water of the tank (~ 500 W.m-2. °C-1), ecal is the calibrator thickness (~ 1 mm), kmetal is the conductivity of the calibrator metallic alloy (~ 400 W.m-1. °C-1), efilm is the thickness of the water film between the pipe and the calibrator (~ 100 mm) and kwater is the conductivity of water (0.63 W.m-1. °C-1). Hence 1/hcal ~ 10-3 + 10-6 + 10-4, which shows that the major role in the heat transfer is played by convection between the calibrator and the water bath. Therefore, hcal can be taken to be identical to the heat transfer coefficient for direct contact between the pipe and the cooling water.

Water of the cooling tank q

Calibrator Water film PA12 tube

Figure 3.30 A simple model for thermal exchanges in the calibrator. Reproduced with permission from J.M. Haudin, A. Carin, O. Parant, A. Guyomard, M. Vincent, C. Peiti and F. Montezin, International Polymer Processing, 2008, 23, 55. ©2008, Hanser [22]

3.5.1.3 Crystallisation Crystallisation has been treated in two different ways. In a first approach, the energy equation was modified to take into account the heat liberated by crystallisation:

w( z ) 96

∂T ( r , z ) k ∂ 2T ( r , z ) DH ∂χ (r , z ) = ( ) + w z ∂z r c p ∂r 2 cp ∂z

(3.12)

Pipe Forming Tools where DH is the actual enthalpy of crystallisation per unit mass and χ(r,z) is the transformed volume fraction (solidified fraction) at point (r, z). The kinetic law χ(t) was described by the extension of Ozawa’s equation to nonconstant cooling rate (see Section 2.4.1). In a second simplified approach, the numerical code directly used an experimental crystallisation curve recorded at an appropriate cooling rate. The gradual release of the latent heat between the onset (Tonset) and the end (Tend) of crystallisation gives rise to an exothermal peak, which is approximated by an isosceles triangle with an area equal to the enthalpy of crystallisation per unit mass DH. At each node (i, j), the temperature is first calculated by solving Equation 3.9, i.e., without taking crystallisation into account. Then, the temperature value is modified to take into account the heat released by the mass element at node (i, j). The transformed fraction c is defined as the ratio of the heat already released to the total heat of crystallisation. In both approaches, the crystallisation temperature is usually defined as the temperature corresponding to c = 0.5.

3.5.1.4 Validation of the Model and Typical Results In the case of PE, the temperature of the internal surface of the pipe could be measured, and the two treatments of crystallisation presented above could be compared. It appeared that the simplified approach gave the better results. Indeed, a good agreement between calculations and experiments is observed both for different line speeds (Figure 3.17) and different pipe thicknesses (Figure 3.31). Therefore, this simplified method was used in subsequent stress calculations, since such calculations require an accurate description of the temperature field. For PA12 pipes, the experimental analysis presented above has shown the importance of drawing in the calibrator on structure development. Therefore, the model should consider the influence of flow on crystallisation kinetics. Unfortunately, no data are available for crystallisation under elongation, and this is the reason why the simplified approach has been retained. This approach works here since it allows, in a practical way, a balance between cooling effects (which tend to decrease the crystallisation temperature), and drawing ones (which tend to increase it). The temperature and transformed fraction profiles obtained by using the model are presented in Figure 3.32 for three locations in the pipe thickness: external surface, centre, and internal surface.

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Design of Extrusion Forming Tools

250 Thickness

Temperature (ºC)

200

1. 0.45 mm 2. 0.65 mm 3. 0.90 mm

150 100

1

2

3

50 0

0

0.2

0.4

0.6

0.8

1

Distance from extrusion die Figure 3.31 Evolution of inner temperature of LDPE pipes for different thicknesses. Comparison between calculations by the simplified crystallisation model (solid lines) and experiments (symbols). Pipe thickness: 0.45 mm (1), 0.65 mm (2), and 0.90 mm (3)

The temperature evolution is calculated from the die exit to the end of the process (8 m). It gives access to the mean cooling rates in the calibrator: 400 °C.s-1 (external surface), 60 °C.s-1 (middle) and 30 °C.s-1 (internal surface). This shows a strong cooling of the pipe external surface during calibration. With the scale used in Figure 3.32a, crystallisation appears as a very small accident on the temperature evolution, which is visible only at the internal surface. Therefore, we also use the transformed fraction curves (Figure 3.32b) and an enlargement of the temperature evolution at the external surface to analyse the crystallisation phenomena in more depth (Figure 3.33). In the external zone, the crystallisation exothermal peak is observed at about 0.11 m from the die exit, at a temperature of 125 °C. Crystallisation ends (transformed fraction equal to 95%) at 0.15 m from the die exit, compared to 1.5 and 1.8 m for the central and internal zones, respectively. Taking into account the distance between the die and the calibrator entrance (0.065 m), this means that most of the crystallisation of the external surface occurs inside the calibrator. The solidified layer at the calibrator exit is about 20-30 mm thick.

98

Pipe Forming Tools

Temperature (ºC)

200 1: External surface 2: Centre 3: Internal surface

150 3 100

2

50

1

0 0

1

2

3

4

6

5

7

8

Distance from extrusion die (m) (a)

Transformed fraction χ

1

0.75

1

0.5

2

1: External surface 2: Centre 3: Internal surface

3

0.25

0 0

0.5

1

1.5

2

2.5

3

Distance from extrusion die (m) (b)

Figure 3.32 Evolution of: (a) temperature, and (b) transformed fraction for three locations in the pipe thickness

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Design of Extrusion Forming Tools

Temperature (ºC)

Calibrator entrance

200 Calibrator exit 150

100

Crystallisation

50 0

0.05

0.1

0.15

0.25

0.2

0.3

Distance from extrusion die (m)

Figure 3.33 Enlargement of the temperature evolution at the external surface of the tube. Location of the crystallisation zone

3.5.2 Stress Development Model

3.5.2.1 General Presentation The mechanical model presented was developed by Parant [14, 22]. It was initially aimed at predicting the development of residual stresses in PE pipes, but it can also be used to describe the distribution of stresses during crystallisation. The problem, which is supposed to be axisymmetric, is treated using a slab method. In cylindrical coordinates, the components [U , V , W ] of the displacement field are in the form , which gives the following strain tensor, within the frame of the small strain assumption:



100

(3.13)

Pipe Forming Tools This implies that for an incompressible material, the axial component of strain εzz is constant:





(3.14)

With the same hypotheses, the stress equilibrium equations reduce to:









(3.15)

(3.16)

where σrr, σθθ and σzz are the diagonal components of the stress tensor in cylindrical coordinates.

3.5.2.2 Boundary Conditions Let us call Ri and Re the internal and external radius of the pipe, respectively. The internal surface of the pipe is subjected to atmospheric pressure (Patm). Hence:





(3.17)

At the external surface, two types of conditions have to be envisaged: (a) the pipe is in contact with the calibrator and there is no radial displacement (Equation 3.18); (b) the contact is lost and a boundary condition involving the pressure in the calibrator ( Pcalibrator ) has to be written (Equation 3.19):





(3.18)

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Design of Extrusion Forming Tools





(3.19)

Finally, the longitudinal stresses have to balance the external loading F, which consists of the stretching force in air and of the friction forces between the pipe and the calibrator:





(3.20)

3.5.2.3 Thermoelastic Model In a first approach, a thermoelastic behaviour is considered. To avoid numerical instabilities in the crystallisation range, the approximate analytical solution proposed by Stanisic and McKinley [23] has been preferred to a numerical one. For radial boundary conditions σσrrrr ( Ri ) = − Pi and σ σrrrr ( Re ) = − Pe , where Pi and Pe are the pressures imposed on the internal and external surfaces of the pipe, respectively, one obtains for the stress field:



(3.21)



(3.22)

where the functions

φ1 (r ) , φ 2 (r ) and φ3 (r ) are defined by:



102

(3.23)

Pipe Forming Tools



(3.24)





(3.25)

and σzz is obtained from:



(3.26)

E is the Young’s modulus, which depends on r, n is the Poisson’s ratio, and aT is the coefficient of thermal expansion. The radial displacement is given by:

(3.27)

The equations of the model can also be written in an incremental form, which will be used in the following. As the elastic modulus in the solid state depends on temperature, an additional stress increment must be introduced at each time increment to take this variation into account. For temperature-independent Poisson’s ratio, it can be written:





(3.28)

3.5.2.4 Viscoelastic Model An incompressible multimode Maxwell model allows us to take into account stress relaxation (see Section 2.4.4). A viscoelastic relaxation modulus E(t) is defined by:

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Design of Extrusion Forming Tools



(3.29)

where (G ( n ) , l( n ) ) are the (shear modulus, relaxation time) couples of a Maxwell spectrum with N elements. H(t) is the Heaviside function. Each stress increment due to external loading and calculated from Equations 3.21, 3.22, 3.26 and 3.28 is distributed into the modes of the relaxation spectrum and relaxed during the time interval t ′ − t with the relaxation time of each element:



(3.30)





(3.31)

The global stress increment is then:



(3.32)

The total stress is obtained by summation of all the stress increments. The influence of temperature is introduced into the model thanks to the time-temperature superposition principle (see Section 2.2.3).

3.5.2.5 Coupling with the Thermal Model The coupled mechanical and thermal problems are solved by a slab method. The stress development is divided into a succession of steps associated with time increments Dt, which correspond to Dz increments (slabs). The temperature variation in the pipe thickness DT(r) combined with the external loading on the slab make it possible to determine the strain and stress increments using the equations presented in the previous sections.

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Pipe Forming Tools

3.5.3 Orientation Development During crystallisation, the external longitudinal forces applied to the tube (stretching and friction forces) are distributed through the thickness. Figure 3.34 shows the distribution of the axial stresses σzzduring crystallisation, as calculated by the model in the case of a PA12 pipe: each σzz value corresponds to the stress distributed in the layer which has just crystallised. The progression of crystallisation is illustrated by the dotted line, which gives the evolution of the noncrystallised thickness in the process. For instance, at the position 0.2 m, the crystallised thickness is about 40 mm (i.e., the zone located between thicknesses 960 and 1000 mm) and the stress frozen in this layer located at 40 mm from the pipe surface is about 1 MPa. This typical stress distribution makes it possible to propose a general interpretation for the generation of orientation.

Thickness (µm)

900

9

800

8

700

7

600

6

500

5 σ zz Noncrystallised thickness

400 300

4 3

200

2

100

1

0

0 0

0.5 1 1.5 Distance from extrusion die (m)

σ zz(MPa)

σ yat 120 ºC

1000

2

Figure 3.34 Distribution of axial stresses σzz during crystallisation (solid line) and noncrystallised thickness (dotted line). PA12 pipe

Let us first recall some experimental results presented above. Due to the value of the extrusion line speed, the polymer may enter the calibrator at a lower velocity and reach the line speed close to the end of the calibrator. Therefore, in the calibrator, the

105

Design of Extrusion Forming Tools polymer experiences both quenching and drawing. Downstream of the calibrator, the polymer velocity is equal to the extrusion line speed and no further drawing occurs. This suggests that the external surface layer is solidified. This is in agreement with the thermal model, which shows that for this type of tube, most of the solidification of the external surface occurs in the calibrator (see Figure 3.33), and justifies a posteriori the simplified crystallisation model. In fact, the polymer is subjected to two types of drawing: drawing in air, between the die and the calibrator, and drawing in the calibrator. As shown in Figure 3.33, the first one occurs at very high temperature and does not generate any specific orientation. Consequently, it will not be considered here. Drawing in the calibrator has been characterised by the draw ratio in the calibration tank (Drtank). The high level of orientation in the skin layer, as measured by birefringence, has been related to Drtank (see Figure 3.26). The mechanical model provides a clear explanation of two major experimental facts: significant birefringence values in a 100 mm-thick superficial zone and a very oriented skin (thickness about 20 mm): (a) Figure 3.34 shows that the axial stresses are concentrated mainly in the first solidified layers: they decrease from 10 to 2 MPa in the first 20 mm. Globally, the mechanical loading is distributed over the outer 100 to 150 mm. This corresponds exactly to the birefringence distribution (see Section 3.4.1) and explains its mechanical origin. (b) According to the thermal model, the crystallisation temperature of the external layer of the pipe is about 125 °C. Previous mechanical testing has given the following values for the yield stress: sy = 9.9 MPa at 120 °C (shown in Figure 3.34) and sy = 6.7 MPa at 140 °C. From these data, it can be inferred that the extreme surface layers have undergone plastic deformation, which is responsible for the high level of orientation observed in these zones. Flow-induced crystallisation cannot explain the birefringence values obtained, especially as the elongation rates are moderate. We must remark that birefringence is in fact related to the stress difference σzz – σrr, but previous calculations [14] have shown that σrr is small compared to σzz. The same type of analysis has been applied to PE pipes. Figure 3.35 shows the fraction of axial loading equilibrated by noncrystallised layers for a LLDPE and a LDPE. Axial stresses rapidly decrease in the liquid phase of LLDPE and, consequently, will concentrate in the first solidified layers, inducing an important elongation in these layers. This explains the high orientation in the external layers (see Figure 3.24), as in the case of PA12.

106

Pipe Forming Tools

Thickness 100

0.4

80

Extrusion direction

0.3

60 %F LLDPE

0.2

%F LDPE

%F

Noncrystallised thickness (mm)

0.5

40 20

0.1

0

0 0

0.1

0.2

0.3

0.4

0.5

Distance from extrusion die (m)

Figure 3.35 Distribution of axial stresses during crystallisation: noncrystallised thickness (left) and fraction (in %) of axial loading equilibrated by noncrystallised layers (right), for a LLDPE and a LDPE In contrast, in LDPE, stresses in the liquid phase progressively decrease during solidification, and no birefringent layer is observed. The differences between the two polyethylenes can be related to the differences in their rheological behaviour: LLDPE has shorter relaxation times than LDPE, which is prepared by radical polymerisation. This is an important factor for the choice of a PE for pipe extrusion, since the presence of oriented layers has been associated with the occurrence of geometrical defects (‘tulip’ shape at the extremities, see Figure 3.18) when the pipes are heated in an oven.

3.5.4 Results of Residual Stress Calculations The computations presented in Section 3.5.2 lead to the stress profiles at the end of cooling [14]. Figure 3.36 shows the stress profile in LDPE tubes, with a viscoelastic multimode Maxwell model (see Section 2.4.4) When the tube is in contact with the calibrator, the friction is taken into account with a Coulomb law and a friction coefficient equal to 0.15. The axial loading also takes into account the drawing force necessary to stretch the pipe between the extrusion die and the calibrator, which has been measured to be around 10 N. A depression in the calibrator tank larger than 107

Design of Extrusion Forming Tools 12 kPa induces superficial tensile stresses in the pipe. The stress profile is similar to the one shown schematically in Figure 3.21. The surface tensile stress values increase with the depression. This is due to the increase in friction force and axial loading, and to associated stresses frozen in the first layer of solidified polymer. The stresses in the rest of the thickness do not depend strongly on the level of vacuum. In association with frozen-in orientation, the tensile surface stresses lead to the ‘tulip’ defect shown in Figure 3.18. The angle after cutting can be calculated and the results agree well with the measurements (Figure 3.37).

30000

24 kPa

σθθ (kPa)

20000

20 kPa

10000

16 kPa 12 kPa

0 0,0

0.1 0,1

0.2 0,2

0,3 0.3

0,4 0.4

0,5 4 kPa 0,6 0 kPa

–10000 Pipe thickness (mm) (a) 30000

24 kPa 20 kPa

σzz (kPa)

20000

16 kPa

10000

12 kPa

0 0

0,1 0.1

0.2 0,2

0.3 0,3

0,4 0.4

0,5 0 kPa

0,6

4 kPa

–10000 Pipe thickness (mm)

(b)

Figure 3.36 Calculated residual stress (a) sqq and (b) szz profiles in LDPE tubes with a viscoelastic model

108

Pipe Forming Tools

Angle after cutting (degree)

300 200 100 0 -100 0

10

20

30

-200 -300 -400 -500 -600 Depression (kPa)

Figure 3.37 Comparison of the calculated (solid line) and measured (dots) angles after cutting the tube along the z axis

3.6 Conclusion With the development of numerical simulation, precise descriptions of flow through extrusion dies, and particularly in pipe dies, are now available. This enables the design of the dies to be optimised. Fewer studies concern the calibration step in pipe manufacturing. A salient feature of this chapter is to show that the calibration conditions may greatly affect the final properties of the pipes, as well as their surface state. Our analysis is based on the combination of on-line measurements, characterisation of pipe microstructure and properties, and numerical simulation. When the velocity of the extrusion line is increased, the pipe may enter the calibrator at a velocity lower than the extrusion line speed. This induces a drawing of the polymer in the calibrator, which leads to a loss of mechanical properties and to a degradation of the surface state. From a practical point of view, the drawing kinematics are greatly influenced by lubrication conditions. Another important result is that, for given processing conditions, the final properties of the pipes depend greatly on the viscoelastic properties of the polymer.

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Design of Extrusion Forming Tools

References 1.

W. Michaeli, Extrusion Dies for Plastics and Rubber, Hanser, Munich, Germany, 2003.

2.

C. Rauwendaal, Polymer Extrusion, Hanser, Munich, Germany, 2001.

3.

B. Vergnes and J.F. Agassant, Advances in Polymer Technology, 1986, 6, 441.

4.

R.B. Bird, W.E. Stewart and E.N. Lightfoot. Transport Phenomena, Wiley, New York, NY, USA, 2007.

5.

P. Saillard, Les Phénomènes Thermiques dans les Outillages d’Extrusion des Matières Plastiques, Ecole des Mines de Paris, 1982. [PhD. Thesis]. [In French]

6.

P. Saillard, B. Vergnes and J.F. Agassant, Polymer Process Engineering, 1984, 2, 53.

7.

J. Phillips and W. Michaeli in the Proceedings of ANTEC’98, Atlanta, GA, USA, p. 2596.

8.

F. Fan Ding and J. Giacomin, Journal of Polymer Engineering, 2000, 20, 1.

9.

K. Yagi and K. Masumoto, inventors; Mitsui Petrochemical Industries Inc., assignee; US 4,159,889, 1979.

10. W. Schreiner, inventor; Farbenfabriken Bayer AG, assignee; GB 1,193,291, 1970. 11. Dynamit Nobel AG, assignee; GB 1,529,370, 1978. 12. T.C. Pearson, inventor; Gas Injection Ltd., assignee; GB 2,354,965, 2001. 13. D. Gneuss, inventor; D. Gneuss, assignee; German Patent 3,243,140, 1984. 14. O. Parant, Etude Expérimentale et Calcul des Contraintes Résiduelles dans des Tubes Extrudés en Polyéthylène, Ecole des Mines de Paris, 2002. [PhD Thesis]. [In French] 15. A. Carin, J-M. Haudin, M. Vincent, B. Monasse, G. Bellet and D. Silagy, International Polymer Processing, 2005, 20, 1. 16. R.G. Treuting and W.T. Read, Journal of Applied Physics, 1951, 22, 130.

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Pipe Forming Tools 17. A. Carin, J.M. Haudin, M. Vincent, B. Monasse, G. Bellet and N. Amouroux, International Polymer Processing, 2005, 20, 305. 18. A. Carin, J.M. Haudin, M. Vincent, B. Monasse, G. Bellet and N. Amouroux, International Polymer Processing, 2006, 21, 70. 19. J.M. Haudin, A. Carin, M. Vincent, N. Amouroux, G. Bellet and F. Montezin, International Journal of Material Forming, 2010, 3, 225. 20. D. Cotto, P. Duffo and J.M. Haudin, International Polymer Processing, 1989, 4, 103. 21. P. Duffo, B. Monasse and J.M. Haudin, Journal of Polymer Engineering, 1991, 10, 151. 22. J.M. Haudin, A. Carin, O. Parant, A. Guyomard, M. Vincent, C. Peiti and F. Montezin, International Polymer Processing, 2008, 23, 55. 23. M. Stanisic and R.M. McKinley, Ingenieur-Archiv, 1959, 27, 227.

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Design of Extrusion Forming Tools

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4

Flat Film and Sheet Dies

J. Vlachopoulos, N. Polychronopoulos, S. Tanifuji, J. Peter Müller 4.1 Film Casting and Sheet Extrusion In flat film or sheet production the first objective is to spread a continuous polymer melt stream coming from an extruder into a die, which terminates in a rectangular and wide cross–section, having a small gap. After the die the molten extrudate is cooled on chilled rollers and solidifies, as shown in Figure 4.1. The figure also shows corona treatment to render the surface more receptive to inks, adhesives or other coatings, beta gauge for measuring the thickness and trim reclaim. Products of less than 0.25 mm in thickness are referred to as films and those over 0.25 mm are referred to as sheets [1].

VIRGIN RESINS

RECLAIM HOPPER

GRINDER

TRIM PIPE

EXTRUDER BETA GAUGE

EXTRUDER EXTRUDER EXTRUDER

CHILL ROLL CHILL ROLL

SLITTER CORONA THEATMENT

FINISHED ROLLS

Figure 4.1 A four-layer cast film line. Adapted from J. Ivey in The SPE Guide on Extrusion Technology and Troubleshooting, Eds. J. Vlachopoulos and J.Wagner, Society of Plastics Engineers, Brookfield, CT, USA, 2001, 12.1 [1]

113

Design of Extrusion Forming Tools The rate at which the extrudate is cooled determines several important properties of the finished product. Longer cooling means there is more time available for crystal growth and thus the crystallites will be larger [2]. Crystallinity affects the density, optical properties, coefficient of friction, impact, barrier and other properties. Compared to blown film, the cast film process shows better optical properties, higher output rate per hour, lower gauge variation and lower converting cost. Most cast film lines manufactured today are coextrusion lines and in fact Figure 4.1 shows a four-layer line. Coextrusion is defined as the process of simultaneous extrusion of two or more materials through a common die. It is used for the purpose of combining material properties and reducing the cost at the same time. Thickness uniformity in monolayer extrusion and layer uniformity in coextrusion are the key measures for quality. Sheet lines, as noted earlier, are lines that produce film with a thickness exceeding 0.25  mm. Typically a sheet line will have a three-roll coating stack after the die, as shown in Figure 4.2, and again the cooling rate plays a very important role in determining the properties of the finished product. A detailed troubleshooting guide of monolayer and coextruded sheet is available in the open literature [3]. Several issues relating to film processing, materials and properties are discussed in a handbook [4].

Extruder Gauge 3 Roll Stack

Figure 4.2 Sheet extrusion line

4.2 Flow Distribution and Channel Design The molten polymer stream coming from an extruder must be distributed as uniformly as possible into a rectangular shaping area so that a thin wide sheet or film of uniform thickness is produced continuously. Between the melt pipe, coming from the extruder, and the rectangular die lips a distribution channel (usually called

114

Flat Film and Sheet Dies a manifold) is needed. The most common dies [5-8] utilise either the simple ‘T-slot’ or the ‘coathanger’ geometry. T-slot dies are the simplest to manufacture. They have a large manifold of usually circular cross-section, which is constant across the entire width of the die, as shown in Figure 4.3. There is very little resistance to flow from the centre (feed) to the side ends of the die and even flow distribution is accomplished by the flow controlling action of the die lips. Such dies are used for low viscosity polymers (high melt flow ndex resins) mainly for extrusion coating applications. A less common type of die is the ‘fishtail’ design, shown schematically in Figure 4.4. Coathanger dies usually involve [5] a manifold, a preland, possibly a flow restrictor (also called a ‘choker bar’), a secondary manifold and finally the primary land (die lips) as shown schematically in Figure 4.5. A picture of the lower half of a modern flat die is shown in Figure 4.6.

Front View

Side View

Figure 4.3 T-slot die with a constant cross-section circular manifold. Adapted from D.R. Garton in Film Extrusion Manual, Eds., T.I. Butler and E.W.Veazey, TAPPI, Atlanta, GA, 1992, p.231 [5]

Figure 4.4 Fishtail die. Adapted from W. Michaeli, Extrusion Dies for Plastics and Rubber, 2nd Edition, Hanser Publishers, Munich, 1992 [6]

115

Design of Extrusion Forming Tools

A

B2

B1

A

A B1

C

Front View

A B2

D

Side View

Figure 4.5 Coat hanger die having a teardrop shaped manifold with a diminishing cross-sectional area from the centre to the sides. A region is the manifold, B1 and B2 are lengths of the preland, C is a secondary manifold and D the land (die lips). Adapted from D.R. Garton in Film Extrusion Manual, Eds., T.I. Butler and E.W.Veazey, TAPPI, Atlanta, GA, 1992, p.231 [5]

Figure 4.6 Picture of the lower half of a modern flat die having a lower sliding lip, which can be adjusted during production. Reproduced with permission from EDS GmbH, Kirchdorf, Austria

116

Flat Film and Sheet Dies The manifold cross-sectional area is frequently teardrop shaped (see Figure 4.7) and is gradually reduced from the centre (feed) to the side ends. Rectangular manifolds (see Figure 4.7) are used in coextrusion and again the cross-sectional area is reduced from the centre to the sides. The function of the manifold is to force the polymer to the sides and downstream at the same time for the generation of a nearly uniform flow distribution by the end of the preland, so that the necessity for subsequent corrections is minimised. The shape and the dimensions of the manifold are crucial in designing a die capable of producing a film or sheet of uniform cross-section from the die lips. Teardrop shaped manifolds have evolved over the years from flat back to curved back, as shown in Figure 4.8. The most common manifold design by far, in the current market, is the straight backline. More sophisticated designs involve a parabolic backline in combination with a parabolic shaped preland, as shown in Figure 4.9. This design is known to reduce what is usually referred to as the ‘M’ or ‘W’ flow output problem of the film or sheet produced, being heavy on each end then having a thin area followed by a thick area in the centre (which can be perceived as having the shape of the letter W or an inverted one).

CIRCLE SHAPE

TEARDROP SHAPE

RECTANGULAR SHAPE

Figure 4.7 Common types of manifold cross-sections

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Design of Extrusion Forming Tools

Figure 4.8 Evolution of the teardrop manifold shape from the older flat back (left) to the modern curved back (right). The highest velocity is in the centre and the lowest on the walls

Figure 4.9 Parabolic backline manifold and preland

Most flat dies include some kind of lip-adjusting systems for fine-tuning of the uniformity. These might be simple adjusting screws or very sophisticated arrangements involving thickness measurement and feedback control. However, these adjusting systems are not capable of correcting large flow nonuniformities which result from poor manifold and preland design. All channel sections must be streamlined, as much as possible, and capable of providing smooth melt flow without any stagnating or recirculating flow regions. A mechanical drawing of a cross-section of a die having a restrictor bar and lip adjustment is given in Figure 4.10.

118

Flat Film and Sheet Dies

Figure 4.10 Cross-sectional area of a die having a restrictor bar and a lip adjustment system. Reproduced with permission from EDS GmbH, Kirchdorf, Austria

Flat die design practitioners also recommend that for film production (especially if transparent) the minimum wall shear rate must not be less than 8 s-1. Low wall shear rates are likely to result in visual defects on the film due to polymer degradation, which may look like brown or black spots, haze bands or even a generalised deterioration of the appearance of the sheet or film. Occasionally, such defects might be confused with sharkskin. The origin, however, is totally different. Sharkskin occurs at the die lip exit as discussed in [9, 10] and Chapter 1 of this book. The low wall shear rate effect originates upstream where the flow channels are deep and consequently the corresponding shear rates may be very low. The previously mentioned minimum wall shear rate value of 8 s-1 has been known and quoted by die designers for several years, even before the extensive use of computer simulation tools, which make possible the accurate determination of the shear rate for a given geometry and given flow rate. Due to long residence times and some sort of sticking of the polymer melt at the die surface, chain scission, cross-linking or other thermal degradations may occur. Some temperature sensitive polymers, notably ethylene-vinyl alcohol (EVOH), polyvinyl chloride (PVC), polyvinylidene chloride and ethylene-vinyl acetate (EVA), are particularly susceptible to this defect. For such materials, the minimum required wall shear rate value to avoid degradation is probably higher, but there have not been any published studies about this available in the open literature. In coextrusion, the EVOH or the EVA layer may contain defects as a result of this sort of degradation,

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Design of Extrusion Forming Tools but the other layers could be defect-free, even though low wall shear rates might be encountered in all layers. In such a case, the degradation of the temperature sensitive layer might be confused with interfacial stability [10, 11], which is discussed later in this chapter. In addition to good flow channel design, it is important that the die body be free from temperature variations during production. Locally higher temperature is likely to produce a heavy-gauge band, due to higher flow rate, while a locally lower temperature is likely to produce a thin-gauge band. Insulation and temperature control of the die body are essential for achieving film or sheet with low thickness tolerances. Flat dies are usually manufactured in widths ranging from 700 mm to 3,500 mm but may occasionally exceed 5000 mm. Film or sheet thicknesses usually range from 10 μm to 30 mm. Deckling systems are used to reduce the width of film or sheet produced. As Garton [5] put it, they are considered a necessary evil in the industry. They compromise the flow distribution because of the restrictions on the two sides of the die. They should definitely be avoided when extruding thermally degradable polymers. Garton [5] recommends that no more than 25% of the total die width should be deckled. Despite the fact that deckling systems do not produce anything resembling a streamlined flow (which is dictated by rheology), many dies are deckled down to almost 50% of the original slot width. Due to the large forces that may develop during extrusion and because a flat die is clamped together at the edges, deflection of the die may occur with the largest magnitude at the centre. This is usually referred to as clamshelling. It results in increased flow in the central region, which must be compensated for through lip adjustments.

4.3 Mathematical Modelling Carley [12] was the first to develop design equations for T-shaped dies assuming Newtonian flow behavior. Pearson [13] extended the design equations to power law fluids. McKelvey and Ito [14] proposed as the design objective the uniformity of flow rate across the die width. These early approaches are elucidated and explained by Tadmor and Gogos [15]. A design method focusing on the distribution problem was proposed by Winter and Fritz [16]. Vlcek and co-workers [17] developed a control volume approach and a software package for flow simulation as the polymer melt spreads laterally and flows downstream. This approach enabled the examination of alternatives, such as the shape of the manifold, flow restrictions and temperature effects. These authors also presented comparisons to experimental data for a small laboratory, die.

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Flat Film and Sheet Dies For simulation of polymer melt flow through the channels of a flat die, the equations of conservation of mass, momentum and energy under creeping flow conditions (Reynolds number