Technology of Thermoforming

James L. Throne Technology of Thermoforming Hanser Publishers, Munich Vienna New York Hanser/Gardner Publications, Inc...

Author: James L. Throne

681 downloads 2261 Views 13MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

James L. Throne

Technology of Thermoforming

Hanser Publishers, Munich Vienna New York Hanser/Gardner Publications, Inc., Cincinnati

The Author: Dr. James L. Throne, Sherwood Technologies, Inc., 158 Brookside Blvd., Hinckley, OH 44233-9676, USA

Distributed in the USA and in Canada by Hanser/Gardner Publications, Inc. 6600 Clough Pike, Cincinnati, Ohio 45244-4090, USA Fax: (513) 527-8950 Phone: (513) 527-8977 or 1-800-950-8977 Distributed in all other countries by Carl Hanser Verlag Postfach 86 04 20, 81631 Miinchen, Germany Fax: +49 (89) 98 12 64 The use of general descriptive names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. While the advice and information in this book are believed to be true and accurate at the date of going to press, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

Library of Congress Cataloging-in-Publication Data Throne, James L., 1937Technology of thermoforming / James L. Throne. p. cm. Includes bibliographical references and index. ISBN 1-56990-198-8 1. Plastics-Molding. I. Title. TP1150.T48 1996 668.4' 12-dc20

96-24175

Die Deutsche Bibliothek-CIP-Einheitsaufnahme Throne, James L.: Technology of thermoforming / James L. Throne.-Munich ; Vienna ; New York : Hanser ; Cincinnati : Hanser / Gardner, 1996 ISBN 3-446-17812-0 All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying or by any information storage and retrieval system, without permission in writing from the publisher. © Carl Hanser Verlag, Munich Vienna New York, 1996 Typeset in Ireland by Datapage International Ltd., Dublin Printed and bound in Germany by Kosel, Kempten

Preface I completed the monograph, Thermo forming, in 1985 and Carl Hanser Verlag published it in 1987. At the time it was written, there were no up-to-date monographs devoted solely to thermoforming. Two monographs devoted to thermoforming appeared about the same time as the Thermoforming monograph. They are: J. Florian, Practical Thermoforming: Principles and Applications, Marcel Dekker, Inc., New York (1987). G. Gruenwald, Thermoforming: A Plastics Processing Guide. Technomic Publishing Co., Inc., Lancaster PA (1987). Reviewers of the three monographs remarked at the time that while there was some overlap, the three monographs provided separate and unique insights to the industry and that technologists would do well to have access to all three. In the preface to that monograph, I stated that: "Thermoforming is not an easy process. It just looks easy."

In many respects, the industry has verified this over and over in the last decade. I further stated that: " . . . [Thermoforming] is becoming challenging as newer process variations, newer materials, tighter sheet and part tolerances, more critical applications and more sophisticated controls are developed."

In the last decade, the industry has successfully tackled many of these very difficult problems. The core business has been helped greatly by improved heaters, more accurate process controls, cooperative interaction between extruders and formers, more easily thermoformable polymers, advanced trimming techniques, and to a great degree, the acceptance of thermoforming as a process by OEMs. In addition, thermoforming has gained the attention of many universities, technical writers, consultants and software companies. And the industry has grown dramatically in size. As an example, the Society of Plastics Engineers Thermoforming Division held a fall conference in Wisconsin in 1991. About 100 people attended. In Midland MI in 1992 more than 200 attended. In South Bend IN in 1993, more than 400 attended, in Atlanta in 1994, more than 600 attended, and in Cleveland in 1995, nearly 700 attended. And plans are being made for nearly 800 in Cincinnati in 1996. There are more than 200 custom thermoformers in the US today and some estimate that a new thermoforming company is born every week. The industry seems to have come of age in the last ten years. This work was intended to be a revision and update of the 1987 monograph. Readers of that monograph will note that this book is much larger. This book is also an overview of the technical aspects of thermoforming and generally follows the outline of the Thermoforming monograph. However, I have included worked-out examples and many guidelines to illustrate and support the technical aspects. As with the 1987 monograph, the material in each chapter of this book moves from relatively simple concepts to more technical, in-depth considerations. This book has ten chapters:

Chapter 1, "Thermoforming—Definitions, History, Methods and Equipment", is a proper introduction to the subject. The chapter includes some history, some market information, a glossary of definitions, the traditional methods of forming and some technical considerations about the machinery. Chapter 2, "Polymeric Materials", briefly reviews the nature of thermoformable polymers and their adducts. General concepts of polymer response to applied loads and temperatures are considered. Some important new information on infrared energy absorption is detailed. Chapter 3, "Heating the Sheet", reviews the three general ways of heating sheet—conduction, convection and radiation. Since infrared radiation is the most popular and efficient means of heating sheet, fundamental aspects are discussed. This chapter becomes quite technical with general guidelines for determining heating cycles and a new section on computer-genera ted prediction of sheet temperature. Chapter 4, "Stretching the Sheet", is concerned with fundamentals of multiaxial sheet deformation. Polymer hot strength is related to tensile characteristics of rubbery solids and viscosity of elastic liquids. Sheet sag is shown to be strongly related to polymer hot strength. Again, the material is quite technical. Chapter 5, "Cooling and Trimming the Sheet", deals with heat removal from the sheet while against the mold surface. Some new material on computer-generated mold temperature prediction is given. The mechanics of cutting the molded part from the web are considered in detail. Chapter 6, "Thermoforming Molds", considers mold materials and mold designs. Vacuum or vent hole sizes and numbers are arithmetically determined and plug materials and designs are also discussed. Chapter 7, "Parts Design", first considers the economics of parts design. Draw ratios are then defined and wall thickness prediction methods discussed. Regrind and material property loss are considered in detail, and an extensive section on part design guidelines follows. Chapter 8, "Producing Sheet and Film", is a new chapter. Since the thermoformer is the customer of the extruder, he/she should know some rudimentary extrusion concepts. This chapter is a brief summary of the extrusion process, with emphasis on sheet quality and quality control. A sheet quality checklist is discussed in detail. Chapter 9, "Newer Thermoforming Technologies", is also a new chapter, written in response to many requests for forming information in several new processing areas. In the 1987 monograph. I said: " . . . [EJngineers seeking the latest information on pressure forming, the heating of foam sheet or forming crystallizing PET will be disappointed."

This is no longer the case. Chapter 9 presents vignettes on the following forming techniques: • • • •

CPET, Pressure forming, Forming filled and reinforced polymers, Laminated sheet forming,

• • • •

Twin-sheet forming, Forming PP, Thermoforming foam sheet, and Other semi-thermoforming technologies.

Chapter 10, "Set-Up Protocols, Troubleshooting, and the Economics of Thermoforming", is an assemblage of production and economic issues that were scattered through several chapters of the 1987 monograph. Guidelines to setting up a new mold, forming a new polymer and troubleshooting both the thin-gage and heavygage forming process are found here. Thermoforming is an energy intensive process that uses only a portion of its raw material, sheet, to make the part. To be competitive, the thermoformer must know what things cost, in detail. This chapter focuses on this theme. Caveats are in order. Some of the engineering details are quite technical. The monograph is designed to provide a technical foundation for the industry. Nevertheless, the casual reader should find ample guidelines, protocols, tips and rules-ofthumb to help him/her with specific processing problems or new product planning. The decade or so since I wrote the Thermoforming monograph has been marked by retirement and deaths of many thermoforming leaders. Most notably, Dr. Herman "Dick" Osmers, SPE Thermoformer-of-the-Year, fellow PhD chemical engineer-consultant-teacher and critical reviewer of the 1985 book, died shortly after it was published. I will always miss his technical accuracy, his thoroughness, and his up-beat enthusiasm. This book is dedicated to his memory. January 1996

James L. Throne, PhD

Contents

Preface ......................................................................................... 1.

v

Thermoforming – Definitions, History, Methods and Equipment ............................................................................

1

1.1

Introduction ...............................................................................

2

1.2

History .......................................................................................

3

1.3

Markets .....................................................................................

4

1.4

Some Definitions .......................................................................

11

Gage .................................................................................

12

Clamping of Thin-Gage Sheet ...........................................

12

Clamping of Heavy-Gage Sheet ........................................

13

Heating of Thin-Gage Sheet ..............................................

13

Heating Heavy-Gage Sheet ...............................................

14

Shaping Thin-Gage Sheet .................................................

14

Shaping Heavy-Gage Sheet ..............................................

14

Trimming the Thin-Gage Sheet ..........................................

15

Trimming the Heavy-Gage Sheet .......................................

15

Depth-of-Draw ...................................................................

15

Methods of Forming ..................................................................

16

One-Step Forming .............................................................

17

Two-Step Forming with Prestretching ................................

19

Multi-Step Forming ............................................................

22

Other Variations .................................................................

25

1.5

This page has been reformatted by Knovel to provide easier navigation.

ix

x

Contents 1.6

2.

Thermoforming Machinery .......................................................

29

Heating Source ..................................................................

29

Forming Platform ...............................................................

31

Vacuum System ................................................................

32

Pressure System ...............................................................

36

Process Control .................................................................

36

Trimming and Cut Parts Handling ......................................

38

1.7

Heavy-Gage Thermoforming Machinery Specifics ..................

38

1.8

Thin-Gage Thermoforming Machinery Specifics .....................

46

1.9

References ...............................................................................

52

Polymeric Materials .............................................................

54

2.1

Introduction ...............................................................................

55

2.2

Network Nature of Polymers ....................................................

55

2.3

Addition and Condensation Polymerization .............................

57

2.4

Aromatic and Aliphatic Polymers .............................................

58

2.5

Molecular Weight and Molecular Weight Distribution ..............

58

2.6

Molecular Weight and Properties .............................................

61

2.7

Morphology and Properties ......................................................

63

2.8

Molecular Orientation ...............................................................

70

2.9

Chain Mobility and Polymer Stiffness .......................................

70

2.10 Stress-Crack Resistance ..........................................................

72

2.11 Gas Permeation ........................................................................

72

2.12 Copolymerization ......................................................................

73

2.13 Blends .......................................................................................

74

2.14 Adducts .....................................................................................

74

Plasticizers ........................................................................

74

Other Additives ..................................................................

76

Fillers and Reinforcing Fibers ............................................

77

2.15 Laminates .................................................................................

78


Contents

xi

2.16 Stress-Strain Behavior of Plastics ............................................

78

2.17 Thermal Properties ...................................................................

82

Heat Capacity ....................................................................

82

Thermal Conductivity .........................................................

84

Thermal Diffusivity .............................................................

84

Thermal Expansion Coefficient ..........................................

87

2.18 Infrared Spectra ........................................................................

87

2.19 Summary ..................................................................................

96

2.20 References ............................................................................... 103

3.

Heating the Sheet ................................................................ 105 3.1

Introduction ............................................................................... 106

3.2

Energy Absorption by Sheet ..................................................... 106

3.3

Heat Transfer Modes ................................................................ 110

3.4

Incorporating Formability and Time-Dependent Heating ......... 115

3.5

Conduction ................................................................................ 121

3.6

Convection Heat Transfer Coefficient ...................................... 124

3.7

3.8

3.9

The Biot Number ...............................................................

125

Effective Radiation Heat Transfer Coefficient ....................

126

Constant Heat Flux ............................................................

127

Radiation Heating ..................................................................... 128 Black Body Radiation .........................................................

129

Gray Body – Emissivity ......................................................

134

Radiant Heater Efficiency – Constant Heat Flux Application ...................................................................

138

Real Heaters – Efficiencies ...................................................... 140 Radiative Heat Transfer Coefficient ...................................

144

Convection and the Heat Transfer Coefficient ...................

145

Rod Heaters ......................................................................

150

Long-Term Radiant Heater Efficiencies ................................... 151


xii

Contents 3.10 Edge Losses – View Factor ...................................................... 152 Local Energy Input .............................................................

155

Pattern Heating ..................................................................

159

Zone, Zoned or Zonal Heating ...........................................

161

Heater to Sheet Distance ...................................................

162

3.11 Thin-Gage Sheet – Approximate Heating Rates ..................... 164 Constant Environmental Temperature Approximation ........

164

Constant Heat Flux Approximation ....................................

167

Thin-Gage Approximations – Comments ...........................

167

3.12 Heavy-Gage Sheet – Internal Temperature Control ................ 168 Constant Environmental Temperature ...............................

168

The Constant Heat Flux Case ............................................

172

The Thickness Effect .........................................................

174

Summary ...........................................................................

175

3.13 Equilibration .............................................................................. 176 Convection Heating ...........................................................

177

Constant Heat Flux ............................................................

179

Computed Equilibration Times ...........................................

180

The W-L-F Equation ..........................................................

181

The Arrhenius Equation .....................................................

182

Relating Shift Factors to Sheet Thickness .........................

182

3.14 Infrared-Transparent Polymers ................................................ 182 3.15 Computer-Aided Prediction of Sheet Temperature ................. 188 The Radiant Boundary Condition .......................................

192

3.16 Guidelines for Determining Heating Cycles ............................. 192 The Biot Number ...............................................................

193

Thin-Gage Guidelines ........................................................

193

Heavy-Gage Guidelines .....................................................

193

Intermediate-Gage Guidelines ...........................................

194

3.17 References ............................................................................... 194


Contents 4.

xiii

Stretching the Sheet ............................................................ 197 4.1

Introduction ............................................................................... 198

4.2

The Stretching Concept ............................................................ 202

4.3

Polymer Hot Strength ............................................................... 208

4.4

4.5

Standard Tensile Tests ......................................................

208

Hot Tensile Tests ...............................................................

212

Hot Creep Tests ................................................................

213

Other Stretching Tests .......................................................

215

Temperature-Dependent Viscosity for Amorphous Polymers ......................................................................

216

Dynamic Mechanical Testing .............................................

222

Stress-Strain-Rate of Strain – Theory ...................................... 226 Elasticity – a Rationalization ..............................................

233

Strain Energy Function ......................................................

235

The Rivlin Form for the Strain Energy Function .................

236

The Ogden Form for the Strain Energy Function ...............

239

Viscoelastic Models ...........................................................

240

Available Stress-Strain Data .................................................... 242 Sensitivity of Models ..........................................................

247

4.6

The Importance of Polymer Material Properties ...................... 248

4.7

Practical Aspects of Stretching ................................................. 257 Funnel Test .......................................................................

260

4.8

Bursting Conditions .................................................................. 264

4.9

Sheet Sag ................................................................................. 266 Initial Sag ..........................................................................

267

Tensile Sag .......................................................................

268

The Catenary Sag .............................................................

269

Parabolic Sag ....................................................................

270

Relating Sag to Hot Sheet Strength ...................................

271

Sag – a Comment ..............................................................

276


xiv

Contents 4.10 References ............................................................................... 277 Appendix 4.I: Biaxial Stretching of an Elastic Membrane ................... 282

5.

Cooling and Trimming the Part .......................................... 284 5.1

Introduction ............................................................................... 285

5.2

Overall Cooling Heat Balance .................................................. 285

5.3

Cooling the Formed Shape ...................................................... 287

5.4

Steady State Heat Balance ...................................................... 288

5.5

5.6

Interfacial Resistance ........................................................

289

Shape Factor .....................................................................

291

Convection Heat Transfer Coefficient ................................

293

Cyclic Heat Balance ................................................................. 302 Cooling the Free Surface of the Sheet ...............................

303

Cooling Thin Sheet in Ambient Air .....................................

303

Transient Heat Removal from the Sheet ............................

305

Quiescent Ambient Air .......................................................

305

Moving Ambient Air ...........................................................

307

Cooling on Nonmetallic Molds ...........................................

309

Transient Heat Transfer during Sheet Cooling on the Mold Surface – Computer Models ............................................ 314 Interfacial Air .....................................................................

5.7

5.8

5.9

318

Shrinkage .................................................................................. 320 Unconstrained Shrinkage ..................................................

321

Constrained Shrinkage ......................................................

325

Trimming ................................................................................... 329 Trimming Heavy-Gage Parts .............................................

331

Trimming Thin-Gage Parts .................................................

333

Mechanics of Cutting ................................................................ 334 The Trim Region ................................................................

336

Registering the Trim Site ...................................................

337


Contents

xv

The Nature of the Cut ........................................................

338

Fracture Mechanics ...........................................................

340

Mechanical Chipping .........................................................

340

Multiple-Edged Tool or Toothed Saw Performance ............

342

Abrasive Cut-Off Wheel .....................................................

344

Toothless or Shear and Compression Cutting ...................

347

Fracture Mechanics in Trimming ........................................

347

Nibbling .............................................................................

355

Brittleness, Orientation and Trim Temperature ..................

358

5.10 Steel Rule Die ........................................................................... 359 Resharpening ....................................................................

362

Tabbing and Notching ........................................................

363

5.11 Punch and Die Trimming .......................................................... 364 Forged and Machined Dies ................................................

367

5.12 Drilling ....................................................................................... 367 5.13 Other Cutting Techniques ........................................................ 372 Thermal Cutting .................................................................

372

Water Jet Cutting ...............................................................

374

5.14 Trimming – a Summary ............................................................ 376 5.15 References ............................................................................... 379

6.

Thermoforming Molds ......................................................... 382 6.1

Introduction ............................................................................... 383

6.2

Prototype Molds ........................................................................ 384 Wood .................................................................................

385

Fiberboard .........................................................................

386

Plaster ...............................................................................

387

Plastic ................................................................................

389

White Metal .......................................................................

397

Nickel ................................................................................

400


xvi

Contents 6.3

6.4

6.5

6.6

Production Molds ...................................................................... 400 Aluminum ..........................................................................

401

Steel ..................................................................................

402

Other Metals ......................................................................

403

Mold Coolant Channels ............................................................ 404 Mold Channel Flow ............................................................

405

Expansion ..........................................................................

406

Contraction ........................................................................

406

Sharp-Edged Orifice ..........................................................

406

Vent Holes ................................................................................ 411 Sizing Vacuum Systems – Steady State ............................

413

Sizing Vacuum Systems – Dynamic ..................................

414

Solenoid Valve Flow Resistance ........................................

416

Vent Hole Resistance to Flow ............................................

417

Vent Hole Diameter ...........................................................

423

Other Types of Vents .........................................................

427

Vent Hole Placement .........................................................

430

Surface Treatments .................................................................. 433 Surface Texture .................................................................

6.7

436

Plug Design Considerations ..................................................... 439 Plug Materials ....................................................................

439

Wood Plugs .......................................................................

441

Plastic Plugs ......................................................................

441

Metal Plugs ........................................................................

441

Plug Design Concepts .......................................................

445

6.8

Sheet Clamping ........................................................................ 448

6.9

Sag Bands and Sheet Supports ............................................... 451

6.10 Other Aspects of Mold Design ................................................. 451 Undercuts ..........................................................................

452

Encapsulation ....................................................................

453


Contents

xvii

Moving Elements ...............................................................

454

Stripper Plates/Bars ...........................................................

455

Mold Releases ...................................................................

456

Web Breakers, Catchers and Chasers ...............................

457

Moats, Dams and Double Steps ........................................

458

Chamfers and Radii ...........................................................

459

Prestretching Restraints ....................................................

461

6.11 Efficient Use of Sheet ............................................................... 461 Heavy-Gage Sheet ............................................................

461

Thin-Gage Sheet ...............................................................

463

6.12 References ............................................................................... 467

7.

Parts Design ......................................................................... 470 7.1

Introduction ............................................................................... 471

7.2

Elements of Parts Design ......................................................... 471 Material Testing and Its Relevance to Part Performance ................................................................

473

Philosophy of Parts Design ................................................

476

Minimizing the Amount of Sheet to Be Reground ...............

478

Rules for Part Layout on Heavy-Gage ...............................

479

Rules for Multiple Part Layout on Thin-Gage .....................

481

Economics of Buying Sheet of Specific Size ......................

483

7.3

Prototyping as a Justification for Thermoformmg .................... 486

7.4

Draw Ratio ................................................................................ 488 Areal Draw Ratio ...............................................................

489

Linear Draw Ratio ..............................................................

494

H:D ....................................................................................

496

Rim and Lip Sheet for Female Cavities ..............................

499

Draw Ratio Usage – a Rationale ........................................

503

Mechanical Assists – Some Design Features ....................

503


xviii

Contents Preblowing or Inflation – Comments ..................................

503

Plug Assist – Comments ....................................................

507

7.5

Computer-Aided Design in Thermoforming ............................. 507

7.6

Wall Thickness Prediction – a Justification .............................. 510

7.7

7.8

7.9

Geometric Element Analysis or GEA .................................

513

Finite Element Analysis .....................................................

521

General Comments on Plug Design ...................................

527

Plug Assist Analysis ..........................................................

533

Plug Design – Geometric Element Analysis .......................

536

Plug Design – Finite Element Analysis ..............................

539

Regrind ..................................................................................... 544 Material Property Deterioration on Regrind ........................

545

Property Value Loss – Experiment and Protocol .......................................................................

549

Cascading 100% Regrind ..................................................

553

General Guidelines for Part Design ......................................... 555 General Tips ......................................................................

555

Process Tips ......................................................................

556

Mold Tips ...........................................................................

557

Prestretch Tips ..................................................................

559

Part Design Tips ................................................................

561

Rim and Edge Designs ......................................................

569

Design – a Comment .........................................................

573

References ............................................................................... 574

Appendix 7.I: Draw Ratios for Truncated Cone .................................. 577 Appendix 7.II: Mechanical Property Loss in Regrind .......................... 579

8.

Producing Sheet and Film .................................................. 582 8.1

Introduction ............................................................................... 583

8.2

Forming Thin Films ................................................................... 584


Contents 8.3

xix

Forming Sheet .......................................................................... 587 Single-Screw Extrusion ......................................................

588

Filtering the Polymer ..........................................................

591

Flow Improvement Devices ................................................

594

Pressure and Temperature in an Extruder .........................

596

Sheet Die Concepts ...........................................................

598

Gage Thickness Monitoring and Control ............................

602

Twin-Screw Extrusion ........................................................

604

8.4

Roll Stacks ................................................................................ 607

8.5

Sheet Trimming ........................................................................ 610

8.6

Take-Off and Take-Up Rolls ..................................................... 611

8.7

Residence Time and Residence Time Distribution through Extruder and Die ......................................................... 614

8.8

Drying ........................................................................................ 617

8.9

Producing Biaxially Oriented Sheet .......................................... 620

8.10 Multilayer Sheet Formation ...................................................... 623 Coextrusion .......................................................................

624

Lamination .........................................................................

628

8.11 Sheet Quality and Quality Control ............................................ 632 Sheet Dimensions .............................................................

637

Orientation .........................................................................

637

Sheet Squareness and Flatness ........................................

639

Moisture ............................................................................

640

Sheet Appearance .............................................................

641

Annoyance Factors ............................................................

642

Lamination .........................................................................

643

8.12 References ............................................................................... 644

9.

Newer Thermoforming Technologies ................................ 648 9.1

Introduction ............................................................................... 649


xx

Contents 9.2

9.3

Thermoforming Crystallizing Polyethylene Terephthalate ............................................................................ 649 PET Crystallinity ................................................................

650

CPET Patents ....................................................................

652

Characterizing Polyethylene Terephthalate .......................

652

The Effect of Temperature on Crystallization during Sheet Extrusion .................................................

659

Cooling CPET on Chill Rolls ..............................................

662

Heating CPET in Roll-Fed Thermoformers ........................

666

Forming the Sheet .............................................................

666

Cooling the Formed Part ....................................................

668

Trimming Parts from Web ..................................................

670

Troubleshooting CPET Forming ........................................

670

Pressure Forming ..................................................................... 670 Thin Gage ..........................................................................

675

Heavy Gage ......................................................................

677

9.4

Forming Filled and Reinforced Polymers ................................. 679

9.5

Laminated Sheet Thermoforming ............................................. 687

9.6

9.7

Heating Multilayer Sheet ....................................................

687

Forming Multilayer Sheet ...................................................

691

Twin-Sheet Thermoforming ...................................................... 696 Simultaneous Twin-Sheet Forming ....................................

701

Sequential Twin-Sheet Forming .........................................

702

Seal Area – Adhesion ........................................................

704

Seal Area – Compressive Form .........................................

706

Seal Area – Design ............................................................

707

Polypropylene Thermoforming ................................................. 710 Sag Test ............................................................................

711

Modified Polypropylenes ....................................................

717


Contents 9.8

9.9

xxi

Thermoforming Foam Sheet .................................................... 722 Cell Architecture – Actual v. Ideal ......................................

722

Radiant Energy Transmission ............................................

726

Internal Cell Gas Pressure .................................................

731

Forming Window for Foam .................................................

733

The Forming Equipment ....................................................

736

Other Forming Technologies .................................................... 738 Interdigitation .....................................................................

738

Sealed Air Cushion/Dunnage .............................................

740

9.10 References ............................................................................... 741

10. Set-Up Protocols, Troubleshooting, and the Economics of Thermoforming ............................................ 748 10.1 Introduction ............................................................................... 749 10.2 Setting up a Thermoforming Machine – Protocols .................. 753 A New Polymer ..................................................................

753

Setting up a New Mold .......................................................

758

Setting the Mold Stops .......................................................

761

Dry-Cycling the Mold .........................................................

761

Checking the Vacuum ........................................................

762

Attaching Cooling to the Mold ............................................

763

The Sheet Delivery System ...............................................

764

Setting the Oven Conditions ..............................................

765

Forming Step – Simple Vacuum Forming ..........................

766

Changing Temperature Conditions ....................................

767

Activating the Assists .........................................................

767

Pressure Boxes .................................................................

768

General Objectives ............................................................

768

10.3 Troubleshooting the Forming Process ..................................... 769


xxii

Contents 10.4 Energy and Materials Cost ....................................................... 783 The Energy Audit ...............................................................

783

Cost of Extrusion ...............................................................

791

Cost of Regrind ..................................................................

793

Competitive Costs of Polymers ..........................................

793

10.5 General Processing Economics ............................................... 794 Rules of Thumb .................................................................

796

Global Production Costs ....................................................

798

Manufacturing Efficiencies .................................................

801

The Learning Curve ...........................................................

806

10.6 Isolated Venture Costs ............................................................. 808 10.7 New Venture Economics .......................................................... 819 Entrepreneurial Risks ........................................................

825

10.8 The Incremental Operation ....................................................... 831 10.9 Comparative Process Economics ............................................ 833 10.10 References ............................................................................... 843

Appendices Appendix A: Abbreviations for Thermoformable Polymers Referred to in Text .................................................................... 844 Appendix B: Typical Conversion Factors Used in Thermoforming (US Customary to Metric, Metric to US Customary) ............................................................................... 847 Appendix C: Glossary of Thermoforming Terms ................................ 849

Author Index ............................................................................... 855 Index ............................................................................................ 862


1 Thermoforming—Definitions, History, Methods and Equipment

1.1 Introduction 1.2 History 1.3 Markets 1.4 Some Definitions Gage Clamping of Thin-Gage Sheet Clamping of Heavy-Gage Sheet Heating of Thin-Gage Sheet Heating Heavy-Gage Sheet Shaping Thin-Gage Sheet Shaping Heavy-Gage Sheet Trimming the Thin-Gage Sheet Trimming the Heavy-Gage Sheet Depth-of-Draw 1.5 Methods of Forming One-Step Forming Two-Step Forming with Prestretching Multi-Step Forming Other Variations 1.6 Thermoforming Machinery Heating Source Forming Platform Vacuum System Pressure System Process Control Trimming and Cut Parts Handling 1.7 Heavy-Gage Thermoforming Machinery Specifics 1.8 Thin-Gage Thermoforming Machinery Specifics 1.9 References

1.1

Introduction

Thermoforming is a generic term encompassing many techniques for producing useful plastic articles from flat sheet. In its simplest concept, thermoforming is simply the manual draping of a temporarily softened sheet over a simple mold shape. In one of its more advanced forms, it involves automatic high-speed indexing of a freshly extruded sheet having very accurately known temperature into a forming and in-situ trimming station, with integral web regrind and automatic parts counting, packaging and shipping. In another, it involves automatic placement, plug and/or pneumatic stretching and pressure forming, with multi-axis router trimming. Thermoforming is one of a family of processes that deal with the pressing or squeezing of pliable plastic into final shape. More than a century ago, celluloid or camphor-solvated cellulose nitrate, was the only malleable semi-synthetic plastic [I]. It was cut or rolled into sheet and made pliable with steam. When soft, it was squeezed into shape in matched dies or rolled into tubes and inflated against metal walls to produce parts. It was also draped over wooden forms. Cosmetic cases, baby rattles, and piano keys were typical of earliest thermoformed parts. The earliest processing of synthetic plastics, included common types of modern thermoforming, albeit restricted to one polymer and very limited processing conditions. Thermoforming always begins with a contiguous sheet of rubbery plastic. The sheet is produced from: • •

Resin liquid by casting as with PMMA, or Pellets or powder by: Calendering, as with PVC, Biaxially blowing film as with PE and PP, Extruding as with PS and ABS, Compression molding as with cellulosics and high temperature polyimides,

or by similar plastics processing techniques1. Thermoforming is differentiated by other processes in the following ways: •

•

1

From injection molding, where the initial resin state ispellet or powder and the shaping is done on the polymer as a liquid. As a result, the patented Monoblow process that injects a disk of plastic that is then compressed to a shape is not considered as a thermoforming variant [2]. From compression molding, where pressures are substantially higher than those employed in traditional vacuum or low pressure thermoforming and where the polymer is shaped as a liquid between matched metal molds. This differentiation blurs when thermoforming technologies are used to form continuous fiber-reinforced composites. The Allied process described in Chapter 9 is a classic example of this blurring. Glass-reinforced nylon 6 or PA-6 sheet is heated to temperatures Throughout the text, plastics are referred to in abbreviations. Thus PMMA is polymethyl methacrylate and so on. A list of common polymer abbreviations is given as Appendix A, at the end of the book.

above the polymer melt temperature, then forged at relatively high pressures of 150 to 1500 lbf/in2 or 0.1 to 1 MPa [3]. Nontraditional thermoforming processes include the Dow scrapless thermoforming process or STP, where the extruded sheet is cut into squares that are then heated and forged into shapes. Other non-traditional and newer processes are discussed in Chapter 9.

1.2

History

Keratin, as a component in tortoise shell, was probably the first material to be thermoformed [4,5]. Keratin is also found in animal horn and hoof. It can be softened by immersing it in boiling water or oil. The sheet is then manually draped over a form and held until cool. Natural cellulose, primary element in tree bark, was shaped in a similar fashion by native Americans. Although others experimented with natural and extracted cellulosics in the 1800s, J. W. Hyatt is credited with first recognizing the full commercial potential of camphorsolvated cellulose nitrate, which he called "celluloid". Most nineteenth century semi-synthetic plastic products were produced of celluloid or products of similar recipes simply by drape forming softened sheet. Sharps piano keys, formed over captive wooden blocks, are an example. A chronology of the early days in plastics as they pertain to thermoforming is given in Table 1.1. Modern thermoforming began about 60 years ago, shortly before, during, and shortly after the second world war, with major developments in two important areas. Research in thermoplastic resin chemistry led to commercialization of extrusion grade flexible PYC or FPVC, CA and PS and the development of cell-cast PMMA. And continuous forming was achieved with the invention of the screw extruder and the roll-fed thermoformer. These breakthroughs allowed a wide variety of prewar domestic products, particularly thin-gage packages, to be developed. This in turn prepared formers for war product developments such as airplane canopies and war survey relief maps. The packaging industry adopted thermoforming as a basic process in the late 1940s to such an extent that the thermoformed package was considered the most significant packaging development of the 1950s decade [6]. In the 1970s, demand for convenience food containers, ovenable portion servings, and more ductile disposable drink cups spurred development of foam PS, CPET, and PP pressure forming processes. Shower stalls, tub surrounds, and refrigerator liners were thermoformed from heavy-gage sheet. The development of the interstate system led to production of large, light-weight illuminated plastic signs and fast food franchises adopted thermoformed plastics as a way of producing such signs. Engineers developed ways of forming plastic sheets for the transportation industry from reinforced and fire-retardant polymers. Ahead lies further opportunities to replace injection molded plastics in many applications such as packaging lids and containers, welded steel in food cans, glass in jars, aluminum in beverage cans, and hand-laid thermoset

Table 1.1 Chronology on Thermoforming1 Period

Event

Prehistory—Egypt

Heating tortoise shell, keratin, in hot oil, then shaping to produce food containers. Heating tortoise shell, keratin, in hot water, then shaping to produce bowls. Heating tree bark, natural cellulose, in hot water, then shaping to produce bowls, boats, canoes. Extrusion process commercialized from forerunners of today's plastics. Alfred P. Critchlow, Florence MA develops molding presses, dies for gutta-percha, shellacs. Gutta-percha replaces ivory for billiard balls. First moldable plastic, fibrous cellulosic pulp and gum shellac by Peck. Cellulose nitrate solvated with camphor to produce "Parkesine" by Alexander Parkes. Celluloid, molding grade Pyroxylin by John Wesley Hyatt. Hydraulic planer for cutting thin sheets by Charles Burroughs Co., MJ, USA. Celluloid tubes steam-heated, placed in metal form, inflated with steam pressure yielding blow molding, pressure forming. Compression molded phenolic bobbin ends, Richard Seabury, Boonton Rubber, Boonton NJ. Sharps piano keys drape-formed over captive wooden cores. Bottle formed from two thermoformed halves by Fernplas Corp. Relief maps for US Coast & Geodetic Survey. Blister pack of cellulose acetate. Roll-fed automatic thermoformer developed by Clauss B. Strauch Co. Cigarette tips, ice-cube trays automatically thermoformed. Cast PMMA acrylic thermoformed for fighter/bomber windows, gun closures, windscreens. Standard mold bases, Detroit Mold Engineering (D-M-E). Cast PMMA acrylic bathtubs thermoformed by Troman Bros., England. Skin-packaged products shown at Hardware Manufacturers Association, Chicago. Thermoformed ABS automobile body by Borg-Warner.

Prehistory—Micronesia Prehistory—Americas 1845 1850s 1856 1862 1868 1870s 1870s 1907 1910 1930 1930s 1938 1938 1938 1942 1942 1948 1954 1970 1

Adapted from [4,5,32]

composites in aircraft and other transports. The prospects for advanced thermoforming systems are discussed in detail in Chapter 9.

1.3 Markets A best estimate of the entire US consumption of thermoformed shapes in 1992 is about 1181 Mkg or 2600 MIb. About 800 Mkg or 1760 MIb is thermoformed into disposables. The fraction of plastics thermoformed into disposables has dropped steadily in the past decade from about 74% in 1983-1984 to about 68% today [7-9].

Table 1.2 US Polymers Converted via Thermoforming Polymer

Amount thermoformed (Mkg) 1962

1969

1977

1983-1984

ABS PMMA Cellulosics LDPE HDPE PP** PS PVC PET Other*

NA NA 8 NA NA NA 69 4 NA 1

40 21 5 0.7 7 1 166 13 NA NA

70 36 8 0.4 10 8 392 24 NA NA

127 38 4 0.6 26 22 480 45 40 NA

Totals

82

254

544

782

1992 202 42 5 0.7 59 72 640 60 80 20 1181

1995 240 50 5 0.8 87 115 720 72 110 50 1450

Annual growth (%) 1984-1992 6 2 0 0 11 16 4 4 9

5

* Includes K-resin, PAN, XT, TPO polymers ** Does not include stampable PP for automotive applications NA = Not available (not included)

An estimate of the amounts of plastic consumed in thermoforming in the US in the 30 years from 1962 to 1992 is given in Table 1.2. The data are obtained from several sources [8-10]. In 1969, the industry was projected to grow at about 8.5% to 9% per year during the 1970s. It was then estimated that 527 Mkg or 1180 MIb would be consumed by 1978 (Fig. 1.1). About 550 Mkg or 1230 MIb was actually consumed in 1977. In 1969, it was predicted that consumption in 1984 would be 780 to 830 Mkg or 1750 to 1860 MIb. In 1986, it was determined that about 773 Mkg or 1732 MIb were actually consumed in 1983-1984 [H]. In 1986, it was predicted that consumption would reach 1045 Mkg or 2300 MIb in 1992. The prediction is about 13% below actual consumption (Fig. 1.1). The 1986 projection also indicated that 1290 Mkg or 2840 MIb would be thermoformed in the year 2000. From Fig. 1.1, current, 1994 projection indicates 1860 Mkg or 4100 MIb will be consumed in the year 2000. This is a continuing annual growth rate of more than 5%. As seen in Table 1.2, polystyrene accounts for more than half the plastic thermoformed into parts. The growth in this established polymer is less than 3% per year. As expected, PP and PET are experiencing double digit annual growth as new polymers and forming processes are developed. Surprisingly, HDPE growth is also very high, with outdoor products contributing to the growth. As an example, there were no HDPE truck bed liners used in the early 1980s. In 1992, the size of this market alone is approaching 23 Mkg or 50 MIb. Cellulosics continue to lose market share to PP, PVC and PS, particularly in packaging. In 1984, 7 Mkg or 16 MIb thermoplastic polyesters or PETs were thermoformed. Only a very small fraction, less than 1%, was crystallized PET. Since the development of CPET thermoforming processes in the 1980s, CPET has become a staple in the disposable food serving and heating tray and dish markets. In 1984, it was estimated that the development of CPET would spur the consumption of PET to double in five years, to 14 Mkg or

1994 Projection (This Edition)

US Production, Million Ib

1986 Projection (First Edition)

Year Figure 1.1 US Production of thermoformed products. Adapted from [7-11]

Table 1.3 Approximate Conversion Level of Thermoplastic Polymers to Thermoformed Produces1 Polymer

Amount of polymer converted to sheet (%)

Amount of sheet thermoformed (%)

Total amount of polymer formed into product (%)

ABS PMMA Cellulosics HDPE LDPE PP PS PVC

25 50 25 2 0.25 1 30 10

60 30 50 30 5 25 65 20

15 15 12.5 0.6 0.01 0.25 19.5 2.0

1

Adapted from [7,8]

32 MIb. In actuality, in 1992, this market is about 70 Mkg or 154 MIb, a ten-fold increase in less than a decade. In 1992, the consumption of CPET is estimated to be about 19 Mkg or 42 MIb, or about 27% of the total thermoformed PET consumption. About half of all sheet stock is converted to product by thermoforming [12]. The amount of resin converted into sheet varies from about half of all PMMA polymers to less than 1% of all LDPE polymers (Table 1.3). Only about 5% of all LDPE sheet and film is converted to product by thermoforming whereas 65% of all polystyrene sheet is converted. These are mid-1980s data but the percentages have held relatively constant since the 1960s. Similarly, major markets have remained relatively obvious for nearly two decades. Major new packaging markets continue to be in disposables, as illustrated by the development of PP and CPET food packages and the commercial realization of convenience food foam carryout containers. Applications have also broadened to include packaging of heavier items such as power tools and medical items such as critical care emergency packages. Nearly all packaging applications rely on roll-fed thin-gage sheet stock. In heavy-gage discrete sheet forming, the market traditionally has focussed on economic production of a few, large pieces. As a result, applications include: • • •

Major appliance components, such as refrigerator cabinet and door liners, Recreation products such as swimming and wading pools, Vehicles such as snow mobiles and all-terrain vehicles, automotive inner-door panels, truck and tractor cab kick panels, • Home products such as tube and shower stalls that are backed with GR-UPE, luggage shells, and • Display items such as advertising, exterior signs and point-of-purchase stands.

Table 1.4 Markets for Thermoformed Products Packaging and related items Blister packs, point-of-purchase containers Bubble packs—slip sleeve, vacuum carded containers Electronics—audio/video cassette holders Tool cases—hand, power Cosmetics—cases, packages Foams—meat, poultry trays Unit serving—foodstuffs Convenience—carry-out, cooking-box trays Convertible-oven food serving trays Wide-mouth jars Vending machine hot drink cups Cold drink cups—beer, soda Egg cartons Wine bottle protectors Produce separators—apples, grapefruit Portion—medical unit dose Form-fill-seal—jelly, crackers, nuts and bolts (Continued)

Table 1.4 (Continued) Vehicular Automotive door innerliners, headliners Automotive utility shelves, liners Automotive instrument panel skins Aircraft cabin wall panels, overhead compartment doors Snowmobile shrouds, windshields Motorcycle windshields, farings, scooter shrouds, mudguards All-terrain vehicle exterior components Golf cart shrouds, seats, trays Tractor shrouds, door fascia Camper hardtops, interior components such as doors, cabinet tops Truck cab door fascia, instrument cluster fascia Recreational vehicle interior components, window blasters Industrial Tote bins Pallets, single deck, double deck Parts trays, transport trays Equipment cases Building products Shutters, window fascia Skylights, translucent domes Exterior lighting shrouds Storage modules—bath, kitchen, pantry Lavys Bath and shower surrounds, GR-UPE backed Soaking tubs, GR-UPE backed Retrofit shower components, shower trays Others Exterior signs Advertising signs, lighted indoor signs Swimming and wading pools Tray, baskets, hampers, carrying cases Luggage Gun cases, golf club cases Boat hulls, surf-boards, with PUR foams Animal containers Prototype concepts for other plastic processes

The recent rediscovery of pressure forming of sheet against a female mold offers an entre into the economically important market of business machine housings [13]. Some major markets are listed in Table 1.4 [14]. The sizes and annual percent growth rates or APR are given in Table 1.5 for several of these markets. Most of the fully developed, mature thermoforming markets are developed around amorphous polymers such as PVC, PS, ABS and PMMA. These polymers are processed quite successfully over rather wide temperature ranges. Amorphous polymers are usually quite forgiving in this respect. Crystalline polymers and reinforced amorphous polymers, on the other hand, have narrower forming

Table 1.5 Selected Growth Markets for Thermoforming Market

77

Thermoforming machinery (US$ x 106) [35]

22

78 79

80

81 82

83

84

85

86

87

88

89

90

91

91

Rigid plastic packaging (US$ x 109) [36]

92

93

94

95

96

97

98

99 Average annual growth (%) (years)

121

5.9 (94/89)

6.6

Plastic containers ( x l O 9 units) [37]

27.2

38

51

6.5 (95/85)

Multilayer containers ( x l O 9 units) [37]

0.3

11

29

58.0 (95/85)

160

231

Business machine housings ( x l O 6 Ib) [38]

55

115

178

Retortable plastic containers ( x l O 9 units) [39] Automotive sheet stamping ( x 106 Ib) [40]

0.25

4

0.3

0.6

1.1

25

8.6 (90/85) 2.8

62 (91/86)

60

19 (91/86)

Polystyrene in packaging ( x l O 9 16) [41]

2.38

Nucleated PP—sheet extrusion ( x l O 6 Ib) [42]

3.3

2.74

3.1

10

2.8 (95/90)

25 (96/91)

Barrier packaging polymers ( x l O 6 Ib) [43]

940

1475

9.4 (95/90)

Modified atmosphere packaging ( x l O 6 Ib) [44]

800

1400

11.8 (95/90)

Controlled atmosphere packaging ( x 106 Ib) [44]

400

800

14.9 (95/90)

(Continued)

Table 1.5 (Continued) Market

77

78 79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

Average annual growth (%) (years)

PP sheet—non-automotive ( x l O 6 Ib) [45]

44

141

21 (90/84)

PP sheet—auto/appliance ( x l O 6 Ib) [45]

35

125

24 (90/84)

Plastics in appliances ( x l O 6 Ib) [46] Microwavable plastic containers ( x l O 6 units) [47]

96

Non-disposable polyolefin foam ( x l O 6 Ib) [48]

20

Medical plastics ( x l O 9 Ib) [49]

0.65

Plastics in packaging (US$xl0 9 ) [52]

310

1.20

4.8 (92/89) 2.86

1.16

1.19

1.55

6.9 (94/89)

5.4 (94/89)

1.30

7.3

4.1 (97/92)

26 (93/88)

23

2.05

Transport plastics ( x l O 9 Ib) [50] Plastics in appliances (US$xl0 9 ) [51]

1047

888

2.4 (91/86) 8.1

2.1 (94/89)

windows, usually require higher forming pressures, and/or usually cannot be consistently formed on conventional thermoforming equipment. Only recently has the forming industry begun to apply fundamental processing principles to the development of machinery capable of holding accurate sheet temperatures and forming pressures.

1.4

Some Definitions

The technology of thermoforming is rapidly changing. Old ideas about process limitations are being challenged daily. In 1992, one machinery manufacturer said that: "More innovations have been made in thermoforming machinery in the last half-dozen years than in all the years before." [15].

As noted below, machinery innovations include: • • • •

Improved and more reliable ways of clamping sheet, Improved heaters, Improved heat distribution patterns, Adaptation of infrared sensors for monitoring and controlling sheet residence time in ovens, • Improved clamping systems, • New plug assist materials, • Better control of stretching forces and pressures, • Improved trim dies, and • Many new mold materials and mold making techniques. Of course, innovations are not just restricted to machinery. The past decade has seen extensive efforts to produce uniform-property extruded sheet, newer polymers with superior sag resistance and oxidative resistance, regrind monitoring techniques and computer programs to predict time-dependent sheet temperature and local wall thickness. However, there are certain guidelines about the thermoforming process that are relatively generic. For example, thermoforming has become economically important since it offers processing advantages over competitive processes such as: • • • •

Accumulator blow molding, Injection blow molding, Rotational molding, and Injection molding.

Relatively low forming pressures are needed and so mold costs are low and products of relatively large size are fabricated economically. Parts with very small thicknessto-area ratio are fabricated. For thin-walled products, fabrication time is very short, making the process economical for products requiring high multiplication factors.

For a few thick-walled parts, molds are made of wood, plaster or other easily shaped, relatively inexpensive materials. Mold fabrication time and thus lead time are very short. Thermoforming is the method most usually selected for prototype and display products to be made of other processes.

Gage Typical thermoforming steps are: • • • • •

Clamping, Heating, Shaping, Cooling, and Trimming.

The general process of thermoforming is loosely separated by sheet thickness or gage. There are two broad categories—thin-gage and heavy-gage. Recently, sub-categories have been introduced to further classify the process. Thin-gage thermoforming means that the sheet thickness is less than about 0.060 in or 1.5 mm. This category is further classified into: • •

Film forming, where the sheet thickness is less than about 0.010 in or 0.25 mm. Thin sheet forming, where the sheet thickness is between about 0.10 in or 0.25 mm and 0.060 in or 1.5 mm.

Heavy-gage thermoforming means the sheet thickness is greater than about 0.120 in or 3 mm. This category is further classified into: • •

Heavy sheet forming, where the sheet thickness is between about 0.120 in or 3 mm and about 0.400 in or 10 mm. Plate forming, where the sheet thickness is greater than about 0.400 in or 10 mm.

There is a gray area between thin-gage (< 0.060 in or 1.5 mm) and heavy-gage (> 0.120 in or 3 mm). In some instances, sheet in this mid-gage range of 0.060 in to 0.120 in or 1.5 mm to 3.0 mm behaves as if it is thin-gage, and in others, as if it is heavy-gage. This gray area is most apparent during handling of the sheet, heating the sheet and trimming the part from the web. Polystyrene and polyolefln foam sheet thicknesses are usually greater than 0.120 in or 3 mm but these foams are usually treated as thin-gage sheet stock.

Clamping of Thin-Gage Sheet Thin-gage sheet is usually supplied to the former in rolls. The majority of packaging applications such as blister pack, form-fill-seal packaging, foam sheet forming and biaxially oriented forming uses thin-gage sheet. Formers that use roll sheet stock are

called roll-fed or continuous sheet formers. Clamping is by parallel continuous loop chain-fed pins or pin-chains that pierce the sheet at 1 in or 25 mm intervals at about 1 in or 25 mm from each edge. For high-temperature forming of CPET or PP, the rails are shielded from the sheet heating source or are actively cooled. The edges of the sheet and the plastic between the formed product are trimmed, reground and reextruded into sheet for forming. Clamping of Heavy-Gage Sheet Heavy-gage sheet is usually supplied as cut, stacked and palleted sheet. The demand for large numbers of large, heavy-wall parts such as refrigerator door liners and vehicle interior components, has led to the development of an in-line heavy-gage sheet extrusion and thermoforming concept (Fig. 1.2) [16]. The sheet extruder is placed in-line with the thermoformer, thus obviating problems associated with handling cut-sheet materials. Formers that use discrete heavy-gage sheet are called cut-sheet formers. Cut-sheet clamping frames are usually held closed with mechanical spring-loaded toggle clamps or pneumatic hold-down bars. Shuttle clamps are also used. Heating of Thin-Gage Sheet There are three ways of heating sheet: Conduction, where the sheet is placed in direct contact with the heating medium, such as a hot plate, Convection, where the sheet is heated with hot air, and Radiation, where infrared heat from metal wires, ceramic plates, or gas-fired combustion is the primary means of heating the sheet. Thin-gage, roll-fed sheet is usually heated by passing the sheet between banks of infrared radiant heaters. Combinations of radiation and convection heating are used

Figure 1.2 In-line heavy-gage sheet extrusion and thermoforming line—Cannon-Shelly [16]

as well. These are detailed in Chapter 4. Most plastics formed today, such as PS, PVC and ABS, are formed at relatively low temperatures such as 2500F to 4500F or 1200C to 2300C.

Heating Heavy-Gage Sheet Intense energy input from radiant heaters is unwarranted and is potentially a problem for heavy-gage sheet. Conduction of the energy from the surface of the sheet to its interior controls the heating time. As a result, heavy-gage sheet is frequently heated with forced convection hot air or reradiated energy from fine mesh metal screens or hot plates. Heater types and temperatures are selected on the basis of optimizing the amount of energy transferred to the sheet per unit time.

Shaping Thin-Gage Sheet In the earliest days of thermoforming, the rubbery sheet was manually stretched or draped over a male mold. Drape forming or male mold forming, requires no forming pressure and no difference in pressure across the sheet thickness. Certain aspects of drape forming are used today, whenever male portions of female molds are employed. Vacuum forming is the application of differential pressure across the sheet thickness of up to 0.1 MPa or 15 lbf/in2 or one atmosphere absolute. In modern commercial machines, the applied vacuum is on the order of 0.05 to 0.09 MPa or 7.3 to 14 lbf/in2. Since thin-gage sheet surface-to-volume ratio is very large, thin-gage sheet loses heat to its surroundings very rapidly. As a result, thin-gage sheet is usually formed very rapidly (in seconds). Furthermore, more than one part is formed at a time. Special clamps called cavity isolators are used to minimize nonuniformity in individual part wall thickness due to polymer pulling from one cavity to another during forming. Foams are normally very resilient or stiff at their forming temperatures and so matched die molding is used.

Shaping Heavy-Gage Sheet Drape forming and vacuum forming are common ways of stretching heavy-gage sheet as well. Usually the mold has a single cavity. Sheet manipulation or prestretching with air or with mechanical assists called plugs is a common way of redistributing the plastic across the mold surface to minimize thin spots. In addition, pressure forming has been rediscovered. Historically, steam pressure was used to force celluloid against mold surfaces [17]. Air is used today in place of steam [18]. Although forming pressures to 3.5 MPa or 5001bf/in2 are used when shaping continuous fiber-reinforced composites, the pressure forming range of 0.14 to 0.56 MPa or 20 to 80 lbf/in2 is commercial with 1.4 MPa or 200 lbf/in2 considered the practical upper limit. Pressure forming is combined with vacuum forming to gain additional differential

Figure 1.3 Roll-fed thin-gage thermoforming line with in-line trimming station—BattenfeldGlencoe

pressure and to minimize air pockets between the mold and the sheet. Pressure forming is best suited to heavy-gage sheet forming into female molds. Higher forming pressures require more substantial mold construction. This increases the mold cost and the lead time. Trimming the Thin-Gage Sheet Thin-gage sheet can be trimmed in the mold or in a separate in-line hydromechanical trimming device. A camel-back or hump-back trimmer is shown in line with a roll-fed former in Fig. 1.3. Trimming the Heavy-Gage Sheet On occasion, heavy-gage sheet is also trimmed in the press. The more common trimming scenario is the transfer of the sheet from the forming press to a trimming fixture. Trimming follows one of the following: • • • •

Manual with a hook-knife or hand-held router, Manual with a bandsaw, Automatic with a multi-axis trimming fixture, or Automatic with a water jet or laser.

Depth-of-Draw Historically, thermoforming rules-of-thumb are based on the depth-of-draw of a given polymer into a given mold configuration. The depth of draw concept is loosely

understood among formers as the ratio of the depth a sheet could be drawn into a female mold to the minimum dimension at the rim. It is frequently given the notation h\d. Unfortunately, this definition is easily misinterpreted, is vague for non-cylindrical shapes, and does not truly describe the stretching process. Other definitions based on areal draw ratio and linear stretching are more accurate, as discussed in Chapter 7. For reference, the areal draw ratio is the ratio of the area of the formed sheet to that of the unformed sheet. The linear draw ratio is the length of an imaginary line drawn on the formed sheet to its original length.

1.5

Methods of Forming

In its simplest form, thermoforming is the stretching of a heated rubbery sheet into a final shape. As the sheet is stretched against the mold surface, it stops drawing. As a result, the final part has thick walls where the sheet touched the mold first and thin walls where it touched last (Fig. 1.4). In many thin-gage forming applications, the extent of sheet that is stretching is small and so the areal draw ratio is small. The package integrity and durability are therefore uncompromised. Typical applications are in packaging areas such as blister and bubble packs, form/fill/seal packages and fast food containers and picnic plates. If a high degree of stretching is needed, as with disposable drink cups or for heavy-gage sheet forming, simple stretching techniques are insufficient. The methods of forming are divided into the number of sequential steps needed to form the part once the sheet is at the forming temperature.

Figure 1.4 Wall thickness variation during drawdown in simple female vacuum forming

Sheet

Mold

Figure 1.5 Drape forming onto a positive or male mold

Male or Positive Forming

One-Step Forming There are at least five types of one-step forming: •

In drape forming, Fig. 1.5, the clamped, heated rubbery sheet is either lowered onto the male mold or the mold is raised into the sheet. The sheet in contact with the mold does not stretch. For modern drape forming, the air trapped between the sheet and the mold is evacuated as the mold penetrates and stretches the sheet against the mold flange. Either vacuum or air pressure is used to produce the differential pressure needed to force the sheet against the male mold. In drape forming, the formed part has a thick bottom and thin sidewalls. The part is thinnest at the rim. Drape forming is also called male molding. • In vacuum forming, Fig. 1.6, the clamped, heated rubbery sheet is sealed against the rim of the female mold. Vacuum is then applied. The differential pressure presses the sheet against the mold surface. As noted before, the formed part has

Mold Sheet

Figure 1.6 Vacuum forming into a negative or female cavity

Negative or Female Forming

Air Pressure Pressure Box Sheet Clamp Mold

Evacuation or Vacuum Pressure Forming Figure 1.7 Pressure forming

a thick rim and is thinnest in the bottom corners. This is also called cavity forming or female molding. • Pressure forming, Fig. 1.7, is similar to vacuum forming. A pressure box is fitted over the sheet as it is held against the mold rim and positive air pressure is used to push the sheet into the mold corners. Since air pressure to 1.4 MPa or 200 lbf/in2 is used, the pressure box must seal against the free surface of the sheet. Pressure forming is used for difficult-to-form roll-fed thin-gage polymers such as PP and for production of highly detailed heavy-gage parts [18]. • In free blowing, Fig. 1.8, the clamped heated rubbery sheet is stretched with air into a free-form shape. The amount of air pressure is controlled with a photocell that senses the height of the expanding bubble. Since the environmental air is slightly cooler than the sheet, the sheet cools in the free-form shape. This

Lamp

Bubble

Sheet

Photoelectric Eye

Clarnp Mold

Inflating Air Free-Blowing Sheet Figure 1.8 Free-blowing to produce domes

Top Mold Half

Sheet

Clamp

Bottom Mold Half

Matched Molds Advancing on Hot Sheet

Pressure Applied to Shape Part Figure 1.9 Matched die molding with trapped sheet formed at applied pressures less than 1 MPa or 150 lbf/in2

technique was pioneered for aircraft gun enclosures. Since the sheet does not touch a solid surface during forming, it remains mar-free. The bubble wall thickness is quite uniform except near the clamping area. Freely blown roll-fed thin-gage bubbles are used for blister packs. • Matched die molding, Fig. 1.9, is a common way of forming shapes from relatively stiff polymers, such as PS foam or filled polymers. The clamped heated rubbery sheet is positioned between two mold halves. As the mold halves close, vacuum is applied to the female half of the mold to assist with forming. Part wall thickness depends on the mating tolerances of the two mold halves. Appreciable material movement is possible if applied forces are relatively large. Usually the applied pressures do not exceed about 1 MPa or 150 lbf/in2 and are usually about 0.34 MPa or 50 lbf/in2. Two-Step Forming with Prestretching Multi-step forming was developed primarily for heavy-gage sheet where single parts are often quite complex and deep and where cost considerations make wall thickness uniformity a significant design parameter. In thin-gage thermoforming, forming times are very short and shapes are relatively simple. Until a few years ago, thin-gage

forming was restricted to one of the one-step techniques described above. In the 1970s, the development of plug assisted pressure forming of PP below its melting temperature led the way to highly automated multi-step forming of thin-gage roll-fed sheet. The first step in multi-step forming is usually a form of sheet stretching, such as plug assist or billowing. The prestretched sheet is then pressed against the mold surface. Some examples of multi-step forming follow: •

There are many variations of bubble or billow prestretching. The first step is to pneumatically inflate the clamped heated rubbery sheet to a controlled height with internal air pressure. Typically the differential pressure is 0.014 to 0.055 MPa or 2 to 8 lbf/in2 gauge. The bubble height is controlled either by touching a microswitch or by intercepting a photoelectric eye. At this point, the mold can interact with the stretched sheet in one of several ways: • In billow drape forming, Fig. 1.10, the male mold is pressed into the top of the prestretched sheet. This technique yields a part with wall thicknesses that are much more uniform than that obtained with straight drape forming. • When a female mold is used, in billow vacuum forming, Fig. 1.11, the differential pressure that has inflated the bubble is reversed. This causes the prestretched sheet to snap into the female mold. Again, the part wall thickness

Top Platen

Mold

Sheet

Vacuum or Exhaust

Bilow Clamp

Pressure Box

Inflating Air Billow Prestretching With Mold Motion

Pressurized Air Vacuum/Pressure Forming

Figure 1.10 Billow drape forming, with either vacuum or applied air pressure shaping the sheet against the mold surface

Bubble Sheet Gasket Clamp

Mold

Vacuum Inflating Air Billow Prestretching

Vacuum Forming

Figure 1.11 Billow vacuum forming

is much more uniform than that obtained with conventional vacuum forming. The eversion of the bubble can be tricky, so this technique is difficult. • If vacuum is used to pull the bubble, a vacuum box is needed (Fig. 1.12). The male mold is immersed or plunged into the prestretched bubble and the vacuum released and air pressure applied. The bubble then snaps against the mold. The technique is called vacuum snap-back forming. The billow drape forming and the vacuum snap-back forming methods work well as pressure forming techniques as well. • The heated rubbery sheet can also be stretched with a mechanically driven plug. There are several plug assisted methods: Prestretching Into Vacuum Box

Mold Plunged Into Billow

Mold

Clamp

Sheet

Draw Box Billow

Vacuum Figure 1.12 Vacuum snap-back forming

Pressurized Air

Platen Plug

Sheet Clamp Mold Plug Moving Into Hot Sheet

Plug Bottoming Out

Vacuum Forming Vacuum

Figure 1.13 Plug-assisted vacuum forming into a female mold

•

• •

The most common form of plug-assisted thermoforming is plug assisted vacuum forming with a female mold (Fig. 1.13). The sheet is prestretched by pressing the plug into it and forcing the sheet toward the bottom of the female mold cavity. Vacuum is then applied to pull the sheet against the mold surface. If the sheet is forced against the female mold surface with air pressure applied through the plug, the technique is known as plug assist pressure forming (Fig. 1.14). Plug-assisted drape forming onto a male mold (Fig. 1.15) is used when the draped sheet must be tucked into three-dimensional corners or into an undercut. It is also used to stretch plastic sheet away from a male portion of a female mold to minimize webbing.

Multi-Step Forming Billow forming and plug assist forming are occasionally combined with drape forming and vacuum forming to obtain unique wall thickness distributions. One example is reverse draw forming with plug assist. Fig. 1.16 illustrates the multiple step process, where a bubble is blown first. The plug is then plunged into the bubble, everting it under control. Once the plug has stretched the bubble nearly to the female mold bottom, vacuum or pressure is applied to force the sheet against the mold surface. This technique requires much patience since a stable bubble without excessive stretching is key to uniform and consistent part wall thickness distribution. Fig. 1.17 illustrates the male mold variation of this, with the mold initially acting as the plug. This technique is sometimes called pressure bubble immersion forming or just immersion forming. Again, the bubble can be formed with vacuum in a vacuum box that also serves as a pressure box for pressure forming. Table 1.6 [19] summarizes many aspects of the thermoforming process.

Plug Platen

Air Pressure

Gasket Ring Plug Sheet Clamp Mold Exhaust Ring Plug Moving Into Hot Sheet

Pressure Balance During Plugging

Air Pressure

Vacuum Vacuum/Pressure Forming Figure 1.14 Plug-assisted pressure forming, using a ring plug

Individual Plug Travel

Clamp Travel

Male Mold

Figure 1.15 Plug-assisted drape forming, using a wire frame plug

Plug

Bilow Sheet Clamp Mold

Vacuum Inflating Air Plug Immersion Into Billow

Vacuum Forming

Figure 1.16 Reverse-draw forming with plug assist

Mold Perimeter Clamp Sheet Clamp Sheet Pressure Box Sheet Clamp-Down Hot Sheet Between Blow Box and Mold

Inflating Air

Prestretching

Pressurized Air Vacuum/Pressure Forming

Figure 1.17 Pressure bubble immersion forming or immersion forming

Table 1.6 Characteristics of the Thermoforming Process1 Mold configuration

Process

Male Vacuum forming Drape forming Matched mold Inflation-plug assist vacuum forming Plug assist Vacuum snap-back Inflation snap-back Trapped sheet, pressure Slip forming 1

Plug assist

Inflation

X X

X

Female X

X X Plug Plug X

X X X Vacuum Pressure X X

X X Mold Rises

Adapted from [19]

Other Variations In order to form certain types of polymers, techniques other than those discussed above have been devised: •

Trapped sheet forming, Fig. 1.18, is used when: The polymer is thermally sensitive, such as certain types of PVC, The polymer is excessively saggy, as with PP and certain types of LDPE and LLDPE, The polymer sheet is highly oriented, as with oriented PS and PP, The sheet is flocked or metallized on one side, The sheet is laminated with a temperature-sensitive adhesive, Vacuum Blowing Air Electrically Heated Platen

Slotted or Porous Blow Plate

Sheet Gasket

Clamp Mold

Air Pressure Sheet Held Against Heater

Exhaust or Vacuum Sheet Drawn/Blown Into Mold

Figure 1.18 Trapped sheet forming with heating against a slotted or porous heated blow plate

The sheet contains wires or printed circuits that are temperature-sensitive, The sheet is very thin and printed or embossed on one side, or The sheet is less than about 0.005 in or 0.13 mm. The clamped sheet is held against a heated plate until the polymer reaches its forming temperature. Pressure is then applied through holes drilled in the plate, forcing the sheet away from the plate and against the female mold. Alternately, vacuum is applied, sucking the sheet away from the heater and against the mold. The plates are heated with electric rod heaters. Although drilled plate is usually used, there is growing interest in porous bronze or stainless steel plates. Zonal or pattern heating is possible by using insulating plate sections that are heated to different temperatures. • In slip forming, the heated rubbery sheet is not tightly clamped. As the differential pressure is applied, instead of the sheet being stretched, it is drawn from the clamp, over the mold rim and into the cavity. At a predetermined time, the sliding is stopped by increasing the clamp force. This is done by compressing springs as shown in Fig. 1.19, or by using cam-type rockers that squeeze against the sheet after a predetermined amount of rotation. Slip forming parallels deep-draw metal forming practice. It is used to form continuous-fiber reinforced polymer composite sheet. This is described in Chapter 9. • Splitty plastics such as PET and PA or nylon and certain multilayer structures are best formed without splitting using diaphragm forming (Fig. 1.20). A warmed thick-walled neoprene bladder or diaphragm is placed against the clamped heated rubbery plastic sheet. The bladder is inflated with air or with a liquid such as hydraulic fluid or hot water. The inflating bladder stretches the plastic sheet into a female mold. Very uniform wall thicknesses and relatively deep draws are obtained for plastics that cannot be formed in other ways.

Pressure Box Sheet Slip Clamp Spring Spring-Clamp

Cammed Clamp Mold Cam-Clamp

Slip Sheet Moving Onto Mold Sheet Slipping From Clamps Figure 1.19 Slip forming. Spring-loaded sheet clamp on the left and cam-loaded sheet clamp on the right

Female Mold Sheet Clamp Diaphragm

Deformed Diaphagm

Pressure Box Air or Hydraulic Fluid Mold Moving Onto Diaphragm/Sheet

Applied Pressure Diaphragm-Stretched Sheet

Figure 1.20 Diaphragm forming for splitty or weak polymers

•

Twin-sheet thermoforming has been a technically viable process for many years. There are several variations on this process, described in greater detail in Chapter 9. One approach is called simultaneous twin-sheet forming. Two sheets are kept separate while heating, then brought together in a double female mold arrangement (Fig. 1.21). Blow pins are inserted between the sheets. Air inflation begins as the sheets are clamped together and the mold halves close. The air pressure keeps the sheets from initially touching and then provides the force needed to press the sheets against the mold surfaces. Twin-sheet forming produces a relatively flat hollow part that with proper design of kiss-offs and pinch-ofls is light weight and very strong. The hollow cavity can also be filled with PUR foam for additional stiffness and insulation. Tack-Off or Kiss-Off

Kiss-Off Completed

Moving Kiss-Off

Female Mold Sheet Clamp Gasket Female Mold

Blow Pin or Hypodermic Needle Moving Kiss-Off

Heated Sheets Clamped Between Molds

Simultaneous Forming

Inflating Air Actuated PlugLike Kiss-Offs

Figure 1.21 Heavy-gage, simultaneous twin cut sheet thermoforming. Here, blow pins extend through sheet surface and kiss-offs are pneumatically driven during forming

Perimeter Cavity Clamp or Grid

Top Sheet Bottom Sheet

in-Situ Trim

Clamp

Top Sheet Trim Bottom Sheet Trim

Female Mold Vacuum Heated Sheets

Vacuum Draw on Clamped Sheets

In-Place Trimming

Figure 1.22 Thin-gage, simultaneous twin roll-fed sheet thermoforming. Here, sheets are heated separately and brought together at the forming station

For large, heavy-gage parts, the technique competes with blow molding and rotational molding. In roll-fed thermoforming, twin-sheet forming is used in a different way (Fig. 1.22). Packages that provide oxygen and moisture barrier are sought. The most effective barrier materials such as PAN, EVOH and PVDC are usually quite expensive and so are used as thin films between layers of less expensive but durable polymers such as PP, HDPE, PET and PS. The results are usually laminates of essentially incompatible plastics glued together with polymers such as EVA that behave as hot-melt adhesives. The web and trim from coextruded laminates of some of these polymers cannot be reprocessed successfully without degradation and gel formation. As a result, a variation on the twin-sheet thermoforming process was developed wherein separate sheets of the candidate polymers are fed from individual rolls, through sandwich heaters and then brought together right at the forming station. Once the multi-layer formed product has been trimmed, the individual layers in the web and trim are stripped from one another. Each polymer web is therefore "clean" and is recycled to produce new sheet. The diverse layers of material in the formed product are only contact-adhered and so will delaminate through misuse. If the force applied to the sheet increases, thermoforming begins to mimic metal forming techniques. At pressures of 1.73 MPa or 250 lbf/in2, the process is similar to tin metal embossing. At pressures of 6.9 MPa or 1000 lbf/in2, the process resembles coining. At pressures of 13.8 MPa or 2000 lbf/in2, the process is like compression molding or forging [20]. It has been adequately demonstrated in the Dow scrapless thermoforming or STP process, that useful products can be fabricated with high-speed impact forming. High pressure forming is described in detail in Chapter 9. The incentive to achieve more uniform part wall thickness even in very deep draw parts continues to spur development of these multi-step procedures. The more

Next Page

sophisticated these procedures become, the more difficult it is to adapt them to high-speed forming without exacting process controls. And yet many new packaging applications, for example, are seeking just such designs. Thermoforming process innovation remains a lively art.

1.6

Thermoforming Machinery

As noted, there are two general thermoforming categories. Typically, heavy-gage sheet is handled as discrete cut sections and the forming equipment are called cut-sheet thermoformers. Thin-gage sheet is handled in continuous rolls and the forming equipment is usually called roll-fed thermoformers. The equipment in both categories includes: • • •

Some form of sheet handling device, A way of moving the sheet from one station to another, A means of controlling the various elements that allow the sheet to be heated, formed and moved from station to station, • A sheet heating oven, • A vacuum system, • A forming press, and • A formed part removal region. In addition, the equipment may include: •

• • •

Some form of prestretching such as: preblowing or plug assist, A pressure system, A trimming press, and Some form of trim removal.

Certain guidelines pertain to both categories of forming equipment. Table 1.7 gives an overview for thermoforming equipment in general [20]. Some of these are summarized below. Heating Source The various heating methods are detailed in Chapter 3. Sheet temperature should be controlled to within + 5°C or + 1O0F. During transfer to the forming station, the sheet temperature drop should not exceed 5 to 100C or 10 to 200F. Infrared heating is most popular today. The various heating methods include [21]: • •

Simple nickel-chrome heating wires, Metal resistance rods, sometimes called calrods,

Previous Page

sophisticated these procedures become, the more difficult it is to adapt them to high-speed forming without exacting process controls. And yet many new packaging applications, for example, are seeking just such designs. Thermoforming process innovation remains a lively art.

1.6

Thermoforming Machinery

As noted, there are two general thermoforming categories. Typically, heavy-gage sheet is handled as discrete cut sections and the forming equipment are called cut-sheet thermoformers. Thin-gage sheet is handled in continuous rolls and the forming equipment is usually called roll-fed thermoformers. The equipment in both categories includes: • • •

Some form of sheet handling device, A way of moving the sheet from one station to another, A means of controlling the various elements that allow the sheet to be heated, formed and moved from station to station, • A sheet heating oven, • A vacuum system, • A forming press, and • A formed part removal region. In addition, the equipment may include: •

• • •

Some form of prestretching such as: preblowing or plug assist, A pressure system, A trimming press, and Some form of trim removal.

Certain guidelines pertain to both categories of forming equipment. Table 1.7 gives an overview for thermoforming equipment in general [20]. Some of these are summarized below. Heating Source The various heating methods are detailed in Chapter 3. Sheet temperature should be controlled to within + 5°C or + 1O0F. During transfer to the forming station, the sheet temperature drop should not exceed 5 to 100C or 10 to 200F. Infrared heating is most popular today. The various heating methods include [21]: • •

Simple nickel-chrome heating wires, Metal resistance rods, sometimes called calrods,

Table 1.7 General Specifications for Thermoformers1 Platen size, W x L (in x in or mm x mm) Maximum depth of draw (in or mm) Forming process (vacuum, pressure, matched mold, plug assist, twin-sheet capability) Platen power drive (pneumatic, hydraulic, mechanical, electric) Indexing power drive (pneumatic, hydraulic, mechanical, electric) Floor space (ft2 or m2) Heater type (metal rod, quartz, ceramic, radiant, gas, nichrome wire) Heater controls (proportioning, percentage timers, timer controlled, zone controlled, programmable controlled, machine controlled) Maximum heater output (kW/ft2 or kW/m2) Special features (purpose, type such as shuttle or rotary, number of stations, number of ovens, etc.) 1

• • • • • • • •

[20] with permission of John Wiley & Sons

Ceramic bricks or tiles, Quartz heaters in rod, spiral or square plate form, Direct gas-fired burners, Indirect gas-fired catalytic burners, Heat lamps, Quartz glass plates, Halogen bulbs, and Wire or rod heated metal plates.

Heater surface temperature is usually measured with thermocouples, thermistors or infrared pyrometric devices. Heating wires and resistance wires are inexpensive but oxidize rapidly and so lose heating efficiency. Quartz heaters are quite efficient, can be turned on and off like light bulbs but are quite expensive and fragile. Quartz is preferred for high temperature and "shaped heating" needs as described in Chapter 3. Sheet is also heated by direct contact with a hot metal plate (trapped sheet heating), by placing the sheet in a hot air oven (convection heating), or by passing it through a high-frequency electromagnetic field (RF or microwave heating). In the last case, the plastic must absorb the high-frequency energy. PVC is heated by radio-frequency energy in the flow-molding embossing process. Other polymers must be doped with "lossy" substances such as inorganic hydrates or even carbon black. There are certain elements that pertain to the ovens for all forming presses. For example: • •

There must be a way of separating the sheet from the heater source at shut-down. Baffles and dampers are used for heavy-gage sheet and fly-open and extracting shuttles are used for thin-gage sheet. There should be adequate means for rapid replacement of burned-out heater elements on both top and bottom heater banks.

Forming Platform The forming station should include all elements necessary to prestretch the sheet, form it, cool it and eject it from the mold. Some of these elements include: • • • •

• • • • • • • •

Substantial guide-rods are needed. For vacuum molds greater than 12 in or 300 mm by 24 in or 600 in dimension and for all pressure forming applications, four guide-rods are recommended. Clamp tonnage should be in proportion to mold size, that is: Typically >100 lbf/in2 or 0.7 MPa for straight vacuum forming, and Typically > 200 lbf/in2 or 1.4 MPa for pressure forming. The press frame pit should not be deeper than 4 ft or 1.3 m. In many areas, pits deeper than this must be entered by people wearing self-contained breathing apparatus. The overhead press frame structure should be robust enough to support such elements as: The entire weight of the mold, if it is desired to mold in an "up" position, In-mold trimming components including cylinders and framework, Ancillary mold elements such as: Plug assist cylinders and frames, Ejector cylinders and frames, and Cavity isolator cylinders and frames, and Pressure boxes. The press should allow for easy mold removal and maintenance. The press platens should allow for easily adjustable mold daylight. There should be adequate headroom for overhead ancillary equipment. There should be adequate space around the presses for vacuum lines, vacuum and air pressure lines, and adequate mold temperature hoses and manifolds. There should be adequate provision for prestretching and billowing, sheet lockdown and stripping and ejecting. There should be allowance for free surface temperature control, including chilled air. The frame should be adequately reinforced and gusseted to carry heavy molds, ancillary hardware, and day-to-day vibration and shaking. There should be adequate provision for part removal and trim or web takeaway.

The drive system that raises and lowers the platens is the key to forming station performance. Depending on the application, the drive unit can be as inexpensive as a simple air cylinder or as complex as the hydromechanical clamps used on injection molding presses. Table 1.8 [21] rates some of the drives used in thermoforming presses. Many high-speed presses use electric toggle clamps and cams. For high-speed pressure forming of polypropylene, double toggle clamps are used. Some straight hydraulic clamps are used in high pressure applications. Electrically driven clamps are being developed that are touted to be more accurate with less maintenance than conventional clamp systems. Most vacuum forming drive systems are designed to close and clamp at maximum pressures of 20 to 40 lbf/in2 or 0.15 to 0.30 MPa. For

Table 1.8 Comparison of Forming Table Drive Units1 Characteristic

Hydraulic Pneumatic Eccentric Pneumatic Geared motor-cams driven motorair toggle cylinder cogwheel

Uniformity Stroke limit control Timing control Repeatability Speed control

0 1 2 0 3/4

5 5 5 5 0/2/5

4 5 4 4 5

0 3 1 0 1

Forming force Clamping force Stability—tracking Energy consumption Trouble-free nature

3 3 0/3 0 3

3 3 3/4 5 5

5 5 0/3 1 0

4 Maintenance 0 Noise level Construction quality 1 5/2 Cost Preblowing capacity 5

5 5 3 4/1 5

0 3 1 0 5

61/65

42/45

Total

30/31

Motor-driven spindle 5 5 4 5 2

0 5 4 0 2

3 3 2 3 0 1 5 4 3 4

2 0 5 3 0

3 4 5 1 0

3 1 5 1 0

26

41

50

5 4 5 2 3

1 From [21: Table 8] by permission of Carl Hanser Verlag Key: 0 = Low or poor; 3 = Moderate or average; 5 = High or excellent

pressure forming, the drive system must clamp against the forming air pressure. Safety factors of 3 to 4 are recommended. If 30 lbf/in2 or 2 MPa air pressure is used in forming, the drive system should be designed to remain closed against 90 to 120 lbf/in2 or 6 to 8 MPa pressure. For a typical mold forming area of 20 x 10 in or 500 x 250 mm, the drive system clamping load is 9 to 12 T or 32 to 107 Mkgf. Platens and guide-rods must be designed to accommodate high bending forces. For forming pressures in excess of 100 lbf/in2 or 0.7 MPa, forming station designs begin to resemble those used in thermoplastic structural foam injection molding [22]. For forming pressures in excess of 200 lbf/in2 or 1.4 MPa, the presses begin to resemble those used in compression molding. Vacuum System The vacuum systems for both categories of forming equipment are quite similar. For stand-alone shuttle and rotary formers, the vacuum pump, surge tank and plumbing is usually an integral portion of the machine (Fig. 1.23). For installations of several forming presses, regardless of the sheet gage, a centrally located vacuum system is frequently used. Even the least expensive vacuum forming press must have adequate means of rapidly drawing the sheet against the mold surface. One critical factor in efficient vacuum draw-down lies in an unencumbered, adequately sized line between the

Pneumatic/Hydraulic Plug Assist

Cooling Fan

Vacuum Tank Platform

Heater Cabinet Clamp Frame

Mold Platen

Forming Table

End View

Side View

Electric/Electronic Cabinet

Figure 1.23 Single station cut sheet shuttle press—Drypoll/Zimco

vacuum surge tank and the mold cavity. Proper vacuum system design requires a vacuum pump capable of drawing down to 710 to 735 mm Hg vacuum1 in the surge tank prior to the beginning of the forming cycle. The path between the surge tank and the cavity between the hot sheet and the mold should have as few constrictions as possible. Long pipes, flow constrictors, quick-disconnects, restrictive valves and large L/D vent holes should be eliminated. Fast-acting rotary ball valves are recommended for vacuum shut-off [23]. Section 6.5 details a method for determining pressure drops through each of the constrictions from the mold cavity to the vacuum pump inlet. A good estimate of the time required to evacuate a mold cavity is obtained from:

where 0 is the pump-down time, V is the total volume of the system to be evacuated, P1 is the initial system pressure (absolute), pf is the final system pressure, and p o is the vacuum tank pressure. S0 is the volumetric evacuation rate. The evacuation rate of the vacuum pump is usually specified by the pump manufacturer, as Sp. The evacuation rate, S0 is given as:

+

h-k h

x

(2.9)

where I 0 is the incident wavelength-dependent radiation, oc is the absorptivity and x is the thickness of the plastic sheet. Figures 2.19-2.30 show the effect of sheet thickness on IR transmission. As is apparent, as the sheet increases in thickness, the amount of IR energy absorbed increases but the general shape of the IR spectra remains the same. For very thin films of less than 1 urn, surface molecular orientation may distort the absolute values of the IR spectra and therefore the energy absorption characteristics of the films.. The primary effect of organic colorant on polymer should be in the visible wavelength range [0.38 jim to 0.71 um]. Solid inorganic particles such as TiO 2 , carbon black, and talc act as opacifiers by increasing surface absorption of visible light and minimizing the amount of visible light that is transmitted into or through

Transmission

Polytetrafluoroethylene, PTFE

Wavelength, pm Figure 2.32 Infrared transmission spectrum for polytetrafluoroethylene, PTFE

Table 2.14 Characteristic Polymer Infrared Absorption Bands in Wavelength (Values in Parentheses Represent Weak Absorption Bands) Polymer

Primary (|im)

HDPE LDPE

3.2-3.9 3.2-3.9

PP

3.2-3.6

PS ABS PVC

3.2-3.6 2.8-3.6 3.2-3.6

PMMA

3.2-3.6

PA 6

3.0-3.2

Cellulose acetate PET

5.5-6.0 7.8-10.0 3.3-3.6

FEP PEI

7.4-9.0 2.7-3.0

PC

3.2-3.6

Secondary (jim)

(7.0-8.0) 6.7-7.1 7.0-8.0 6.6-7.0 7.1-7.3 (8.4-8.7) (9.8-10.1) 6.4-7.3 6.4-7.3 (1.65-1.8) 2.2-2.5 5.7-6.0 6.8-11.0 1.4-2.2 1.1-1.25 5.7-6.0 6.2-9.5 19.-2.8 (6.0-7.8) 2.7-2.9 5.9-6.0 7.0-9.2 (4.2-4.4) 5.8-6.0 (6.9-9.2) 5.5-6.2 6.6-7.7 7.8-9.5

the polymer. Thus, effective opacifiers should have particle sizes in the 0.1 to 10 jim range. Opacifiers with particle sizes of 3 to 12 (im will also act to block incident infrared radiation and thus change the absorption characteristics of the polymer. The effect of colorant dosage on the polymer IR spectrum is shown in Fig. 2.33 [50]. The general effect is to gradually increase the IR absorptivity with increasing dosage. The nature of the colorant also affects the polymer IR spectrum (Fig. 2.34) [51]. Organic dyes and tints are designed to affect the polymer electromagnetic radiation spectrum in the visible wavelength range (Fig. 2.35), and are usually used in small quantities. Although the spectra of these organics overlay those of the polymers, the small dosages usually do not materially affect the energy absorption efficiencies of the polymers. As discussed in Chapter 3, a sound measure of the amount of energy absorbed by a plastic is obtained by integrating the wavelength-dependent absorption curve over the wavelength range of the incident radiation. This is shown in

Penetration Depth, cm

0.25 wt % Red Pigment

Wavelength, urn Figure 2.33 Effect of colorant dosage on absorption characteristics of polymethyl methacrylate, PMMA [50]

Fig. 2.36 [52] for several colorants. Simply put, IR spectra offer substantial information about relative processing effects such as: The effect of increasing heater temperature on sheet heating rate, The effect of sheet downgaging on energy absorption, and thus on cycle time, sheet surface temperature, and discoloration, • The effect of increasing sheet thickness on sheet surface temperature, • The effect of thin cap-sheeting or film on heating rate of sheet, • The effect of changing pigment type and dosage, and • The effect of printing on sheet heating characteristics.

Penetration Depth, cm

• •

Natural

Translucent Blue

Opaque White

Wavelength, jjm Figure 2.34 Effect of pigment type on absorption characteristics of polystyrene, PS [51]

Absorptivity, cm"1

Undyed Red Dyed

Wavelength, jum

Figure 2.35 Absorptivity of natural and red dyed polymethyl methacrylate, PMMA

Integrated Absorption

Temperature, 0K

Opaque White PS Blue PS PP

Amorphous PET Transparent PS

PMMA

Temperature, 0F Figure 2.36 Heater temperature-dependent total absorption for several natural and pigmented thermoplastics [52]

2.19 Summary Although it was stated at the beginning of this chapter that "If a polymer can be produced as a sheet, it can be thermoformed into a product", this does not mean that all polymers thermoform with equal effort. Typically, amorphous polymers have broader forming windows than crystalline ones. Again, the forming temperatures given in Table 2.5 represent extremes and ideal conditions. Actual forming temperature ranges are usually only a few degrees. Table 2.15 gives some general forming characteristics for many of the polymers listed in Tables 2.5 and 2.12. Special, more expensive polymeric homologs are being developed to circumvent the forming inadequacies of certain polymeric classes.

Table 2.15 Thermoforming Processing Characteristics of Some Formable Polymers. (These characteristics are generic unless otherwise noted) Polymer

PS

Process temperature range (0C)

(0F)

150-190

300-375

Char, maximum draw ratio

Major draw limitation

Comments

Tears at high temperatures

Some yellowing at higher temperatures, long oven times. Trim dust is tenacious. Parts are brittle in 3D corners at deep draw. Sheet is easily marked off in plug assist.

Elongation is limited at high rubber content.

Yellowing at higher temperatures. Tends to be dimcult to form into sharp 3D corners. Needs to be held on mold longer than PS or ABS. This is particularly true for high rubber content. Sheet is easily marked off in plug assist.

8:1 HIPS

163-204

325-400 8:1

ABS

150-204

300-400

10:1

Elongation at high temperature.

Moisture causes pits, bubbles, blisters. Discolors at higher temperatures, long oven times. Can be splitty at low temperatures.

mPPO

163-218

325-425

6:1

Stiff at moderate temperature.

Thermoforms like HIPS but stiffer. Odor can be objectionable. Can yellow at high forming temperature. Sharp corners dimcult to form at modern temperatures. Heavy gage must be trimmed cold. Trim dust can be tenacious. Ideal candidate for pressure forming.

OPS

127-160

260-320

5:1

Very stiff at low temperature. Can lose orientation at high temperature.

Must be very carefully heated to maintain orientation. Birefringence can be used to monitor orientation. Sheet best heated by direct contact. Sharp corners dimcult. Very tough, splitty to trim cold. Superior surface gloss, opticals, impact strength. (Continued)

Table 2.15 (Continued) Polymer


(0F)


PMMA

150-204

300-400

12:1

Elongation at high temperature for lightly crosslinked PMMA.

Sheet can scorch, blister at high energy input. Surface can change from glossy to matte at high energy input. Highly stretched sheet can be brittle in 3D corners. Sheet is easily scratched during handling, trimming. Sheet frequently thermoforming with protectivefilmin place. Trim dust can be tenacious, statically charged.

PMMA/PVC

150-190

300-375

8:1

Elongation at high temperature.

Sheet can blister, scorch, yellow at high energy input. Sheet can be brittle during trimming. Sheet can be pressure formed with good results.

FPVC

107-150

225-300

10:1

General weakness at higher temperatures.

Upper temperature limit is discoloration. Plasticizer odor objectionable. Rubbery sheet requires longer mold times to set. Embossings wash at high temperature or draw ratios > 5:1.

General weakness at higher temperatures.

Upper temperature limit is discoloration. Long oven times cause yellowing. Decomposition product is HCl. Difficult to prestretch as heavy gage. Thin gage transparency not as good as PS, ABS. Tends to be tougher in 3D corners than PS, PMMA. Virgin transparent has bluish tint to balance yellowish regrind color.

Melt elasticity low at forming temperatures.

Usually processed above its melt temperature of 115°C. Can exhibit excessive sag very quickly. Sag bands recommended for thin gage. Increased haze at higher temperatures.

121-177 250-350

RPVC

260-350

Comments

6:1

127-177 LDPE


6:1

HDPE

138-193

280-380

8:1

Melt elasticity.

Usually processed above its melt temperature of 135°C. Can exhibit excessive sag. Black sheet heats much faster than white. Thin gage sheet can excessively sag very quickly. Increased haze at higher temperatures.

PP

143-166

290-330

6:1

Excessive sag and narrow forming window.

Very difficult to control sag with straight homopolymer. Very narrow forming temperature range. Sag bands should be slipcoated to prevent sticking. Draw frequently shows necking. Sheet easily marked by plug. Thin gage can have high residual stress, can pull out of pins during heating. High energy input increases haze. Sheet does not draw well into sharp corners when cold. Best parts are pressureformed. Trim blades must be sharp to avoid forming whiskers, angel hair. Parts formed at higher temperature onto cold mold are frequently brittle. Can be pressure-formed in thin gage below melt temperature.

EP Copoly.

132-177

270-350

8:1

Sags, necks at high forming temperature.

Forms like HDPE in thin gage. Good elongation at lower forming temperature range. Can be very difficult to trim when very cold.

OPP

143-166

290-330

5:1

Rapidly loses orientation, sages at upper temperature.

Very low haze, excellent surface gloss, high impact strength. Must be heated very carefully to maintain orientation. Thin-gage sheet heated best with direct contact.

EVA

135-177

275-350

8:1

Tears at high temperature.

Not normally used alone. As a tie layer, usually draws well with little resistance. (Continued)



(0F)


PP-20% talc

149-204

300-400

5:1

Elongation low at moderateto-high temperature.

Stiff at forming temperature. Elongation limited byfillerloading. Deep draw restricted. Matte surface may not be acceptable. Sharp corners difficult to form without plug assist. Plug assist useful primarily at low forming temperature. Pressure forming desired heavy-gage sheet.

PP-40% GR

149-232

300-450

4:1

Very stiff at all forming temperatures.

Best pressure-formed. Deep draws cause splits, polymer-rich areas in corners. Plug assist not always effective owing to low melt strength of base polymer.

PC

177-232

350-450

8:1

Stiffness even at upper forming temperature.

Tends to be stiff at all forming temperatures. Can discolor at high temperature. Moisture causes pinholes, pock marks, bubbles, brittleness. Transparent sheet harder to heat than acrylic. Sharp corners hard to form at low forming temperature. Very tough to trim cold. Routering recommended.

APET

127-166

260-330

6:1

Sags, necks, anneals, orients rapidly at high temperature.

Slowly crystallizing polymer. Rate increases with decreasing molecular weight. Low molecular weight heat sets rapidly. Residual stress problem in thin gage. Necking difficult to avoid in deep drawn parts. Plug assist enhances necking. Bubble prestretching enhances orientation, reduces maximum draw ratio. Hard to form sharp corners on hot mold. Trimming is difficult cold, can cause whiskers or angel hair.


Comments

Trim registry difficult owing to high polymer movement. Very thin gage sheet and film best heated by direct contact. 185-199

365-390

5:1

Very stiff if sheet too hot. Sheet tears if too cold.

Requires careful oven temperature control. Mold must be heated to 1800C or 3600F or so. Excessive draw-down leads to brittle corners. Crystallinity in 20% range gives optimum properties. Difficult to control crystallinity in very thin sheet. Normal gage range of 1 mm or 0.040 in or more.

135-177 TPE/TPO (Depends on polymer)

275-350

6:1

Spring-back.

Many versions of thermoplastic elastomers. Those with high natural rubber content more difficult to maintain part shape. Very cold molds required. Hard to maintain long-term dimensions. Pre-stretch must remain on during forming to minimize spring-back.

93-121

200-250

4:1

Cell collapse at high temperature, very stiff at low temperature.

Thin laminatingfilmimproves surface appearance, increases maximum draw, stiffensfinalproduct. Parts mostly restricted to shallow draw. Matched die molding usually required for low temperature forming, preferred to control sheet thickness expansion from oven. Trim dust is statically charged, very tenacious. Sharp corners not desired owing to splittiness, poor impact of foam.

CPET

PS Foam

(Continued)



(0F)


PP foam

149-166

300-330

4:1

Cell collapse, stickiness, mushiness can occur very rapidly.

Matched die molding preferred. Compression molding sometimes recommended. Processing window very narrow. Cell collapse catastrophic. Crosslinked PP preferred. Deep draw difficult without prestretching. Material elasticity at lowest temperature prevents sharp corners.

XLPE foam

149-204

300-400

4:1

Cell collapse, mold stick at high temperature.

Sheet tends to be rubbery-elastic, like TPE. Some spring-back. Cell rupture at high energy input. Difficult to get deep draw even with aggressive plugging. Foam very soft at low density. Best for shallow draw.

PET foam

185-199

365-390

4:1 (?)

Hot mold and other CPET forming limitations.

Early results show 400 to 900 kg/m3 PET form like unfoamed CPET. Foam sheet crystallinity 1 mm or 0.040 inch).


Comments

2.20 References 1. J. Frados, Ed., Plastics Engineering Handbook, 4th Ed., Van Nostrand Reinhold Co., New York, 1976, p. 274. 2. J.L. Throne, "Polymer Properties", in M. Bakker, Ed., Encyclopedia of Packaging Technology, John Wiley & Sons, New York, 1986. 3. J.-M. Charrier, Polymeric Materials and Processing: Plastics, Elastomers and Composites, Hanser Publishers, Munich, 1991. 4. R. Gachter and H. Muller, Eds., Plastics Additives Handbook: Stabilizers, Processing Aids, Plasticizers, Fillers, Reinforcements, Colorants for Thermoplastics, 2nd Ed., Carl Hanser Verlag, Munich, 1985. 5. H.R. Simonds, Source Book of the New Plastics, Vol. II, Van Nostrand Reinhold Co., New York, 1961, p. 21. 6. J.L. Throne, Thermoplastic Foams, Sherwood Publishers, Hinckley OH, 1996. 7. J.L. Throne, "Thermoforming—A Look Forward", SPE ANTEC Tech. Papers, 29 (1983), p. 464. 8. R.D. Deanin, Polymer Structure, Properties, and Applications, Cahners Books, Boston, 1972, p. 154. 9. J.L. Throne, Plastics Process Engineering, Marcel Dekker, New York, 1979, p. 65. 10. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich, 1993, Chapter 1. 11. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich, 1993, Table 2.4, pp. 90-91. 12. T. Alfrey and E.F. Gurnee, Organic Polymers, Prentice-Hall, New York, 1967, p. 51. 13. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich, 1993, p. 129. 14. A. Ziabicki, Fundamentals of Fibre Formation: The Science of Fibre Spinning and Drawing, John Wiley & Sons, New York, 1976, p. 112. 15. J.L. Throne, "Thermoforming Crystallizing Polyethylene Terephthalate (CPET)", Adv. Polym. Tech., £(1988), pp. 131-171. 16. J.L. Throne, "Thermoforming Crystallizing PET", SPE ANTEC Tech. Papers, 27(1981), p. 598. 17. J.A. Brydson, Plastics Materials, IHfTe, London, 1966, p. 58. 18. R.E. Dempsey et al., US Patent 4,127,631, Assigned to Amoco Chemicals Corp., Chicago IL, 28 Nov 1978. 19. H. Saechtling, International Plastics Handbook for the Technologist, Engineer and User, 2nd Ed., Hanser Publishers, Munich, 1987, Figure 82, p. 166. 20. LJ. Gibson and M.F. Ashby, Cellular Solids: Structure & Properties, Pergamon Press, Oxford, 1988, Figure 3.2, p. 47. 21. J.A. Brydson, Plastics Materials, Iliffe, London, 1966, p. 34. 22. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich, 1993, Figure 2.45, p. 135. 23. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich, 1993, Figure 2.46, p. 136. 24. L.E. Nielsen, Mechanical Properties of Polymers, Reinhold, New York, 1962, p. 244. 25. R.D. Deanin, Polymer Structure, Properties, and Applications, Cahners Books, Boston, 1972, p. QQ

oo.

26. J.L. Throne, "Polymer Properties", in M. Bakker, Ed., Encyclopedia of Packaging Technology, John Wiley & Sons, New York, 1986, p. 533. 27. R.M. Ogorkiewicz, Ed., Thermoplastics: Properties and Design, Wiley-Interscience, Ltd., London, 1974, p. 28. 28. J.B. Howard, "Fracture—Long Term Testing", in N.M. Bikales, Ed., Mechanical Properties of Polymers, Wiley-Interscience, New York, 1971, p. 73.

29. R.L. Baldwin and K.E. Van Holde, Fortschr. Hochpolym. Forsch., 1 (1960), p. 451. See also J.L. Throne, Plastics Process Engineering, Marcel Dekker, New York, 1979, p. 765. 30. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich, 1993, p. 12. 31. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich, 1993, Figure 2.47, p 142. 32. F. Brinken and H. Potente, "Some Considerations of Thermodynamics in Thermoforming", SPE ANTEC Tech. Papers, 29 (1983), p. 467. 33. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich, 1993, Table 1.4. 34. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich, 1993, Table 1.3, p. 13. 35. J.L. Throne, "Polystyrene Foam Sheet Expansion During Heating", SPE ANTEC Tech. Papers, 29(1983), p. 1328. 36. J.A. Brydson, Plastics Materials, Uiffe, London, 1966, p. 58. 37. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich, 1993, Figure 133, p. 208. 38. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich, 1993, Figure 246, p. 316. 39. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich, 1993, Figure 241, p. 311. 40. Material adapted from W.K. McConnell, Jr., handout material, "Thermoforming Technology for Industrial Applications", SPE Seminar, 12-14 March 1985, Arlington TX. 41. J.L. Throne, Plastics Process Engineering, Marcel Dekker, New York, 1979, pp. 714-735. Taken from an extensive series of tables published by R.G. Griskey in Modern Plastics, 1966-1967. 42. H. Voigt, "Lehrgang fur Thermoformung", Paul Kiefel Thermoformmaschinen GmbH, Freilassing, Germany, undated. 43. Z. Tadmor and CG. Gogos, Principles of Polymer Processing, John Wiley & Sons, Inc., New York, 1979, pp. 697-703. 44. B.C. Sakiadis and J. Coates, AIChE J., 2 (1956), p. 88. See also J.L. Throne, Plastics Process Engineering, Marcel Dekker, New York, 1979, p. 737. 45. R.C. Progelhof, J.L. Throne and R.R. Ruetsch, "Methods for Predicting the Thermal Conductivity of Composite Systems: A Review", Polym. Eng. Sci., 1(5(1976), p. 615. 46. J.P. Holman, Heat Transfer, 4th Ed., McGraw-Hill Book Company, New York, 1976, Figure 8-1, p. 274. 47. F. Kreith, Principles of Heat Transfer, 2nd Ed., International Textbook Co., Scranton, 1965, p. 199. 48. Anon., "Plastic Film Measurement", Technical Note TN100, IRCON, Inc., Niles IL, 1993, Figures 2-5. 49. I. Kossler, "Infrared-Absorption Spectroscopy", N.M. Bikales, Ed., Characterization of Polymers, Wiley-Interscience, New York, 1971, p. 125. 50. D. Weinand, "Modellbildung zum Aufheizen und Verstrecken beim Thermoformen", DoktorIngenieurs Dissertation, Rheinisch-Westfalische Technische Hochschule, Aachen, Germany, 16 JuIi 1987, BiId 3.10. 51. D. Weinand, "Modellbildung zum Aufheizen und Verstrecken beim Thermoformen", DoktorIngenieurs Dissertation, Rheinisch-Westfalische Technische Hochschule, Aachen, Germany, 16 JuIi 1987, BiId 3.9. 52. D. Weinand, "Modellbildung zum Aufheizen und Verstrecken beim Thermoformen", DoktorIngenieurs Dissertation, Rheinisch-Westfalische Technische Hochschule, Aachen, Germany, 16 JuIi 1987, BiId 2.14.

3 Heating the Sheet 3.1 3.2 3.3 3.4 3.5 3.6

3.7

3.8

3.9 3.10

3.11

3.12

3.13

3.14 3.15 3.16

3.17

Introduction Energy Absorption by Sheet Heat Transfer Modes Incorporating Formability and Time-Dependent Heating Conduction Convection Heat Transfer Coefficient The Biot Number Effective Radiation Heat Transfer Coefficient Constant Heat Flux Radiation Heating Black Body Radiation Gray Body—Emissivity Radiant Heater Efficiency—Constant Heat Flux Application Real Heaters—Efficiencies Radiative Heat Transfer Coefficient Convection and the Heat Transfer Coefficient Rod Heaters Long-Term Radiant Heater Efficiencies Edge Losses—View Factor Local Energy Input Pattern Heating Zone, Zoned or Zonal Heating Heater to Sheet Distance Thin-Gage Sheet—Approximate Heating Rates Constant Environmental Temperature Approximation Constant Heat Flux Approximation Thin-Gage Approximations—Comments Heavy-Gage Sheet—Internal Temperature Control Constant Environmental Temperature The Constant Heat Flux Case The Thickness Effect Summary Equilibration Convection Heating Constant Heat Flux Computed Equilibration Times The W-L-F Equation The Arrhenius Equation Relating Shift Factors to Sheet Thickness Infrared-Transparent Polymers Computer-Aided Prediction of Sheet Temperature The Radiant Boundary Condition Guidelines for Determining Heating Cycles The Biot Number Thin-Gage Guidelines Heavy-Gage Guidelines Intermediate-Gage Guidelines References

3.1

Introduction

The thermoforming process is neatly segmented into four steps: • • • •

Heating the sheet, Stretching the sheet, Cooling the sheet on the mold surface, and Trimming the part from its surroundings.

During the forming and trimming steps, the sheet dimensions are changing but the sheet is essentially at constant temperature. During the heating and cooling steps, the sheet dimensions are essentially constant, but the sheet temperature is changing. Thus the heat transfer process and the mechanical deformation process are best treated separately. This chapter focuses on the ways in which sheet is heated to the stretching or forming temperature. Chapter 4 concentrates on the technical details of sheet stretching. Chapter 5 considers the process of cooling and the trimming step. The material in these chapters is quite technical. However, newer thermoforming technologies mandate a thorough understanding of the basic concepts underlying the general process. And many of the troubleshooting solutions to processing problems are apparent once these concepts are understood. Roll-fed formers have used infrared heating for years owing to its efficiency in heating thin-gage sheet. Heavygage cut sheet formers have used forced convection hot air ovens for heating, in order to minimize sheet surface degradation. There are technical and management reasons behind these decisions. In certain instances, these are not necessarily the optimum choices. Pattern or zone heating is used extensively to produce more uniform part wall thicknesses and so parts designers should be aware of some of the details of this technique. Again, what is important is an adequate understanding of the interaction between the plastic sheet, initially at room temperature but being heated to its proper forming temperature, and the chosen heating medium. The material in this chapter begins with a review of the basic types of heating methods. Then thin-gage heating, particularly infrared heating, and forced convection hot air heating of heavy-gage material are considered in detail. Equilibration time is discussed. This is the time it takes for the sheet to achieve a uniform temperature across its cross-section, once the heating source is removed. Computeraided models are also outlined. And finally, guidelines for determining heating cycle times for both thin-gage and heavy-gage sheet are presented.

3.2

Energy Absorption by Sheet

Thermoforming is an energy intensive plastics process. Economics require the most efficient use of energy. The amount of energy needed to heat a unit mass of sheet

PP HDPE

PS

PB

Specific Heat, kcal/kg-°C

PA-6

PC RPVC PMMA PSO2

PTFE Air

Diamond

Temperature, 0C Figure 3.1 Temperature-dependent heat capacities or specific heats of several thermoplastics

from room temperature, RT, to the forming condition, Tf, is enthalpy increase, Ah, and is obtained from: Q or A h =

c p (T)dT

[cal/g] = [cal/g 0C] • [0C]

(3.1)

JRT

[Btu/lb] = [Btu/lb°F] • [0F] cp(T) is the temperature-dependent heat capacity (Table 2.5 and Fig. 3.1) [I]. The forming temperature is assumed to be the average sheet temperature at the normal forming temperature:

Tf

K0f T ( x ) d x

(32)

-

T(x) is the temperature at position x across the sheet half-thickness, 0 < x < L. If the temperature dependency of heat capacity is unknown, an average value will give a reasonable estimate. If the heat capacity of a specific polymer is unknown, a value of a homologous polymer can be used as a first approximation. The change in enthalpy is a much better method of determining the amount of energy uptake by the sheet. Figure 2.17 provides enthalpy values for several thermoformable polymers. H is the enthalpy at a given temperature, as [cal/g], [kcal/kg], or [Btu/lb]. The energies needed to heat typical polymers to forming temperatures are given in Table 3.1.

Table 3.1 Thin-Gage Heating Efficiencies (Heater Temperature, T 0 0 = 7600C; Heater Output, Q/A = 40 kW/m2) Polymer

LOPE PMMA PVC PS PTFE/FEP2 PA 662 1 2

]

Heating rate

Energy absorbed

Efficiency

Maximum effective heating transfer coefficient

(cal/g°C)

(s/mm)

(kW/m2)

(%)

(kW/m2 • 0C)

(Btu/ft2 • h • 0F)

0.498 0.559 0.265 0.341 0.342 0.479

25 27 21 13-15 13 27

11.5 19.44 15.88 19.03-21.97 25.86 19.84

27.6 48.6 39.7 47.6-54.9 64.7 49.6

0.0182 0.0333 0.0255 0.0310-0.0358 0.0548 0.0370

3.21 5.87 4.50 5.46-6.30 9.65 6.52

Normal forming temperature (0C)

Enthalpy

Density

C

(cal/g)

(g/cm3)

129 177 138 146 288 224

71.7 104.1 55.9 65.0 66.9 105.0

0.92 1.2 1.4 1.05 2.2 1.2

Used only for effective heat transfer coefficient calculation Values from Fig. 3.26

Some of these values are based on assumed values for heat capacity and some are approximate. Example 3.1 compares energy uptake for ABS and HDPE, representing amorphous and crystalline polymers, respectively. As detailed in Chapter 2, additional energy is required to melt a crystalline polymer such as HDPE. If the energy input to both sheet is the same, it requires substantially longer to heat HDPE than ABS. Example 3.2 illustrates this point. Example 3.1 Energy Absorbed by Plastic Sheet Calculate the amount of energy required to heat ABS and HDPE to their respective normal forming temperatures. If the energy input is the same to each sheet, calculate the relative times to heat each sheet to the forming temperature.

From Table 2.5, the normal forming temperature of both of these polymers is 295°F or 146°C. The amount of energy required to heat the plastic from room temperature (say 77°F or 25°C) to the forming temperature is obtained from Fig. 2.17. HDPE: 128 kcal/kg @ 146°C - 8 kcal/kg @ 25°C = 120 kcal/kg ABS: 51 kcal/kg @ 146°C - 7 kcal/kg @ 25°C = 45 kcal/kg The densities of these polymers are obtained from Table 2.12. HDPE: 960 kg/m3 ABS: 1050 kg/m3 The energy required per unit volume is given as: HDPE: 120 kcal/kg • 960 kg/m3 = 0.115 x 106 kcal/m3 ABS: 45 kcal/kg • 1050 kg/m3 = 0.0473 x 106 kcal/m3 Relative to the heating time for ABS, HDPE takes 0.115/0.0473 - 2.43 times longer to heat to the normal forming temperature. Example 3.2 Time to Heat Plastic Sheet Consider the two polymers of Example 3.1. If the plastic is 0.100 inch thick (2.5 mm) and the sheet receives 12.9 W/in2 (2 Wjcm2) heating energy1, determine the time required to heat the sheet to the forming temperature. Assume that the sheet heats uniformly throughout its thickness. IW= 3.413Btujh.

The energy required to raise each polymer to the forming temperature is given as: HDPE: 120 kcal/kg • 1.8 = 216 Btu/lb ABS: 45 kcal/kg • 1.8 = 81 Btu/lb 1

Correctly, this example assumes that the net energy interchange between the heater and the sheet is constant with time. See the section on basic concepts in radiation heat transfer for additional details on this constant heat flux assumption.

HDPE: 216^-0.96-624-0-1 - ^ S = 0 7 4 9 S 1 ABS:

4 5 ^ - 1.05 • 62.4 ^ • 0.1 in- J - ^ = ( U O 7 * ? Ib ft3 1728 in3 in2 Energy input per unit time x time = Total energy uptake HDPE: 12.9 ^ x t (time in h) • 3.413 - ^ - = 0.749 ? z^ nr Wh m HDPE: 12.9 ^ 2 x t (time in h) • 3.413 - ^ - = 0.307 ? 2^ in W •h m Solving for time: HDPE:

t =

0.749 3.413- 12.9 =

0

ABS:

t =

3.4133012.9 =

Q 0Q697h = 25 1S

-017h ~ '

6 L 2 S

'

There are many ways of heating sheet to the forming temperature. No heating process is 100% efficient. Regardless of the nature of the polymers, all heating systems must input more specific energy than the amount indicated in Table 3.1. Economics dictate a balance between the efficiency of net energy interchange between the source of heat and the sheet and the net rate of heating to the forming conditions. Where the heating rate controls the cycle time, process optimization usually calls for lowered energy efficiencies. To a large degree, sheet thickness dictates the type of heating that is most effective. Thin sheets are heated quite efficiently with radiant heaters. Thick sheets are best heated in forced convection hot air ovens.

3.3 Heat Transfer Modes There are three ways of exchanging energy between objects of different temperatures: Conduction Conduction is solid phase contact heat transfer. Conduction is the primary way energy moves through plastic sheet and metal molds. Three thermal properties are important in conduction: • • •

Density, Specific heat, heat capacity or enthalpy, and Thermal conductivity.

These properties have been reviewed in Chapter 2. In addition, thermal diffusivity is important in time-dependent heat conduction. Thermal diffusivity was also reviewed in Chapter 2.

Convection

Convection is fluid phase contact heat transfer. Throughout the thermoforming process, the sheet contacts ambient air. Energy is transferred when the air temperature differs from the sheet temperature. Energy transfer depends on the extent of air movement. As expected, energy transfer is low in quiescent air and relatively high when the air is actively moved across the plastic surface. The proportionality between thermal driving force or temperature differential and the amount of heat transferred is called the convective heat transfer coefficient. Convection is important when water mist or fog is used to cool the free surface of a formed part1. Otherwise, the effect of convection on overall heat transfer is secondary to conduction and radiation. Radiation

Radiation is electromagnetic energy interchange between an energy source or hot element and an energy sink or cold element. Radiation pervades nature. Electromagnetic energy is usually characterized by the wavelength of the energy. As seen in Fig. 3.2 [2], X-rays and gamma rays are characterized by very short wavelengths. Ultraviolet rays have wavelengths less than 0.4 um. Visible light wavelength range is 0.4 to 0.7 jam. Near infrared wavelength range is 0.7 to about 2 um. Far infrared wavelength range is from about 2 um to 8 um. Longer wavelength electromagnetic energy includes microwave, short wave radio frequency, long radio frequency and ultrasonic frequency. Heated metal or ceramic surfaces are used throughout thermoforming to radiantly heat plastic sheet. The majority of energy transfer takes place in the 2 um to 8 um wavelength range, or the far infrared region. There are several aspects of radiant heat transfer that require careful attention. For example, the efficiency of energy transfer depends on the relative abilities of the source and sink to transfer

Thermoforming Region

log [Wavelength, m]

log [Frequency, s1]

Radio Hertzian Waves

X-Rays

Infrared Ultraviolet

Cosmic Rays

Gamma Rays

Visible Figure 3.2 Electromagnetic radiation spectrum, showing radio waves, atomic energy, visible light, ultraviolet and infrared domains. Thermoforming region is also shown [2] 1

Cooling the formed part against the mold is the subject of Chapter 5.

energy efficiently. Absorptivity and emissivity are terms used to describe this efficiency. For most thermoforming applications, the energy interchange is between the heater surface and the plastic sheet surface. In some cases, energy is transmitted into or through the polymer. In addition, even though the sheet is sandwiched between the heaters, the interchanging elements are not infinite in extent. As a result, the efficiency of energy interchange depends on geometric factors as well as material properties. Efficient heating and cooling of thermoplastic sheet depends on the balance between the rate of energy input to the sheet surface and the rate of energy conduction from the sheet surface to the centerline. There are two classic cases of time-dependent conduction that illustrate this. Step Change in Surface Temperature

Consider contact or trapped sheet forming (Fig. 1.5). When the sheet is placed against the isothermal hot plate, its surface temperature immediately increases to the plate temperature (Fig. 3.3). As time increases, energy is conducted to the interior of the sheet. If the sheet is held against the hot surface long enough, the sheet temperature will eventually equal the hot surface temperature everywhere throughout the sheet. If the sheet is heated on both sides as shown in Fig. 3.4, the temperature profile through the sheet will be symmetric about the centerline. Constant Energy Input to the Sheet Surface

The amount of energy the heaters interchange with the sheet surface per unit area is called heat flux (kW/m2 or Btu/h • ft2). If the energy input is constant, the time-dependent temperature profile of Fig. 3.5 is obtained. This case illustrates some basic concepts in radiation heat transfer to plastic sheet. If the energy input is equal on both sides of the sheet, the time-dependent temperature profile through the sheet is symmetric about the centerline (Fig. 3.6). The temperature profile is again symmetric about the centerline when the sheet is heated on both sides. Unlike the previous case, the sheet surface temperature continues to increase with time. Unlike the previous case, the sheet temperature never reaches a constant value.

T

Temperature

oo

Increasing Time

initial Surface

Thickness

Figure 3.3 Time-dependent temperature profile for conduction into polymer sheet, constant surface temperature indicative of contact heating

Temperature

Increasing Time

"•"initial Surface.

Centerline

Surface2

Thickness

Temperature

Figure 3.4 Time-dependent temperature profile for two-sided conduction into polymer sheet, constant surface temperature indicative of contact heating

Temperature

Figure 3.5 Time-dependent temperature profile for conduction into polymer sheet, constant surface heat flux, indicative of radiation heating

Increasing Time

initial

Surface

Thickness

Increasing Time

T

initial Surface ^

Centerline

Surface2

Thickness Figure 3.6 Time-dependent temperature profile for two-sided conduction into polymer sheet, constant surface heat flux, indicative of radiation heating

Temperature

Increasing Time

initial Thickness Figure 3.7 Time-dependent temperature profile for conduction into polymer sheet, very low heat flux, very high polymer thermal conductivity or very thin sheet

Temperature

It is apparent from these two cases that there is an interrelationship or coupling between the energy input to the sheet surface and energy conducted to interior of the sheet. Consider this coupling in concept. Figure 3.7 is an extreme example of very low heat flux to the sheet surface coupled with very high thermal conductivity or diffusivity or very thin sheet (or both). The temperature profile through the sheet is essentially flat and the centerline temperature essentially equals the surface temperature. If the energy input to the sheet surface is very high and the polymer thermal conductivity or diffusivity is very low or the sheet is very thick, the sheet surface temperature will appear to reach a fixed temperature nearly instantaneously, as shown in Fig. 3.8 or Fig. 3.3. Figure 3.9 shows a more typical coupling between sheet surface and internal temperatures.

Increasing Tm ie

T

initial Thickness

Figure 3.8 Time-dependent profile for conduction into polymer sheet, very high energy input to the surface, low polymer thermal conductivity or very thick sheet

Temperature

Increasing Time

T

Figure 3.9 Time-dependent profile for conduction into polymer sheet for typical energy flux, polymer properties and nominal sheet thickness ranges

3.4

inttial Thickness

Incorporating Formability and Time-Dependent Heating

Formability is a key aspect of thermoforming. As seen in Table 2.5, all thermoformable polymers have forming windows, defined by the lower, normal and upper forming temperatures (LFT, NFT and UFT). As noted, the lower and upper forming temperatures form the absolute boundaries on formability. Whether a specific polymer can be formed into a specific shape at temperatures near these boundaries depends on: •

• • • •

The sheet characteristics such as: Intrinsic orientation, Hot strength, Sag tendencies, Thermal sensitivity of the polymer, Sheet geometry and Thickness, Uniformity of heating, Depth of draw, General mold geometry, and Other mechanical aspects such as: Transfer time, Ambient air temperature, Plugging geometry, Plug rate, Plug temperature, etc.

Despite these limitations, the upper and lower forming temperatures are useful in defining the nature of the temperature within the sheet. The upper forming temperature relates to the sheet surface. For example, if the upper forming temperature is set because the polymer is prone to blistering or color change above this temperature, the sheet surface temperature must never exceed this value during forming. If the

Forming Range T

Temperature

upper

normal

lower

Surface

Average Centerline

initial Optimum Heating Time Time Figure 3.10 Ideal relationship between polymer forming temperature range [shaded area] and time-dependent sheet surface, average and centerline temperatures

lower forming temperature is set because the polymer is too stiff to be formed or because it forms microcracks below this temperature, the centerline temperature must exceed this temperature before the sheet can be formed. The forming temperature range is shown as a time-independent band in Fig. 3.10. Figure 3.10 also shows the superimposition of the time-dependent local temperatures from Fig. 3.8 for the ideal case where the surface temperature reaches the upper forming temperature at the same time the centerline temperature reaches the lower forming temperature. And the average sheet temperature just equals the normal forming temperature at the same time1. As an example of the interplay between the sheet characteristics and the time-dependent energy input to the sheet, consider the following examples: Thin-Gage Sheet

As noted, when the sheet is very thin, energy input to the sheet controls. If the sheet of Fig. 3.10 is dramatically reduced in thickness, the temperature profiles of 3.11 are obtained. Although this profile is entirely acceptable, it is apparent that the rate of heating can be increased substantially without affecting the formability of the polymer (Fig. 3.12). 1

The shapes of the temperature curves for sheet surface, average and centerline are representative of profiles for sheet heated by radiant or radiant/convective means. The actual shapes depend on the nature of energy input to the sheet surface and the conductive and geometric characteristics of the sheet. See Section 3.15 on predicting temperature profiles for more details.

Temperature

Forming Range T

upper

^normal T

lower Surface, Average, Centerline

Figure 3.11 Relationship between polymer forming temperature range [shaded area] and timed-dependent sheet surface, average and centerline temperatures for very thin sheet

Time

Heavy-Gage Sheet

When the sheet is very thick, conduction from the sheet surface to the centerline controls. If the sheet of Fig. 3.10 is dramatically increased in thickness, the

Temperature

Forming Range

Time

Increasing Heating Rate

Temperature

Forming Range

Time

Minimum Heating Time

Figure 3.12 Effect of increasing energy input rate on the relationship between polymer forming temperature range [shaded area] and time-dependent sheet surface, average and centerline temperatures for very thin sheet

Temperature

Forming Range

S

A

Time

C

Figure 3.13 Relationship between polymer forming temperature range [shaded area] and time-dependent sheet surface, average and centerline temperatures for very heavy gage sheet

temperature profiles of Fig. 3.13 are obtained. To get the temperatures back into the forming window, the rate of heating must be decreased (Fig. 3.14). Changing Polymer Characteristics Without Changing Sheet Thickness

If the new polymer has a broader forming window than the old polymer, the rate of heating can be increased without affecting the formability characteristics. If the new polymer has a narrower forming window, the heating rate must be decreased as seen in schematic in Fig. 3.15. If the new polymer has a higher thermal conductivity than the old polymer, as is the case when filled polymers are used, the rate of heating can be increased without affecting the formability characteristics. If the new polymer has a lower thermal conductivity, as may be the case when foamed polymers are used, the heating rate must be decreased. Changing Other Aspects of the Forming Process

If the new mold requires greater depth of draw than the old one, the sheet may need to be formed at higher temperatures than before. As a result, the forming window may need to be narrowed or the lower forming temperature value increased. The result is that the rate of energy input may need to be reduced and the time to the forming condition extended. If the sheet requires prestretching, time between exiting the oven and completing the stretching may be longer than with straight forming. As a result, the sheet may need to be heated to higher temperatures than before. The rate of heating may need to be reduced and the time to the forming condition extended to increase the average temperature without exceeding the upper forming temperature. Example 3.3 illustrates the interaction of forming temperatures with heating cycle times for ABS and HDPE.

Temperature

Forming Range

Temperature

Decreasing Heating Rate

Time

Forming Range

Mn im i um Heating Time Time Figure 3.14 Effect of decreasing energy input rate on the relationship between polymer forming temperature range [shaded area] and time-dependent sheet surface, average and centerline temperatures for very heavy gage sheet

Example 3.3 Minimum and Maximum Forming Times Consider the two polymers of Examples 3.1 and 3.2. Determine the times required to reach lower and upper forming temperatures.

The upper and lower forming temperatures are obtained from Table 2.5 for ABS and HDPE. As is apparent, the lower forming temperature for both is 2600F (127°C) and the upper forming temperature is 3600F (182°C). The enthalpy increases to these temperatures are obtained from Fig. 3.1, as before: HDPE: LFT: 62 kcal/kg = 112 Btu/lb UFT: 142 kcal/kg = 256 Btu/lb ABS: LFT: 36 kcal/kg= 65 Btu/lb UFT: 60 kcal/kg= 108 Btu/lb Since all other factors are equal, the times are obtained by ratio with the values of Example 3.2. The values are tabulated here:

Polymer

HDPE ABS

Time (s) to reach Lower forming temperature

Normal forming temperature

Upper forming temperature

31.7 20.1

61.2 25.1

72.5 33.5

Temperature

Forming Range

Time

Minimum Heating Time

Reduction in Forming Range

Old Forming Range

Temperature

New Forming Range

Mn im i um Heating Time Time Figure 3.15 Effect of decreasing forming window on the relationship between polymer forming temperature range [shaded area] and time-dependent sheet surface, average and centerline temperatures

These various interactions and the general concepts of coupling of energy input to the sheet surface and conduction into the sheet interior are bundled into a predicting method in as described in Section 3.15. The various elements of this protocol are discussed below and the details of the protocol follow these discussions.

3,5

Conduction

As noted, conduction is solid phase energy transfer on an atomic or molecular level. Owing to high vibrational and rotational mobility of electrons and regular crystallographic structure, metals achieve high levels of conduction energy transfer. Organic materials, on the other hand, have relatively immobile atomic structures and so are poor thermal and electrical conductors. Polymers have even less molecular mobility and in addition, have high free volumes, allowing chain segments to move without contacting other segments. Polymers are therefore very poor thermal and electrical conductors. This was noted in Chapter 2. Classically, thermoformable sheet is considered as a two-dimensional planar surface with lateral dimensions far greater than its thickness dimension. In the bulk of the analyses that follow, the sheet is assumed to be planar to incident energy. One-dimensional steady-state heat conduction across the sheet thickness is given as: (3.3)

a/A

Steady-State Conduction

Q/A

Transient Conduction

Radiation Through Semitransparent Polymer Figure 3.16 Classical temperature profiles through plastics

Q/A

where Q/A is the heat flux, k the polymer thermal conductivity. AT is the temperature difference and Ax is the sheet thickness (Fig. 3.16). AT/Ax is the thermal gradient across the sheet thickness. The dimensions of these terms are: cal/cm • s •0 C cm 0 C cal/cm2 • s or kW/m2

k Ax AT Q/A

Btu/ft • h • 0 F ft 0 F Btu/ft2 • h

Example 3.4 shows the importance of material thermal conductivity in conduction heat transfer. It is apparent that energy conduction through plastic sheet is an important effect. Example 3.4 Relative Steady-State Temperature Differential Compare the steady-state temperature difference for 0.3 cm (O.Olft) thick polystyrene and aluminum for a thermal heat flux of 0.21 cal/cm2 • s [8.8 kW/m2]. The thermal conductivity for polystyrene =5.8 x 10~ 4 cal/cm • s • 0 C. That for aluminum is 5.8 x 10" 1 cal/cm • s • 0 C. The temperature difference for polystyrene is: AT =

^

k

= 0

Cm . 2 1 ^ 2L ' • g ; ° C -0.3Cm=IO 3 X cm -s 5.8 x 10~4 4 cal

For aluminum, the temperature difference is: AT =

(^) k

= 0

. 2 1 - ^2L - ^ * , — , ° C -0.3Cm = (UO-C cm -s 5.8 x 10" 1 cal

As noted, conduction of energy from the sheet surface to its interior controls the heating rate. The rate at which energy transfer occurs is called transient one-dimensional heat conduction. The time-dependent net energy increase or decrease equals the change in heat flux within the plastic sheet [3,4]: . . 6H 6T 6 / 8T\ 6 /Q\ t , Net enthalpy change per unit time = — « pcp - = - ^k - J = - ^ J

(3.4)

The polymer temperature is now a function of time and position across the sheet thickness, T(6,x). Three boundary conditions are needed to solve this equation in most applications: The Initial Condition The initial temperature throughout the sheet, T(G = 0, x = L) is needed, where L is the half-sheet thickness for equal two-sided heat flux energy input to the sheet surface. Usually the initial sheet temperature is not dependent of thickness for equal two-sided heat flux energy input to the sheet surface. Thus: T(O9L) = T0

(3.5)

A Symmetry Condition When the sheet is heated uniformly from both sides, the centerline forms a plane of symmetry (Fig. 3.5). The energy conducted from one side just equals that conducted from the other side. Thus the heat flux at the symmetry plane is zero, Q/A = 0. The condition at the centerline is described as: ^

=0

(3.6)

The Surface Condition The condition at the sheet surface in contact with the heating environment is also required. There are three characteristic conditions: Conduction, where the sheet directly contacts the heating source. For this condition: T(9, x = L) = TL(0)

(3.7)

Depending on the nature of the heating source, TL(0) can be time-dependent or constant. Convection, where the sheet contacts a fluid environment. For this condition: § A

= " 6,x = L

k

= h [ T ( ° ) " Too(e)l

TT 0X

(3-8)

0,x = L

The term h is the convective heat transfer coefficient. T(G) is the sheet surface temperature and T00(G) is the temperature of the environment. These temperatures can be time-dependent. Radiation, where there is energy interchange between the plastic sheet at absolute temperature T*(6) and the heating source at absolute temperature TJ3(G). The general form for this boundary condition is: = fU*(e),T* (9)] = G[T*4 - T*4]

^ A

(3.9)

e,x = L

The function f [ - ] is highly nonlinear in absolute temperature. The third equality is one representation showing the typical radiation fourth-power relationship, with G including geometry and radiation characteristics of both the heating source and the sheet. Quantification of the term G is given in the radiation section below. Numerical solution of the one-dimensional transient heat conduction equation with the nonlinear radiant heat flux boundary condition is difficult. As discussed below, certain approximations are made to simplify the arithmetic. These approximations also allow more direct comparison of radiation and convetion effects. Of course, combinations of these boundary conditions are significant as well. Figure 3.16 illustrates some of the characteristics of the temperature profiles through plastics for these various modes of energy transfer.

3,6

Convection Heat Transfer Coefficient

The convection heat transfer coefficient, h, is defined in Equation 3.8 as a proportionality constant1. When the energy source is a fluid, energy is transferred between the bulk moving fluid at temperature T00 and the solid surface at temperature T across a thin near-stagnant fluid layer. The heat transfer coefficient is a measure of the resistance to heat transfer across this layer. As the bulk fluid motion increases, the resistance to heat transfer decreases and the value of h increases. Representative ranges for heat transfer coefficients are given in Table 3.2. As is apparent, air is a poor convective heat transfer medium, water is more efficient than air and condensing steam is an excellent heat transfer medium. Example 3.5 shows the linearity between fluid temperature and energy transmitted to the plastic sheet. Table 3.2 Range in Values for Convection Heat Transfer Coefficient Conventional heat transfer coefficient (Btu/ft2 • h • 0F)

(10- 3 W/cm 2 - s-°C)

Fluid Quiescent air Air moved with fans Air moved with blowers Air and water mist Fog Water spray Oil in pipes Water in pipes Steam in pipes, condensing

0.5-1 1-3 3-10 30-60 30-60 30-90 30-180 60-600 600-3,000

0.8-2 2-5 5-20 50-100 50-100 50-150 50-300 100-1,000 1,000-15,000

Example 3.5 Convection Heat Transfer to Plastic Sheet For a sheet at temperature T0 = 1000F in hot air at temperature T00 = 2000F, the heat flux, Q/A, is 200 Btu/ft2 • h. What is the heat flux when the air temperature, T00 = 3000F? The solution uses a ratio of heat flux to temperature difference: (QZA)3OQ = / T 0 0 - T o \ ^_ (300 - 100) (Q/A) 200 V T 0 0 - T 0 ; (200-100)

=

Or: (Q/A) 300 = 2 x 200 = 400 Btu/ft 2 • h.

1

The convection heat transfer coefficient is also important in mold cooling analysis, as detailed in Chapter 5.

The Biot Number An important interrelationship between conduction and convection is useful here in determining the relative importance of convection energy. The Biot number, a dimensionless group, is defined as: Bi = ^

(3.10)

where Bi is the Biot number and L is a characteristic sheet dimension, typically the half-thickness. The range on the Biot number is zero to infinity, 0 < Bi < oo. Consider the following cases: •

Small Biot number occurs when the sheet thickness is very small, the convection heat transfer coefficient is very small, the sheet thermal conductivity is very large or combinations of these are in effect. Convection controls energy transfer into the sheet. For these conditions, Bi < 0.1 or so. • Large Biot number occurs when the sheet thickness is very great, the convection heat transfer coefficient is great, the sheet thermal conductivity is small or combinations of these are in effect. Conduction into the plastic controls energy transfer into the sheet. For these conditions, Bi > 1 or so.

Example 3.6 illustrates the use of this dimensionless group. As expected, the Biot number value for very heavy gage sheet is usually very large. Similarly that for very thin sheet is very small. Usually if the energy transfer between the environment and the sheet surface controls, that is, if Bi is very small, a more efficient means of heating should be sought. Example 3.6 The Biot Number in Convection Heating Consider a 0.240 inch (=2L) sheet heated in a high-velocity forced air convection oven where h = 10Btu/ft2 • h • F. The thermal conductivity, k= 0.1 Btu/ft • h • F. What is the Biot number? Does convection or conduction heat transfer control? Then consider a 0.024 inch (=2L) sheet heated in natural convection air where h = 2 Btu/ft2 • h - F. The thermal conductivity, k = 0.2 Btu/ft • h • F. Does convection or conduction heat transfer control the heating rate of this sheet?

For the first case, from Equation 3.10, hL = i 0 . O 1 2 k 0.1 12 Since Bi is large, conduction probably controls the heating rate. For the second case: .

Bl=s

hL

2 0.012

T = OT "IT =0-01

Since Bi is small, convection controls the heating rate.

Effective Radiation Heat Transfer Coefficient As noted, the radiant heat flux boundary condition, Equation 3.9, is nonlinear. Example 3.7 shows this strong nonlinearity. In certain instances, the nonlinear radiation condition can be approximated by a pseudo-convection condition: Q/A = f(T,T J « hr(T - T00)

(3.11)

where hr is a pseudo-convection heat transfer coefficient or radiation heat transfer coefficient. Methods for obtaining values of hr and ways of combining the value of the radiation heat transfer coefficient with the convection heat transfer coefficient value are detailed below. This approximation is best for high radiant heater temperatures or where T00 > T. Thin-gage roll-fed sheet formed into products such as cookie trays, blister packs and live plant containers are examples where this approximation is useful1. Example 3.7 Radiation Heat Transfer to Plastic Sheet For a sheet at temperature T0 = 1000F being heated radiantly from a hot plate at temperature T00 = 2000F, the heat flux, Q/A, is 200Btu/ft2 • h. What is the heat flux when the plate temperature, T00 = 3000F? The absolute sheet temperature, T0= 100+460= 5600R. The absolute plate temperature is either T00 = 200 + 460= 66O0R or T00 = 300+ 460= 7600R. The solution uses a ratio of heat flux to temperature difference: (QZA)3QO _ (T*J - T4A = (7604 - 5604) _ (Q/A)200 V T r - T 0 - ; (660 4 -560 4 ) ' Or: (Q/A)300 = 2.6 x 200 = 520 Btu/ft2 • h.

1

In even more specific cases, the difference between the heater temperature, T00, and the sheet surface temperature is so large that the difference in the fourth powers of their absolute temperatures is essentially independent of time. In that case, the heat flux, Q/A, to the sheet surface is assumed to be constant. The resulting equation is: ^ f ( T , T J « ^T 00 only)

(3.12)

As seen in Equation 3.9, the temperature gradient at the sheet surface is then constant: - k ^- ^f(T00 only) (3.13) Sx The constant heat flux condition represents one of the two ideal cases described earlier. This condition is approximated when roll-forming very thin-gage sheet exposed to very high radiant heater temperatures.

Constant Heat Flux In certain instances, T00 » T and the heat flux can be considered constant, for at least a short portion of the heating cycle: Q/A ^f(TYT00) ^f(T00)

(3.14)

The determination of radiant heater output efficiency is a practical use for the concept of constant heat flux. This technique is' detailed below. For very thin sheets that are truly opaque to incident radiation, the lumpedparameter approximation is important. If the energy transfer through the sheet is secondary to the energy transfer to the sheet, the partial differential equation, Equation 3.4, is replaced with a simple ordinary differential equation based on a simple time-dependent heat balance. The lumped-parameter approximation is discussed later in this chapter. Plastic sheet can be heated to forming conditions by conduction and/or hot air convection energy transfer. In trapped sheet forming [5], the sheet contacts a heated, porous blow plate only on one side (Fig. 3.17). The energy is conducted through the sheet and convected to the ambient air on the free surface. Trapped sheet forming is used in thin-gage form-fill-seal operations, when the plastic sheet is very thin, 0.13 mm or 0.005 in or less, when the sheet requires very high forming temperature, and/or when the plastic thermally degrades. The arithmetic for predicting sheet temperature is given below. Thick PMMA and PC sheet is held vertically on rails in large forced convection hot air ovens prior to being drape-formed into aircraft canopies [6] and whirlpool spas. Slow convection heating allows very thick sheets to

Vacuum

Slotted or Porous Blowing Plate

Hot Plate Sheet Clamp Urethane or Silicone Gasket Heating

Mold Blowing Air

Forming Air Exhaust or Vacuum

Figure 3.17 Trapped sheet forming, an example of conduction heating of plastic sheet [5]

thoroughly dry, anneal and stress relieve prior to forming. This gentle treatment minimizes distortion, spring-back and impact crazing. Contact heating accounts for about 15% of the surface area of sheet formed. Convection heating accounts for about 5% with radiant heating representing the remaining 80%.

3.7

Radiation Heating

Radiation is electromagnetic energy transfer between a hot source and a cold sink that it sees. Radiation energy transfer does not depend on the distance separating the source and the sink. It is the most energy efficient way of heating planar surfaces but misuse can lead to surface scorching or burn, very uneven temperature distribution through the thickness of the sheet, and energy waste. Most roll-fed and many shuttle thermoformers now heat with radiant sources. Common heating sources are: • • • • • • • • • •

Nichrome spiral wires, Steel rod heaters, Steel or Nichrome tapes, Halogen tube heaters, Ceramic plates with embedded resistance wires, Ceramic bricks with embedded resistance wires, Quartz tube heaters, Steel plates that reradiate combustion energy from gas flames, Steel wire grids that reradiate combustion energy from gas flames, and Direct gas combustion.

Primary radiant heat transfer is correctly a net energy interchange between an energy source and energy sink(s). It is apparent that the hot source radiates energy toward the sink, but the sink also radiates energy, albeit weakly, toward the hot source. The primary radiant energy impinging on any surface is either absorbed, reflected or transmitted (Fig. 3.16). If the incident radiant energy is either reflected or absorbed on the surface, the sink is opaque. Other radiation characteristics of materials are given in Table 3.3. The thermal radiation wavelength range is normally from about 0.1 urn to 20 um. The ultraviolet or UV region is 0.1 urn to 0.38 um. The visible light region is 0.38 um to 0.7 um. Near-infrared is 0.7 um to about 3 um and far-infrared is about 3 um to 20 um. The important wavelength range for most radiant thermoforming processes is a portion of the far-infrared range from about 3 um to about 20 um. As reference, the sun at an effective surface temperature of about 55000C [10,0000F] emits more than 90% of its radiation in the wavelength range of 0.1 um (UV) to 3 um, or the near-infrared region. The efficiency of radiant energy interchange depends on several attributes of the source and sink relationship. Some of these are:

Table 3.3 Radiation Characteristics of Bodies Nature

Definition

Surface reflection, diffuse Surface reflection, specular

Incident radiation reflected evenly in hemisphere Incident radiation reflected preferentially in a given steradian segment of hemisphere Unreflected incident radiant energy absorbed on surface, no transmission All unreflected incident radiant energy transmitted through and out of material All unreflected incident radiant energy transmitted through material, partially reflected back from second surface— sometimes specular Properly, semitransparent Nonreflected incident radiant energy partially absorbed volumetrically, partially transmitted through sheet All radiant energy totally absorbed at all wavelengths, no energy reflected, no energy transmitted Also, a radiant source that emits the maximum amount of energy at all wavelengths In contrast to black body, no radiant energy absorbed at any wavelength, can be either ideally transparent or perfectly reflecting A fixed fraction of radiant energy absorbed, independent of wavelength

Opaque Transparent Transparent with internal reflection—light-piping Translucent Semitransparent Black body—ideal

White body—ideal Gray body—ideal

• • • •

The The The The

efficiencies of the sink and the source in absorbing and emitting radiation, wavelength dependencies of these efficiencies, geometry of the sink and the source and their relative proximities, and absolute temperatures of the sink and the source.

The objective of the sections that follow is to quantify the proportionality constant G in Equation 3.9. An understanding of basic elements of radiation energy interchange is needed to achieve this objective. Black Body Radiation The maximum total energy emitted by any source at all wavelengths at a given absolute temperature T* is that emitted by a black body: Eb = aT*4 (3.15) T* is the source temperature in K = 0C + 273 or 0R = 0 F + 460. a is the StefanBoltzmann constant having the following units: kW [a] = 0.5674 x \Q-">—— m •K -0.1714 x 1 0 -

^

^

Monochromatic Black Body Energy Intensity, E^ Visible

Solar, 65000C

Infrared Wavelength, JJm

Figure 3.18 Temperature- and wavelength-dependent monochromatic black body energy intensity

E b is the total energy emitted for all wavelengths, in kW/m 2 or Btu/ft2 • h. All thermoforming radiant sources are referenced to the amount of energy emitted by a black body source. The wavelength-dependent radiant energy emitted by a black body at temperature T* is given as:

where X is the monochromatic wavelength in jim. The values for C 1 and C 2 are given as: [C1]

=

3.743 x 10« k

W

J

m 4

, 1.187 x 10»

B

4

^

[C2] = 1.439 x 104 K • jam = 2.59 x 104 0 R • jam The wavelength-dependent energy emitted by a black body source at temperature T* is given in Fig. 3.18. The wavelength at which the maximum energy is emitted is given as: U

= a/T*

(3.17)

The specific energy emitted at this wavelength is: Eb,,,max = C 3 -T* 5

(3.18)

Appropriate values for a and C 3 are: [a] = 2897.6 \im • K = 5215.6 jim • 0 R IcW [C 3 ]= 1.287 x 10- 1 4

"Rtii

, *Y = 2.161 x 1 0 ~ 1 3 r 7 , o , L 3J 2 5 m -K -îm ft2-h-°R5-|im Example 3.8 illustrates the energy output from a black body. Nearly 20% of the total emitted energy occurs within 0.5 jim of the peak wavelength value (Equation 3.17). Typical radiant heater temperatures ranges from 400 0 F to 1500 0 F or 200 0 C to 815°C. The total energy emitted by a black body source, the wavelength at m a x i m u m energy

emission and the energy emitted at that wavelength are given in Table 3.4 for several temperatures in this range. Example 3.8 Energy Output from a Black Body A black body is at 8000C. Determine the total amount of energy emitted and the amount emitted at the wavelength of maximum energy emission. The total amount of energy is obtained from Equation 3.15:

Eb,0_ „ = 0.5674 x 1 0 - . (800 + 273)< = ^

= * f £ *

The wavelength of maximum energy emission is given as: ^max = 2897.6/(800 + 273) = 2.7 um EÛÔ-^ +W

=^

=

^

Approximately (5.82/75.2) = 7.7% of the total radiant energy emitted over the entire wavelength spectrum is emitted at exactly 2.7 um. Most plastics absorb radiant energy preferentially in specific wavelength ranges as discussed in Chapter 2 and as seen in Table 3.5. To maximize the energy absorbed by the plastic, thermoformer heater temperatures should be set to those correspondTable 3.4 Wavelength of Maximum Energy Transmission Black Body Radiation Temperature

Wavelength

Specific energy at peak wavelength1

(0F)

(0C)

(um)

(kW/m 2 • u m )

(Btu/ft2 • h • Jim)

400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

204 260 316 371 427 482 538 593 649 704 760 816 871 927 982 1038 1093

6.06 5.43 4.92 4.50 4.14 3.84 3.57 3.34 3.14 2.96 2.80 2.66 2.53 2.41 2.31 2.21 2.11

0.32 0.55 0.91 1.43 2.16 3.16 4.52 6.28 8.57 11.5 15.1 19.7 25.2 32.0 40.1 49.8 61.3

102 176 289 454 686 1005 1435 2000 2720 3650 4810 6250 8020 10,200 12,700 15,800 19,500

1

E b X m a x = C 3 T* 5 where C 3 = 1.287 x 10~ 1 4 kW/m 2 • k 5 • um = 2.161 x IO" 1 3 Btu/ft 2 • R 5 • um

Table 3.5 Ideal Radiant Heater Temperature Ranges for Several Thermoformable Plastics Plastic

LDPE HDPE PS PVC PMMA PA-6 PET Cellulose acetate

Ideal wavelength (um)

Temperature range (0C)

(0F)

3.2-3.9 3.2-3.7 3.2-3.7 (6.4-7.4) 3.2-3.6 (5.7-6.0) 3.2-3.6 3.0-3.2 3.3-3.6 (7.0-9.2 5.5-6.0 7.8-10.0

470-630 510-630 510-630 120-180 530-630 210-235 530-630 630-690 605-630 42-140 210-255 15-100

877-1170 950-1170 950-1170 245-355 990-1170 410-455 990-1170 1170-1280 1120-1170 107-285 410-490 60-210

Total Emissive Power Fraction

ing to these wavelength ranges. The practical upper limit for thermoformer heater temperature is about 16500F or 9000C. Above this temperature, special materials of construction are needed for the heaters, special reflectors are required, and the high energy level to the polymer sheet makes its temperature control very difficult. The fraction of energy emitted by a black body source at a given temperature over a given wavelength range is obtained by subtracting values from Fig. 3.19 or Table 3.6 [7]. Examples 3.9 and 3.10 illustrate the usefulness of this approach.

^ M max

Wavelength-Temperature Term, AT, um-°R Figure 3.19 Fraction of total emissive power, E0 _ JE0 _ ^, at or below wavelength

Table 3.6 Radiation Functions [7] Eb,0 - XT*

ax-* 1 0 5

CTT* 4

O 1000 1200 1400 1600 1800 2000

0 0.000394 0.001184 0.01194 0.0618 0.2070 0.5151

0 0 0 0 0.0001 0.0003 0.0009

2200 2400 2600 2800 3000

1.0384 1.791 2.753 3.872 5.081

0.0025 0.0053 0.0098 0.0164 0.0254

3200 3400 3600 3800 4000

6.312 7.506 8.613 9.601 10.450

0.0368 0.0506 0.0667 0.0850 0.1051

4200 4400 4600 4800 5000

11.151 11.704 12.114 12.392 12.556

0.1267 0.1496 0.1734 0.1979 0.2229

5200 5400 5600 5800 6000

12.607 12.571 12.458 12.282 12.053

0.2481 0.2733 0.2983 0.3230 0.3474

6200 6400 6600 6800 7000

11.783 11.480 11.152 10.808 10.451

0.3712 0.3945 0.4171 0.4391 0.4604

7200 7400 7600 7800 8000

10.089 9.723 9.357 8.997 8.642

0.4809 0.5007 0.5199 0.5381 0.5558

1 0 5

CFT* 4

8200 8400 8600 8800 9000

8.293 7.954 7.624 7.304 6.995

0.5727 0.5890 0.6045 0.6195 0.6337

9200 9400 9600 9800 10000

6.697 6.411 6.136 5.872 5.619

0.6474 0.6606 0.6731 0.6851 0.6966

10200 10400 10600 10800 11000

5.378 5.146 4.925 4.714 4.512

0.7076 0.7181 0.7282 0.7378 0.7474

11200 11400 11600 11800 12000

4.320 4.137 3.962 3.795 3.637

0.7559 0.7643 0.7724 0.7802 0.7876

12200 12400 12600 12800 13000

3.485 3.341 3.203 3.071 2.947

0.7947 0.8015 0.8081 0.8144 0.8204

13200 13400 13600 13800 14000

2.827 2.714 2.605 2.502 2.416

0.8262 0.8317 0.8370 0.8421 0.8470

14200 14400 14600 14800 15000

2.309 2.219 2.134 2.052 1.972

0.8517 0.8563 0.8606 0.8648 0.8688

Example 3.9 Energy Emitted in a Narrow Wavelength Range Consider an 8000C= 1472°F black body emitter. Determine the fraction of total energy emitted in the wavelength range of 2 jum to 4 jum. Repeat this for a 7000C= 1292°F source.

The term ^T* = 2 • (1472 + 460) = 3864. From Table 3.6, the fraction of energy emitted between 0 and 2 jim is given as: E b , 0 _ 2 = 9.14% For AT* at 4 jim, the fraction of energy emitted between 0 and 4 jim is: E b , 0 _ 4 = 53.14% Thus the amount of energy emitted between 2 jim and 4 jim is (53.1 — 9.1) = 44% of the total amount. For a 7000C source, the energy fractions at 2 \xm and 4 urn are, respectively: Eb,0_2 = 5.9% E 1 ^ 4 = 45.3% And the amount of energy emitted between 2 jim and 4 jim is (45.3 — 5.9) = 39.4% of the total. Example 3.10 Energy Emitted in The Thermoforming Wavelength Range Consider an 8000F ceramic heater. Assume it emits as a black body. Determine the fraction of total energy emitted in the normal thermoforming wavelength range of 3.5 jum to 9 jum. The term ^T* = 3.5 • (800 + 460) = 4410. From Table 3.6, the fraction of energy emitted between 0 and 3.5 urn is given as: Eb^3.5=15.1% For XT* at 9 urn, the fraction of energy emitted between 0 and 9 um is: E 1 ^ 9 = 76.2% Thus the amount of energy emitted in the 3.5 jim to 9 jim thermoforming wavelength range is (76.2 — 15.1) = 61.1% of the total amount.

Gray Body—Emissivity No practical material emits at black body energy levels. Many materials, including nearly all polymers, emit at 80%) to 95%o of the maximum level, however. A gray body is one that emits energy at a fixed fraction of the total black body energy level: Eg = e-aT* 4 (3.19) where e is emissivity, 0 < e < 1. If the fraction of energy emitted by the material is wavelength-dependent, e = e(k), Emissivities are usually wavelength-dependent for real surfaces (Fig. 3.20). The total energy is obtained from integration: (3.20)

log Monochromatic Black Body Energy Intensity, E^

Figure 3.20 Comparison of wavelength-dependent monochromatic energy intensity for black, gray and real bodies

Black Body, e = 1 Gray Body, e < 1 Real Body, e = f(A)

Wavelength

As an approximation, emissivity can be considered constant over specific ranges in wavelength. The individual energies in each of these range segments are then summed to obtain the total energy: T

Eg.,,,*, = «

*4 n

V^N

7

N-I -

W/

I

e A i

+

. - * i )

L (P tc p/ 2 ) J

,3.26,

For very short times, 6 < 10 s, the plate temperature increases linearly with time, with the slope of the temperature curve containing information about the efficiency of the radiant heating source. Example 3.13 illustrates the use of this equation in determining the energy output of radiant heaters. Heaters are rated in 'watt density' or Watts per unit area of heater surface. The units of watt density are W/cm2 or W/in2. This is a direct measure of the consumption of electric power. The efficiency of conversion of electricity to heat is illustrated in Example 3.13 as well. The application of the concept of constant heat flux illustrates a practical way of measuring and monitoring radiant heater performance. Example 3.13 Radiant Heater Efficiency Consider a 6-in x 6-in aluminum plate, 0.125 in thick. It is heated on both sides from 100 to 125°F in 9.43 s by exposure to ceramic heaters having a measured temperature of 6000F. Determine the radiant heater efficiency. Determine the efficiency of energy conversion if the heaters are rated as 8 W/in2 at 6000F. The plate mass is given as: m = p • L2t = 0.439 Ib =199 g. The increase in energy in the plate is: q =m . c p d T = 1 9 9 g ^ . < i M

The rate of increase is: q/9 = 619/9.43 s = 65.7 cal/s

-C-,,, cd

Next Page

The heat flux to the plate is: f

= 65.7^- - - > 0 . 1 4 2 - ^ L = 0 . 5 9 5 ^ = 1 8 8 5 - ^ = 3.83 ^ A s 2 • 15.22 cm2 cm2 • s cm2 ft2 • h m2 The ideal heat flux from this source temperature is obtained from: § = o T * « = 0.164 - ^ 2i - = 0.685 ^ 5 2 = 2172 ^ 2 = 4 . 4 2 ^2 A cm • s cm ft • h m The radiant heater efficiency is given as: Efficiency = 100 • — = 87% The energy conversion efficiency is given by the ratio of the energy actually emitted by the heater to the rated heater efficiency. For the heater rating of 8 W/in2, the energy conversion efficiency is: Energy Conversion Efficiency = 100 • - ^ - = 47.9% 8

3.8

Real Heaters—Efficiencies

As noted earlier, only a fraction of the energy supplied by utility companies to the thermoforming machine is converted to radiant energy to heat the sheet (Fig. 3.22) [9]. Efficiencies of actual radiant heating sources are given in Table 3.8. The efficiencies of various types of heating sources for various polymers are given in Table 3.9. These values represent net efficiencies. The energy conversion from power source to radiant thermal energy at the heater surface is relatively efficient (Example 3.13). Quartz heaters are more efficient at higher temperatures (Fig. 3.23). About 50% of the electrical power input is converted to radiant energy at 316°C or 6000F. Essentially all is converted at 9000C or 16500F. As seen in Table 3.8, tubular and spiral wire heaters have similar efficiencies at about 50% when new. Gas combustion efficiency at 9000C or 16500F for one type of surface infrared burner is reported to be 82% to 84% [10], with an average heat flux at this temperature of 236.5 kW/m2 or 75,000 Btu/ft2 • h. The ideal black body energy emitted at this temperature is 107.4 kW/m2 or 33,970 Btu/ft2 • h. Other types of surface burners show efficiencies somewhat lower than this. Note in Table 3.8 that the effective surface heat fluxes for most gas-fired burners operating at very high temperatures are substantially greater than the values predicted by black body radiation. Convective energy transfer is apparently a major factor with these burners. Since all radiant heaters operate in an air environment, convection losses from heater surfaces reduce heater efficiency, sometimes by as much as 30% to 50%.

Previous Page

The heat flux to the plate is: f

= 65.7^- - - > 0 . 1 4 2 - ^ L = 0 . 5 9 5 ^ = 1 8 8 5 - ^ = 3.83 ^ A s 2 • 15.22 cm2 cm2 • s cm2 ft2 • h m2 The ideal heat flux from this source temperature is obtained from: § = o T * « = 0.164 - ^ 2i - = 0.685 ^ 5 2 = 2172 ^ 2 = 4 . 4 2 ^2 A cm • s cm ft • h m The radiant heater efficiency is given as: Efficiency = 100 • — = 87% The energy conversion efficiency is given by the ratio of the energy actually emitted by the heater to the rated heater efficiency. For the heater rating of 8 W/in2, the energy conversion efficiency is: Energy Conversion Efficiency = 100 • - ^ - = 47.9% 8

3.8

Real Heaters—Efficiencies

As noted earlier, only a fraction of the energy supplied by utility companies to the thermoforming machine is converted to radiant energy to heat the sheet (Fig. 3.22) [9]. Efficiencies of actual radiant heating sources are given in Table 3.8. The efficiencies of various types of heating sources for various polymers are given in Table 3.9. These values represent net efficiencies. The energy conversion from power source to radiant thermal energy at the heater surface is relatively efficient (Example 3.13). Quartz heaters are more efficient at higher temperatures (Fig. 3.23). About 50% of the electrical power input is converted to radiant energy at 316°C or 6000F. Essentially all is converted at 9000C or 16500F. As seen in Table 3.8, tubular and spiral wire heaters have similar efficiencies at about 50% when new. Gas combustion efficiency at 9000C or 16500F for one type of surface infrared burner is reported to be 82% to 84% [10], with an average heat flux at this temperature of 236.5 kW/m2 or 75,000 Btu/ft2 • h. The ideal black body energy emitted at this temperature is 107.4 kW/m2 or 33,970 Btu/ft2 • h. Other types of surface burners show efficiencies somewhat lower than this. Note in Table 3.8 that the effective surface heat fluxes for most gas-fired burners operating at very high temperatures are substantially greater than the values predicted by black body radiation. Convective energy transfer is apparently a major factor with these burners. Since all radiant heaters operate in an air environment, convection losses from heater surfaces reduce heater efficiency, sometimes by as much as 30% to 50%.

Energy Supplied to Heaters Energy Loss During Conversion to Radiant Heat Energy Convected From Heaters Radiation Loss to Surroundings Reradiation From Surroundings Reradiation From Heaters Radiation From Sheet to Heaters Convection Heat Loss to Surroundings

Energy Absorbed by Sheet Figure 3.22 Schematic of heat transfer energy distribution in thermoforming operation [9]. Figure used by permission of Society of Plastics Engineers, Inc.

An estimate of the convection heat loss is detailed below. Heaters radiant energy to all visible surfaces, including: • • • • • • • • •

Plastic sheet, Reflectors, Other heaters, Rails, Sheet clamping devices, Heater guards, Objects outside the oven edges, Oven sidewalls, and Shields and baffles.

As much as 20% to 30% of the energy emitted by radiant heaters is lost to the environment in this way. Further, a fraction of net radiant energy absorbed by the plastic sheet is convected to the cooler air environment from the hot sheet itself. Thus, only about 20% to 50% of the power supplied by the utility is converted into increasing the enthalpy of the sheet. The actual efficiency depends on: • • •

Matching source temperature with plastic radiation absorption range, Minimizing all thermal sinks other than the plastic sheet, and Controlling the convective energy losses from heater and sheet surfaces.

Table 3.8 Efficiencies of Commercial Radiant Heating Sources1 Radiant source

Maximum energy

Maximum temperature

Black body energy

Maximum efficienty

(0C)

(0F)

(kW/m2)

(Btu/ft 2 • h)

(%)

Heating (kW/m2) Bulb: R-40 reflector

(Btu/ft 2 • h)

Comments

Response time Cooling

16.1

5120

2200

4000

2140

678,000

74

. Q 1 7 4 . [ L 0 7 3 4 - 0.7734] = 9.56 kW/m2

For the screen-to-sheet interchange: ^ = 56.74 • 0.164 • [0.7734 - 0.4734] = 2.86 kW/m2 For the heater-to-sheet interchange: ^ = 56.74 • 0.136 • [1.0734 - 0.4734] = 9.84 kW/m2 The total energy transfer is: Y = 9.56 + 2.86 + 9.84 = 22.26 kW/m2 This compares with the unscreened energy transfer: 5

= 56

J 4 . o.7O • 0.777 • [1.0734 - 0.4734] = 31.57 kW/m2

The screen provides a 29.5% reduction in the amount of radiant energy interchange between the heater and the sheet.

Zone, Zoned or Zonal Heating With the advent of discrete heating elements, the effect of shielding or screening certain areas of the sheet has been, for the most part, replaced with local heating element energy output control. The earliest heating stations employed manually set proportional controllers on every heating element. Computer-aided controllers are now used. In certain circumstances, the energy output from every heating element is controllable. For very large ovens and very many heating elements, individual control is impractical. Regional banks of heating elements have a single controller and thermocouple. Thus, for an oven with 100 x 100 elements, top and bottom, requiring 20,000 controlling elements, the oven may have 40 zones, top and bottom. In certain circumstances, individual elements may be transferred from one zone to another electronically. In other cases, hard rewiring is necessary. Usually, zonal conditions are displayed on a CRT screen. As noted in the equipment section, most ceramic and metal plate heaters use PID-based controls and thermocouple temperature is the indicating readout variable. Quartz heaters operate on percentage of the time on and percentage is the indicating readout variable. Technically, of course, these variables are simply measures of intrinsic energy output of the heater or bank of heaters. Zone heating or zonal heating is used to change local energy input to the sheet in much the same way as pattern heating. With pattern heating, the pattern must be some distance from the sheet surface to minimize a sharp edge effect, shadowing or spotlighting where the pattern ends. In zone heating, the heaters are some distance from the sheet surface to begin with. As a result, energy change in a local heater or heater bank not only affects the sheet directly below it but also changes the energy input to the sheet elements in the vicinity. This is seen in the

Uniform Heating Pattern, Local Thickness Shown in Insert Temperature,0F

Top Surface Bottom Surface

Temperature, 0F

Time, s Top Center Top Side

Top Corner

Bottom Center Bottom Side Bottom Corner

Time, s Heating Pattern, Shown in Insert Figure 3.37 Effect of patterning on thermoforming part wall thickness and temperature for polystyrene, PS [16,17]. Initial sheet thickness = 2.1 mm. In lower figure, up to four layers of tissue paper are used as screening. In lower figure, thickness ratio, t/t0 = 0.29 to 0.32 over entire part. Figure used by permission of Krieger Co.

computer-generated energy input scheme of Fig. 3.38 [14]. Increasing a specific heater element energy output 14% results in a 6% increase in energy input to the immediate sheet element neighbors and lesser amounts elsewhere even though energy outputs from neighboring heater elements have not changed. If this is an undesirable effect or the effect sought requires greater focus, the bank of heaters making up the specific zone must be reduced in number. Heater to Sheet Distance As stated earlier, radiation does not depend on fluid or solid medium. Relative heater-to-sheet spacing does affect radiant energy interchange however. This was demonstrated in Fig. 3.29 and is apparent in Equation 3.40. Figure 3.39 shows the effect of a 50% increase in heater-to-sheet spacing relative to the optimum energy

11 O 0.1% 21 0 0.1% 31 0 O 41 0 0 51 0 0 61 0 0 71 0 0

12 0 0.2% 22 0 0,2% 32 0 0.1% 42 0 0.1% 52 0 0.1% 62 0 0 72 0 0

13 0.2% 0.4% 23 0.2% 0.5% 33 0.2% 0.4% 43 0.1% 0.3% 53 0 0.1% 63 0 0.1% 73 0 0

14 0.8% 1.6% 24 1.1% 2.2% 34 0.7% 1.5% 44 0.4% 0.7% 54 0.1% 0,3% 64 0 0.1% 74 0 0

15 3.1% 6.2% 25 6.9% 14.0% 35 3.1% 6.3% 45 0.7% 1.5% 55 0.2% 0.4% 65 0 0.1% 75 0 0

16 7.0% 14.1% 26 28.2% 56.4% 36 6.9% 14.0% 46 1.1% 2.2% 56 0.2% 0.5% 66 0.1% 0.2% 76 0 0.1%

17 3.1% 6.3% 27 7.0% 14.1% 37 3.1% 6.2% 47 0.8% 1.6% 57 0.2% 0.4% 67 0.1% 0.2% 77 0 0.1%

Figure 3.38 Spotlighting effect from two-fold and four-fold increases in heater output at [2,6]. Percentage represents local increase in heat absorption [14]

11

12

13

14

15

73%

83%

85%

85%

85%

16 QQO/

17 73%

21

22

23

24

25

26

27

85%

95%

97%

98%

97%

95%

83%

32

33

34

35

36

37

97%

99%

99. 5%

99%

97%

85%

41

42

43

44

45

46

47

85%

98%

100%

99.5%

98%

31

85%

99. 5%

54

55

56

57

QQ R°/

99%

97%

85%

51

52

85%

97%

61

62

63

64

65

83%

95%

97%

98%

97%

53

99%

85%

W, O/o

66

95%

67

83%

71

72

73

74

75

76

77

73%

83%

85%

85%

85%

83%

73%

Figure 3.39 Effect of heater-to-sheet spacing on energy received by sheet elements. Local percentage of initial energy input for Z = 1.5 as given in Figure 3.35 for Z = I [14]

input profile of Fig. 3.35 [14]. As expected, energy input to edges and corners are most affected. But the overall energy input to the sheet also substantially decreases. The energy output from each of the heaters must be changed to compensate for the change in gap distance. Again, the arithmetic in Equation 3.40 is a most useful aid in this process.

3.11 Thin-Gage Sheet—Approximate Heating Rates For thin-gage sheet, especially roll-fed film for packaging and blister-pack applications, the time-dependent heating model can be significantly simplified. The net enthalpic change in the sheet is simply equated to the rate at which energy in the sheet is interchanged with its environment. As a first approximation, the temperature gradient through the plastic film thickness is assumed to be zero. There are two general approaches to this lumped-parameter approximation—constant environmental temperature and constant heat flux to the sheet surface.

Constant Environmental Temperature Approximation Consider T00 to be the constant environmental temperature. The lumped-parameter approximation then becomes: d(VH) = V • pcp dT = !1A(T00 - T) d9

(3.44)

V is the sheet volume, V = At, A is the sheet surface area and t is its thickness. T is the sheet temperature. T00 can be the radiant heater temperature, with h being the approximate radiation heat transfer coefficient. Or T00 can be air temperature with h being the convection heat transfer coefficient. This ordinary differential equation is written as:

cffM£)" If t0 = T(G = 0), and T00 is constant: ln(IÎ)

VT00 - T 0 ;

=

Z^

( 3. 46 )

tpCp

or:

This is a first-order response of a system to a change in boundary conditions. This lumped-parameter transient heat transfer model is valid only where conduction through the sheet thickness is less significant than energy transmission from the environment to the sheet surface. There are two dimensionless groups that define the

Table 3.13 Lumped-Parameter Maximum Sheet Thickness Moving air heat transfer coefficient, Table 3.2: 0.0014cal/cm 2 -s-°C 1 Btu/ft2 • h • 0 F Plastics thermal conductivity, Table 3.12: 4.1 to 8.3 x 10- 4 cal/cm- s • 0C 0.1 to 0.2 Btu/ft2 • h • °F/ft Maximum thickness for Bi = 0.1: 0.025 to 0.5 cm 0.010 to 0.100 in 10 to 100 mils

limits of the lumped-parameter model. One is the Biot number, Bi = ht/k, which is the ratio of internal to external heat transfer. The second is the Fourier number, Fo = k9/pcpL2 = oc0/L2, where a is the thermal diffusivity, a = k/pcp, and L is the half-thickness of the sheet when heated equally from both sides1. The lumpedparameter model should be applied only when Bi < 0.1, or when the internal resistance is low. For air moving over plastic sheet, the sheet thickness should be less than about 0.010 in or 0.3 mm or so, Table 3.13 [19], but can be more than this for higher thermal conductivity and higher air velocity. Figure 3.40 [20] expands the limits of Table 3.13 by demonstrating the relative sensitivity of the sheet thickness to the assumed temperature difference from the sheet surface to its centerline. Example 3.21 explores the use of this figure in determining the appropriateness of the lumped-parameter model for convectively heating one side and both sides of a thin-gage sheet. Practical heating times for various thin-gage polymers over a wide range in sheet thickness are given in Fig. 3.26 [21]. The linear relationship is apparent. The energy source temperature and the sheet temperature at forming time is not given for these data. A radiant heater at T00 = 7600C or 14000F produces an energy spectrum with a peak wavelength of about 2.8 um. The energy source output at this temperature is 40 kW/m2. This energy input produces a near-linear heating rate. A lower source temperature, T00 = 5100C or 9500F, does not produce a forming time that is linear with sheet thickness.

1

Note throughout the discussion on sheet heating that the half-thickness of the sheet is used if the sheet is heated equally on both sides. If the sheet is heated on only one side, as is the case with trapped sheet heating, contact heating, or single-side radiant heating, and if the free surface can be considered as insulated or without appreciable energy transfer to the surroundings, then the proper value for L is the total sheet thickness. If the sheet is unevenly heated on both sides or if one side of the sheet is heated in one fashion, such as contact heating and the other side is heated in another fashion, such as forced convection heating, then the proper value for L is the total sheet thickness. More importantly, models describing non-symmetric heating or one-side heating with an insulated free surface cannot be applied. The proper model requires appropriate boundary conditions on each surface of the sheet.

Fourier Number, Fo

1% Difference

5% Difference

10% Difference

Biot Number, Bi Figure 3.40 Sensitivity of sheet thickness to temperature difference between sheet surface and centerline [20]. Dimensionless time, Fourier number = a6/L2 and relative surface resistance, Biot number = hL/k

Example 3.21 The Limits on the Lumped-Parameter Model A 0.020-in (0.5 cm) PET sheet is radiantly heated equally on both sides, from room temperature, 800F to its forming temperature, 3800F. The combined convection and radiation heat transfer coefficient is 10Btu/ft2 • h •0^F. The thermal dijfusivity of the sheet is 0.002ft2/h and its thermal conductivity is 0.08Btujft • h 0F. Determine the heating time for a 1% difference in temperature between the sheet surface and center. Repeat for a 10% difference. What is the heating time for onesided heating and a 1% or 10% temperature difference? Comment on the relative times. For Fig. 3.40, values for Bi and Fo are required.

Fo - .6,L= - 0.002 £ . J L „ ) . J ± L ft-= = 0.8« « From Fig. 3.40, Fo = 5.2 at 1% AT. Therefore 6 = 5.2/0.8 = 6.5s. From Fig. 3.40, Fo = 2.1 at 10% AT. Therefore 0 = 2.1/0.8 = 2.6s. In other words, to keep the centerline essentally at the surface temperature, the heating rate must be adjusted to achieve the forming temperature in about 6.5 seconds. At the forming temperature of 3800F, the centerline temperature will be 0.99 • (380 - 80) + 80 = 377°F, or 3°F below the surface temperature. If the heating rate is faster than this, the centerline temperature

will lag the surface temperature by more than 1%. If the heating rate is such that the sheet reaches the forming temperature in about 2.6 seconds, the centerline temperature will lag the surface temperature by about 10%. At the forming temperature of 3800F, the centerline temperature will be 0.90 • (280 - 80) + 80 - 3500F, or 300F below the surface temperature. For one-sided heating, L = 0.020 in.

ft2 6 Fo = aB/U = 0.002 _ • _

(s )

144 • _ ft-

= 0.29 (s)

From Fig. 3.40, Fo = 4.05 at 1% AT. Therefore 0 = 4.05/0.2 = 20.25s. From Fig. 3.40, Fo = 1.8 at 10% AT. Therefore 0 = 1.8/0.2 = 9.0s. It takes 20.25/6.5 = 3.1 times longer to heat the one-sided sheet to 1% temperature difference and 3.5 times longer to heat it to 10% temperature difference.

Constant Heat Flux Approximation If the heat flux to a thin sheet is constant, Q/A = constant, then: O dT ^ = constant = tpcp —

(3.48)

dT = ^ — d 9 Atpcp

(3.49)

T T

(3 50)

Rearranging:

Integrating this yields:

- «=xi^

-

For a given set of processing conditions, the constant heat flux approximation indicates that the time to heat a very thin sheet of plastic to a given forming temperature is proportional to the sheet thickness. The data of Fig. 3.26 indicate this linearity, even though no values for forming temperature or heat flux are given. Thin-Gage Approximations—Comments The heating efficiencies for several polymers can be determined by using the normal forming temperatures from Tables 3.1 or 2.5. For a given polymer, the enthalpic change between room temperature and the normal forming temperature is determined from Fig. 2.17. The individual heating rate is determined from the slope of the curve of Fig. 3.26, for example. The net energy increase is then calculated. As seen in

Next Page

Table 3.1, most thin-gage polymers absorb 40% to 60% of the energy supplied by the heating source. The relatively low efficiency of LDPE is unexplained. PP heating efficiency is also reported to be low [22]. This indicates that the 7600C source temperature used in the calculation may be improper for efficient heating of olefin materials. This is discussed below. This analysis is restricted to one very specific processing area—thin-gage polymers—and to very stringent conditions—lumped-parameter with linear approximation of the logarithmic function. But it serves to illustrate that only a fraction of the energy emitted by the source, about half in the cases examined, is actually taken up by the polymer sheet. The rest is lost to the environment or passes completely through the sheet unabsorbed.

3.12 Heavy-Gage Sheet—Internal Temperature Control For thin-gage sheet and film, energy transmission to the sheet controls the heating cycle time. Radiant heating is far more efficient than convection heating and so is preferred for thin-gage thermoforming. For heavy-gage sheet however, energy absorbed on the sheet surface must be conducted through the thermally insulating plastic to its centerline1. For very thick sheets, the overall heating cycle time is controlled by the sheet centerline temperature and so the overall heating rate must be controlled to prevent surface overheating. As with the thin-gage discussion earlier, there are two general cases to be considered—constant environmental temperature, T00 = constant, and constant heat flux to the sheet surface, Q/A = constant. Constant Environmental Temperature Usually hot air is used as a heating medium for very heavy sheet. As a result, the T00 = constant case prevails. As with all transient heating problems, the centerline temperature lags the surface temperature. This is seen by reviewing the graphical solution to the one-dimensional time-dependent heat conduction equation with a convection boundary condition (Figs. 3.41 and 3.42) [23,24]. Figure 3.41 gives the conditions at the sheet centerline. Figure 3.42 gives the equivalent conditions at the sheet surface. Similar figures for intermediate points throughout the thickness of the sheet are found in standard handbooks [25]. As with the thin-gage approximation, the dimensionless temperature dependency, Y, for heavy-gage sheet is a function of two dimensionless groups, the Biot number and the Fourier number:

1

Again, symmetric heating is assumed throughout this discussion. The general arithmetic described herein must be modified if the sheet is heated in an unsymmetric fashion or if it is heated only on one side.

Previous Page

Table 3.1, most thin-gage polymers absorb 40% to 60% of the energy supplied by the heating source. The relatively low efficiency of LDPE is unexplained. PP heating efficiency is also reported to be low [22]. This indicates that the 7600C source temperature used in the calculation may be improper for efficient heating of olefin materials. This is discussed below. This analysis is restricted to one very specific processing area—thin-gage polymers—and to very stringent conditions—lumped-parameter with linear approximation of the logarithmic function. But it serves to illustrate that only a fraction of the energy emitted by the source, about half in the cases examined, is actually taken up by the polymer sheet. The rest is lost to the environment or passes completely through the sheet unabsorbed.

3.12 Heavy-Gage Sheet—Internal Temperature Control For thin-gage sheet and film, energy transmission to the sheet controls the heating cycle time. Radiant heating is far more efficient than convection heating and so is preferred for thin-gage thermoforming. For heavy-gage sheet however, energy absorbed on the sheet surface must be conducted through the thermally insulating plastic to its centerline1. For very thick sheets, the overall heating cycle time is controlled by the sheet centerline temperature and so the overall heating rate must be controlled to prevent surface overheating. As with the thin-gage discussion earlier, there are two general cases to be considered—constant environmental temperature, T00 = constant, and constant heat flux to the sheet surface, Q/A = constant. Constant Environmental Temperature Usually hot air is used as a heating medium for very heavy sheet. As a result, the T00 = constant case prevails. As with all transient heating problems, the centerline temperature lags the surface temperature. This is seen by reviewing the graphical solution to the one-dimensional time-dependent heat conduction equation with a convection boundary condition (Figs. 3.41 and 3.42) [23,24]. Figure 3.41 gives the conditions at the sheet centerline. Figure 3.42 gives the equivalent conditions at the sheet surface. Similar figures for intermediate points throughout the thickness of the sheet are found in standard handbooks [25]. As with the thin-gage approximation, the dimensionless temperature dependency, Y, for heavy-gage sheet is a function of two dimensionless groups, the Biot number and the Fourier number:

1

Again, symmetric heating is assumed throughout this discussion. The general arithmetic described herein must be modified if the sheet is heated in an unsymmetric fashion or if it is heated only on one side.

Dimensionless Temperature

Biot Number, Bi

Fourier Number, Fo Figure 3.41 Time-dependent temperature profile at center of plastic sheet with relative surface resistance as parameter [23,24]. Biot number, Bi = hL/k, Fourier number, Fo = a0/L2. Dimensionless temperature = (T - Tinitial)/(Theater - Tinitia])

The Biot number, Bi = hL/k. It represents the convective boundary condition and is the ratio of external to internal resistance at the environment-sheet interface.


•

Biot Number, Bi

Fourier Number, Fo Figure 3.42 Time-dependent temperature profile at plastic sheet surface with relative surface resistance as parameter [23,24]. Biot number, Bi = hL/k, Fourier number, Fo = aO/L2. Dimensionless temperature = (T - Tinitial)/(Theater - Tinitial)

• •

The Fourier number, Fo = ocO/L2. It represents a dimensionless time for conduction of energy. Y is the dimensionless temperature, Y = (T - T0)Z(T00 - T 0 ). Y(G = O) = O. Y(G = oo)= 1.

Examples 3.22 and 3.23 show how these figures are used with heavy-gage sheet. As noted, forced air convection heating yields a lower energy input per unit time than does radiation heating. This lowered energy input enables the centerline temperature to reach the lower limit on the forming window before the sheet surface temperature exceeds the upper forming temperature.

Example 3.22 Transient Heating with Convective Boundary Conditions—I Consider a 0.250-in or 6.4 mm sheet placed in a hot air forced convection oven with an air temperature, T00 = 5000F= 2600C. The thermal diffusivity of the sheet is 0.002 ft2 jh and its thermal conductivity is 0.08Btu/ft - h 0F. The convection heat transfer coefficient is 2Btujft2 - h -0F. Determine the time to heat the centerline from 800F to 3200F, the lower forming temperature. What is the surface temperature at this time? Increase the oven temperature to 6000F and repeat the exercise. The Biot and Fourier numbers are needed for Figs. 3.41 and 3.42. 2 Btu

Bl = h L / k

Fo

0.250 ,

1 ft • h • 0 F

^^

= fH^-il2 'ao8-Bfr = a 2 6

0 002

= -

ft2

ft

Q (s)

144

ir-^ -(o^(V2)^= 0 - 005e(s)

For the condition at the centerline: Y

(320-80) (500-80)

From Fig. 3.41, Fo - 3.6. Therefore 0 = 3.6/0.005 = 720 seconds = 12 minutes. From Fig. 3.42, for the sheet surface at this Fourier number, Y = 0.626. Therefore: Ts = 80 + 0.626 • (500 - 80) = 343°F The surface is 343 - 320 = 23°F hotter than the centerline. For the 600 0 F oven temperature, Y = 0.46, and from Fig. 3.41, Fo = 2.78. The time to heat the centerline to this temperature, 0 = 2.78/0.005 = 556 s = 9.3 minutes. From Fig. 3.42, the dimensionless surface temperature at Fo = 2.78 is Y = 0.52. The surface temperature is now 3500F or 30 0 F above the centerline temperature.

Example 3.23 Transient Heating with Convective Boundary Conditions— II Consider the sheet of Example 3.22. What is the centerline temperature when the sheet surface reaches the upper forming temperature of 3800F? Use an oven temperature of 6000F.

For a surface temperature of 3800F: Y =

(380-80) (600-80) = °- 58

The Fourier number at this value of Y from Fig. 3.42 is Fo = 3.15. The centerline temperature from Fig. 3.41 for this value of Fo is Y = 0.53. The centerline temperature is 0.53 • (600 - 80) + 80 = 355°F or 25°F below the surface temperature.

The average sheet temperature is usually obtained by interpolation for x/L values between zero and one. Table 3.14 shows interpolated values for two cases. The second case also includes an empirical equation that accurately predicts the internal sheet temperatures. The average temperature can also be approximated by: Taverage,approx ~ (1 /3)T surface + (2/3)T centerline

(3.51)

The approximate average temperature for the first case of Table 3.14 is 221°C, or within 2%. The approximate average temperature for the second is 1610C or within 3%.

Table 3.14 Convective and Conduction Heat Transfer Through Heavy-Gage Plastic Sheet At T00 = 8000C Fo = 0.34, 0 = 67.6 s Bi =1.04

At T00 = 7600C Fo = 0.5, 0 = 89 s Bi = 0.52

x/L

Y

Temperature (0C)

Y

Temperature (°C)

Temperature1 (0Q

0 0.2 0.4 0.6 0.8 1.0

0.833 0.82 0.78 0.74 0.65 0.56

150 160 192 223 293 363

0.865 0.84 0.82 0.80 0.75 0.70

120 138 153 168 205 242

120 131 147 169 200 242

= 225°C

-*- average

= 169°C

= 166°C

T x

1

average

Calculated from T = 151.65 • e-°-51x + 90.35 where 0 < x < 3.2 cm

The Constant Heat Flux Case For sheet that is not very heavy or for sheet having a large forming window, the energy input to the sheet surface is supplied by forced convection hot air ovens and radiant heater ovens where indirect radiation is used. For these ovens, the concept of constant heat flux may be applied. In Equation 3.27, the heat flux to the sheet surface was given as: ^f(T 5 T 005 G 5 F)

(3.52)

where G is a geometric factor that might include radiation view factor and F is a gray-body factor. The heat flux to the sheet surface can be considered constant if: • • •

The time of exposure to the heat flux is short, The environmental temperature, T00 » T, or The environmental temperature changes with heating rate such that Q/A is constant.

Figure 3.43 [26] shows the graphical representation for the effect of constant heat flux input to the sheet surface on surface and internal temperatures in a non-thin sheet. The arithmetic again depends on the dimensionless group, Fourier number, Fo = oc6/ L2 and on a dimensionless temperature, Q*, given as:

Q =fr T

(153)

* - °>-yb:]

The dimensionless distance, x/L, is zero at the surface and one at the centerline. Examples 3.24 and 3.25 illustrate the use of this graphic for determining the heating cycle time in a moderately heavy-gage sheet. Note that the time-temperature curves are linear beyond Fo = \. The following equation describes these curves:

Q

*= F o + (£f-(iM :

Fo

^

(3 54)

-

Example 3.24 Heating Rate of Thick Sheets in Constant Heat Flux—I A 0.250-in sheet of PET is heated at the rate of 3000 Btu/ft2 • h1. Determine its surface temperature when its centerline temperature is 3200F. The thermal diffusivity of the sheet is 0.002ft2jh and its thermal conductivity is 0.08 Btu/ft - h - 0F. For Fig. 3.43, the Fourier number is: ft2

B Fo = a9 = 0.002 ¥ • — (S) • ^ 1

144 ^

= 0.0059 (s)

Compare this value with the heat flux, !1(T00 - T2) for Example 3.22. For T00 = 5000F, T = 8O0F, and h = 2 Btu/ft2 • h • 0F, Q/A = 840 Btu/ft2 • h.

The dimensionless temperature Q* is: Q* = (320 - 80) • L0QQ8 Q1^25] = (320 - 80) • 0.00256 = 0.614 From Fig. 3.43, Fo - 0.756. 0 = 0.756/0.005 = 151 seconds or 2.5 minutes. The dimensionless sheet surface temperature at this time is given as Q* = 1.12. Thus Tsurface = 1.12/0.00256 + 80 = 518°F. Thus for this example, the temperature difference between the surface and the centerline is nearly 2000F. At this heating rate, the sheet surface may scorch or discolor before the centerline temperature reaches the minimum forming temperature value.

Example 3.25 Heating Rate of Thick Sheets in Constant Heat Flux—II Consider the data of Example 3.24. Assume that the constant heat flux is 840 Btu/ ft2 • h. Using Equation 3.54, determine the surface temperature when the centerline temperature is 3200F. Compare the results with the constant ambient air case of Example 3.22.

The Fourier number remains 0.0050 (s). The dimensionless temperature, Q* is: Q* = (320 - 80) • [ 8 4 0 ° 8 0 | 2 5 ] = (320 - 80) • 0.00914 = 2.19 This value is beyond the curves in Fig. 3.43. As a result, Equation 3.54 is used:

For x/L = 1, Fo = 2.19 - 1 + 1 - 1/3 = 1.86. Thus 0 = 1.86/0.005 = 371 seconds or 6.2 minutes. At this Fourier number, the dimensionless temperature Q* at x/L = 0 is given as Q* = 2.19 + 1/3 = 2.52. The temperature at the surface is 2.52/ 0.00914 +80 = 356°F. In Example 3.22 for constant 5000F ambient air and a convection heat transfer coefficient, h = 2 Btu/ft2 • h •0 F, the sheet heated to the 3200F centerline temperature in 12 minutes. The surface temperature was 343°F at that time.

When comparing constant heat flux and constant ambient temperature, the discussion of Section 3.12 must be kept in mind. For constant heat flux, the sheet surface temperature increases linearly with time beyond Fo = \, as seen in Fig. 3.43. There is no upper limit to the surface temperature for this concept. For constant ambient temperature, the sheet surface temperature asymptotically approaches the ambient temperature.


Fourier Number, Fo Figure 3.43 Time-dependent temperature profile through plastic sheet with constant heat flux to the sheet surface [26], Fourier number, Fo = a0/L2. Dimensionless temperature = (T — Tinitial)/ (Theater-Tinitial)

The Thickness Effect If conduction controls, the effect of sheet thickness on time-dependent temperature is quite dramatic. First note that for the boundary condition, the dimensionless group, Bi = hL/k, doubling the sheet thickness doubles the value of the Biot number. The dimensionless time is given as the Fourier number, Fo = a9/L2. Doubling the thickness increases the Fourier number by a factor of four. Reviewing Fig. 3.42, if the Biot number doubles and the Fourier number is constant, the effect is a dramatic increase in surface temperature1. If the Fourier number increases by four at constant Biot number, there is a dramatic increase in surface temperature2. If the Biot number doubles and the Fourier number increases by four, the result is an even more dramatic increase in surface temperature. Example 3.26 focuses on the relative effect of thickness on temperature. The conduction heat transfer axiom appears applicable here: AU other things equal, heating times increase in proportion to the square of the sheet thickness. 1 2

This has the effect of doubling the heat transfer coefficient. The energy input to the surface doubles. This has the same effect as increasing the cycle time by a factor of four. The same energy input results in an increased surface temperature.

Table 3.15 is an experimental corroboration of this axiom for PMMA and a PVC/PMMA blend. Example 3.26 Thickness Effect—Heavy-Gage Sheet Consider a sheet having a Biot number of 0.2 and a Fourier number of 5.0 at the end of the heating cycle. If the initial sheet temperature is 800F and the centerline temperature at the end of the heating cycle is. 3800F, determine the surface temperature. Now increase the sheet thickness by 20%, keeping the heating cycle the same and recalculate the sheet surface temperature. What is the percentage increase in cycle time needed to get the centerline back to 3800F? From Fig. 3.42, Y = 0.592. Therefore the heater temperature is:

T00 = SSl0F from this expression. From Fig. 3.41, Y = 0.64. The surface temperature is: (587-80) orT s u r f a c e = 404°F. For a 20% increase in sheet thickness, Bi = 0.24 and Fo = 3.47. From Fig. 3.42, Y = 0.50. The ambient temperature remains at 587°F. Therefore the new centerline temperature is: Tcenterlme = 0.5 • (587 - 80) + 80 = 334°F The surface temperature is given from Fig. 3.41 as Y = 0.60, or T surface = 0.6 • (587 - 80) + 80 = 384°F. From Fig. 3.42, the Fourier number when Bi = 0.24 and Y = 0.592 is Fo = 4.0. The cycle time increase is 100 • (4.0 - 3.47)/3.47 = 15%. The sheet surface temperature at this Fourier number from Fig. 3.41 is Y = 0.64 and the surface temperature, Tsurface = 0.64 • (587 - 80) + 80 = 404 0 F. There is essentially no effect on the thermal driving force into the sheet. The major effect is increased cycle time.

Summary Thus if Bi < 0.1, the sheet heating process is probably controlled by the rate at which energy is delivered to the sheet surface. Increasing the rate of energy input will result in reduced heating cycle times. On the other hand, if Bi > 1, the heating process is probably controlled by the rate at which heat is conducted to the interior of the sheet. Increasing the rate of energy input may result in surface melting, scorching, yellowing or blistering. Although the range in values for plastic thermal conductivity

Table 3.15 Thermoforming Heating Cycle Times for Heavy-Gage PVC/PMMA and PMMA Sheet1 Sheet thickness (cm)

X/Xo

0.102 0.152 0.203 0.236 0.254 0.318 0.475 0.635 0.953

1.0 1.49 1.99 2.31 2.49 3.12 4.66 6.23 9.34

1

(X/X0)2

time (s) 1.0 2.22 3.96 5.35 6.20 9.72 21.7 38.8 87.3

PMMA

PVC/PMMA

15 35 61 82 150 324 594

[e/e0]

time (s)

[0/O0]

1.0 2.33 4.07 5.47

13 29 51

1.0 2.23 3.92

80 126 276 495 1122

6.15 9.69 21.2 38.1 86.3

10.0 21.6 39.6

Adapted from [44], with copyright permission. Sheet heated on both sides. Note: Heating time controlled by conduction through plastic is proportional to square of sheet thickness. Compare columns 3, 5, and 7

is relatively narrow, usually no more than a factor of two or so, the effective heat transfer coefficient value range is more than 20. Thus the range in sheet thicknesses where neither conduction nor convection dominate the heating rate is quite broad. Further, this range spans most of the common sheet thermoforming thicknesses. As a result, care must be taken in applying any of the above approximations. Case-bycase analysis is always recommended.

3.13 Equilibration When the sheet has reached its forming temperature, it is removed from the heating environment. Immediately the surface temperature begins to decrease. The centerline temperature continues to increase, albeit at a rate slower than before. A schematic of the time-dependent temperature gradient through the sheet is shown as Fig. 3.44. The sheet temperature approaches the average value. This effect is known as equilibration. The relationship between equilibration time, sheet temperature and the forming window is shown in Fig. 3.45. The time to equilibration is sometimes called the soaking time. Equilibration time values are strongly dependent on the sheet thickness and the sheet temperature profile through the sheet at the time the sheet exits the oven. The shape of the temperature profile through the sheet depends on the method of heating. The exponential form for the temperature profile is the best representation of the shape when the sheet is convectively heated. A more linear form is best when the sheet is heated at constant flux. These cases are detailed below.

Temperature

Ts u rf ace

Increasing Time

"âverage T

centerline

Surface

Centerline

Sheet Thickness

Temperature

Figure 3.44 Schematic of time-dependent temperature profile through plastic sheet during equilibration

Forming Range

Surface

Average

Heating

Equilibration

Centerline Time

Transfer

Forming Time

Figure 3.45 Relationship between polymer forming temperature range [dashed area] and time-dependent sheet surface, average and centerline temperatures for heating and equilibration of polymer sheet

Convection Heating When convection heat transfer controls the energy input to the sheet, approximate equilibration times are obtained either from Fig. 3.46 or from: (3.55)

Dimensionless Temperature, (T-a)/c

Thermal Time Constant, bL

Equilibration Line x/L = 0

Fourier Number, Fo Figure 3.46 Adiabatic time-dependent equilibration temperature through sheet thickness [27]. Initial sheet temperature profile, T(x) = a • ebx + c, where a, b, c are curve-fit parameters. Fourier number, Fo = oc0/L2. Figure used by permission of McGraw-Hill Book Co.

This equation and the curves of Fig. 3.46 are determined by assuming that the sheet has a temperature profile T(x) = a • exp(bx) + c at the instant the heat source is removed. This empirical profile has been used previously with good success to calculate the local temperature for heavy-gage sheet in Table 3.14. Theoretically, this empirical profile does not exactly mirror typical heat conduction temperature profiles, but practically it yields useful approximate information. Example 3.27 illustrates the method of calculating equilibration times. Note in Fig. 3.46 that the dimensionless equilibrium time Foeq is essentially constant over a wide range of values of bL (0.5 \, the temperature profile is best given by Equation 3.54. The equilibration curves are shown in Fig. 3.47 [27], where the equilibration times begin at specific values of Fo. Example 3.28 compares times for constant heat flux with those for step change in surface temperature.

Fourier Number at End of Heating

Fourier Number From Beginning of Heating, Fo Figure 3.47 Time-dependent temperature profile through plastic sheet with constant heat flux to the sheet surface [27], showing equilibration. See Figure 3.43. Fourier number, Fo = a6/L2. Dimensionless temperature = (T - Tinitial)/(Theater - Tinitial)

Example 3.28 Equilibration Time—II Consider constant flux energy input to the sheet. Assume that the energy input is 1700 Btu/ft2 - h. Determine the surface temperature when the centerline temperature reaches 3000F. Then determine the time required to get the surface and centerline temperatures to within 5°F. Finally determine the time required to get full equilibration.

The dimensionless heat flux, Q*, is given as:

Q* = (300 - 80) - [ y ^ l ^ ] = 1 -035 From Equation 3.54, the Fourier number at x/L = 1 is: Fo = 1.035-1 + 1-1/3 = 0.702 The sheet surface temperature, when x/L = 0, is given as: Q

^ ( T -- 8 0 ) [l70^0r]^ L 3 6 8

Or Tsurf=370°F. From Fig. 3.47, the Fourier number when AQ* less than 5% of the average value for Q* is Fo = 0.333. Fo > 0.5 when AQ* = 0. As a result, the time to reach 5% of average and the full equilibration time are given as:

05% = Fo • ^ = j j m • (0.01)2 • 3600 = 59.9 s T2 Q CQQ O0 = Fo • - = — - • (0.01)2 • 3600 = 90 s OC U.UUZ

Computed Equilibration Times In Section 3.5, the finite difference form for the one-dimensional heat conduction equation was presented, together with appropriate boundary conditions. The simplest way of determining the temperature profile through the sheet during equilibration is to simply "switch off" the surface energy boundary condition. For the effective convective boundary condition, this is written in terms of Equations 3.8 and 3.11 as: § = IwP(T 0 0 - T )

(3.56)

where heff is the combined convection and effective radiation heat transfer coefficients, heff = h + h r and P is the switching factor. So long as the sheet is in the heating portion of the cycle, P = I . When the sheet is removed from the oven and

transferred to the forming press, P = 0. In a practical sense however, heat transfer between the sheet surface and its environment does not cease when the sheet is removed from the oven. Instead, the environmental temperature T00 simply becomes the cool air temperature surrounding the forming press. The effective heat transfer coefficient, heff, becomes just the convection heat transfer coefficient between the environmental air and the hotter sheet1. Computer models of this arithmetic are usually designed to accommodate sheet surface cooling during equilibration. For heavy-gage sheet, the calculated times needed to achieve temperature differences across the sheet of ten degrees (100C) or less are usually longer than the practical time to transfer the sheet to the forming station and begin the forming process. This means that at the time of forming, the sheet temperature and hence the sheet strength varies across the sheet cross-section. As seen in Chapter 4, particularly Fig. 4.5, the sheet strength is quite nonlinear with temperature. As detailed in Chapter 4, temperature dependency is modeled with either the W-L-F equation or an Arrhenius equation. The W-L-F Equation The W-L-F equation is given as: logl0aT=-^r(T~TT8)T

(3.57)

where aT, the shift factor, is the effect of temperature on polymer response to applied load, C lg and C2g are experimentally determined constants, and Tg is the glass transition temperature of the polymer. Table 3.16 gives W-L-F coefficients for several thermoformable polymers. As a first approximation, C lg = 17.44 and C2g = 51.6 are adequate "universal constants" [28]. The W-L-F equation is suitable in the temperature range of Tg < T < Tg + 1000C. Table 3.16 Williams-Landel-Ferry or WLF Constants for Several Polymers—Universal Constants Also Given [45]

1

Polymer

C1

C2

T g (K)

Polyisobutylene (PIB) Natural rubber Polyurethane elastomer Polystyrene (PS) Polyethyl methacrylate (PEMA) Polycarbonate (PC) Universal constant

16.6 16.7 15.6 14.5 17.6 16.14 17.44

104 53.6 32.6 50.5 65.6 56 51.6

202 202 238 373 335 423 -

Technically, the sheet surface also interchanges radiant energy with its surroundings. The linear form for the radiant heat transfer coefficient, Equation 3.29, works well here. Example 3.29 illustrates the relative values for the convective and radiative heat transfer coefficients during the equilibration step.

The Arrhenius Equation For polymers being processed at temperatures in excess of Tg + 1000C, the Arrhenius equation is used: 1, the sheet is considered as heavy-gage. Thin-Gage Guidelines When Bi < 0.1, the sheet is considered thin from a heat transfer viewpoint. The sheet surface temperature is not significantly greater than either the centerline or average sheet temperature. Surface blistering, degradation and burning are not considered primary concerns. Energy input to the sheet surface controls the cycle time. Therefore, radiant heating provides the most efficient means of heating1. To minimize cycle time, the maximum net energy exchange should occur in a wavelength range where the polymer has the greatest absorptivity. This usually implies high heater temperatures and short cycle times. Technically, the most efficient mode of heating is constant heat flux. The effective heat transfer coefficient is typically in the range of 0.06 to 0.12 kW/m2-°C or 10 to 20 Btu/ft2 • h •0 F. Energy transfer is dominated by radiation that is as much as 10 times the natural convection contribution. As a first approximation, convection effects can be ignored. For thin sheet such as those used in blister and bubble packaging, the heating cycle time is proportional to the film thickness. The lumped-parameter model discussed in Section 3.11 is used to predict average sheet temperature during thin-gage heating. It ignores temperature gradients through the sheet and so yields only average temperature values. Cycle time is determined when the average temperature reaches the normal forming temperature. From net energy balances, about 50% of the electric power supplied to the radiant heaters is actually used to heat the sheet to its forming temperature.

Heavy-Gage Guidelines The rate of heating of heavy-gage sheet is governed by the maximum allowable surface temperature. For heavy-gage sheet, Bi > 1. Energy transmitted to the surface from the environment must be conducted to the sheet interior. If the rate of conduction is low compared with the input energy rate, the sheet surface temperature may reach an undesirable level. Thus, highly efficient heating methods may in fact be undesirable. Constant surface temperature or declining heat flux methods are preferred. Forced hot air convection ovens are used to heat very heavy-gage sheet. 1

However, for very thin sheet, radiant transmission through the sheet may be a controlling feature. As a result, alternate heating means, such as direct conduction, contact or trapped sheet heating, are recommended.

Relatively low surface temperatures imply relatively low input energy levels, manifested as relatively low values of the effective convection heat transfer coefficient. Typical values are on the order of 0.005 to 0.03 kW/m2 •0C or 1 to 5 Btu/ft2 • h •0 F. Cycle times are controlled by conduction into the sheet and therefore are proportional to the square of the sheet thickness. Traditional distributed parameter one-dimensional transient heat conduction equations are used to predict the heating rates. Overall energy efficiency is relatively low at about 20% to 30%. Heat losses to surroundings and low conversion efficiencies when heating air are considered the primary reasons. And equilibration times are also proportional to the square of the sheet thickness. Intermediate-Gage Guidelines If the Biot number value is between 0.1 and 1, the sheet has intermediate gage with regard to heat transfer. Typically, the heating cycle time, 9, is proportional to the sheet thickness, L, to a power, 0ocLn

(3.74)

where l < n < 2 1 . Radiant heating dominates the heating methods, with heater temperatures set substantially below the region where optimum energy absorption occurs. Heater temperature values are usually dictated by the upper forming temperature, sheet blistering, discoloration, degradation and/or burning. The energy conversion efficiency is usually between 30% and 50%. Distributed parameter one-dimensional transient heat conduction equations are used to predict heating cycle times.

3.17 References 1. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 24, p. 47. 2. F. Kreith, Principles of Heat Transfer, 2nd Ed., International Textbook Co., Scranton PA (1965), Figure 5-1, p. 199. 3. F. Kreith, Principles of Heat Transfer, 2nd Ed., International Textbook Co., Scranton PA (1965), p. 142. 4. F. Kreith, Principles of Heat Transfer, 2nd Ed., International Textbook Co., Scranton PA (1965), pp. 128-132. 5. J. Frados, Ed., Plastics Engineering Handbook, 4th Ed., Van Nostrand Reinhold. New York (1976), pp. 278-281. 6. W. McConnell, "Industrial Thermoforming Symposium and Workshop", SPE Thermoforming Division, Arlington TX, 12-14 March 1985, Distributed Handout. 7. F. Kreith, Principles of Heat Transfer, 2nd Ed., International Textbook Co., Scranton PA (1958), Appendix A-4. 1

Again, for thin-gage heating, n = 1. For heavy-gage heating, n = 2.

8. F. Kreith, Principles of Heat Transfer, 2nd Ed., International Textbook Co., Scranton PA (1965), p. 215. 9. F. Brinken and H. Potente, "Some Considerations of Thermodynamics in Thermoforming", SPE ANTEC Tech. Papers, 24 (1978), p. 65. 10. J.L. Throne, Plastics Process Engineering, Marcel Dekker, New York (1979), pp. 714-715. 11. R.R. Kraybill, "Emission Efficiency of Reflector Materials for an Infrared Tubular Heater", SPE ANTEC Tech. Papers, 29 (1983), p. 466. 12. D. Weinand, "Modellbildung zum Aufheizen und Verstrecken beim Thermoformen", Doctoral Dissertation, Institut fur Kunststoffverarbeitung (IKV), Aachen, 16 JuI 1987, BiId 4.9. 13. J.P. Holman, Heat Transfer, 4th Ed., McGraw-Hill Book Co., New York, 1976, Figure 8.8, pp. 284-288. 14. J.L. Throne, "Radiant Heat Transfer in Thermoforming", SPE ANTEC Tech. Papers, 41 (1995), p. 000. 15. D. Weinand, "Modellbildung zum Aufheizen und Verstrecken beim Thermoformen", Doctoral Dissertation, Institut fur Kunststoffverarbeitung (IKV), Aachen, 16 M 1987, BiId 4.10. 16. N. Platzer, "Rigid Thermoplastic Sheeting", Mod. Plastics, 37:11 (Nov 1954), pp3, 144. 17. N. Platzer, "Sheet Forming", in E.C. Bernhardt, Ed., Processing of Thermoplastic Materials, Reinhold, New York (1959), p. 485. 18. J.L. Throne, Plastics Process Engineering, Marcel Dekker, New York (1979), p. 657. 19. F. Kreith, Principles of Heat Transfer, 2nd Ed., International Textbook Co., Scranton PA (1965), pp. 129-135. 20. PJ. Schneider, "Conduction", in W.M. Rohsenow and J.P. Hartnett, Eds., Handbook of Heat Transfer, McGraw-Hill Book Co., New York, 1973, Figure 21, pp. 3-39. 21. A. Hoger, Warmformen von Kunststoffen, Carl Hanser Verlag, Munich (1971), pp. 36-37. 22. H. Gross and G. Menges, "Influence of Thermoforming Parameters on the Properties of Thermoformed PP", SPE ANTEC Tech. Papers, 28 (1982), p. 840. 23. PJ. Schneider, "Conduction", in W.M. Rohsenow and J.P. Hartnett, Eds., Handbook of Heat Transfer, McGraw-Hill Book Co., New York, 1973, Figure 22e, pp. 3-39. 24. PJ. Schneider, "Conduction", in W.M. Rohsenow and J.P. Hartnett, Eds., Handbook of Heat Transfer, McGraw-Hill Book Co., New York, 1973, Figure 22a, pp. 3-39. 25. V.S. Arpaci, Conduction Heat Transfer, Addison-Wesley Publishing Co, Reading MA, 1966. See also references [2,13,20]. 26. PJ. Schneider, "Conduction", in W.M. Rohsenow and J.P. Hartnett, Eds., Handbook of Heat Transfer, McGraw-Hill Book Co., New York, 1973, Figure 40a, pp. 3-67. 27. PJ. Schneider, "Conduction", in W.M. Rohsenow and J.P. Hartnett, Eds., Handbook of Heat Transfer, McGraw-Hill Book Co., New York, 1973, Figure 40b, pp. 3-67. 28. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes and Tests for Design, Hanser Publishers, Munich (1993), Table 3.4, pp. 250-257. 29. R.C. Progelhof, J. Quintiere, J.L. Throne, "Temperature Distribution in Semitransparent Plastic Sheets Exposed to Symmetric, Unsymmetric, and Pulsed Radiant Heating and Surface Cooling", J. Appl. Polym. ScL, /7(1973), p. 1227. 30. R.C. Progelhof, J. Franey, T.W. Haas, "Absorption Coefficient of Unpigmented Poly(methyl Methacrylate), Polystyrene, Polycarbonate, and Poly(4-methylpentene-l) Sheets", J. Appl. Polym. ScL, 75(1971), p. 1803. 31. D. Weinand, "Modellbildung zum Aufheizen und Verstrecken beim Thermoformen", Doctoral Dissertation, Institut fur Kunststoffverarbeitung (IKV), Aachen, 16 JuI 1987, BiId 3.13. 32. R.C. Progelhof, J. Franey, T.W. Haas, "Absorption Coefficient of Unpigmented Poly(methyl Methacrylate), Polystyrene, Polycarbonate, and Poly(4-methylpentene-l) Sheets", J. Appl. Polym. ScL, 25(1971), p. 1806. 33. A. Hoger, Warmformen von Kunststoffen, Carl Hanser Verlag, Munich (1971), Table 2, p. 31. 34. D. Weinand, "Modellbildung zum Aufheizen und Verstrecken beim Thermoformen", Doctoral Dissertation, Institut fur Kunststoffverarbeitung (IKV), Aachen, 16 JuI 1987, BiId 3.14. 35. J. Florian, Practical Thermoforming: Principles and Applications, Marcel Dekker, New York (1987), p. 39.

36. J. Florian, Practical Thermoforming: Principles and Applications, Marcel Dekker, New York (1987), Chapter 2, "Components of the Thermoforming Process". 37. J.L. Throne, TFl, Integrated Design Engineering Systems, Inc., Laramie WY 82070, 1991. 38. D.R. Croft and D.G. Lilley, Heat Transfer Calculations Using Finite Difference Equations, Applied Science Publishers, London (1977). 39. D.R. Croft and D.G. Lilley, Heat Transfer Calculations Using Finite Difference Equations, Applied Science Publishers, London (1977), pp. 92-94. 40. F. Kreith, Principles of Heat Transfer, 2nd Ed., International Textbook Co., Scranton PA (1965), Table 5-1, pp. 215-216. 41. A.I. Brown and S.M. Marco, Introduction to Heat Transfer, McGraw-Hill Book Co., New York (1951), p. 68. 42. R.W. Singleton, "Electric Infrared: Textile Applications in the 1980's", Proceedings, AATCC National Technical Conference (1980), p. 201. 43. J.H. Perry, Ed., Chemical Engineers' Handbook, 4th Ed., McGraw-Hill Book Co., New York (1963), pp. 10-11 to 10-13. 44. H.R. Osmers, "Industrial Thermoforming Symposium and Workshop", SPE Thermoforming Division, Arlington TX, 12-14 March 1985, Distributed Handout. 45. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes and Tests for Design, Hanser Publishers, Munich (1993), Table 6.17, p. 651.

4 Stretching the Sheet 4.1 Introduction 4.2 The Stretching Concept 4.3 Polymer Hot Strength Standard Tensile Tests Hot Tensile Tests Hot Creep Tests Other Stretching Tests Temperature-Dependent Viscosity for Amorphous Polymers Dynamic Mechanical Testing 4.4 Stress-Strain-Rate of Strain—Theory Elasticity—A Rationalization Strain Energy Function The Rivlin Form for the Strain Energy Function The Ogden Form for the Strain Energy Function Viscoelastic Models 4.5 Available Stress-Strain Data Sensitivity of Models 4.6 The Importance of Polymer Material Properties 4.7 Practical Aspects of Stretching Funnel Test 4.8 Bursting Conditions 4.9 Sheet Sag Initial Sag Tensile Sag The Catenary Sag Parabolic Sag Relating Sag to Hot Sheet Strength Sag—A Comment 4.10 References Appendix 4.1 Biaxial Stretching of an Elastic Membrane

4.1

Introduction

Once the plastic sheet temperature is within the forming window, it is ready to be stretched. There are many ways of stretching and prestretching the sheet, as detailed in Chapter 1. Vacuum, air pressure, mechanical aids such as plugs, rubber diaphragms, and combinations of these are used to shape the rubbery sheet against the mold surface. The extent to which a given polymer at a given temperature can be stretched limits the ways in which it can be thermoformed. Part design, especially local part wall thickness, depends on the extent of polymer deformation. Deep drawing, drawing into sharp corners, and replication of mold surface details such as patterns or lettering, require polymers that can be rapidly and uniformly stretched. Chapter 9 examines part design in greater detail. This chapter focuses on the interaction between the forces available to stretch the rubbery sheet and the inherent nature of the rubbery plastic to resist these forces. This interaction, noted in Chapter 2, is directly related to the stress-strain-rate of strain behavior of the plastic at its forming temperature. Typically, at low temperatures, the plastic is quite stiff, does not stretch easily, and will not faithfully replicate the mold details under the modest forces available with simple vacuum forming. Higher forming forces are required when sheet temperatures are low. On the other hand, cycle times are short when sheet temperature is low, so economics favor low sheet temperatures. At high forming temperatures, the sheet is quite limp, is easily stretched and replicates the mold surfaces well at very modest forming pressures. But sagging and sheet surface discoloration can be serious problems, cycle times are increased, and part wall thickness uniformity is usually compromised. Intuitively knowing when a sheet is hot enough for processing is an acquired skill. Sheet sag and sheet smoke are frequently considered as first-line visual indicators. As a plastic sheet is heated, it undergoes several phases of motion [I]: •

The sheet may momentarily tighten in a drum-head fashion. This tightening may be accompanied by some off-gassing from the sheet surface. The gas is usually vaporizing surface or adsorbed water. Since the sheet temperature at this point is usually only a few degrees above room temperature, the tightening is usually attributed to the last steps in sheet extrusion. In the case of heavy-gage sheet, the last step involves palletizing where the just-extruded and guillotined sheet is stacked on older sheet. Heat retention in the palletized stack can cause residual stresses in the individual sheets that are not attributable to the extrusion process, per se. A similar situation occurs with thin-gage sheet that is rolled. The rolling action may cause residual stresses called curl. This curl is relaxed out when the sheet is first heated. In addition, heat retention in the roll can also cause residual stresses that are not attributable to the extrusion process. • As heating continues, the drum-tight state of the sheet is rapidly replaced by sheet rippling or "swimming". The sheet may also exhibit very rapid droop or sag and the sheet texture may change from glossy to matte at this time. During this time, the sheet temperature is passing through the glass transition temperature for

Specific Volume, cm 3 /g

PS PMMA PC

30% DOP FPVC

RPVC

Figure 4.1 Temperature-dependent specific volume for several thermoplastics

Temperature, 0 C

amorphous polymers. For certain crystalline-tendency polymers such as PET, some surface crystallization may take place1. The swimming effect is apparently due to nonuniform residual stresses imparted in the sheet during the last stages of cooling the sheet. The sag is probably due to the decrease in density as the sheet passes through the glass transition temperature (Fig. 4.1) [3]. • As heating continues, the rippling state is frequently replaced with a second tautness in the sheet. Off-gassing or "smoking" may begin at this time, as well. The sheet temperature is usually above the glass transition temperature at this time. This state is directly attributable to sheet orientation during extrusion. At this time, the sheet may pull from the clamps. Careful observation of sheet tautness at this point may yield clues as to the balance in residual stresses in the MD and TD directions. In heavy-gage sheet forming, this balance may dictate the orientation of the sheet to the mold. The liberated gases may be external lubricants, processing aids, and other low boiling adducts. Since the sheet surface is now substantially hotter than the center, moisture and dissolved gas bubbles appear. • Continued heating produces sheet sag. At this point, the tensile strength of the hot sheet is dropping rapidly and the force of gravity pulls the sheet into a catenary-like shape. As stated above, sag is universally used as an early indicator

1

This crystallization is sometimes called "cold crystallization" [2]. It occurs above the glass transition temperature but below the crystalline melting point.

Heater Sheet Clamp

Heater

Sag

Figure 4.2 Schematic of sheet sag in thermoforming oven

that the sheet is ready to be formed. As seen below, sag is a measure of the temperature-dependent tensile strength of the polymer, in combination with sheet geometry. Since sag is so important to the forming process, it is discussed in detail in Section 4.9. Care must be taken when heating sheet beyond the early stages of sag. As seen in schematic in Fig. 4.2, sag dramatically changes the relationship of the sheet to the top and bottom heaters. The general effect is to change the local view factor between the heaters and the sheet. The center of the sag moves away from the top heaters and toward the bottom heaters. The heat flux becomes unbalanced, with more energy being transferred from the bottom heater. The results are different for thin-gage and heavy-gage sheet. • For heavy-gage sheet, conduction from the sheet surface to its interior controls the general heating condition. As discussed in Chapter 3, energy inputs to the two surfaces of the sheet are usually controlled independently to ensure uniform energy distribution throughout the sheet thickness. Therefore, energy input to each sheet surface must be carefully controlled to prevent overheating or inadequate heating. When the sheet sags, it approaches the lower heater and retracts from the upper heater. Local energy input to the middle of the underside of the sheet increases dramatically and that to the top side decreases dramatically. This shifts the center of energy symmetry toward the lower surface of the middle of the

Temperature Profile

Sheet

Sheet Sag

Figure 4.3 Characteristic shift in temperature profile through heavy-gage sheet during sagging

sheet toward the lower surface while the center of symmetry remains near the center of the sheet (Fig. 4.3). • For thin-gage sheet, where energy input to the sheet surface controls the general heating condition of the sheet, the top-to-bottom energy uniformity is less important than the local change in view factor in the sagging center of the sheet. As the sagging sheet nears the lower heater, the view factor increases toward unity. Energy input is more intense and that area of the sheet heats more rapidly. The local sheet temperature may exceed the upper forming temperature of the polymer. If the polymer melt strength is particularly temperature sensitive, the sheet will flow apart and may drip into the heater. If the polymer degrades rapidly at this temperature, the sheet may blacken or ignite. Since sheet energy uptake is governed by the heater energy output, excessive sag and overheating may occur so quickly with certain polymers that degradation or fire cannot be prevented. Again, sheet distortion and sag during heating is the result of the polymer responding to external forces of temperature and stress. This chapter lays the foundation for a technical understanding of the fundamental nature of polymer material deformation. Thermoplastic sheet at its forming temperature is considered as: • • •

A rubbery elastic solid, A highly viscous liquid, or Something in between.

Owing to the commercial interest in deforming polymers near their melt or softening temperatures, an extensive body of knowledge has been created on this topic. The stretching characteristics of thermoformable sheet are important since understanding the characteristics leads to a very basic understanding of polymer behavior during elongation.

In this chapter, elongational deformation of polymer sheet will be examined in great technical detail. The objective of this examination is a better understanding of the thermoforming process and how that relates to intelligent selection of the proper polymer for a given application. While the parts designer does not necessarily need to understand the specifics of the concepts presented to design quality parts, the material should be reviewed for general concepts.

4.2

The Stretching Concept

Stretching is elongational material deformation. All real materials, such as polymers, steel, wood and even concrete, stretch to some extent when forces are applied. For modest stretching, the extent of stretching or elongation is called strain, e. For solids, strain is the polymer response to the applied force per unit cross-section, or stress, a. For a thin membrane, stretching can be in one direction, uniaxial deformation. Or it can be in two directions or biaxial deformation. If the amount of biaxial stretching is the same in both directions, stretching is equal biaxial deformation. The simplest uniaxial relationship is Hooke's law: G =E - e

(4.1)

The proportionality is the elastic modulus, often called Young's modulus [4]. The units on a and E are the same, either MPa or lbf/in2. The relationship in biaxial stretching is written as:

where v is Poisson's ratio and i is the strain direction (i=l,2). Simply put, a Hookean material responds instantaneously to the applied load. So long as the load remains constant, the material retains a constant strain or elongation. When the load is removed, the material instantaneously returns to its unstrained state. Hooke's law adequately describes the small deformation response of most traditional materials such as metals and wood and is often pictured as a simple spring. For modest deformation levels, most solid materials respond in some fashion similar to that of Equation 4.1. When solids are strained to high levels, they may simply fracture, they may exhibit deviation from the Hookean relationship, or they may yield (Fig. 4.41). Most polymers below their glass transition or melting temperatures show these general characteristics.

1

As an example, if the solid material deforms in a Hookean manner to a given level of deformation, then yields to produce a continuous deformation under constant load, the material is called an elastoplastic solid [5].

Stress

Hookean/Spring

Rigid Tough Yield Point Ductile

Break

Strain Figure 4.4 Characteristic stress-strain curves for thermoformable thermoplastics, compared with the classic purely elastic Hookean spring

Elongation, e

Figure 4.5 Tensile elongation under load [5]

Load, F

For extensive deformation, consider the example of tensile loading (Fig. 4.5) [6]. The weight of the hanging sheet is given as: W = p - hbL

(4.3)

where p is the density of the sheet, h is its thickness, b is its width, henceforth assumed to be unity, and L is its length. The tensile strength is given as: W ao = x (4.4) where A = bh. Therefore the initial stress applied to the top of the sheet at the clamp is: (4.5)

The local deformation, e, per unit length of the sheet is given as: e=

AL i r

(4.6)

The differential strain is uniform everywhere along the vertical sheet axis. Engineering strain, eeng, is given in terms of the initial sheet length, L0:

eeng = ^

= f = X-1

(4.7)

where X is the ratio of instant to initial length, L/L o . Hencky strain or true strain, etrue, is obtained by summing all differential strains over the sheet length:

£true=

T T = l n ( r ) = l n {X)

(4 8)

-

For relatively low values of strain, e « L 0 , engineering strain is a good approximation to true strain. At an engineering strain level of 20%, the error in approximating true strain is about 10%. Example 4.1 illustrates other values for engineering and true strain. The engineering strain value is always smaller than the true strain value. Example 4.1

True and Engineering Strain

Determine values for true and engineering strain for L = 1.1 L0 and L = 2 L0. True strain is given by Equation 4.8. Engineering strain is given by Equation 4.7. For the first case, L = L l L 0 . eeng = 0.1 and e true = 0.095. eeng/etrue = 1.15. For the second case, L = 2 L 0 . eeng=\.O a n d e true = 0.693. eeng/etrue = 1.44. Engineering strain is always greater than true strain. As with strain, there are two ways of defining stress. Engineering stress is the applied force per initial unit area:

A 85'2Q-Q8 ' - " C 2 T 8 5 - A " 347.7 +85 A = 8.953 or 55 00008866- AA

r|e o(Tg) = 0.897 x 109 Pa • s = 0.897 GPa • s For the Arrhenius-like results, the hypothetical viscosity at T g = 1050C is: rje 105 = 4400 • exp[230 - 105)/13.8] r| el05 = 37.8 MPa • s = 0.038 GPa • s [It is reported that r|e o (T g ) « 1 GPa • s for all amorphous polymers at their glass transition temperatures. It is apparent that the WLF equation yields a viscosity value similar to the expectation. The Arrhenius-like expression does not.] Keep in mind that in thermoforming, stretching is primarily a solid polymer deformation action. The elastic character of the polymer dominates. Nevertheless, the hot rubbery strength of the polymer is frequently compared with its hot melt strength. The actual viscosity value of a polymer is less important than the temperature dependency of the viscosity. If the value of l/p d is very small, the polymer viscosity drops very rapidly with temperature. Since a wide rubbery plateau is sought for thermoforming, polymers with small 1/(3^ values should have narrower forming windows than those with large 1/(3^ values. Examples 4.3, 4.4 and 4.5 illustrate how the zero-state viscosity might be used for thermoforming. Example 4.3 Determination of the Viscosity for Polystyrene in the Thermoforming Window Determine the zero-state viscosity of polystyrene at its lower, average, and upper forming temperatures. Use the Arrhenius-like Equation 4.25. The Arrhenius-like equation is: TWi = 1 V 2 ' exp[P,(T2 - T1)] The relevant temperatures for PS are: T g = 1050C, T L = 127°C, T A = 149°C, T 0 = 182°C. The viscosity of PS at 2100C is 9000 Pa • s and \j% for polystyrene from Table 4.3 is 73.

Tle,i27 = ^ 2 1 0 • exp[(210 - 127)/73] = 28,060 Pa • s 1

IcI 49 = i!e,2io ' exp[(210 - 149)/73] = 20,760 Pa • s

TU.182 = 1Ic2Io • exp[(210 - 182)/73] = 13,210 Pa • s

Example 4.4 Comparison of Polystyrene and ABS Viscosities in the Thermoforming Window Determine the zero-state viscosity of ABS at its lower, average, and upper forming temperatures. Use the Arrhenius-like Equation 4.45. Then compare the results with Example 4.3 for PS. The Arrhenius-like equation is: Tlci = ^,2 * exp[P,(T2 - T1)] The relevant temperatures for ABS are: T g = 1050C, T L = 127°C, T A = 146°C, Tu = 182°C. The viscosity of ABS at 1900C is 43,000 Pa • s and l/|3e for ABS from Table 4.3 is 65. Tlc.127 = Tie,i9o • exp[(190 - 127)/65] = 113,300 Pa • s TIcI46 = 1IcI9O • exp[(190 - 146)/65] = 84,600 Pa • s Tle,i82 = Tle,i9o " exp[(190 - L82)/65] = 48,600 Pa • s ABS has about four times the viscosity of PS across the entire forming window. This implies that the forming forces need to be about four times greater for ABS and that ABS sag should be less of a problem than PS sag. Example 4.5 Predicting the Forming Window From Viscosity Measurements A new polymer is known to have a IJp1 value of 200C. If its viscosity at 3000C is 2,000 Pa • s, determine its approximate forming temperature range. Assume that "best forming viscosity" is 40,000 Pa • s and that the "forming range for viscosity" is a factor of 2. From the information given, the approximate viscosity at the upper forming temperature is about 25,000 Pa • s and that at the lower forming temperature is about 50,000 Pa • s. The Arrhenius-like equation is: 1

Hd = Û2 ' exp[(3,(T2 - T1)]

Applying this equation three times and solving for the temperatures: 2,000 = 25,000 • exp[(Tu - 300)/20] 2,000 = 40,000 • exp[(TA - 300)/20] 2,000 = 50,000 • exp[(TL - 300)/20] 0

T1J = 250 C, T A = 2400C, T L = 235°C. Potentially, this polymer has a very narrow forming window of 15°C.

E

1

1e1

1

E

W

Stress

2

Figure 4.19 Maxwell-Voigt mechanical analog of linear viscoelasticity [29]

Dynamic Mechanical Testing Dynamic mechanical testing is used to determine the relative importance of the elastic and viscous aspects of polymers [28]. If the polymer response to applied load can be considered as linear viscoelastic, then simple spring-and-dashpot models serve to illustrate the response. The spring represents the elastic or fully recoverable portion of the response and the dashpot represents the viscous or fully dissipative portion of the response. Figure 4.19 [29] is an example of a fourparameter element, having the Maxwell viscoelastic model of a spring and dashpot in series, in series with a Voigt-Kelvin model of a spring and dashpot in parallel. The response of the four-element model to an instantaneously applied constant stress, a, is shown in Fig. 4.20 [30]. When the load is applied in a periodic, sinusoidal fashion, the elastic portion of the model responds instantaneously. The phase angle between the input and response is therefore zero. The phase angle for the purely viscous portion is always TI/2 radians or the viscous portion is always 90° out of phase (Fig. 4.21) [31]. The four-element model, representing linear viscoelastic response, shows a response with a phase angle that is somewhere between 0° and 90°. The sinusoidal strain displacement of the polymer is given as: a = ao sin (©9)

(4.27)

Stress

Time

Strain

Retarded Elastic Strain Viscous Flow

Permanent Set Elastic Strain Time Figure 4.20 Response of Maxwell-Voigt mechanical analog of Fig. 4.19 to step-change in applied tensile load [30]

where a o is the amplitude of the displacement, co is the frequency, and 0 is time. The response to the strain displacement is usually written in complex terms as: T* = T ' + i • T"

(4.28)

where T* is the complex stress, T' is the real component of the stress and x" is called the imaginary component of the stress. Four functions are associated with polymer response to sinusoidal load.The complex modulus, G* is given as: G* =

(T*/OC) =

(T'/a) + i • (x'Voc)

(4.29)

This is also written as: G* = G' + i • G"

(4.30)

The real or in-phase portion of the modulus, G', is called the storage modulus. It represents that portion of the inputted energy that is elastically recovered. The imaginary or out-of-phase portion of the modulus, G", is called the loss modulus. It represents that portion of energy that is dissipated. The ratio of the loss modulus to storage modulus is the loss tangent, loss factor or tan S. It is written as: (4.31)

Y

COt

Applied Sinusoidal Strain GY' cot

Elastic Element Response ricoY1

cot

Viscous Element Response Figure 4.21 Response of elastic and viscous portions of Maxwell-Voigt mechanical analog to sinusoidal tensile load [31]

At constant temperature, polymer response changes with changing frequency. At very high loading frequencies, co-» large, most polymers appear glassy. Thus, the storage modulus, G' is large, the loss modulus, G" is small and tan 8 is small. At very low loading frequencies, oo-> small, many polymers appear rubber-like. Thus, G' is small, G" is small, and tan 5 is moderately small. At intermediate frequencies, the storage modulus, G' is decreasing with increasing frequency. The loss modulus, G' on the other hand, goes through a maximum. The value of tan 6* also goes through a maximum, as shown in Fig. 4.22 [32]. This analysis holds for constant frequency, changing temperature conditions as well. At very low temperatures, polymers appear glassy. At elevated temperatures, polymers appear rubbery. At intermediate temperatures, polymers exhibit loss in rigidity and increased viscous dissipation. The test used to obtain the temperaturedependent complex modulus is called dynamic mechanical analysis or thermomechanical analysis, DMA or TMA [33]. Typically, only G', the storage modulus and tan 5, the loss factor are measured. G", the loss modulus is obtained from Equation 4.31. Figure 4.23 shows classic TMA curves for polycarbonate [34]. As is expected, 30% glass-reinforced PC has a much greater modulus than that for unreinforced PC. However, at or about 1400C, both materials experience rapid drops in G'. Simultaneously, tan 5 for both polymer species shows a rapid increase, indicating a

Loss and Storage Moduli, GPa

Rubbery

Viscoetastic

Glassy

Frequency, O), Log Scale Figure 4.22 Frequency-dependent response of elastic and viscous portions of Maxwell-Voigt mechanical analog to applied sinusoidal tensile load [32]. The loss tangent, tan 5 = G"/G'

PC

Loss Factor

Shear Modulus, G1, MPa

30% GR PC

PC

30% GR PC

Temperature,0C Figure 4.23 Temperature-dependent shear modulus and loss factor for unreinforced and 30% glassreinforced polycarbonate, PC. Figure redrawn from [34] and used by permission of copyright owner

rapid increase in G". In a word, in this temperature range, the polymer is becoming more viscous and less elastic. The glass transition temperature of PC is listed as 1500C. Figure 4.24 shows the effect of molecular weight on transitions for PS [35] and Fig. 4.25 shows the effect of crystalline level on transitions for polyethylene [36].

4.4

Stress-Strain-Rate of Strain—Theory

The time-dependent elastic nature of polymers at the thermoforming temperature is understood in terms of solid or fluid behavior. There are two acceptable ways of including time dependency in the typical stress-stain analysis of a solid. One method is to alter conventional rheological stress-strain rate viscosity models to include solid-like behavior at high strain rates [37]. The other is to include some time dependent factor in a typical stress-strain relationship of a solid. Both are simplified approaches to the general cases of viscoelastic mechanical analyses [38-40]. The common methods for determining polymer strain-strain rate response to applied stretching stresses include extensional rheometry [37,41], biaxial or bubble inflation of a tube or sheet [21-23,42], biaxial stretching of a blown film [43], free blowing of a preform [44], uniaxial stretching of fibers [41], and creep experiments [9,17,45]. Creep experiments are the easiest tests to conduct. These tests yield information on polymer response to constant low-level load at isothermal conditions [9,14,45-47]. Rate-dependent terms are considered negligible. At room temperature, many polymers follow a near-ideal strain-hardening ductile material creep rupture response to constant load: o = a0 exp(m • e) = ao • em • X

(4.32)

where a is instant stress, a o is the initial stress, e is the elongational strain, and m is the straining-hardening factor. If the polymer is ideally ductile, m = 1. It has been shown that 0.92 < m < 1.6 for many polymers at strain-rate levels of less than 0.0333 s"1, Table 4.4 [47]. At high loading levels and/or elevated temperatures, creep rates are so high that measuring and conditioning errors make accurate interpreta-

Table 4.4 Strain Hardening Constants for Several Polymers1 a - a o e me Polymer

POM, Delrin POM, Delrin PA-66, nylon mPPO, Noryl PVC PE PP PE PTFE PTFE 1

Strain rate

From plot [47]

(S-1)

(MPa)

(lbf/in2)

0.00027 0.0027 0.0027 0.0027 0.0027 0.0333 0.0133 0.0333 NR NR

67.9 68.95 49.6 48.3 38.6 7.93 12.93 13.5 10.34 12.41

9850 10000 7200 7000 5600 1150 1875 1960 1500 1800

Adapted from [47], by permission of copyright owner NR = Not reported

From least squares m 1.12 1.19 0.919 1.061 0.974 1.182 1.103 1.182 1.58 1.203

(MPa)

(lbf/in2)

65.8 68.3 51.5 48.8 38.0 8.26 13.0 13.4 10.3 12.51

9539 9910 7468 7079 5518 1198 1889 1943 1495 1815

m 1.312 1.257 0.894 1.06 1.029 1.144 1.102 1.169 1.603 1.203

MFl = 15 MFI = 9 and 26

Loss Factor

Shear Modulus, G', GPa

MFI = 15

MFI = 9 and 26

Loss Factor MFI = 15

Temperature,0C Figure 4.24 Temperature-dependent shear modulus and loss factor for two molecular weights of polystyrene, PS. Figure redrawn from [35] and used by permission of copyright owner

tion difficult or impossible [9,17]. Time-dependent behavior has been added to the creep model as [48-50]: G = G0- f(e) • g(G)

(4.33)

For amorphous polymers such as PMMA and HIPS, at normal forming temperatures, the data favor an ideal elastic or non-strain-hardening, model, Table 4.5 [49]: a = GO • em • 6n

(4.34)

HIPS appears to have little time-dependent behavior, with n « 0. On the other hand, ABS/PVC and PVC exhibit substantial strain-hardening at processing temperatures (Fig. 4.26) [17]. Recent studies on polypropylene show that the time-dependent coefficient can be either positive or negative, depending on the nature and size of spherulites (Fig. 4.27) [51]. So long as the polymer deforms uniformly during uniaxial stretching, its isothermal ultimate tensile strength is obtained from: (4.35)

Dynamic Shear Modulus, G1, GPa

UHMPE

Loss Factor

HDPE

LDPE

LDPE HDPE UHMWPE

Temperature,0C Figure 4.25 Temperature-dependent shear modulus and loss factor for three types of polyethylene. Figure redrawn from [36] and used by permission of copyright owner

where ef is the true strain at fracture. If m « I5 as is the case for most polymers in Table 4.3, T* « ao. As noted in Figs. 4.9 and 4.10, polymer tensile strength decreases with temperature. Tensile strength values for most polymers at normal vacuum forming temperatures are in the 0.07 to 0.7 MPa or 10 to 100 lbf/in2. For truly elastic polymers, the classic temperature-dependent stress-strain curves usually appear as shown in Fig. 4.28. At low temperatures, the polymer is purely elastic. Its modulus is very high and its ultimate strain is very low. As the temperature increases, a small amount of plastic deformation occurs before the polymer breaks. The modulus decreases with increasing temperature. At a slightly higher temperature, the polymer may show a distinct yielding. The higher strain regions beyond the yield point are characterized by localized drawing or necking.

Table 4.5 Stress-Strain Behavior of Two Plastics in Biaxial Extension1 -.

G

=

_

c mnn

CJQE

U

Polymer

n

m

PMMA HIPS

-0.05 -0.33

1.0 1.1

1

Adapted from [49], with permission

Strain, %

Temperature = 116°C

Time, s Figure 4.26 Temperature-dependent strain rate for rigid polyvinyl chloride, RPVC [17]

Growth Function > 1

Viscosity, GPa*s

Growth Function = 1

Growth Function < 1

Time, s Figure 4.27 Schematic of time-dependent viscosity for polyolefin polymers that exhibit various strain rate effects [51]

Increasing Temperature

Stress

Break

Yield Point

Elongational Strain Figure 4.28 Characteristic temperature-dependent stress-strain curves

Not all polymers neck. As the temperature increases, the ultimate elongation increases rapidly. There is an upper limit to the temperature of course. When the polymer cannot sustain any applied force without extensive plastic deformation and fracture, it is considered a fluid. Figure 4.29 shows the interrelationship between the polymer response to applied load and the forming window. As expected, the amount of force required to draw the polymer sheet to a given extent is highest at the lower forming temperature. This implies that the depth of draw or areal draw ratio increases with increasing

Stress

Increasing Temperature

Forming Region

Elongational Strain Figure 4.29 Characteristic overlay of forming temperature on temperature-dependent stress-strain curves

Viscosity, MPa»s

Strain Rate = 1.15 s

Time, s Figure 4.30 Strain-rate dependent extensional viscosity for high-density polyethylene, HDPE, at 18O0C. Figure adapted from [37]

temperature. At the upper forming temperature, draw uniformity gives way to localized flow and the areal draw ratio then abruptly decreases with increasing temperature. When the processing temperature is substantially above Tg for an amorphous polymer or Tm for a crystalline one, the polymer is a fluid. Behavior under load is correctly considered in terms of elongational viscosity. Isothermal elongational viscosity usually increases with increasing time (Fig. 4.30). At very low strain rates, r|eoc0n, where n < 1, Table 4.6 [49]. Isochronous biaxial elongational viscosities of olefins at forming temperature and very low strain rates of 0.000015 to 0.006 s" 1 are inversely proportional to strain rate [46]. As the strain rate increases, the apparent viscosity deviates from the asymptote at earlier and earlier times. As seen in Fig. 4.31 [17], for LDPE at 2 seconds, the polymer has a viscosity about 15 times greater at a strain rate of e = 1 s"1 than at e = 0. At this same rate at 10 s, the elongational viscosity is about 1000 times greater. In short, the polymer is rapidly becoming solid-like in its response to applied load. Example 4.6 continues this analysis. In terms of true stress and true strain, e = e • 0, the data show an initial linear region, a yield region, then strain hardening and fracture (Fig. 4.32) [37]. In other words, both amorphous and crystalline polymers behave as elastic liquids at typical thermoforming temperatures [52]. The extent of elasticity is important in determining the formability of the polymer in question.

Table 4.6 Biaxial Extensional Viscosities for Olefins at Very Low Strain Rates1 Polymer PP, 0.003 in Unoriented2 PP, 0.0015 in Unoriented PP, 0.003 in Ethylene-propylene copolymer, 0.003 in 1

Viscosity, MPa-s

2

(XlO-6S-1)

Viscosity (GPa • s)

2310 196 19.1 2260 219 15.1 5110 262 28.8 181 24.3

4.99 71.6 762 6.51 50.9 719 28.1 496 4150 890 6310

Strain rate, e

Adapted from [49], with permission Least squares fit, r) = 0.015/e

Strain Rate = 1.0 s"1

Time, s Figure 4.31 Comparison of experimental and theoretical strain-rate dependent extensional viscosity for low-density polyethylene, LDPE. Solid lines are theory. Dashed lines are experiment. Figure adapted from [39], and used with permission of John Wiley and Sons, Inc.

Example 4.6 Time-Dependent Strains for LDPE From Fig. 4.31, at a strain rate of ' £= 0.1 s~\ determine the time required to achieve the same level of stiffness as is achieved for e= 1 s"1 in 2s. Repeat for a strain rate of e=0.01 s-L From Fig. 4.31, the stress at e = 0.1 s" 1 is about 15 times greater than that for e = 0 at 12 s. The stress at e = 0.01 s" 1 is about 15 times greater than that for e = 0 at about 100 s.

Stress

Fracture Strain Hardening Yield Linear

Figure 4.32 Schematic of various stages in the straining of a ductile polymer. Figure adapted from [37], and used with permission of John Wiley and Sons, Inc.

True Elongational Strain

Elasticity—A Rationalization When a thermoformed shape is placed in an environment having a temperature substantially greater than Tg and a temperature typically approaching that of its forming temperature, the shape returns to a flat sheet. A recovery rate of 90% in less than 0.002 s has been measured [53]. It is argued therefore that thermoforming is a solid phase deformation process. A contrary argument [40,54] is that this response is proper for a highly strained elastic liquid as well as a purely elastic membrane. Sheet stretching behavior is best viewed in terms of relative orders of magnitude of process times and polymer memory. Consider the simple series spring-and-dashpot model of a linear viscoelastic material, the Maxwell fluid (Fig. 4.33) [55]. If this simple model is strained to a fixed value, e = eo, and e = 0, the Maxwell element response is: a = ao-exp[-E6/Tie]

(4.36)

where E is the tensile or elastic modulus of the spring, r|e is the elongational viscosity of the dashpot and a o = E • eo. The retardation time, 9p = r|e/E, is a characteristic of the polymer. Creep data are used to obtain values for this retardation time. At low strain rates, e « 0 , the tensor stress-strain-rate-of-strain elastic liquid equation reduces to a simple relationship between stress and retardation [56]: a = 6G9p • e • [1 - exp( - 9/9p)]

(4.37)

For long times, 9 -» oo and: a-+6G9 p -e = î-e

(4.38)

The model yields the Trouton-Newton form for a purely viscousfluid.At high strain rates, on the other hand, e-*oo: a -> G | 7 I + ^V exp(2e9) - ^ • exp(4e9)l

(4.39)

The term 5 is proportional to the ratio of second to first normal stress difference of the polymer and its value is always negative or zero [40]. The model predicts that

Maxwell Model Stress Off

Strain

Stress On Elastic Element Viscous Element Voigt Model

Strain

Time

Stress Off Stress On

Elastic Element Viscous Element Time Figure 4.33 Responses of the Maxwell series mechanical analog of polymer linear viscoelasticity [top] and Voigt parallel mechanical analog of polymer linear viscoelasticity [bottom] to instantaneous change in applied tensile load

under constant deformation rate, stress increases exponentially with time. In other words, at high strain rates, the rate of stress increase is greater than the rate of internal material stress relaxation. The polymer therefore behaves as if it is an elastic solid [40,57]. It is highly unlikely that constant biaxial deformation can be sustained or is desirable in conventional thermoforming. Deformation rates have been measured that, for the most part, decrease with time [40]. For constant velocity stretching, deformation rate decreases with time [40]. This helps stabilize the initially rapidly growing stress. As noted, retardation times are obtained from creep experiments. Usually these values decrease monotonically with increasing temperature, as seen schematically in Fig. 4.34. discontinuity in the retardation time curve for PP occurs at 1100C or 2300F [45]. This is attributed to a deformation mechanism change on the molecular level. The importance of creep data in parts design has produced a substantial library of information [58]. If temperature-dependent retardation times are not available for a given polymer, approximate values can be obtained at any temperature [48] from: Gp(T)=

§m

(4 40)

-

where r\o is the zero-shear viscosity and G is the tensile modulus or the initial slope of the stress-strain curve. If the processing time is less than 0p by a factor of about 10, the material behaves as an elastic membrane. Snap-back thermoforming depends on elastic membrane response, for instance. If the processing time is greater than 0p by a factor of about 10, the polymer should behave as an elastic liquid. Pressure

Logarithmic Retardation Time Figure 4.34 Characteristic temperature-dependent retardation time for linear viscoelastic polymers

T

m

Temperature

forming, coining, and high surface replication depend to some degree on plastic or anelastic polymer response. Example 4.7 illustrates some of these aspects. As a point of reference, instantaneous stretching rates of 2.4 s~l are recorded for 0.100 in or 2.5 mm HIPS sheet [59] and rates up to 26.8 s" 1 are reported for 0.060 in or 1.25 mm PP [60]. Example 4.7

Stretching Rate and Retardation Time for ABS

Determine the retardation time for ABS at 1200C. The tensile modulus of ABS at 1200C is 80 lbf/in2 = 0.55 MPa. The viscosity for ABS is obtained from Example 4.4: Tle,i2o = 1Ie5I90 ' exp[(190 - 120)/65] = 126,200 Pa • s The retardation time, 0p(120°C) = 126200/(0.55 x 106) = 0.23. If the stretching rate is substantially greater than l/0 p = 4.4 s" 1 , ABS should behave as an elastic solid. If the stretching rate is substantially less than 4.4 s" 1 , ABS should behave as an elastic liquid.

Strain Energy Function1 Principal stresses in elastic solids are defined in terms of the strain energy function, W:

«-w

(4 41)

-

The stress-strain analysis that follows is applicable to large elongational levels. As a result, the analysis is sometimes called hyperelastic analysis, in contrast with linear models used to describe elastic solid response to low elongation levels typically found in structural analysis.

where X{ is the extension ratio in the ith direction (i = 1,2,3). 9W is the incremental amount of work done by the solid when it is stretched an incremental amount dX under stress, a. In general, the strain energy function is written in terms of three principal invariants of the Cauchy strain tensor [40,45,61], as: W = W(I9II9III)

(4.42)

where: I = X2 + Xl + Xl

(4.43)

H = V 2 + K2 + K2 2

2

III = X -X -X

(4.44)

2

(4.45)

A stress-strain energy expression is written in terms of these invariants as: /6WVdI \

/6W\/6II\

/8WV8III\

,A

Ars

F o r a n incompressible solid, X1 -X2 -X3 = 1, o r I I I = 1. T h e last t e r m of E q u a t i o n 4.46 is zero. F o r uniaxial stretching, X1 = X, X2 = X3 = X~l/2. Therefore, this e q u a t i o n is written as: < * - H ) [ < £ H © ] In this equation, <JX is the tensile stress and a is the force per unit area of unstrained cross-section. For equal biaxial stretching, X1 = X2 = X, X3 = X~2. The equation is written as:

The bracketed terms in Equations 4.47 and 4.48 represent the specific elastic solid response, in this case polymeric response, to applied load. The exact form for W(IJI) depends to a great degree on curve-fitting elongational data. Several models follow.

The Rivlin Form for the Strain Energy Function Fifty years ago, the following simple power-law form for W was proposed, to predict elongational response of rubber to stress [62,63]: W(IJI) = X QJ (I - 3)1' (H - 3 ^

(4.49)

The neo-Hookean solid yields one of the simplest forms for W: W = 0^(1 - 3 ) = C10(I - 3 )

(4.50)

The second strain invariant is usually added in one of several ways. The Rivlin-Saunders1 version is the most general form [64]: 1

This is sometimes just called the Rivlin model.

W = Qj(I-S)H-F(II-B)

(4.51) 1

The Mooney version assumes a linear function for F(II) : W = C 0 1 (I-3)+ C 10 (II-3)

(4.52)

Other Rivlin-type forms are found in Table 4.7. Representative strain energy function coefficients for several polymers are given in Table 4.8. The uniaxial and biaxial forms for the elastic stress-strain equation are obtained by differentiating the W-function and substituting into Equations 4.47 and 4.48, respectively. Consider the simple Mooney version. 9W/9I = C01 and 9W/9II = C10. The Mooney stress-strain functions for uniaxial and equal biaxial extension are:

aX = (^2 - i y ^2 • C01 + ^ • C10J GX

= ( ^ - ^ V P • C01 + 2 • ^2 • C10]

(4.53) (4.54)

These equations are used to curve-fit rubbery elastic sheet deformation, and C01 and C10 are the curve-fitting constants. Further, since stress-strain relationships are temperature-sensitive, C01 = C01(T) and C10 = C10(T). In the limit, as the strain goes to zero, the constants are defined in terms of an elastic modulus [59]: E 9W 9W

6 = ^r + 9ir

(4 55)

I = C 0 1 + C10

(4.56)

-

For the Mooney form for W:

For the Schmidt model, Table 4.7, E/6 = C01, where the prime denotes a different value for the first constant [59,131,132]. For HIPS and PS, predicted modulus values range from 427 to 1192 MPa or 62 to 173 lbf/in2. Measured values range from 310 to 3900 MPa or 45 to 566 lbf/in2, with errors ranging from -100% to 4-50%. If the temperature dependency of the polymer modulus is known or is accurately measured, a good approximation of the temperature dependencies of the strain function coefficients can be obtained. For many polymers, 9W/9I»9W/9II [69]. For the Mooney model, as an example, C01 » C10. The curves are approximated best with the new-Hookean model (Fig. 4.35). For vulcanized natural rubber, the value for C10 is about 0.05 to 0.15 times the value for C01 [70]. If C01 = 0, C10 equals the elongational, tensile or elastic modulus, G. The neo-Hookean model works best at low levels of deformation, at or just above the linear viscoelasticity region [59,71]. The Schmidt, Mooney and higher order versions of the Rivlin model work best at very high levels of deformation. Schmidt recognized that although his sheet was being deformed rapidly to large deformation, it was not isothermal and was not being deformed at a constant rate. Funt, on the other hand, correlated PP isochronous creep data best with C10 = 0 [45]. 1

This is sometimes called the Mooney-Rivlin model.

Table 4.7 Strain Energy Density Forms for Polymers and Elastomers Name

Form

Material

Source of data

Comments

Neo-Hookean

W = C 10 (I-3) W = C10(I -3) + C01(II -3)

Funt Schmidt

W = C01 (II-3) W = C10(I -3) + C02(II -3)2

Treloar [63] Williams [65] Mooney [66] Treloar [63] Schmidt [59] Funt [5] Schmidt [59]

Uniaxial stretching

Mooney

Natural rubber PMMA Vulcanized natural rubber HIPS PP HIPS

Mooney-Rivlin Three-parameter Signiorini

W = C10(I -3) + C01(II -3) + C 11 (I-3) (II-3) W = C 1 0 (I-3)+ C 01 (II-3) + C 20 (I-3) 2 W = C 1 0 (I-3)+ C 01 (II-3) + C11(I - 3) - ( H -3) + C20(II -3)2

PVC

Warnecke/Frankenhauser [67]

Creep Bubble inflation (gross) Uniaxial tensile

EPDM


Uniaxial tensile

PVC, EPDM


Uniaxial tensile

Third-order

Uniaxial stretching

Table 4.8 Typical Strain-Energy Function Coefficients for Rubbery Solids C 1 = C 10 = 9W/SI

C2 = C01 = ew/en C 3 = C 02 - 2(6W/8II) • (II - 3) C2 (MPa)

C 3 x 106 (MPa)

Source Comments

Polymer

Temperature C 1 (0C) (MPa)

HIPS HIPS HIPS Vulcanized Natural rubber PP Cellulose acetate

123.31 124.7 124.4 25.0

0.0758 0.08598 0.14445 0.1618 0.01472

0.001179] 0.001331 f [68] 0.002234J [63]

A=165-T NR

0.0 0.1236

3400

0.080 • A

[45] [68]

No representative values for Mooney C2 given Data on rubber, by Rivlin and Saunders Creep data is source No temperature given, C3 value seems high

1

About 29°C higher at pole than at edge Average value. Range is 0.010 to 0.022. Value decreases with increasing value of II NR = Not reported 2

Strain Rate = 2.5 min "•

Figure 4.35 Tensile stress-strain behavior of polymethyl methacrylate, PMMA, at 1600C. Theory assumes tensile modulus =130 lbf/in2 or 0.9 MPa and Mooney-Rivlin model with C2 = 0. Figure adapted from [69], and used with permission of Ellis Horwood Ltd., copyright owner

Theory

The Ogden Form for the Strain Energy Function It is apparent that the more constants that are available for curve-fitting, the more accurate the model will be in imaging the data. Figure 4.36 compares several Rivlin-type models with experimental data on PVC and EPDM [72, 129-130]. The agreement with experimental data is only satisfactory, even with a three-constant model. Ogden [61,73-75] proposed replacing the general Rivlin strain energy function model, Equation 4.49, with:

Stress, MPa

Model 1 Model 2 Model 3 Experimental

Compression

Tension

Strain, % Figure 4.36 Comparison of experimental stress-strain data with several types of Rivlin constitutive equations. Figure adapted from [72]

(4.57)

Here ocn and [In are the Ogden curve-fitting constants. Although m is unbounded, its value is practically restricted to no more than 3, thus yielding 2, 4 or 6 constants. It has been shown that bubble dynamics are stable for n = 1 when Oc1 > 3. When n = 2, (X1 = 2 and oc2 = —2, the result is the Mooney model (Equation 4.52). Although the Ogden model is based on the strain energy function relationship to principal invariants, the values of ocn are not restricted to integer values, as with the Rivlin version. There is some theoretical justification for the integer values of ocn [63,76]. However, the primary justification for the Rivlin version of a sum of principal invariant effects on the strain energy function is "[a] considerable simplification of the theory..." [77]. Certainly if the linearized Rivlin model is acceptable, the semi-empirical Ogden model is also acceptable.

Viscoelastic Models Models for viscoelasticity are much more complex than elastic models. Most models employ coordinates that translate, rotate and distort with the fluid element under stress. As noted above, Equations 4.33 and 4.34 describe one simple way of including time-dependent polymer properties with traditional stress-strain relationships [48-50]: (4.33) or: (4.34)

Correctly, the constitutive equation for a viscoelastic fluid must contain the concept of an imperfect or fading memory [78]. The general integral form for fading memory viscoelasticity is [79]: re CT(9) =

n(0 - 9') h(I,II) B(9,9') d9'

(4.58)

Jo where JLX(0 — 6') is the relaxation factor or memory function: p.(9 - 9') = Z ^ exp[-(9-9')/XJ

(4.59)

i = i î

and G1 and X{ are material parameters. h(I,II) is a damping function of the two strain invariants, usually written as the Wagner form [80]: h(I,II) = [1 + a y ( I - 3 ) - ( I I - 3 ) ] - 1 / 2

(4.60)

In biaxial stretching, h(I,II) is written as: h(e) = [a exp(2e) + (1 - a)exp(me)]-{

(4.61)

where e(0) = In L(O), a = exp( — 2eo), and eo and m are measured constants for a given polymer. L(O) is the stretch ratio, related to time 0'. B(0,0') is the Finger strain tensor for deforming coordinates [81]. Temperature is included in the G1 material parameters as: G1(T) = G1(T0) • exp[ - P(T - T0)]

(4.62)

where T 0 is a reference temperature and p is an Arrhenius-like parameter. The K-BKZ constitutive equation is used to describe the biaxial deformation of a viscoelastic sheet [82,83,119-123]. For biaxial plane stretching, the principal stresses in the i = 1,2 directions are given as: CJ11 =

fe

|4.(9 - 9') • h(e) • [Lf(9,9') - L|(9,9')] d9'

Jo + h(e(9)) • [L?(9) - L|(9)] • P *i(9 - 9') d9' J-OO

(4.63)

where L1(O9O') is the stretch ratio at time 0 related to time 0'. The K-BKZ model is used to shape the time-dependent elongational viscosity curves of the type shown in Figs. 4.26 and 4.30. Typically, only the first term of the memory function is needed: ji(9 - 90 = ^ exp[ - (9 - 90A]

(4.64)

A,

This simplifies Equation 4.63 and allows strain recovery experiments to be used to obtain the necessary parameters [84].

Next Page

As will be seen in Chapter 9, the viscoelastic characteristics of polymers are secondary to their pure elastic characteristics when predicting wall thickness variation in thermoforming1.

4.5

Available Stress-Strain Data

Stress, 1000 lbf /in2

Although there is a plethora of models to predict large deformation of solid and viscoelastic membranes, there is a dearth of temperature-dependent stress-strain-rate of strain data. This section records some of the available data. Typical room temperature data are shown in Fig. 4.37 [87]. The relationship between creep data and stress-strain curves is shown in Fig. 4.38 [88]. Typically, tensile strain or sample elongation is determined as a function of time for a given load or stress. When the data are replotted in terms of stress and strain, time is the parameter and the data are referred to as isochronous stress-strain. Figures 4.39 [89], 4.40, 4.41, and 4.42

PMMA

PA-6 PC ABS

PP HDPE PUR LDPE

Strain, % Figure 4.37 Room-temperature stress-strain data for several thermoplastics [87]

1

Wineman [85,86] notes that if the membrane is an elastoviscous solid under fixed applied pressure, the membrane dimensions eventually reached fixed equilibrium values. If it is a viscoelastic fluid, on the other hand, the polymer will continue to creep. Since practical processing times are usually very small when compared with viscoelastic material times, longterm fluid effects are usually ignored in all but certain plug-assist conditions.

Previous Page

As will be seen in Chapter 9, the viscoelastic characteristics of polymers are secondary to their pure elastic characteristics when predicting wall thickness variation in thermoforming1.

4.5

Available Stress-Strain Data

Stress, 1000 lbf /in2

Although there is a plethora of models to predict large deformation of solid and viscoelastic membranes, there is a dearth of temperature-dependent stress-strain-rate of strain data. This section records some of the available data. Typical room temperature data are shown in Fig. 4.37 [87]. The relationship between creep data and stress-strain curves is shown in Fig. 4.38 [88]. Typically, tensile strain or sample elongation is determined as a function of time for a given load or stress. When the data are replotted in terms of stress and strain, time is the parameter and the data are referred to as isochronous stress-strain. Figures 4.39 [89], 4.40, 4.41, and 4.42

PMMA

PA-6 PC ABS

PP HDPE PUR LDPE

Strain, % Figure 4.37 Room-temperature stress-strain data for several thermoplastics [87]

1

Wineman [85,86] notes that if the membrane is an elastoviscous solid under fixed applied pressure, the membrane dimensions eventually reached fixed equilibrium values. If it is a viscoelastic fluid, on the other hand, the polymer will continue to creep. Since practical processing times are usually very small when compared with viscoelastic material times, longterm fluid effects are usually ignored in all but certain plug-assist conditions.

Strain

Creep Curve

Strain

Time of Load Time Diagram

Stress

Stress

Isochronous Stress-Strain Curve

Time of Load Figure 4.38 Interrelationship between isochronous stress-strain, creep and time-dependent stress [88]

[90-92] show isochronous stress-strain curves for amorphous RPVC and PMMA, and crystalline HDPE, LDPE and PP, respectively. These curves are for temperatures below Tg for the amorphous polymers and Tm for the crystalline ones. Stress-strain curves at elevated temperatures for several polymers are given in the attached figures: • • • • • • • • • • • • •

SAN in Fig. 4.43 [93], P homopolymer in Fig. 4.44 [94], PMMA in Fig. 4.45 [95], PET in Fig. 4.46 [96], ABS in Fig. 4.47 [97], PS in Fig. 4.48 [98], ASA terpolymer in Fig. 4.49 [99], PTFE in Fig. 4.50 [100], FEP in Fig. 4.51 [101], Nylon 6 or PA-6 in Fig. 4.52 [102], Nylon 66 or PA-66 in Fig. 4.53 [103],' PBT in Fig. 4.54 [104], and Polyimide in Fig. 4.55 [105].

As is apparent, most of the data are for engineering and high performance polymers. There are few data for commodity polymers such as PVC and PE. Unfortunately, commodity polymers make up the bulk of the polymers thermoformed today.

Temperature = 400C

Temperature

Temperature = 20°C PMMA Flow Zone

Flow Zone

RPVC

Strain, % Elongation, % Figure 4.39 (left) Time-dependent isochronous stress-strain for rigid polyvinyl chloride, RPVC. (right) Polymethyl methacrylate, PMMA. Figures redrawn from [89] and used with permission of copyright owner

Stress, MPa

HDPE

Failure Curve

Elongation, %

Stress, MPa

Figure 4.40 Time-dependent stress-strain curve for high-density polyethylene, HDPE at 65°C. Figure redrawn from [90] and used with permission of copyright owner

LDPE

Elongation, % Figure 4.41 Time-dependent stress-strain curve for low-density polyethylene, LDPE at 400C. Figure redrawn from [91] and used with permission of copyright owner

Stress, MPa

PP

Failure Curve

Elongation, % Figure 4.42 Time-dependent stress-strain curve for polypropylene, PP at 1100C. Figure redrawn from [92] and used with permission of copyright owner

Stress, MPa

SAN

Elongation, %

Figure 4.43 Temperature-dependent stress-strain curves for SAN. Figure redrawn from [93] and used with permission of copyright owner

Stress, MPa

PP Homopolymer

Strain Rate = 0.025 s~1

Elongation Figure 4.44 Temperature-dependent stress-strain curves for polypropylene, PP, homopolymer [94]

The method of fitting a model to the stress-strain data depends on the method used to obtain the data [106,124-128,133]. It is apparent that fitting uniaxial data with a neo-Hookean model is substantially easier and less arduous than fitting nonuniform biaxial data with a 4- or 6-constant Ogden model.

Sensitivity of Models Two aspects of polymer response to applied load remain for discussion: •

The first deals with the sensitivity of the values of the curve-fit constants for any model to stress-strain prediction and then to the variation in wall thickness of the formed part. This will be addressed in Chapter 9 on the design of thermoformed parts.

Stress, MPa

PMMA

Elongation, % Figure 4.45 Temperature-dependent stress-strain curves for polymethyl methacrylate, PMMA. Figure redrawn from [95] and used with permission of copyright owner

•

The second deals with the accuracy required of the models in the prediction of wall thicknesses of production quality thermoformed parts. This will be addressed in Chapter 10, on production quality control.

Obviously, if the day-to-day forming process conditions are not under control, the accuracy of wall thicknesses of formed parts will be poor. As a result, substantial effort to achieve great accuracy in the prediction of wall thicknesses is unwarranted. However, even with the most carefully controlled forming operation, it appears that no single rubbery solid model can describe the behavior of a polymer sheet undergoing nonisothermal high-speed, large scale deformation. Some guidelines are obtained, however, by beginning with the simplest model, the neo-Hookean model, for illustration.

4.6

The Importance of Polymer Material Properties1

A typical thermoforming process applies near-instantaneous, near-constant differential pressure to the rubbery sheet to deform it. If the pressure is insufficient or if the 1

In [135], this section was titled "A Material Parameter, (J)(T)". Although 4>(T) was identified as a material parameter related to the derivative of the strain energy function with respect to the first principal invariant of the Cauchy strain tensor, the analysis that followed used the neo-Hookean version of this derivative. More importantly, the relationship between (T) and the Mooney constants was not clear.

Stress, MPa

PET

Strain Rate = 0.025 s ~1 Figure 4.46 Temperature-dependent stress-strain curves for polyethylene terephthalate, PET [96]

Elongation

sheet is not soft enough, the sheet will not distort fully to fill the mold or will not replicate the mold details. Earlier sections focused on the basic polymer response to applied load. There is a logical solid mechanistic approach to development of the parameters that are used to determine proper processing conditions for a given material [69,107]. As seen in Appendix 4.1, the neo-Hookean relation between the elongational or tensile elastic modulus, G, and the inflation pressure, P, for a uniform disk of radius a and initial thickness ho, forming a dome of 5 units above the horizontal is [107]: Pa 4C10(8/a) 2h o ~l+(5/a) 2

4G(5/a) l+(5/a) 2

l

"

;

The modulus G is temperature-dependent, G(T), and has the units of MPa or lbf/in2. G(T) is determined by heating a sheet of radius a and initial thickness ho to a fixed, uniform temperature, then measuring the extent of bulging, (5/a), as a function of applied pressure. The analysis is dependent on all neo-Hookean assumptions and is inaccurate at the clamped sheet edge [107]. If the deforming sheet has a constant thickness everywhere:

Stress, MPa

ABS

Elongational Strain, % Figure 4.47 Temperature-dependent stress-strain curves for ABS. Figure redrawn from [97] and used with permission of copyright owner

Stress, 1000 lbt /in2

PS

Strain, % Figure 4.48 Temperature-dependent stress-strain curves for polystyrene, PS. Figure redrawn from [98] and used with permission of copyright owner

Stress, MPa

ASA T e r p o l y m e r

Elongation, % Figure 4.49 Temperature-dependent stress-strain curves for ASA terpolymer. Figure redrawn from [99] and used with permission of copyright owner

V = [l+(5/a)- 2 ]

(4.65)

^ = [l+(5/a)2]

(4.66)

At a maximum value of ^ = 2.59, (5/a)=1.26, and an approximate relationship between the applied pressure and the neo-Hookean modulus becomes:

pmax-y3-G(T)(v)

(4 67)

-

Unfortunately, experiments indicate that biaxially stretched sheet does not have constant thickness, a fundamental assumption in Equation 4.1.10 [50,68]. This analysis is extended to constrained biaxial deformation as sheet draw-down into a cone or funnel of wall angle oc. The polymer not in contact with the wall is biaxially stretching as A1 = XQ and ro = a: (4.68) The thickness is: (4.69)

The pressure is: (4.70) where: (4.71) Note that g(oc) is a geometric factor. If s is the distance down the cone side from the cone opening to the point where the sheet leaves the cone wall (Fig. 4.56): (4.72) (4.73) (4.74)

Examples 4.8 and 4.9 derive the expressions for commercial funnels where a = 60° and cylindrical or straight walled molds, where a = 90°. Example 4.8 Neo-Hookean Draw-Down into a 60° Funnel Determine the pressure-modulus relationship for a neo-Hookean polymer being drawn into a 60° funnel. From Equation 4.71, g(oc) = 0.5. The dimensionless radius is given by Equation 4.72 as:

I = 1-0.577 P ) The wall thickness is given by Equation 4.73 as:

And the pressure-modulus relationship from Equation 4.74 is:

Example 4.9 Neo-Hookean Draw-Down into a Straight-Wall Can Determine the pressure-modulus relationship for a neo-Hookean polymer being drawn into a can having 90° walls.

The wall thickness is given by Equation 4.73 as: h

ex

/

2s\

hr VirJ And the pressure-modulus relationship from Equation 4.74 is:

Stress, MPa

PTFE

Elongation, % Figure 4.50 Temperature-dependent stress-strain curves for polytetrafluoroethylene, PTFE. Figure redrawn from [100] and used with permission of copyright owner

The general form for the pressure equation is: P= G(T)-^Vg

(4.75)

where g is the general form for the geometry of the mold. The deformation pressure is directly proportional to both the neo-Hookean modulus and the relative sheet thickness, ho/a. If the pressure is fixed by the process, as with vacuum forming where P 0.5. As a result, the equation is written as:

p* 4GCn-(I) For simple vacuum forming, P < 0 . 1 MPa or 15 lbf/in2. As a result: G(T) = ^ 1 ^

= 0.626 MPa = 91.9 lbf/in2

As seen in Table 4.9, the modulus of PVC at 1100C is 93.6 lbf/in2 or 0.65 MPa and the measured elongation at this temperature is substantially greater than that at 1000C. Example 4.11 The Minimum Forming Temperature for PVC Determine the minimum forming temperature for PVC for the data in Example 4.10.

The maximum applied pressure occurs when the geometric parameter, g, is maximum. This occurs when s/a = 0. At this condition: Pmax = 4G(T)(^) -0.875 Again, for the vacuum forming case, Example 4.10, P 1 or so. A cross-plot of the ratio of S/L to Y/L for parabolic and catenary sag shows essentially identical shapes for values of Y/L greater than about 0.1 (Fig. 4.69). S/L approaches a value of 2(Y/L) for sag levels greater than Y/L = 2.

Relating Sag to Hot Sheet Strength The catenary relationship between Tmax, the tension at the gripped edge of the sheet, and T0, the tension at (x = 0,y = 0) is obtained from: Tmax = T0 + nS = T0 + ( ^ )

•( ^ )

= T0 + phoL

(4.112)

or Tmax is a constant factor greater than T0, regardless of the extent of stretching. On the other hand, for the parabolic relationship: U2T2

TL x = T2 + ^ -

(4.113)

Again, JI = jdo • (h/ho) and (h/ho) = (S/L)"1 as before. Since Tmax is assumed to be constant and equal to the tensile stress of the polymer, this equation is solved for T0 as: T 2

/ T \ 2

T2 = T2 -'u 2 — I -

(4 114)

Unlike Equation 4.112 for the catenary, T0 is not constant but a function of the length of the sagged sheet, which in turn is a function of the vertical extent of sag, Y/L, according to Equation 4.107. So long as S/L is small, the effect of sag on the value of T0 is small and u «jao. As the sag becomes more significant, the basic premises used to develop the parabolic equations are no longer valid.

Dimensionless Sag, y/L or Dimensionless Sagged Sheet Length, s/L

s/L, Catenary y/L, Catenary

s/L, Parabola

y/L, Parabola

Dimensionless Weight, WL/2TO Figure 4.68 Dimensionless extent of sag as function of sheet dimensionless weight for catenary and parabolic sag

From Equation 4.113: (4.115) Again, since T max is constant, Equation 4.114 is written as: (4.116) This relationship also shows T 0 as a function of Y/L. The hot strength of the polymer, in terms of the engineering stress at the grip, is written as a = T max /A where A is the initial cross-section of the sheet, A = h o • b. As a result, a measured property, the tensile strength of the polymer, is directly related to the measured response, sag, for both the catenary and parabolic models [134]. Example 4.14 illustrates the relationship between the catenary and parabolic models. Examples 4.15 and 4.16 illustrate the effect of sheet temperature on sheet sag.

Dimensionless Sagged Sheet Length,S/L

Dimensionless Sag,Y/L Figure 4.69 Dimensionless sheet length as function of dimensionless extent of sag. Catenary and parabolic curves coincide

Example 4.13 Initial Sag of Polypropylene Sheet Consider a 0.250-in thick polypropylene sheet at 1000C. The modulus of the sheet is 10 MPa or 1500 lbf/in2 and its Poisson's ratio is 0.35. Determine the extent of sag if a Wx 10 in sheet is clamped on all sides and then only on two sides. Determine the equivalent sag if a 10-in disk is being heated. The extent of sag is obtained from Equation 4.92. The density of PP is 0.91 g/cm3 = 56.8 lb/ft3 = 0.0329 lb/in3. q = 0.0329 • h. Therefore for the sheet clamped on all sides, (3 = 0.0444: y =

0.0444 • 0.0329 • h • 104 0.0444 • 0.0329 • 104 = 1500^ 1500-0.25*

= a i 5 6 m

For the sheet clamped on two sides, P = 0.1421, and y = 0.500 in1. The extent of sag of a disk is obtained from Equation 4.93: y

1

3 • 0.0329 • h • 100 • (5 + 0.35) . = 542 m = 16 • 1500-h' °-°

Correctly, when the value of y calculated using this equation exceeds the half-thickness of the sheet, the equation should not be used to predict sag.

Example 4.14 Parabolic and Catenary Sag Consider the sag of a 0.100 in thick by 48 in wide plastic sheet having a density of 62.4 Ib I ft3. The tensile stress on the sheet at the grip is 10 Ib/in2. Determine the sag as given by the parabolic and catenary sag equations. Consider the sag in catenary terms first. The tension in the sheet at the grip is given in Equation 4.112 in terms of the tension, T 0 at (x = 0, y = 0), written as:

For the catenary sag model, (Y/L) c a t « 0.0265 and Y cat = 1.27 in. For the parabolic sag concept:

As seen in Fig. 4.72, the S/L-to-Y/L ratio for the catenary model has nearly the same dependency on Y/L as the parabolic model. If (Y/L) para « 0.0265 and if \i « \io:

Note that 2Tmax/jioL - 2 T O / J I L = 11.54 - 9.43 = 2.11 or approximately the same value as for the catenary problem 1 . In other words, for this case, there probably is not a substantial difference in the sag predicted by parabolic and catenary equations.

1

The reader is warned that this may be a self-fulfilling prophecy in that an incorrect value is used for Y/L. Please review the comments in the text at this point.

Example 4.15 Effect of Temperature on Sheet Sag Consider the polymer of Example 4.14. Consider the stress given in the example to be at 2000C. The polymer elongational energy of activation is 20,000 kcal/mol. Determine the tensile stress at 2300C and determine the extent of sag at 2300C. Repeat the analysis for a sheet temperataure of 2400C. The Arrhenius form is given as:

where the temperatures are absolute. From the values given:

° - = 1 0 • e x p [ i ^ ferbs - 200T273)]=10 •exp(-L269) = 2.81 lb/in2 ^ HL

2 2 81 " -2=1.243 0.0361 • 48

^=(TKT'=0-

8 0 4

For catenary sag, Y/L « 0.215 and Ycat « 10.3 in. From Fig. 4.73, S/L « 5.12 (Y/L) or S/L « 1.10 = (Wh0)-1. For parabolic sag, assume: ^ = £ * 1.243 ^L 4Y Therefore, Y/L « 0.201 and Ypara « 9.65 in1. The sheet sag has increased by about 700% in 300C. Now consider the effect at 235°C. Q235 = 2.31 lb/in2. For catenary sag, 2To/îL = 0.663 and r| = 1.508. For catenary sag, Y/L « 0.463 or Ycat = 22.2 in. For parabolic sag, Y/L « 0.36 or Ypara «17.3 in. The extent of sag has more than doubled in 5°C. 1

This value is suspect since \i has not been adjusted by the 10% decrease in sheet thickness

Example 4.16 The Importance of Temperature-Dependent Hot Strength on Sag Consider two polymers exhibiting the same extent of sag of 9.65 in for a 48-in sheet span at 2300C as given in Example 4.15. Assume that Polymer A has an elongational energy of activation value of 20,000 and B has a value of 10,000. Determine the extent of sag at 235°C.

a

A,235 = 2.31 in Example 4.15. For catenary sag, 2To/uL = 0.663 and r| = 1.508. For catenary sag, Y/L ^ 0.463 or Ycat = 22.2 in. <jB235 = 2.55 from Example 4.15

For catenary sag, Y/L w 0.305 and Ycat « 14.6 in. The results are tabulated as follows:

Polymer A Polymer B

(Y/L) 230

(Y) 230

(Y/L) 2 3 5

0.201 0.201

9.65 9.65

0.463 0.305

(Y) 235 22.2 14.6

It is apparent that Polymer B is substantially stiffer at 235°C than Polymer A and that this should be visibly apparent in the relative extents of sag. Some interesting insights to sheet sag are obtained by reviewing the parabolic and catenary models. For example, the tension on the sheet increases in proportion to the sheet thickness—through the value for JJ,—and in proportion to the square of the span1. S/L varies between 1.0 and 2Y/L as Y/L varies between 0 and oo (Fig. 4.69). Note that the thickness of the sheet decreases from a value of h for Y/L = 0 to a value of h/2 or S/L = 2 at a value of Y/L « 0.82. When Y/L « 1.9, S/L « 4 and the sagged sheet thickness is about 1/4 its original thickness. But, minor sagging does not appreciably reduce the sheet thickness. Even when the sheet has sagged Y/L = 0.2, S/L is just 1.1 and the sagged sheet thickness is still more than 90% of its original thickness. Sag—A Comment The parabolic model is simpler to use than the catenary model and satisfactory for small values of Y/L. For larger values of Y/L, the catenary model is more accurate. The catenary and parabolic sag models are simplistic. They assume that the sheet is 1

2

As noted in Equation 4.85 for initial sheet sag, the extent of sag was proportional to the fourth-power of the span. Note that this assumes that the sheet is thinning uniformly across the sagging span. In reality, the tension on the sheet is varying from the minimum at mid-span to maximum at the clamped edges. As a result, an isothermal sheet should stretch more in the clamped edges. In general, the sheet tends to be cooler in the vicinity of the clamped edges, and so uniform sheet stretching is a reasonable assumption.

essentially infinite in length. Large deformation of two-dimensional sheet and axisymmetric sheet is best solved using finite element analysis, as described in Chapter 7, on part design.

4.10 References 1. J. Florian, Practical Thermoforming: Principles and Applications, Marcel Dekker, Inc., New York (1987), pp. 122-125. 2. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich (1993), p. 104. 3. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich, (1993). 4. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich (1993), p. 196. 5. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich (1993), p. 293. 6. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich (1993), Figure 3.4, p. 193. 7. C.J.S. Petrie, Elongational Flows, Pitman, London (1979), p. 93, p. 4. 8. C D . Denson, "Implications of Extensional Flow in Polymer Fabrication Processes", Polym. Eng. Sci., 13 (1973), p. 125. 9. V.E. Malpass and C H . White, "Laboratory Comparison of the Thermoforming Properties of Two ABS Materials", in P.F. Bruins, Ed., Basic Principles of Thermoforming, Gordon and Breach, New York (1973), p. 103. 10. V.E. Malpass and J.T. Kempthorn, "Comparison of Polyolefin Thermoforming by Hot Tensile Testing", SPE ANTEC Tech. Papers, 32 (1986), p. 63. 11. H. Saechtling, International Plastics Handbook for the Technologist, Engineer and User, 2nd Ed., Hanser Publishers, Munich (1987), Figure HOA, p. 417. 12. H. Saechtling, International Plastics Handbook for the Technologist, Engineer and User, 2nd Ed., Hanser Publishers, Munich (1987), Figure HOB, p. 417. 13. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), p. 506. 14. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), p. 208. 15. V.E. Malpass and C H . White, "Laboratory Comparison of the Thermoforming Properties of Two ABS Materials", in P.F. Bruins, Ed., Basic Principles of Thermoforming, Gordon and Breach, New York (1973), p. 105. 16. R.C Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich (1993), Section 6.5, "Solid Mechanical Properties: Long-Time Tests". 17. R.L. Harris and P.F. Bruins, "A New Technique for Predicting Optimum Thermoforming Conditions", in P.F. Bruins, Ed., Basic Principles of Thermoforming, Gordon and Breach, New York (1973), p. 81. 18. D. Weinand, "Modellbildung zum Aufheizen und Verstrecken beim Thermoformen", DoktorIngenieur Dissertation, Rheinisch-Westfalische Technische Hochschule Aaachen (Institut fur Kunststoffverarbeitung) (1987), BiId 5.1. 19. D. Weinand, "Modellbildung zum Aufheizen und Verstrecken beim Thermoformen", DoktorIngenieur Dissertation, Rheinisch-Westfalische Technischen Hochschule Aaachen (Institut fur Kunststoffverarbeitung) (1987), pp. 74-79. 20. J.M. Dealy, "On the Relationship Between Extensional Viscosities for Uniaxial and Biaxial Extension", Trans. Soc. RheoL, 77(1973), p. 255.

21. C D . Denson and R J . Gallo, "Measurement on the Biaxial Extension Viscosity of Bulk Polymers: The Inflation of a Thin Polymer Sheet", Polym. Eng. ScL, 11 (1971), p. 174. 22. D.D. Joye, G.W. Poehlein and C D . Denson, "A Bubble Inflation Technique for the Measurement of Viscoelastic Properties in Equal Biaxial Extensional Flow. II", Trans. Soc. Rheol., 17 (1973), p. 287. 23. D.D. Joye, G.W. Poehlein and C D . Denson, "A Bubble Inflation Technique for the Measurement of Viscoelastic Properties in Equal Biaxial Extensional Flow", Trans. Soc. Rheol., 17 (1973), p. 421. 24. R J . Arenz, R.F. Landel and K. Tsuge, "Miniature Load-Cell Instrumentation for Finite-Deformation Biaxial Testing of Elastomers", Exp. Mech., 15 (1975), pp. 114-120. 25. S. Peng and R. Landel, "Viscoelastic Response of a Highly Filled Polymer", NASA Tech. Brief., 16:3 (Mar 1992), p. 135. 26. N.G. McCrum, C P . Buckley, and CB. Bucknall, Principles of Polymer Engineering, Oxford University Press, Oxford (1988), Figure 7.13, p. 281. 27. J.M. McKelvey, Polymer Processing, John Wiley & Sons, New York (1962), p. 45. 28. R.C Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich (1993), pp. 565-569. 29. R.C Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich (1993), Figure 3.28. 30. R.C Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich (1993), Figure 3.29. 31. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich (1993), Figure 3.40. 32. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich (1993), Figure 3.42. 33. G. Kampf, Characterization of Plastics by Physical Methods: Experimental Techniques and Practical Application, Hanser Publishers, Munich (1986), Chapter 4, "Thermoanalytic Methods". 34. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 371, p. 427. 35. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 113, p. 194. 36. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 11, p. 39. 37. CW. Macosko and J.M. Lorntson, "The Rheology of Two Blow Molding Polyethylenes", SPE ANTEC Tech. Papers, 19 (1973), p. 461. 38. Z. Tadmor and C G . Gogos, Principles of Polymer Processing, John Wiley & Sons, New York (1979), "Elongational Flows", Section 6.8, p. 184. 39. R.B. Bird, R.C Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. 1, John Wiley & Sons, New York (1977), p. 451. 40. J.L. White, "Theoretical Considerations of Biaxial Stretching of Viscoelastic Fluid Sheets with Application to Plastic Sheet Forming", Rheol. Acta, 14 (1975), p. 600. 41. CJ.S. Petrie, Elongational Flows, Pitman, London (1979), p. 93, Section 3.3, "Spinning and Related Flows". 42. J. Meissner, "A Rheometer for Investigation of Deformation-Mechanical Properties of Plastic Melts Under Denned Extensional Straining", Rheol. Acta, 8 (1969), p. 78. 43. C D . Han and J.Y. Park, "Studies on Blown Film Extension. I: Experimental Determination of Elongational Viscosity", J. Appl. Polym. Sci., 19 (1975), p. 3257. 44. L. Erwin, H. Gonzolez and M. Pollock, "Analysis and Experiments in Blow Molding of Oriented PET Bottles", SPE ANTEC Tech. Papers, 29 (1983), p. 807. 45. J.M. Funt, "An Analysis of the Rheology of Polypropylene Below Its Melting Point", Polym. Eng. Sci., 75(1975), p. 817. 46. K.C Hoover and R.W. Tock, "Experimental Studies on the Biaxial Extensional Viscosity of Polypropylene", Polym. Eng. Sci., 1(5(1976), p. 82. 47. S. Bahadur, "Strain Hardening Equation and the Prediction of Tensile Strength of Rolled Polymers", Polym. Eng. Sci., 13 (1973), p. 266.

48. P.I. Vincent, "Short-Term Strength and Impact Behaviour", in R.M. Ogorkiewicz, Ed., Thermoplastics: Properties and Design, John Wiley & Sons, Ltd., London (1974), p. 72. 49. M.O. Lai and D.L. Holt, "The Extensional Flow of Poly(Methyl Methacrylate) and High-Impact Polystyrene at Thermoforming Temperatures", J. Appl. Polym. Sci., 19 (1975), p. 1209. 50. M.O. Lai and D.L. Holt, "Thickness Variation in the Thermoforming of Poly(Methyl Methacrylate) and High-Impact Polystyrene Sheets", J. Appl. Polym. Sci., 19 (1975), p. 1805. 51. D. Hylton, Laboratory Techniques for Predicting Material Thermoformability: A Review, SPE ANTEC Tech. Papers, 57(1990), pp. 580-583. 52. A.S. Lodge, Elastic Liquids, Academic Press, London (1964), p. 99. 53. L.R. Schmidt and J.F. Carley, "Biaxial Stretching of Heat-Softened Plastic Sheets: Experiments and Results", Polym. Eng. Sci., 75(1975), p. 51. 54. T. Alfrey, Jr., "Plastics Processing and Fabrication Problems Involving Membranes and Rotational Symmetry", SPE Trans., 5:4 (Apr 1965), p. 68. 55. J.A. Brydson, Flow Properties of Polymer Melts, Van Nostrand Reinhold, New York (1970), pp. 18-19. 56. R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. 1, John Wiley & Sons, New York (1977), p. 83. 57. C.J.S. Petrie, Elongational Flows, Pitman, London (1979), p. 65. 58. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich (1993), p. 655. 59. L.R. Schmidt, Biaxial Stretching of Heat-Softened Plastic Sheets, PhD Thesis, University of Colorado, Boulder CO (1972). 60. H. Gross and G. Menges, "Influence of Thermoforming Parameters on the Properties of Thermoformed PP", SPE ANTEC Tech. Papers, 28 (1982), p. 840. 61. R.W. Ogden, "Large Deformation Isotropic Elasticity—on the Correlation of Theory and Experiment for Incompressible Rubberlike Solids", Proc. Roy. Soc. London A326 (1912), pp. 565-584. 62. L.R.G. Treloar, "Elasticity of a Network of Long-Chain Molecules", Trans. Faraday Soc, 39 (1943), pp. 241f. 63. L.R.G. Treloar, The Physics of Rubber Elasticity, Oxford University Press, Oxford (1958). 64. R.S. Rivlin and D.W. Saunders, "Free Energy of Deformation of Vulcanized Rubber", Trans. R. Soc. Lond., A243 (1951), pp. 25If. 65. J.G. Williams, "A Method of Calculation for Thermoforming Plastics Sheets", J. Strain Anal., 5(1970), p. 49. 66. M. Mooney, "Theory of Large Elastic Deformations", J. Appl. Phys, 11 (1940), pp. 582-592. 67. HJ. Warnecke and B. Frankenhauser, "Montage von Kunststoffschlauchen mit Industrierobotern", Kunststoffe, 78 (1988), pp. 440-444. 68. L.R. Schmidt and J.F. Carley, "Biaxial Stretching of Heat-Softened Plastic Sheets Using an Inflation Technique", Int. J. Engng. Sci., 13 (1975), p. 563. 69. J.G. Williams, "A Method of Calculation for Thermoforming Plastics Sheets", J. Strain Anal., 5(1970), p. 52. 70. L.R.G. Treloar, The Physics of Rubber Elasticity, Oxford University Press, Oxford (1958), p. 162. 71. L.R.G. Treloar, The Physics of Rubber Elasticity, Oxford University Press, Oxford (1958), p. 169. 72. HJ. Warnecke and B. Frankenhauser, "Montage von Kunststoffschlauchen mit Industrierobotern", Kunststoffe, 78 (1988), Figure 5, pp. 440-444. 73. R.W. Ogden, Non-Linear Elastic Deformations, Ellis Horwood Ltd., West Sussex, England (1984). 74. W. Song, "Large Deformation Finite Element Analysis for Polymer Forming Processes", PhD Dissertation, McMaster University, Hamilton ON Canada (1993), Section 4.2.1. 75. K. Kouba, O. Bartos, and J. Vlachopoulos, "Computer Simulation of Thermoforming in Complex Shapes", Poly. Eng. Sci., 32 (1992), pp. 699-704. 76. A.E. Green and J.E. Adkins, Large Elastic Deformations and Non-Linear Continuum Mechanics, Oxford (1960), p. 26. 77. A.E. Green and J.E. Adkins, Large Elastic Deformations and Non-Linear Continuum Mechanics, Oxford (1960), p. 85.

78. B.D. Coleman, H. Markovitz and W. Noll, Viscometric Flows of Non-Newtonian Fluids, Springer, New York (1966), pp. 37-41. 79. K. Kouba and J. Vlachopoulos, T-FORMCAD: A Finite Element Software Package for Thermoforming and Blow Molding, Accuform Co., Ltd, and CAPPA-D, McMaster University, Hamilton ON Canada L8S 4L7. 80. M.H. Wagner, "A Constitutive Analysis of Extensional Flows of Polyisobutylene", J. Rheol., 34 (1990), pp. 943-958. 81. R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids. Volume 1. Fluid Mechanics, John Wiley & Sons, New York (1977), p. 430. 82. K. Kouba and J. Vlachopoulos, "Modeling of 3D Thermoforming", SPE ANTEC Tech. Papers, 3* (1992), pp. 114-116. 83. R.I. Tanner, Engineering Rheology, Clarendon Press, Oxford (1985). 84. R.B. Bird, R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids. Volume 1. Fluid Mechanics, John Wiley & Sons, New York (1977), p. 107. 85. A. Wineman, "On Axisymmetric Deformations of Nonlinear Viscoelastic Membranes", J. Non-Newt. Fluid Mech., 4 (1978), pp. 249-260. 86. A. Wineman, "On the Simultaneous Elongation and Inflation of a Tubular Membrane of BKZ Fluid", J. Non-Newt. Fluid Mech., 6(1979), pp. 111-125. 87. H. Saechtling, International Plastics Handbook for the Technologist, Engineer and User, 2nd Ed., Hanser Publishers, Munich (1987), Figure 109, p. 416. 88. H. Saechtling, International Plastics Handbook for the Technologist, Engineer and User, 2nd Ed., Hanser Publishers, Munich (1987), Figure 111, p. 418. 89. H. Saechtling, International Plastics Handbook for the Technologist, Engineer and User, 2nd Ed., Hanser Publishers, Munich (1987), Figure 112, p. 419. 90. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 14. 91. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 15. 92. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 42. 93. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 129, p. 206. 94. D. Weinand, "Modellbildung zum Aufheizen und Verstrecken beim Thermoformen", DoktorIngenieur Dissertation, Rheinisch-Westfalische Technische Hochschule Aaachen (Institut fur Kunststoffverarbeitung (1987), BiId 5.2. 95. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich (1993), Figure 6.25. 96. D. Weinand, "Modellbildung zum Aufheizen und Verstrecken beim Thermoformen", DoktorIngenieur Dissertation, Rheinisch-Westfalische Technische Hochschule Aaachen (Institut fur Kunststoffverarbeitung (1987), BiId 5.3. 97. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 153, p. 228. 98. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, Munich, 1993, Figure 6.29. 99. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 171, p. 245. 100. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 242, p. 313. 101. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 260, p. 328. 102. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 295, p. 374. 103. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 296, p. 374.

104. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 403, p. 456. 105. H. Domininghaus, Plastics for Engineers: Materials, Properties, Applications, Hanser Publishers, Munich (1993), Figure 492, p. 534. 106. R.H. Finney and A. Kumar, "Development of Material Constants for Nonlinear Finite-Element Analysis", Rubber Chem. Tech., 61 (1988), pp. 879-891. 107. J.G. Williams, Stress Analysis of Polymers, John Wiley & Sons, New York (1973), pp. 211-220. 108. L.R.G. Treloar, The Physics of Rubber Elasticity, Oxford University Press, Oxford (1958), p. 170. 109. CJ.S. Petrie, Elongational Flows, Pitman, London (1979), p. 90, p. 210. 110. S. Middleman, Fundamentals of Polymer Processing, McGraw-Hill Book Co., New York (1977), p. 48. 111. R. Allard, J.-M. Charrier, A. Ghosh, M. Marangou, M.E. Ryan, S. Shrivastava and R. Wu., "An Engineering Study of the Thermoforming Process: Experimental and Theoretical Considerations", Paper presented at First Annual Meeting, Polym. Proc. Soc, Akron OH, 28-29 March 1984. 112. MJ. Stephenson and M.E. Ryan, "Experimental Study of Thermoforming Dynamics", SPE ANTEC Tech. Papers, 40 (1994), pp. 844-849. 113. S. Middleman, Fundamentals of Polymer Processing, McGraw-Hill Book Co., New York (1977), p. 49. 114. R J . Roark and W.C. Young, Formulas for Stress and Strain, Fifth Ed., McGraw-Hill Book Co., New York (1975), Table 26, p. 386. 115. R J . Roark and W.C. Young, Formulas for Stress and Strain, Fifth Ed., McGraw-Hill Book Co., New York (1975), p. 363, Example 10b. 116. J.L. Meriam, Mechanics: Part I Statics, John Wiley and Sons, Inc., Chapman & Hall, Ltd., London (1952), Figure 52, p. 184. 117. J.L. Meriam, Mechanics: Part I Statics, John Wiley and Sons, Inc., Chapman & Hall, Ltd., London, (1952), Figure 51. 118. J.M. McKelvey, Polymer Processing, John Wiley & Sons, New York (1962), p. 41. 119. M.C. Boyce and E.M. Arruda, "An Experimental and Analytical Investigation of the Large Strain Compressive and Tensile Response of Glassy Polymers", Polym. Eng. ScL, 30 (1990), pp. 1288-1298. 120. H.G. DeLorenzi and H.F. Nied, "Finite Element Simulation of Thermoforming and Blow Molding", in A.I. Isayev, Ed., Modeling of Polymer Processing, Carl Hanser Verlag, Munich (1991), Chapter 5. 121. W.N. Song, F.A. Mizra and J. Vlachopoulos, "Finite Element Analysis of Inflation of An Axisymmetric Sheet of Finite Thickness", J. Rheol., 35(1991), pp. 93-111. 122. K. Kouba, O. Bartos and J. Vlachopoulos, "Computer Simulation of Thermoforming in Complex Shapes", Polym. Eng. Sci., 32 (1992), pp. 699-704. 123. H.F. Nied, CA. Taylor and H.D. DeLorenzi, "Three-Dimensional Finite Element Simulation of Thermoforming", Polym. Eng. Sci., 30(1990), pp. 1314-1322. 124. S.D. Batterman and J.L. Bassani, "Yielding, Anisotropy, and Deformation Processing of Polymers", Polym Eng. Sci., 30 (1990), PP. 1281-1287. 125. S. Bahadur, "Strain Hardening Equation and the Prediction of Tensile Strength of Rolled Polymers", Polym. Eng. Sci., 13 (1973), pp. 266-272. 126. MJ. Miles and N J . Mills, "The Deep Drawing of Thermoplastics", Polym. Eng. Sci., 17 (1977), pp. 101-110. 127. G.W. Halldin, "Solid-Phase Flow Behavior of Polymers", Polym. Eng. Sci., 25 (1985), pp. 323-331. 128. K. Nakamura, K. Imada and M. Takayanagi, "Solid State Extrusion of Isotactic Polypropylene Through a Tapered Die. I. Phenomenological Analysis", Int. J. Polym. Mat., 2(1972), pp. 71-88. 129. G. Menges and D. Weinand, "Modellierung des Verstreckprozesses beim Warmformen", Kunststoffe 78 (1988), pp. 456-460. 130. K. Kouba, M.O. Ghafur, J. Vlachopoulos and W.P. Haessly, "Some New Results in Modelling of Thermoforming", SPE ANTEC Tech. Papers, 40 (1994), pp. 850-853.

131. L.R. Schmidt and J.F. Carley, "Biaxial Stretching of Heat-Softened Plastic Sheets: Experiments and Results", Polym. Eng. Sci., 75(1975), pp. 51-62. 132. L.R. Schmidt and J.F. Carley, "Biaxial Stretching of Heat-Softened Plastic Sheets Using an Inflation Technique", Int. J. Eng. Sci., 13 (1975), pp. 563-568. 133. D. Hylton, "Laboratory Techniques for Predicting Material Thermoformability: A Review", SPE ANTEC Tech. Papers, 37(1990), pp. 580-583. 134. C. Cruz, Jr., "The Sag Process in Modified Polypropylene", SPE ANTEC Tech. Papers, 40 (1994), pp. 854-858. 135. J.L. Throne, Thermo forming, Carl Hanser Verlag, Munich (1987), pp. 110-112.

Appendix 4.1 Biaxial Stretching of an Elastic Membrane1 Consider the biaxial extension of a rubbery solid membrane of initial thickness ho and radius a, inflated with a differential pressure P. The extensional stress-strain equations for a Mooney-type polymer are:

a, = 2(C10+ Q 1 - A l ) - ( V - J ^ i ) CT6 = 2(C10+ C 0 1 - V ) - ( ^ - 5 ^ 1 )

(4.1.1) (4-1-2)

C01 and C10 are the Mooney constants, and 1 and 0 are the in-sheet or meridional and angular or hoop direction coordinates (Fig. 4.70). N is the force acting on the membrane, N = a • h, a is the local stress and h is the local thickness. The radius from the center axis is r, ^h = h/ho, Xe = r/ro, and X1 = (kh- Xd)~l. The angle of the membrane from the horizontal is p and 5 is the cap height above the horizontal plane. Thus r/R = sin (3, where R is the radius of the spherical cap (Fig. 4.71). At the top of the dome: (4.1.3) For C01 « C 1 0 and Xh Xh» 1, or for large deformations of a neo-Hookean solid: (4.1.4) (4.1.5) The forces acting on the membrane are: (4.1.6) (4.1.7) Now N1 • N 0 = (2C10ho)2 = N 0 , a constant. Further, stress equilibrium yields N1 = N 0 . Therefore, N 0 = N1. A hoop stress condition is: (4.1.8) 1

Adapted from Williams [107], by permission of Ellis Horwood, Ltd., copyright owner.

Figure 4.70 Membrane stretching geometry. Adapted from [107] and used with permission of Ellis Horwood, Ltd., copyright owner

Figure 4.71 Geometry of stretched cap or dome. Adapted from [107] and used with permission of Ellis Horwood, Ltd., copyright owner

As a result: (4.1.9)

And since R = R(a,5) above: (4.1.10)

where G is the elastic modulus of the new-Hookean polymer, G = C10. This is the neo-Hookean relationship between applied pneumatic pressure, P, and the extent of deformation, given as 5/a.

5 Cooling and Trimming the Part 5.1 5.2 5.3 5.4

5.5

5.6 5.7

5.8

5.9

5.10

5.11 5.12 5.13

5.14

Introduction Overall Cooling Heat Balance Cooling the Formed Shape Steady State Heat Balance Interfacial Resistance Shape Factor Convection Heat Transfer Coefficient Cyclic Heat Balance Cooling the Free Surface of the Sheet Cooling Thin Sheet in Ambient Air Transient Heat Removal From the Sheet Quiescent Ambient Air Moving Ambient Air Cooling on Nonmetallic Molds Transient Heat Transfer During Sheet Cooling on the Mold Surface—Computer Models Interfacial Air Shrinkage Unconstrained Shrinkage Constrained Shrinkage Trimming Trimming Heavy-Gage Parts Trimming Thin-Gage Parts Mechanics of Cutting The Trim Region Registering the Trim Site The Nature of the Cut Fracture Mechanics Mechanical Chipping Multiple-Edged Tool or Toothed Saw Performance Abrasive Cut-Off Wheel Toothless or Shear and Compression Cutting Fracture Mechanics in Trimming Nibbling Brittleness, Orientation and Trim Temperature Steel Rule Die Resharpening Tabbing and Notching Punch and Die Trimming Forged and Machined Dies Drilling Other Cutting Techniques Thermal Cutting Water Jet Cutting Trimming—A Summary

5.15 References

5.1

Introduction

Once the part has been heated and formed to shape by contact with the cool mold surface, it must be cooled and rigidified. Then the part and its web must be separated by trimming. When thermoforming a reactive polymer such as thermosetting polyurethane or a crystallizing one such as nucleated CPET1, the formed shape is rigidified by holding it against a heated mold to continue the crosslinking reaction or crystallization. In most cases, rigidifying implies cooling while in contact with a colder mold. For automatic thin-gage formers, the molds are usually actively cooled with water flowing through channels. Free surfaces of medium- and heavy-gage sheet are frequently cooled with forced air, water mist or water spray. For amorphous polymers, cooling of the formed part against a near-isothermal mold rarely controls the overall thermoforming cycle. For crystalline and crystallizing polymers such as PET, PP and HDPE, the cooling cycle can be long and can govern overall cycle time. When the sheet has cooled sufficiently to retain its shape and, to a large extent, its dimension, it is stripped from the mold and transferred to a trimming station, where the web is separated from the product. Requisite holes, slots and cut-outs are drilled, milled or burned into the part at this time. Thin-gage roll-fed sheet can be either trimmed on the mold surface immediately after forming or on an in-line mechanical trimming press. Heavier-gage formed sheet is usually removed from the mold and manually or mechanically trimmed on a remote station. Trimming is really a solid phase mechanical process of crack propagation by brittle or ductile fracture. Care is taken when trimming brittle polymers such as PS or PMMA to minimize microcracks. With brittle polymers, the very fine sander, saw or microcrack dust generated by mechanical fracture can be a serious problem. Other polymers such as PP and PET are quite tough and require special cutting dies. Dull steel-rule dies cause fibers or hairs at the cutting edge of thin-gage fiber-forming polymers such as PET and PP. Slowly crystallizing polymers pose registry problems when in-line trimming presses are used. The speed of trim cutting and the nature of the cutting surface control the rate of crack propagation through the plastic. There are no exhaustive studies of the unique trimming and cutting characteristics of thermoformed polymers and so much information must be inferred from other sources including extensive studies on the machining of plastics.

5.2

Overall Cooling Heat Balance

As a first approximation, as the sheet touches the mold, it is assumed to be at an average, equilibrated temperature, Tequil, as described in Section 3.13. When the 1

The unique processing conditions for crystallizing polyethylene terephthalate (CPET) are described in detail in Chapter 9, Advanced Thermoforming Processes.

average sheet temperature reaches the set temperature, Tset, as given in Table 2.5, the sheet is assumed to be sufficiently rigid to be removed from the mold surface. Typically, the amorphous polymer set temperature is about 200C or 400F below its glass transition temperature, Tg. The crystalline polymer set temperature is about 200C or 400F below its melting temperature, Tmelt. During the cooling time, the mold temperature is assumed to be essentially constant at Tmold. The amount of heat to be removed by the coolant flowing through the mold is given as:

(5.1) where V* is the volume of plastic, p is its density and cp is its heat capacity1. Heat is removed from the free sheet surface by convection to the environmental air. Heat is removed from the sheet surface against the mold surface by conduction through the mold to the coolant. The coolant fluid removes the heat by convection. The heat load at any point on the mold surface depends on the sheet thickness, tlocal, at that point. Sheet thickness, as noted in Chapter 4, is not uniform across the mold surface. The heat load at any point is given as: (5.2)

This is the heat to be removed from a given region during the cooling portion of the total cycle, 9cool. The local heat flux then is: (5.3)

The units on q['ocal are kW/m2 or Btu/ft2 • h •0 F. The total heat load during this time is given as: (5.4)

The total heat load per unit time on a steady-state process is given as: (5.5) where N is the number of parts produced per unit time. The units on Qsteady state are kW or Btu/h. At steady-state conditions, this energy is removed by the coolant system and by convection to the environmental air. Mold, coolant and air temperatures increase until the steady state is reached.

1

The last equality relates the amount of heat removed to the differential enthalpy, AH, of the polymer. This expression should be used if the polymer is crystalline and molten during the forming step and is solidifying during the cooling step.

5.3

Cooling the Formed Shape

Consider a typical cooling step. The sheet of variable thickness but known temperature is pressed against a slightly irregular surface of a mold. The properties of the mold material are known and uniform throughout its volume, Chapter 6. Coolant of known properties flows through uniformly spaced channels in the mold. The free surface of the part is also cooled. At any instant, the temperature profile through the various layers of material is as shown in schematic in Fig. 5.1. The rate at which the energy is removed from the plastic to ambient air and coolant depends on the sum of resistances to heat transfer through each of these layers. Heat removal by the coolant is the primary way of cooling the formed part. Transfer to the environmental air is a secondary method but it can be quite important when trying to optimize cycle time. There are two aspects to heat removal from the formed polymer sheet to the coolant. The first deals with the overall heat transfer at steady state conditions. The second focuses on certain aspects of cyclical transient heat transfer. Mold Surface Irregularities

Plastic Sheet

Coolant Film Resistance

Air Gap

Coolant * Tp

T0

Free Surface Film Resistance

T

L

o

Ambient Environment Mold

Series Thermal Resistances Figure 5.1 Schematic of various thermal resistances for sheet cooling against thermoforming mold

Baffle

Mold Insert

Flow Channel

Mold Base

Coolant

Figure 5.2 Typical serpentine coolant flow channel through thermoforming mold

5.4

Steady State Heat Balance

Consider the limiting case where all the energy is transferred directly to the mold and thence to the coolant. Typically, coolant lines are drilled or cast on a discrete, regular basis parallel to the mold surface (Fig. 5.2). Heat removal by coolant depends on convection heat transfer, or fluid motion1. The total amount of energy removed by the coolant embedded in the mold is given as: Q = UA AT

(5.6)

where U is the overall heat transfer coefficient, A is the coolant surface area and AT is the increase in coolant temperature between inlet and outlet portions of the flow channel. The overall heat transfer coefficient includes all flow resistances between the sheet and the coolant. It is usually written as:

where R1 represents the ith resistance to heat transfer. As seen in Fig. 5.1, for flowing fluids, there is a convective film resistance at the conduit surface, l/7iDhc, where TTD is the circumference of the conduit. If the coolant fluid is not kept clean, the coolant channel can become coated with residue, thus increasing thermal resistance. This 1

This section deals only with the convection heat transfer coefficient of coolant flowing through lines in the mold. Section 6.4 considers coolant pressure drop-flow rate relationships for the specification of coolant line size.

Table 5.1 Fouling Factors for Coolant Lines1 Coolant

Condition

Fouling factor (Btu/ft-h^F)-1 Velocity < 3 ft/s

Velocity > 3 ft/s

Treated make-up cooling tower water

Water < 125°F

0.001

0.001

Treated make-up cooling tower water City water City water River water River water Treated boiler feedwater Treated boiler feedwater Industrial heat transfer oil Ethylene glycol Glycerine-water Brine Brine Steam

Water > 125°F

0.002

0.002

Water Water Water Water Water

125°F 125°F 125°F 125°F 125°F

0.001 0.002 0.002 0.003 0.001

0.001 0.002 0.001 0.002 0.0005

Water > 125°F

0.001

0.001

0.001

0.001

0.001 0.002 0.002 0.003 0.001

0.001 0.001 0.001 0.002 0

1

< > < >
125°F

Information extracted from [1] by permission of copyright holder

resistance is called a fouling factor, ff. Fouling factors are given in Table 5.1 [I]. The resistance through the mold depends on the relative shapes of the mold surface and the coolant channels, and is usually described as 1/Skm, where S is a shape factor and km is the thermal conductivity of the mold material. Since polymer sheet does not press tightly against the mold surface, there is a conductive resistance owing to trapped air, l/ha. Although there may be other thermal resistances, these are the primary ones. So the overall heat transfer coefficient, U, is written as:

Interfacial Resistance In most heat transfer processes, intimate or perfect contact between the hot and cold solids is assumed. Imperfect contact causes resistance to heat flow. Mold surface waviness and microscopic roughness or asperities reduce physical contact (Fig. 5.3). Increasing pressure against the sheet increases physical contact and reduces the resistance to heat transfer. In general, energy is transmitted across the interstices by a combination of:

Cold, Rigid or Heavy-Gage Sheet Trapped Air

Cold Mold

Hot, Flexible or Thin-Gage Sheet

Moid Asperities

Hot Mold

Figure 5.3 Interfacial resistance between sheet and thermoforming mold surface. Left shows substantial thermal resistance owing to large air gap. Right shows reduced thermal resistance

• • •

Conduction at the asperities, Conduction through the interstitial fluid, and Radiation.

The resistance is thus a function of: •

• •

The contacting material properties such as Relative hardness, Thermal conductivity, Surface roughness and Flatness, The conductivity and pressure of the interstitial fluid, and The pressure applied against the free surface of the sheet.

The interface coefficient, ha, is a measure of thermal resistance across the gap. It is similar in concept to the confection heat transfer in that resistance to heat flow decreases with increasing value of ha. For perfect contact, h a -» oo. In thermoforming, the interstitial fluid is air, perhaps at a substantially reduced pressure. If the interface is a uniform air gap of 8 = 0.025 cm or 0.010 in and air thermal conductivity is kair = 0.029 W/m •0C or 0.0167 Btu/ft • h •0 F, the value for ha is about 114 W/m2 •0C or 20 Btu/ft2 • h •0 F. Contact heat transfer coefficient values between flowing polymer melts and mold surfaces of about ha = 568 W/m2 •0C or 100 Btu/ft2 • h •0 F have been reported in injection molding [2-4]. Similar values are expected here. For two surfaces in contact, ha = h ao p n , where p is the applied pressure and hao depends on the relative waviness and roughness of the two surfaces, Table 5.2 [5]. As seen, n has a value of about 2/3 for both rigid-rigid and rigid-flexible material contact in vacuum. Values for hao are typically 50 times greater in air than in hard vacuum. There are no available data for interfacial resistance during thermoforming of softened plastic sheet against various types of mold surfaces. For hot sheet pressed against a relatively smooth, heated mold surface at a relatively high differential pressure, it is expected that an appropriate value for ha would be about 568 W/m2 •0C or 100 Btu/ft2 • h •0 F. For rapidly cooling and rigidizing plastic sheet pressing against a highly textured, cold mold surface with a modest differential pressure, an appropriate value range for ha of 114 to 284 W/m2 •0C or 20 to

Table 5.2 Contact Resistance and Conductance [5] Ji0 = Ii00P11

(Units on h c = W/m 2 • 0 C or Btu/ft2 • h • 0 F) (Units on p are MPa or lbf/in2) Material contact

Inner layer

Elastic deformation theory Hard-to-hard Hard-to-hard Hard-to-soft

None Vacuum Air Vacuum

n

Contact coefficient, Ji00 (W/m2 • C)

(Btu/ft2 • h • 0F)

35.8 113.6 5680 170

6.3 20 1000 30

2/3 2/3 1/6 2/3

50 Btu/ft2 • h •0 F should be considered for first cooling time estimates. Section 5.6 on computer simulation of the cooling process explores the relative effect of interfacial resistance on time-dependent sheet cooling. Shape Factor For thin metal molds, heat is conducted very rapidly from the plastic to the coolant. For relatively thick molds of: • • • • • •

Plaster, Wood, Epoxy, Glass fiber-reinforced unsaturated polyester resin (FRP), Pressed fiberboard, or Any other nonmetallic material,

heat transfer is slowed by low mold material thermal conductivity. If the coolant system is considered to be coplanar with the mold surface (Fig. 5.4), the resistance to heat transfer per unit length (L = 1), across a mold D units thick is given as: Rn, = ^ -

(5.9)

where km is the mold material thermal conductivity. Thermal conductivity values for many mold materials are given in Table 2.7. For round discrete conduits (Fig. 5.5) [6], a shape factor, S, is used: (5.10) where S is given by: (5.11)

Mold

Mold

Coolant

Figure 5.4 Schematic of coplanar coolant flow channel and mold surface, concept frequently called flooded cooling

Hot Sheet

Coolant Channel

P

D

Mold

d Figure 5.5 Geometric factors for mold shape factor analysis

Figure 5.6 gives this equation in graphic form. It is apparent that the thermal resistance of the mold decreases with increasing value of S, which is achieved with many large-diameter coolant lines placed relatively close to the mold surface. The typical value range for S is 2 < S < 3. Example 5.1 illustrates the relative effects of these parameters on mold thermal resistance. Example 5.1 Shape Factors and Mold Thermal Resistance Determine the relative thermal resistances for the following two molds: Mold 1: Thin-walled aluminum mold with km = 131 Btu/ft h 0F, having d= 1/2-in water lines on P = 2 in centers, with the center line being D=I in from the mold surface.

Mold 2: Thick-walled plaster mold with km= LO Btu/ft h 0F, having d= 1/2-in water lines on P = 4 in centers, with the centerline being D = 2 in from the mold surface. Mold 1 P/d = 4, D/d = 2. From Fig. 5.6, S = 2.

Mold 2 P/d = 8, D/d = 4. From Fig. 5.6, S = 1.6.

^ i = U T T o = 0-625 The plaster mold has more than 160 times the thermal resistance to heat transfer than the aluminum mold.

Shape Factor, S

D/d = 1

Coolant Channel Spacing to Diameter Ratio, P/d Figure 5.6 Effect of coolant line location on mold shape factor

Convection Heat Transfer Coefficient The metal molds on automatic, roll-fed thin-gage thermoformers and on many heavy-gage forming operations are actively cooled, with water being the primary

coolant. There is a thermal resistance between the cool bulk flowing fluid and the warmer tube wall (Fig. 5.1). The primary dimensionless group used in fluid mechanics is the Reynolds number, Re: Re = ^ P

(5.12)

where D is the tube diameter, v is the fluid velocity, p is the density of the fluid and |i is its Newtonian viscosity [7]. The Reynolds number is the ratio of inertial to viscous forces for the fluid. Slowly moving fluids are laminar when Re < 2000. Convection heat transfer to slowly moving fluids is poor. Rapidly flowing fluids are fully turbulent when Re > 10,000 and heat transfer is very rapid. Example 5.2 illustrates the interaction between flow rate and Reynolds number. Example 5.2 Water as Coolant—Flow Rates Consider 21° C or 700F water flowing through 0.5-in or 1.27-cm diameter coolant channels. Determine the Reynolds number and the flow characteristic if the velocity is a) 0.52 ft I s or 0.16 m/s and b) 2.6 ft/s or 0.79 m/s. What are the volumetric flow rates at these velocities?

The water density is 62.4 lb/ft3. The viscosity is 0.658 x 10"3 lbm/ft • s. The Reynolds number is:

«,^_£ft.4^._l_£_3*,.v For v = 0.52 ft/s: Re = 2050 and the water is laminar. For v = 2.6 ft/s: Re = 10,300 and the water is turbulent. The volumetric flow rate is given as: V =^ ! . 4

v = 0 .00136-v^

s

= 0.612-v-^mm

For v = 0.52 ft/s, the flow rate is 0.32 GPM. For v = 2.6 ft/s, the flow rate is 1.6 GPM.

As discussed in Section 3.6, energy interchange between solid surfaces and flowing fluids is by convection. The proportionality between heat flux and thermal driving force is the convection heat transfer coefficient. The convection heat transfer coefficient is obtained from standard heat transfer theory and experiments. There are many methods for calculating values of hc. In general, however, the Chilton-Colburn analogy between resistance to fluid flow and resistance to thermal energy flow yields adequate results [8]. The analogy states: St

.Pr2/3

=

f / 8

( 5

1 3 )

where St is the Stanton number, Pr is the Prandtl number and f is the coefficient of friction or friction factor. The Stanton and Prandtl numbers are:

Table 5.3 Prandtl Number Values For Several Coolants [9,10] Prandtl no.

Coolant

Temperature (0F) (0C)

Air Air

32 100

0 38

0.72 0.72

Steam Steam

212 400

100 204

0.96 0.94

Water Water Water Water Water

32 70 100 150 200

0 21 38 66 93

13.7 6.82 4.52 2.74 1.88

SAE SAE SAE SAE SAE SAE

60 100 150 200 250 300

16 38 66 93 121 149

1170 340 122 62 35 22

Glycerine Glycerine Glycerine Glycerine Glycerine

50 70 85 100 120

10 21 29 38 49

31,000 12,500 5,400 2,500 1,600

Air Air Air

32 300 600

0 150 315

105 150 250 300

40 65 120 150

Light Light Light Light

30 Oil 30 Oil 30 Oil 30 Oil 30 Oil 30 Oil

Oil Oil Oil Oil

0.72 0.71 0.685 340 62 35 22

(5.14) (5.15) where h is the convective heat transfer coefficient, p is the fluid density at the appropriate temperature, cp is the fluid heat capacity, v is the average fluid velocity, v = u/p, is the kinematic viscosity, and a = k/pcp, is the fluid thermal diffusivity. In essence, Pr is the ratio of inertial to thermal properties and St is the ratio of fluid to thermal resistances. Table 5.3 gives appropriate Prandtl number values for several coolants [9,10]. As examples, Pr ^ 7 for room temperature water and Pr « 300 for

10O0C oil coolant. Examples 5.3 and 5.4 show the relative effectiveness of convection heat transfer to water and oil, respectively, as coolants 1 . Example 5.3 Water as Coolant—Temperature Increase A 3 ft x 3 ft x 0.125-in thick plastic sheet is initially at 31 5°F and is to be cooled to 2000F. The sheet specific heat is 0.5 Btu/lb • 0F and the density is 70 Ib I ft3. Twelve sheets are thermoformed per hour. Determine the increase in coolant temperature. Heat load from sheet: Q (per cycle) = pcpV*AT = 574 Btu/cycle Q = 574 • 12 = 6888 Btu/h to be removed by coolant. 0

Consider 70 F water flowing through a 3/4-in diameter coolant channel at v = 4 ft/s. From Example 6.1, the Re = 50,000. The flow is turbulent and f = 0.0248. Pr = 7.02. St • Pr2/3 = f/8 h

^ R ^ = 770Btu/ft2 - h -° F

Consider a 3 ft x 3 ft aluminum mold containing four waterlines spaced evenly. Consider the water lines to be I^ in from the mold surface. Thus P/d = 12 and D/d = 2. From Fig. 5.6, the shape factor, S « 3. The thermal conductivity of aluminum, from Table 2.12, km = 72.5 Btu/ft • h • 0 F. Consider h a = oo. As a result, the total thermal resistance, U is given as: U = 0.75Tr.770 + 3 ^ 7 l 5 2

= 0 0112

-

0

or: U = 89.3 Btu/ft • h • F. Surface area of coolant channels, A = TtDLN, where D is the coolant diameter ( = 3/4-in), L is the channel length ( = 1 0 ft), and N is the number of channels ( = 4). A = 7.85 ft2. The increase in coolant temperature is given as: _ ^ = _6888_ UA 89.3-7.85 This is considered to be a typical temperature increase for turbulent coolant flow.

1

These examples use fluid flow data of Examples 6.1 and 6.2 discussed in detail in Chapter 6 on mold design.

Example 5.4 Oil as Coolant—Temperature Increase A 3 ft x 3 ft x 0.125-in thick plastic sheet is initially at 375°F and is to be cooled to 2000F. The sheet specific heat is 0.5 Btu/lb •0 F and the density is 70 Ib I ft3. Twelve sheets are thermoformed per hour. Determine the increase in coolant temperature. Comment on the results. Heat load from sheet: Q (per cycle) = pcpVAT = 574 Btu/cycle Q = 574 • 12 = 6888 Btu/h to be removed by coolant. Consider 1500F SAE 10-like oil flowing through a 3/4-in diameter coolant channel at v = 4 ft/s. From Example 5.2, the Reynolds number, Re = Dvp = 1925. The flow is laminar and f = 0.0333. From Table 5.3, the Prandtl number, Pr = 290 and Pr 2/3 = 44. St • Pr2/3 = f/8 hc - (pCpV) 2 3 - 38 Btu/ft2 • h • 0 F ~ 8 • (29O) / "" ' Consider a 3 ft x 3 ft aluminum mold containing four waterlines spaced evenly. Consider the water lines to be \\ in from the mold surface. Thus P/d = 12 and D/d = 2. From Fig. 5.6, the shape factor, S « 3. The thermal conductivity of aluminum, from Table 2.12, k m = 72.5 Btu/ft • h • 0 F. Consider h a = oo. As a result, the total thermal resistance, U is given as: U = 0.757T • 38 + 3 • 72.5 = ° ' ! 3 9 or: U = 7.22 Btu/ft2 • h • 0 F. Note that this value is less than one tenth that of water flowing at the same velocity. Surface area of coolant channels, A = 71DLN, where D is the coolant diameter ( = 3/4-in), L is the channel length ( = 1 0 ft), and N is the number of channels ( = 4). A = 7.85 ft2. The increase in coolant temperature is given as:

±_

6888

UA

7.22-7.85

Since the sheet must be cooled to 200 0 F, this is an unacceptable increase in oil temperature. One solution is to increase the oil flow rate. Doubling the flow rate will increase the pressure drop by a factor of just less than four. A second alternative is to increase the number of coolant lines. Doubling the number of coolant lines will increase the coolant surface area but the oil velocity will drop to half its current value, thus driving the flow even deeper into the laminar region. Since the friction factor is inversely proportional to the fluid velocity, the friction factor will double. However, since the heat transfer coefficient is essentially independent of velocity, the heat transfer coefficient will not be affected. In addition, the decreased spacing will

increase the shape factor value, S. As a result, doubling the number of coolant lines results in a more than double increase in the overall heat transfer coefficient and a resulting more than halving of the increase in oil temperature. More exact calculations require knowledge of the roughness and geometric characteristics of the flow channel. In general, the Nusselt number, Nu, a dimensionless convective heat transfer coefficient is a product of the Reynolds number and the Prandtl number, as: Nu = ^ = C Rem Prn

(5.16)

For fully developed turbulent flow in very smooth tubes, C = 0.023, m = 0.8 and n = 0.4, and the equation is called the Dittus-Boelter equation. For developing turbulent flow in smooth tubes, m = 0.8, n = 0.33 and C is given as: / D \0.055

C = 0.036 -

(5.17)

where D is the channel diameter, L is the channel length and 10 < L/D < 400. This equation is usually called the Nusselt equation. For laminar flow in smooth tubes, the Sieder-Tate version is frequently used. It is much more dependent on entrance length effects and so has m = 1/3, n = 1/3 and C as: /DY/3 C =1.86 (5.18) where Re • Pr • (d/L) > 10. Example 5.5 illustrates heat transfer coefficient values obtained from these equations for a relatively simple flow channel design. For noncircular channels, the diameter is replaced with the hydraulic diameter, Dh, given as: Dh=^

(5.19)

where A is the cross-sectional area of the flow channel and P is the wetted perimeter. Example 5.5 Convective Heat Transfer Coefficients for Serpentine Mold Channel Figure 5.7 shows an example of a serpentine mold channel that has a roughness value, e = 0.001 D. Two coolants are to be evaluated—water at 700F and oil at 1500F. Determine the relative heat transfer effectiveness of these coolants in this flow channel. Water as a coolant

v, kinematic viscosity= 1.06 x 10~5 ft2/s p, density = 62.4 lb/ft3

Plug

Mold Base Coolant

Mold Cavity Flow Channel

Figure 5.7 Coolant flow channel around mold insert

k, thermal conductivity = 0.35 Btu/ft • h • 0 F v, average velocity = 4 ft/s Prandtl number = 7 Reynolds number, R e = 15,700 Flow is turbulent The friction factor-Reynolds number equation is: f = 0.0204 + 4.212 • R e " 0 6 4 2 = 0.0289 •

Stanton/Colburn analogy: (P^)_ f p r 2 /3 8 h, convective heat transfer coefficient = 886 Btu/ft2 • h • 0 F Nu, Nusselt number = hD/k = 106 Dittus-Boelter fully developed flow: Nu = 0.023 • Re0-8 • Pr0-4 = 114. h = 958 Btu/ft2 • h • 0 F Nusselt developing flow (L = 20): h =

• e

/ I \0.055

Nu = 0.036 • Re 0 8 • Pr0-33 • —

\4oy

• •

h = 1061 Btu/ft2 h - 0 F Nusselt developing flow (L = 6): N u = 136, h = 1137 B t u / f t 2 - h - 0 F Weighted average: Nu = 131, h = 1095 Btu/ft2 • h • 0 F

= 127.

The effect of developing flow is higher convective heat transfer. Oil as a coolant v, p, k, v,

kinematic viscosity= 1.35 x 10~ 4 ft2/s density = 54.5 lb/ft3 thermal conductivity = 0.082 Btu/ft • h • 0 F average velocity = 4 ft/s

Prandtl number, Pr = 290 Reynolds number, Re = 1235 Flow is laminar The friction factor-Reynolds number equation is: f = 16/Re = 0.0518 •

Stanton/Colburn analogy: 8 h, convective heat transfer coefficient = 58 Btu/ft2 • h • 0 F Nu, Nusselt number = hD/k = 29.5

• • •

Sieder-Tate for developing flow (L = 20): Nu = 1.86 • [Re • Pr • (D/L)] 2/3 = 20.8 h = 41 Btu/ft2 • h - 0 F Sieder-Tate for developing flow (L = 6): Nu = 46.4, h = 91 Btu/ft2 • h • 0 F Averaged developing flow: Nu = 32.1, h = 63 B t u / f t 2 - h - 0 F

Again, convective heat transfer in developing flow is higher than with fully developed flow. The arithmetic is considerably simplified for water at 210C or 70 0 F. For laminar flow, the heat transfer coefficient is obtained from [H]:

hlam = 3.52 ^-j Re 1 / 2 (j-J

(5.20)

where k is the thermal conductivity of the fluid, and D and L are the diameter and length of the flow channel, respectively. For turbulent flow in water at 210C or 70 0 F, the heat transfer coefficient is: /k\

/ r > \ 0.055

hturb = 0.068(^-jRe O8 ^ r J

(5.21)

The relative values for laminar and turbulent water flow in a typical mold are given in Example 5.6.

Example 5.6 Laminar and Turbulent Heat Transfer Determine the relative effect on heat transfer coefficient when the flow rate is increased so that flow moves from laminar to turbulent flow. Consider a serpentine coil imbedded in an epoxy mold, where Lid— 100. From Equations 5.20 and 5.21, determine the relative effect on the Biot number, the relative heat transfer coefficient. Then determine the actual heat transfer coefficients for water flowing in a 0.500-in or 1.27 cm diameter coil. The thermal conductivity of water is 0.35 Btu/ft - h -0F. Then, determine the heat transfer coefficient in the transition region, 2000 < Re < 10,000. For laminar flow, Re < 2000. Consider Re = 2000. From Equation 5.20, the Biot number, Bi = hD/k is given as: hD /D\1/2 Bi = — = 3.52 • Re1/2 • = 3.52 • 20001/2 • (0.01)1/2 = 15.7

k

Vw

¥ or fully turbulent flow, Re > 10,000. Consider Re = 10,000. From Equation 5.21, the Biot number is: hD /£)\0.055 Bi = —- = 0.068 • Re0-8 • = 0.068 • 10000° 8 • (0.0I) 0 - 055 = 83.7

k

VW

Note that increasing the flow rate by a factor of five increases the relative heat transfer coefficient by a factor of 83.7/15.7 = 5.3. The dimensional convection heat transfer coefficient for the coolant is given as: l W i a m = 15.7 - 0.35

ft

. h . o F • Qjjt = 132

hwater,turb = 83.7 • 0.35 • 12 = 700

ft2

^

ft2 fa o p

op

= 750 ^ - ^

= 4000

^ ^

Approximate values for heat transfer coefficients in the transition region, 2000 < Re < 10,000 are obtained by averaging. For example, for Re = 5000, the approximate heat transfer coefficient is:

Since the heat is conducted perpendicular to the mold plane, the fluid resistance must be corrected for the flow channel diameter and the number of flow channels in a given mold area (Fig. 5.1). If there are N flow channels of diameter d, then the area ratio is TTDNL/L 2 , and the effective resistance is:

R

»=s^

0, the free surface is approximated by an insulated surface. This is usually the case for natural convection of heat to quiescent air. Figure 5.11 is used

T Mold, Conduction


T

o

Free Surface, Convection T

o

Biot Number, hL/k

Dimensionless Time, Fourier Number Figure 5.11 Time-dependent average sheet temperature as a function of the rate of heat loss from the free surface. Dimensionless temperature, Y = (T — T0V(Tj — T0) where Tj is the initial sheet temperature and T 0 is the mold surface temperature and ambient air temperature. Biot number, Bi = hL/k, where h is the convection heat transfer coefficient and k is polymer thermal conductivity. Fourier number, Fo = a0/L2 where a is the thermal diffusivity, 9 is time and L is sheet thickness

to obtain the dimensionless time, Fo = a9/L 2 , with the dimensionless temperature, Y, given as: ( T - Tmold) CTi-T mold ) where T1 is the average sheet temperature. Since h conv = O, Bi = hL/k, the Biot number is zero. L is the actual thickness of the sheet. This is illustrated in Example 5.7.

Example 5.7 Cooling Time for Heavy-Gage PS Sheet—I A polystyrene sheet 0.438 in or 1.11 cm thick initially at 375°F or 191°C is cooled by pressing it against a mold at 75° F or 24° C. Determine the time required to cool the sheet to an average temperature of 175°F or 79° C. PS thermal conductivity is 0.073 Btu I ft -h-°F or 0.0003 cal/g • s 0C. PS density is 65.5 Ib /ft3 or 1.05 g/cm3. PS specific heat is 0.5 Btu jib • 0F= 0.5 cal/g • 0C. The free surface is exposed only to ambient air. Assume Zz00 = O.

The dimensionless average temperature, Y is given as: Y =

(175-75) (375-75) = °- 3 3 3

From Fig. 5.11, the Fourier number at this value of Y for Bi = O is Fo = 0.345. The polymer thermal diffusivity is given as: a=— = p-cp

5.7x 1 0 - 4 — = 2.2 x 10~3 ^ s h

The cycle time is given as: r/fl

1

F o = j - 2 = 5.7 XlO" 4 - ^

5

- 6[S] = 0.345

or 8 = 746 s. This is shown as the top line of Table 5.5. Moving Ambient Air Figure 5.11 is also used to approximate the relative effect of air movement on the cooling time, so long as T a i r « T mold . Now the dimensionless temperature, Y, is given as: v —

~

mold

)

(T — Tair)

(T,-T m o l d r(T,-T a i r )

p 2y)

-

Note that as Bi increases, through increase in the value for the convection heat transfer coefficient, the cooling time drops rapidly. When Bi = oo, the dimensionless time to reach the same dimensionless temperature drops to one-fourth that for the quiescent case (Bi = 0). This directly supports the discussion in Chapter 3 that the value of L should be half the actual sheet thickness when energy interchange is symmetric across the sheet. Example 5.8 illustrates the effect of heat transfer coefficient value on cooling time. Example 5.8 Cooling Time for Heavy-Gage Polystyrene Sheet—II Using the data from Example 5.7, determine the reduction in cooling time if the following free surface coolants are used: a) Ambient air with a heat transfer coefficient of h = 1 Btu/ft2 -h'°F

or 5.68 W/m2 • 0 C,

b) Forced air with a heat transfer coefficient of h=10

Btu/ft2 -h

0

F or 57 W/m2 • 0 C, and

c) Water mist with a heat transfer coefficient of h = 100 Btu/ft2 -h-°F

or 570 W/m2 • 0 C.

For each case, the Biot number Bi = hL/k. 1 Btu 0.438 1 ft • h •0 F Ac a)Bl= F l ^ - I ^ f t ' 0 ^ - B t u - = 0-5 .

The value of Fo for Y = 0.333 and Bi = 0.5 from Fig. 5.11 is Fo = 0.256 or 0 = 554 s. This is about 26% reduction in cycle time from the insulated example, Bi = 0 of Example 5.7. b) Bi = 5. The value of Fo from Fig. 5.11 is 0.123 and the cycle time 9 = 266 s. This is a 64% reduction in cycle time. c) Bi = 50. The value of Fo from Fig. 5.11 for Bi = oo is 0.086. The best cycle time possible is 0 = 187 s. This is a 75% reduction in cycle time from the insulated example. These values are tabulated in Table 5.5. The effect of altering free surface cooling conditions is seen in Fig. 5.12 [12]. The nature of the polymer and the details about the processing conditions are unknown. For L = 0.15 cm or 0.060 in sheet, water spray reduces the cooling time by about 60%. For thicker sheet, L = 0.45 cm or 0.180 in, water spray reduces the cooling time by about 65%. As expected, the cooling time increases with increasing sheet thickness. As expected, the effect is most obvious with ambient air cooling. On the other hand, if conduction heat transfer through the plastic sheet controls the cooling time, the effect of sheet thickness is easily determined. At any given average sheet temperature, the dimensionless time, Fo, is a simple function of the Biot number: Fo = f(Bi) (Y

fixed)

(5.30)

Ambient Air

Time, s

Forced Air

Water Spray Calculated

Sheet Thickness, ft

Figure 5.12 The effect of various cooling techniques on cooling time as function of sheet thickness, theory from Fig. 5.11. Experimental data from [12]

Table 5.5 Calculated Cooling Times for Polystyrene Sheet (Combined Convection/Conduction Heat Transfer) Type of free surface cooling

Convection heat transfer coefficient (W/m 2 • 0C)

Thin-gage sheet

Heavy-gage sheet

L - 0 . 1 1 1 c m - 0 . 0 4 3 8 in

L = 1.11 cm = 0.438 in

Biot no.

Biot no.

(Btu/ft2 • h • 0 F) Fourier no.

Cycle time

Fourier no.

(S)

Insulated Ambient air Forced air Water spray Direct contact or matched die

0 5.7 57 570

0 1 10 100

0 0.05 0.5 5.0

OO

OO

OO

0.345 0.323 0.256 0.123 0.086

7.46 6.98 5.54 2.66 1.87

Cycle time (S)

0 0.5 5.0 50 OO

0.345 0.256 0.123 0.086 0.086

746 554 266 187 187

Initial sheet temperature, T1 = 190.60C or 375°F Mold ambient temperature, T 0 = 23.90C or 75°F Final average temperature, T a = 79.4°C or 175°F Thermal conductivity = 0.0003 cal/gm • s • 0 C or 0.073 Btu/ft • h • 0 F Thermal diffusivity = 0.00057 cm2/s or 0.0022 ft2/h From these values, the following are found: Y = 0.333 Bi = hL/k Fo = oc9/L2 = 0.0863[l + 3 exp( - 0.667 Bi0-667)]

As seen in Fig. 5.11, the approximate form for Equation 5.30 is: Fo = Fo00[I + p exp(-aBia)]

(5.31)

Fo00 is the value of Fo when Bi = oo, the limiting case where both sides of the sheet contact solid surfaces. When Bi = 0, Fo = 4Fo00. Since Fo = oc9/L2, the cooling time 9 is proportional to the square of the sheet thickness for both limiting cases. It is apparent then, that when conduction from the plastic sheet controls heat transfer, doubling the sheet thickness always increases the cooling time by a factor of four. See also Table 5.5 where Equation 5.31 is used to compare cycle times for various free surface cooling modes. As seen in Fig. 5.13, actual cooling times show a relationship that is more linear with thickness. The observed values are substantially higher than the values obtained from Fig. 5.12. This indicates that, at least in this case, the experimental sheet is not being cooled in a conduction-controlled environment. Again, the exact conditions used to obtain the experimental data of Fig. 5.13 are unknown. Cooling on Nonmetallic Molds For metal molds, the time-dependent conduction resistance through the mold during the heat removal process from the sheet is usually negligible. Heavy-walled, non-

Ambient Air

Tjme, s

Forced Air

Water Spray

Sheet Thickness, in

Figure 5.13 Measured sheet thickness-dependent cooling times for various cooling techniques. Figure adapted from [23]

metallic molds alternately store and liberate heat during each forming cycle. An estimate of the rate at which heat penetrates the mold is obtained by integrating the time-dependent heat conduction equation, as discussed below. As with energy input to plastic sheet, Chapter 3, there are two limiting conditions that are important. If the energy transfer is constant heat flux and the mold is considered as a semi-infinite slab of thermal diffusivity, oc, the depth of penetration of energy, 5, is approximately: (Constant Heat Flux)

6 = v/6oc6

(5.32)

2

The penetration dimensionless time, Fo = aO/L = 1/6 [13]. If the surface is raised instantaneously to a constant temperature, the depth of penetration of energy, 5, is approximately: (Constant Surface Temperature)

5 = ^/24ae

(5.33)

2

and the penetration dimensionless time, Fo = a0/L = 1/24. In thermoforming, the actual condition is closer to the constant flux approximation. If D is the effective thickness of the mold, then the time for the heat to be felt at the coolant interface is: D2 A D2 2^

(5.77)

where h is the cut-off height or the distance between the saw centerline and the bottom of the plastic sheet [m], and b is the sheet thickness [m]. Example 5.15 illustrates these relationships.

Figure 5.37 Toothed saw trimming geometry [30]. Figure used with permission of copyright owner

Table 5.12 Classification of Plastic Machining Chips1 Classification

Nature of chip

Continuousflow

Continuous

Continuousshear

Continuous

Discontinuous simple shear

Discontinuous

Discontinuouscomplex

Discontinuous

Discontinuouscrack

Discontinuous

Discontinuouscomplex (shear with cracks)

Discontinuous

Thicknessto-cut depth

>1

>1

Cutting force fluctuation

Surface roughness

Nature of deformation

Material type

Cutting speed

Typical plastics

Comments on cause

Small

Small

Elastic

High elongation, rubber-like

Slow

PE, PTFE, FEP, PP

High elastic deformation

Small

Irregular, shear marks

Plastic

Irregular

Plastic

Moderate

Brittle

Irregular

Very irregular, wavy

Elastic

Very large

Hackle marks

Brittle, elastic fracture

Very large

Gouges

Brittle fracture

Large

PMMA High, PMMA, PS

Brittle sticky

>1 Irregular Chips

PS, ABS

Medium Brittle

Irregular

1

Mediumhigh

Adapted from [26], with permission of author

Brittle

Slippage continuously by shear stress Plastic fracture by simple shear shear Plastic fracture by shear with compressive and/or tensile stress

PMMA, PS

Elastic fracture, brittle fracture

High modulus, Low-elongation

Plastic fracture by shear with compressive and/or tensile stress

High Brittle High

Example 5.15 Cutting with a Saw Consider a D = 15.2 cm or 6 in diameter saw having four teeth per in. The blade revolves at 1000 RPM. The sheet is 0.100 in or 0.254 cm thick. The cut-off height is 2 in or 5.1 cm. A minimum tooth depth, g, is selected to be 0.004 in or 0.1 mm to minimize gumming or burning. Determine the maximum feed rate.

From Equation 5.76: v

_

u-g

p sin (j) p is the tooth spacing or 1/4 (teeth/in) = 0.25 in or 0.637 cm. The peripheral speed of the blade, U = TCDN. For this example: U = Ti • 15.2 cm

— = 47,750 cm/min mm

The angle is given from Equation 5.77 as: . *=

l C0S

(h-b/2) - D / ^ = C0S

1

(5.1-0.254/2) 15.2/2

(j) = 49 degrees Therefore: 47,750 • 0.01 cm2/min , . v= — — = 990 cm/mm =16.5 cm/s = 6.5 m/s 0.637 cm • 0.756 This is the maximum feed rate of this stock into the saw.

Note that the feed rate is proportional to blade speed and diameter and inversely proportional to tooth spacing. Thermal damage to the plastic is minimized by: • • • • •

Wide tooth spacing, Coarse toothed blades, Small blades, Low speed blades, and High feed rates.

However wide tooth spacing causes relatively rough cut edges with many brittle polymers such as PS, ABS, SAN, PMMA, and RPVC. Hollow-ground blades with no tooth set and wide-kerf carbide blades yield smooth cut edges. Spring-set and swag-set teeth also produce quality cut edges (Fig. 5.38). Abrasive Cut-Off Wheel Abrasive wheels with 30- to 200-grit surfaces produce relatively smooth cut surfaces at high cut-off rates with about one-half to one-third the heat generated by toothed

Carbide Tip

Hollow Ground

Spring-Set

Swaged

Tangential Cutting Force, Ft , kgf

Figure 5.38 Typical saw tooth designs

36 Grit Silicon Carbide Abrasive Hollow-Ground Tooth Saw, No Set 36 Grit Aluminum Oxide Abrasive

Hollow-Ground Tooth Saw, Spring-Set

Feed Rate, v, mm/min Figure 5.39 Comparison of abrasive disk and toothed saw trimming forces as function of polymer feed rate [30]. Figure used with permission of copyright owner

saws [26]. Typical abrasives include aluminum oxide or alumina, silicon carbide, tungsten carbide and diamond. Diamond abrasive wheels are the most expensive but last longest. Abrasives are held together with thermosetting binders such as phenolics, ureas and epoxies. Cutting forces are usually higher for abrasive wheels than for toothed wheels at the same feed rate (Fig. 5.39). Abrasive wheel cut surface roughness is usually smaller (Table 5.13). Finer grit wheels produce smoother surfaces but at reduced cut-off rates (Table 5.14). The machinability of a plastic, r\, is written as [26]: (5 78)

^V^s{

-

3

where V1n is the volume of polymer cut per unit time [mm /min], Vw is the amount of tool wear per unit time [mm3/min], HP is power consumption [kg • m/min], and Sf is the cut surface roughness [urn]. For properly selected cutting wheels, the amount of tool wear is essentially negligible, and machinability is redefined as:

Table 5.13 Cutting-Off Operation1 Peripheral speed 2500 m/min Polymer

Surface roughess (urn)

SAN ABS PA-610 (a nylon) PC

Abrasive wheel

Circular saw

A

B

C

D

32 28 34 8

23 16 16 6

8 10 6 5

200 200 36 25

1

After [26], with permission of author Key: A: 36 Grit silicon carbide, resinoid—medium grade B: 36 Grit aluminum oxide, resinoid—medium grade C: Saw, hollow-ground, zero-set teeth, 300 mm, diameter, 2 mm thick, 2.3 teeth/cm or 5.8 teeth/in, 0° rake-angle, 60° relief angle D: Saw same as C except 0.2 mm set to teeth

Table 5.14 Finishing Operations Sanding Belt Surface Roughness1 Silicon carbride sanding belt 2000 m/min, 0.5 kg/cm 2 applied force, 20 min Polymer

Surface roughness (um) 60 Grit 240 Grit

Amount removed (g) 60 Grit 240 Grit

PMMA RPVC PC

36 39 41

295 360 235

1

2 2 2

28 8 9

Removal rate (mg/s) 60 Grit 240 Grit 246 300 196

23.3 6.7 7.5

Adapted from [26], with permission of author

(5.79)

For cutting wheels: (5.80)

where b is the sheet thickness, B is the wheel thickness and F t is the tangential component of the cutting force. The ratio of efficiencies of abrasive and toothed wheels operating at the same feed rates and peripheral speeds is written as: (5.81)

Abrasive wheels are typically 2 to 5 times thicker than toothed wheels. Both produce about the same surface roughness on the cut edge at the same cut-off speeds (Table 5.13). The cut-off forces for abrasive wheels are about 2 to 5 times those for toothed wheels. Therefore:

Although the cut-off efficiencies of abrasive and toothed wheels are about the same, the cost of operating abrasive wheels is about 5 to 10% that of toothed wheels. When loaded, abrasive wheels are redressed with a gum block or a wire brush. Toothed wheels require resharpening and tooth resetting.

Toothless or Shear and Compression Cutting Saw cutting depends on brittle tensile fracture of the plastic under the force of the tooth. In solids, Young's modulus, E, is the proportionality between pure elastic tensile stress and strain: G =E - e

(5.83)

When a solid is sheared, the proportionality between pure shear stress and strain is the modulus of rigidity or shear modulus, G: O5 = G • es

(5.84)

The bulk modulus, B, is the ratio of hydrostatic pressure to solid volume change per unit volume. Solid compressibility is the reciprocal of the bulk modulus. These moduli are related through Poisson's ratio, v, the ratio of unit width change to unit length change [31]: E = 2G • (1 + v) = 3B • (1 - 2 • v)

(5.85)

Poisson's ratio, v = 0.5 for a material with constant volume under stress. For most plastics, 0 . 3 < v < 0 . 4 . Table 5.15 gives representative values for Poisson's ratio. A value of v = 0.35 is used if none is available for a given polymer. At this value, E/G = 2.7 and E/B = 0.9. Thus the expected resistance to shearing forces is only about 40% of the resistance to stretching or tensile forces. When this is applied to traditional thermoform trimming operations, shear cutting should require lower specific energy than, say, saw cutting.

Fracture Mechanics in Trimming Fracture mechanics is the study of crack propagation in plastics under stress. There are three general types of fracture, as shown in Fig. 5.40 [32,52]. Mode I is a tensile mode where fracture surfaces are spread apart by the stress. Mode II is a shear mode, where stress forces the fracture surfaces to slide perpendicular to the advancing crack. Mode III is a tearing mode, where the fracture surfaces are forced apart

Table 5.15 Poisson's Ratio for Several Thermoformable Polymers [50] Polymer

Poisson's ratio

LDPE HDPE PP PIB PS

0.49 0.47 0.43 0.47 0.38

Rigid PVC PCTFE PTFE PMMA mPPO

0.42 0.44 0.46 0.40 0.41

PPS PET PBT PA 66 PA 6

0.42 0.43 0.44 0.46 0.44

PC Polysulfone Polyimide

0.42 0.42 0.42

by the stress in the direction parallel to the crack [32]. Shear cutting is considered as Mode III, antiplane shear. This fracture mode is also found in torsion of notched rods. Mode I fracture, cleavage or tensile-opening, dominates classical fracture analysis since it is the most common form of material failure and since it is the easiest to study in the laboratory. If the plastic is not held tightly, Mode III crack propagation control is difficult to maintain. The advancing crack tends to meander uncontrollably, and secondly, tangent cracks can form. The amount of force required to propagate a crack in any mode and the rate at which a stable crack is propagated can only be estimated from the extensive studies of Mode I failures. The amount of energy needed to initiate a crack in any polymer is substantially less than its theoretical cohesive strength. Cracks begin at flaws or defects in the polymer. They propagate when the decrease in elastic strain energy equals or exceeds the energy needed to create a new crack surface. When a tensile specimen with a small horizontal crack, a in length, is stressed, it is in plane stress. The stress needed to propagate that crack is: (5.86) where E is Young's modulus and G* is the fracture energy: (5.87)

Mode I, Tensile Fracture

Mode II, In-Plane Shear or Sliding Fracture

Mode III, Anti-Plane Shear or Tearing Fracture

Figure 5.40 Characteristic fracture modes [32]. Redrawn Figure used with permission of PrenticeHall Publications, Inc.

where P is the plastic work done during yielding and y is the surface energy of the polymer. In Equation 5.86, Kc is the fracture toughness. The stress needed to initiate a crack is frequently far greater than that needed to sustain crack propagation [33]. For PMMA for example, the ratio of stress levels is about 1000. For vulcanized natural rubber, it is about 325. The plastic deformation stretching energy is usually much greater than the surface energy. For ductile plastics such as PTFE, PP, PET and TPO, P » y. Even for very brittle plastics such as PS and PMMA, P > 2 • y. Example 5.16 illustrates these relative values. One measure of the fracture toughness of a polymer is the area under its tensile stress-strain curve. If the area is large, the polymer is tough. Polymers that show great plastic flow after yielding, such as HDPE, PP and PET, have high fracture toughness. If the area under the tensile stress-strain curve is small, as with PS and PMMA, the polymer is brittle. The stress level that produces fracture is analogous to the crack tip stress intensity level that produces sustained fracture. As noted in the second part of Equation 5.86, fracture toughness or stress intensity factor, Kc is written as: (5.88) Characteristically, the stress intensity factor is written as: (5.89)

Example 5.16 Fracture Energies for Ductile and Brittle Polymers Consider vulcanized rubber and PMMA as typical of ductile and brittle polymers, respectively. Determine the relative ratio of yielding work to surface energy of each of these polymers. The plastic work during yielding of PMMA is P = 0.185 ft-lbf/in2 = 0.211 kJ/m 2 . The surface energy for PMMA is about y = 0.039 kJ/m 2 = 0.0342 ft-lbf/in2. The fracture energy is given as: G* = 2(P + Y) = 2(0.211 + 0.039) = 0.5 kJ/m2 = 0.44 ft-lbf/in2 The ratio of plastic work to surface energy, P/y = 0.211/0.039 = 5.4. This is a strong indication of a very brittle polymer. The plastic work for vulcanized rubber, P, is unknown. However, the fracture energy, G* = 13 kJ/m 2 = 11.4 ft-lbf/in2 and the surface energy for vulcanized rubber is about y = 0.012 kJ/m 2 = 0.010 ft-lbf/in2. The calculated value of plastic work is: P = ^ - - Y = y - 0.012 = 6.49 kJ/m2 = 5.67 ft-lbf/in2 Thus the ratio of plastic work to surface energy, P/y « 6.49/0.012 = 567. This is a strong indication of a very ductile polymer. The coefficient C depends on the geometry of the crack and the surface being fractured. One example of C is given in Fig. 5.41 [32], for an edge crack of length a in a sheet of width W under uniaxial tension. For unreinforced polymers, values for Kc range from about 0.5 to 10. Typical values for Kc for a few polymers are given in Table 5.16. The fracture stress given in Equation 5.88 is written symbolically as:

Stress Intensity Factor

a =

Reduced Crack Width, a/W

Material Parameter : Geometric Parameter

^_ (5.90)

Figure 5.41 Geometric parameters for stress concentration factor for mode I, tensile fracture [32]. Redrawn Figure used with permission of Prentice-Hall Publications, Inc.

Table 5.16 Stress Intensity Factors for Some Plastics and Other Materials1 Values in parentheses obtained from K c — ^JE • G* Materials

Young's modulus, E (GPa)

(1000 lbf/in2)

Vulcanized rubber Polyethylene PS HIPS PMMA

0.001 0.15 3.0 2.1 2.5

21.8 435 305 363

Epoxy Rubber-modified epoxy FRP Glass Wood Aluminum

2.8 2.4

406 348

7 70 2.1 69

0.145

1,015 10,150 305 10,000

Fracture energy, G*

Stress intensity factor, K

(kJ/m2)

(ft-lb/in2)

(MN/m3/2)

(1000 lb/in3/2)

13

11.4

( 0.114)

( 2.05)

20* 0.4 15.8* 0.5

17.5 0.35 13.8 0.48

( 1.73) 1.1) ( 5.76) 1.1

( 31.2) 19.8 (104) 19.8

0.1 2

0.087 1.75

0.5 2.2

9.0 39.8

7 0.007 0.12 20

6.12 0.0061 0.105 17.5

7 0.7 0.5 37

126 12.6 9.0 666

1 Adapted from [33], with permission of copyright holder * J-Contour Integral. See [33: p. 82]

In the tensile Mode I, an infinite stress is needed to initiate a crack of zero length. Once the crack is propagating, the stress diminishes rapidly. The rate of crack propagation is also important. For fatigue failure where the load is applied in cyclic fashion, the following relationship is used: ^

= A f -AK»

(5.91)

where N is the number of cycles, AK is the stress intensity factor range, AK = Kmax — Kmm, and m and Af are polymer material properties [33]. Some values for Af and m are given in Table 5.17. AK is proportional to Kc, the fracture toughness. For many polymers [33,34], 0.5 < [AK/KJ < 0.67. Cyclic crack propagation rate is used only as a guide to determine crack speed in Mode I tensile fracture. Cyclic fatigue crack growth occurs at substantially lower stress levels than those needed to sustain crack growth in continuous loading. In turn, this guideline can be used only as an estimate of the shear stress needed to control crack propagation in Mode III antiplane shear fracture. As noted, Mode III fracture is the apparent mode occurring during shear cutting of plastics such as guillotining, diagonal brake cutting, nibbling and paper cutting. An appropriate relationship for the rate of crack propagation, a = da/d9 is: da a = — = P • am • ocm/2 d(3

(5.92)

where a is the shear stress and (3 includes geometric factors and material constants. If the crack length ahead of the shear is to remain stable, or a is to be constant, the

Table 5.17 Crack Propagation Parameters da/dN = Af • AKm da/dN units are mm/cycle AK units are MN/m 3/2 Af units are [MN/m 1 / 2 ]- m Note: For crystalline polymers, the general relationship is: da/dN = A* • (AK/E) 7 where 0.5 < A* < 3 (AK/E) units are m 1/2 Polymer

Af x 1000

m

Range of AK

PS PMMA PES HDPE PC

2.65 99 1.5 0.35 0.118

3.73 10.0 9.5 5.22 4.81

0.5 0.4 0.6 1.0 1.0

to to to to to

1.2 1.0 1.2 2.5 3.0

6.2 2.2 3.63 3.2

1.0 0.5 1.5 1.5

to to to to

3.0 1.0 8.0 8.0

mPPO RPVC PA 66 PVF

0.0365 0.164 0.00728 0.0087

rate of shear is approximately proportional to the applied force to the mth power. For RPVC in Table 5.17, m = 2.2 and the shear rate should increase about four times when the applied load is doubled. On the other hand, for PMMA, m = 10 and the shear rate should increase about four times with only a 15% increase in applied load. Compression cutting occurs when the steel rule die is pressed perpendicularly into plastic sheet that is resting on an unyielding surface. Load compression follows the true material stress-strain curve. A modification of compression molding that employs uniaxial plane-strain compression is used to determine stress-strain curves for polymers that neck or fracture easily [35]. Compression cutting is particularly useful when the polymer yields in compression but fractures brittlely in tension or shear. RPVC, PC and PMMA are typical polymers that lend themselves well to compression cutting. Compression yield stresses are usually higher than tensile or shear yield stresses (Table 5.18). Thus more force per unit cutting area is required to die cut a plastic than to shear cut it. Comparison of typical stress-deformation curves, Fig. 5.42 [35], shows that the area under the compression curve continues to increase with increasing strain. As a result, crack propagation in compression is more stable for polymers that are brittle or neck badly. Compression cutting is the preferred method for cutting LDPE and should be considered for trimming PET, PA or nylon, POM or acetal, low-density foams and thin-gage OPS and PMMA. For a very brittle polymer being trimmed with a very sharp, wide die, a perpendicular crack is created ahead of the blade. The crack can propagate as a Mode I fracture, as shown in schematic in Fig. 5.43. The crack can be uncontrollable with an irregular reverse side cut surface, microcracking and crazing. The stress required to cut through the material is given in terms of the stress intensity factor,

Table 5.18 Yield Stresses and Cutting Shear Stresses Polymer

Tensile yield stress (MPa)

(10001bf/in2)

HDPE LDPE PP PET PC

28 10 34

4 1.5 5

62

9

RPVC PS HIPS ABS mPPO

41

6

21 34 48

3 5 7

PA 66* CA CAB CAP PMMA

55

8

POM**

10 * Dry nylon 69 ** Polyoxymethylene, acetal homopolymer

Compressive stress (yield) (MPa)

(10001bf/in2)

Flexure stress (yield) (MPa)

(10001bf/in2)

20

3

41 83 86

6 12 12.5

45 98 94

6.5 14 13.5

69 83

10 12

41 76

6 11

77 77 35 35 84

11 11 5 5 12

103

15

17 6 5 4 13 14

21 83

12

119 42 35 28 91

110

16

98

Sharp knife cutting stress (MPa)

(10001bf/in2)

91-126 110

13-18 15.8

60 45-50

8.6 6.4-7.1

Stress

Uniaxial Compression

Uniaxial Tension

Tension on a Notched Specimen

Figure 5.42 Stress-strain curves for various types of fracture in a notch-sensitive ductile polymer [35]. Figure used with permission of John Wiley & Sons, Inc.

Deformation

Trim Die

Polymer

Figure 5.43 Characteristic mode I fracture with wedge effect of trim die into brittle polymer sheet

Equation 5.89, and is frequently much less than the compression yield strength. One way of partially controlling crack propagation in very brittle polymers such as PS and PMMA is to place a very slightly resilient mat between the plastic sheet and the table. Very hard rubber or heavy-gage UHMWPE works well. If the plastic or rubber is too soft, the plastic may bind the cutting blade and chipping, splitting and uncontrolled fracture may be aggravated. Typically the force required to cut through a polymer sheet is proportional to the cut length and the sheet thickness to some power a, Force a (cut length) x (sheet thickness)a

(5.93)

Since the crack proceeds the cutting blade tip, the force required to cut through a brittle polymer may be only weakly dependent on sheet thickness. For a ductile polymer and a blunt die, the deformation stress is obtained directly from the stress-strain curve. Since most polymers strain harden to some extent in

Trim Die

Polymer

Shear Anvil Pinch Gap Figure 5.44 Characteristic combined shear and compression punch- and die-cutting of tough or ductile polymer

compression, the force required to cut through a tough polymer increases with the depth of cut [36] and a « 1: Force oc (cut length) x (sheet thickness)1

(5.94)

For ductile polymers, a combination shear and compression die cut is used (Fig. 5.44). With this type of cut, the lower force of a shear cut is combined with the stable crack propagation at increasing stress-strain of a compression cut. Heavy-gage plasticized PVC or FPVC, HDPE and PP are cut in this way. Combination shear cutting forces for several cutter blade designs are shown in Figs. 5.45 and 5.46 [37] for RPVC and PS, respectively. It is apparent that cutter force is proportional to sheet thickness, or a = 1 in Equation 5.93. Cutting forces for dull knives at 200C or 680F are measurably higher for polymers that yield, such as PVC, as seen in Table 5.19. These values compare well with blanking force guidelines for thin-gage sheet (Table 5.20) [38]. Values for effective shear stress or cutting force per unit thickness and cutter length are about the same as compressive yield stress values in Table 5.18. Cutting forces at 600C or 1400F are about 10% lower than those at 200C or 68°F. Nibbling Nibbling is a cyclic trimming process. An estimate of nibbling force is made from Mode I tensile fracture crack propagation. Example 5.17 illustrates this. As noted earlier, the force required to initiate a crack is as much as 1000 times greater than that needed to sustain it. Nibbling is a process requiring crack initiation at each stroke. Further additional force is required to overcome friction between the nibbler

Force, 100 kgf

Dulled Cutter

Cutter Design 4 Cutter Design 3 Sandwich Cutter

RPVC

Thickness, mm

Figure 5.45 Experimental sheet thickness-dependent shear cutting force for rigid polyvinyl chloride, RPVC, for several trim dies [37]

Table 5.19 Shear Cutting Forces Data from Figs. 5.45 and 5.46 [37] Blade length = 10 in or 250 mm Sheet thickness = 0.039 in or 1 mm Polymer

Blade design

Cutting stress

Total force (kgf)

(Ib)

(MPa)

(10001bf/in2)

PVC

#3 #4 Sandwich Dull*

2600 2710 3100 3900

5720 5960 6820 8580

104 108 124 156

15.1 15.8 18.0 22.6

PS

#3 #4 Sandwich Dull**

2680 2680 2680 2900

5900 5900 5900 6380

107 107 107 116

15.5 15.5 15.5 16.8

* Exhibits a "zero thickness" resistance of 1360 kgf or 2990 lbf, or an equivalent shear stress of 54 MPa or 7900 lbf/in2 ** Exhibits a "zero thickness" resistance of 460 kgf or 1000 lbf, or an equivalent shear stress of 18 MPa or 2700 lbf/in2. In addition, the force-thickness curve is nonlinear

Dulled Cutter Sandwich Cutter

Force, 100 kgf

Cutter Design 4 Cutter Design 3

PS Figure 5.46 Experimental sheet thickness-dependent shear cutting force for polystyrene, PS, for several trim dies [37]

Thickness, mm

Table 5.20 Blanking Force Guidelines for 0.25 mm or 0.010 in Sheet [38] Nature of polymer

Type of polymer

Force

Total force in 10 in or 25 cm

(kgf/cm)

(lbf/in)

(kgf)

(lbf)

Soft

Polyolefins, cellulosics

27

150

680

1500

Medium

Flexible PVC

46

250

1140

2500

Hard, Tough

OPS, PET, PS PMMA, RPVC

91

500

2280

5000

blade and the plastic and to push the cut-off plastic piece from the kerf. As with compression cutters, a good first approximation of the value of the cutting-off shear stress is the yield strength value of the plastic (Table 5.18). Example 5.17 Nibbling Force for PMMA Thin-gage PMMA sheet, 0.025 in or 0.1 cm thick, is to be cut in one cycle. Determine the force needed to sustain a crack. For PMMA, Af= 0.1 (MN/m1/2)~m and m = 10.

From Equation 5.91, the stress intensity factor range is obtained: AK = (da/dN)1/m-Af-1/m Now da/dN>0.1 cm. Assume da/dN=l. Then AK =1.26 MN/m1/2. If AK = 0.5 Kc, the critical stress intensity factor for crack propagation and C = 2, the tensile stress needed to sustain the crack is given as: a = ^ • V ^ = 1-26 V*-0.001 =

Qm

M N / m 2 = 1Q 3

^

2

This is the tensile stress needed to propagate the crack. The tensile yield stress of PMMA is about 45 MN/m2 or 8500 lbf/in2. Since nibble-cutting should require about the same expenditure of force as compression-shear cutting, the apparent force needed to sustain a crack in PMMA is only about 0.1% of the total force needed to cut the plastic. Brittleness, Orientation and Trimming Temperature Many brittle plastics exhibit uncontrolled fracture, secondary crack propagation, reverse-side chipping, crazing and splitting when compression or shear cut. Brittle fracture occurs when the brittle strength of the polymer is less than its yield strength. The brittle strength of a polymer below its glass transition temperature, Tg, is weakly dependent on temperature over a wide range of temperature, 1000C or more. On the other hand, polymer yield stress is essentially a linearly decreasing function of temperature (Table 5.21) [35]. As the polymer temperature increases, the probability of fracture at stresses below the yield stress decreases. The problems associated with brittle fracture trimming diminish as well. At elevated temperatures approaching Tg, the polymer is ductile and high-speed cutting-ofT techniques that depend on brittle fracture crack propagation such as routering and sawing, become inefficient. Table 5.21 Temperature Effect on Yield Stress For Various Polymers1 Polymer

Linear region

Temperature coefficient of yield stress (0F)

(0C) PMMA PTFE PE* RPVC PP

50 -250 -60 -80 -40

PA 66 APET PC

-100 to 60 - 2 0 to 60 - 4 0 to 120

to to to to to

100 -140 20 60 20

122 -420 -76 -112 -40

to to to to to

212 -220 68 140 68

-150 to 140 - 4 to 140 - 4 0 to 250

3 Adapted from [35], with permission of copyright owner * Chlorinated polyethylene

(MPa/°C)

(lbf/in2 - 0F)

0.97 0.85 0.82 0.80 0.74

78.2 68.5 66.1 64.5 59.6

0.74 0.49 0.32

59.6 39.5 25.8

Next Page

Thermoformed shapes frequently have high degrees of orientation in the trim areas. Part shape can enhance nonuniform orientation. Cutting relieves local stresses. Although this aids crack propagation to some degree, particularly in shear cutting, it can also cause: • • • •

Nonuniform part dimensions, Binding of the cutting tools, Part warping, and Part distortion.

Secondary fracture effects such as crazing can result. Uncontrolled crack propagation can proceed in the orientation direction rather than in the desired cut path, or the crack can meander. This results in splitting and splintering, dust and angel hair. Highly nonuniformly oriented parts of brittle polymers are frequently formed on hot molds and then the parts with the trim attached are partially annealed prior to trimming. The trim is then cut away while the parts are still warm. Heavy-gage amorphous polymers such as RPVC and ABS and easily oriented crystalline polymers such as PA 66 and PP benefit by this trimming approach. As a general rule for all polymers, dust and angel hair problems and part warpage and distortion may be minimized by increasing the part temperature at the time of trimming. Increasing trim die temperature is usually ineffective. Part of the difficulty in trimming with hot dies is in conduction heat transfer from the heating source to the cutting tip. The die is envisioned as a metal fin. The effectiveness of conduction heat transfer to fins decreases rapidly with the length-to-thickness ratio of the fin [39]. This is due primarily to convection heat transfer to the cool ambient air:

where mL is given as: mL = / ^ L

(5.96)

where h is the convection heat transfer coefficient between the heated trim die and the ambient air (Table 5.4), k is the thermal conductivity of the trim die steel, L is the distance between the heater and the cutting tip and t is the thickness of the trim die. As is apparent in Fig. 5.47 [40], the energy efficiency of heated dies decreases rapidly with L/t ratio, for both rectangular and tapered dies. In words, since heated dies are very inefficient heat transfer devices, it is usually difficult to maintain cutting tips at the proper temperature.

5.10 Steel Rule Die As noted, the steel rule die is the most common method of cutting prototype sheet having thickness less than about 0.100 in or 2.5 mm. Steel rule dies are most effective

Previous Page

Thermoformed shapes frequently have high degrees of orientation in the trim areas. Part shape can enhance nonuniform orientation. Cutting relieves local stresses. Although this aids crack propagation to some degree, particularly in shear cutting, it can also cause: • • • •

Nonuniform part dimensions, Binding of the cutting tools, Part warping, and Part distortion.

Secondary fracture effects such as crazing can result. Uncontrolled crack propagation can proceed in the orientation direction rather than in the desired cut path, or the crack can meander. This results in splitting and splintering, dust and angel hair. Highly nonuniformly oriented parts of brittle polymers are frequently formed on hot molds and then the parts with the trim attached are partially annealed prior to trimming. The trim is then cut away while the parts are still warm. Heavy-gage amorphous polymers such as RPVC and ABS and easily oriented crystalline polymers such as PA 66 and PP benefit by this trimming approach. As a general rule for all polymers, dust and angel hair problems and part warpage and distortion may be minimized by increasing the part temperature at the time of trimming. Increasing trim die temperature is usually ineffective. Part of the difficulty in trimming with hot dies is in conduction heat transfer from the heating source to the cutting tip. The die is envisioned as a metal fin. The effectiveness of conduction heat transfer to fins decreases rapidly with the length-to-thickness ratio of the fin [39]. This is due primarily to convection heat transfer to the cool ambient air:

where mL is given as: mL = / ^ L

(5.96)

where h is the convection heat transfer coefficient between the heated trim die and the ambient air (Table 5.4), k is the thermal conductivity of the trim die steel, L is the distance between the heater and the cutting tip and t is the thickness of the trim die. As is apparent in Fig. 5.47 [40], the energy efficiency of heated dies decreases rapidly with L/t ratio, for both rectangular and tapered dies. In words, since heated dies are very inefficient heat transfer devices, it is usually difficult to maintain cutting tips at the proper temperature.

5.10 Steel Rule Die As noted, the steel rule die is the most common method of cutting prototype sheet having thickness less than about 0.100 in or 2.5 mm. Steel rule dies are most effective

Fin Efficiency, % Characteristic Fin Length, Lf2(h/kAm)1/2 Figure 5.47 Heat transfer efficiency for heated dies with rec tangular and triangular cross-sections. L is the length of the die from the heating source and A is its cross-sectional area. Redrawn from [40]

if the sheet thickness at the point of trimming is less than about 0.025 in or 0.64 mm. Typically, the steel used in thermoforming trim dies is A2D2. Depending on the method of hardening, this steel has a hardness of 48 to 52 Rc. It is considered a hard steel. The steel hardness range depends on the particular application, the polymer and the ease of sharpening the die: •

Steels with 45 to 48 Rc hardness are considered maleable or peenable. They are easily bent and sharpened and are easily dulled by cutting against compression plates having equal hardness. • Steels with 48 to 52 Rc hardness hold excellent edge but are not normally honed. They form the base stock for advanced hardening techniques. Steels with this hardness dull when cutting toughened polymers. • Steels with 52 to 56 Rc hardness are considered hardened steels. They hold excellent edge even when single- and double-honed, as described below. If they are chipped or dulled, they are difficult to resharpen but can be honed. • Steels with 56 Rc and higher hardness are usually produced by locally hardening lower Rockwell steels. Dies with 60 Rc hardness are used to trim filled, reinforced, and composite thermoformed parts. Although there are many ways of hardening steel, steel rule dies are usually hardened by: • •

Case hardening, Induction hardening,

• •

Shot peening, particularly the cut edge, and Cubic boron nitriding.

Nitriding yields the hardest cut edge. Typically, nitrided dies are rarely resharpened at the thermoforming press. Steel rule dies are usually 1 in or 25 mm to 2 in or 50 mm in depth. The die thickness depends on the sheet thickness, its toughness and the depth of draw of the part. Although dies are standardized in the following thicknesses: • • •

Thin-gage die with 0.028 in or 0.71 mm thickness, Medium-gage die with 0.042 in or 1.07 mm thickness, and Heavy-gage die with 0.056 in or 1.42 mm thickness,

steel rule dies are fabricated from sheet steel of any thickness if necessary. The die steel is available in strip form or in continuous roll. The cutting edge shape is ground in one of several configurations shown in Fig. 5.48. The single-sided cutting edge is most common, since it is inexpensive to grind and to resharpen. The double-sided cutting edge is recommended if the die is quite long, since the forces are centered directly over the fracture zone. The double-sided cutting edge is more difficult to align for trim line accuracy than the single-sided cutting edge. Honed cutting edges are used when a very sharp, very hard cutting edge is needed. These edges tend to feather and nick easily however. Recently, there has been interest in the single-sided double bevel cutting edge, which requires less frequent sharpening than the single bevel cutting edge.

Double Bevel

Single Hone

Single Bevel

Double Hone

Figure 5.48 Characteristic cutting edges on trim dies

Double Angle

Blunt Nose

Trim Knife

Rubber Particles

Brittle

Rubberized

Polymer

Dull Knife

Feathered Edge

Figure 5.49 Generation of dust and angel hair from various trim die cutting edges

Resharpening In time, all steel rule dies dull. The edge either flattens from repeated contact with the compression plate or the edge rolls or feathers (Fig. 5.49). As noted, blunt dies require greater force to penetrate the plastic than sharp dies. Blunt dies are considered to be the primary source of dust, microcracks, crazing and angel hair. With the possible exception of nitrided dies and dies having hardness values in excess of 56 Rc, all dies can be resharpened. Usually the die must be removed from the trim press for resharpening. There are several ways to resharpen the dies: •

Stropping, or smoothing the cutting edge with a leather strap. Stropping is most effective when the cutting edge shows a very light feather. The technique removes very little if any steel, typically < 0.0005 in or 0.46 0.500 12.7 Polyester-glass J 0.30 0.750 19.1 0.13 1.000 25.4

63 30 18 12 5

5.14 Trimming—A Summary The selection of a trimming technique depends primarily on the stress-strain nature of the polymer at its trimming temperature. However many processing elements influence the choice including: • • • • • • • • • • • •

The The The The The The The The The The The The

gage of the sheet, part size, overall draw ratio, nonplanar nature of the trim line, complexity of the trim line, acceptable level of cut surface roughness, required dimensional tolerance, economically required speed of trimming, extent of fixturing or hold down, number of secondary or ancillary piercing steps required, skill of the operator or pressman, availability of the desired trim equipment,

Table 5.26 Suitability of Trimming Techniques (Heavy-gage in parentheses) O = Unsuitable 9 = Preferred, best Trim category Type of polymer

Typical polymer

Die cut

Shear cut

Nibble cut

Router

Circular saw

Abrasive wheel

Band saw

Hot wire

Hot gas jet

Water Jet

Grit blast

Laser cut

Very brittle

PS, PMMA, SAN

9(0)

6(2)

4(4)

0(4)

2(6)

2(9)

3(4)

6(7)

4(6)

5(2)

8(8)

7(8)

Brittle

ABS, RPVC CA, CPET, CAB

9(0)

7(2)

5(4)

2(5)

3(7)

4(9)

3(4)

4(7)

5(4)

6(2)

8(8)

7(8)

Tough

mPPO, CAP, PPS, PA 6, PA 66, PET OPS

5(2)

9(6)

6(8)

3(8)

5(9)

5(7)

4(7)

3(8)

7(6)

5(2)

8(2)

7(8)

8(2)

9(8)

7(8)

5(3)

7(6)

7(5)

2(3)

2(8)

3(6)

3(5)

0(0)

7(9)

Ductile, rubbery

LDPE, PP, HDPE, FPVC, TPE, TPO, PTFE, FEP

Table 5.27 Trim Quality—Thin Gage1 Process

Trimming type

Tooling type Pressure/vacuum Parts per shift Quality A = highest Scrap rate

Time for delivery of tooling—weeks

Manual/Steel rule Manual/Steel rule In-line/Steel rule Off-line/Punch & die Roll-fed/Contact heat Trim-in-place Roll-fed Trim-in-place

Epoxy Aluminum Aluminum Aluminum

Vacuum Pressure Both Both

Lowest Medium High Highest

D D C B

High Low Low Low

3-4 5-6 5-6 12-15

Aluminum Aluminum

Pressure Pressure

Medium Low

B A

Low Low

5-6 15-20

Process type

Setup time

Tooling cost/ Skill required cavity

Machine cost

Per Part cost

Number formed Comments parts per shift— comparative

Roll-fed/Cut sheet Epoxy tool Roll-fed/Cut sheet Aluminum Roll-fed/In-line Trim Roll-fed/OfT-line Punch & die

Lowest

Lowest

Lowest

Lowest

Highest

Low

Low

Low

Low

Medium

Medium

Medium

High

High

Low

Medium

Roll-fed/Cut sheet Roll-fed/Cut sheet Roll-fed Roll-fed

Form-trim-inplace— Contact heat Form-trim-inplace— Oven heat 1

800

Manual die cut

High

4000

Manual die cut

Medium

Low

4000

High

High

Lowest

8000

Medium

Low

Medium

Low

5000

Highest

Medium

Highest

Low

8000

Adapted from [51], with permission of copyright owner

Large number of cavities, maximum number of parts Female, nonplugged molds Fewer cavities, fewer formed parts

• •

The availability of resharpening methods, Among others.

As noted, not all trimming devices are suitable for all types of plastics in all thicknesses. For example, automatic programmed laser cutting is clean and accurate but is impractical for trimming thin-gage roll-fed APET. Die cutting is quicker, cheaper and can be installed as part of the in-mold forming process and therefore becomes the trimming process of choice. Some techniques such as abrasive grit cutting are dusty and work best on high modulus, brittle polymers. Abrasive and toothed wheel cutting and routering generate dust. Heavy-gage parts of PMMA, ABS and PS should be sprayed with an antistat prior to trimming to minimize the almost-impossible-to-remove cutter dust. The rank of trimming techniques in Table 5.26 is meant only as a guideline, in terms of matching the intrinsic natures of the cuts and the material stress-strain behavior. Economics and availability may dictate a less-than-optimum choice for trimming. Table 5.27 illustrates some of the economic considerations needed for thin-gage trimming [51].

5.15 References 1. D.Q. Kern, Process Heat Transfer, McGraw-Hill Book Co., New York (1950), Table 12 of Appendix, p. 845. 2. M.R. Kamal and S. Kenig, "The Injection Molding of Thermoplastics. Part II: Experimental Test of the Model", Polym. Eng. ScL, 12 (1972), pp. 302-308. 3. M.K. Liao and CS. Li, "Investigation of Thermal Conduction Between Mold and Melt During Injection Molding Process", SPE ANTEC Tech. Papers 40 (1994), pp. 501-505. 4. B.O. Rhee, CA. Hieber and K.K. Wang, "Experimental Investigation of Thermal Contact Resistance in Injection Molding", SPE ANTEC Tech. Papers 40 (1994), pp. 496-500. 5. J. Schneider, "Conduction", in W.M. Rohsenow and J.P. Hartnett, Eds., Handbook of Heat Transfer, McGraw-Hill Book Co., New York (1973), Section 3, pp. 3-14 to 3-16. 6. J.L. Throne, Plastics Process Engineering, Marcel Dekker, Inc., New York (1979), Figure 10.5-6A, p. 518. 7. S.W. Churchill, Viscous Flows: The Practical Use of Theory, Butterworths, Boston (1988), pp. 12-13. 8. R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, John Wiley & Sons, Inc., New York (1960), pp. 401-404. 9. CP. Kothandaraman and S. Subramanyan, Heat and Mass Transfer Data Book, 3rd Ed., John Wiley & Sons, Inc., New York (1977), pp. 13-16. 10. F. Kreith, Principles of Heat Transfer, 2nd Ed., International Textbook Company, Scranton PA (1965), Appendix IL 11. J.P. Holman, Heat Transfer, 4th Edition, McGraw-Hill Book Co., New York (1976), p. 207. 12. W.K. McConnell, Jr., Handout, SPE Industrial Thermoforming Symposium and Workshop, Arlington TX, 12-14 March 1985. 13. T.R. Goodman, "Application of Integral Methods to Transient Nonlinear Heat Transfer", in T.F. Irvine, Jr., and J.P. Hartnett, Eds., Advances in Heat Transfer, Vol. 1, Academic Press, New York (1964), p. 54, p. 59. 14. D.R. Croft and D.G. Lilley, Heat Transfer Calculations Using Finite Difference Equations, Applied Science Publishers, London (1977).

15. G.M. Dusinberre, Heat-Transfer Calculations by Finite Differences, International Textbook Co., Scranton PA (1961). 16. P.C. Powell, Engineering with Polymers, Chapman & Hall Ltd., London (1983), pp. 254-257. 17. R. C. Progelhof and J. L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, New York (1993), p. 342. 18. G. Gruenwald, Thermoforming: A Plastics Processing Guide, Technomic Publishing Co., Inc., Lancaster PA (1987), pp. 34-36. 19. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, New York (1993), Figure 4.34. 20. G.L.F. Ehlers, "Thermal Stability", in E. Baer, Ed., Engineering Design for Plastics, Reinhold, New York (1964), p. 402. 21. Z. Tadmor and C G . Gogos, Principles of Polymer Processing, John Wiley & Sons, New York (1979), p. 163. 22. D. Burgess, "Vacuum Forming of High-Density Polythene", British Plast., 32 (1959), pp. 195-6, 223. 23. W.K. McConnell, Jr., "Thermoforming Plastic Sheet and Film", from Tool and Manufacturing Engineers Handbook, Vol. 2., 4th Ed., Part of handout at SPE Industrial Thermoforming Symposium and Workshop, Arlington TX, 12-14 March 1985. 24. G. Beall, Glenn BealVs Design Guide II for Pressure Formed Plastic Parts, Arrem Plastics Inc., Addison IL (1985), pp. 11-12. 25. J.H. Schut, "Make Way for Lots of Firsts", Plast. World, 52:6 (Jun 1994), pp. 57-61. 26. A. Kobayashi, Machining of Plastics, McGraw-Hill Book Co., New York (1967), Chapter 1, "Fundamental Considerations". 27. J. Florian, Practical Thermoforming: Principles and Applications, Marcel Dekker, Inc., New York (1987), pp. 112-113. 28. G. Gruenwald, Thermoforming: A Plastics Processing Guide, Technomic Publishing Co., Inc., Lancaster PA (1987), p. 57. 29. J. Florian, Practical Thermoforming: Principles and Applications, Marcel Dekker, Inc., New York (1987), p. 112. 30. A. Kobayashi, Machining of Plastics, McGraw-Hill Book Co., New York (1967), p. 32. 31. L.E. Nielsen, Mechanical Properties of Polymers, Reinhold, New York (1962), p. 6. 32. R. W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, John Wiley & Sons, New York (1976), p. 262. 33. AJ. Kinloch and R J . Young, Fracture Behaviour of Polymers, Applied Science Publishers, London (1983), p. 87. 34. R.W. Hertzberg and J.A. Manson, Fatigue of Engineering Plastics, Academic Press, New York (1980), p. 136. 35. P.I. Vincent, "Fracture—Short-Term Phenomena", in N.M. Bikales, Ed., Mechanical Properties of Polymers, Wiley-Interscience, New York (1971), p. 125. 36. J. Florian, Practical Thermoforming: Principles and Applications, Marcel Dekker, Inc., New York (1987), pp. 103-106. 37. H. Voigt, Lehrgang fur Thermoformung, Paul Kief el Thermoformmaschinen GmbH, Freilassing, Germany, undated, p. 1.4.2. 38. G. Gruenwald, Thermoforming: A Plastics Processing Guide, Technomic Publishing Co., Inc., Lancaster PA (1987), p. 59. 39. J.P. Holman, Heat Transfer, 4th Edition, McGraw-Hill Book Company, New York (1976), pp. 38-39. 40. J.P. Holman, Heat Transfer, 4th Edition, McGraw-Hill Book Company, New York (1976), p. 40. 41. J. Florian, Practical Thermoforming: Principles and Applications, Marcel Dekker, Inc., New York (1987), pp. 106-112. 42. M. Bakker, Ed., The Wiley Encyclopedia of Packaging Technology, John Wiley & Sons, New York (1986). 43. J. Florian, Practical Thermoforming: Principles and Applications, Marcel Dekker, Inc., New York (1987), Figure 31-1.

44. J. Florian, Practical Thermofowning: Principles and Applications, Marcel Dekker, Inc., New York (1987), p. 116. 45. J. Florian, Practical Thermoforming: Principles and Applications, Marcel Dekker, Inc., New York (1987), p. 114. 46. J. Frados, Ed., Plastics Engineering Handbook, 4th Ed., Van Nostrand Reinhold, New York (1976), p. 701. 47. O.M. Griffin, "Thermal Transport in the Contract Melting of Solids", Polym. Eng. Sci., 12 (1972), pp. 265-271. 48. J.L, Throne, Plastics Process Engineering, Marcel Dekker, Inc., New York (1979), p. 824. 49. J. Korican, "Water-Jet and Abrasive Water-Jet Cutting", in CA. Dostal, Ed., Composites, Volume 1, Engineered Materials Handbook, ASM International, Metals Park OH, 1987, pp. 673-675. 50. R.C. Progelhof and J.L. Throne, Polymer Engineering Principles: Properties, Processes, Tests for Design, Hanser Publishers, New York (1993), p. 551. 51. S.R. Rosen, "Trimming Basics", paper presented at SPE Thermoforming Conference, Midland MI (Sept. 1992). 52. R.M. Caddell, Deformation and Fracture of Solids, Prentice-Hall, New York (1980), p. 209.

6 Thermoforming Molds 6.1 6.2

Introduction Prototype Molds Wood Fiberboard Plaster Plastic White Metal Nickel 6.3 Production Molds Aluminum Steel Other Metals 6.4 Mold Coolant Channels Mold Channel Flow Expansion Contraction Sharp-Edged Orifice 6.5 Vent Holes Sizing Vacuum Systems—Steady State Sizing Vacuum Systems—Dynamic Solenoid Valve Flow Resistance Vent Hole Resistance to Flow Vent Hole Diameter Other Types of Vents Vent Hole Placement 6.6 Surface Treatments Surface Texture 6.7 Plug Design Considerations Plug Materials Wood Plugs Plastic Plugs Metal Plugs Plug Design Concepts 6.8 Sheet Clamping 6.9 Sag Bands and Sheet Supports 6.10 Other Aspects of Mold Design Undercuts Encapsulation Moving Elements Stripper Plates/Bars Mold Releases Web Breakers, Catchers and Chasers Moats, Dams and Double Steps Chamfers and Radii Prestretching Restraints 6.11 Efficient Use of Sheet Heavy-Gage Sheet Thin-Gage Sheet 6.12 References

6.1

Introduction

The thermoforming mold can be as simple as a smoothed block of wood over which a heated sheet is draped. Many production molds may be as complicated as injection molds and may include: • • • • • • •

Moving elements, Articulated plugs, Sophisticated means for isolating sheet over multicavities, Means for ejecting formed sheet from mold cavities, In-mold trimming of formed sheet, Means for retrieving trimmed parts, and Down-stream filling and sealing means.

Selection of suitable mold materials depends to a great degree on the severity and length of service. If only a few parts are to be made and the hot plastic sheet is deformed at relatively low temperatures with vacuum alone, wood, plaster or reinforced or filled thermosetting resin is usually used as a mold material. On the other hand, if thousands of parts are needed and high temperatures and pressures are needed to achieve final part design criteria, durable mold materials such as aluminum and even steel are used. Regardless of the materials used, every mold must meet specific criteria and must therefore have certain features: • • • • • • •

Provisions are needed for clamping the sheet against the mold surface, Vent or vacuum holes are needed in those areas into which the sheet is drawn last, The mold surface should be relatively nonadhesive during sheet contact and static-free after the sheet is removed, The mold surface must be sufficiently hard to retain its shape and texture for the lifetime of the part, The mold must not change in dimensions during fabrication through thermal expansion, The mold must not change in dimensions during storage through moisture absorption, and The mold material should be relatively free from chemical and moisture attack to minimize: Swelling, Cracking, Rusting, Corrosion, and Patina development.

Appropriate mold material selection depends on economic considerations as well. Some of these are: • •

Initial cost of the material, Ease of fabrication,

• • • • • • • • •

Ease of repair, Ease of maintenance, Ease of serviceability, Quick-change suitability, Requirements for safe storage, Mold weight, Availability of mold materials, Shop personnel familiarity with the material, and Material toughness against accidental abuse while in use and in storage.

Some of these intrinsic factors frequently rule out an otherwise acceptable mold material [I]. In this chapter, prototype and production mold materials are discussed. Design of coolant flow channels and vacuum holes are then described. Some design features regarding surface treatment and surface texture are detailed. Plug design is described. Clamping methods, sheet supports, rapid mold change concepts and some aspects of break-away molds are discussed.

6.2

Prototype Molds

Strictly speaking, prototype molds are designed to produce a few parts. These parts may be used to demonstrate part design concepts that are to be subsequently either production thermoformed or injection or blow molded. The parts may be used for product testing and prototype assembly schemes. Frequently, however, prototype molds are used to produce tens to hundreds of parts in a classical production run. Some of the many heavy-gage sheet applications that are made exclusively on prototype molds include: • • • • • • • • • •

Indoor and outdoor signs, Point-of-purchase displays, Horticultural pool shells, Swimming pools, Spas, Plant containers, Custom dunnage, Bathtubs, Equipment housings, and Medical furniture.

Wood, plaster, glass fiber-reinforced plastic, filled epoxy and zinc and white metal alloys are materials commonly used for prototype molds. Each has many advantages and some serious limitations. A review of each follows.

Table 6.1 Finishing Qualities of Selected Hardwoods Used in Prototype Thermoform Molds [2] Wood

Planing

Shaping

Drilling

Sanding

Resistance to splitting

Ash Hickory Hard maple Red oak Walnut White oak Mahogany

C C E A D D C

E H C H G G F

A A A A A A A

C B G B G G E

C D E C E E F

A = 90-100 C = 70-79 E = 50-59 G = 30-39 J = 10-19 B = 80-89 D = 60-69 F =40-49 H = 20-29 K = 0-9

Wood Hardwoods are used for prototype and short production runs. The wood must be thoroughly kiln-dried before shaping to minimize stress relief in the form of warping, cracking and checking during fabrication, forming and storage. Properties of typical hardwoods used in thermoforming are shown in Tables 6.1 and 6.2 [2]. Woods such Table 6.2 Minimum Mechanical Properties of Selected Hardwoods Used in Prototype Thermoform Molds [2] Property

Unit

Wood type Shagbark hickory

Sugar maple

Pinoak red oak

White oak

Walnut

40 640 1.57 10.8 23.7 1.67 4580 31.6

35 560 1.55 10.7 13.3 0.94 4020 27.7

36 580 1.32 9.10 14.0 0.99 3680 25.4

37 600 1.25 8.62 11.6 0.82 3560 24.5

32 510 1.42 9.79 14.6 1.03 4300 29.6

670 4.62 5.96

840 5.79 5.45

640 4.41 6.28

720 4.96 5.11

670 4.62 5.31

490 3.38 8.78

38 0.97

74 1.88

40 1.02

48 1.22

42 1.02

37 0.94

White ash Density Flexural modulus Work to maximum load Maximum crushing strength parallel to grain Fiber stress at proportional limit Parallel-toperpendicular compression ratio Impact bending height to complete failure

lb/ft3 34 kg/m3 550 x 106 lbf/in2 1.44 GPa 9.93 in-lbf/in3 16.6 cm-kg/cm3 1.17 lbf/in2 3990 MPa 27.5 lbf/in2 MPa

in m

as walnut or pecan are oily. The exuding oil may interfere with adhesives during fabrication and may be undesirable during forming. Wood such as balsa, boxwood and certain second-growth pines are too soft for mold materials. Repeated forming pressure crushes the cell structure, producing poor surfaces and loss of dimension. Conventional woodworking techniques are used to fabricate the molds. Wood sections are usually assembled with fluted dowels and resorcinol, hot melt or epoxy adhesives. Hot melt adhesives allow faster assembly but yield weaker joints than other adhesives. Standard wood-filling techniques are used to remove surface defects during mod fabrication. Vent holes are usually drilled through the primary surface first, then enlarged by drilling from the back of the mold. Final sanding to 200 grit is usually sufficient for most low-pressure forming techniques. It is very important that the assembled wood be carefully and thoroughly dried prior to finishing. During forming, the rapid, cyclical heating from contact with the hot plastic will pull moisture from the wood. If the wood is not thoroughly dried, the wood can crack or check. Wood grain is characterized by alternating hard and soft bands. Occasionally, the softer portions preferentially shrink. This leads to an unacceptable texture that is transferred to the formed shape. After thorough drying, the surface may need to be sealed with temperature-resistant enamel or varnish1. Recently, epoxy enamels and varnishes have been developed that protect wood surfaces for hundreds of cycles without refinishing. For low-temperature forming of polymers such as cellulose acetate, flexible PVC and PS foam, polyurethane varnishes work satisfactorily. If mold release is needed, paraffin wax, carnuba wax, Treewax, Vaseline or a light coating of white grease is used. For food contact and medical parts, vegetable oil spray is used. Wood is also used as a component in molds made of more permanent materials. For example, since wood is rapidly and easily shaped, temporary plugs, web catchers and web breakers can be formed during early mold shake-down trials to determine optimum material distribution. Owing to the high compression strength and low density of end-grain woods such as oak and walnut, these are used for permanent plugs in many production runs. Fiberboard Fiberboard is produced by mixing wood fibers with thermosetting resins such as urea or phenolic, then pressing the mass into sheets at elevated temperature to crosslink the resin. The sheet product so produced is typically 0.125 to 3 in or 3.2 to 75 mm in thickness and is available in at least three grades:

1

It may not always be necessary or desirable to seal the wood surface. As an example, the mold is not usually sealed when it is used to produce parts of slightly different designs. When a wood that is not thoroughly dried is sealed, then cyclically heated, the moisture diffusing to the surface will crack or blister the sealant. Excessive blistering leads to unacceptable surface texture in the formed shape. Sometimes it is more feasible economically to rebuild the mold than to strip the sealant from the surface.

Low-density fiberboard, with a density range of 0.1 to 0.4 g/cm3 or 6 to 25 lb/ft3. This product is used in construction as moderately good insulation sheathing. One brand is known in the US as Cellotextm, • Medium-density fiberboard, with a density range of 0.5 to 0.7 g/cm3 or 30 to 45 lb/ft3. This product is relatively new and is produced by radio-frequency curing of the thermosetting resin. It finds use in the thermoforming industry as a mold material, as described below. One brand is known in the US as Moldboardtm [45], and • High-density fiberboard, with a density range of 0.8 to 1.3 g/cm3 or 50 to 90 lb/ft3. This product is used in construction as grainless dense paneling. One brand is known in the US as Masonitetm. •

Physical properties of medium-density fiberboard used as a mold material are given in Table 6.3 [44]. Pressure is not very high during fabrication of this density fiberboard. As a result, the board is somewhat porous. This allows vacuum to be drawn everywhere through the mold surface, as shown in schematic in Fig. 6.1. The fiberboard is worked using conventional woodworking saws and shapers. Overheating the cutting surface will result in gum deposit on the cutting surface, however. The porosity of the fiberboard is not exceptional. As a result, fiberboard molds work best for relatively shallow-draw male parts. Although primary surfaces are usually semi-gloss or matte, there are no vent hole nibs or nipples. Plaster Most commercial molding plasters are not strong or tough enough to be used for prototype molds. The properties of the few that are strong enough are given in Table 6.4 [3-5]. Plasters are inorganic calcicious materials that hydrolytically react and

Table 6.3 Medium-Density Pressed Fiber Board for Prototype Tooling [44] Property

Density Internal bond Modulus of rupture Modulus of elasticity Linear expansion from 50% to 90% RH at room temperature Moisture content Thickness tolerance Length/width tolerance Corner to corner tolerance

Unit

Thickness 3/16 to 5/16 in

3/8 to 7/8 in

1 to \\ in

lb/ft3 lbf/in2 lbf/in2 lbf/in2 %

50 125 5,500 500,000 40

48 110 4,500 450,000 30

47 90 4,000 400,000 30

% Wt

5 to 7 ±0.005 ±0.0625 ±0.0156

5 to 7 ±0.005 ±0.0625 ±0.0156

6 to 8 ±0.005 ±0.0625 ±0.0156

in in in/ft

Air

Porous Mold

Air

Vacuum Hole

Vacuum Hole

Vacuum Box

Vacuum Line to Surge Tank Figure 6.1 Schematic of mold, vacuum holes, vacuum box and corrugated line to surge tank. Mold material here is porous fiberboard through which air will pass

Table 6.4 Molding Plasters [3,4] Commercial

Pattern shop hydrocal (Hydrocal A-Il) Industrial white Hydrocal Ultracal 30 Densite K5 Super X Hydro-Stone

(PPh)

Setting time (min)

Dry compressive strength (MPa) Obr/in2)

US Gypsum

54-56

20-25

22.1

3,200

US Gypsum

40-43

20-30

37.9

5,500

US Gypsum Georgia Pacific US Gypsum

35-38 27-34

25-35 15-20

50.3 65.5

7,300 9,500

21-23

17-20

96.5

14,000

Source

Water ratio

harden when mixed with water. Molds are formed by casting the newly prepared water-plaster mixture against a part pattern. Since the hydrolytic reaction is exothermic to 1000C or 2000F or so, the pattern should not be fragile. Typically, pattern surfaces are coated with a release agent such as a water soluble polyvinyl alcohol or PVOH. Soaps such as Murphy's oil soap are also used1. Vents are designed in by placing release-agent-coated wires perpendicular to the pattern surfaces prior to casting. For very thick molds made in multiple pours, soda straws are placed over the wires after the first pour. This allows the wire to be easily extracted later and also provides for a modicum of "back-drafting". For very thick molds made in a single pour, release-agent-coated tapered pins should always be used. 1

Murphy's oil soap is available in leather goods stores and laundry and cleaning areas of grocery stores. It is used primarily to clean porous surfaces such as wood and leather. To use it as a water-soluble release agent, coat the pattern with a generously thick coat and allow it to dry thoroughly. Add a second coat to those areas that are not at least semi-glossy.

A very hard void-free surface on the mold is achieved by "splatting" a thin layer of relatively high water content plaster slurry against the pattern. This thin layer is allowed to harden before further casting continues. This technique is adapted from that used by artisans working in fine plaster art [3]. To ensure adequate dispersion, the plaster must be very carefully mixed into the water. Once the plaster and water are thoroughly mixed, the mass is vibrated for several seconds to dislodge large air bubbles. Sisal, straw and glass fibers are excellent reinforcing fibers for plaster1. To ensure adequate wet-out and adhesion, the organic fibers are soaked in water for several minutes before being mixed into the water-plaster mixture. Typically, fiber reinforcement levels are 10% to 20% (vol). The mixture becomes harsh with higher reinforcement levels and air bubbles and voids are easily entrapped. Additional water reduces the strength of the plaster matrix. Typically, 35 parts water per hundred parts of plaster is a useful starting recipe [4]. Cure against the pattern usually takes about 30 min, although slower reacting plasters and those with fibers or fillers need as much as 1 h to reach demolding or green strength given in Table 6.4. While casting the primary mold, the moldmaker usually pours a simple test mold. This test mold is used to judge the level of exotherm and the degree of cure. After the pattern has been removed and the vent wires pulled, the mold is set aside in a dry, warm area for several days. This time is needed to develop the final properties and to stabilize the water content. The mold can crack if it is used before being thoroughly dried. The mold surface is then finished with open grit paper. Epoxy floor varnish is used if a smoother surface is needed. However, plaster molds are usually used for expediency and a fine surface is of secondary concern. Plaster molds are surprisingly durable. They can withstand cyclic forming temperatures of most commercial plastics quite well. They are quite heavy however. The molds should never be subjected to flexural loading since failure is catastrophic. Plaster breaks are repaired with epoxy, as are spalled surfaces. Auto body putty also works well. However, broken molds are rarely repaired and so patterns are usually preserved to produce a new mold. Glass- and polymer-fiber reinforced plasters and cementitious products such as water-curing hydraulic cement are also prototype mold materials, but material costs are up to 5 times those of conventional molding plasters. Plastic Great advances have been made in the last few years in formulating thermosetting plastic resins for thermoforming mold. In particular, plastic molds are economically preferred for: • •

Heavy-gage sheet forming, Part forming where heat buildup on the mold surface is intermittent,

1

Other fibers are used to reinforce plaster. As an example, polypropylene and polyethylene fibers are used to produce impact-resistant construction grade plaster board. Normally, these fibers are too expensive to be used for prototype thermoforming tooling.

• •

• •

Parts where the mold does not need to be heated to facilitate forming, Very large surface area parts such as, Outdoor signs, Camper tops, Outdoor swimming pools, Drape and low-pressure vacuum forming, and Prototype forming for less than 100 parts.

For applications where mold surface temperatures do not exceed 600C or 1400F and where drape and vacuum forming are used, epoxy and unsaturated polyester resin or UPE are materials of choice. These resins are usually combined with various forms of glass fibers. The exact manner of mold fabrication depends on the mold shop, but the general procedure usually followed is described below. Since thermosetting resins usually cure most efficiently at relatively high temperatures of 1000C to 125°C or 2000F to 2600F, the pattern must be thoroughly dried before beginning mold fabrication. Plaster and wood patterns should be thoroughly air oven dried at 500C or 1200F for at least 24 h [5]. The pattern surfaces are then sealed with at least two coats of air-sprayed thinned automotive lacquer or industrial polyurethane or PUR. The hard surfaces are then waxed twice with a hard paste wax such as carnuba or Treewax. The surfaces are power-buffed after each wax application. PVC plastisol is recommended as a release agent [5]. If the patterns are thoroughly dried, aqueous polyvinyl alcohol or PVOH can also be used. These release agents should be air-sprayed to ensure uniform coverage. Two coats are recommended. Generally, epoxy and UPE molds are fabricated in similar ways, although UPE mold construction is more complex and requires more steps. For glass fiber-reinforced UPE or GR-UPE molds, a 0.3 to 0.4 mm or 0.010 to 0.015 in layer of special resin known as gel-coat is first sprayed against the prepared pattern. A typical gel-coat recipe is given in Table 6.5. This layer is allowed to air-cure, either at room temperature or under infrared lamps, until tacky. An optional second layer can then be applied. The surface should then be carefully inspected for bubbles or other defects. Care at this point can obviate expensive surface repairs later on. A layer of very fine glass fabric known as C-veil is then placed against the gel-coat. A thin layer of UPE laminating resin is sprayed over the fabric and gently squeegeed into it1. Once the C-veil layer has cured to a tacky state, the first layers of reinforcing glass are laid. The glass fabric normally used is a plain weave of about 4 to 10 threads per cm or 10 to 25 threads per in [tpi]. The fabric weight is about 500 g/m2 or 20 oz/yd2. Plain weave is used because: • • • •

It It It It

has good strength in both directions, can be easily oriented and shifted to fit tight curves, has an open weave to minimize air entrapment, and is an expensive fabric structure.

1

An alternate method involves dipping the C-veil in catalyzed resin and applying the wetted fabric to the gel-coated surface. Since the wet fabric is quite weak and the resin drains readily from it, this approach is quite messy and consumes more resin than spraying. However, it leads to far fewer dry pockets between the gel-coat and C-veil.

Table 6.5 Typical Recipe for Unsaturated Polyester Resin Gel-Coat for Prototype Thermoforming Molds Resin recipe

Molar ratio

Isophthalic acid Maleic anhydride Neopentyl glycol, glycol excess, 2%

1 1 2.04

Acid number « 18 Hydroxyl number « 30 Gel time % 5 min Peak exotherm « 225°C or 4370F Formulation (polystyrene in polyester at 40 wt %)

Parts by weight

Polyester, as above TiO2 pigment Styrene diluent Fumed silica Cobalt octoate promoter Methyl ethyl ketone peroxide catalyst

48 20 32 1.5 0.3 to 0.6 0.2 to 0.3

Clear casting properties on 0.32 cm or 0.125 in thick sample Flexural strength Flexural modulus Elongation Tensile strength Heat distortion temperature

145 MPa 3.9 GPa 1.9% 65.5 MPa 112°C

21,000 lbf/in2 570,000 lbf/in2 9,500 lbf/in2 234°F

Strips are cut 50 mm by 500 mm or 2 in by 20 in, hand-dipped in catalyzed resin and hand-applied to critical high-stress areas of the mold such as corners and rim. The strips are then hand-rolled to express air and ensure intimate contact with the C-veil layer. Usually the resin bath is catalyzed for a 1 h gel time1. The moldmaker needs to know approximately how much resin can be applied in that time. A typical resin recipe is given in Table 6.6. Since UPE resins exotherm when curing, the thickness of the built-up layer of uncured resin on the mold must be restricted to about 6 mm or 0.25 in. Excessive thickness will cause the resin to crack during curing. Thicker mold sections are fabricated by building atop the mostly cured mold substrate. Once the critical stress areas on the mold have been constructed, reinforcing elements such as thoroughly dried, untreated wood or plaster are added. Scrap fully cured GR-UPE pieces are also used. These are held in place with resin and resin-wetted woven glass fabric. Automotive body putty, a filled UPE product, can also be pressed into irregular areas. These materials are then cured in place. The rest of the mold surface area is then built, either by hand dipping squares of fabric in catalyzed resin and hand applying, or by applying dry fabric to the surface and 1

The gel time is that time when the resin becomes stringy and jelly-like. Beyond the gel time, no further manipulation such as rolling or expressing, is possible.

Table 6.6 Typical Recipe for Unsaturated Polyester Resin With Fiberglass Mat Reinforcing for Prototype Thermoforming Molds Resin recipe

Molar ratio

Isophthalic acid Maleic anhydride Propylene glycol

1 1 2.2

Acid number « 8 Hydroxyl number « 45-50 Gel time « 8 - 1 2 min Peak exotherm « 225°C or 437°F Formulation (polystyrene in polyester at 40 wt %) [recipe depends on wet-out method, type of fabrication—hand layup, spray-up, etc.]

Parts by weight

Polyester, as above TiO2 pigment Styrene diluent Cobalt octoate promoter Methyl ethyl ketone peroxide catalyst

42 28 30 0.3 0.15 to 0.3

Total physical properties of fiberglass-reinforced polyester Flexural strength Flexural modulus Elongation Tensile strength Heat distortion temperature

138 MPa 7.0 GPa 1.0% 82.8 MPa 1900C

20,000 lbf/in2 1,000,000 lbf/in2 12,000 lbf/in2 375°F

squeegeeing resin into it. These layers are then cured to tackiness. Once the entire pattern surface has been covered with at least one layer of nearly-cured fabric and resin, additional layers are added in rapid succession. Wet fabric does not stick well to vertical surfaces. As a result, the mold orientation during fabrication usually dictates the total mold construction time. After a solid reinforced layer is built over the entire mold surface to 3 mm or 0.125 in or so, chopped glass and reactive resin is sprayed onto it. Since the sprayed material resembles wet hay, it must be carefully rolled to express air. Rolling is done with special spaced-disk rollers. The reason for restricting spray-up techniques to supporting roles for hand lay-up is that spray-up laminates have only about 70% to 80% of the flexural strength and modulus of hand lay-up laminates [5]. The spray-up technique is much less labor intensive than hand lay-up and so mold costs are reduced by using it in a supporting role. Once the minimum mold thickness of about 6 mm or 0.25 in is reached, the inner mold structure is constructed. The moldmaker must keep in mind that molds built this way are very large. Even the low pressures used in vacuum forming generate substantial forces that can crush or buckle an unreinforced plastic mold. A standard inner structure is an egg-crate of 2 to 2.5 cm or 0.75 to 1 in thick exterior plywood or laminated wood having 10 to 15 cm or 4

Figure 6.2 Egg-crate cradle that fits into male mold

to 6 in openings. For a male mold, the egg-crate is shaped to fit inside the formed mold structure (Fig. 6.2). For a female mold, the egg-crate is fashioned into a cradle to support the flat sides and bottom of the mold (Fig. 6.3). The plywood is held in place with reactive resin and glass fiber tape or automotive body putty. The plywood core allows for easy access to vent holes. For additional reinforcement, aluminum or phenolic-impregnated paper honeycomb sheet up to 5 cm or 2 in thick is used. This is just laid into the wet resin against the mold back and held in place until the resin cures. PVC, aluminum or GR-UPE pipe is also used for reinforcing. Pipe sections are joined to each other and to the mold back with resin-wet fabric strips and held until the resin sets. The rim of a very

Figure 6.3 Egg-crate cradle into which female mold fits

Figure 6.4 Pipe reinforcement in castable reinforced resin mold

large mold is commonly reinforced in this manner (Fig. 6.4). If the mold must withstand large buckling forces although not necessarily large pressures, lightweight organic cement can be cast against the mold back. Recently, many cements have been developed where a low density aggregate such as slag or Perlite expanded material is mixed with 5 to 10% (wt) UPE or epoxy resin as a binder, then troweled in place. Expanded polystyrene or EPS beads can be mixed with low viscosity epoxy resin as a low-density backing agent1. It is necessary to ensure adequate vent hole placement and access before the mold back is sealed in several cm or inches of cement. Once the mold construction is complete, it must be cured thoroughly, preferably at room temperature. Curing normally takes 24 to 48 h, although more time may be required if some areas are quite thick. To ensure a hard, thorough cure, the mold and pattern is placed in a warm air oven at 500C or 1200F for an additional 24 to 48 h. It is difficult to cure UPE in thin cross-sections to a tack-free state at room temperature, however. Oxygen inhibits reactivity on exposed surfaces. One way around this is to spray a thin layer of air-drying, film-forming PVOH solution on the exposed curing UPE surface. Once the UPE is fully cured, the resulting thin film is then stripped from the mold. If the pattern has been properly prepared, it should release easily from the mold. UPEs and epoxies shrink on curing. If sufficient taper has not been provided on female molds and female portions of male molds, the mold and pattern can become locked together. Even with adequate pattern preparation, considerable manpower and time may be required to release very large molds from patterns. Mold release is best accomplished with combinations of air, water and weight. The entire assembly is suspended slightly above the shop floor with the heavier element nearest the floor. Air and water from separate sources are forced between the edges of the mold and the pattern at the interface. In desperate cases, the pattern must be destroyed to free the mold. The mold surface is then adequately cleaned, waxed and buffed prior to use. Even though plastic molds can last for hundreds of cycles, surface deterioration can begin 1

EPS cannot be used with unsaturated polyester resin, since UPE contains styrene monomer, a solvent for EPS.

in a very few cycles. The most common problem is pinholing, due to collapse of small bubbles trapped in and behind the gel-coat during fabrication. Some dimples and dents are caused by problems in applying the release coat to the pattern. Delamination and blistering are also problems. Usually surface repairs are straightforward. The defect area is sanded or ground down to good material. A patch of catalyzed resin containing a filler such as fumed silica is then troweled into the defect. Cure is by infrared lamp. The region is then finished by feathering into the surrounding good mold material. A cave-in, break-out or development of star cracks during forming require heroic repair efforts. Growing numbers of these usually indicate that the mold was improperly reinforced. Fabrication of a new mold is preferred over attempted repair. Although this section has focused on UPE mold fabrication, epoxy molds are fabricated in a similar fashion [5]. Most epoxies are mineral- or aluminum-powder filled [6]. The extent of shrinkage varies with the filler content. There is very little exotherm generated with the higher loaded epoxies. Glass fabric is used with epoxies in a manner similar to that with UPE. Some molding epoxies have very high viscosities, however, and substantial effort is required to thoroughly wet out the glass fabric. Water lines are easily cast in place in molds made using either UPE or epoxy. GR-UPE is economically preferred for very large molds and water lines can be PVC pipe or aluminum electrical conduit. Epoxy is more expensive than GR-UPE and so is used for small molds where water lines are aluminum or copper tubing. Filled thermosetting polyurethanes are now available for prototype tooling [46]. The PUR systems are typically two-part 1:1 mixtures. Most of the systems are heavily loaded, typically with calcium carbonate or other inorganics. By varying the catalyst concentration, prepared recipes have been developed that allow pot life times of a few minutes to several hours. Table 6.7 gives properties of some typical prototype PURs. Two other approaches for producing prototype molds are being developed. The first uses the concepts of rapid prototyping, detailed in Section 7.3. Rapid prototyping or RP is the generic term used to describe several methods for producing prototype parts [47]. The process uses the mold shape as the computer-generated image. Stereolithography or SLA has been used to produce small prototype molds from UV-curable polyurethane. The cured SLA sheet was back-filled with aluminumfilled epoxy for rigidity [48]. Syntactic foams are also being used to produce prototype tools. The computer-generated mold image is fed to a multi-axis trimming station, Chapter 5. The syntactic foam is then machined into the prototype mold. Both techniques provide rapid turn-around times. These allow the designer to evaluate specific part details and to make necessary changes in a very short time. SLA and syntactic foam molds are not expected to produce more than a few parts. Recently, aerospace applications have included thermoforming of high temperature polymers1. The typical GR-UPE and GR-epoxy molds described above are restricted to about 600C or 1400F use temperature. Higher temperature epoxies are 1

Composite thermoforming is covered in detail in Chapter 9 on advanced thermoforming techniques.

fet7] sPl°P\[ uwojouiiaqx adijojojj JOJ opsBfj §ui[oox auBqjajnifoj jo sapjadojj //9 ajqcx Property

Gel time Demolding time Specific gravity Hardness Shrinkage Compressive strength Flexural strength Flexural modulus Tensile strength Tensile modulus Izod impact strength Heat deflection temperature @ 66 lb f /in 2 @ 264 lb f /in 2 Mixed viscosity Volumetric yield

Unit

Reprotm Identifier Fast

10

83

Slow

Light

min @ 770F min g/cm3 Shore D % lbf/in2 lbf/in2 x 1000 lb f /in 2 lbf/in2 x 1000 lbf/in2 ft-lbf/in

4-5 15-30 1.90 83-85 0.05 6,470 5,140 939 3,130 941 0.31

5-6 30-60 1.90 83-85 0.04 6,470 5,140 939 3,130 941 0.31

6-7 60-90 1.90 83-85 0.02 6,470 5,140 939 3,130 941 0.31

14-16 180-240 1.90 83-85 0.01 6,470 5,140 939 3,130 941 0.31

6-8 90-120 0.90 68 0.05 3,980 2,620 347 1,530 350 0.15

0

57 55 1,100 16

57 55 1,100 16

57 55 1,100 16

57 55 1,100 16

56 48 1,500 30

C C cp in3/lb 0

used to produce molds with use temperatures to 1500C or 3000F. Molds of these materials require much high curing temperatures and much greater care in pattern making and preparation. One manufacturer [7] recommends graphite cloth with aerospace-grade epoxy for fabricating molds that see sheet temperatures of 2000C or 3900F or more. These molds are used to form reactive reinforced resin sheet at low temperatures with curing of the formed sheet against the heated mold at temperatures of 2000C or 3900F. These sophisticated molds include aluminum or copper pipe so that hot oil can be used as a coolant. A mold life of 500 parts is claimed.

White Metal The metal welding industry relies on the establishment of an intensely hot arc that is drawn between electrically isolated metal surfaces energized with high amperage, low voltage DC power. The arc causes most common metals to melt. When high velocity air is blown into the arc, the molten metal breaks into very fine drops. The molten drops can be transported by the air for short distances before cooling below their fusion temperature. One system, the TAFA system, is shown in Fig. 6.5 [8,50]. The TAFA system establishes the electrical arc between metal wires that are fed in a controlled fashion into the spray zone. By accurate control of the metal wire speed, a uniformly fine metal spray issues. The spray is used to coat a surface. Although most metals can be sprayed, zinc and zinc-alloy metals offer good balances of: • • • • • •

Flexibility in spraying, Relatively low molten metal temperatures, thus preserving pattern surfaces, Small drop sizes, Good densification and low porosity, Good hardness in the metal mold, and Good strength in the metal mold.

Typically, zinc has a spraying temperature of 4100C or 7700F and is dispensed at 5 kg/h or 10 lb/h at a power level of 50 to 150A, 20 V, 2 kW. A typical surface hardness is 70 Rb. Other physical properties are given in Table 6.8 [8]. The spraying process begins by securely fastening the pattern, if small, to a turntable and then coating it and the table edges with aqueous PVOH release agent. Once the pattern is secured and dry, the metal spray unit is activated and an arc established1. The spraying rhythm is similar to paint spraying. The objective is to build up a uniform coat over the entire surface. It is necessary to begin spraying at the edges of the pattern. This ties the edges of the metal layer tightly to the edges of the turntable. If this is not done correctly, the air carrier can infiltrate a loose edge, lift the thin metal layer, and catastrophically tear it from the pattern. The process

1

Care must be taken at this point. If the release coating is not dry, the hot metal will cause steam bubbles in the finished mold surface. If the metal is focused too long against one portion of the PVOH-coated mold, the PVOH may degrade and pits will form in the finished mold surface.

Figure 6.5 Rendering of tafa metal spray technique for thermoform mold fabrication. Photo by permission of TAFA Metallisation, Dow (Concord) NH

must then be restarted beginning with pattern preparation. It is recommended that spraying be done in a well-ventilated hood and the operator should wear a breathing helmet. The pattern can be made of temperature sensitive materials such as plastic, foam or paper. However, care must be taken to prevent concentrated hot metal spraying in a small area. If the pattern is small and spraying is continuous, the interfacial temperature between the pattern and the mold can easily exceed 1000C or

Table 6.8 Properties of Sprayed Zinc [8] Hardness Tensile strength Melting point Temperature on contact with pattern Density, % theoretical Shrinkage Cost Weight Melting power Spray droplet size

70 Rc 128 MPa 4100C 2400C

18,500 lbf/in2 7700F 6500F

92 to 95% 0.1 to 0.2% $102/m2 24.3 kg/m2 0.44 kW/kg 50 to 150 ^m

$9.50/ft2 5.0 lb/ft2 0.2 kW/lb

2100F. Temperatures to 2000C or 3900F have been measured. Typically, spraying continues until a uniform layer of metal of at least 1.5 mm or 0.060 in has been deposited. Reinforcing structures, copper water lines, and other features are then placed on the molded shell and these are encapsulated by spraying additional metal (Fig. 6.6). For greater metal rigidity in stress areas such as edges, corners, and deep recesses, the zinc layer is built to 6 mm or 0.25 in or more. To complete the mold fabrication, epoxy cements are cast behind the metal skin. If the mold needs to be a good heat sink, metal-filled epoxies of the types described above are used. These typically have continuous use temperatures to 1200C or 2500F. Note that certain epoxy formulations depolymerize at local temperatures of 2000C or 3900F or more. Although spray metal molds have been used in production for vacuum forming for years without appreciable wear, pressure forming dramatically shortens mold life.

TAFA Spray Head Molten Metal Droplets Mold

Pipe

Pattern

Figure 6.6 Schematic of metal coolant line or metal pipe rein forcement for spray-up white metal mold

When molding sheet molding compound or SMC for example, pressures to 1.5 MPa or 200 lbf/in2 and temperatures to 1200C or 2500F are common. For SMC, mold life is reduced to 100 pieces or so. The primary mode of failure is surface flaking or spalling. In SMC applications, chrome surface diffusion plating is used to harden the zinc surface. The chrome surface is 0.08 to 0.188 or 0.003 to 0.007 in thick and its hardness is 55 to 65 Rc. This treatment increases the cost and delivery time of the mold and should be used only when spray molds are used in pressure thermoforming.

Nickel Nickel molds are also used for prototype thermoforming. A thin electroformed nickel surface is reinforced with sprayed zinc metal, cast white metal or cast aluminum- or nickel-filled epoxy. The nickel used is a very pure electroplating grade of 99.95% with a trace of cobalt. The pattern surface must be conductive. For wood, plaster, ceramic and plastic patterns, a coating of PUR varnish is sprayed over the pattern surface. While the varnish is still tacky, a very fine coating of powdered graphite is air-blown onto it. The PUR is then cured, either at room temperature or in a free convection oven at 500C or 1200F. The pattern is then immersed in a cold plating bath. Nickel is laid against the surface at 4 jim/h until a uniform layer of about 1.5 mm or 0.060 in thick has been deposited over the entire pattern surface. The pattern and nickel layer assembly is then removed from the bath. The dried surface is then backed with sprayed white metal or cast epoxy. Alternately, the pattern and nickel assembly is immersed in a second plating bath where copper is added to a thickness of 10 mm or 0.400 in thick or so. Hot plating techniques lay nickel against the pattern surface at the rate of 10 to 20 um/h, but may produce a coarse-grained porous surface. Normally this surface is dull and matte. It cannot be polished to semi-gloss. But the surface is quite satisfactory for production of non-appearance low-pressure thermoformed parts. Nickel produced by hot plating has about 50% of the toughness of cold plated electroformed nickel. It also costs about 50% of the cost of cold plated electroformed nickel. Recently, a vapor deposition technique has been developed that achieves dry nickel vapor deposition of 0.25 mm/h or 0.010 in/h. Thicknesses of 1 mm or 0.040 in to 25 mm or 1-in are possible without secondary backing. Surface hardness of 40 to 42 Rc with Class A and texture finishes are possible [9].

6.3

Production Molds

Roll-fed sheet forming economics dictate high production rates and long mold life. For this reason, metals yielding low maintenance, good surface hardness, reasonably low cost and low wear are selected. Aluminum has been the choice for many years for the following reasons:

• • • • • • •

Aluminum Aluminum Aluminum Aluminum Aluminum Aluminum Aluminum

molds wear well, has outstanding heat transfer characteristics, is relatively lightweight, is easy to machine, is easy to cast, is moderate in cost, and has excellent strength-to-weight ratio.

Of these, low cost, ease of fabrication and lightweight are most important. Aluminum molds do not appreciably increase the inertial factor of mechanically acting forming presses. More details are given below. Many newer forming processes involve increased pressures, temperatures and stresses. Processes such as: • • • • • •

High speed forming, Matched die forming, In-mold punch-and-die trimming and cutting, Pressure forming, Heated mold forming for crystallizing polyethylene terephthalate (CPET), and Composite sheet forming,

are challenging the ability of aluminum to meet long-term high-production requirements. Steels that are normally used in injection molding are sometimes used in severe conditions in thermoforming. Molds for these high-demand applications may be quite expensive. The expense is justified only on the basis of faithfulness and accuracy over millions of cycles and other features, such as sliding action or in-mold punch-and-die trimming and only if these features cannot be achieved in any other mold material. Aluminum As noted, aluminum is the material of choice for nearly all thermoforming molds. It is easily fabricated, it has a very high thermal conductivity so sensible heat from the formed plastic sheet is rapidly removed and it is a lightweight, tough metal. Thermoforming molds can be made of either machined plate or cast metal. Typically, molds are fabricated from plates of either 2024-T4 or 6061-T651, aircraft grade. Table 6.9 gives the chemical analysis of these aluminums. Typical machined aluminum is 130 Brinnell. Since tool steel hardness is as high at 300 Brinnell, care must be taken during mold changing and alignment with steel members to avoid marring, gouging and dinging the softer aluminum. Aluminum also has a relatively high thermal expansion coefficient of 19XlO - 6 0 C" 1 when compared with steel at 11 x 10^ 60 C" 1 . Care must be taken to minimize abnormal stress and potential buckling or bowing when heating large tightly clamped molds. Molten aluminum is cast at 550 to 6000C or 1000 to 11000F against ceramic, dried and fired gypsum plaster or foundry sand patterns. Usually atmospheric casting is used to produce prototype of short production-run molds. Atmospheric or foundry casting yields molds that may have surface porosity and nonuniform surface hardness. Porosity

Table 6.9 Chemical Analysis of Machined Aluminum 2024-T4 or 6061-T651 Cu Mg Mn Si Fe Ni Ti Zn

1 to 2% 0.5 to 1% 0.5% 4 to 8% 1% [maximum] T% trace trace

can be a problem when pressure forming or if drilled coolant lines pass close to the mold surface. A uniformly high finish is very difficult with foundry cast molds. Pressure casting produces a much denser mold with much more uniform surface hardness, albeit at a higher cost. Patterns must be rugged enough to withstand the thermal shock of molten metal and 7 MPa or 1000 lbf/in2. Fired ceramic patterns are recommended. A casting ceramic of ethyl silicate and quartz powder can be fused into patterns that will withstand pressure casting conditions [10]. Steel The most severe service of any thermoforming process cannot match, say, typical mold temperatures, pressures, erosiveness, or number of required parts of a standard injection molding using filled or reinforced thermoplastics. As a result, thermoforming molds do not need to be constructed of the same tool steel as that specified for injection molds. However, mold shops that specialize in injection molds are accustomed to working in specific types of steels. They will therefore frequently quote jobs in these materials. Prehardened steels such as AISI P20 are recommended for large molds and molds with low demands on wear resistance. P20 is more difficult to machine and polish than, say, S7 or Hl3 steels. Since the last two must be air-hardened after machining, the mold shop does not risk mold dimensional change or distortion with P20. The chemical analyses of Hl3 and P20 are given in Table 6.10. H13 hardness is 30 to 36 Rc. Sophisticated techniques have been developed for machining and hobbing molds and for hardening steel surfaces. Although these heroic methods are not really needed for the thermoforming mold itself, hardened steel is used for mold elements such as: • • • • • •

Slides, Collapsing cores, Rails, Guides, Platen frames, Sheet clamps,

Table 6.10 Chemical Analysis of Tool Steels Oil-hardened P20 tool steel

Air-hardened Pl3 tool steel

• • •

C Cr Mo V

0.35% 5.0% 1.5% 1.0%

C Cr Mo

0.35% 1.25% 0.4%

Hardness

50 to 54 Rc

Hardness

30 to 36 Rc

Hold-downs, Pins, and Pin-chains.

Similarly, investment casting of molten steel against ceramic patterns is also not needed [H]. Other Metals In injection molding, beryllium/copper and Kirksite are frequently used for prototype tooling [11,12]. Typically, molds of these materials yield more parts than aluminum, sprayed metal and plastic molds. Steel molds yield more parts than all other materials (Table 6.11). Low-pressure thermoforming molds see much less severe service than injection molds. As a result, these exotic materials are rarely used. Nevertheless, the service guides given in Table 6.11 hold for low-pressure forming molds as well, with mold lifetimes extended by a factor of at least 10. The mold

Table 6.11 Guide to Tooling Materials [12] Material

Epoxy Sprayed metal Kirksite Beryllium/copper Aluminum Cast Machined Steel 1 2

Delivery time, (weeks) 2 2 4 4

to to to to

4 4 20 12

3 to 8 12 to 20 12 to 26

Number of parts, x 1000 Injection mold 0.1 0.05 100 500 50 100 1,000

Repairs1

Changes, texture

Surface2 finish

3 3 1 2

No No Yes Yes

C/D C/D B A/B

2 2 1

Yes Yes Yes

A/B B/C A

Thermoform 0.1 to 10 0.1 1,000 1,000 200 1,000 10,000

1 = Easy, 3 = Difficult A = Diamond, B = Buff polish, C = 400-600 Grit, D = 350-400 Grit

lifetimes for high-pressure pressure forming and composite forming at elevated temperatures are much closer to those for injection molding.

6.4

Mold Coolant Channels

Many prototype molds and all production molds are actively cooled. Recommended mold temperatures are given in Table 2.5. In production molds, the coolant lines are usually gun-bore drilled in a manner similar to those drilled in injection molds [13]. There are many ways of embedding coolant lines in prototype molds, as noted above. Plastic, aluminum or copper pipe is cast in place in spray-up GR-UPE molds and cast epoxy molds. Copper or aluminum tubing is fastened in place in spray-up white metal and electroformed nickel molds. The location, diameter and number of coolant lines depends on the size and complexity of the mold, and the heat removal potential of the mold material. The determination of heat removal rates is given below. It is recommended that a mold with a properly designed coolant channel pattern should exhibit no more than 3°C or 5°F temperature variation across its surface at steady-state production and that the coolant temperature increase is no more than 3°C or 5°F from inlet to outlet [14]. It is apparent from Table 2.5 that for most commodity polymers, the coolant of choice is hot water. Figure 6.7 shows a simple closed circulating system for treated water as a coolant in thermoforming. To cool polycarbonate, polysulfone and other engineering polymers, the water coolant system must be pressurized to 0.4 MPa or 50 lbf/in2 or more. Hot oil is recomManifold

Mold

Manifold

Coolant

Chiled Water Circulating Pump Shell-and-Tube Heat Exchanger Figure 6.7 Mold coolant plumbing schematic

mended for filled and fiber-reinforced engineering polymers and for higher temperature polymers. Mold Channel Flow Figure 6.7 shows an example of a closed circuit coolant system for a thermoform mold assembly. Pressure losses in complex plumbing systems such as that shown in Fig. 6.7 are best analyzed in terms of Darcy-Weisbach "head losses" for each of the various elements in the system [15]. For simple flow in a constant-area pipe, the head loss is given as:

where L/D is the pipe length-to-diameter ratio, v is the average fluid velocity, gc is an appropriate conversion factor, and f is the friction coefficient. The value of f depends on the nature of fluid flow in the pipe. There are three types of fluid flow that can occur in mold channel flow. These are identified in terms of the fluid Reynolds number, defined as: Dv • p Re, Reynolds number = (6.2) where D is the pipe diameter, v is the average fluid velocity, p is the fluid density, and |i is the fluid viscosity. Re is dimensionless and the other variables are in appropriate units. The fluid is called laminar when Re ^ 2000. The friction factor for laminar flow is given as:

The fluid is in transition when 2000 < Re < 10,000 or so. And the fluid is considered to be turbulent when Re > 10,000. For transition and turbulent flow, the friction factor is given in terms of a relative pipe wall roughness factor, e/D [16,17]. An approximate correlation for e/D < 0.04 and Re > 10,000 is [18,19]: f=a + b R e - c (6.4) where a = 0.094 [e/D]° 225 + 0.53 [e/D] b = 88 [e/D]044 c =1.62 [e/D]0134 For flow through non-constant area elements, the head loss equation is written as: h= K|-

(6.5)

An appropriate value for K, the head loss coefficient, is needed for each flow area change. These are summarized as follows:

Expansion For flow from a small diameter pipe, D1, to a large diameter pipe, D2, K is obtained from: * - [ . - ( D 7

When the pipe discharges into a reservoir, K = I . Contraction For flow from a large diameter pipe to a small diameter pipe, the head loss coefficient is given as:

K = (I-lJ

(6.7)

where c, the contraction coefficient, is a function of the area ratio of the pipes (Fig. 6.8). Note that c asymptotically approaches c = 0.61. This value is frequently used when the exact area ratios are unknown, such as flow through pinched tubing. Sharp-Edged Orifice

Contraction Coefficient, c

On occasion, short constrictions are used to control fluid flow. Approximate values for K for valves, elbows, tees and other constrictions are given in Table 6.12 [20,21]. In certain instances, quick-disconnects and other devices that allow for rapid

Asymptote

Area Ratio, A1 /A 2 Figure 6.8 Head loss contraction coefficient for coolant flow

Table 6.12 Head Loss Coefficients for Various Plumbing Fittings [20,21] Type of constriction

K, Head loss coefficient

Elbow Tee 180° bend Globe valve fully open Check valve fully open Gate valve fully open Gate valve 3/4 open Gate valve 1/2 open Gate valve 1/4 open Diaphragm or butterfly Diaphragm or butterfly Diaphragm or butterfly Diaphragm or butterfly

0.9 1.8 2.2 10.0 2.5 0.19 1.15 5.6 24.0 2.3 2.4 4.3 21.0

valve valve valve valve

fully open 3/4 open 1/2 open 1/4 open

disassembly are employed. These devices can be considered as sharp-edged orifices and the head loss coefficients obtained from: K = (l-l)

(6.8)

where co is obtained from Fig. 6.8, it is advisable to use a single velocity when making head loss calculations for complex systems. The complex system head loss equation would appear as:

In certain cases, as with multiple flow channels in a conventional mold, head losses in the manifolded section should be calculated separately. Example 6.1 is an analysis for water as the coolant. Example 6.2 shows the effect of using oil as the coolant. In Section 5.5, heat transfer between the sheet, its ambient air environment and the coolant medium was considered in detail. As noted, heat transfer efficiency is highest when coolant flow is turbulent, that is, when Re > 10,000 or so. This is usually the case for most dedicated water coolant recirculating systems. High Reynolds number is achieved through high fluid velocity, small pipe dimension, and low fluid viscosity. Turbulent flow is best achieved by high volumetric flow rates although it is apparent from the head loss equation, Equation 6.9, that pressure drop increases roughly in proportion to the square of fluid velocity. Increasing coolant viscosity reduces coolant effectiveness. Thus, reducing coolant temperature, changing from water to brine, water-ethylene glycol, or water-glycerine, and changing to oil usually results in increased pressure drop and a reduction in heat transfer efficiency.

Example 6.1 Water as a Coolant Consider the mold coolant system of Fig. 6.7 using water with the following dimensions: Waterline diameter = i | in Water line length (total)= 100ft Water temperature = 700F Water kinematic viscosity= 1.06x 10~5ft2js Water density = 62.4 Ib/ft3 Fluid velocity in 1\ in water line is 4.1 ft Is. Pipe and mold roughness ratio, e/D = 6x 10~5. The friction factor\ f determined from empirical Equation 6.4, to be: f= 0.01058+ 1.221 Re-044 Plumbing to the mold includes: 6 elbows 1 fully open gate valve Ij open gate valve 1 expansion into a 2\ in manifold 1 contraction from a 2\ in manifold Consider pressure loss through manifold negligible. Mold conditions include: Coolant diameter = 3/4 in Coolant length = 10 ft 4 parallel coolant lines in mold Plumbing in mold includes: 6 90° turns or elbows 1 expansion into 2j in manifold 1 contraction into 2\ in manifold 2 quick disconnects considered to be orifices Determine pressure drop through plumbing complex. Flow in Ij in waterline Reynolds number, Re = Dvp/ji = 50,000. Flow is turbulent. Friction factor, f= 0.021. Head Head Head Head Total

loss coefficient, fL/D = 16.83 loss for 6 elbows, 6 x 0.9 = 5.4 loss for valves = 5.79 loss for expansion, contraction = 1.5 head loss 29.52

Total pressure drop in \\ in waterline: v2 P = 29.52 — x p = 3.31 lbf/in2 2gc Flow in mold channels With 4 parallel lines, velocity in mold channels same as that in \\ waterline from pump. However, Re decreases: Reynolds number, Re = Dvp/ji = 25,000. Flow is turbulent. Friction factor, f = 0.0248. Head Head Head Head Total

loss coefficient, fL/D = 3.96 loss for 6 elbows, 6 x 0.9 = 5.4 loss for 2 sets of orifices = 3.374 loss for expansion, contraction = 1.5 head loss 10.27

Total pressure drop in 3/4 in mold coolant channel: P =10.27 ^ - x p =1.59 lbf/in2 Thus the total pressure drop for the plumbing system of Fig. 6.7 is given as: P total = 3.31 +1.59 = 4.9 lbf/in2 Pumping rate: Q=

v 1 | p 2 _|_ = 225gal/m . n

Example 6.2 Oil as a Coolant Consider the mold coolant system of Fig. 6.7 using oil with the following dimensions: Oil coolant line diameter — Ij in Oil coolant line length (total)= 100ft Oil temperature = 1500F Oil kinematic viscosity1 = 1.3 x 10~4 ft2/s Oil density = 54.5 Ib/ft3 Fluid velocity in Ij in oil coolant line is 4.1 ft Is. Pipe and mold roughness ratio, e/D = 6 x IO~5. The friction factor, f is determined from empirical Equation 6.4, to be: / = 0.01058+ 1.221 Re-044 1

Equivalent to SAE 10 oil

Plumbing to the mold includes: 6 elbows 1 fully open gate valve 1 open gate valve 1 expansion into a 2\ in manifold I contraction from a 2-$ in manifold Consider pressure loss through manifold negligible. Mold conditions include: Coolant diameter = 3/4 in Coolant length = 10 ft 4 parallel coolant lines in mold Plumbing in mold includes: 6 90 turns or elbows 1 expansion into 2j in manifold 1 contraction into 2\ in manifold 2 quick disconnects considered to be orifices Determine pressure drop through plumbing complex. Flow in l\ in oil coolant line

Reynolds number, Re = Dvp/ja = 3850. Flow is in transition. Friction factor, f= 0.0429. Head Head Head Head Total

loss coefficient, fL/D = 34.31 loss for 6 elbows, 6 x 0.9 = 5.4 loss for valves = 5.79 loss for expansion, contraction = 1.5 head loss 47.00

Total pressure drop in \\ in oil coolant line: v2 p = 47.00 — XD = 4.60 lbf/in2 2gc Flow in mold channels

With four parallel lines, velocity in mold channels is same as that in \\ oil coolant line from pump. However, Re decreases: Reynolds number, Re = Dvp/}i = 1925. Flow is laminar. Friction factor, f= 0.0333. Head Head Head Head Total

loss coefficient, fL/D = 5.33 loss for 6 elbows, 6 x 0.9 = 5.4 loss for 2 sets of orifices = 3.374 loss for expansion, contraction = 1.5 head loss 15.60

Next Page

Total pressure drop in 3/4 in mold coolant channel: P =15.60 ^- x p=1.52 1bf/in2 Thus the total oil coolant pressure drop for the plumbing system of Fig. 6.7 is given as: Ptotai = 4.60+1.52 = 6.12 lbf/in2

6.5

Vent Holes

As the hot plastic sheet is drawn into the mold, the trapped air must be evacuated. Small holes are drilled into the mold surface in the regions where the last portion of the drawing sheet contacts the mold. The number and diameter of these vent or vacuum holes is determined prior to mold design. If the vent hole diameter is too large, hot plastic is drawn into it, producing an unsightly bump, nipple or nib on the finished part. If the nib is excessive, the sheet can rupture. If too few holes are provided or if the vent area is too small, the rate of draw-down will be controlled by the rate of air flowing from the cavity. If the rate is very slow, the sheet may cool so much during draw-down that it can no longer be stretched into the corners and full mold shape replication is not achieved. Vent hole design is coupled to vacuum system design. Figure 6.9 shows a schematic of a typical thermoformer vacuum system. The basic system consists of: •

Vacuum pump, capable of achieving 28 to 28.5 in mercury (28 in Hg to 28.5 in Hg) vacuum,

Mold Cavity

Vacuum Hole Back-Drill Vacuum Channel

Vacuum Box

Vacuum Gauge

Vacuum Line

Shut-Off Valve

Vacuum Pump Surge Tank

Figure 6.9 Typical plumbing between mold cavity and vacuum pump

Previous Page

Total pressure drop in 3/4 in mold coolant channel: P =15.60 ^- x p=1.52 1bf/in2 Thus the total oil coolant pressure drop for the plumbing system of Fig. 6.7 is given as: Ptotai = 4.60+1.52 = 6.12 lbf/in2

6.5

Vent Holes

As the hot plastic sheet is drawn into the mold, the trapped air must be evacuated. Small holes are drilled into the mold surface in the regions where the last portion of the drawing sheet contacts the mold. The number and diameter of these vent or vacuum holes is determined prior to mold design. If the vent hole diameter is too large, hot plastic is drawn into it, producing an unsightly bump, nipple or nib on the finished part. If the nib is excessive, the sheet can rupture. If too few holes are provided or if the vent area is too small, the rate of draw-down will be controlled by the rate of air flowing from the cavity. If the rate is very slow, the sheet may cool so much during draw-down that it can no longer be stretched into the corners and full mold shape replication is not achieved. Vent hole design is coupled to vacuum system design. Figure 6.9 shows a schematic of a typical thermoformer vacuum system. The basic system consists of: •

Vacuum pump, capable of achieving 28 to 28.5 in mercury (28 in Hg to 28.5 in Hg) vacuum,

Mold Cavity

Vacuum Hole Back-Drill Vacuum Channel

Vacuum Box

Vacuum Gauge

Vacuum Line

Shut-Off Valve

Vacuum Pump Surge Tank

Figure 6.9 Typical plumbing between mold cavity and vacuum pump

•

Vacuum surge tank, typically having 6 to 20 times the volume of the combined mold cavity volumes, • A solenoid-actuated shut-off valve, • A flow control valve, • Large-diameter plumbing between the surge tank and the vacuum box, • A vacuum box or chamber fixed between the mold platens, • A mold or molds containing: Vacuum or vent holes, and Vacuum slots or drilled holes between the vent holesand the vacuum box, and • A mold cavity or mold cavities. As discussed in Chapter 1, two types of vacuum systems are currently used. For very large installations with many thermoforming machines and other uses for evacuated air such as vacuum-routering, a central vacuum system is used. Very large diameter (4 in minimum with 6 in typical) vacuum lines are plumbed to each application. This is shown in schematic in Fig. 6.10. For small installations, each thermoforming machine is equipped with a dedicated vacuum system. This is shown in schematic in Fig. 1.23 for a shuttle press1.

Surge Tank

Vacuum Pump

Vacuum Box

Shut-Off Valve

Figure 6.10 Central vacuum system schematic

1

Even though a central vacuum system is preferred for a multiple former shop, it is important to reserve at least one, dedicated, working vacuum pump and surge tank for emergencies or for auxiliary evacuation for deep cavity jobs.

Table 6.13 Gas-Law Constant Numerical value

Units

1.987 1.987 82.06 0.08205 10.731 0.7302

cal/g-mol-K Btu/lb-mol • 0 R cm3 • atm/g-mol • K liter • atm/g-mol • K ft3 • lbf/rn3 • lb-mol • 0R ft3 • atm/lb-mol • 0R

Sizing Vacuum Systems—Steady State Low pressure air is compressible. As a result, the arithmetic needed to size vacuum lines and vent holes depends on compressible fluid flow. For all intents, air can be considered as an ideal gas. The ideal gas equation is: PV = nRT

(6.10)

where P is pressure, V is volume, T is absolute temperature, n is a function of the molecular weight of the gas, and R is the gas constant (Table 6.13). For most cases, air flow can be considered as isothermal. That is, the gas does not change in temperature as it flows through various constrictions in the piping system. In this case, the relationship between gas pressure and volume is given as: P 0 -V 0 = P 1 -V 1

(6.11)

where P is pressure (absolute), V is volume and " o " and " 1 " represent the various states of the air. Example 6.3 illustrates how this expression is used to size the surge tank. Typically, the minimum volume of any vacuum system, including the piping and vacuum box, should be 6 to 20 times that of the mold cavity.

Example 6.3 Surge Tank Volume Determine the surge tank volume, Vs, needed to evacuate a mold of Vmft3 volume initially at Pm pressure (absolute). The minimum desired pressure on the hot sheet against the mold is to be Pmin. The vacuum pump is capable of achieving Pv pressure (absolute) and the air temperature is constant. The working equation is: Y-P v

s

x

+

v ~

Y v

. p _ (Y _|_ y \ . p . m

x

m

V

v

s ~

v

m)

x

mm

Consider the case where: Pv = 28 in Hg = 0.94 lbf/in2 (absolute) p min = 24 in Hg = 2.9 lbf/in2 (absolute)

P m = 14.7 Ib2/in2 (absolute) = atmospheric pressure (14.7- 2.9) s m m (2.9-0.94) In other words, the vacuum surge system, including all the piping and the vacuum box, should be at least six times the volume of the mold cavity. Consider the case where blow air is used to prestretch the thermoformed sheet. For this case, P m is greater th,an atmospheric pressure. The typical prestretching pressure for 10% talc-filled polypropylene from Table 6.13 is 8 to 10 lbf/in2. Repeat the above calculation, using P m = 24.7 lbf/in2 to determine the minimum vacuum surge system volume. .(24.7-2.9) Vs

V1n

( 2 9

_

0 9 4 )

11-1 V m

Typically, vacuum tank volumes range from six to 20 times the expected cavity volume.

Sizing Vacuum Systems—Dynamic The above discussion does not consider the time required to evacuate a mold cavity. The fundamentals of compressible fluid flow are given elsewhere [22,23]. As a first approximation, air flow in thermoforming vacuum systems can be considered as isothermal. Correctly, gas flow is adiabatic for high velocities through constrictions. That is, the pressure-volume relationship is: PoVJ = P1VI

(6.12)

where y = 1.4 for air. The speed of an adiabatic compression wave moving through stagnant air is given as: c2 =

Y• P

1

— (6.13) P where p is the density of the gas. c is usually called the velocity of sound. As shown in Example 6.4, the velocity of sound at 70 0 F and atmospheric pressure is about 1100 ft/s. The Mach number, Ma, is the ratio of gas velocity to sonic velocity: Ma = c

(6.14)

Example 6.4 The Velocity of Sound in Air Determine the velocity of sound at one atmosphere and 700F and at 0.05 atmospheres and 700F.

The velocity of sound, c, at 14.7 lbf/in2 is: c =1078 ft/s p = 0.05 • 0.082 = 0.0041 lbm/ft3 The velocity of sound, c, at 0.74 lbf/in2 is: c = 241 ft/s Gas velocities equal to the velocity of sound are sonic and those less than the velocity of sound are subsonic. Usually, air flow in vacuum systems is always subsonic. However, sonic conditions can exist at the vent hole inlet, as discussed below. If the fluid velocity is low in certain portions of the vacuum system, that region can be considered as incompressible and traditional incompressible fluid mechanics can be used. If the velocity is high—that is, if the Mach number is on the order of 0.3 or more—the flow must be considered compressible. Sonic flow is established so long as the absolute pressure differential across the vent hole is less than 0.53. For incompressible air flow in vacuum systems, as detailed in Section 6.4, an appropriate pressure drop-flow rate relationship is written in "head loss" terms:

where f is the friction factor, L is the pipe length, D is the pipe diameter, V is average air velocity, gc is a unit correction term, and K is the effective resistance through a specific constriction (Table 6.12). The pressure drop, dPi9 through the ith segment of the vacuum system is given as: dPi

=

V

PM^811

(616)

where h{ is the head loss through the ith segment and pi5inean is the average or geometric mean air density through that segment. The total pressure drop is then: CiPtOtHi = Z d P 1

(6.17)

Another less accurate method is to obtain an effective pressure drop, given as: dP = h t o t a l -p m e a n

(6.18)

Note that p mean is an appropriate mean value of the density of the air. It can be the simple geometric mean of the density of the inlet and exit gas stream or the average of the densities of the air at every segment. Considering air as an incompressible gas in order to determine pressure drop is satisfactory for all the piping with the exceptions of: • •

Flow through solenoid valves, and Flow through vent holes.

Orifice Coefficient, c

A0ZA1

Pipe Reynolds Number, Re Figure 6.11 Flow rate-dependent orifice coefficient for compressible air in vacuum systems. Reynolds number, Re = Dv • p/|i where D is pipe diameter, v is air velocity, p is air density and [i is air viscosity

Solenoid Valve Flow Resistance Solenoid valves act to isolate the surge tank-vacuum pump system from the vacuum box-mold system. Although vacuum system solenoid valves are designed to offer minimum resistance to the flowing air, their shapes are such that the pressure loss is best considered as the result of compressible flow through a sharp-edged orifice. Figure 6.11 gives the orifice coefficient for incompressible flow. The pressure dropflow rate equation is written as: in = C 0 YA 0 ^dP) 1 / 2

(6.19)

where rh is the mass flow rate, co is the orifice coefficient, A 0 is the minimum area of the orifice, A 0 = 7iDo/4, P1 is the density of air in the pipe ahead of the orifice and dP is the pressure drop across the orifice. The compressibility correction factor, Y of Fig. 6.12, is dependent on the pressure drop across the orifice and the diameter ratio of the orifice and pipe. Example 6.5 illustrates the use of this equation. Example 6.5 Compressible Air Pressure Drop Across Solenoid-Orifice Air at 5 Iby/'in2 (absolute) pressure and 700F flows at 1 lbm\s through a 2-in diameter pipe. A solenoid valve, acting as an orifice, has an internal constriction, A0/A1 = 0.5. What is the pressure drop across the valve? p (14.7 & 700F) = 0.082 l b j f t 3

Re > 100,000 From Fig. 6.11, C(AJA1 = 0.5) = 0.69 D J D 1 =0.707 Equation 6.19 is used to obtain the result: rh = C0YA0 (2P1ClP)1/2 The air density at 5 lbf/in2 is: p = 0.0812 • (-^~\

= 0.0276 lb/ft3

Since Y is a function of pressure drop, initially assume Y = 0.9 (P 2 /Pi = I)Solving for AP: 4 I2 144

[

0.695-1.0-4,J 2-0.0276-32.3-= ° - 5 3 1 b f / m 2 P 2 = 5 - 0 . 5 2 = 4.48 lbf/in2. P2ZP1 = 0.89. The value for Y is then iterated using this value of P2ZP1. When Y = 0.96, AP = 0.57 and P 2 = 5.43 lbf/in2.

Expansion Factor, Y

Orifice-Flange Taps or Vena-Contracta Taps

Venturi

Pressure Ratio Figure 6.12 Expansion factor for vacuum system compressible air flowing through orifices and Venturis

Vent Hole Resistance to Flow When the solenoid is actuated to open the valve, the compression shock wave propagates from the valve upstream to the vent holes. Since the vent holes offer very high flow resistance, sonic velocity may be established at the entrances to the vent

holes. If this is the case, the maximum rate of evacuation of the mold cavity is calculated from: rhmax = NAV • c • p

(6.20)

where A v is the area of the vent hole, N is the number of vent holes, c is the sonic velocity and p is the density. If the evacuation time of a mold cavity having volume v m is t, and the diameter of a given vent hole is D, the air velocity is:

v =

(6 21)

- 'Wm

-

If the velocity of air through the hole is sonic, the number of vent holes of diameter D is given as: N=

— (6 27) K } c(;rD 2 /4)t Example 6.6 illustrates the relationship between rate of cavity evacuation and the absolute minimum number of vent holes. The analysis above assumes that the vent hole has essentially zero length. As expected, the pressure drop increases with increasing vent hole length. The effect of vent hole length is determined from compressible "head loss" calculations. In essence, the friction factor is determined in standard fashion, as outlined earlier in this chapter. The head loss factor, fL/D, is calculated and the reductions in velocity and pressure are determined from Table 6.14. Example 6.7 illustrates the way in which this is determined.

Example 6.6 Vent Hole Sizing and Evacuation Rate What is the absolute minimum number of vent holes needed to evacuate a I ft3 mold cavity in 10 si D = 1/32-in. Individual vent hole area, A = TID 2 /4 = 0.000767 in2 Mass flow rate = 1 ft3/10 s = 0.1 ft3/s. Sonic velocity = 1000 ft/s Mass flow rate, a single vent =

— = 0.00533 ft3/s

Total minimum number of vents, N = 0.1/0.00533 = 18.8 « 20

Example 6.7 Effect of Vent Hole Length on Pressure Drop and Air Velocity Consider the case where 700F air is exhausting at sonic velocity through a 0.0313-in diameter vent hole. Determine the pressure drop and velocity reduction for vent hole lengths of 0.125 in and 0.250 in. Consider the drilled hole to be smooth-bored. The kinematic viscosity of air is 2.15 x 10~4ft2/s.

The Reynolds number is given as:

From the Moody friction factor-Reynolds Number diagram, Fig. 6.13 (see Streeter, pp. 288-289), f = 0.029. Thus: fL/D = 0.029 • 0.125/0.0313 = 0.116 Interpolating from Table 6.14, v/c = 0.789 and P/P c = 1.365. At the exit of the vent hole, the velocity is 1000 x 0.789 = 789 ft/s. And the pressure is 14.7/1.365 = 10.77 psi. For L = 0.250 in, fL/D = 0.232. Interpolating from Table 6.14, v/c = 0.694 and P/P c = 1.519. At the exit of the longer vent hole, the velocity is 694 ft/s and the pressure is 9.69 psi.

Table 6.14 Effect of Vent Hole Length on Pressure Drop fL/D

V/c

P/Pc

0 0.0033 0.0145 0.0363 0.0723 0.1273 0.2081 0.3246 0.4908

1 0.958 0.915 0.870 0.825 0.779 0.732 0.684 0.635

1 1.061 1.129 1.205 1.289 1.385 1.493 1.618 1.783

Note that the Reynolds number through a vent hole is relatively small, even though the air velocity is sonic. This is due to the very small vent hole diameter. The volumetric flow rate through the piping between the vent holes and the vacuum surge tank is determined from: (6.23) where N is the number of vent holes and c is sonic velocity. As a first approximation, the air velocity everywhere else is given as: (6.24) Thus the Reynolds number in any pipe of diameter D p is related, on first approximation, to the sonic velocity Reynolds number:

(6.25) Example 6.8 explores the relative values of these variables. Example 6.8 Relative Velocities and Reynolds Numbers in Vacuum Pipe Flow Assume the air velocity through a 0.0313-in diameter vent hole is 1000 ft Js1. There are 100 vent holes. Calculate the volumetric flow rate, the Reynolds number through the vent hole, and the Reynolds number through the 2-in diameter pipe to the vacuum surge tank.

Volumetric flow rate = If the volumetric flow rate is constant, the Reynolds number in the 2-in diameter pipe is given as:

If the density is constant, the velocity in the 2-in diameter pipe is given as:

1

This assumes that the vent hole length is negligible. If it is not, it is necessary to reduce this value to the proper one as shown in Example 6.7.

It is apparent from the examples for flow through the solenoid valve and the vent holes that such constrictions can severely restrict the rate at which air is exhausted from the mold cavity. Before summarizing the evacuation flow characteristics, consider the replacement of traditional hard piping for vacuum lines with corrugated flexible tubing. Corrugations act as localized expansions and contractions along the entire length of the tubing. The worst case scenario is obtained by multiplying the orifice head loss coefficient K by the number of corrugations in the tubing length. The best case scenario assumes that the pipe has great roughness. As seen in Fig. 6.13, the friction factor approaches a constant value at relatively low Reynolds numbers for very rough pipe. For a pipe of e/D = 0.05, for example, the friction factor approximates 0.072 for Re > 30,000 or so. Computation of flow rate-pressure drop through the vacuum system requires trial-and-error. The following protocol will enable rapid iteration: 1. Assume the air pressure in the mold cavity to be atmospheric. 2. Assume sonic velocity at the inlet to the vent holes.

Complete Turbulence, Rough Pipe

e/D=0.05

Friction Factor, f

Transition

e/D=0.000001

Reynolds Number, Re Figure 6.13 Flow rate-dependent friction factor for flow in a pipe. e/D is measure of roughness, Reynolds number, Re = Dv • p/ji where D is pipe diameter, v is fluid velocity, p is fluid density and u is fluid viscosity

3. 4. 5. 6. 7.

Determine the velocity, density, and pressure at the exit of the vent holes. Calculate the Reynolds number for air flow in the vent hole. Estimate the practical number of vent holes to be used1. If a solenoid valve is used, estimate the pressure loss from the orifice equation. Using the "head loss" approach, calculate the total head losses for the piping, elbows, vacuum box, back-drilled region on the mold, expansion loss into the surge tank and the pressure loss across the solenoid. 8. Add to this the pressure loss across the vent hole. 9. If the calculated pressure loss is greater than that determined from the available pressure differential between the surge tank and atmosphere, the assumed flow rate is too great. In other words, there is too much resistance to flow for the chosen velocity. Since the controlling factor is the assumption of sonic flow at the inlet to the vent hole, it is necessary to reduce this velocity to a value below the velocity of sound. Estimate this velocity by multiplying the ratio of available to calculated pressure drops by the velocity of sound. Continue to iterate until there is reasonable agreement on pressure drop values. 10. If the calculated pressure loss is less than that determined from the available pressure differential between the surge tank and atmosphere, the assumed flow 1

This number is usually 3 to 5 times greater than the minimum number of vent holes calculated. One method is to assume that the Reynolds number in the vacuum pipe equals that in the vent hole. From this, the number of vent holes can be estimated.

rate is too small. Unfortunately, the greatest flow rate that can be assumed is sonic flow. This means that the cavity volume could be evacuated even faster than the current mold design allows. One way of increasing evacuation rate is to increase the number of vent holes. Another is to increase the diameter of the vent holes. Both of these options can lead to part design problems, as detailed later. Example 6.9 illustrates a portion of this iterative procedure. Example 6.9 Pressure Drop Through Vacuum System Consider the mold vent hole and vacuum line assembly outlined in Examples 6.7 and 6.8, with the vent hole length of 0.250 in, the vacuum line being 2-in smooth bored pipe 100ft in length and having eight elbows, two tees, one solenoid valve and one check valve and one globe valve. Assume that air flow through the vacuum box represents 10 equivalent ft of smooth bored pipe, and that there are two flow expansions and two flow contractions. Use a mean air density between the surge tank and the vent hole outlet. Assume the flow to be isothermal, for the most part. Density of air at the exit of the vent hole is directly proportional to the pressure at that point. P vent exit

Patmospheric vêxit/ *atm/

Pventexh = 0.082 • (1/1.519) = 0.054 lb m /ft 3

The pressure in the surge tank is assumed to be 0.05 atmospheres (absolute). The density at 70 0 F is therefore: Psurge = 0.082 • (0.05/1) = 0.004 lbm/ft3 The mean density is: Pmean = Vpsurge ' Pvent exit = 0.015 lb m /ft 3

Assume the Reynolds number in the 2-in pipe = 18,900. From Fig. 6.13 for smooth-bored pipe, f =0.0251. So: fL/D = 0.0251 • (100 + 10) • 12/2 = 15.67 The head losses from the rest of the system, with the exception of the vent hole and solenoid losses, include: 2 2 8 2 1 1

expansions x 1.0 = contractions x 0.5 = elbows x 0.9 = tees x 1.8= check valve = globe valve =

2.0 1.0 5.4 3.6 2.5 1O1O 24.5

Thus the total head loss value is given as: Ktotai= 15.67 + 24.5 = 40.2 The velocity through the 2-in pipe is given as 24.5 ft/s:

The total head loss, htotal is then given as: htotal = 40.2 -(v2/2gc) = 374.7 ft The pressure drop is then: AP = htotal • p mean = 374.7 • 0.015/144 = 0.04 lbf/in2 The pressure drop through the solenoid is given in Example 6.5 as AP = 0.57 lbf/in2 and that through the vent holes in Example 6.7 as AP = 5.01 lbf/in2. Thus the total pressure drop is: Pressure drop through vent holes = Pressure drop through solenoid= Pressure drop through piping = Total pressure drop =

5.01 0.57 0.04 5.62 lbf/in2

Since the calculated pressure drop is less than the maximum available pressure drop (1 — 0.05) • 14.7 lbf/in2 = 14.0 lbf/in2, the rate of evacuation is restricted by sonic velocity at the vent holes. Increased evacuation is possible by increasing the number of vent holes or increasing the vent hole diameter.

Vent Hole Diameter Mold vent hole diameter depends on several factors, including the modulus of the hot plastic sheet, the sheet thickness over the hole and the allowable draw depth into the vent hole. The maximum deflection 8, of a sheet of thickness h and modulus E into a hole diameter d is given as [24]:

where the maximum deflection is proportional to h, with oc as the proportionality or acceptable nib or nipple height, q is the applied pressure, and v is Poisson's ratio (Table 5.15). For many plastics, 0.35 < v < 0.5. If v = 0.5:

« = 0.516-|-(|y

(6.27)

The maximum nib or nipple height is usually specified by the user or the designer. This establishes an upper limit for a = 5/h. Rearranging Equation 6.27:

This is shown in Fig. 6.14. As discussed in Chapter 4, the secant modulus or (|>(T) is about 100 lbf/in2 or 0.7 MPa for vacuum forming where the pressure is about 15 lbf/in2 or 0.1 MPa. Thus the E/q is 7. A typical range for E/q is 2 < E/q < 10. A good range for d/h in Equation 6.28 is: 1 . 4 a 1 / 4 < £ < 2 . 1 oc1/4 n

(6.29)

Hole Diameter to Local Sheet Thickness Ratio, d/t

Characteristic Sheet Modulus, dE/q Figure 6.14 Specific vent hole design parameter, d/t, as function of modulus-to-pressure ratio, E/q, and relative nib height, a. This is for initial draw-down into vent hold

Example 6.10 further explores this relationship. If the nib or nipple height is restricted to no more than the sheet thickness, the vent hole diameter must be less than 2.1 times the sheet thickness. This value is compared with Gruenwald's rule [25]: "... vacuum hole sizes smaller than the (local) material thickness will not become visible..."

That is, holes where d/h < 1 yield acceptable nib heights. Example 6.11 tests the sensitivity of Equation 6.28 as regards local sheet thickness and sheet temperature. As illustrated, locally thin sheet produces a much greater effect on nib or nipple size than does locally hot sheet. It is recommended that the vent hole diameter be based on the first common drill size below the calculated value. The vent holes should be drilled perpendicular to the plane of the mold. The backs of the vent holes must be enlarged by machining vacuum channels or drilling large holes on the reverse side or back of the mold (Fig. 6.15). Since the objective is to reduce flow resistance through the long vent hole, the counterboring, "back-drilling" or "back-drafting" should be taken within 0.5 in or 13 mm of the mold surface or closer if possible. This counterboring must be done carefully to minimize locally weakening the mold in the Example 6.10 Relative Nipple Height Determine the maximum vent hole diameter for 0.040 in PS sheet drawn 4:1 against the vent hole area. Assume E/q < 10. Consider the maximum nipple height to be equal to the local sheet thickness.

From Equation 6.29, d/h < 2.1 oc1/4 for E/q < 10. At 4:1 draw ratio, the local sheet thickness is 0.010 in. The maximum nipple height is 0.010, with a = 1. The hole diameter must be less than 2.1 • h or dmax = 0.0210 in.

vent hole region. Drilling too closely or milling a channel that is too wide invites long-term metal fatigue and possible local collapse [26]. Experienced mold-makers avoid drilling too few vent holes of diameters that are too large. If the vent hole diameters are too large, objectionable nibs or nipples result, even though the volume of entrapped air is exhausted in a reasonable time. Large diameter vent holes force the fabricator to lower sheet temperature. If stretching forces are limited or the sheet is particularly stiff, poor part replication results.

Example 6.11 The Relative Sensitivity of Nipple Height to Process Parameters Consider polymers having the following temperature-dependent elastic moduli:

Determine the relative effects on nipple height on: ± 10% change in pressure, ±10% change in sheet thickness, and ±20° C change in sheet temperature for PMMA where 1/0 = 200C'1 and PE where 1/0 = 850C'1.

Equation 6.28 is differentiated as follows:

Now dd/d = 0. The expression for dE is:

Therefore:

For PMMA:

The sheet temperature affects the PMMA nipple height the most. For PE:

I The sheet thickness variation affects the PE nipple height the most.

Vent Hole

Mold Thickness

Vent Hole

L

d. L

Mold

Vacuum Slot or Vacuum Box

D Vacuum Slot or Vacuum Box Back-Drill or Back-Draft

Figure 6.15 Characteristic geometric factors for vacuum or vent hole without back-drilling (left) and with back-drilling (right)

Occasionally a mold that is functional for one polymer yields unacceptable nipple or nib heights with another. Peening the vent hole area will reduce the vent hole size, albeit at the risk of deglossing the mold in that area1. Note that Equation 6.28 is also acceptable for determining vent hole size for pressure forming. As discussed in Chapter 9, pressure forming pressures are typically four to six times greater than vacuum forming pressure. As a result, the lower inequality in Equation 6.29 is operative. A properly dimensioned vent hole for vacuum forming would yield a relative nib or nipple height that is four times greater with pressure forming. One saving aspect is that pressure forming sheet temperatures are usually somewhat lower than equivalent vacuum forming temperatures. Cooler sheet implies higher elastic modulus and shorter nibs or nipples.

Vacuum Slot

Vacuum Slot

Vacuum Slot

Vacuum Channel or Vacuum Box Figure 6.16 Vacuum or vent slots for two-piece female mold (left) and one-piece female mold (right) 1

Plugging and redrilling the hole is always an option.

Figure 6.17 Vacuum or vent slots for male mold

Other Types of Vents Slot vents are used when linear protruding elements occur inside a female mold (Fig. 6.16) or along the bottom rim of a male mold (Fig. 6.17). Slots are end-milled or cut with a wire EDM along the two-dimensional corner. Exit vents are then drilled from the reverse side. The width of the slot vent is determined in a manner similar to that for round holes. From Fig. 6.18, let d be the slot width. The maximum deflection, 5, is given as [27]: (6.30) Rearranging: (6.31) Since this equation is of the same form as Equation 6.28 for a circular hole, Fig. 6.14 is used for slot vents if: (6.32)

y d

Figure 6.18 Vacuum or vent hole geometric factors

Table 6.15 Porous Metal Product Manufacturers (Specialties in parentheses) Allied Sinterings, Inc. (small parts) 29 Briar Ridge Rd. Danbury CT 06810 203-743-7502 Arrow Pneumatics, Inc. (bronze) 500 Oakwood Rd. Lake Zurich IL 60047 708-438-9100 Astro Met, Inc. (stainless steel, powder metallurgy) 9974 Springfield Pike Cincinati OH 45215 Atlantic Sintered Metals (powder metallurgy) 12 Cushing Dr. Wrentham MA 02993 508-384-3100 Helsel, Inc. (stainless steel) State Rd. 60 W. r K°n K Txr ym™ Campbellsburg IN 47108 813-755-4501 . f Mott Metallurgical Corp. (stainless steel) Farmington Industrial Park 84 Spring Lane Farmington CT 06032-3159 203-677-7311

National Sintered Alloys, Inc. (parts, powder metallurgy) 10 Heritage Park P.O. Box 332 Clinton CT 06413 203 669 8653 Newmet Krebsoge, Inc. (stainless steel, bronze, titanium) P.O. Box 68 Terryville CT 06786 800-426 0977 8 °° 4 2 6 °977 Pacific Sintered Metals, Inc. (bronze, stainless ^ 0 Permaflow? Inc at the same address] 14002 s

Ayalon

^ 1 CA 310-715 9800

Los

p T I T e c h n o l o ies 950 R a n c h o

Blyd

^

6 1

Inc

ônejo

Blyd

Newbury Park CA 91320 OQQ ^Q, Q^g1 SSI Technologies, Inc. (stainless steel, bronze) ^^32 Palmer Dr Janesville WI 53566 608-755-1900 The Wakefield Corp. (stainless steel, bronze) 35 Foundry St. Wakefield MA 01880 800-548-9253

For the same allowable depth of draw into the vent, slot vent widths need to be only 83% of circular vent diameter. For a given dimension, slot vents have two to four times greater venting area. Reverse side exit holes must be large enough to allow unrestricted air flow from the cavity. And slot vent width must be small enough to minimize part undercutting into the vent or the formation of an unsightly line. Porous metal plugs are used in flat areas in place of multiple vent holes. Sintered powders of bronze, brass or stainless steel of about 50% open area are commercially available (Table 6.15). These plugs are machined to fit an existing vent hole region or can be custom ordered to meet a specific design. A hole large enough to vent the area, usually 6.4 mm or 0.250 in or more, and deep enough to accommodate the plug, usually 6.4 mm or 0.250 in or more, is drilled in the primary mold surface (Fig. 6.19).

Porous Plug Mold

Vacuum Box Figure 6.19 One method of fastening porous plug in mold cavity for evacuation

Exit holes are then drilled from the reverse side to facilitate venting. Another approach is to use a tapered hole and a tapered plug that is inserted from the reverse side and held in place with a simple spring ring or screw (Fig. 6.20). The effective porous metal venting area width is as much as ten times greater than the venting area from clustered vent holes. Nipples cannot form when porous vent plugs are used. However, the open areas or pores in the plugs tend to fill with detritus, particularly if in-mold trimming is used. Other sources of detritus include sander and router dust, atmospheric dust and contamination on the incoming sheet. Venting efficiency gradually decreases with time, necessitating periodic plug replacement. Fine stainless steel, nickel and brass welded wire screen is also commercially available. Open area for ^100 mesh screen is usually 35% to 65%. Screen must be supported to minimize bending under sheet draw-down forces. Screen may also leave an undesirable pattern on the part. The poppet valve is usually mechanically operated and is used in injection molding to break the suction between a part and the mold surface on flat, large-area parts. The poppet valve can be used in this fashion for parts thermoformed onto male molds. In venting, poppet valve is used to rapidly exhaust cavity air in large-volume deep-draw female molds (Fig. 6.21). The evacuating poppet value is spring loaded and is normally open. It is closed either on a clock timer or by the sheet draw-down force. Example 6.12 illustrates the high evacuating efficiency of a poppet valve.

Porous Plug Mold

Vacuum Box Figure 6.20 One method of fastening porous plug in mold cavity for evacuation

Poppet Valve Open Poppet Valve Closed

Mold Spring Vacuum Channel

Vacuum Channel

Spring

Figure 6.21 Spring-loaded poppet valve for evacuating large-volume or deep female mold cavities

Example 6.12 Poppet Valve in Mold Evacuation A 6-in diameter x 8-in deep female mold cavity is currently being evacuated in LOs through 18 vent holes, each 1/16-in in diameter. Determine the maximum evacuation rate if the number of vent holes is doubled. Then compare this with the evacuation rate if a 3-in diameter poppet valve with an open gap of 0.020 in is installed in the cavity bottom. The cavity volume is 7iD2L/4 = 226.2 in3. The current volumetric evacuation rate is 226.2 in3/1.0 s = 226.3 in3/s. The area of current vacuum holes is: N

' ^ = 18 • 0.7854 • (0.0625)2 = 0.0552 in2

The current air flow is 226.2/0.0552 = 4096 in/s = 341 ft/s or subsonic flow. If the number of holes doubles and the air flow remains the same, the cavity is evacuated in 1.0/2 = 0.5 s. The vent area of the open gap of 0.020 in on the poppet valve is 7idp • 0.020 = 0.1885 in2. The total vent area is then 0.0552 + 0.1885 = 0.2437 in2. If the air flow remains the same, the cavity is evacuated in 1.0(0.0552/0.2437) = 0.23 s.

Vent Hole Placement It is apparent that there must be venting in areas where the plastic last contacts the mold surface. These areas are usually two- and three-dimensional corners. There is substantial verification that adequate parts can be made by vacuum forming into a five-sided female mold having venting only in the four three-dimensional corners.

Vent Holes on Flat Surface

Figure 6.22 Vent hole location on flat or planar mold surface for both male and female molds

The wall thickness prediction programs of Chapter 7 are most useful in defining where other potential late evacuation regions are. Other areas where vent holes are desirable include: •

Flat, large-area horizontal surfaces on both male and female molds (Fig. 6.22). The sheet usually touches the mold surface here first. Subsequent shrinkage can tightly lock the sheet to the surface. The vent holes are used primarily to blow the cool sheet from the mold surface. • Vertical walls and vertical two-dimensional corners, in particular (Fig. 6.23). These vents serve primarily to lock the sheet against the mold surface during stretching. These vents are particularly important for low coefficient of friction polymers such as polypropylene, polyamide and PTFE. • In and around lettering, bosses, ribs, and corrugations. Since it is not always apparent how the plastic will form around and into these details, venting must be adequate and vent holes regularly spaced (Fig. 6.24). • In the rim regions for female molds, particularly when dams or moats are used (Fig. 6.25). These help lock the sheet against the mold and minimize the amount of web area sheet drawn over the rim and into the mold cavity. Vent Holes on Vertical 2D Corner

Vent Holes to Minimize Slip

Figure 6.23 Vent hole location on vertical two-dimensional female mold surfaces

Boss Raised Letter With Insert Vent Hole

Rib

Vent Hole

Mold

Figure 6.24 Vent hole location around bosses, raised sections and ribs Dam

Vent Hole

Shelf

Moat/Dam

Vent Hole

Vent Hole

Mold

Mold Mold

Figure 6.25 Venting around moat (left), moat/dam (center), and shelf rims (right) for female molds

• •

Along the double step for male molds (Fig. 6.26). These help lock the sheet against the mold before most of the forming occurs. This locking effect minimizes air bleed under the sheet during the final stages of draw-down. Along two- and three-dimensional corners on male molds (Fig. 6.27).

Although usually not necessary, vent holes are regularly spaced along horizontal two-dimensional corners on female molds (Fig. 6.28). These regularly spaced holes

Sealing Off

Mold

Figure 6.26 Two-step hold-down for male mold, showing seal-off effect on right

Mold

2D Corner Vents

2D Corner Vents Sealing Vents

Figure 6.27 Vent hole locations on two-dimensional vertical and horizontal male mold surfaces and sealing locations

are used in place of vent holes clustered in three-dimensional corners. Functionally, they do not perform as well as clustered vent holes but are more esthetically pleasing.

6.6

Surface Treatments

Unlike injection molds, thermoform molds rarely require intermittent topical applications of mold release in areas where the formed shape is particularly difficult to release from the mold surface. There are several reasons for this:

Mold Side

Vent Holes on Horizontal 2D Corners Mold Bottom

Figure 6.28 Uniformly spaced vent hole locations on horizontal two-dimensional corners in female mold

• •

The pressures used in thermoforming are modest, The plastic is a rubbery solid rather than a sticky fluid and so polymer adhesion is minimal1, • The plastic shape is usually quite flexible when stripped from the mold and so vacuum pockets usually do not form, and • The mold surfaces are usually vented in precisely thoseareas where sticking might occur. Surface treatments on thermoform molds are used for other reasons. When biaxially stretched, hot sheet contacts a cold high energy surface such as metal mold and stretching ceases. Walls of simple vacuum-formed female parts thin with draw depth as described in Chapters 4 and 7. In order to avoid excessive draw-down, or to effect preferential draw-down into a specific region, permanent surface treatments are used. Baked-on surface treatments of low-friction materials such as PTFE and FEP are common. These treatments produce surfaces that are essentially integral parts of the mold surface. In certain mold designs, the sheet must not slide against the surface. Olefins tend to alternately slip and stick when vacuum formed. This causes visible ridges in the part surface. These are sometimes wrongly attributed to plug mark-off. Roughening the mold surface does little to prevent sliding and may aggravate the problem. Roughening may actually reduce contact area as shown in schematic in Fig. 6.29. A better alternative is to treat the surface with a high frictional coefficient substance such as a curable polyurethane, polybutadiene or silicone rubber. Also additional forming pressure is used to minimize sliding. The relationship between frictional force Fs and normal force F n is: (6.33)

Sheet Mold Asperities Sheet

Mold Figure 6.29 Aspects of mold surface texture replication by thermoformed sheet. Matte surface at top, rough textured surface at bottom

1

There are exceptions to this, of course. Parts can stick if the polymer contains an adduct that blooms to the surface and/or becomes tacky, if there are sufficient undercuts or if the mold is very hot.

W z

N

D Figure 6.30 Geometric factors for extensional drawdown v. sliding of sheet into female mold

where c is the kinetic frictional coefficient and takes the experimental range of 0.67 to 1.0. Larger values of a indicate that the deformation of contact area of the plastic is determined by its viscoelastic properties. Values of c are in the range of 0.1 to 0.25 for typical smooth-surface metal molds and most solid plastics [49]. To better understand the surface effect on forming, consider a simple example of draw-down of a sheet of initial thickness to and tensile strength T into a cylindrical female mold of diameter D with a differential pressure P. As shown in Fig. 6.30, the sheet has already contacted the cylinder to a depth Z and has thinned to a thickness h. The normal force holding the sheet against the surface is: F n = P • TiDZ

(6.34)

The second part of the term on the right is the contact area of the sheet on the mold. If the sheet is to slide on the mold rather than stick to the mold, it must be pulled with a force greater than this. This is written as: F s > c • F n = c • P n • (7iDZ)n

(6.35)

The force required to draw the sheet is the polymer tensile strength times the cross-sectional area, A: F s = T • A = T • TiDh

(6.36)

Note that the cross-sectional area is at the point where the sheet last contacts the mold surface. If the force needed to slide the sheet exceeds that needed to draw it, the sheet will tend to draw. This is written as: F s > F d -• c • (PTiDZf > T • TiDh

(6.37)

Consider the example where oc = 1: (6.38) or: (6.39)

If the force needed to draw the sheet exceeds that needed to slide it, the sheet will slide. Thus:

z
644 > 645> 717 > 745 Rosen > S L - 2O6f> 377-378T, 379, 381 Rosenzweig, N. 499, 514, 515, 574, 704T, 706 > 747 Ruetsch, R.R. 84, 104, 727, 746 R an Y > M E - 262 > 263 > 281

Saechtling, H. 65, 103, 210, 210F, 21 IF, 242, 242F, 243F, 244F, 277, 280, 679, 680F, 728T, 743, 746 Sakiadis, B.C. 83, 104 Salazkin, K.A. 513, 558, 559F, 576 Saunders, D.W. 236, 281 Savonuzzi, A. 675, 742 Scheibelhoffer, A.S. 717, 718F, 719, 745, 746 Scheiner, L.L. 675, 711, 712, 712F, 743 Scherbak, V.V. 513, 577 Scherer, R. 683, 743 Scheryschew, M.A. 513, 577 Schmidt, H J . 513, 577 Schmidt, L.R. 233, 235, 237, 238T, 239T, 249, 279, 282, 489, 490, 495, 499, 503505, 574 Schneider, P J . 165, 166F, 168, 169F, 172, 174F, 178F, 179, 179F, 195, 290, 291T, 379, 444, 469

Italic = Reference location, F = Figure, f = Footnote, T = Table

Schrage, D.A. 403, 403T, 467 Schreiber, H.P. 547, 549, 576 Schrieffer, J. 678, 743 Schuetz, M.A. 729, 746 Schuster, A. 689, 744 Schut, J.H. 333, 380 Sebastian, D.H. 717, 745 Semerdjiev, S. 437, 437F, 469 Shapiro, A.H. 414, 468 Shea, J.W. 547, 549, 576 Sheryshev, M.A. 513, 558, 559F, 576 Shlyakhova, T.G. 513, 577 Shrivastava, S. 262, 281 Shutov, F.A. 623, 625T, 647

742

X S £ '1 7 4 i "gletn R W 42Lf43T i t t

dingieion, r^.w. i^-z 1101, iyo ^^^ c. n A, Sinofsky, M . 727, 7/f/C 746 Skee, S. 387, 388T, 389, 467

oi i_i o T^ crvr ^^^ Skoblar, S.K. 595, 645 Skochdopole, R.E. 727, 746 Smets, G. 658, 742 Smoluk, G. 595, 645 Song, W.N. 240, 241, 279, 281, 521, 523, 534, 536F, 543, 544F, 575 Sors, L. 384, 467 Souders, M. 405, 406, 407T, 467, 468 Steinberg, A.H. 3, 52, 675, 742 Stephenson, M J . 263, 281 Stern, C H . 729, 746 Stewart, W.E. 294, 379 Storr, L.D. 623, 647 Streeter, V.L. 405, 406, 407T, 414, 467, 468 Subramanyan, S. 295, 295T, 379 Suh, N.P. 436, 468 Sung, N.H. 436, 468

Tadmor, Z. 82, 82F, 104, 226, 278, 327, 380, 499, 514, 515, 574, 654, 741, 836, 843 Takayanagi, M. 244, 281 Tanner, R.I. 241, 280, 539, 579 Taylor, C A . 241, 281 Taylor, G.A. 527, 529F, 576

Thorp, M.E. 397, 399T, 467 Timoshenko, S. 504, 574 Tock, R.W. 226, 231, 260F, 266, 278 Treloar, L.R.G. 236, 237, 238T, 239T, 240, 257, 279, 281, 527, 529F, 575 Tsuge, K. 216, 278 Turner, S. 475, 476f, 574, 576

Valenzuela, J.A. 727, 746 ' K E ' 7 3 ' 104 Vll f M A 513 ^ ' ' 576 ^5^^380^ ^

Van Holde

^^?rSvf V

^

^

l f m 28h 521

'

ôc/;i7

Technology of Thermoforming

Technology of Thermoforming

Fundamentals of Plastics Thermoforming

Update on Troubleshooting in Thermoforming

Thermoforming: A Plastics Processing Guide, Second Edition

Thermoforming: A Plastics Processing Guide, Second Edition

Management of Technology : Key Success Factors for Innovation and Sustainable Development (Management of Technology) (Management of Technology) (Management of Technology)

Principles of Modern Technology

The Future of Technology

Technology of Quantum Devices

Encyclopedia of Technology Terms

Technology of Bottled Water

Handbook of Sealant Technology

Technology of Bottled Water

Encyclopedia of Bioprocess Technology

Principles of modern technology

Encyclopedia of Corrosion Technology

Encyclopedia of corrosion technology

Encyclopedia Of Corrosion Technology

Dictionary of Information Technology

Dictionary Of Bio-technology

Encyclopedia of chemical technology

Fundamentals of DSL Technology

Principles of Fermentation Technology

Handbook of Hydrothermal Technology

Technology of Fluoropolymers

Technology of the Guitar

Handbook of Evaporation Technology

Technology of Fluoropolymers

Technology of fluoropolimers

Technology of Fluoropolymers

Technology of Thermoforming