The SGTE Casebook: Thermodynamics at Work, Second Edition

Citation preview

i

The

casebook

Thermodynamics at work

ii

iii

The casebook Thermodynamics at work Second edition Edited by K. Hack

Woodhead Publishing and Maney Publishing on behalf of The Institute of Materials, Minerals & Mining

CRC Press Boca Raton Boston New York Washington, DC

WOODHEAD

PUBLISHING LIMITED

Cambridge, England

iv Woodhead Publishing Limited and Maney Publishing Limited on behalf of The Institute of Materials, Minerals and Mining Woodhead Publishing Limited, Abington Hall, Abington Cambridge CB21 6AH, England www.woodheadpublishing.com Published in North America by CRC Press LLC, 6000 Broken Sound Parkway, NW, Suite 300, Boca Raton FL 33487, USA First published 2008, Woodhead Publishing Limited and CRC Press LLC © 2008, Institute of Materials, Minerals and Mining The authors have asserted their moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the authors and the publishers cannot assume responsibility for the validity of all materials. Neither the authors nor the publishers, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress. Woodhead Publishing ISBN 978-1-84569-215-5 (book) Woodhead Publishing ISBN 978-1-84569-395-4 (e-book) CRC Press ISBN 978-1-4200-4458-4 CRC Press order number WP4458 The publishers’ policy is to use permanent paper from mills that operate a sustainable forestry policy, and which has been manufactured from pulp which is processed using acid-free and elementary chlorine-free practices. Furthermore, the publishers ensure that the text paper and cover board used have met acceptable environmental accreditation standards. Typeset by Replika Press Pvt Ltd, India Printed by TJ International Limited, Padstow, Cornwall, England

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Contents

Contributing authors Software packages used for the case studies Member organisations of SGTE

xvii xx xxiii

Editor’s acknowledgements

xxv

Foreword

xxix

Introduction

xxxi

Part I:

Theoretical background

I.1

Basic thermochemical relationships

I.1.1 I.1.2 I.1.3 I.1.4 I.1.5 I.1.6

Introduction Thermochemistry of stoichiometric reactions Thermochemistry of complex systems Activities of stable and metastable phases Extensive property balances References

3 4 7 9 12 13

I.2

Models and data

14

I.2.1 I.2.2 I.2.3 I.2.4 I.2.4.1 I.2.4.2 I.2.4.3 I.2.4.4

Introduction Gibbs energy data for pure stoichiometric substances Conclusion Relative data Advantages of the use of relative Gibbs energies Solution phase systems Disadvantages Conclusion

14 16 20 20 22 22 24 25

3

vi

Contents

I.2.5 I.2.5.1 I.2.5.2 I.2.5.3 I.2.5.4 I.2.5.5 I.2.5.6 I.2.5.7 I.2.5.8 I.2.5.9 I.2.5.10 I.2.6

Solution phases Substitutional solutions Partial properties for the substitutional solution The sublattice model Partial properties for the sublattice model Interstitial solutions Lattice defects Ionic solid solutions The ideal gas Liquid solutions Magnetic effects in solution phases References

25 25 28 28 29 30 31 31 32 34 37 41

I.3

Phase diagrams

43

I.3.1 I.3.2 I.3.2.1 I.3.2.2 I.3.3 I.3.4 I.3.4.1 I.3.4.2 I.3.4.3

Introduction: types of phase diagrams Zero-phase-fraction lines Special cases of zero-phase-fraction intersections Conclusions on zero-phase-fraction lines Beyond classical phase diagrams The phase rule The ammonium chloride–gas equilibrium The water–phosphoric acid equilibrium The 2CaO·SiO2–3CaO·MgO·2SiO2–CaO·MgO·SiO2 equilibrium Conclusions References

43 50 53 55 55 66 69 70 70 71 72

Summarising mathematical relationships between the Gibbs energy and other thermodynamic information

73

I.4.1

Reference

74

Part II:

Applications in material science and processes

II.1

Hot salt corrosion of superalloys

77

II.1.1 II.1.2 II.1.3 II.1.4 II.1.5

Introduction Data used for the calculations The gas–salt equilibrium The interaction of gas and salt with Cr2O3 Limitations of the data and calculated results

77 77 79 82 88

I.3.5 I.3.6 I.4

Contents

vii

II.1.6 II.1.7 II.1.8 II.1.9

Extension to higher-order systems Future developments Acknowledgements References

88 89 89 90

II.2

Computer-assisted development of high-speed steels

91

II.2.1 II.2.2 II.2.3 II.2.4 II.2.5

Introduction Background Calculation Discussion Reference

91 91 91 92 97

II.3

Using calculated phase diagrams in the selection of the composition of cemented WC tools with a Co–Fe–Ni binder phase

98

II.3.1 II.3.2 II.3.3 II.3.4 II.3.5 II.3.6

Introduction: background to the problem The region of favourable carbon contents Effects of replacing Co by Fe and Ni Favourable carbon contents of a family of alloys Conclusions References

98 98 99 102 103 105

II.4

Prediction of loss of corrosion resistance in austenitic stainless steels

106

II.4.1 II.4.2 II.4.3 II.4.4 II.4.5 II.4.6 II.4.7

Introduction Theory Results Discussion Method of plotting diagrams Database References

106 106 107 111 112 113 113

II.5

Prediction of a quasiternary section of a quaternary phase diagram

114

Introduction Solid phases Modelling Results References

114 114 115 115 117

II.5.1 II.5.2 II.5.3 II.5.4 II.5.5

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Contents

II.6

Hot isostatic pressing of Al–Ni alloys

118

II.6.1 II.6.2 II.6.3 II.6.4 II.6.5 II.6.6

Introduction Generalised Clausius–Clapeyron equation Application to the Ni–Al equilibrium Conclusions Acknowledgement References

118 119 120 121 121 122

II.7

Thermodynamics in microelectronics

123

II.7.1 II.7.2

Introduction Thin-film deposition of SrTiO3 and interface stability with Si Reactive ion etching of HfO2 dielectric films Annealing of amorphous Ru–Si–O and Ir–Si–O thin films Conclusions References

123

128 130 131

Calculation of the phase diagrams of the MgO–FeO–Al2O3–SiO2 system at high pressures and temperatures: application to the mineral structure of the Earth’s mantle transition zone

132

II.7.3 II.7.4 II.7.5 II.7.6 II.8

II.8.1 II.8.2 II.8.3 II.8.4 II.8.5 II.8.6 II.8.7 II.9

II.9.1 II.9.2 II.9.3 II.9.4 II.9.5 II.9.6

Introduction Phases and models Phase equilibria in subsystems Phase diagrams for selected subsystems of the FeO–MgO–Al2O3–SiO2 system Phase diagram of the mantle composition at pressures up to 30 GPa Acknowledgements References

124 127

132 134 135 137 141 142 142

Calculation of the concentration of iron and copper ions in aqueous sulphuric acid solutions as functions of the electrode potential

144

Introduction Fe–H2SO4–H2O subsystem Cu–H2SO4–H2O subsystem The complete system Cu–Fe–H2SO4–H2O Conclusions and further developments References

144 145 146 148 152 154

Contents

II.10

ix

Thermochemical conditions for the production of low-carbon stainless steels

155

II.10.1 II.10.2 II.10.3 II.10.4 II.10.5 II.10.6

Introduction The mass action law approach The complex equilibrium approach Engineering conclusions Acknowledgements References

155 156 156 159 160 160

II.11

Interpretation of complex thermochemical phenomena in severe nuclear accidents using a thermodynamic approach

161

II.11.1 II.11.2 II.11.2.1 II.11.2.2 II.11.2.3 II.11.2.4 II.11.2.5 II.11.3 II.11.4 II.11.4.1 II.11.4.2 II.11.5 II.11.6 II.12

II.12.1 II.12.2 II.12.3 II.12.4 II.12.5 II.12.6 II.12.7

Introduction The nuclear thermodynamic database Pure elements and oxide components History Thermodynamic modelling of substance and solution phases Critical assessment of binary and ternary subsystems (calculation-of-phase-diagrams method) Content, assessed sub-systems, solution and substance phases Equilibrium calculation software Complex thermochemical phenomena in severe nuclear accidents In-vessel applications Ex-vessel applications Conclusions References

161 162 162 163 164 166 167 168 169 169 171 173 174

Nuclide distribution between steelmaking phases upon melting of sealed radioactive sources hidden in scrap

178

Introduction List of relevant nuclides Preparation of a suitable set of thermochemical data Calculated partition ratios Realistic distribution ratios Conclusions References

178 179 179 182 185 186 187

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II.13

Contents

Pyrometallurgy of copper–nickel–iron sulphide ores: the calculation of the distribution of components between matte, slag, alloy and gas phases

188

II.13.1 II.13.2 II.13.3 II.13.4 II.13.5 II.13.6 II.13.7

Introduction Blowing the matte Phase separation in the matte Solidification and recrystallisation Thermodynamic models and data Acknowledgements References

188 188 191 191 193 195 198

II.14

High-temperature corrosion of SiC in hydrogen– oxygen environments

200

II.14.1 II.14.2 II.14.3 II.14.4 II.14.5 II.14.6 II.14.7

Introduction Models for the corrosion process Thermodynamic analysis Si–C–H system Si–C–O–H system Discussion References

200 201 204 204 206 206 211

II.15

The carbon potential during the heat treatment of steel

212

II.15.1 II.15.2 II.15.3 II.15.4 II.15.5 II.15.6

Introduction The carbon potential The carbon activity in industrial furnace atmospheres The carbon activity of multicomponent steels Summary References

212 212 213 219 220 223

II.16

Preventing clogging in a continuous casting process

224

II.16.1 II.16.2 II.16.3 II.16.4

Introduction Setting up the calculation Solution Final remarks

224 224 225 226

II.17

Evaluation of the EMF from a potential phase diagram for a quaternary system

228

Introduction Theory

228 229

II.17.1 II.17.2

Contents

xi

II.17.3

Results

230

II.18

Application of the phase rule to the equilibria in the system Ca–C–O

231

Thermodynamic prediction of the risk of hot corrosion in gas turbines

239

II.19 II.19.1 II.19.2 II.19.3 II.19.4 II.19.5 II.19.6 II.19.7 II.20

Introduction Hot corrosion Thermodynamic modelling Hot-corrosion risk in second generation circulating pressurised fluidised-bed combustion Hot-corrosion risk in pressurised pulverised coal combustion Conclusions References

239 240 240 241 244 246 246

The potential use of thermodynamic calculations for the prediction of metastable phase ranges resulting from mechanical alloying

248

II.20.1 II.20.2 II.20.3 II.20.4 II.20.5

Introduction Calculation principles and previous related work Results Summary and conclusions References

248 249 251 261 262

II.21

Adiabatic and quasi-adiabatic transformations

263

II.21.1 II.21.2 II.21.3 II.21.4 II.21.5

Introduction Theory Numerical calculations Discussion References

263 263 264 265 266

II.22

Inclusion cleanness in calcium-treated steel grades

267

II.22.1 II.22.2 II.22.2.1 II.22.2.2

Introduction Choice of the most adapted sample to qualify the calcium treatment Nature and composition of the inclusions obtained by scanning electron microscopy Thermodynamic modelling

267 268 268 270

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Contents

II.22.3 II.22.4

Conclusion References

272 272

II.23

Heat balances and CP calculations

273

II.23.1 II.23.2 II.23.3 II.23.4

Introduction Practical calculations Calculational example References

273 276 281 281

II.24

The industrial glass-melting process

282

II.24.1

Introduction to some fundamentals of industrial glass melting Description frame for the thermodynamic properties of industrial glass-forming systems Description frame for one-component glasses and glass melts Description frame for multicomponent glasses and glass melts Heat content of glass melts The batch-to-melt conversion Heat demand of the batch-to-melt conversion; simple raw materials Dolomite and limestone as examples of complex raw materials Modelling the batch-to-melt conversion Conclusions References

295 297 301 302

Relevance of thermodynamic key data for the development of high-temperature gas discharge light sources

304

II.25.1 II.25.2 II.25.3 II.25.4 II.25.5

Introduction Operation principle of high-intensity discharge lamps Thermochemical modelling Conclusions References

304 305 306 310 310

II.26

The prediction of mercury vapour pressures above amalgams for use in fluorescent lamps

312

Introduction Use of amalgams in compact fluorescent lamps

312 312

II.24.2 II.24.2.1 II.24.2.2 II.24.2.3 II.24.3 II.24.3.1 II.24.3.2 II.24.3.3 II.24.4 II.24.5 II.25

II.26.1 II.26.2

282 284 284 286 290 293 293

Contents

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II.26.3 II.26.4 II.26.5

Calculation of phase equilibria for amalgam systems Conclusions References

314 321 321

II.27

Modelling cements in an aqueous environment at elevated temperatures

322

II.27.1 II.27.2 II.27.3 II.27.4 II.27.5 II.27.5.1 II.27.5.2 II.27.5.3 II.27.6 II.27.7

Introduction Previous modelling studies MTDATA Modelling approach Results and discussion C–S–H solubility at room temperature C–S–H solubility at higher temperatures Leaching simulation Conclusions References

322 323 323 325 328 328 331 333 334 335

Part III:

Process modelling – theoretical background

III.1

Introduction

341

III.2

The Gulliver–Scheil method for the calculation of solidification paths

343

III.2.1

Reference

346

III.3

Diffusion in multicomponent phases

347

III.3.1 III.3.2 III.3.3 III.3.4

Introduction Phenomenological treatment Analysis of experimental data: the general database References

347 347 349 350

III.4

Simulation of dynamic and steady-state processes

351

III.4.1 III.4.2 III.4.3 III.4.4 III.4.5 III.4.6

Introduction Concept of modelling processes using simple unit operations General description of the reactor model The control of material flows Conclusions References

351 351 353 355 357 357

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Contents

III.5

Setting kinetic controls for complex equilibrium calculations

359

III.5.1 III.5.2 III.5.3 III.5.4

Introduction The basic concept Simple equilibrium calculations References

359 359 364 367

Part IV:

Process modelling – application cases

IV.1

Calculation of solidification paths for multicomponent systems

371

IV.1.1 IV.1.2 IV.1.3

Introduction: description of the phase diagram Solidification paths References

371 372 374

IV.2

Computational phase studies in commercial aluminium and magnesium alloys

375

IV.2.1 IV.2.2 IV.2.2.1 IV.2.2.2 IV.2.2.3 IV.2.2.4 IV.2.2.5 IV.2.3 IV.2.4 IV.2.5

Introduction Thermodynamic calculations for ternary subsystems Al–Mg–Zn system Cu–Mg–Zn system Al–Cu–Mg system Al–Cu–Zn system Quaternary Al–Cu–Mg–Zn system Conclusions Acknowledgement References

375 375 376 381 381 382 382 383 384 384

IV.3

Multicomponent diffusion in compound steel

386

IV.3.1 IV.3.2

386

IV.3.4 IV.3.5

Introduction Numerical calculation of diffusion between a stainless steel and a tempering steel Calculation of partial equilibrium between a carbon steel and an alloy steel Summary References

IV.4

Melting of a tool steel

392

IV.4.1 IV.4.2 IV.4.3

Introduction Calculation Discussion

392 393 393

IV.3.3

386 389 391 391

Contents

xv

IV.4.4 IV.4.5 IV.4.6

Conclusions Acknowledgement References

396 397 397

IV.5

Thermodynamic modelling of processes during hot corrosion of heat exchanger components

398

IV.5.1 IV.5.2 IV.5.3 IV.5.3.1 IV.5.3.2 IV.5.4 IV.5.5 IV.6

Introduction Database work Calculational results Two-dimensional mappings (phase diagrams) for alloys in corrosive atmospheres Model calculations for gas-phase corrosion Conclusions References

398 398 399 399 400 403 404

Microstructure of a five-component Ni-base superalloy: experiments and simulation

405

IV.6.1 IV.6.2 IV.6.3 IV.6.4 IV.6.5 IV.6.6 IV.6.7

Introduction Experimental work Microstructure simulation Discussion Conclusions Acknowledgement References

405 406 408 412 413 414 414

IV.7

Production of metallurgical-grade silicon in an electric arc furnace

415

IV.7.1 IV.7.2 IV.7.3 IV.7.4 IV.7.5

Introduction The stoichiometric reaction approach The complex equilibrium approach The countercurrent reactor approach Reference

415 416 416 418 424

IV.8

Non-equilibrium modelling for the LD converter

425

IV.8.1 IV.8.2 IV.8.3 IV.8.4 IV.8.5 IV.8.6 IV.8.7

Introduction Process model development Modelling tool Simulation results Conclusions List of symbols References

425 426 429 430 434 435 436

xvi

IV.9 IV.9.1 IV.9.2 IV.9.3 IV.9.4

Contents

Modelling TiO2 production by explicit use of reaction kinetics

437

Introduction Anatase–rutile transformation – a simple example of the constrained Gibbs energy method Model for the TiCl4 burner: comparison with the image component technique References

437

441 445

Index

447

437

xvii

Contributing authors

Editor and author: Klaus Hack, GTT-Technologies, Herzogenrath, Germany Authors • John Ågren, Royal Institute of Technology, Stockholm, Sweden • Fritz Aldinger, Max-Planck-Institut für Metallforschung and Institut für nichtmetallische Anorganische materialien, Universität Stuttgart, Germany • Ibrahim Ansara, Institute National Polytechnique de Grenoble, Grenoble, France • Tom I. Barry, Amethyst Systems, Hampton, UK • Claude Bernard (emeritus), Institute National Polytechnique de Grenoble, Grenoble, France • Bernd Böttger, ACCESS e.V., Aachen, Germany • Pierre-Yves Chevalier, Thermodata, Grenoble, France • Bertrand Cheynet, Thermodata, Grenoble, France • Reinhard Conradt, RWTH Aachen (GHI), Aachen, Germany • R. Hugh Davies, National Physical Laboratory, Teddington, UK • Alan T. Dinsdale, National Physical Laboratory, Teddington, UK • Nathalie Dupin, Calcul Thermodynamique, 3 rue de l’avenir, 63670 Orcet, France • Gunnar Eriksson, GTT-Technologies, Germany • Olga Fabrichnaya, Max-Planck-Institut für Metallforschung (PML), Stuttgart, Germany • Françoise Faudot, Centre d’Etudes de Chimie Métallurgique, CNRS, Vitry-sur-Seine Cedex, France • Armando Fernandez Guillermet, Consejo Nacional de Investigationes Cientificas y Technicas, Centro Atomico Bariloche, Bariloche, Argentina • Evelyne Fischer, INPG – ENSEEG BP75, Saint Martin d’Hères, France • Graham M. Forsdyke, GE Lighting Europe, Leicester, UK • Suzana G. Fries, SGF Scientific, Consultancy, Aachen, Germany • John A. Gisby, National Physical Laboratory, Teddington, UK • Per Gustafson, Royal Institute of Technology, Stockholm, Sweden • Anke Güthenke, Daimler–Chrysler, Stuttgart, Germany

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• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

Contributing authors

Bengt Hallstedt, RWTH Aachen (MCh), Aachen, Germany Mireille G. Harmelin, Centre d’Etudes de Chimie Métallurgique, CNRS, Vitry-sur-Seine Cedex, France Ulrike Hecht, ACCESS e.V., Aachen, Germany Mats Hillert, Royal Institute of Technology, Stockholm, Sweden Torsten Holm, AGA AB Innovations, Lidingö, Sweden Michael H.G. Jacobs, Technische Universität Clausthal, ClausthalZellerfeld, Germany Tamara Jantzen, GTT – Technologies, Herzogenrath, Germany Stefan Jonsson, Royal Institute of Technology, Stockholm, Sweden Jürgen Korb, GTT-Technologies, Herzogenrath, Germany Pertti Koukkari, VTT, Espoo, Finland Ulrich Krupp, Fachhochschule Osnabrück, Osnabrück, Germany Ping Liang, Hans Leo Lukas and Fritz Aldinger, Max-Planck-Institut für Metallforschung and Institut für Nichtmetallische Anorganische Materialien, Universität Stuttgart, Germany Hans Leo Lukas (emeritus), Max-Planck-Institut für Metallforschung and Institut für Nichtmetallische Anorganische Materialien, Universität Stuttgart, Germany Jean Lehmann, ArcelorMittal, Maizières, France Dexin Ma, Foundry Institute of the RWTH Aachen, Aachen, Germany Torsten Markus, Forschungszentrum Jülich, Jülich, Germany Raymond Meilland, ArcelorMittal, Maizières, France Susan M. Martin, National Physical Laboratory, Teddington, UK Michael Modigell, RWTH Aachen (IVT), Aachen, Germany Peter Monheim, SMS-Demag, Düsseldorf, Germany Stuart A. Mucklejohn, GE Lighting Europe, Leicester, UK Erik M. Mueller, University of Florida, Department of Materials Science and Engineering, Gainesville FL, USA Michael Müller, Forschungszentrum Jülich, Jülich, Germany Dieter Neuschütz, RWTH Aachen (Men), Aachen, Germany Klaus G. Nickel, Universität Tübingen, Tübingen, Germany Ulrich Niemann, Philipps Forschungszentrum, Aachen, Germany Risto Pajarre, VTT, Espoo, Finland Karri Pentillä, VTT, Espoo, Finland Stefan Petersen, GTT-Technologies, Herzogenrath, Germany Günter Petzow (emeritus), Max Planck Institut für Metallforschung (PML), Stuttgart, Germany Alexander Pisch, Institute National Polytechnique de Grenoble, Grenoble, France Caian Qiu, Royal Institute of Technology, Stockholm, Sweden Hans-Jürgen Seifert, Technische Universität Freiberg, Freiberg, Germany Malin Selleby, Royal Institute of Technology, Stockholm, Sweden

Contributing authors

• • • • • • •

Philip J. Spencer, Spencer Group, Trumansburg, New York, USA Bo Sundman, University of Toulouse, Toulouse, France Jeff R. Taylor, Johnson Matthey Technology Centre, Reading, UK Vicente Braz de Trindade Filho, Technische Universität Siegen, Siegen, Germany Mark Tyrer, Imperial College, London, UK Colin Walker, Natural History Museum, London, UK Nils Warnken, University of Birmingham, Birmingham, UK

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xx

Software packages used for the case studies

ChemSheet GTT-Technologies and VTT Gesellschaft für Technische Thermochemie und -physik mbH Kaiserstraße 100 D-52134 Herzogenrath, Germany www.chemsheet.com

DICTRA Thermo-Calc Software AB Stockholm Technology Park Björnnäsväge 21 S-11347 Stockholm, Sweden www.thermocalc.se

FactSage GTT-Technologies and Thermfact Gesellschaft für Technische Thermochemie und -physik mbH Kaiserstraße 100 D-52134 Herzogenrath, Germany Thermfact Ltd/LTEE 447 Berwick Montreal H3R 1Z8, Canada www.factsage.com

GEMINI Thermodata 6 rue du Tour de l’Eau F-38400 Saint-Martin d’Hères, France http://thermodata.online.fr/

Software packages used for the case studies

InCorr/ChemApp University of Siegen and GTT-Technologies Institut für Werkstofftechnik (IWT) FB 11 Lehrstuhl für Werkstoffkunde und Materialprüfung Universität Siegen 57068 Siegen, Germany http://www.mb.uni-siegen.de/ifw2/ Gesellschaft für Technische Thermochemie und -physik mbH Kaiserstraße 100 D-52134 Herzogenrath, Germany http://www.gtt-technologies.de/chemapp

Micress ACCESS e.V. Intzestraße 5 D-52072 Aachen, Germany http://www.micress.de/

MTDATA National Physical Laboratory Hampton Road, Teddington Middlesex TW11 0LW UK www.npl.co.uk/mtdata/

SimuSage GTT-Technologies Gesellschaft für Technische Thermochemie und -physik mbH Kaiserstraße 100 D-52134 Herzogenrath, Germany http://www.gtt-technologies.de/simusage

Thermo-Calc Thermo-Calc Software AB Stockholm Technology Park Björnnäsväge 21 S-11347 Stockholm, Sweden www.thermocalc.se

xxi

xxii

Software packages used for the case studies

CEQCSI Process Engineering Department, Arcelor Research J. Lehmann Voie Romaine, BP 30320 F-57283 Maizières-lès-Metz, France [email protected]

xxiii

Member organisations of SGTE

www.sgte.org Arcelor Research Maizières-lès-Metz, France Forschungszentrum Jülich GmbH Institute for Energy Research (IEF-2 : Materials Microstructure and Properties) Jülich, Germany GTT-Technologies Gesellschaft für Technische Thermochemie und -physik mbH Herzogenrath, Germany Institute National Polytechnique de Grenoble Laboratoire de Science et Génic des Matériaux et Procédés Saint-Martin-d’Hères, France Max-Planck-Institut für Metallforschung und Institut für Nichtmetallische Anorganische Materialien der Universität Stuttgart Stuttgart, Germany National Physical Laboratory Thermodynamics and Process Modelling Teddington, UK Royal Institute of Technology Department of Materials Science and Engineering Stockholm, Sweden

xxiv

Member organisations of SGTE

RWTH Aachen Materials Chemistry (MCh) Aachen, Germany Thermfact Ltd/LTEE Montreal, Canada Thermo-Calc Software AB Stockholm, Sweden Thermodata Saint-Martin-d’Hères, France The Spencer Group Trumansburg, New York, USA

xxv

Editor’s acknowledgements

I would like to express my thanks to all contributing authors without whom this book never could have been realised. Special thanks are due to those new members of SGTE who have been willing to contribute at rather short notice. I would like to thank Wajahat Murtuza Khan for his very patient and extensive help in the preparation of the manuscript. Klaus Hack

xxvi

xxvii

Dedication

This book is dedicated to Professor E. Bonnier and Dr Himo Ansara. Professor Bonnier, the first chairman of SGTE, provided tremendous enthusiasm, vision and patience during the creation, development and implementation of a European-based structure for the Scientific Group Thermodata Europe (SGTE). His wise leadership through the initial years of SGTE as a European group, first as a project supported by the French CNRS and afterwards by DG XIII of the European Community, was largely responsible for the establishment of the present wide-reaching joint activities of SGTE members. Himo Ansara was the first manager of the SGTE Pure Substance Database. He had an infectious love for the application of Gibbs energy thermodynamics to practical problems which led to the generation of valuable databases as well as to the development of fundamental Gibbs energy models for nonideal solutions. His many insights have made thermodynamics such a valuable thermodynamic tool to the materials industry for the development and optimisation of materials and processes. Their colleagues in SGTE, present and past, will always remember their contributions with affection

xxviii

xxix

Foreword

The major purpose of this book is to illustrate how thermodynamic calculations can be used as a basic tool in the development and optimisation of materials and processes of many different types. Since the first edition of this book was published in 1996, the field of ‘computational thermochemistry’ has exploded as the reliability and scope of commercial databases have grown, as software packages have been developed to cover kinetic considerations and as more scientists have become acquainted with the potential that the field offers for understanding and modelling industrial and environmental processes. The examples selected in this book are, to a large extent, real case studies dealt with by members of SGTE and their collaborators in the course of their work. SGTE is a consortium of European and North-American research organisations working together to develop high-quality thermodynamic databases for a wide variety of inorganic and metallurgical systems. SGTE has been at the forefront of the broader international effort to unify thermodynamic data and assessment methods by promoting use of standard reference data for the elements and binary systems, and generic models to represent the variation in thermodynamic properties with temperature and composition. SGTE data can be obtained via members and their agents for use on personal computers with commercially available software, to enable users to undertake calculations of complex chemical and phase equilibria efficiently and reliably. The case studies presented in the book have been treated using SGTE data in combination with such software. Members of SGTE have played a principal role in promoting the concept of ‘computational thermochemistry’ as a time- and cost-saving basis for the control and modelling of various types of materials processes. In addition, such calculations provide crucial process-related information regarding the nature, amounts and distribution of environmentally hazardous substances produced during the different processing stages. While further developments in data evaluation techniques, in the modelling of Gibbs energies of the different types of stable and metastable phases, in

xxx

Foreword

the coupling of thermodynamics and kinetics and in the scope of application software are still needed, the case studies presented in this volume demonstrate convincingly that thermochemical calculations have great potential for providing a sound and inexpensive basis for materials and process development in many areas of technology. Alan Dinsdale (Chairman)

xxxi

Introduction

‘The real raison d’être for the continuation of extensive experimental research in metallurgical thermochemistry is the potential application of its principles and data to practical, in particular industrial, problems. For this purpose the gathering of raw experimental data is obviously not enough. Missing numerical information must be supplemented by estimates … Raw data must be sifted and critically evaluated to provide for every chemical system a consistent set of thermochemical properties … In practice, it is true, the knowledge of reaction rates is as important as that of equilibrium, if not more so, but the kinetic problems can only be tackled when the thermodynamic ones have been settled. It is also true that, in practice, metallurgical reactions are quite involved … but with some effort it will be found that even complicated chemical processes may be broken up into simpler reactions which are accessible to normal thermodynamic evaluation.’ The above points are made in the 5th edition of Metallurgical Thermochemistry by Kubaschewski and Alcock in 1979 [79Kub]. Elsewhere in the same book the term databank is used, albeit in quotation marks. Most of the statements are still relevant; computer-supported calculations provide an enormous potential for the application of thermodynamic principles to the solution of practical problems. There is still the need for good estimates arising from the lack of data in certain fields of interest, and critical evaluation of raw experimental results to obtain consistent thermodynamic data sets for complete chemical systems is still of paramount importance. Nevertheless, the development of software for treating thermochemical problems has made considerable advances in the past two decades and the questions that remain open can be tackled in a much more comprehensive way. The enormous effort involved in data collection and evaluation as carried out for example by Kubaschewski for pure substance data and by Kaufman [78Kau] in the field of alloy phases is now a somewhat less arduous task because of the availability to thermochemists of the computer. This has made it possible to treat thermochemistry in a completely new way. The

xxxii

Introduction Experimental thermodynamic properties and phase equilibria

Assessment programs Gibbs energy database

←

augmented by estimation techniques based on (a) experimental trends (b) ab initio calculations

Application programs

Calculations of: Thermodynamic properties Phase equilibria Process simulation

I.1 Flow sheet of the work procedure, from data assessment to an application calculation.

computer, because of its data storage and management and its ‘numbercrunching’ capabilities, has enabled us to look at the thermochemistry of a system as a whole, i.e. in many cases the user needs nothing more than a list of elements in his system and the values of the global variables temperature, pressure and element concentrations to carry out a theoretical study. Calculations can then be made of the phases stable at equilibrium, their amounts and compositions, and even information about the degree of instability of the phases not present at equilibrium can be provided. The flow sheet shown in Fig. I.1 may be used to illustrate the work procedure entailed in the application of computational thermochemistry. The purpose of the present volume is to present some examples of such calculations and thus to demonstrate the enormous potential of this technique. The computerised databases are still limited but a considerable effort is ongoing to expand them. SGTE is making a major effort to provide comprehensive high-quality self-consistent computerised thermodynamic databases both for pure substances and for mixtures of all types and is playing a leading role in establishing methods for data evaluation and modelling of solution phases. Software for the storage and retrieval of assessed data has been developed and there are a number of application programs to treat different aspects of chemical equilibrium [70Kau, 80Bar, 83Tur, 84Sch, 85Sun, 85Tho, 85Tur, 87Bar, 88Che, 88Din, 88Roi, 88Sun, 88Tho, 002CAL].

Introduction

xxxiii

References 70Kau

L. KAUFMAN and H. BERNSTEIN: Computer Calculation of Phase Diagrams, Academic Press, New York, 1970. 78Kau L. Kaufman and H. Nesor: Calphad: Comput. Coupling Phase Diagrams Thermochem. 2, 1978, 55–80. 79Kub O. KUBASCHEWSKI and C.B. ALCOCK: Metalurgical Thermochemistry, 5th edition, Pergamon, Oxford, 1979. 80Bar I. Barin, B. Frassek, R. Gallagher and P.J. Spencer: Erzmetall 33, 1980, 226. 83Tur A.G. TURNBULL: Calphad 7, 1983, 137. 84Sch E. SCHNEDLER: Calphad 8,1984, 265–279. 85Sun B. SUNDMAN, B. JANSSON and J.-O. ANDERSSON: Calphad 9, 1985, 153. 85Tho W.T. THOMPSON, A.D. PELTON and C.W. BALE: F*A*C*T facility for the analysis of chemical thermodynamics, guide to operations, McGill University Computing Centre, Montreal, 1985. 85Tur A.G. TURNBULL and M.W. WADSLEY: The CSIRO–SGTE THERMODATA System, Institute of Energy and Earth Resources, CSIRO, Port Melbourne, 1985. 87Bar T.I. Barry, A.T. Dinsdale, R.H. Davies, J. Gisby, N.J. Pugh, S.M. Hodson and M. Lacy: MTDATA Handbook: Documentation for the NPL Metallurgical and Thermochemical Databank, National Physical Laboratory, Teddington, 1987. 88Che B. CHEYNET: Proc. Int. Symp. on Computer Software in Chemical and Extractive Metallurgy, Montreal, Quebec, Canada, 1988, in Proc. Metall. Soc. CIM 11, 1988, 87. 88Din A.T. DINSDALE, S.M. HODSON, T.I. BARRY and J.R. TAYLOR: Proc. Int. Symp. on Computer Software in Chemical and Extractive Metallurgy, Montreal, Quebec, Canada, 1988, in Proc. Metall. Soc. CIM 11, 1988, 59. 88Roi A. ROINE: Proc. Int. Symp. on Computer Software in Chemical and Extractive Metallurgy, Montreal, Quebec, Canada, 1988, in Proc. Metall. Soc. CIM 11, 1988, 15. 88Sun B. SUNDMAN: Proc. Int. Symp. on Computer Software in Chemical and Extractive Metallurgy, Montreal, Quebec, Canada, 1988, in Proc. Metall. Soc. CIM 11, 1988,75. 88Tho W.T. THOMPSON, G. ERIKSSON, A.D. PELTON and C.W. BALE: Proc. Int. Symp. on Computer Software in Chemical and Extractive Metallurgy, Montreal, Quebec, Canada, 1988, in Proc. Metall. Soc. CIM 11, 1988, 87. 002CAL Calphad 26(2), A special edition on integrated thermodynamic databank systems, 2002, Elsevier.

xxxiv

Part I Theoretical background

1

2

I.1 Basic thermochemical relationships KLAUS HACK

I.1.1

Introduction

Since the publication of Gibbs last paper [878Gib] in the series ‘On the equilibrium of heterogeneous substances’ in 1878, all terms necessary to describe (chemical) equilibrium are defined. The chemical potential had been introduced, and the relation governing the different types of phase diagram (the Gibbs–Duhem equation) had been derived. Furthermore the different work terms in what we now rightly call Gibbs fundamental equation had been discussed far beyond the contribution of chemical or electrical work and included already, e.g. the contribution of surface tension or the gravitational potential. Gibbs also stated clearly that it is only the relative magnitude of each of these terms that permits omission for practical purposes; in principle, all possible contributions are always present. Most problems dealt with in equilibrium thermochemistry are those with constant temperature and pressure and where the other work terms, except for the chemical contribution, are usually omitted. Electrochemistry, of course, can only be treated if the electrical work term is also explicitly included. It is important to keep this in mind since the entire database derived under these conditions is a Gibbs energy, rather than a Helmholtz, enthalpy or internal energy database. Problems with constant temperature and volume, for example, have thus to be treated in an indirect way, which is, of course, no problem for the computer. Using the Maxwell relations, one can easily derive a diagrammatic scheme (Fig. I.1.1) to relate the Gibbs energy in its natural variables (G(T, P)) with the other state functions and their natural variables, i.e. the Helmholtz energy F(T, V), the enthalpy H(S, P) and the internal energy U(S, V). The arrows in the scheme indicate the signs of the derivatives that one has to take of the respective state function with respect to the chosen natural variable, e.g. (∂G/∂P)T = V or (∂G/∂T)P = – S. It will suffice here to say that a complete change from one state function to another can be obtained by application of a mathematical procedure, called the Legendre transformation 3

4

The SGTE casebook S

U

H P

V F

G

T

I.1.1 Diagram representing the Maxwell relations.

[71Hit]. Such transformations have also been introduced by Gibbs himself. Equilibrium is established if the potential function of the system for the conditions chosen has reached an extremum; in the case of the Gibbs energy as a function of T, P, the mole numbers, etc., it is a minimum as expressed by the following equations: G = min or dG = 0 and d2G > 0 dG = – S dT + V dP + ∑µi dni + ∑zjFΦj dnj…

(I.1.1) (I.1.2)

with total entropy S, temperature T, total volume V, pressure P, chemical potential µ, molecular number n, charge number z, Faraday constant F and electric potential Φ. From Equations (I.1.1) and (I.1.2), two different routes for a quantitative approach to equilibrium are possible. These are described in the following two sections.

I.1.2

Thermochemistry of stoichiometric reactions

The historical route, established experimentally before Gibbs, is the method of stoichiometric reactions. For isothermal and isobaric conditions, disregarding electrical and other work terms in a system, one obtains dG = ∑µi dni

(I.1.3)

The mass balance of a stoichiometric reaction can generally be written as ∑ νiB i = 0

(I.1.4)

with ν being positive for products and negative for reactants. Thus the changes dni of the absolute mole numbers ni of the substances Bi are defined by the change dξ in the extent of reaction ξ and the stoichiometric coefficients νi of the mass balance equation, and Equation (I.1.4) becomes: dni = νi dξ

(I.1.5)

After splitting the chemical potential µ into the reference potential µ° and the activity contribution RT ln a (R = general gas constant)

µ = µ° + RT ln a

(I.1.6)

one obtains the well-known law of mass action expressed in the equation for equilibrium:

Basic thermochemical relationships

(

νi

∆G ° = Σν i µ i° = – RT ln Π a i

) = – RT ln K

5

(I.1.7)

This equation permits the derivation of most informative relations between the activities of the products and reactants: ai = func (aj, T) with j ≠ i

(I.1.8)

It should be noted that the temperature dependence of this relationship is contained solely in the Gibbs functions of the pure substances (µi° = Gi°(T)) that are involved in the reaction. However, in practice, one is usually interested in a relationship between concentrations rather than activities. The derivation of such a relation based on a stoichiometric reaction approach is perfectly feasible but is subject to two pitfalls, one mathematical, the other a chemical. Firstly, the use of numerical methods cannot be avoided except in the simple case of ideal homogeneous systems, e.g. gas equilibria. In general, one has to deal with transcendental equations and even in the simple case an auxiliary equation for the total pressure of the system has to be employed. It is, in other words, not a question of straight linear algebra. Secondly, and much more importantly, all the independent reactions in a system must be known before starting the calculation. In other words, one must either make assumptions on the complete set of independent reactions in the system or analyse these experimentally before a reasonable calculation can be carried out. Such assumptions can easily lead to simplifications with very striking differences in the results, e.g. in a phase diagram. Figure I.1.2 and Fig. I.1.3 show phase stability diagrams for Ni in sulphurand oxygen-containing atmospheres with additions of H2O(g). This is a 0 NiS

NiSO4

log pSO2 (bar)

–2

–4

NiO Ni3S2

–6

–8

Ni –20

–10 log pO2 (bar)

0

I.1.2 Phase stability diagram for Ni as a function of the partial pressure of O2 and SO2 at 873 K. Use of this diagram could give a misleading impression of the dependence of coexistence lines on log p O 2 at high and low oxygen potentials (cf. Fig. I.1.3).

6

The SGTE casebook 0 NiS NiSO4

log Σ pi mi (bar)

–2 Ni3S2 –4 NiO –6

Ni

–8 –24

–20

–16

–12 –8 log pO (bar)

–4

0

2

(a) 0 NiSO4

log Σ pi mi (bar)

–2 NiS –4 NiO –6 Ni3S2 –8

Ni –20

–10 log pO (bar)

0

2

(b)

I.1.3 Phase stability diagrams for Ni as a function of log p O 2 and log ∑ pimi, where pi is the partial pressure of each species containing S and mi is the stoichiometry number of S in species i at 873 K for two different pressures of H2O: (a) p H 2O = 10–5 bar; (b) p H 2O = 1 bar.

typical case for the application of stoichiometric reactions in the derivation of an equilibrium diagram. In Fig. I.1.2, only O2 and SO2 are considered to be important gas species, thus leading to the well-known straight line phase boundaries, whereas Fig. I.1.3(a) and Fig. I.1.3(b) show the influence of the entire equilibrated gas phase with a fixed potential of H2O. Comparison of Fig. I.1.2 and Fig. I.1.3(a) shows that, for some conditions, low p H 2 O and oxygen pressures between 10–20 and 10–4, the results are in good agreement, but outside the appropriate pressure range for oxygen the behaviour of the phases is quite different. However, if the partial pressure of

Basic thermochemical relationships

7

H2O is raised to a much higher level (Fig. I.1.3(b)), there is very little similarity left between Fig. I.1.2 and Fig. I.1.3(b) because of the severe changes in the gas phase. These cannot be taken into account if one bases all reactions on the assumption that SO2 and O2 are the only important gas species under all conditions.

I.1.3

Thermochemistry of complex systems

The above example leads directly to the second method. Equilibria in complex systems, i.e. systems with many components and many phases (some or all of which may be non ideal mixtures), can only be treated safely by minimisation of the total Gibbs energy of the system under some constraints. This requires the compulsory use of numerical methods. As indicated, computer programs for the solution of multivariable transcendental equation systems have to be developed. Now, the equilibrium condition is written as G = ∑niµi = minimum

(I.1.9)

Here the chemical potentials µi refer to the entire set of chemical species in the system, no matter whether they are, for example gas species or aqueous species and thus part of one particular phase, or condensed stoichiometric substances such as Al2O3 or CaCO3 and thus one phase each. A clearer way to write the same sum is given by putting a greater emphasis on the phases of the system. After all, it is the phases that can come into equilibrium and some species used for the description of condensed phases might well be artefacts of the model used for the Gibbs energy of the particular phase. Now

G = Σ ( ΣniΦ ) GmΦ = minimum Φ

i Here, GmΦ

(I.1.10)

is the molar integral Gibbs energy of phase Φ, and niΦ are is used. the mole numbers of the phase constituents i of this phase. Thus the inner sum refers to the respective phase amounts, and the outer sum runs over all phases. The mass balance equations ∑niai,j = bj with j = 1 to m

(I.1.11)

are subsidiary conditions and, with the introduction of Lagrangian multipliers Mj, one obtains at equilibrium the simple relationship: G = ∑bjMj

(I.1.12)

In this set of equations the ai,j are the stoichiometric coefficients of the n species i with respect to the m independent system components j (normally but not necessarily the elements), the bj are the mole numbers of the system components j, and the Mj the chemical potentials of the system components at equilibrium. Note that there are usually far more species i than system

8

The SGTE casebook

Table I.1.1 Example of a stoichiometric matrix for the gas–metal–slag system Fe– N–O–C–Ca–Si–Mg Phase

Components

System components Fe

N

O

C

Ca

Si

Mg

Gas

Fe N2 O2 C CO CO2 Ca CaO Si SiO Mg

1 0 0 0 0 0 0 0 0 0 0

0 2 0 0 0 0 0 0 0 0 0

0 0 2 0 1 2 0 1 0 1 0

0 0 0 1 1 1 0 0 0 0 0

0 0 0 0 0 0 1 1 0 0 0

0 0 0 0 0 0 0 0 1 1 0

0 0 0 0 0 0 0 0 0 0 1

Slag

SiO2 Fe2O3 CaO FeO MgO

0 2 0 1 0

0 0 0 0 0

2 3 1 1 1

0 0 0 0 0

0 0 1 0 0

1 0 0 0 0

0 0 0 0 1

Liquid Fe

Fe N O C Ca Si Mg

1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

components j (n » m) in a system. As an example the matrix ai,j for a gas– metal–slag system with the elementary components Fe–C–Ca–Mg–N–O–Si is given in Table I.1.1. It is also interesting to note that Equation (I.1.12) is a simple linear equation which defines the tangential hyperplane of dimension m – 1 to the transcendental Gibbs energy surface of the system of dimension m. At equilibrium the chemical potentials µi of the species can all be calculated from

µi = ∑ai,jMj

(I.1.13)

It is clear from the above that methods for calculation of complex chemical equilibrium require, on the one hand, models for the description of the molar integral Gibbs energy of ideal and non-ideal mixture phases and, on the other, robust and reliable numerical algorithms to solve the equation systems indicated. From Equation (I.1.10) it is obvious that for each phase an expression for GmΦ , the integral molar Gibbs energy of the phase Φ is required. Two cases can occur: either the phase is treated as a pure stoichiometric substance

Basic thermochemical relationships

9

(compound), e.g. alumina with the formula Al2O3, or the phase is a solution (mixture) with variable content of its phase components, e.g. a body-centred cubic (bcc) alloy of iron and chromium, {Fe, Cr}bcc. In the first case the Gibbs energy only needs to be known as a function of T and P, GmΦ (T, P), whereas the second case requires the Gibbs energy to be known as a function of T, P and the mole ni of the phase components, GmΦ (T, P, ni). It must be noted that for the modelling of the molar Gibbs energy it is preferable to use concentrations rather then absolute mole numbers. However, as will be demonstrated in Chapter I.2 the choice of the concentration variable, e.g. mole fraction, site fraction or equivalent fraction, is already intimately related to the Gibbs energy model used in a particular case. It may therefore suffice here to indicate that in general for solution phases the modelling requires the Gibbs energy to be described with at least the following three explicit terms: G(T, P, ni) = Gref(T, P, ni) + Gid(T, ni) + Gex(T, P, ni)

(I.1.14)

The first term contains the contribution of the pure phase components, the second term gives the contribution due to the ideal mixing of the chosen phase components, whereas the third term contains non-ideal (excess) contributions with respect to the chosen ideal mixing. For an overview of the most widely used Gibbs energy models and their mathematical representation see Chapter I.2.

I.1.4

Activities of stable and metastable phases

To obtain a better understanding of a calculation, which also includes solution phases, the following Gibbs energy diagrams for an example system A–B will be considered. The system consists of a solution phase (liquid), a stoichiometric compound AB and the pure end members A and B. The composition for which equilibria are considered has xB = 0.333. Figure I.1.4 shows the situation for an equilibrium between pure A and pure B. The linear combination of 0.6667µA + 0.3333µB = 0.6667µ A° + 0.3333µ B° represents the minimum Gibbs energy for the fixed system composition with xB = 0.333, as is given by Equation (I.1.12) G = ∑ biµi. This equation is obviously course the mathematical equivalent of the common tangent. Now it is possible to compare the Gibbs energies of the two other phases, AB and liquid, with the equilibrium Gibbs energy found. One finds that all other phases have Gibbs energies which are more positive than the common tangent line, i.e. they are not stable. One can even give a quantitative measure for the distance from equilibrium. For the compound phase AB this is a straightforward task as the Gibbs energy difference µA + µB – µAB = ∆ = RT ln a can be read directly from Fig. I.1.5. Note, however, that the diagram is for molar Gibbs energies. Thus one reads from the diagram half the value of ∆ as defined above.

10

The SGTE casebook

Gm

Liquid

° * µAB/2 µB = µ°B

µA° = µA

A

B

0.333

xB

I.1.4 The equilibrium situation between pure A and B in a binary system exhibiting the phases A, AB, B and liquid.

Gm

*

° mAB/2 mB

D (mA + mB)/2 mA

A

0.333

xB

B

I.1.5 The driving force or activity of pure AB.

For the solution phase it is slightly more complicated to find a value for ∆. As the composition in this phase is variable, there are infinitely many possibilities to calculate a difference between the common tangent and the Gibbs energy curve. However, there is only one point which is closest to the common tangent. This point is easily found by drawing a tangent to the curve which has the same slope as the common tangent (Fig. I.1.6). Now a value of ∆ (and with it the value x B′ ) can be calculated even for a solution phase. It must be noted that the interpretation of ∆ = RT ln a, i.e. as an activity, which is straightforward for a stoichiometric compound, may seem unusual in conjunction with a solution phase. The term ‘driving force’, applied to both cases, has also been suggested but so far the quantity has no unique name. Essentially, it is the amount of Gibbs energy that would have to be added to the phase to make it stable if all other conditions remain unchanged.

Basic thermochemical relationships

11

Liquid

Gm

mB

D = RT ln a

mA

xB′

A

B

xB

I.1.6 The driving force or activity of liquid.

So far the discussion has always been aimed at the stable equilibrium state, i.e. all activities are 1 or less than 1. In reality, and of course in the computations, it is quite possible to obtain states in which activities greater than one can occur. In terms of the diagrams above the consequences are obvious. The Gibbs energy value of the compound AB and at least a section of the Gibbs energy curve of the liquid phase must fall below the common tangent. Thus ∆ values greater than zero can be obtained. In reality this can occur, for example, during undercooling of liquids without human interference. A computer program, however, must be told deliberately by the user to search for such a state. All good complex equilibrium programs therefore have a facility by which the user can suppress or suspend a particular phase or even a species within a phase. Thus it is possible to calculate for example metastable extensions of phase boundaries or equilibria in systems with very strong kinetic inhibitions. Note that for the calculation of for example the non-equilibrium state of a gas mixture with H2, O2 and H2O this technique is not suitable. In this case the number of independent system components will have to be increased from 2 to 3, i.e. instead of a stoichiometric matrix of the system with three lines and two columns one now has to use three lines and three columns (Table I.1.2). In such a case, H2O is not formed by way of an equilibrium reaction from H2 and O2 but is fed into the system as an independent non-reacting further species. Table I.1.2 Change in stoichiometric matrix from free equilibrium to frozen equilibrium

H2 O2 H 2O

H

O

2 0 1

0 2 1

→

H

O

H 2O

2 0 0

0 2 0

0 0 1

12

The SGTE casebook

Such a change in stoichiometric matrix is usually not part of the equilibrium programs. However, a method for the explicit incorporation of reaction kinetic data into a general equilibrium environment based on the above reasoning has been recently designed [01Kou] as is discussed in more detail in Part III.

I.1.5

Extensive property balances

The extensive properties of a system such as enthalpy, entropy, Gibbs energy, internal energy, free energy and volume are all state functions and thus path independent. In Fig. I.1.7 the two-dimensional state space is depicted using the variables temperature and extent of reaction. It is assumed that all other variables such as pressure (or volume) of the system are kept constant. In such a representation the classical form of an extensive property balance based on a stoichiometric mass balance formula can easily be understood. The mass balance is given by ∑νrBr = ∑νpBp with Br and Bp the reactant and product substances respectively, and νr and νp their stoichiometric coefficients. In Fig. I.1.7, four different states are marked and interconnected by straight lines: I (= ∑ νr Br(Tr)) and I′ (= ∑ νr Br(Tp)) define the state of reactants at the initial temperature Tr and the final temperature Tp. II′ (= ∑ νp Bp(Tr)) and II (= ∑ νp Bp(Tp)) correspond to the products at the initial and final temperature. The straight lines indicate the two routes which are normally used to carry out an extensive property balance. Route 1 assumes that the reactants will be heated to the product temperature and then the isothermal formation reaction takes place to obtain the products. Route 2 is based on the idea that an isothermal reaction takes place at the reactants temperature and the products are then heated to their final temperature. In particular, the second route is often used to explain how an exothermal

I′

II

I

II′

Tp

Tr

0

ξ

I.1.7 Two-dimensional state space.

1

Basic thermochemical relationships

13

reaction takes place. (In the first step the reaction releases heat which is absorbed by the products in the second step; there is a clear-cut distinction between isothermal enthalpy of reaction (state I to state II′) and heat content (state II′ to state II).) However, the path independence permits any path between the initial state I and the final state II to be used, e.g. the complicated curved line, but also, and most important, the shortest possible line, i.e. the diagonal. This latter method results in a very simple mathematical expression for all extensive property balances ∆ Z (Z = H, S, G, U, F, ...). It resembles the stoichiometric mass balance

∆Z = ∑νpZp(Tp) – ∑νrZr(Tr)

(I.1.15)

From this equation it is for example possible to work out what influence a preheating, i.e. a rise in Tr, would have on ∆Z (= f(Tr)). On the other hand, an inverse question could be answered such as: what temperature will the products have if an adiabatic process takes place? For Z = H, one would have to find TP such that ∆H = 0. For a calculation of this type it is useful to employ numerical methods. There is, however, one large danger in the approach discussed above. In the mass balance, both sides need to be defined – not only the reactants but also the products! What if the reaction does not take place as assumed by the mass balance? The value of ξ could be less than 1, or worse the assumed products are not formed at all. A very good example for such a case is the simple reaction SiO2 + 2C → Si + 2CO(g). It is stoichiometrically sound but does not take place at all. Instead SiC and SiO(g) together with CO(g) will be formed in a wide temperature range. In order to avoid such a grave error it is necessary to base the calculations of the extensive property balances on the determination of the products with the execution of a complex equilibrium calculation. Only the initial state needs to be fully defined by the user in terms of the substances, their amounts and their temperature(s). The final state (TP and the products as well as their amounts) is a result of the equilibrium calculation thus avoiding assumptions. Of course it is also possible to calculate the inverse problems using such an approach. An adiabatic flame temperature can easily be carried out without prior knowledge of the resulting gas species.

I.1.6

References

878Gib J.W. GIBBS: ‘On the equilibria of heterogeneous substrances’, Trans. Conn. Acad. 3, 1878, 176. 71Hit O. HITTMAIR and G. ADAM: Wärmetheorie, Vieweg, Braunschweig, 1971. 01Kou P. KOUKKARI, R. PAJARRE and K. HACK: Z. Metallkde. 92(10), 2001, 1151.

I.2 Models and data KLAUS HACK

I.2.1

Introduction

The assessment of the thermodynamic properties of individual phases, i.e. their Gibbs energy as a function of temperature, composition and possibly pressure, is the basis for the successful establishment of a thermodynamic databank. Gibbs energy data have to be made available for phases with a wide variety of properties such as the following: –

– – – – – – –

–

Pure stoichiometric phases, e.g. metallic elements, stoichiometric oxides or gas species in their standard state. Pure stoichiometric condensed substances under a high pressure, e.g. real or synthetic geological phases or pure metals. Ferromagnetic, antiferromagnetic or paramagnetic pure substances, e.g. magnetic elements or oxides. Species forming solutions, e.g. ideal or non-ideal gases or aqueous solutions. Condensed substitutional solutions, e.g. alloys with metallic components. Interstitial solutions, e.g. carbon and gases in alloys. Solutions exhibiting chemical defects, e.g. non-stoichiometric oxides or salts with differently charged ions. Solutions with several sublattices, e.g. alloys with metallic components occupying different sites of lattice or solid salts with equally charged ions. Solutions exhibiting ordering transformations, e.g. magnetically or chemically ordered alloys with metallic components.

The following chapter will give an overview of models used in the assessment of the Gibbs energies of such phases. Once the Gibbs energies are known, all other thermodynamic properties can be derived such as the following: 14

Models and data

15

molar volume V =  ∂G  ∂P T,n

(I.2.1)

molar entropy S = –  ∂G  ∂T P,n

(I.2.2)

molar enthalpy H = G – T  ∂G  ∂T P,n

(I.2.3)

internal energy U = G – T  ∂G  – P  ∂UG  ∂T P,n ∂P T,n

(I.2.4)

Helmholtz energy F = G – P  ∂G  ∂P T,n

(I.2.5)

heat capacity at constant volume  2   ∂ 2 Gm  2  ∂ Gm  Cν = – T   +     ∂P∂T    ∂T 2  P,n   

 ∂ 2 Gm      ∂P 2  

   T,n 

(I.2.6)

heat capacity at constant pressure C P = – T  ∂ G2   ∂T  P,n 2

(I.2.7)

chemical potential (partial Gibbs energy)

µ =  ∂G  ∂n T,P

(I.2.8)

We know about the thermodynamic properties of phases through information gathered by experience or, more often, by intentional experiments. This information is formed into a physical picture or model, e.g. oscillating atoms on lattice sites with a restricted number of degrees of freedom to describe the maximum value of the heat capacity. To be able to quantify the measured properties the next step is to put the physical model into a mathematical form, which can subsequently be used for making interpolations or even predictions. Before accepting the prediction given by a model it is important to test the model by comparing a number of such predictions with experimental information already available or obtained by new experiments. Even after a

16

The SGTE casebook

successful passing of such a test it must be realised that the physical model behind the mathematical description may not give a correct picture of the real world, e.g. the description of carbon in iron or other metals as a substitutional solution. The mathematical description may thus result in incorrect predictions outside the range in which it has been tested.

I.2.2

Gibbs energy data for pure stoichiometric substances

Although the Gibbs energy is the central function, it is still customary to store and apply the data of a pure stoichiometric substance in the form of the enthalpy of formation and entropy at standard conditions (T = 298.15 K and Ptot =1 bar) as well as a temperature function of the heat capacity. The latter is integrated over temperature to derive the temperature dependence of H and S:

∫

H = H ref +

T

C P dT

(I.2.9)

CP dT T

(I.2.10)

Tref

and

S = S ref +

∫

T

Tref

The Gibbs energy is then calculated from the Gibbs–Helmholtz relation G = H – T S. Solid-state physics show that the temperature dependence of the heat capacity CV is best explained by a quantum-mechanical picture of lattice vibrations [07Ein, 12Deb]. Thus one obtains the Debye function

( )

CV = D Θ Τ

(I.2.11)

where Θ is the Debye temperature which is a material-dependent constant. Although this approach is theoretically sound and describes the heat capacity of a series of elements in their crystalline state very well, it is not suitable for assessing all experimentally known values for solid substances within their respective error limits. For a more detailed discussion see [98Hil]. Furthermore, for most applications it is not necessary to recur to 0 K as the reference temperature. Thus a system of thermochemical data has been established on the basis of the standard element reference (SER) state. As indicated above, room temperature (298.15 K) and a total pressure of 1 bar are introduced as standard conditions, and the enthalpy H298 of the state of the elements which is stable under these conditions is set to zero by convention. The entropy S298 is given by its absolute value and the heat capacity Cp, at constant pressure is described

Models and data

17

by a polynomial, commonly by the polynomial introduced by Meyer and Kelley [49Kel]: C P = c1 + c 2 T + c 3 T 2 +

c4 T2

(I.2.12)

This approach permits assessment of the thermal properties of most substances within their experimental error limits as there are sufficiently many adjustable parameters. In some exceptional cases it may be necessary to split the temperature ranges of the fit to stay within the experimental error limits. As an example, see Fig. I.2.1 for the heat capacity of O2(g). It should be noted that, instead of splitting the temperature range, some workers prefer to add further terms such as T1/2, T–3 or the inverse of these into the above equation. This has become customary particularly among geochemists [94Sax, 78Rob]. From the standard CP polynomial and the known values of ∆H298 and S298, one obtains the Gibbs energy equation as G = C1 + C2 T + C3 T ln T + C 4 T 2 + C5 T 3 +

C6 T

(I.2.13)

The coefficients Ci are now the data to be stored in a Gibbs energy databank. Note, that the first two coefficients contain contributions from both ∆H298 and CP as well as S298 and CP respectively, whereas the latter four can be directly related to the four coefficients of the standard CP equation. In this way, all properties, especially the enthalpy and entropy values at room temperature, are the result of a calculation and cannot be simply read from coefficient tables as in the standard compilations of, for example, JANAF [85Cha] or Barin et al. [77Bar]. For use in computer programs, however, there is the advantage that the above equation can be interpreted as a scalar product between a

Heat capacity (J mol–1 K–1)

44

I 24

0 RT 1000

II 3300 Temperature (K)

III 6000

I.2.1 Heat capacity of O2 (g): RT, room temperature.

18

The SGTE casebook

r r substance vector a = (C1, C2, C3, C4, C5, C6) and a temperature vector TG = (1, T, T ln T, T2, T3, 1/T). Taking the appropriate derivatives of the elements of this temperature vector with respect to T, all other properties H, S and CP can also be calculated as scalar products using the same substance vector: r r (I.2.14) Z = a • Tz with Z = CP, H, S, G Phase transitions of first order can easily be integrated into this data system, once the temperature and enthalpy of transition and the coefficients of the CP equation of the phase at higher temperature are known. The G function for the higher range is again derived from the integrals of enthalpy and entropy, but now based on the transition temperature instead of room temperature. Furthermore, the changes in enthalpy and entropy on phase transition need to be added. The standardised treatment described above has also been used for substances which exhibit magnetic (second-order) phase transitions. To be able to handle the anomaly in the heat capacity that arises in such a case (Fig. I.2.2), it was customary to split the temperature range around the Curie temperature into several small intervals such that the standard expression for CP could be used. This procedure creates an unnecessarily large number of coefficients (e.g. eight times four CP parameters for Ni) and it also causes numerical difficulties because of the unusually large values of the parameters. It is furthermore not suitable for solution phases where the Curie temperature as well as the magnetic moment depend on composition. SGTE has therefore adopted an approach suggested by Inden [76Ind1, 76Ind2] which simplifies the situation considerably.

Magnetic heat capacity (J mol–1 K–1)

30

0

0

1600 Temperature (K)

I.2.2 Magnetic contribution to the heat capacity of body centred cubic (bcc) Fe.

Models and data

19

Lattice heat capacity (J mol–1 K–1)

40

20 0

1600 Temperature (K)

I.2.3 Lattice contribution to the heat capacity of bcc Fe.

Total heat capacity (J mol–1 K–1)

60

20 0

1600 Temperature (K)

I.2.4 Total heat capacity of bcc Fe.

The magnetic contribution is treated separately, thus leaving a well-behaved curve for the non-magnetic contribution to the heat capacity which can usually be described by one set of standard parameters for the entire temperature range. For the magnetic part of CP the critical temperature (Tc, either the Curie or the Néel temperature), the crystalline structure of the phase and the magnetic moment ß per atom in this particular structure are the prerequisites. One obtains for 1 mol of magnetic element Gmagnetic = RT f  T  ln( β + 1)  Tc 

(I.2.15)

20

The SGTE casebook

f is a structure-dependent function of temperature. It is different for the ranges above and below the critical temperature (for the details of this as used by SGTE see [91Din]). Figure I.2.2, Fig. I.2.3 and Fig. I.2.4 show the two distinct contributions from magnetism and lattice to the heat capacity and the resulting total curve respectively. A further additive contribution to the Gibbs energy, which is usually ignored because of its negligibly small value, stems from the pressure dependence of the molar volume. However, recent technical developments such as hot isostatic pressing but also more detailed research in geochemical phenomena have created a need to be able to handle this extra contribution. SGTE has adopted the Murnaghan equation [44Mur] for its mathematical description. This equation uses explicit expressions for the molar V° volume at room temperature, its thermal expansion, α(T), the compressibility K(T), at 1 bar and the pressure derivative of the bulk modulus n (bulk modulus = 1/compressibility).  G pressure = V ° exp  

 [1 + nK ( T ) P ](1–1/n )–1 α (T) dT  (I.2.16) ( n – 1) K ( T )  298 K

∫

T

with α(T) and K(T) polynomials of the temperature: α(T) = A0 + A1T + A2T 2 + A3T 2

(I.2.17)

K(T) = K0 + K1T + K2T 2

(I.2.18)

The necessary parameters have so far been assessed for a few substances, mainly metallic elements and some oxide phases of geological interest. Examples of the resulting P – T phase diagrams, are given in Fig. I.2.5 and Fig. I.2.6.

I.2.3

Conclusion

For pure stoichiometric substances the Gibbs energy is a function of only temperature and, if appropriate, total pressure: G(T, P). Different additive contributions can be treated separately. Thus one obtains Gm = latticeG + magneticG + pressureG

(I.2.19)

where latticeG depends upon ∆H298, S298 and a CP polynomial, magneticG depends upon lattice structure, critical temperature and magnetic moment, and pressureG depends upon standard molar volume, compressibility, thermal expansion and pressure derivative of the bulk modulus.

I.2.4

Relative data

If only an equilibrium state is to be discussed, it is sufficient and therefore often customary to work with relative Gibbs energies. This method is used

Models and data

21

Liquid

Temperature (K)

2500 1980 K 5.14 GPa 2000

Fcc A1 Bcc A2

1500 757 K 10.4 GPa

1000

Bcc A2

Hcp A3

500 0

10

20 30 Pressure (GPa)

40

50

I.2.5 Calculated P–T phase diagram for Fe. 2200 Liquid 2000 Cristobalite

Temperature (K)

1800

[67 Coh] [13 Fen] [78 Gra] [62 Ken] [66 Ost] [76 Jac] [76 Jac]

Tridymite 1600

1400

1200 β-quartz 1000 α-quartz 800

0

1

2

3

4 5 6 7 Pressure (108 Pa)

8

9

10

I.2.6 Calculated P–T phase diagram and experimental phase equilibria for SiO2 [94Sax].

for systems which can be treated by the stoichiometric reaction approach as well as solution phase systems. It has some advantages but great care must be taken not to confuse data from a relative scale with those based on the SER state.

22

I.2.4.1

The SGTE casebook

Advantages of the use of relative Gibbs energies

Pure substances and stoichiometric reactions As an example the simple gas system H2–O2–H2O will be discussed. This 1 system is governed by the stoichiometric reaction H2 + O2 with 2

 PH 2 O  ∆ G° = – RT ln K = – RT ln    PH 2 PO 2 

(I.2.20)

The standard method is to calculate ∆G° from the standard chemical potentials µ i° (= Gi° ) as

∆G° = µ H° 2 O – µ H° 2 – 1 µ O° 2 2

(I.2.21)

However, the same result can be obtained by postulating that the chemical potentials of the independent system components hydrogen (H2) and oxygen (O2), are identical with zero and the chemical potential of H2O is given relative to these two, i.e.

µ H° 2 ≡ µ O° 2 ≡ 0 and µ H° 2 O ≡ ∆G °

(I.2.22)

For the calculation of the equilibrium constant this postulation has no impact. Additionally, it can often be justified experimentally that ∆G° has a simple linear temperature dependence in a particular temperature range, resulting in constant values for the enthalpy change ∆H° and the entropy change ∆S° of the reaction, i.e. ∆G° = ∆H° – T ∆S°. This leads to a simple and easy-to-use thermochemical description of systems which are governed by only a few stoichiometric reactions. The experimental finding is the major reason for the relatively late appearance of generalised thermodynamic databases for pure substances. Furthermore, it is of course possible to use the simple numbers in complex equilibrium software which is based on Gibbs energy minimisation techniques. The equilibria calculated using relative Gibbs energies are the same as those calculated from SER data!

I.2.4.2

Solution phase systems

Although solution-phase systems cannot be based on a stoichiometric reaction approach, it is even for such systems customary to work with relative Gibbs energies. The thermodynamic reason is that in the equation for the molar Gibbs energy of a solution phase (Gm = refGm + idGm + exGm; see below) the first term, refGm, is linearly dependent upon the contributions of all phase constituents:

Models and data ref

G Φ = Σ X iref G Φi

23

(I.2.23)

Thus a choice of suitable reference states for each of the independent system components will only result in a linear transformation of the Gibbs energies of all phases in the system and does not affect any equilibria in the system. Graphically this can be seen from the two diagrams for the three solution phases present in the system Pb–Sn at the eutectic temperature and 1 bar pressure. Figure I.2.7 shows the Gibbs energy of the liquid, body-centred tetragonal (bct) (Sn) and face-centred cubic (fcc) (Pb) on the SER scale. Figure I.2.8 shows the Gibbs energies of the same phases relative to pure bct Sn and fcc Pb. There is of course no difference in the composition of the points of common tangency. The advantage of the relative scale representation is again found in the mathematical simplicity of the so-called lattice stabilities. For the liquid phase of the Pb–Sn system, one finds for example ref

liq liq bct fcc G liq = x Pb ( o GPb – o GPb ) + x Sn ( o GSn – o GSn )

(I.2.24)

Both terms in parentheses can again be represented by simple linear expressions in T, since only the enthalpy and the entropy of melting for the two elements are needed in the first approximation: ref

melt melt melt melt (I.2.25) G liq = x Pb ( ∆Η Pb – T ∆S Pb ) + x Sn ( ∆Η Sn – T ∆SSn )

–22000

Total Gibbs energy (J)

Liquid phase Bct A5 phase Fcc A1 phase

–32000

0

0.26

0.74

0.98

Mole fraction of Sn

I.2.7 Gibbs energy on the SER scale for 1 mol of phase (T = 454.56 K; SER).

24

The SGTE casebook 5000

Total Gibbs energy [J]

Liquid phase Bct A5 phase Fcc A1 phase

0

–5000

0

0.26

0.74

0.98

Mole fraction of Sn

I.2.8 Gibbs energy relative to bct Sn and fcc Pb for 1 mol of phase (T = 454.56 K; relative data).

I.2.4.3

Disadvantages

There are some severe disadvantages of the method of relative Gibbs energies which need careful consideration. 1

2

3

Once a data set for a system or even a complete database has been established, it is not possible to add any species or phases to this package unless the same reference state is chosen. Thus any data on the SER scale must be completed or augmented by other data on this scale. The same holds of course for data on a relative scale. Since it is possible to choose a different reference state for each class of substances (e.g. metals, salts, oxides and gases), data from two (or more) different sources need very careful checking with respect to the chosen reference state of each source before they can be combined. Combination is only possible after transformation to a common reference state! Extensive property calculations as well as property balances, especially heat balances, are normally not possible for systems which are described by relative Gibbs energies since in almost all cases the simple linear temperature dependence is chosen. However, proper extensive property balances can only be carried out if the contribution of the heat capacity is fully integrated in the Gibbs energy.

Models and data

I.2.4.4

25

Conclusion

The best possible representation of Gibbs energies is given using the SER state. Data on this scale permit any possible type of calculation. For restricted use relative Gibbs energies as discussed above may be useful. Great care must be taken when integrating relative Gibbs energies into data sets or databases which use the SER state. In particular, the application of the Neumann–Kopps rule for the derivation of missing heat capacities of compounds or phase constituents needs careful consideration.

I.2.5

Solution phases

I.2.5.1

Substitutional solutions

The properties of solutions are usually described relative to the properties of the pure substances in the same structure (ϕ) and at the same temperature: G ref = Σ x i o Giϕ

(I.2.26)

where xi is the mole fraction of constituent i. In addition at least one further term needs to be added which contains the contribution from ideal mixing: Gid = RT ∑ xi ln xi

(I.2.27)

The deviation of the real solution from this ideal solution must be described by a further additive term. It is important that the mathematics used for this term are such that they permit estimation the properties of higher-order systems from assessed data for the lower-order systems. Redlich and Kister [48Red] proposed that one should estimate the properties of a ternary solution from the three component binaries by applying the following type of expression to each binary and should evaluate the parameters by fitting to binary experimental information: ex GAB = x A x B Σ LνAB ( x A – x B ) ν

(I.2.28)

The first term in the Redlich–Kister series is xA xB L°AB and it is identical with the energy parameter in the so-called regular solution model. That model can be justified by the Bragg–Williams approach based on random mixing and a consideration of the energies of different bonds between like (AA and BB) and unlike (AB) next-nearest neighbours. The higher LνAB parameters may be regarded as describing the composition dependence of the energy parameter. If there is experimental information on the ternary solution range, this can be used to evaluate deviations from the predictions obtained by the Redlich– Kister sum for the binaries. One would first try to describe these deviations with a further regular term xA xB xC LABC or, if necessary, one may use a power

26

The SGTE casebook

series for LABC based on similar principles as the Redlich–Kister series. Experimental information from all four ternaries making up a quaternary system and from the quaternary itself is often not available. However, if that is the case, one may even introduce a quaternary regular term xA xB xC xD LABCD. The whole expression for the Gibbs energy of a solution phase can thus be given as GmΦ = Σ x i °GiΦ + RT Σ x i ln x i + Gmex,Φ

= Gmref + Gmid + Gmex,Φ

(I.2.29)

with mij

Gmex,Φ = ΣΣ x i x j Σ Lνij ( x i – x j ) ν + ΣΣΣ x i x j x k Lijk ν =0

i< j

i< jshow ac(o).w(liquid,mn) AC(O).W(LIQUID,MN)=–8.092E–ll command>show ac(o).w(liquid,si) AC(O).W(LIQUID,SI)=–2 .543E–10 The symbol AC is used to denote activity for the component given within the parentheses. The symbol W is used to denote mass fraction and this can be

226

The SGTE casebook

indexed with just a component, if one means the overall composition or, as in this case, a phase name and a component if one means the composition of a phase. The negative values show that the oxygen activity will decrease by increasing the weight fraction of Mn or Si in the liquid alloy. Changing the Mn or Si content will also affect the chromium activity. This effect can be calculated in the same way: command>show ac(cr).w(liquid,mn) AC(CR).W(LIQUID,MN) =5.805E–2 command>show ac(cr).w(liquid,si) AC(CR).W(LIQUID,SI) =1.824E–1 This shows that an increase in the Mn or Si content will increase the chromium activity, which will tend to make the Cr2O3 oxide more stable. However, considering that the oxygen activity is raised to 3 and the chromium activity to 2 in order to obtain the solubility product of Cr2O3, the overall effect will be to decrease its stability. One may try different Mn or Si fractions manually but again Thermo-Calc offers a facility to calculate the desired answer directly. The fact that one is interested in is the Mn or Si content that will make the Cr2O3 oxide unstable. The limiting value would be when Cr2O3 is just stable and one may specify this as a condition. At the same time, one releases the condition on the Mn fraction and allows the program to determine this itself. The same calculation can then be repeated with the Si fraction set free. In the present case the formation of Cr2O3 could be prevented by increasing the alloy content of Mn from 0.4 to 0.55 wt% or the content of Si from 0.2 to 0.28 wt%. The necessary change in the Si content is smaller, which is in agreement with the derivatives listed above. One may even calculate the curve giving the solubility of Cr2O3 for various Mn and Si contents in the liquid metal. This is shown in Fig. II.16.1.

II.16.4

Final remarks

Many processes in steelmaking are well established and, if there is a problem, a qualified engineer can often easily determine the cause and find a remedy. However, skilled personnel are scarce and costly and it is advantageous if some of the experience that it takes years to gather from practical work can be stored into a computerised databank. Such databanks can be operated on a routine basis if properly adopted. The current interest in new materials for which there is little or no previous experience and the demand for less pollution and the rising cost for energy also make it necessary to develop unorthodox methods for solving manufacturing problems. Today, thermodynamic databanks such as Thermo-Calc can handle most types of

Preventing clogging in a continuous casting process

227

0.32 0.30

Si in metal (wt%)

0.28 Liquid + slag 0.26 0.24 Liquid + slag + Cr2O3 0.22 0.20 0.18 0.35

0.40

0.45 0.50 Mn in metal (wt%)

0.55

II.16.1 This curve shows the solubility of crystalline Cr2O3 in liquid metal and slag for various Si and Mn contents in the metal when the Cr content of the liquid metal is 25 wt%.

problem and generate almost any kinds of diagram for systems where consistent thermodynamic data are available. However, the amount of carefully assessed and consistent thermodynamic data for solution phases is very small and a collective and financially strong effort in this field is necessary.

II.17 Evaluation of the EMF from a potential phase diagram for a quaternary system M AT S H I L L E R T

II.17.1

Introduction

When studying the EMF of a certain electrolytic cell at 1 bar and 1000 K, one needs to check the results by comparing with the information given in the form of potential phase diagrams. The cell had one electrode which was a mixture of MnS, MnO, Cu2S and Cu and the other was a mixture of MnO, Mn3O4, Cu2S and Cu. The electrolyte was solid zirconia stabilised with calcia. The available potential phase diagrams are presented in Fig. II.17.1 and Fig. II.17.2.

5.0 2.5 0 Cu2S

log PSO2

–2.5 Cu2O

–5.0 –7.5 Cu

–10.0 –12.5 –15.0 –25

–20

–15 log pO2

–10

–5

II.17.1 Potential phase diagram for the Cu–O–S system at 1000 K.

228

Evaluation of the EMF from a potential phase diagram

229

5.0 MnSO4

2.5 0

MnS

log PSO2

–2.5 –5.0 Mn3O4

–7.5 MnO

–10.0 –12.5 –15.0 –25

–20

–15 log PO2

–10

–5

II.17.2 Potential phase diagram for the Mn–O–S system at 1000 K.

II.17.2

Theory

By assuming that there is no solubility of Cu in the Mn phases and no solubility of Mn in the Cu phases, one can construct a potential phase diagram for the quaternary Cu–Mn–O–S system by plotting all the lines in the same diagram. For a quaternary system the phase diagram at constant T and P should actually have three dimensions and the diagram now obtained is thus a projection, obtained by projecting in the µCu direction (or the µMn direction depending upon what potential is chosen as the dependent one). As an example, the line representing MnS + MnO in Fig. II.17.2 is now the projection of a planar surface, parallel to the µCu axis and it terminates at the three-phase line MnS + MnO + Cu2O where µCu is so high that Cu2S forms. All twophase fields are two dimensional but all the old fields only appear as onedimensional projections, as the line representing MnS + MnO. All the new fields appear two-dimensionally and they are marked in the new diagram, Fig. II.17.3. All the lines are of the same kind but, in order to show clearly which lines came from the Cu–O–S diagram, they are given as dashed lines. The point of intersection between the lines representing equilibrium between MnS and MnO in Fig. II.17.2 and between Cu2S and Cu in Fig. II.17.1 now represents a four-phase point with the four three-phase lines radiating in different directions. The one-phase fields for Cu2S and for Cu are situated above those for MnS and MnO if the µCu axis is plotted upwards. The points representing the two electrodes are labelled 1 and 2 respectively.

230

The SGTE casebook 5.0 MnSO4 +Cu2S

2.5 0

MnS + Cu2S

log PSO2

–2.5

MnO + 2 Cu2S

Cu2O + MnSO4 Mn3O4 + Cu2S

–5.0 –7.5 –10.0

MnS + Cu

Mn3O4 + Cu2O

MnO + Cu 1

–12.5 –15.0 –25

Mn3O4 + Cu –20

–15 log PO2

–10

–5

II.17.3 Projection of the potential phase diagram for the Cu–Mn–O–S system at 1000 K.

The electrical current can pass through the electrolyte mainly by the diffusion of O2– ions. The EMF will thus be an expression of the difference between the oxygen potentials of the two electrodes and it can be estimated from the difference between RT ln PO 2 for the two points in the new diagram representing the electrodes. Because oxygen ions are divalent, we obtain the EMF from E = ∆µO/2F where F is Faraday’s constant (equal to 96 486 C mol–1).

II.17.3

Results

From the two points we read ∆µO = 0.5 ∆ µ O 2 = 0.5RT lnl0(–10.9 + 20.8) = 11.4RT and obtain E = 0.49 V. The diagrams presented here were calculated from the SGTE Pure Substance database using the Thermo-Calc databank.

II.18 Application of the phase rule to the equilibria in the system Ca–C–O KLAUS HACK

Complex equilibria are calculated by minimising the total Gibbs energy under some constraints, usually for a given temperature and total pressure as well as a given system composition. Although this procedure is not tied to a specific reaction between the system constituents, it can be useful to consider the equilibrium in terms of a stoichiometric reaction as will be shown below. Equilibria between CaCO3, CaO and the gas phase consisting of CO2 and, perhaps, other gas species show how the phase rule is used to gain some understanding for the results from complex equilibrium calculations among these substances. The dissociation of CaCO3 is governed by the stoichiometric reaction CaCO3 ↔ CaO + CO2 (g)

(II.18.1)

This reaction represents a three-phase equilibrium between two different solids and the gas. In terms of the phase rule one obtains the following for the coexistence of all three phases: Number of elementary components: ε = 3 (C, Ca and O) Number of phases Φ = 3 (CaCO3, CaO, gas = {CO2}) Number of stoichiometric constraints s = 1; nC / nO = 1/2 in the gas Number of Gibbsian components c = ε – s = 3 – 1 = 2 (one may choose for convenience CaO and CO2) Number of degrees of freedom f = c + 2 – Φ = 2 + 2 – 3 = 1 For the condition of coexistence of the three phases CaCO3, CaO and gas (CO2), one obtains one degree of freedom. One can express either the total pressure as a function of temperature or vice versa: P = P(T) or T = T(P)

(II.18.2)

This result is reflected in the equilibrium tables given below. 231

232

The SGTE casebook

However, it is possible to increase the degrees of freedom in this system. From the above tabular summary of the system it can be seen that there are two different options for doing this: – –

By increasing the number of components to 4 adding an arbitrary amount of argon (here 10–5 mol) to the gas phase. By removing the stoichiometric constraint nC/nO=1/2 in the gas phase adding an arbitrary amount of O2 (here 10–5) to the gas phase.

How can equilibrium tables be used to find for example the value of the dissociation pressure pCO 2 over CaCO3 for a temperature of 1000 K and a total pressure of 1 bar ? Entering the amount of CaCO3 as 1 mol, setting T = 1000 K and p = 1 bar leads to equilibrium table, Table II.18.1. In this table it is shown that CaCO3 will dissociate into CaO and CO2. The activity of CaO is calculated to be 1. However, a gas phase with the single species CO2 cannot form since the total pressure that was set to be 1 bar is greater than the partial pressure of CO2 according to the given temperature. Because no CO2 is transferred to the gas phase, the amount of CaO is also calculated to be zero. In terms of the stoichiometric reaction above, the equilibrium is completely on the left-hand side; i.e. although equilibrium is calculated, the extent of reaction is 0. Although neither CaO nor CO2 has been generated, the CO2 partial pressure has the correct equilibrium value for the coexistence of the three phases CaCO3, CaO and gas (CO2) since the activities of the two stoichiometric condensed phases are both 1. In the next step the total pressure P is set equal to the partial pressure of CO2 in Table II.18.1 and the equilibrium calculation is executed. In Table II.18.2 it is shown that under the condition that the ‘total pressure of the gas’ is equal to the ‘partial pressure of CO2’, the above-stated stoichiometric reaction will come to completion. In other words, the equilibrium is now completely on the right-hand side, i.e. the extent of reaction is 1. However, the result still relates to the three-phase equilibrium since the activities of the two stoichiometric condensed phases are again equal to 1 (see Table II.18.2). Now let us run a calculation with constant volume in which the value of the (gas) volume is set to be less than that calculated in the previous calculation (Table II.18.2), e.g. 1000 l. The result as given in Table II.18.3 shows that, for these new conditions (1 mol of CaCO3; T = 1000 K, V = 1000 l) all three equilibrium phases are present in a finite amount. It follows that the extent of reaction has a value between 0 and 1. The actual value (read from the equilibrium mole numbers of CaO or CO2, 7.5663 × 10–1) is directly related to the value fixed for the gas volume (the finite volume of the condensed phases is not considered in the calculation). Also note that the total pressure is now a calculated (!) quantity, and of course it is equal to the partial pressure of CO2.

Application of the phase rule to the equilibria in Ca–C–O system

233

Table II.18.1 Equilibrium table for 1 mol of CaCO3 at T = 1000 K and P = 1 bar; edited output data from Factsage T = 1000.00 K P = 1.00000E+00 bar V = 0.00000E+00 dm3 STREAM CONSTITUENTS CaCO3

PHASE: gas_ideal CO2 CO O2 O O3 CaO Ca C2O C3O2 C Ca2 C2 TOTAL:

EQUIL AMOUNT mol 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

AMOUNT/mol 1.0000E+00 MOLE FRACTION bar 1.0000E+00 4.8962E-07 2.4481E-07 3.1021E-13 3.0192E-22 1.5610E-25 6.4944E-27 2.2401E-40 1.5019E-41 7.0687E-43 1.3078E-56 5.4967E-61 1.0000E+00

FUGACITY 6.2910E-02 3.0802E-08 1.5401E-08 1.9515E-14 1.8994E-23 9.8201E-27 4.0856E-28 1.4093E-41 9.4484E-43 4.4469E-44 8.2273E-58 3.4580E-62 6.2910E-02

mol ACTIVITY 1.0000E+00 1.0000E+00 CaCO3 CaO 0.0000E+00 1.0000E+00 CaO2 T 0.0000E+00 7.9011E-06 C 0.0000E+00 8.5732E-15 Ca 0.0000E+00 1.5978E-24 CaC2 0.0000E+00 3.3334E-48 ***************************************************************** Cp_EQUIL H_EQUIL S_EQUIL G_EQUIL V_EQUIL J.K-1 J J.K-1 J dm3 ***************************************************************** 1.24535E+02 -1.12943E+06 2.20214E+02 -1.34964E+06 0.00000E+00 Mole fraction of system components: gas_ideal Ca 5.4197E-26 O 6.6667E-01 C 3.3333E-01 The cutoff limit for phase or gas constituent activities is 1.00E-70 *Data on this species has been extapolated.

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Table II.8.2 Equilibrium table for 1 mol of CaCO3 at T = 1000 K and P = 6.2910 × 10–2; edited output data from Therma-Calc T = 1000.00 K P = 6.29100E-02 bar V = 1.32165E+03 dm3 STREAM CONSTITUENTS CaO3C_CaCO3(s) Ar/gas_ideal/

PHASE: gas_ideal O2C OC O2 O O3 CaO Ca OC2 O2C3 C Ca2 C2 TOTAL:

EQUIL AMOUNT mol 1.0000E+00 4.8962E-07 2.4481E-07 3.1021E-13 3.0192E-22 1.5610E-25 6.4944E-27 2.2401E-40 1.5019E-41 7.0687E-43 1.3078E-56 5.4967E-61 1.0000E+00

AMOUNT/mol 1.0000E+00 0.0000E+00 MOLE FRACTION bar 1.0000E+00 4.8962E-07 2.4481E-07 3.1021E-13 3.0192E-22 1.5610E-25 6.4944E-27 2.2401E-40 1.5019E-41 7.0687E-43 1.3078E-56 5.4967E-61 1.0000E+00

FUGACITY 6.2910E-02 3.0802E-08 1.5401E-08 1.9515E-14 1.8994E-23 9.8201E-27 4.0856E-28 1.4093E-41 9.4484E-43 4.4469E-44 8.2273E-58 3.4580E-62 1.0000E+00

mol ACTIVITY CaO_CaO(s) 1.0000E+00 1.0000E+00 CaO3C_CaCO3(s) 0.0000E+00 1.0000E+00 CaO2_CaO2(s) T 0.0000E+00 7.9011E-06 C_C(s) 0.0000E+00 8.5732E-15 Ca_Ca(s) 0.0000E+00 1.5978E-24 CaC2_CaC2(s) 0.0000E+00 3.3334E-48 ***************************************************************** Cp_SUM_PHASES H_EQUIL S_EQUIL G_EQUIL V_EQUIL J.K-1 J J.K-1 J dm3 ***************************************************************** 1.05542E+02 -9.60379E+05 3.89263E+02 -1.34964E+06 1.32165E+03 Mole fraction of system components: gas_ideal Ca 5.4197E-26 O 6.6667E-01 The cutoff limit for phase or gas constituent activities is 1.00E-70 *Data on this species has been extrapolated.

Application of the phase rule to the equilibria in Ca–C–O system

235

Table II.18.3 Equilibrium table for 1 mol of CaCO3 at T = 1000 K and a (gas) volume of 1000 l; edited output data from Factsage T = 1000.00 K *P = 6.29102E-02 bar V = 9.99999E+02 dm3 STREAM CONSTITUENTS CaO3C_CaCO3(s) Ar/gas_ideal/

PHASE: gas_ideal O2C OC O2 O O3 CaO Ca OC2 O2C3 C Ca2 C2 TOTAL:

EQUIL AMOUNT mol 7.5663E-01 3.7046E-07 1.8523E-07 2.3471E-13 2.2844E-22 1.1811E-25 4.9139E-27 1.6949E-40 1.1364E-41 5.3484E-43 9.8951E-57 4.1589E-61 7.5663E-01

AMOUNT/mol 1.0000E+00 0.0000E+00 MOLE FRACTION bar 1.0000E+00 4.8962E-07 2.4481E-07 3.1021E-13 3.0192E-22 1.5610E-25 6.4944E-27 2.2401E-40 1.5019E-41 7.0687E-43 1.3078E-56 5.4967E-61 1.0000E+00

FUGACITY 6.2910E-02 3.0802E-08 1.5401E-08 1.9515E-14 1.8994E-23 9.8201E-27 4.0856E-28 1.4093E-41 9.4484E-43 4.4469E-44 8.2273E-58 3.4580E-62 1.0000E+00

mol ACTIVITY CaO 7.5663E-01 1.0000E+00 CaCO3 2.4337E-01 1.0000E+00 CaO2 T 0.0000E+00 7.9011E-06 C 0.0000E+00 8.5732E-15 Ca 0.0000E+00 1.5978E-24 CaC2 0.0000E+00 3.3334E-48 ***************************************************************** Cp_SUM_PHASES H_EQUIL S_EQUIL G_EQUIL V_EQUIL J.K-1 J J.K-1 J dm3 ***************************************************************** 1.10163E+02 -1.00152E+06 3.48122E+02 -1.34964E+06 1.00000E+03 Mole fraction of system components: gas_ideal Ca 5.4197E-26 O 6.6667E-01 C 3.3333E-01 The cutoff limit for phase or gas constituent activities is 1.00E-70 *Data on this species has been extrapolated.

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Table II.18.4 Equilibrium table for 1 mol of CaCO3 at T=1000 K and P=1; bar with 1 × 10–5 mol of Ar; edited output data from Factsage T = 1000.00 K P = 1.00000E+00 bar V = 8.87270E-04 dm3 STREAM CONSTITUENTS CaO3C_caco3(s) Ar/gas_ideal/

PHASE: gas_ideal Ar O2C OC O2 O O3 CaO Ca OC2 O2C3 C Ca2 C2 TOTAL:

EQUIL AMOUNT mol 1.0000E-05 6.7133E-07 3.2870E-13 1.6435E-13 2.0825E-19 2.0269E-28 1.0479E-31 4.3599E-33 1.5039E-46 1.0083E-47 4.7454E-49 8.7796E-63 3.6901E-67 1.0671E-05

AMOUNT/mol 1.0000E+00 1.0000E-05 MOLE FRACTION bar 9.3709E-01 6.2910E-02 3.0802E-08 1.5401E-08 1.9515E-14 1.8994E-23 9.8201E-27 4.0856E-28 1.4093E-41 9.4484E-43 4.4469E-44 8.2273E-58 3.4580E-62 1.0000E+00

FUGACITY 9.3709E-01 6.2910E-02 3.0802E-08 1.5401E-08 1.9515E-14 1.8994E-23 9.8201E-27 4.0856E-28 1.4093E-41 9.4484E-43 4.4469E-44 8.2273E-58 3.4580E-62 1.0000E+00

mol ACTIVITY CaCO3 1.0000E+00 1.0000E+00 CaO 6.7134E-07 1.0000E+00 CaO2 T 0.0000E+00 7.9011E-06 C 0.0000E+00 8.5732E-15 Ca 0.0000E+00 1.5978E-24 CaC2 0.0000E+00 3.3334E-48 ***************************************************************** Cp_SUM_PHASES H_EQUIL S_EQUIL G_EQUIL V_EQUIL J.K-1 J J.K-1 J dm3 ***************************************************************** 1.24535E+02 -1.12943E+06 2.20216E+02 -1.34964E+06 8.87270E-04 Mole fraction of system components: gas_ideal Ca 9.0855E-27 Ar 8.3236E-01 O 1.1176E-01 C 5.5879E-02 The cutoff limit for phase or gas constituent activities is 1.00E-70 *Data on this species has been extrapolated.

Application of the phase rule to the equilibria in Ca–C–O system

237

Table II.18.5 Equilibrium table for 1 mol of CaCO3 at T = 1000 K and P = 1 bar with 1 × 10–5 mol of O2 added; edited output data from Factsage T = 1000.00 K P = 1.00000E+00 bar V = 8.87270E-04 dm3 STREAM CONSTITUENTS CaOC3 O2 (ideal gas)

PHASE: gas_ideal O2 CO2 O O3 CO CaO Ca C C2O C3O2 Ca2 TOTAL:

EQUIL AMOUNT mol 1.0000E-05 6.7133E-07 1.6245E-15 9.6202E-17 4.2139E-17 1.0479E-31 5.5893E-37 7.7991E-57 3.1685E-58 2.7234E-63 1.4429E-70 1.0671E-05

AMOUNT/mol 1.0000E+00 1.0000E-05 MOLEFRACTION bar 9.3709E-01 6.2910E-02 1.5223E-10 9.0150E-12 3.9488E-12 9.8201E-27 5.2377E-32 7.3084E-52 2.9692E-53 2.5521E-58 1.3521E-65 1.0000E+00

FUGACITY 9.3709E-01 6.2910E-02 1.5223E-10 9.0150E-12 3.9488E-12 9.8201E-27 5.2377E-32 7.3084E-52 2.9692E-53 2.5521E-58 1.3521E-65 1.0000E+00

mol ACTIVITY CaCO3 1.0000E+00 1.0000E+00 CaO 6.7133E-07 1.0000E+00 CaO2 T 0.0000E+00 6.1632E-02 C 0.0000E+00 1.4090E-22 Ca 0.0000E+00 2.0484E-28 CaC2 0.0000E+00 1.1543E-67 ***************************************************************** Cp_EQUIL H_EQUIL S_EQUIL G_EQUIL V_EQUIL J.K-1 J J.K-1 J dm3 ***************************************************************** 1.24535E+02 -1.12943E+06 2.20217E+02 -1.34964E+06 8.87270E-04 Mole fraction of system components: gas_ideal Ca 4.7603E-27 O 9.6950E-01 C 3.0496E-02 The cutoff limit for phase or gas constituent activities is 1.00E-70 *Data on this species has been extraplated.

238

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Now let us see how the constraints mentioned above can be broken. First let us set the total pressure to 1 bar again but add an arbitrary amount of Ar (e.g. n(Ar) = 10–5) to the system. By adding Ar a gas phase is always present in the system. The result in Table II.18.4 shows, in fact, that the amount of Ar now defines the volume of the gas phase into which CO2 can be transferred as a result of the dissociation of CaCO3. Thus the extent of reaction (here 6.7133 × 10–7) can be controlled by the amount of Ar, the additional (!) Gibbsian component. In the second step let us use a total pressure of 1 bar but add an arbitrary amount of O2 (n(O2) = 10–5) instead of Ar to the system. Adding O2 means that we do not (!) change the number of elementary components in the system. From Table II.18.5 it is obvious that adding O2 yields exactly the same result as adding Ar. The number of degrees of freedom is now also two, but we have reached this result not by increasing the number ε of elementary components of the system, but instead by taking away the stoichiometric constraint s that the ratio of O to C in the gas phase be equal to 2. With the additional O2 amount this ratio can now take on any value. However, the free oxygen does not take part in the solid–gas equilibrium according to the stoichiometric reaction above. The amount of O2 only defines the volume of the gas phase into which CO2 can be transferred. Thus the extent of reaction (here 6.7133 × 10–7) can be controlled by the amount of O2 in the same way as by the amount of Ar. Nevertheless, the Gibbsian components are now the three elements C, Ca and O.

II.19 Thermodynamic prediction of the risk of hot corrosion in gas turbines MICHAEL MÜLLER

II.19.1

Introduction

The limitation of fossil fuel resources and the necessity of reducing CO2 emission require an increase in the efficiency of power plants by using combined cycle power systems. Up to now efficiencies in excess of 50% are only achievable by using ash-free fuels, e.g. natural gas or oil in gas and steam power stations. Coal constitutes 80% of the world’s total fossil fuel resources. Today it is mostly fired in steam power stations. Even if supercritical steam parameters are used, these coal-fired power plants only reach efficiencies below 50%, so that further development is essential. Therefore, different types of coal-fired combined cycle power systems are under development. The direct use of hot flue gases originating from coal combustion or gasification for driving a gas turbine requires a hot-gas clean-up to prevent corrosion of the turbine blading. One of the main problems is the alkali release during the coal conversion process. The alkali metals are mainly bound in the mineral matter of the coal as salts and silicates. The alkali release leads to an alkali concentration in the flue gas significantly higher than the specifications of the gas turbine manufacturers (< 0.01 mg Na + K m–3(STP)). During development of hot-gas clean-up it is necessary to check its effectiveness for preventing hot corrosion. On the one hand, however, corrosion tests are expensive and time consuming and need to represent the conditions in a gas turbine closely to generate reliable results. On the other hand, the specifications of the gas turbine manufacturers are mainly based on experiences with ash-free fuels and do not take into consideration the interactions with other gaseous species or ash particles. The use of computational thermochemistry is, however, a means to generate results within a short time and at low cost. It is possible to compare the complex systems occurring during coal conversion with systems occurring in gas- or oil-fired gas turbines. Moreover, previous investigations have already shown the potential of 239

240

The SGTE casebook

thermodynamic calculations for the prediction of condensed and gaseous phases in gas and oil fired gas turbines [90Sin, 004Bor]. In the present work, thermodynamic modelling was used to estimate the risk of hot corrosion in gas turbines driven by cleaned coal-based flue gases. Thus, the effectiveness of the hot-gas clean-up concerning alkali removal was evaluated.

II.19.2

Hot corrosion

Hot corrosion is initiated by deposition or condensation of corrosive species, e.g. sulphates. The condensation of sulphates on gas turbine blades takes place owing to high concentrations of alkalis in combination with high concentrations of sulphur. Even at high concentrations of chlorine, alkali sulphates are formed, because these are the least volatile alkali species. The typical temperature range for hot corrosion in gas turbines is 600– 950 °C. The upper temperature limit is given by the dew point of the alkali sulphates. The lower temperature limit is given by the melting point of eutectics formed by the deposits and the corrosion product scale of the blade material. Hot corrosion has been characterised as type I (high-temperature) corrosion and type II (low-temperature) corrosion. Type I hot corrosion mainly occurs at 800–950 °C. It is caused by the formation of liquid alkali sulphates above their melting points, leading to basic dissolution of the oxide scale of the blade material. Type II hot corrosion mainly occurs at 600–800 °C. It is caused by the formation of a eutectic melt of NiSO4 or CoSO4 and alkali sulphates above the eutectic temperature. NiSO4 and CoSO4 are formed by reaction of the oxide scale of the blade materials with SO3 depending on the SO3 partial pressure in the hot flue gas. As a consequence of the above, in the following estimations, the coexistence of alkali sulphates and NiSO4 or the formation of liquid alkali sulphates was taken as the criterion for a risk of hot corrosion.

II.19.3

Thermodynamic modelling

The aim of the thermodynamic calculations was the prediction of the thermodynamic stability of the sulphates and other species in the gas turbine. For the calculations, the computer program FactSage 5.3.1 together with the FACT Solution Database and SGTE Pure Substance Database [002Bal] was employed for the investigation of two different reactor models. The particular boundary conditions for the calculations are given in the examples. For comparison, the dew point of sodium sulphate was also calculated for a gas turbine burning light fuel oil with 0.2% S and 0.5 ppm Na. There has been much experience obtained about the corrosion behaviour of such turbines.

Thermodynamic prediction of hot corrosion in gas turbines

241

Thus, it is possible to calculate at least a corrosion risk (in a ‘coal’-driven gas turbine) relative to the well-known corrosion risk (in an oil-driven gas turbine), even if the available thermodynamic data are not sufficient to calculate the exact values. In addition, the calculated dew points were compared with a typical temperature–pressure profile of a gas turbine operating under full load to determine the T–P area with hot-corrosion risk. Kinetic aspects, partload operation, starts and shutdowns, and turbine parts with temperatures different from the typical profile were not taken into consideration.

II.19.4

Hot-corrosion risk in second-generation circulating pressurised fluidised-bed combustion

Second-generation circulating pressurised fluidised-bed combustion (CPFBC) is a lignite-fired combined cycle concept which is able to achieve efficiencies in excess of 50% [005Rom]. A schematic flow diagram is given in Fig. II.19.1. The system mainly consists of a two-stage combustion operating at a pressure of 10–16 bar. In the first stage, the coal is gasified under reducing conditions (air-to-fuel ratio λ < 1) at temperatures of 650–750 °C. After leaving the first stage, the flue gases pass a gas-cleaning section, which consists of ceramic filters for ash removal and an alkali sorption unit. The cleaned gas is then mixed with a secondary air stream and either burned in a second combustion chamber or directly inside the gas turbine, in both Gasifier

Hot gas cleaning

l = 0.6 750 °C Ash, alkalis

Combustion chamber

l = 1.2 1250 °C

Coal

Air Steam cycle

II.19.1 Schematic flow diagram of second-generation CPFBC.

242

The SGTE casebook

cases at λ > 1. The residual thermal energy of the gas stream leaving the turbine is finally transferred to a steam cycle. Laboratory investigations were conducted to assess the potential for the reduction of alkali metals from hot gas by different aluminosilicate sorbents, such as silica, bauxite, bentonite and mullite, under reducing atmospheres at a temperature of 750 °C [004Wol]. Using a flow channel reactor, an alkali chloride-laden gas stream was passed through a bed of aluminosilicate sorbents. Qualitative and quantitative analyses of the hot gas downstream of the sorbent bed was performed using high-pressure mass spectrometry (HPMS). The investigations revealed the possibility of reducing the overall alkali concentration in the hot gas under second-generation. CPFBC conditions to values of less than 50 vol. ppb through the use of bentonite and activated bauxite. Based on the experimental results, the thermodynamic stability of the sulphates and other species in the gas turbine was calculated using a fourstage reactor model. The calculations were performed for two types of lignite, ‘Lausitzer Braunkohle WBK 1778’ (53% C, 4% H, 17% H2O, 20% O2, 0.8% S, 5% ash and 0.2% Cl) and ‘Rheinische Braunkohle HKN’ (55% C, 4% H, 15% H2O, 23% O2, 0.4% S, 5% ash and 0.2% Cl). Figure II.19.2 shows a scheme of the reactor model. Thermodynamic equilibrium was calculated for the reactors labelled ‘Gasifier’, ‘Hot-gas cleaning’, ‘Combustion chamber’, and ‘Gas turbine’. The boundary conditions for the calculations are given in the scheme. First of all, in the ‘Gasifier’ the equilibrium between coal and synthetic air (79 vol% N2 and 21 vol% O2) was calculated to obtain an idea of the hot-flue-gas composition leaving the gasifier. Both the gas phase and the condensed phase from ‘Gasifier’ were taken as input for the following calculations. In ‘Hot gas cleaning’, thermodynamic equilibrium was calculated using the corresponding boundary conditions. The alkali partial pressure in the resulting gas phase was manually set to 4E-8 bar according to the Coal Air

Air

Hot-gas cleaning 900 °C 16 bar l = 0.6

Gasifier

750 °C 16 bar l = 0.6

Ash

Alkalis

1200 °C 16 bar l = 1.2

1200–600 °C 15–1 bar l = 1.2

Combustion chamber

Gas turbine

II.19.2 Reactor model used for thermodynamic calculations for second-generation CPFBC.

Thermodynamic prediction of hot corrosion in gas turbines

243

experimentally obtained values. Since all particles should be removed by a filter candle, only the gas phase was used for the following calculations. In the ‘Combustion chamber’, synthetic air was added to reach an overall λ value of 1.2. Finally, the thermodynamic stability of the sulphates results from equilibrium calculations for the ‘Gas turbine’. Via target (search) calculations, the temperature of the first occurrence of the different sulphates was determined in dependence of the pressure. For the calculation of NiSO4 stability, about 0.01 mol% Ni in relation to the sulphur in the gas stream leaving the ‘Combustion chamber’ was added. Thus, the amount of sulphur bound by Ni is too small to have an influence on the dew point of the alkali sulphates. The main results of the thermodynamic calculations are shown in Fig. II.19.3. The calculated dew points of Na2SO4 are much lower in the case of second-generation CPFBC than those in case of a gas turbine burning fuel oil. Therefore, the corrosion risk should be much lower. In the case of HKN, the dew point of Na2SO4 is lower than in the case of WBK owing to the smaller amount of sulphur in the coal and subsequently in the flue gas. The formation of NiSO4 also occurs at lower temperatures in case of HKN, which is not shown in Fig. II.19.3. However, this result indicates that one has not only to look at the alkali concentration in the flue gas to estimate the corrosion risk. Anyway, Na2SO4 condenses much below its melting point of 884 °C, so that there is not any risk of type I hot corrosion. The shaded area marks the region in which both Na2SO4 and NiSO4 are stable above the 1300 Coal WBK, dew point Na2SO4 Coal HKN, dew point Na2SO4 Light fuel oil, dew point Na2SO4 Formation of NiSO4 Gas temperature Blade–vane temperature

1250 1200 1150 Temperature (°C)

1100 1050 1000 950

Tm(Na2SO4)

900 850 800 750

Risk of hot corrosion

700 650

Te(Na2SO4–NiSO4)

600 14 12

10

8

6

4 Pressure (bar)

1

II.19.3 Results of thermodynamic calculations for estimation of the hot-corrosion risk in second-generation CPFBC.

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eutectic temperature of 671 °C, where type II hot corrosion may occur. However, there is no blade operating at these critical conditions under full load. Furthermore, there should be no condensation of Na2SO4 on the gas turbine blades operating under full load at all, because the calculated dew points are lower than the temperature of each blade. Therefore, no hot corrosion should take place during full-load operation. Even during part-load operation no blade should operate at critical conditions. At the worst, the first blades operate at critical conditions. However, these blades are covered by thermal barrier coatings.

II.19.5

Hot-corrosion risk in pressurised pulverised coal combustion

Pressurised pulverised coal combustion (PPCC) is another coal-fired combined cycle concept which is able to achieve efficiencies in excess of 50% [97Han]. A schematic flow diagram is given in Fig II.19.4. Combustion of pulverised coal takes place at temperatures of about 1600 °C under a total pressure of about 15 bar. The produced flue gas is routed through a column of ceramic balls as a liquid-slag separation unit at an average temperature of 1450 °C. A separate alkali removal (T ≤ 1400 °C) is the last clean-up step before the flue gas enters the gas turbine. Here too, the residual thermal energy of the gas stream leaving the turbine is finally transferred to a steam cycle. Laboratory investigations were conducted to find a sorbent material for alkali removal at 1400 °C under PPCC conditions sufficient to fulfil the demands of the gas turbine manufacturers [005Esc]. In laboratory-scale flow channel and HPMS experiments at 1400 °C, similar to those mentioned above, kaolin- and silica-enriched bauxite have shown the best ability to remove the alkalis sufficiently. The alkalis are bound in a melt–glass phase formed during alkali sorption. The total NaCl concentration can be reduced to values less than 30 vol. ppb. Coal

Liquid slag separators

Alkali removal

Furnace

Steam turbine

Slag

T = 1350–1550 °C Air

Gas turbine

II.19.4 Schematic flow diagram of PPCC.

Thermodynamic prediction of hot corrosion in gas turbines Coal

245

Air Hot-gas cleaning

1600 °C 15 bar l = 1.5

1350 °C 15 bar

Air

1200–600 °C 15–1 bar l=2

Gas turbine Combustion chamber

Ash Alkalis

II.19.5 Reactor model used for thermodynamic calculations for PPCC.

Based on the experimental results, the thermodynamic stability of the sulphates and other species in the gas turbine was calculated using a threestage reactor model. The calculations were performed for a typical hard coal (79% C, 5% H, 2% H2O, 7% O2, 1% S, 6% ashes and 0.1% Cl). Figure II.19.5 shows a schematic diagram of the reactor model. Thermodynamic equilibrium was calculated for the reactors labelled ‘Combustion chamber’, ‘Hot-gas cleaning’ and ‘Gas turbine’. The boundary conditions for the calculations are given in the scheme. First of all, in the ‘Combustion chamber’ the equilibrium between coal and synthetic air (79 vol.% N2 and 21 vol.% O2) was calculated to obtain an idea of the hot-flue-gas composition leaving the combustion chamber. Both the gas phase and the condensed phase from the ‘Combustion chamber’ were taken as input for the following calculations. In ‘Hot-gas cleaning’, thermodynamic equilibrium was calculated using the corresponding boundary conditions. The alkali partial pressure in the resulting gas phase was manually set to 2.4E-8 bar according to the experimentally obtained values. Since all particles and slag droplets should be removed by the liquid-slag separators, only the gas phase was used for the subsequent calculations. Since, in the process, cooling air is added at the entrance of the gas turbine, synthetic air (79 vol.% N2 and 21 vol.% O2) was added to the gas stream entering the ‘Gas turbine’ to reach an overall λ value of 2. Finally, the thermodynamic stability of the sulphates results from equilibrium calculations for the ‘Gas turbine’. The same target calculations were performed as for second-generation CPFBC. The results of the thermodynamic calculations are shown in Fig. II.19.6. The dew point curve of Na2SO4 is similar to that for second-generation CPFBC and much lower than that for a gas turbine burning fuel oil. The dew point curve for K3Na(SO4)2, which is the thermodynamically stable sulphate containing potassium, is about 20 K lower. For the same reasons explained for second-generation CPFBC, no hot corrosion should take place in the case of PPCC.

246

The SGTE casebook

1300

Coal, dew point Na2SO4 Coal, dew point K3Na(SO4)2 Fuel oil, dew point Na2SO4 Formation of NiSO4 Gas temperature Blade–vane temperature

1250 1200 1150

Temperature (°C)

1100 1050 1000 950

Tm (Na2SO4)

900 850 800 750

Risk of hot corrosion

700 650

Te (Na2SO4–NiSO4)

600 14

12

10

8

6

4 Pressure (bar)

1

II.19.6 Results of thermodynamic calculations for estimation of the hot-corrosion risk in PPCC.

II.19.6

Conclusions

Computational thermochemistry was successfully employed to estimate the risk of hot corrosion in second-generation CPFBC and PPCC using state-ofthe-art alkali removal. The coexistence of alkali sulphates and NiSO4 or the formation of liquid alkali sulphates were taken as the criterion for the risk of hot corrosion. In addition, the results for the coal-based processes were compared with thermodynamic calculations for a gas turbine burning fuel oil. The results of the thermodynamic calculations are similar for both processes. The calculated dew points of Na2SO4 are much lower in the case of the coalbased processes than those in the case of a gas turbine burning fuel oil. Moreover, there should be no condensation of sulphates on the gas turbine blades at all, because the calculated dew points are lower than typical blade temperatures. Therefore, no hot corrosion should take place in both coalbased processes.

II.19.7 90Sin

97Han

References L. SINGHEISER and H.W. GRÜNLING: ‘Hochtemperaturkorrosion in stationären Gasturbinen bei alternierender Betriebsweise’, Report for Bundesministerium für Forschung and Technologie, Germany, 1990. K. H ANNES , F. N EUMAN , W. T HIELEN and M. P RACHT : ‘Kohlenstaub– Druckverbrennung’, VGB-Kraftwerkstechnik 77, 1997, 393–400.

Thermodynamic prediction of hot corrosion in gas turbines 002Bal

247

C.W. BALE, P. CHARTRAND, S.A. DEGTEROV, G. ERIKSSON, K. HACK, R. BEN MAHFOUD and J. MELANCON: ‘FactSage thermochemical software and databases’, Calphad 26(2), 2002, 189–228. 004Bor B. BORDENET: ‘High temperature corrosion in gas turbines: thermodynamic modelling and experimental results’, PhD Thesis, RWTH Aachen, Germany, 2004. 004Wol K.J. WOLF, M. MÜLLER, K. HILPERT and L. SINGHEISER: ‘Alkali sorption in secondgeneration pressurized fluidized-bed combustion’, Energy Fuels 18, 2004, 1841– 1850. 005Esc I. Escobar, H. Oleschko and M. Müller: ‘Einbindung von Alkalien bei der Druckkohlenstaubfeuerung’, 22. Deutscher Flammentag, VDI-Berichte 1888, VDI-Verlag, Düsseldorf, 2005, pp. 57–62. 005Rom H.B. ROMBRECHT, H. R ISTAU and H.J. KRAUTZ: ‘Ein braunkohlenbasierte Kombikraftwerksprozeß (ZDWSF) – Versuchsergebnisse und -erfahrungen’, 22. Deutscher Flammentag, VDI-Berichte 1888, VDI-Verlag, Düsseldorf, 2005, pp. 49–56.

II.20 The potential use of thermodynamic calculations for the prediction of metastable phase ranges resulting from mechanical alloying P H I L I P J. S P E N C E R

II.20.1

Introduction

Many researchers have reported the appearance of non-equilibrium phenomena in mechanically alloyed materials. Some of the phenomena are found to result in improved properties, which are reflected in the practical use of the alloys concerned. Among the observed phenomena are supersaturated solid solutions, metastable phase formation, and amorphous and nanostructured materials. An excellent summary of the available experimental information is contained in the review by Suryanarayana [001Sur]. One of the most frequent observations made in mechanical alloying is that solid solubilities can be extended in a wide range of binary and higherorder alloys when using the elemental powders as starting material. The observed changes in solid solubility are sometimes very large, and the fact that it is possible to obtain significant solid solubilities even in systems where no observable solubility has been found under equilibrium conditions is particularly surprising. Many different types of mill have been used in the mechanical alloying process, and much attention has been given to investigating the effect of variation in parameters such as milling speed, milling time, powder size, ball-to-powder weight ratio and milling temperature on the final phase constitution of the samples under investigation. In contrast, much less attention has been given to the quantitative determination of the energy imparted to the powder components during the ball-milling process, by carrying out calorimetric measurements of the enthalpy release after milling. The stored energy can be very significant, and the magnitude of the measured enthalpies together with the frequently observed extended solubilities imply that the mechanical energy imparted by the alloying process results in changes to the equilibrium Gibbs energies of the alloy components in question. These changes can be taken into account in attempts to use thermodynamic calculations to simulate the extended, metastable solubility behaviour in a particular alloy system. Such simulations make use of thermodynamic data which have been 248

Prediction of metastable phase ranges

249

critically assessed using the methods described in the journal Calphad [77Liu]. The SGTE Solution Database [002SGT] is an excellent source of such data. The major advantage of assessments based on the calculation of phase diagrams (CALPHAD) is that they allow the Gibbs energy curves (or surfaces) for the phases in a given system to be extended into the composition and temperature ranges of metastability, outside the normally observed equilibrium ranges. The calculations described in this article represent an investigation of the potential of thermodynamic calculations for the prediction of some observed metastable phenomena resulting from mechanical alloying processes.

II.20.2

Calculation principles and previous related work

Previous calculation of metastable phase ranges have been been made for coatings produced using physical vapour deposition coating processes [86Sau, 90Spe, 98Spe, 000Spe, 001Spe]. A main assumption of these calculations [86Sau] is that, owing to the low temperatures of the substrates on which the coatings are deposited, diffusion is insufficient to allow the more complex equilibrium, two-phase, three-phase, or multiphase structures to form, and hence formation of the homogeneous single-phase alloy with the lowest Gibbs energy is preferred. The assumption has enabled good agreement to be obtained between experimental determination and thermodynamic prediction of metastable phase ranges, e.g. in the Cr–Ni [86Sau], Al2O3–AlN and AlN– TiN systems [90Spe, 98Spe, 000Spe, 001Spe]. In the case of mechanical alloying, temperatures during alloying are generally also rather low and again it is to be anticipated that, owing to restricted diffusion, the formation of less complex single-phase structures is favoured. In some cases, the energy imparted to the powder components during the mechanical alloying process has been determined experimentally. For example, Eckert et al. [92Eck] carried out calorimetric measurements on Cu powders, annealed at approximately 200 °C after ball milling, and observed an energy release of 5000 J mol–1. Uenishi et al. [91Uen] used differential scanning calorimetry (DSC) studies of ball-milled powders in the Ag–Cu system to determine the energy release as a function of composition in the temperature range 157–317 °C (Fig. 20.1). The measured values were found to be about 5000 J mol–1 greater than the enthalpy of mixing of face-centred cubic (fcc) Ag–Cu alloys across the entire composition range, which results in good agreement with the work of Eckert et al. [92Eck] for the value associated with pure Cu. Battezzati [97Bat] has used the results of different workers to plot the energy release from ball-milled copper as a function of annealing temperature. His calculated curve, relative to the bulk single crystal, is shown in Fig. II.20.2. The energy of the undercooled liquid is also included in the figure

250

The SGTE casebook

14000 12000

∆H (J mol–1)

10000 8000 6000 4000 2000 0 0

Calculated enthalpy of mixing of fcc alloys 0.1

0.2

0.3

Experiment (DSC)

0.4 0.5 0.6 Mole fraction of Cu

0.7

0.8

0.9

1.0

II.20.1 Exothermic enthalpy output from Ag–Cu alloy powders after ball milling [91Uen].

G (powder) – G (bulk single crystal) (kJ mol–1)

10 9

Undercooled liquid Cu

8 7 6 5 4

Nanocrystalline Cu

3 2 1 0 400

600

800 1000 Temperature (K)

1200

1400

II.20.2 Gibbs energies of nanocrystalline and supercooled Cu as functions of temperature [97Bat].

for purposes of comparison of the energy differences between the different states of Cu. It is evident from the above experimental studies that the energy imparted to powdered metals by the ball-milling process must be taken into consideration in calculations of the phase equilibria resulting from the particular alloying

Prediction of metastable phase ranges

251

process concerned. The energy will vary depending on the process parameters used, but appears to be capable of calibration for a particular mill by use of appropriate DSC measurements. In the thermodynamic calculations discussed below, metastable phases and phase ranges have been predicted assuming the following. –

– –

The component Gibbs energies are modified by quantities corresponding to the energy imparted to the component powders by the mechanical alloying process. The observed phase is the homogeneous single close-packed phase with lowest Gibbs energy for an overall alloy composition. Solubility boundaries are given by the points of intersection of the Gibbs energy curves for the single phases.

For all calculations, the well-established modelling and calculation methods presented in the journal Calphad [77Liu] and in the first edition of The SGTE Casebook [96Hac] have been employed.

II.20.3

Results

Example equilibrium phase diagrams and calculated Gibbs energy curves for the systems Al–Mn, Cr–Co, Cu–Fe, Fe–Ni, Ta–Al and Al–Ti are presented in Fig. II.20.3, Fig. II.20.4, Fig. II.20.5, Fig. II.20.6, Fig. II.20.7 and Fig. II.20.8 respectively. All Gibbs energy curves have been calculated, using assessed thermodynamic data from the SGTE Solution Database [002SGT], for a temperature of 200 °C – a typical ball-milling temperature. For each system, the point of intersection of the solvent-phase Gibbs energy curve with the curve for the primary precipitating phase defines the calculated metastable solubility limit (denoted in the diagrams by a dashed line to the composition axis). A summary of experimental and calculated results for the above systems is presented in Table II.20.1. It can be seen from this table that results from experimental studies vary widely, although observed solubilities are in all cases significantly greater than the equilibrium values. It is likely that the various parameters associated with the different experimental ball-milling processes are in part responsible for the differing results. The selected constant ‘milling energy’ (–5000 J mol–1) used to amend the Gibbs energies of the different powder components may, therefore, also not be a suitable value in all cases. Nevertheless, the general agreement between experimental and calculated metastable solubility boundaries, using the calculation principles listed above, is surprisingly good. An example is the Cr–Co system, in which the equilibrium solubility of Co in Cr is 4.9 at.% at 600 °C. After mechanical alloying, the solubility from experimental measurements [001Sur] increases very significantly to 30 or 40 at.% according to two different studies. The

252

The SGTE casebook 1400

Liquid

1200

Bcc Fcc

Hcp

T (°C)

1000 Al8Mn5 D810 800

Al11Mn4 Cubic A13

600

Fcc A1 Al16Mn Al14Mn

400 Al12Mn

Cbcc

200 0

0.1

0.2

0.3

0.4 0.5 0.6 Mole fraction of Mn (a)

0.7

0.8

0.9

1

10000 5000

∆G (J)

0 –5000 –10000

Fcc

–15000

Cbcc

–20000 Cubic A13 –25000 0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 Mole fraction of Mn (b)

0.8

0.9

1

II.20.3 The calculated equilibrium phase diagram for the Al–Mn system. (b) Calculated Gibbs energy curves for the fcc, cubic A13 and complex body-centred cubic (cbcc) phases of the Al–Mn system at 200 °C with component Gibbs energies changed by –5000 J mol–1 (stoichiometric compounds and the Al8 Mn5 D810 phase, with complex crystallography, are omitted).

metastable solubility limit from thermodynamic calculation, using assessed data for the system [002SGT] is 35 at.%. A difficulty in making a complete comparison of calculated solubilities with the experimental results is that workers have tended to place emphasis

Prediction of metastable phase ranges

253

2100 1900 Liquid

1700 1500

T (°C)

1300

Bcc Fcc

1100

Fcc + fcc 2

900 σ

700

Hcp

500 300 100 0

0.1

0.2

0.3

0.4 0.5 0.6 Mole fraction of Co (a)

0.7

0.8

0.9

1.0

8000 7000 6000 Fcc

5000 4000

∆G (J)

3000 2000 1000 0 –1000 Hcp

–2000 Bcc

–3000 –4000 –5000 0

0.1

0.2

0.3

0.4 0.5 0.6 Mole fraction of Co (b)

0.7

0.8

0.9

1.0

II.20.4 (a) The calculated equilibrium phase diagram for the Cr–Co system. (b) Calculated Gibbs energy curves for the body-centred cubic (bcc), fcc and hexagonal close-packed (hcp) phases of the Cr– Co system at 200 °C with component Gibbs energies changed by –5000 J mol–1 (the σ phase, with complex crystallography, is omitted).

on the observed extended solubilities and in nearly all cases provide very little information on the phases precipitating from the solvent solution. Nevertheless, in many systems and experiments, the extent of the experimentally observed solubilities is such that the compositions of

254

The SGTE casebook 1700 Liquid

1500

Bcc Fcc

1300

T (°C)

1100 Fcc 900 Bcc 700 500 300 100 0

0.1

0.2

0.3

0.4 0.5 0.6 Mole fraction of Fe (a)

0.7

0.8

0.9

1.0

0.7

0.8

0.9

1.0

8000 6000

Bcc

4000

∆G (J)

2000 0 –2000 Fcc –4000 –6000 –8000 –10000

0

0.1

0.2

0.3

0.4 0.5 0.6 Mole fraction of Fe (b)

II.20.5 (a) The calculated equilibrium phase diagram for the Cu–Fe system. (b) Calculated Gibbs energy curves for the fcc and bcc phases for the Cu–Fe system at 200 °C with component Gibbs energies changed by –5000 J mol–1.

intermetallic compound phases observed in the equilibrium diagram are exceeded, which supports the proposition that phases with more complex crystallographic structure are difficult to produce when starting from the pure powder components in mechanical alloying. Examples are the Al–Mn

Prediction of metastable phase ranges

255

1700 Liquid 1500 1300

T (°C)

1100 Fcc 900 700 500

Bcc Ordered fcc

300 100 0

0.1

0.2

0.3

0.4 0.5 0.6 Mole fraction of Ni (a)

0.3

0.4 0.5 0.6 Mole fraction of Ni (b)

0.7

0.8

0.9

1.0

8000 6000 4000

∆G (J)

2000 0 –2000

Fcc

–4000 –6000 –8000 –10000 0

Bcc

0.1

0.2

0.7

0.8

0.9

1.0

II.20.6 (a) The calculated equilibrium phase diagram for the Fe–Ni system. (b) Calculated Gibbs energy curves for the fcc and bcc phases of the Fe–Ni system at 200 °C with component Gibbs energies changed by –5000 J mol–1.

system for which Mn solubilities greater than the Mn concentrations of the phases Al12Mn and Al6Mn have been measured, the Al–Ti system, for which Ti solubilities beyond the 25 and 33 at.% Ti compositions of the phases Al3Ti and Al2Ti have been reported, and the Ta–Al system, with Al solubilities

256

The SGTE casebook 2900 2500 Liquid 2100

T (°C)

Bcc 1700 1300 σ

900 500 100 0

Al3Ta

Al3Ta2 0.1

0.2

0.3

0.4 0.5 0.6 Mole fraction of Al (a)

0.7

0.8

0.9

1.0

7000 3000 –1000

Fcc

∆G (J)

–5000 –9000 –13000

Bcc

–17000 –21000 –25000 0

0.1

0.2

0.3

0.4 0.5 0.6 Mole fraction of Al (b)

0.7

0.8

0.9

1.0

II.20.7 (a) The calculated equilibrium phase diagram for the Ta–Al system. (b) Calculated Gibbs energy curves for the bcc and fcc phases of the Ta–Al system at 200 °C with component Gibbs energies changed by –5000 J mol–1 (stoichiometric compounds and the σ phase, with complex crystallography, are omitted).

greater than the 27–39 at.% Al range of the σ phase. There are some systems which, with careful systematic experimentation, could provide a sensitive test of the calculation principles used here. For example, in the Cu–Si system, in the Cu-rich range, not only are a number of stoichiometric compound

Prediction of metastable phase ranges

257

1900 1700 Liquid 1500 Bcc

Al3Ti D022

1300

Al3Ti3 Al3Ti D022

1100

AlTi

900 AlTi D019

700

Hcp

Fcc 500 Al3Ti 300 100

0

0.2

0.4 0.6 Mole fraction of Ti (a)

0.8

1.0

10000 5000 0

∆G (J)

–5000 –10000 –15000 Fcc

Hcp

Bcc

–20000 –25000 –30000 –35000 0

0.1

0.2

0.3

0.4 0.5 0.6 Mole fraction Ti (b)

0.7

0.8

0.9

1.0

II.20.8 (a) The calculated equilibrium phase diagram for the Al–Ti system. (b) Calculated Gibbs energy curves for the fcc, bcc and hcp phases of the Al–Ti system at 200 °C with component Gibbs energies changed by –5000 J mol–1 (stoichiometric compounds and phases with complex crystallography are omitted).

phases found, but also a bcc and an hcp phase in addition to the fcc Cu-rich solid solution (Fig. II.20.9). The Gibbs energies of the single-phase fcc, hcp and bcc structures have very similar values as shown by the calculated curves for a temperature of

258

Solvent–solute alloy system

Experimental equilibrium solubility (at.%)

Equilibrium precipitated phase

Solubility after mechanical alloying (at.%)

Precipitated phase

Solubility at 200 °C, from calculation (at.%)

Precipitated phase, from calculation

Al–Mn Al–Ti

0.4 (400 °C) 12 as it dissolves in a percolating groundwater. A high pH in a repository is desirable because it helps to minimise the solubility of many radionuclides, metal corrosion and microbial activity. In order to assess the likely performance of the chemical barrier, reliable models are required with which to make predictions beyond the spatial and temporal limits imposed by experiment and observation. The application of thermodynamic modelling to cement chemistry has been advocated by numerous workers over the last two decades [87Atk, 87Gla, 88Ber, 90Rea]. Its development has been driven largely by the need for a predictive capability in modelling the near-field processes, which govern the performance of proposed nuclear waste storage facilities. Early work [87Atk, 87Gla, 88Ber] sought to develop pragmatic models with which to simulate the thermodynamic evolution of cements in the repository environment. In achieving this goal, of critical importance was the development of a robust description of the incongruent dissolution of C–S–H gel because of its major contribution to the longevity of the chemical barrier. It is expected that a nuclear waste repository would experience elevated temperatures, due both to the heat of hydration of the cement and to radiolytic heating, and that this perturbation may last for thousands of years. Longterm experiments at elevated temperatures have been shown to have a marked effect on the solubility behaviour of C–S–H gel [005Gla]. Thus, developing a credible predictive model, which can be used to describe the dissolution of C–S–H gel at room and elevated temperatures, is of critical importance. 322

Modelling cements in an aqueous environment

II.27.2

323

Previous modelling studies

The conventional modelling approach takes either one or two pure solid phases with variable solubility products and the model parameters are arbitrarily adjusted until a match is made with the measured solubility data of the C–S– H system [87Atk, 87Gla, 88Ber]. Whilst these models are pragmatic and effective, they are not based on strong thermodynamic theory, nor readily expanded to include elevated temperatures. Despite these observations, the Berner [88Ber] model and its variants have been the most widely used owing to its elegant simplicity. A more thermodynamically rigorous description of the C–S–H system uses a solid-solution aqueous-solution-based model [96Ker, 97Bor, 99Maz], which can only be applied to Ca to Si ratios greater than 1.0 in the C–S–H gel and is only reliable at room temperature. True free-energy-based models (of which this chapter represents one example) strive for greater flexibility and rely more on thermodynamic realism than earlier methods. They can therefore account for all Ca-to-Si ratios expected in the C–S–H gel and extend the limits of temperature.

II.27.3

MTDATA

A number of computer programs implementing the principles of the calculation of chemical and phase equilibria, reviewed by Bale and Eriksson [90Bal], have been reported over the years. Many of these have been limited to handling problems involving specific systems, types of material or stoichiometric compounds, but this is not the case with MTDATA [002Dav], a reliable and general software tool for calculating phase and chemical equilibria involving multiple solution or stoichiometric phases with ease and reliability. It provides true Gibbs energy minimisation through the solution of a nonlinear optimisation problem with linear constraints using the National Physical Laboratory (NPL) Numerical Optimisation Software Library, which guarantees mathematically that the Gibbs energy reduces each time that it is evaluated. MTDATA has a long history of application dating back to the 1970s in fields such as ferrous and non-ferrous metallurgy [001Wan, 002Hun, 003Put], slag and matte chemistry [93Bar, 002Gas, 004Tas], nuclear accident simulation [93Bal], molten salt chemistry [87Bar] and cement clinkering [000Bar]. The aim of the work described here has been to develop a rigorous model and associated thermodynamic data for the C–S–H system, at room temperature and elevated temperatures, which are compatible with existing NPL oxide and aqueous species databases. The calculation-of-phase-diagrams (CALPHAD) [002Dav] modelling principle embodied within MTDATA is that a database developer derives parameters to represent accurately the Gibbs energy of each phase that might form in a system as a function of temperature, composition and, if necessary,

324

The SGTE casebook

pressure. This is achieved using data assessment tools within MTDATA, which allow parameters to be optimised to give the best possible agreement between calculated thermodynamic properties (such as heat capacities, activities and enthalpies of mixing), phase equilibria (solubilities) and collated experimental properties. Generally, such data assessment is undertaken for ‘small’ systems (one, two or three components combined) allowing predictions to be made in ‘large’ multicomponent (multielement) systems, which may contain a great many species. Examples of calculated phase diagrams for the CaO–H2O and SiO2–H2O binary oxide systems, compared with experimental data, are shown in Fig. II.27.1 and Fig. II.27.2 respectively. In single-point calculations, predictions are made by specifying an overall system composition and temperature. MTDATA then determines the 400

Constraints Pressure 101325 Pa

Portlandite + gas

H2O H2 O CaO

380

1.0000 0.0020

H2O+0.2 wt% CaO

Temperature (K)

360

H 2O CaO

0.9980 0.0020

340

320

Portlandite + aqueous Aqueous

300

280 270 0.00 H2O

0.05

0.10 Proportion of CaO (wt%)

0.15

0.20

Linke [58 Lin] (coarse solid [34 Bas]) Linke [58 Lin] (fine solid [34 Bas]) Shenstone and Crandall [883 She] Pepler and Welis [54 Pep] Weare [87 Wea] Hatches database with Pitzer Debye–Hückel terms [97 Bon] CTT recommended value [87 Gar]

II.27.1 CaO–H2O phase diagram showing the stability field of the aqueous phase in equilibrium with gas and portlandite. Data for Ca(OH)2 from the SGTE Substance Database [002SGT] were adjusted slightly to give correct CaO–H2O equilibria [883She, 34Bas, 54Pep, 58Lin, 87Gas, 87Wea, 97Bon].

Modelling cements in an aqueous environment 400

325

Constraints Pressure 101 325 Pa

Gas + quartz

H2 O

380

H 2O SiO2

H2O + 0.01 wt% SiO2

360 Temperature (K)

1.0000 0.0000

H2 O SiO2

Aqueous

0.9999 0.0001

340 Aqueous + quartz 320 Hatches database with Pitzer Debye–Hückel terms [97 Bon] Selected by Baes and Mesmer [76 Bae] Evaluation by Instituto de Investigaciones Electricas [000Ver]

300

280 270 0.000 H2O

0.002

0.004 0.006 Proportion of SiO2 (wt%)

0.008

0.010

II.27.2 SiO2–H2O phase diagram showing the stability field of the aqueous phase in equilibrium with gas and quartz, compared with experimental data [76 Bae, 97Bon, 000Ver].

combination of phases and phase compositions that gives the lowest overall Gibbs energy, based upon the Gibbs energy parameters in its databases, and reports that as the equilibrium state. A series of calculations of this type can be completed automatically and the results presented in a range of different ways including binary phase diagrams, ternary isothermal sections, cuts through multicomponent systems or general x–y plots, where x and y can be overall composition variables (phase amounts or speciation variables such as molalities), temperature, pressure, thermodynamic properties (such as heat capacity or enthalpy), pH or Eh. All these properties are calculable from the stored Gibbs energy functions.

II.27.4

Modelling approach

In the current work, the well-established compound energy model [92Bar] has been used to represent the Gibbs energy of the C–S–H system as a function of composition and temperature, extending earlier work on C–S–H and SiO2 gels [98Tho, 001Kul] by fully modelling the temperature dependence of the Gibbs energy of the phase, including Cp(T) of the species (or unaries) at its solution limits, consistent with reference states used throughout the world by SGTE [002SGT]. This ensured compatibility between the C–S–H model and a wealth of data already available for oxide phases, alloys, gaseous

326

The SGTE casebook

species and aqueous solutions, including high ionic strength aqueous solutions, available in other MTDATA databases. The compound energy model distributes species among a series of sublattices where they may interact either ideally or non-ideally. In the current model, the C–S–H gel was represented using five sublattices, as in the paper by Thomas and Jennings [98Tho], with occupancies as given in Table II.27.1, where Va indicates a potentially vacant sublattice. The Gibbs energies of unaries formed by taking one species from each sublattice in turn, i.e. (CaO) 7/3(H 2O) 1(SiO 2) 2 (SiO2 ) 1(H 2 O) 6, (CaO) 7/3 (CaO) 1 (SiO 2 ) 2 (SiO 2) 1 (H 2 O) 6 , (CaO) 7/3 (CaO) 1 (SiO 2 ) 2 (Va) 1 (H 2 O) 6 and (CaO)7/3(H2O)1(SiO2)2(Va)1(H2O)6 were described using a standard function of the form G(T) = A + BT + CT ln T + DT2 +ET3 +F/T. The parameters C, D, E and F were obtained from estimates of Cp(T) for each species guided by those for portlandite and notional compounds such as plombierite, for which thermodynamic data could be found in the SGTE Substance Database [002SGT]. Initial values for the entropy-like B parameters and enthalpy-like A parameters were estimated on a similar basis. The A parameters were then adjusted, together with others introduced to represent the excess Gibbs energies of interaction between sublattice species, using MTDATA’s data assessment tools to reproduce measured compositions of the C–S–H gel and aqueous phases in equilibrium from data available in the literature [34Fli, 40Rol, 50Tay, 52Kal, 65Gre, 81Fuj, 87Atk, 89Go, 96Con, 004Che, 006Wal]. The SiO2 gel was represented using two sublattices with occupancies as given in Table II.27.2. This model allows the composition of the SiO2 gel phase to vary within a triangle formed by the unaries (Ca7/3H14O28/3)1(SiO2)3, (Ca10/3H12O28/3)1(SiO2)3 and (SiO2)3 as opposed to a line used by Kulik and Kersten [001Kul]. The first two of these unaries lie along the SiO2-rich side Table II.27.1 Occupancies of the five sublattices of the C–S–H gel model Sublattice

Occupancy

Number and type of sites

1

CaO

1 3

2 3 4 5

CaO, H2O SiO2 SiO2, Va H 2O

1 2 1 6

interlayer site, 2 sites pairing with SiO2 interlayer site sites pairing with CaO site representing bridging SiO2 or vacancy sites

Table II.27.1 Occupancies of the five sublattices of the C–S–H gel model Sublattice

Occupancy

Number of sites

1 2

Ca7/3H14O28/3, Ca10/3H12O28/3, Va SiO2

1 3

Modelling cements in an aqueous environment

327

of the composition parallelogram formed by the four C–S–H unaries. Data for the pure SiO2 unary were based upon those for amorphous SiO2 in the SGTE Substance Database. Parameters representing interactions between species on the first sublattice in this phase were derived, as for the C–S–H gel phase, to represent experimental phase equilibrium and experimental solubility data as closely as possible. Calculations are most flexibly undertaken in MTDATA by specifying the start and end compositions at a fixed temperature or the start and end temperatures with a fixed composition. The program will perform a series of phase equilibrium calculations for conditions varying between the specified extremes and results can be plotted in terms of phase amounts, aqueous species, element distributions or pH values. Thermodynamic equilibria between C–S–H and SiO2 gels, portlandite and an aqueous phase could, for example, be calculated by stepping across the H2O–SiO2–Ca(OH)2 composition triangle between points A and B, shown in Fig. II.27.3, accounting for all expected Ca-to-Si ratios of C–S–H gels in OPC. These points correspond to overall systems containing 1 kg of H2O and 1 mol of SiO2 with 0 mol (A) and 2.5 mol (B) of CaO.

II.27.3 MTDATA calculating a ternary phase diagram of the system Ca(OH)2–SiO2–H2O at room temperature and pressure. The axes of the diagram indicate the mass fractions of the components.

328

The SGTE casebook

Figure II.27.3 takes the form of a screen shot of how MTDATA would actually appear when calculating an isothermal section showing phase equilibria within the H2O–SiO2–Ca(OH)2 composition triangle at 298.15 K. The window to the top left is where the temperature and composition scale (mass or mole fraction) for the diagram is selected. Within the triangle itself, white areas indicate single phase fields, tie lines indicate two-phase fields, and shaded tie triangles indicate three-phase fields. Since the data for the C–S–H and SiO2 phases, for other stoichiometric phases such as quartz and portlandite and for aqueous species (in the MTDATA implementation of the SUPERCRIT-98 Database) all have Gibbs energies modelled as a function of temperature, as already described, equilibria can be calculated at elevated temperatures just as at room temperature using these data. Surprisingly few long-term experiments of C–S–H gels maintained at high temperatures have been reported. Work by Glasser et al. [005Gla] has been chosen with which to compare the results of the predictive calculations. The thermodynamic data derived for the C–S–H and SiO2 gel phases also allow calculations to be carried out in which leaching by pure water is simulated, using MTDATA’s ‘built-in’ process modelling application. A series of calculations was undertaken in which a C–S–H gel with an initial Ca-toSi ratio of 2.7 was equilibrated with successive volumes of pure water at room temperature, the equilibrated aqueous phase being replaced each time. Predicted CaO and SiO2 solubilities and pH values were plotted against the cumulative volume of water for comparison with experimental data published by Harris et al. [002Har].

II.27.5

Results and discussion

II.27.5.1 C–S–H solubility at room temperature Both the predicted and the experimentally derived pH, the calcium concentration and the silicon concentration are shown in Fig. II.27.4, Fig. II.27.5 and Fig. II.27.6 respectively, as functions of the Ca-to-Si ratio in the solid phase(s) (including contributions from portlandite and amorphous silica). The experimental data are well matched by the model predictions for all expected Ca-to-Si ratios in the solid phase(s). The model can therefore be used beyond the compositional limits imposed by other approaches. The thermodynamic rigour inherent in the modelling approach described here means that not only can the pH and composition of the aqueous solution be predicted, but also a wide range of other thermodynamic properties and phase equilibria involving other phases in the C–S–H system, including the effects of altering temperature. The use of reference states compatible with those adopted in existing MTDATA databases for materials such as complex

Modelling cements in an aqueous environment 13.0

329

Constraints

12.5

T = 298.15 K P = 1 atm n(H2O) = 55.50843 mol n(SiO2) = 1 mol

12.0

Stage 1 Number of calculations = 502 Number shown on plot = 502

pH

11.5

11.0

10.5

10.0

9.5

9.0 0.0

0.5

1.0 1.5 CaO/SiO2 by amount (mol)

2.0

2.5

Flint and Wells [34 Fli] Greenberg and Chang [65 Gre] Fuji and Kondo [81 Fuj] Atkinson et al. [87 Atk] Grutzeck et al. [89 Gru] Cong and Kirkpatrick [40] [96 Con] Chen et al. [004 Che] Walker et al. [006 Wal](112 weeks) Walker et al. [006 Wal](80 weeks) Walker et al. [006 Wal](64 weeks)

II.27.4 Calculated pH in solution as a function of the CaO-to-SiO2 ratio in the solid phase(s) (solid curve) compared with relevant experimental data for the aqueous phase in equilibrium with C–S–H and SiO2 gel.

oxide and sulphide solutions, alloys and high ionic strength aqueous solutions allows the interaction of C–S–H gels with such phases to be predicted. Calculations of this type provide a consistency test of the model framework and parameters derived for the gel phases. It should be noted that C–S–H gels are generally considered to be metastable because they continue to react very slowly, eventually forming other CaO– SiO2 phases. MTDATA predicts this true equilibrium state but metastable equilibria can be studied by ‘classifying as absent’ long-term decompostion products, or by removing them from consideration in the calculations.

330

The SGTE casebook

0.022

Constraints T = 298.15 K P = 1 atm n(H2O) = 55.50843 n(SiO2) = 1 mol n(H) = 0 mol Stage 1 Number of calculations = 502 Number shown on plot = 502

Molatility of component CaO (mol kg–1)

0.020 0.018 0.016 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000 0.0

0.5

1.0 1.5 2.0 CaO/SiO2 by amount (mol)

2.5

Flint and Wells [34 Fli] Roller and Ervin [40 Roi] Taylor [50 Tay] Kalousek [52 Kal] Greenberg and Chang [65 Gre] Fuji and Kondo [81 Fuj] Atkinson et al. [87 Atk] Grutzeck et al. [89 Gru] Cong and Kirkpatrick [96 Con] Chen et al. [004 Che] Walker et al. [006 Wal](80 weeks) Walker et al. [006 Wal](112 weeks) Glasser et al. [005 Gla]

II.27.5 Calculated CaO in solution as a function of the CaO-to-SiO2 ratio in the solid phase(s) (solid curve) compared with relevant experimental data for the aqueous phase in equilibrium with C–S–H and SiO2 gel.

It has been observed by several researchers that the experimental data divide into two populations, depending on whether the solid C–S–H phase has been prepared directly from ions in solution (i.e. by direct reaction or double decomposition) or from the hydration and subsequent leaching of tricalcium silicate (the latter route forming the least soluble solid). For the purposes of this model, parameters have been derived to reproduce the lower SiO2 solubility curve, thought to be appropriate to the most stable gel structure, although reproducing either curve is relatively straightforward.

Modelling cements in an aqueous environment –2.0

Constraints T = 298.15 K P = 1 atm nH O = 55.50843 mol 2 nSiO = 1 mol 2 nH = 0 mol Stage 1 Number of calculations = 502 Number shown on plot = 502

–2.5

log10 [molality of component SiO2 (mol kg–1)]

331

–3.0 –3.5 –4.0 –4.5 –5.0 –5.5 –6.0 –6.5 0.0

0.5

1.0 1.5 CaO/SiO2 by amount (mol)

2.0

2.5

Flint and Wells [34 Fli] Roller and Ervin [40 Roi] Taylor [50 Tay] Kalousek [52 Kal] Greenberg and Chang [65 Gre] Fuji and Kondo [81 Fuj] Atkinson et al. [87 Atk] Grutzeck et al. [89 Gru] Chen et al. [004 Che] Walker et al. [006 Wal](60 weeks) Walker et al. [006 Wal](80 weeks) Walker et al. [006 Wal](112 weeks) Glasser et al. [005 Gla]

II.27.6 Calculated SiO2 concentration in solution as a function of the CaO-to-SiO2 ratio in the solid phase(s) (solid curve) compared with relevant experimental data for the aqueous phase in equilibrium with C–S–H and SiO2 gel.

II.27.5.2 C–S–H solubility at higher temperatures The predicted solubility of C–S–H gels at elevated temperature and pH are shown in Fig. II.27.7, Fig. II.27.8 and Fig. II.27.9 with experimental data [005Gla] superimposed in Fig. II.27.8 and Fig. II.27.9. It should be noted that the model predictions were not fitted to the experimental data shown. The order and shape of each curve in Fig. II.27.8 and Fig. II.27.9 are similar

332

The SGTE casebook

13.0

Constraints 25 °C

12.5

T = 298.15, 328.15, 358.15 K P = 1 atm n(H2O) = 55.50843 mol n(SiO2) = 1 mol n(H) = 0 mol

12.0 55 °C

11.5

pH

11.0

Stage 1 Number of calculations = 502 Number shown on plot = 502

85 °C

10.5 10.0 9.5 9.0 8.5 8.0 0.0

0.5

1.0 1.5 2.0 CaO/SiO2 by amount (mol)

2.5

II.27.7 Predicted pH in solution plotted against the CaO-to-SiO2 ratio in the solid phase(s) at different temperatures for the aqueous phase in equilibrium with C–S–H and SiO2 gel at 25, 55 and 85 °C. 0.022

Glasser et al. [005 Gla] 25 °C Glasser et al. [005 Gla] 55 °C Glasser et al. [005 Gla] 85 °C

Molality of component CaO (mol kg–1)

0.020

Constraints 25 °C

T = 298.15, 328.15, 358.15 K P = 1 atm nH O = 55.50843 2 nSiO = 1 mol 2 nH = 0 mol Stage 1 Number of calculations = 502 Number shown on plot = 502

0.018 55 °C

0.016 0.014

85 °C

0.012 0.010 0.008 0.006 0.004 0.002 0.000 0.0

0.5

1.0 1.5 2.0 CaO/SiO2 by amount {mol)

2.5

II.27.8 Predicted and experimental CaO concentrations in solution plotted against the CaO-to-SiO2 ratio in the solid phase(s) at different temperatures for the aqueous phase in equilibrium with C–S–H and SiO2 gel at 25, 55 and 85 °C.

Modelling cements in an aqueous environment

333

Aqueous phase in equilibrium with CSH and SiO2 gel at 28 °C, 55 °C, 85 °C

log10 [molality of component SiO2 (mol kg–1)

–2.0

Constraints

Glasser et al. [005 Gla] 25 °C Glasser et al. [005 Gla 55 °C Glasser et al. [005 Gla] 85 °C

–2.5

T/K = 298.15, 328.15, 358.15 K P/atm = 1 atm nH O = 55.50843 2 nSiO = 1 mol 2 nH = 0 mol Stage 1 Number of calculations = 502 Number shown on plot = 502

–3.0

–3.5

–4.0

–4.5

–5.0

–5.5

–6.0 0.0

0.5

1.0 1.5 CaO/SiO2 by amount (mol)

2.0

2.5

II.27.9 Predicted and experimental SiO2 concentrations in solution plotted against the CaO-to-SiO2 ratio in the solid phase(s) at different temperatures for the aqueous phase in equilibrium with C–S–H and SiO2 gel at 25, 55 and 85 °C.

for both predicted and observed results, reflecting a decreasing solubility of C–S–H gel with increasing temperature. The general magnitude of the changes in solubility with temperature are predicted well, although the match between absolute solubility values at high CaO-to-SiO2 ratios appears poor. When Fig. II.27.5 is taken into account, however, it is clear that the room-temperature solubility results obtained by Glasser et al. [005Gla] for CaO-to-SiO2 ratios greater than 1.5 are not representative of the bulk of the experimental solubility data available. As more high-temperature experimental data become available, there is every reason to believe that the current calculated results will provide a better match.

II.27.5.3 Leaching simulation The pH and the CaO and SiO2 concentrations in aqueous solution, as predicted using MTDATA and experimentally measured [002Har], during the leaching of a portlandite–(C–S–H gel) mixture are shown in Fig. II.27.10 and Fig.

334

The SGTE casebook

13.0

Constraints Initial settings T = 298.15 K P = 1 atm Mass of H2O = 0.2 kg Mass of CaO = 0.572723 kg Mass of SiO2 = 0.227277 kg Mass of O2 = 0 kg Step number = 700 Step type = 2 Application code = 30 Stage 1, Resize = 500 Number of calculations = 94 Number shown on plot = 94

12.8 12.6 12.4 12.2 12.0

pH

11.8 11.6 11.4 11.2 11.0 10.8 10.6 10.4 10.2 10.0 0.0

Experimental data from Harris et al. [002 Har] 0.5 1.0 1.5 2.0 2.5 3.0 3.5 log10 [volume of water added (cm3 g–1)]

4.0

II.27.10 Predicted pH of the aqueous phase plotted against the volume of pure water added at 25 °C in a leaching simulation for a simple mixture of portlandite–(C–S–H gel) mixture with an initial Cato-Si ratio of 2.7.

II.27.11. The experimental data are well matched by the predicted model. The thermodynamic properties of the C–S–H gel used for this leaching simulation were derived from the previous calculations to predict the solubility data shown in Fig. II.27.4, Fig. II.27.5 and Fig. II.27.6. These results provide further evidence for the validity of the model and the assumption that the dissolution of the solid phase(s) in the C–S–H system can be predicted by means of phase equilibrium calculations.

II.27.6

Conclusions

This work describes the use of a sublattice model as a basis for modelling the incongruent dissolution of C–S–H gel at different temperatures. The model correctly predicts the pH and calcium and silicon concentrations as a function of the Ca-to-Si ratio of the solid phase(s) at room temperature and as a function of volume of water added in simple leaching calculations and describes the general trends in the available solubility data at higher temperatures. Rather than imposing constraints, the sublattice model imparts a degree of flexibility that can be used to include other previously unaddressed variables. Ongoing work includes mixing of Al2O3 and alkali metal hydroxides in the

Modelling cements in an aqueous environment 3.0

Key

Experimental data from Harris et al. [002 Har]

log10 [molality of CaO and SiO2 (mmol kg–1)]

2.5

1 H2O 3 SiO2

CaO SiO2

2.0 1.5

335

2 CaO 4 O2

Constraints Initial settings T = 298.15 K P = 1 atm Mass of H2O = 0.2 kg Mass of CaO = 0.572723 kg Mass of SiO2 = 0.227277 kg Mass of O2 = 0 kg Step number = 700 Step type = 2 Application code = 30 Stage 1, resize = 500 Number of calculations = 94 Number shown on plot = 94

CaO

1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0 0.0

SiO2 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

log10 [volume of water added (cm3 g–1)]

II.27.11 Predicted CaO and SiO2 concentrations in the aqueous phase plotted against the volume of pure water added at 25 °C in a leaching simulation for a simple portlandite–(C–S–H gel) mixture with an initial Ca-to-Si ratio of 2.7.

C–S–H sublattice, collating available Pitzer virial coefficients for calculations at high ionic strength, and the solubility behaviour of other cement hydrate phases under a range of physicochemical conditions. Ultimately, the harmonisation of data for cementitious phases with those for pollutant species and radionuclides will increase the range of systems, which may be addressed. This may lead to coupling of MTDATA to other codes, e.g. groundwater transport, in order to assess the likely behaviour of cemented wasteforms and structures for environmental protection.

II.27.7

References

883She W.A. SHENSTONE and J.T. CUNDALL: J. Chem. Soc. (Lond.), 43, 1883, 550. 34Bas H. BASSETT JR: J. Chem. Soc. (Lond.), 1934, 1270–1275. 34Fli E.P. FLINT and L.S. WELLS: ‘Study of the system CaO–SiO2–H2O at 30 °C and of the reaction of water on anhydrous calcium silicates’, J. Res. Natl Bur. Stand. 12, 1934, 751–783. 40Rol P.S. ROLLER and G. ERVIN, JR: ‘The system calcium–silica–water at 30°. The association of silicate ion in dilute alkaline solution’, J. Am. Chem. Soc. 62(3), 1940, 461–471.

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50Tay

H.F.W. TAYLOR: ‘Hydrated calcium silicates. Part I. Compound formation at ordinary temperature’, J. Chem. Soc. 726, 1950, 3682–3690. G. KALOUSEK: ‘Application of differential thermal analysis in a study of the system lime–silica–water’, Proc. 3rd Int. Symp. Chemistry of Cement, London, UK, 1952, pp. 296–311. R.B. PEPLER and L.A. WELLS: J. Res. Nat Bur. Stand. 52, 1954, 75–92. W.F. LINKE (Ed.): Solubilities of Inorganic and Metal-Organic Compounds (A– Ir) Vol. I, 4th edition, van Nostrand, Princeton, 1958. S.A. GREENBERG and T.N. CHANG: ‘Investigation of the colloidal hydrated calcium silicates. II. Solubility relationships in the calcium–silica–water system at 25 °C’, J. Phys. Chem. 69, 1965 182–188. C.F. BAES and R.E. MESMER: The Hydrolysis of Cations, Wiley–Interscience, New York, 1976. K. FUJII and W. KONDO: ‘Heterogeneous equilibrium of calcium silicate hydrate in water at 30 °C’, J. Chem. Soc., Dalt. on Trans. 2, 1981, 645–651. A. ATKINSON, J.A. HEARNE and C.F. KNIGHTS: ‘Aqueous and thermodynamic modeling of CaO–SiO2–H2O gels’ Report AERER12548, UK Atomic Energy Authority, 1987. T.I. BARRY and A.T. DINSDALE: ‘Thermodynamics of metal–gas–liquid reactions’, Mater. Sci. Technol 3, 1987, 501–511. D. GARVIN, V.B. PARKER and H.J. WHITE (Eds): CODATA Thermodynamic Tables, Hemisphere Publishing, New York, 1987. F.P. GLASSER, D.E. MACPHEE and E.E. LACHOWSKI: ‘Solubility modeling of cements: implications for radioactive waste immobilization’, Mater. Res. Soc. Symp. Proc. 84, 1987, 331–341. J.H. WEARE: Reviews in Mineralogy, Vol. 17 (Eds. I.S.E. Carmichael and H.P. Eugster), 1987, Chapter 5, pp. 143–176. U.R. BERNER: ‘Modeling the incongruent dissolution of hydrated cement minerals’, Radiochim. Acta 44–45, 1988, 387–393. M. GRUTZECK, A. BENESI and B. FANNING: ‘Silicon-29 magic angle spinning nuclear magnetic resonance study of calcium silicate hydrates’, J. Am. Ceram. Soc. 72, 1989, 665–668. C.W. BALE and G ERIKSSON: Can. Metal. Q. 29, 1990, 105–132. E.J. REARDON: ‘An ion interaction model for the determination of chemical equilibria in cement/water systems’, Cem. Concr. Res. 20, 1990, 175–192. T.I. BARRY, A.T. DINSDALE, J.A. GISBY, B. HALLSTEDT, M. HILLERT, B. JANSSON, S. JONSSON, B. SUNDMAN and J.R. TAYLOR: ‘The compound energy model for ionic solutions with applications to solid oxides’, J. Phase Equilibria 13(5), 1992, 459–475. R.G.J. BALL, M.A. MIGNANELLI, T.I. BARRY and J.A. GISBY: ‘The calculation of phase equilibria of oxide core–concrete systems’, J. Nucl. Mater. 201, 1993, 238–249. T.I. BARRY, A.T. DINSDALE and J.A. GISBY: ‘Predictive thermochemistry and phase equilibria of slags’, JOM 45(4), 1993, 32–38. X. CONG and R.J. KIRKPATRICK: ‘29Si MAS NMR study of the structure of calcium silicate hydrate’, Adv. Cem. Based Mater. 3, 1996 144–156. M. KERSTEN: ‘Aqueous solubility diagrams for cementitious waste stabililization systems. 1. The CSH solid-solution system’, Environ. Sci. Technol. 30, 1996, 2286–2293.

52Kal

54Pep 58Lin 65Gre

76 Bae 81Fuj 87Atk

87Bar 87Gar 87Gla

87Wea 88Ber 89Gru

90Bal 09Rea 92Bar

93Bal

93Bar 96Con 96Ker

Modelling cements in an aqueous environment 97Bar

97Bon

98Tho 99Maz 000Bar 000Ver 001Kul

001Wan 002Dav

002Gis

002Har

002Hun 002SGT 003Put 004Che 004Tas

005Gla

006Wal

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S. BÖRJESSON, A. EMRÉN and C. EKBERG: ‘A thermodynamic model for the calcium silicate hydrate gel, modelled as a non-ideal binary solid solution’, Cem. Concr. Res. 27, 1997, 1649–1657. K.A. BOND, T.G. HEATH and C.J. TWEED: ‘MTDATA implementation of HATCHES: a referenced thermodynamic database for chemical equilibrium studies’, Nirex Report NSS/R379, December 1997. J.J. THOMAS and H.M. JENNINGS: ‘Free-energy-based model of chemical equilibria in the CaO–SiO2–H2O system’, J. Am. Ceram. Soc. 83(3), 1998, 606–612. M. MAZIBUR RAHMAN, S. NAGASAKI and S. TANAKA: ‘A model for dissolution of CaO–SiO2–H2O gel at Ca/Si > 1’, Cem. Concr. Res. 29, 1999, 1091–1097. T.I. BARRY and F.P. GLASSER: ‘Calculation of Portland cement clinkering reactions’, Adv. Cem. Res. 12(1), 2000, 19–28. M.P. VERMA: Proc. World Geothermal Congr., Kyushu-Tohoku, Japan, 28 May– 10 June 2000. D.A. KULIK and M. KERSTEN: ‘Aqueous solubility diagrams for cementitious waste stabilization systems: II, end-member stoichiometries of ideal calcium silicate hydrate solid solutions’, J. Am. Ceram Soc. 84(12), 2001, 3017–3026. J. WANG and S. VAN DER ZWAAG: ‘Composition design of a novel P containing TRIP steel’, Z. Metallkunde 92, 2001, 1299–1311. R.H. DAVIES, A.T. DINSDALE, J.A. GISBY, J.A.J. ROBINSON and S.M. MARTIN: ‘MTDATA – thermodynamics and phase equilibrium software from the National Physical Laboratory’, Calphad 26(2), 2002, 229–271. J.A. GISBY, A.T. DINSDALE, I. BARTON-JONES, A. GIBBON, P.A. TASKINEN and J.R. TAYLOR: ‘Phase equilibria in oxide and sulphide systems’, in Sulfide Smelting 2002, TMS Meeting (Eds R.L. Stephens and H.Y. Sohn), Seattle, Washington, 2002. A.W. HARRIS, M.C. MANNING, W.M. TEARLE and C.J. TWEED: ‘Testing of models of the dissolution of cements – leaching of synthetic CSH gels’, Cem. and Concr. Res. 32, 2002, 731–746. C. HUNT, J. NOTTAY, A. BREWIN and A.T. DINSDALE: NPL Report MATC(A) 83, National Physical Laboratory, 2002. SGTE: Thermodynamic Properties of Inorganic Materials, Landolt–Börnstein New Series, Group IV (Physical Chemistry), Vol. 19 Springer, Berlin 2002. D.C. PUTMAN and R.C. THOMSON: ‘Modeling microstructural evolution of austempered ductile iron’, Int. J. Cast Metals Res. 16(1), 2003, 191–196. J.J. CHEN, J.J. THOMAS, H.F.W. TAYLOR and H.M. JENNINGS: ‘Solubility and structure of calcium silicate hydrate’, Cem. Concr. Res. 34, 2004, 1499–1519. P. TASKINEN, A.T. DINSDALE and J.A. GISBY: ‘Industrial slag chemistry – a case study of computational thermodynamics’, Proc. Symp. Metal Separation Technologies III, Copper Mountain, Colorado, USA, 2004. F.P. GLASSER, J. PEDERSEN, K. GOLDTHORPE and M. ATKINS: ‘Solubility reactions of cement components with NaCl solutions: I. Ca(OH)2 and C–S–H’, Adv. Cem. Res. 17(2), 2005, 57–64. C.S. WALKER, D. SAVAGE, M. TYRER and K.V. RAGNARSDOTTIR: Non-ideal solid solution aqueous solution modeling of synthetic calcium silicate hydrate, Cem. Concr. Res. 37(4), 2007, 502–511.

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Part III Process modelling – theoretical background

339

340

III.1 Introduction KLAUS HACK

So far all discussions have been related to static equilibrium situations. However, all processes are dynamic in nature. One can claim that, for a description of processes, thermodynamics can be applied and that thermodynamics are a combination of thermostatics (here equilibrium thermochemistry) and kinetics. These two fields of thermodynamics are at present not in a comparable state. As was outlined in Part I, thermochemistry can be used in a rigorous way. The basic theory has been transformed into computer programs, model pictures on the basis of atoms in crystal lattices or sites in molecular structures have been developed, from which the Gibbs energy equations can be derived, and for many substances the necessary data are available. On the other hand, kinetics are not yet in such an advanced state on the theoretical side nor regarding the data for particular substances (phases). When the combination of thermostatics and kinetics for the simulation of processes is considered, different approaches are at present in use, which are outlined in the following chapters. They all work with the concept of local equilibrium. The major difference lies in the treatment of the flow of matter or the control of the way in which reactions proceed. On the one hand, there are cases in which only the Gibbs energy data are needed to run the process model while the kinetic aspects are treated in a simplified way. The Gulliver–Scheil method which is discussed in detail in Chapter III.2 and for which an application case is given in Chapters IV.1 and IV.2, for example, works without explicit diffusion data, and the steady-state calculations for countercurrent reactors described in Chapter III.4 and applied in Chapters IV.7 and IV.8 can be used without explicit reaction kinetic data. On the other hand, there are cases in which all data, thermodynamic as well as kinetic, are needed and available. Thus, with explicit information on the diffusion of the various components in all phases of a system as discussed in Chapter III.3 and applied in Chapters IV.3 to IV.6, a detailed 341

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description can be given of the time dependence of phase formation while, with the explicit knowledge of reaction kinetic data as outlined in Chapter III.5 and applied in Chapter IV.9, the way in which equilibration can proceed in a system in which major reactions are kinetically inhibited can be investigated.

III.2 The Gulliver–Scheil method for the calculation of solidification paths B O S U N D M A N and I B R A H I M A N S A R A

For solidification purposes in multicomponent systems, the crystallisation sequences that occur upon cooling are important for the properties of the material. The nature and the compositions of the various phases which precipitate can affect casting properties, microstructures and hence mechanical properties. It is difficult to determine these sequences by microanalysis because quantities such as partition coefficients are difficult to measure. In addition, the thermal effect associated with the process is a parameter which is difficult to evaluate experimentally. Equilibrium compositions can provide such information if the thermodynamic properties of all the phases involved in the crystallisation process are known. Their compositions are generally derived by minimisation of the Gibbs energy of the system. However, most practical cases of solidification will not follow simple equilibrium cooling. The Gulliver–Scheil approach has been widely used in practice to simulate solidification for slow cooling rates. This approach assumes the following. 1 2 3

Local equilibrium at the solid–liquid interface. Homogeneity of the liquid. No diffusion in the solid phases.

Hence the fraction of the solid phase which precipitates will no longer participate in the solidification process. Thus all calculations can be performed without explicit knowledge of diffusion data, i.e on the basis of the Gibbs energy minimisation alone. Even if some elements can diffuse much more quickly than the others, such as carbon and nitrogen in steels, one can take that into account by assuming that they will distribute themselves according to full equilibrium [02Che]. During solidification, and as long as a two-phase equilibrium occurs, the overall composition of the liquid is continuously modified until a third phase precipitates. If the three-phase equilibrium is peritectic, solidification will continue, precipitating the new solid phase. If it is eutectic and the system is 343

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binary, the remaining liquid will transform isothermally to the two phases. If the system has three or more components, the composition of the liquid will follow a three-phase or multiphase line until no liquid remains. As an example we can use the binary A–B system with a phase diagram according to Fig. III.2.1. Pure A is face-centred cubic (fcc) and pure B is body-centred cubic (bcc); there is a compound A2B with the C14 structure that forms peritectically, and the liquid forms a eutectic with this compound and the bcc phase. The solidification path for an alloy with a composition 1300 1200

Temperature (°C)

1100 Liquid

1000 900 800 700

Fcc

600 Bcc

500 C14 400 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole fraction of B

III.2.1 Phase diagram with the solidification path marked. 1000 950

Temperature (°C)

900

Fcc Equilibrium solidification

850 800 750

C14

700 650 600 550 C14 + bcc 500 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Fraction of solid phase

1.0

III.2.2 Variation in the fraction of the solid phase with temperature T.

Gulliver–Scheil method for calculation of solidification paths

345

indicated by the asterisk in the diagram is shown in the figure and in Fig. III.2.2 the fraction of solid phase is shown both for equilibrium solidification and for a Scheil–Gulliver simulation. According to equilibrium the alloy should solidify completely when the C14 phase is formed but, as this requires diffusion through the layer of C14 formed between the liquid and the fcc phase, it will never occur during normal solidification. In Fig. III.2.1 the

Mole fraction of B in various phases

0.7 0.6 0.5

Liquid

C14

0.4 0.3 0.2 Fcc

0.1 0 500

600

700 800 Temperature (°C)

900

1000

III.2.3 Composition as a function of temperature T. 0 –1

Latent heat (kJ mol–1)

–2 –3 –4 –5 –6 –7 –8 –9 –10 –11 500

600

700 800 Temperature (°C)

III.2.4 Evolution of latent heat.

900

1000

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overall composition of the fcc phase is shown as a dotted line inside the fcc region, the extrapolation of the overall composition is shown as a dotted line inside the fcc + liquid two-phase region. The liquid composition follows the liquidus line down to the peritectic with the C14 phase. There will be segregation also inside the C14 phase and its overall composition is also shown as a dotted line. The compositions in the liquid and solid phases are shown in Fig. III.2.3 as a function of the temperature, and in Fig. III.2.4 the latent heat is shown. All these figures are from the same Scheil–Gulliver simulation. The simulation of the solidification can be performed by considering steps in the temperature, or by fixing the proportion of the solid phase which precipitates, or even by fixing the amount of energy (heat) which is extracted. A case study on a practical system, the solidification of Al-rich liquid Al– Mg–Si, is given in Chapter IV.1.

III.2.1 02Che

Reference Q. CHEN and B. SUNDMAN, Mater. Trans., Jap. Inst. Metals 43, 2002, 551–559.

III.3 Diffusion in multicomponent phases JOHN ÅGREN

III.3.1

Introduction

On the microscopic scale the physical models and their mathematical counterparts available today are much more detailed and elaborate than the relatively coarse approach discussed in the previous section for macroscopic processes. Phase transformation between condensed phases, for example, can be treated by a method which combines an explicit description of diffusion processes with thermochemical calculations assuming local equilibrium. A short description of this method is discussed below.

III.3.2

Phenomenological treatment

One-dimensional diffusion along, for example, the z-axis in a binary system A–B usually obeys the well-known Fick’s law

J B = DB

∂CB ∂z

(III.3.1)

where JB is the diffusive flux (mol m–2 s–1), DB the diffusivity of B and ∂CB/∂z the concentration gradient. If the thermodynamic behaviour of the system is sufficiently well known, one may express the chemical potential µB of B for a given temperature as a function of the B concentration:

µB = µB(cB)

(III.3.2)

One may thus express Fick’s law in terms of the true driving force of diffusion, the gradient of µB, rather than the concentration gradient [48Dar]: J B = – DB

∂µ B 1 dµ B /dc B ∂z

(III.3.3)

In fact, arguments based on the theory of absolute reaction rates and principles of irreversible thermodynamics suggest that the latter expression for the flux would be a more fundamental formulation of an irreversible process. The 347

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reason is that the gradient ∂µB/∂z is really the average force acting on the diffusing species. One should therefore write

∂µ B (III.3.4) ∂z and consider LBB as a purely kinetic parameter indicating how easily the species moves under the action of a force. This relation may be compared with, for example, Ohm’s law for the electric current. It is obvious that the parameter L plays exactly the same role as the electric conductivity. The picture can be even more generalised because one would expect a linear relationship between velocity and force when the motion of a body is opposed by the friction with the medium through which it is moving. One should thus consider L as a basic kinetic parameter characteristic for the case under consideration and, by combining the last two equations, one obtains J B = – LBB

DB =

∂µ B L dc B BB

(III.3.5)

In a multicomponent alloy the chemical potential of a species normally depends on the content of all the different species and the expression for the flux of species i would be J i = – Lii

n –1 ∂µ i ∂µ i ∂c j = – Lii Σ 1 ∂z ∂c j ∂z

(III.3.6)

One can now introduce the multicomponent diffusivity D i j from the relation

∂c j (III.3.7) 1 ∂z where n is the number of components and where the matrix D of the diffusivities is related to the kinetic parameters L: n –1

J i = Σ Dij

Dij =

∂µ i L ∂c j ii

(III.3.8)

In an ideal solution ∂µi/∂cj is different from zero only if i = j and the offdiagonal elements of the matrix formed by all the D ij coefficients will vanish. However, in general, chemical non-ideality prevails and all D ij coefficients have finite values. Moreover, if the interactions are strong, the off-diagonal elements may be of the same order of magnitude as the diagonal elements. A more extensive discussion of this subject has been given by Kirkaldy and Young [87Kir]. Evidently, one would thus expect that the diffusion of a species depends not only on its own concentration gradient but also on the gradients of the other species. From a practical point of view, this means that a very mobile species may be redistributed in a body if there are gradients of less mobile species even if the mobile species itself is homogeneously distributed at the

Diffusion in multicomponent phases

349

beginning. The effect may even be so strong that an element may diffuse to regions of much higher concentration, so-called uphill diffusion. The effect was first demonstrated experimentally by Darken [49Dar]. Another piece of evidence of the correlation of chemical potential, rather than concentration, and diffusion is given by the behaviour of a system with a miscibility gap. The region of spontaneous demixing under the spinodal curve is the region in which the chemical potential drops with increasing concentration, thus driving the components into the region of higher concentration. This is demonstrated for a binary system by Fig. III.3.1 and Fig. III.3.2. The expression for the flux in the multicomponent case (see Equation (III.3.6)) was intended as a first approximation which excludes the possible influence of the chemical potential gradients of other species upon the flux. In general, this possibility should be taken into account and one should write the flux as n –1

J i = – Σ Lik 1

n –1 n –1 ∂µ k ∂µ k ∂c j = – Σ Lik Σ 1 1 ∂c j ∂z ∂z

(III.3.9)

and, for the diffusivity, one thus obtains n –1

Dij = Σ Lik 1

III.3.3

∂µ k ∂c j

(III.3.10)

Analysis of experimental data: the general database

In view of the discussion above, it must be concluded that, in order to make the most efficient use of experimental diffusion data and to obtain reliable extrapolations, one should take into account the thermochemical properties

1100 1000

Temperature (K)

α

α′

900 800 700 600

α + α′

500 400 A

Concentration

III.3.1 Region of demixing.

B

350

The SGTE casebook 700 K 1.2 1.1

800 K

Activity of component B

1.0 0.9

1000 K

0.8 1200 K

0.7 0.6

w

0.5 0.4

t’s ul

la

o

Ra

0.3 0.2 0.1 0 A

Concentration

B

III.3.2 Activities in the region of demixing.

of the system when known. One should evaluate the Lij parameters from diffusion experiments and store those as functions of composition and temperature in a phase-related database similar to the storage which has already be adopted for the thermochemical (Gibbs energy) data. Whenever the diffusivity is needed, it can then be calculated from this coupled store of kinetic and thermochemical data. This database is in a true sense thermodynamic. However, it must be noted that detailed information on the diffusivities is still lacking in many cases and for practical calculations it may often be impossible to evaluate the off-diagonal L parameters mentioned above. It may therefore be necessary in many practical calculations to use approximations by neglecting these terms and only applying Equations (III.3.7) and (III.3.8). This procedure has proven quite acceptable [88Hil]. It should also be emphasised that the simple Lii parameters depend on the composition and that rather detailed experimental information is needed in order to derive a consistent set of Lii parameters for a phase in a particular system.

III.3.4 48Dar 49Dar 87Kir 88Hil

References L.S. DARKEN: Trans. Metall. Soc. AIME 175,1948,184. L.S. DARKEN: Trans. Metall. Soc. AIME 180, 1949, 430–438. J.S. KIRKALDY and D.J. YOUNG: Diffusion in the Condensed State, Institute of Metals, London, 1987. M. HILLERT and J. ÅGREN: in Advances in Phase Transitions (Eds J. D. Embury and G.R. Purdy) Pergamon, Oxford, 1988, pp. 1–19.

III.4 Simulation of dynamic and steady-state processes K L A U S H A C K and S T E F A N P E T E R S E N

III.4.1

Introduction

Many industrial processes are performed at high temperatures using nonisothermal furnaces into which gaseous and condensed material as well as energy are supplied at different levels, e.g. blast furnaces, reverberatory furnaces or electrothermal furnaces. For the optimisation of such a process with regard to product yield and energy consumption, time-consuming trialand-error experiments are still frequently utilised. These could be performed in a more systematic manner if the temperature and composition profiles of a process could be predicted by using chemical equilibrium computations. A theoretical calculation is advantageous, because it can be difficult, if not impossible, to measure these profiles experimentally. A multistage reactor model, to be described below, can give a complete characterisation of a technical process. This has been demonstrated, for example, in the modelling of several alternative carbothermic silica reduction processes [78Eri]. The variations in temperature and composition inside the reactor model were predicted for various values of charge composition and enthalpy input. Conditions were then found which optimise the process investigated so as to obtain a maximum product yield at minimum energy consumption. Another example is the study of cement making by Ginsberg et al. [005Gin].

III.4.2

Concept of modelling processes using simple unit operations

The process modelling software SimuSage [007Pet1] is an innovative tool for process simulation and flowsheeting tasks. Based on the programming library ChemApp [96Eri, 007Pet2] and its rigorous Gibbs energy-minimising technique, it provides a library of reusable and extensible components for the development of highly customised process simulation models. This library of both visual and non-visual components is integrated into Borland’s 351

352

The SGTE casebook

programming environment Delphi and allows for the simulation of both dynamic and steady-state processes. Using a simple set of unit operations is sufficient for the development of general, thermochemically based process models which are only limited by the availability of the relevant thermochemical data, i.e. the Gibbs energies of the phases. These basic unit operations consist of an equilibrium reactor, a heat exchanger, splitters and mixers. The various units are interconnected by streams which transport the matter and energy (enthalpy) from one unit to another. Streams may be single-phase or multi-phase amounts of matter. They also have a temperature and pressure which makes them fully defined states in terms of thermodynamics. While regular streams are used to connect unit operations, input streams are associated with user-defined feed materials for the flowsheet. The user may for example define a stream called air which is defined as consisting of 21 vol% O2 and 79 vol% N2. For further use downstream in the process, a stream that leaves an equilibrium reactor unit usually needs to be split, either by amount or by phase(s). In a practical process the phase split is achieved by particular units such as cyclones (separation of solids from gases) or filters (separation of solids from liquids). In reality, however, cyclones and filters do not perform a 100% perfect phase separation. Thus, in order to give a realistic picture of the cyclone unit, as an example, the condensed phases are first separated from the gas using the phase splitter and in a second step a certain percentage of the condensed matter is removed. In a third step, the small amount of carry-over dust is added again to the gas stream with a physical mixer. In practical processes, chemical inhibitions do unavoidably occur. However, especially with respect to chemical (reaction) kinetics the database for most systems is highly limited. Thus an explicit treatment of the reaction kinetics, although possible even in complex equilibrium calculations (see Chapters III.5 and IV.9), usually cannot be implemented. Nevertheless, the method described above is still applicable since the stream and phase splitters, together with the (physical) mixer, enable the user to introduce material bypasses in his model. These in turn permit the amount of reacting materials to be limited, thus treating the kinetic inhibitions in an indirect way. Another reason why splitters need to be used is found in countercurrent reactors. The stream leaving an equilibrium reactor unit has to be split in order to direct one (set of) phase(s) in one direction while others go the opposite way. This feature is used for the treatment of the silicon arc furnace process which is discussed in Chapter IV.7. Because countercurrent reactors introduce internally closed loops through matter being transported back upstream, they also necessitate the use of auxiliary unit operations which help to manage the calculation of such partially under-defined flowsheet segments. For additional reading on reactor modelling see the work by Traebert

Simulation of dynamic and steady-state processes

353

[001Tra], Modigell et al. [004Mod], Brüggemann [004Brü], and especially Petersen et al. [007Pet1].

III.4.3

General description of the reactor model

The process to be discussed below (silicon arc furnace) is a continuously working, vertical reactor into which raw material and energy can be supplied in principle at any level. Various chemical reactions will occur in different volume segments of the reactor at rates depending on the temperature, and phases formed at one level will flow for further reaction to other levels or will form part of the products leaving the reactor. To simulate such a process, the model reactor is conceptually divided into a number of sequential stages, each considered to be an equilibrium reactor unit. Inside the reactor, gaseous and condensed phases are flowing in opposite or parallel directions. According to these flow directions, gaseous and condensed products leave a given stage to react in neighbouring stages or to exit the reactor. Figure III.4.1 shows schematically the sequence of equilibrium stages for a countercurrent aggregate, taking the silicon arc furnace as an example (see Chapter IV.7). In order to calculate the amounts, compositions and temperatures of the phases leaving a stage, it is assumed that in the stage the chemical reactions occurring proceed to completion. Two different types of stages are possible. An enthalpy-regulated stage is the stage for which the reaction temperature is determined by a given value for the heat balance ∆H. At the reaction temperature, the total enthalpy change within this stage, i.e. the sum of energies absorbed or released when the gaseous and condensed reactant phases are heated or cooled, and the energies absorbed or released while the reaction is completed, counterbalances the enthalpy supplied to the stage from the outside (e.g. by electrical heating) or released from the stage (e.g. by heat losses). For a temperature-regulated stage the reaction temperature is fixed at the outset and the total enthalpy of reaction is calculated when the chemical equilibrium is known. When the silicon arc was first modelled [78Eri], the ‘Reactor’ module of ChemSage was the only available tool. It allowed for the calculation of multistage, linear, cocurrent or countercurrent reactors, while the exchange of matter was described through the use of distribution coefficients for each stage. The modelling paradigm commonly employed by modern process simulation tools, assembling a network of unit operations interconnected by material streams, is quite different. To verify that both paradigms lead to the same results, the silicon arc furnace was remodelled with SimuSage [007Pet3], and the results were found to be identical. The solution technique employed is one of internal iteration in two cycles which treat stages 1 and 2 (leftmost in Fig. II.4.1) as one closed loop while

354

Condensed stream in

Phase splitter EQ

All

(∆H1 = 0)

Gas stream out

Condensed stream

The SGTE casebook

Gas stream

Gas stream Equilibrium reactor EQ (∆H2 = 0)

All

EQ

EQ

(∆H3 = 0)

(∆H4 > 0)

Gas stream

III.4.1 Sequence of equilibrium (EQ) stages for a countercurrent aggregate.

Condensed stream out

Simulation of dynamic and steady-state processes

355

stages 3 and 4 (rightmost in Fig. II.4.1) are combined into a second closed loop. It can be shown using graph theory [92Sed] that this is an optimum split. These two closed loops are, as mentioned before, controlled via an auxiliary unit operation which provides the necessary functions for an iterative handling of flowsheet segments. A process model consisting of only temperature-regulated stages is primarily suitable for processes having a known temperature profile. The reaction temperature is then a process parameter (e.g. fixed from the outside), and much less computing time will generally be required to find the equilibrium composition of a stage and the steady-state conditions than for an enthalpyregulated model. However, convergence cannot be guaranteed in this case. Divergent results can, for example, be found if stage temperatures are chosen such that all condensed phases entering a stage are transformed into gas species. The amounts of these condensed phases will then accumulate for each iteration cycle.

III.4.4

The control of material flows

As indicated above, in the standard case all phases produced in a stage move to the adjacent stages and simple phase splitters are sufficient (see also Fig. III.4.1). For the simulation of incomplete reactions, i.e. kinetic inhibitions, it is assumed that the gaseous and condensed intermediary reaction products leaving one stage may bypass the adjacent stages without being cooled or heated and can be distributed over several stages before reacting. In the case of the silicon arc furnace, split factors in the gas stream have been set such that 80% of the gas leaving a specific stage reacts in the next stage, 15% bypasses one stage before reacting and 5% bypasses two stages before reacting. In general, different distribution coefficients may be used for each of the possible phases, i.e. gases, liquids or solids. The implication for the top stage is that not only products formed in it leave the reactor but also those which might bypass the end stage according to their distribution coefficients. Figure III.4.2 gives a summary of the use of distribution coefficients. These need to be retranslated into split factors for use in the calculation. Figure III.4.3 shows the application of amount splitters for the gases leaving stage 4. For a countercurrent reactor, the use of chemical and thermal equilibria in each stage combined with distribution coefficients as described above will give the compositions and temperatures of the gaseous and condensed flows at each stage boundary. These flows are never in equilibrium with each other and the departure from equilibrium is proportional to the thickness of the stages. For a reactor with one stage and no flow distribution, only the exit flows will have a composition and temperature corresponding to the chemical and thermal equilibria inside the stage. If, on the other hand, the number of

356

The SGTE casebook Stage

n–3 n–2 Condensed phase

n–1

Upper 100%

80% 15% 5%

Stage boundary

n Condensed phase

Lower

n+1

100%

80%

n+2

15%

n+3

15% 5% 5%

5%

III.4.2 A sketch showing the flows within the nth segment of a counter-current reactor (the other segments are indicated by straight lines). The distribution factors are 80%, 15% and 5% for the gaseous species whereas the condensed phases are assumed to be distributed in full to the immediately adjacent stage. The arrows reach the segments where the species react.

Amount splitter

Amount splitter Gas (from EQ stage 4)

20%

25% total from 4 to 1 = 20% × 25% = 5%

75% total from 4 to 2 = 20% × 75% = 15%

80%

Stage 1

Stage 2

Stage 3

III.4.3 The application of amount splitters for the gases leaving stage 4.

stages goes to infinity, thermochemical equilibrium will be attained at every point of the reactor. Kinetic limitations in mass and energy transfer can thus be simulated to some extent by the choice of the number of stages and the distribution coefficients. Nothing can be said about the geometrical size of the stages as the kinetics of the reactions occurring and the heat transfer are unknown. The size should, however, decrease with increasing reaction temperature as the reaction rates and the heat transfer increase with increasing temperature.

Simulation of dynamic and steady-state processes

III.4.5

357

Conclusions

Based on exact thermochemical calculations and the possibility of treating kinetic constraints by a number of empirical parameters, a modelling concept has been devised to simulate multistage chemical processes. Applying this modelling approach can be conceptually divided into the following steps. 1 2 3

Preparing a set of thermochemical data (the Gibbs energies) of all the phases possible in the process. Conceptually dividing the entire process space into a limited number of equilibrium reactors and other unit operations. Connecting the unit operations with streams, taking in particular bypass streams into account in order to model kinetic inhibitions in the process based on experimental data.

It is always advisable to carry out initial individual equilibrium calculations for the various stages (equilibrium units) of a process model with an interactive program for the calculation of complex equilibria. This will enable the user to initiate the possible iteration process with proper values of the reaction products or the reaction temperatures for the enthalpy-regulated stages.

III.4.6 78Eri 92Sed 96Eri

001Tra

004Mod

004Brü

005Gin

007Pet1

References G. ERIKSSON and T. JOHANSSON: Scand. J. Metall. 7, 1978, 264–270. R. SEDGEWICK: Algorithms, Addison-Wesley, Reading, Massachusetts, 1992. G. E RIKSSON , K. H ACK and S. P ETERSEN : ‘ChemApp – a programmable thermodynamic calculation interface’, in Werkstoffwoche ’96, Symp. 8: Simulation, Modellierung, Informationssysteme (Ed. J. Hirsch), DGM Informationsgesellschaft, Frankfurt, 1997, pp. 47–51. A. TRAEBERT: ‘Berichte aus der Verfahrenstechnik, Methodik zur Modellierung von Hochtemperaturprozessen’, Dissertation RWTH Aachen, Shaker Verlag, Aachen, 2001. M. MODIGELL, A. TRAEBERT, P. MONHEIM and K. HACK: ‘A modelling technique for non-equilibrium metallurgical processes applied to the LD converter’, in Chemical Thermodynamics for Industry (Ed. T.M. Letcher), Royal Society of Chemistry, London, 2004. P. BRÜGGEMANN: Prozess-Simulation mit SimuSage, Großer Beleg, Institut für Physikalische Chemie, Technische universität Bergakademie Freiberg, Freiberg, 2004. T. GINSBERG, D. LIEBIG, M. MODIGELL, K. HACK and S. YOUSIF: ‘Simulation of a cement plant using thermochemical and flow simulation tools’, in Proc. Eur. Symp. Computer Aided Process Engineering – 15 (Eds L. Puigjaner and A. Espuña), Elsevier, Amsterdam, 2005, pp. 361–366. S. PETERSEN, K. HACK, P. MONHEIM and U. PICKARTZ: ‘SimuSage – the component library for rapid process modeling and its applications’, Int. J. Mater. Res. 98(10), 2007 pp. 946–953.

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007Pet2 S. PETERSEN and K. HACK: ‘The thermochemistry library ChemApp and its applications’, Int. J. Mater. Res. 98(10), 2007 pp. 935–945. 007Pet3 S. PETERSEN and K. HACK: ‘SimuSage – the component library for rapid process modeling, in Proc. Eur. Metallurgical Conf. 2007, Vol. 4, Düsseldorf, Germany, 11–14 June 2007, volume 4, GDMB Medienverlag, Clausthal-Zellerfeld, 2007, pp. 1849–1862.

III.5 Setting kinetic controls for complex equilibrium calculations P E R T T I K O U K K A R I, R I S T O P A J A R R E and K L A U S H A C K

III.5.1

Introduction

Many methods exist which cover only the kinetic aspects of a stoichiometric reaction and how it proceeds in time. Only a few attempts have been made so far to link equilibrium aspects of multicomponent systems with kinetic inhibitions or even single reaction rates (e.g. the paper by Korousic et al. [95Kor] on the NH3 kinetics in nitriding gases or the introduction of the image component approach by Koukkari et al. [97Kou]). None of these has led to a generally applicable link between the terms used in reaction kinetic equations and the Gibbs energy minimisation method available for general equilibrium calculations. Mostly dedicated solutions for special cases have been established. The image component method, although practical, is not fully consistent thermodynamically when used for solution phases. This subject will be more closely discussed below. In the present chapter a method will be described which combines multicomponent multiphase equilibrium thermodynamics with reaction kinetics. This method will be applied to some example cases in later chapters in order to demonstrate its ease of use.

III.5.2

The basic concept

In his publication ‘On the equilibrium of heterogeneous substances’ J.W. Gibbs [878Gib] refers in many places to the components of a system, most importantly in conjunction with the chapter in which the phase rule is derived. However, the phase rule is derived for the case of equilibrium ‘which does not depend upon possible resistances to change’, i.e. which is not inhibited in any way by kinetic restrictions. On the other hand, Gibbs states: ‘ ... in respect to a mixture of vapour of water and free hydrogen and oxygen (at ordinary temperatures) we may not write 9Saq = 1SH + 8SO

(III.5.1) 359

360

The SGTE casebook

[note the use of mass units] but water is to be treated as an independent substance, and no necessary relation will subsist between the potential for water and the potentials for hydrogen and oxygen’. In other words, normally the equation H 2 O = H 2 + 12 O 2

(III.5.2)

holds together with the equation

µ H 2 O = µ H 2 + 12 µ O 2

(III.5.3)

but in the case of kinetic inhibitions they do not; H2O, H2 and O2 can coexist in a metastable state in which H2 does not react with O2 to give additional H2O, and the chemical potential of water is an independent quantity. In terms of complex equilibrium calculations this can easily be expressed by two stoichiometric matrices the columns of which contain ‘the independent substances’ (see above) of the system and the rows of which contain the chemical species in the system. The matrix in Table III.5.1(a) is for a system Table III.5.1 Stoichiometric matrices for the system H2–O2–H2O (a) free equilibrium. (b) Passive resistance according to Gibbs. (c) Passive resistance with liquid H2O phase included. (d) Matrix with species H2O*, gas phase only. (e) Matrix with kinetic constraint. (f) Matrix with kinetic constraint and artificial stoichiometric condensed species R(H2O) (a)

H 2O H2 O2

(d) H

O

2 2 0

1 0 2

H2 O H2O* H2 O2

(b)

H 2O H2 O2

O

H2 O

2 0 2 0

1 0 0 2

0 1 0 0

H

O

H2O*

2 2 0

1 0 2

1 0 0

H

O

H2O*

2 2 0 2 0

1 0 2 1 0

1 0 0 1 1

(e) H

O

H 2O

0 2 0

0 0 2

1 0 0

(c)

H 2O H2 O2 H2O(l)

H

H2 O H2 O2 (f)

H

O

H 2O

0 2 0 0

0 0 2 0

1 0 0 1

H2O(g) H2(g) O2(g) H2O(l) R(H2O)

Setting kinetic controls for complex equilibrium calculations

361

in free equilibrium and the matrix in Table III.5.1(b) for a system where a ‘passive resistance’ inhibits Reaction (III.5.2) given above. It should be noted that the number of columns in the matrices relate to the components that have to be counted in the phase rule, and not the number of rows. The matrix of Table III.5.1(c) includes a simple example of how these resistances can be used to calculate unstable states with a physical meaning. If water is introduced to the system as H2O(l), this species is allowed to transfer to the respective gaseous constituent, and thus, for example, the vapour pressure of water in a mixture of O2, H2 and H2O can be correctly calculated with this method (Fig. III.5.1). The matrices in Tables III.5.1(b) and III.5.1(c) are for a system where Reaction (III.5.2) is fully inhibited. What if a partial reaction was to be permitted? By combining the matrices in Tables III.5.1(a) and III.5.1(b), one would end up with a new matrix that contains two different variants of H2O in the gas phase, one (H2O) which can form from H2 and O2 and the other (H2O*), which would only exist if it was added to the system as a substance of it is own (Table III.5.1(d)). This matrix containing a so-called image constituent H2O* for water and also the free reactive water H2O can be used to handle the case in which the initial water vapour H2O* is mixed with H2 and O2 (just as in Gibbs case), but the reaction can still take place. The image component method offers a practical means for simulating chemical states in systems, which have one important kinetically controlled reaction. By algorithmically changing at the point of the system input some of the amount of the inert image species to the corresponding reactive form between two simulation steps, the degree of advancement of the critical Partial pressures of H2/O2/H2O mixture

0.7 0.6

O2(g)

P (bar)

0.5 0.4

H2(g)

0.3 0.2

H2O*(g)

0.1 0 20

40

60

80

T (°C)

III.5.1 Partial pressures of oxygen, hydrogen and water between 25 and 70 °C, for inputs 1 kg of H2O, 3 mol of O2 and 2 mol of H2.

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The SGTE casebook

reaction can be controlled while otherwise the composition and energy balances of the system can be calculated using the standard Gibbs energy minimisation. A practical example (TiCl4 burner) that has been successfully modelled using the image component method is presented in Chapter IV.9. However, with the matrix of Table III.5.1(d), there is a problem now for all cases in which significant amounts of both water (H2O) and its image component (H2O*) occur in the gas phase. The entropy of mixing will not be calculated correctly. Since the two different kinds of water used in the calculation have no distinct physical meaning, only the total mole fraction x H 2 O of water should be used to calculate the entropy of mixing. However, normally in a complex equilibrium program, each species in a mixture phase is taken separately to contribute to the entropy of mixing and thus two contributions will come from H2O and H2O* in the case above. It is obvious that with x H 2 O = x H′ 2 O + x H′′ 2 O

(III.5.4)

one obtains x H 2 O ln x H 2 O ≠ x H′ 2 O ln x H′ 2 O + x H′′ 2 O ln x H′′ 2 O

(III.5.5)

The entropy of mixing is (in ideal systems)

∆SMIX = – R∑ xi ln xi

(III.5.6)

It is obvious that the presence of an image component in a reactive system impedes the presence of the respective reactive substance in the mixture phase for a thermodynamically consistent calculation. The entropy of mixing is not affected if the input amount of the reactive substance becomes entirely consumed during each calculation step. (It is also not affected if the image component can be treated as a stoichiometric condensed pure substance, as in the case of rutile formation from anatase; see Chapter IV.9.) Also the introduction of an image component fixes the direction of the kinetically controlled reaction. For example, in the system above, the choice to use an image component for H2O (instead of H2 or O2) means that the modelling can be done only for compositions where the amount of H2O is equal to or greater than that at equilibrium (which is realistic only for systems with temperatures above 2000 K). Furthermore, the simultaneous modelling of several kinetically controlled reactions can in general not be executed satisfactorily. Nevertheless, it is obvious that, if kinetic inhibitions are to be taken into account, a modification of the number of columns in the stoichiometric matrix, i.e. the addition of further components in the Gibbsian sense, is a feasible method. This has been shown in detail for example by Smith and Missen [91Smi] and Alberty [91Alb]. The method has also recently been

Setting kinetic controls for complex equilibrium calculations

363

demonstrated to be suitable for modelling the evolution of complex reactive systems [91Paj]. In the present case, we introduce a new system component H2O* with zero mass and Gibbs energy and rewrite the matrix as in Table III.5.1(e). From the list of species (i.e. the number of lines of the matrix) it is obvious that no problems with the entropy of mixing will occur, since only three species will contribute just as in the free equilibrium. On the other hand, with the introduction of the system component H2O* we have a means of interfering with the mass balances of the species H2O since the presence of the H2O* column conserves the input amount of water during the Gibbsian calculation. If there is no other species than H2O which contains H2O*, then H2O cannot react at all; it cannot dissociate nor can it be formed by reaction between H2 and O2. If, however, the matrix is extended by addition of an artificial stoichiometric condensed substance R(H2O) with zero Gibbs energy to the list of species, then the mass balance of the reaction can be satisfied again: H 2 + 12 O 2 + R(H 2 O) = H 2 O

(III.5.7)

The new matrix that is required for this system is given in Table III.5.1(f). If R(H2O) is not allowed to form in the calculated pseudo-equilibrium state, i.e. it is set dormant in the calculation, the advancement of the reaction is controlled by the input amount of R(H2O). The conversion from H2 and O2 to H2O can then be algorithmically controlled by the value of this input amount. The inclusion of liquid water (subject to same constraint R(H2O)) allows the calculation of the independent vapour–liquid equilibrium of water with the same matrix structure. The key element of the new matrix structure is the massless new component, which can be applied to set appropriate physical constraints to the multicomponent calculation. In this example, the massless component together with the additional ‘pseudo-constituent’ is used to oppress the metastable or reaction-controlled state between oxygen, hydrogen and water. It has been recently shown that additional constraints can be initialised to include other physical phenomena such as surface energies and electrochemical effects into the multiphase domain [006Kou1, 006Kou2]. It should be noted that adding R(H2O) does not change the Gibbs energy balance because it has zero Gibbs energy. In fact, none of the extensive property balances is changed because the enthalpy, entropy and heat capacity of R(H2O) are implicitly zero too. It should also be noted that, since we are not changing the number of species present in calculated (pseudo)equilibrium states, there is no difficulty in extending the method to situations where there is more than one kinetically controlled reaction. Use of R(H2O) also offers a simple way of calculating the true equilibrium state while using the restricted matrix. Based on Equation (III.5.6), we have in any calculated pseudoequilibrium state the equality

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The SGTE casebook

µ H 2 + 12 µ O 2 + µ R(H 2 O) = µ H 2 O

(III.5.8)

Since in the true equilibrium state we have µ H 2 + 12 µ O 2 = µ H 2 O

(III.5.9)

a necessary and sufficient condition for the system including R(H2O) to be in equilibrium is µ R(H 2 O) = 0, µ °R(H 2 O) = 0 ⇔ a R(H 2 O) = 1

(III.5.10)

This condition can be used to equilibrate the restricted chemical system as is shown in Table III.5.2 and Table III.5.3.

III.5.3

Simple equilibrium calculations

The equilibrium states shown in Table III.5.2 and Table III.5.3 have been calculated for the simple system H 2 –H 2 O–O 2 using an augmented stoichiometric matrix as discussed above. Table III.5.2 shows the free

Table III.5.2 Unrestricted equilibrium in the system H2–H2O–O2: output data T = 1000.00 K P = 1.00 bar V = 236.971 dm3 STREAM CONSTITUENTS H2(gas_ideal) O2(gas_ideal) *R(H2O)

AMOUNT/mol 2.3000 1.7000 2.3000

EQUIL AMOUNT MOLE FRACTION FUGACITY mol bar 2.3000 0.80702 0.80702 0.5500 0.19298 0.19298 0.0000 0.00000 0.00000 2.8500 1.00000 1.00000 mol ACTIVITY R(H2O) 0.0000 1.00000 *************************************************************** Cp_EQUIL H_EQUIL S_EQUIL G_EQUIL V_EQUIL J.K-1 J J.K-1 J dm3 *************************************************************** 114.068 -483932 680.887 -1164820 236.971 PHASE: gas_ideal H 2O O2 H2 TOTAL:

Setting kinetic controls for complex equilibrium calculations

365

Table III.5.3 Restricted equilibrium (50%) in the system H2–H2O–O2: output data T = 1000.00 K P = 1.00 bar V = 284.781 dm3 STREAM CONSTITUENTS AMOUNT/mol H2(gas_ideal) 2.3000 O2(gas_ideal) 1.7000 R(H2O) 1.1500 EQUIL AMOUNT MOLE FRACTION FUGACITY mol bar 1.1500 0.33577 0.33577 1.1500 0.33577 0.33577 1.1250 0.32847 0.32847 3.4250 1.00000 1.00000 mol ACTIVITY R(H2O) 0.0000 0.00000 *************************************************************** Cp_EQUIL H_EQUIL S_EQUIL G_EQUIL V_EQUIL J.K-1 J J.K-1 J dm3 *************************************************************** 121.398 -198886 764.105 -962991 284.781 PHASE: gas_ideal H 2O H2 O2 TOTAL:

equilibrium which is calculated using T = 1000 K, and P = 1 bar together with arbitrary overall amounts of H2 (2.3 mol) and O2 (1.7 mol) as well as the fixed activity for R(H2O) (a = 1). Note that the corresponding input amount of R(H2O) is calculated by the Gibbs energy minimisation. The reaction proceeds to completion, i.e. essentially the entire amount of free H2 is consumed and the excess amount of O2 together with the amount of H2O formed in the reaction establishes the equilibrium composition of the system. In Fig. III.5.2, the matrix in Table III.5.1(f) has been used for two corresponding calculations. Table III.5.3 shows a restricted equilibrium for the same values of T, P and overall amounts of H2 and O2 but using a fixed input amount of R(H2O), here 1.15. This forces the formation reaction of water to advance to only half the amount of H2O of the unrestricted equilibrium. Entering an amount of zero for R(H2O) would of course inhibit the reaction completely. From the above the actions necessary for the calculation of inhibited equilibria are obvious. 1

Introduce an appropriate addition to the stoichiometric matrix of the system.

The SGTE casebook

Amount (mol)

2.5

0.3 0.25

2

0.2 1.5 0.15 1

Total H2O

p/bar

366

H2O(aq)

p(H2O(g))

0.1

0.5

0.05 0

0 10

30

50

70

T (°C) (a) 0.3

3.5

0.25

2.5

0.2

2

0.15

1.5

0.1

1

Total H2O

p/bar

Amount (mol)

3

H2O(aq)

p(H2O(g))

0.05

0.5 0 10

30

50

0 70

T (°C) (b)

III.5.2 Vapour pressures of water in the O2–H2–H2O system: (a) fully restricted; (b) partially restricted.

2 3

4

Carry out an unrestricted equilibrium calculation by using a = 1 for the ‘new species’. Use any value between 0 and the value calculated in step 2 for the input amount of the ‘new species’ in order to fix the desired or known degree of inhibition. If explicit reaction kinetic data are available, use the kinetic equation(s) to derive the appropriate sequence of input amounts as a proper function of time. (See the example for TiO2 in Chapter IV.9.)

In Fig. III.5.2(a), the thermostatic O2–H2–H2O system, where no reaction is assumed between the oxygen and hydrogen, allows for calculation of the VL equilibrium in terms of temperature. In Fig. III.5.2(b), with each increasing temperature step, a constant increment of water is formed from O2 and H2. The vapour pressures of water are the same in the two systems. Instead, the effect of forming water is visible in the other two curves. The amount of aqueous water increases until the vaporisation at each temperature step exceeds the reaction increment.

Setting kinetic controls for complex equilibrium calculations

III.5.4

367

References

878Gib J.W. GIBBS: ‘On the equilibrium of heterogeneous substances’, Trans. Conn. Acad. 3, 1978, 176. 91Alb R.A. ALBERTY: J. Phys. Chem. 95, 1991, 413. 91Paj R. PAJARRE: Master’s Thesis, Helsinki University of Technology, Helsinki, 1991. 91Smi W.R. SMITH and R.W. MISSEN: Chemical Reaction Equilibrium Analysis: Theory and Algorithms, Robert E. Krieger, Malabar, Florida, 1991. 95Kor B. KOROUSIC and M. STUPNISEK: Steel Res. 66, 1995, 349. 97Kou P. KOUKKARI, I. LAUKKANEN and S. LIUKKONEN: Fluid Phase Equilib. 136, 1997, 345. 006Kou1 P. KOUKKARI and R. PAJARRE: Calphad 30, 2006, 18–26. 006Kou2 P. KOUKKARI and R. PAJARRE: Computers Chem. Eng. 30, 2006, 1189–1196.

368

The SGTE casebook

Part IV Process modelling – application cases

369

370

IV.1 Calculation of solidification paths for multicomponent systems B O S U N D M A N and I B R A H I M A N S A R A

IV.1.1

Introduction: description of the phase diagram

Figure IV.1.1 represents an isothermal section at 580 °C. There are no ternary intermetallic compounds in this system. The Al-rich corner is shown in Fig. IV.1.2(a) on a larger scale. The dotted lines correspond to the monovariant lines. A maximum temperature equal to 594.2 °C occurs on the line representing the composition of the liquid phase in equilibrium with the Al solid solution and Mg2Si. The eutectic temperature corresponding to the four-phase equilibrium between Si, (Al), Mg2Si and the liquid phase is 553 °C.

Mg

Al

Si

IV.1.1 Isothermal section of the Al–Mg–Si system at 580 °C.

371

372

The SGTE casebook

IV.1.2

Solidification paths

Figure IV.1.2(b) shows the solidification paths for two alloys for which the content of Mg and Si differ very little. The intersection of these lines with the monovariant line are on either sides of the maximum. Hence, the final alloy will present different microstructures, a eutectic structure containing silicon for the alloy (A) and Al3Mg2 for the alloy (B). Liquid (Al) + liquid

14

(Al) + Mg2Si

12

Mg (wt%)

10

Liquid + Mg2Si

594.2 °C 8 6

553 °C A

4 Liquid + (Al)

Liquid

2 Liquid + Si 0

0

4

14

8 12 Si (wt%) (a)

16

20

(A) 0.75 wt% Mg, 0.25 wt% Si (B) 0.80 wt% Mg, 0.20 wt% Si

12 Mg2Si

Mg (wt%)

10 594.2 °C 8 6

(B) (A)

553 °C

(A)

4

0 0 Al

(Si)

(Al)

2

4

8

12 Si (wt%) (b)

16

20

IV.1.2 (a) Isothermal section of the Al–Mg–Si system at 853 K in the Al-rich region. (b) Projection of the liquidus surface and solidification paths for two alloys.

Calculation of solidification paths for multicomponent systems

373

Figure IV.1.3(a) represents the fraction of the remaining liquid versus temperature for the alloy (B). The break point corresponds to the precipitation of the Mg2Si. In the Gulliver–Scheil treatment, the partition coefficient is defined as the ratio of the composition of an element in the liquid and solid phases k i = X il / X is and is often assumed to be constant, even in multicomponent systems. This

750 0.75 wt% Mg, 0.25 wt% Si

Temperature (°C)

700

650 Liquid + (Al) 600

← Liquid + (Al) + Mg2Si

0.0

0.2

0.4 0.6 0.8 Liquid phase fraction (a)

1.0

0.5 0.75 wt% Mg, 0.25 wt% Si

Partition coefficient

0.4

0.3

0.2

kMg

0.1

kSi

0.0 500

550

600 650 Temperature (°C) (b)

700

IV.1.3 (a) Fraction of the liquid phase during solidification. (b) Partition coefficients for Mg and Si during solidification.

374

The SGTE casebook 750 0.3 wt% Cu, 0.75 wt% Mg, 0.25 wt% Si

Temperature (°C)

700

650 Liquid + (Al) 600

550

Liquid + (Al) + Mg2Si

Liquid + (Al) + Mg2Si = Al2Cu 500 0.0

0.2

0.4 0.6 0.8 Liquid phase fraction

1.0

IV.1.4 Solidification path of an alloy containing 0.3 wt% Cu.

quantity is difficult to obtain experimentally because information on the phase composition is often lacking. As the phase diagram is calculated thermodynamically, a consequence is that the partition coefficients of all the elements in a multicomponent system can be derived. Figure IV.1.3(b) shows the evolution of the partition coefficients of Mg and Si versus temperature. If kSi is constant, kMg varies slightly with temperature. Note that the partition coefficients change when the precipitation of Mg2Si starts. Similar characteristics are observed in a quaternary alloy where Cu is added, as shown in Fig. IV.1.4. The thermodynamic data used in these calculations are those existing in the SGTE Database [86Ans]. The examples shown in this contribution have all been calculated using Thermo-Calc, a computer software for phase diagram and complex equilibrium calculations [85Sun].

IV.1.3 85Sun 86Ans

References B. SUNDMAN, B. JANSSON and J.O. ANDERSON: Calphad 9(2), 1985, 153–190. I. ANSARA and B.SUNDMAN: ‘Computer Handling and Dissemination of Data’, in Proc. 10th Codata Conf. (Ed. P.S. Glaeser), Ottawa, Canada, July 1986, Elsevier, Amsterdam, 1986.

IV.2 Computational phase studies in commercial aluminium and magnesium alloys H A N S - J Ü R G E N S E I F E R T, E R I K M. M U E L L E R, P I N G L I A N G, H A N S L. L U K A S, F R I T Z A L D I N G E R, S. G. F R I E S, M I R E I L L E G. H A R M E L I N, F R A N Ç O I S E F A U D O T and TAMARA JANTZEN

IV.2.1

Introduction

The quaternary system Al–Cu–Mg–Zn is one of the key systems of high-strength aluminium alloys (e.g. the 7000 series), which are extensively used in aircraft construction and in other high-strength applications. Additionally, commercial magnesium alloys of the AZ and CZ series are based on this system. For successful development of new materials and improvement of existing Al and Mg alloys, knowledge of phase diagrams and thermodynamic data is crucial. Thermodynamic computer calculations applying the Calculation of Phase Diagrams (CALPHAD) method [98Sau, 07Luk] were combined with selected experimental investigations to establish an analytical thermodynamic data set for the Al–Cu–Mg–Zn system. Two software packages, BINGSS/BINFKT [77Luk, 92Luk] and Thermo-Calc [85Sun, 93Jan], were used in this work. The development of the data set included the thermodynamic optimisation of the ternary systems Al–Mg–Zn [98Lia1], Cu–Mg–Zn [98Lia2] and Al–Cu–Mg [98Buh, 98Lia3] and of the Al-rich corner of the ternary system Al–Cu–Zn [98Hur]. A thermodynamic description for the Al–Cu–Zn system has also been reported [98Lia4, 97Che]. The main achievements of the thermodynamic assessment of the quaternary Al–Cu–Mg–Zn system have been summarised by Fries et al. [98Fri]. This work presents the way of modelling the solution phases of the ternary systems and their extension to describe the quaternary system. Additionally, the computer simulation of a Scheil–Gulliver solidification for the magnesium alloy AZ91 is presented to describe the microstructure of the casting alloy.

IV.2.2

Thermodynamic calculations for ternary subsystems

Figure IV.2.1 shows the calculated isothermal sections at a temperature of 673 K (400 °C) for the ternary subsystems Al–Mg–Zn, Cu–Mg–Zn, Al–Cu–Mg and Al–Cu–Zn. The ternary homogeneity ranges of solutions in the solid elements, the binary phases and the liquid phase are indicated. Additionally, the ternary 375

376

The SGTE casebook Zn 1.0 Liquid

0.9

Mo le

fra c

tio n

of Zn

0.8 0.7

C14

0.6 C35

0.5

0.4

0.3 0.2 0.1 Cu

(Cu)

Mg2Cu

C15

(Cu) C35 C14

V Bcc

Mg Mg

γ τ

Q

5

Liquid T ε β

T

C14

Hcp A3 Liquid

(Al)

(Al)

Zn

Al

Liquid Zn

IV.2.1 Calculated isothermal sections at 673 K (400 °C) for the Al–Mg–Zn, Cu–Mg–Zn, Al–Cu–Mg and Al–Cu–Zn systems.

stoichiometric and solution phases are shown. An overview on the thermodynamic modelling of the solution phases is given below.

IV.2.2.1

Al–Mg–Zn system

Experimental investigations by electron probe microanalysis were specifically carried out on ternary Al–Mg–Zn alloys [98Lia1] to provide missing data for the ternary solubilities of the Al–Mg and Mg–Zn phases as well as to improve knowledge of the extensions of the homogeneity ranges of the ternary T and φ phases. A thermodynamic description for the Al–Mg–Zn system was obtained by taking into account these experimental data together with the phase diagram, thermodynamic and crystallographic information from the literature [98Lia1]. The binary intermetallic phases were modelled to consider the ternary solubilities. The ternary T phase was modelled according to its cubic crystal structure [57Ber] as (Mg)26 (Mg, Al)6 (Al,Zn,Mg)48 (Al)1 in the compound energy formalism. The φ phase (space group Pbc21 or Pbcm; stable up to 663 K (390 °C); does not

Computational phase studies in commercial Al and Mg alloys

377

appear in Fig. IV.2.1) was described by the sublattice formula Mg6(Al, Zn)5 according to its homogeneity range [97Don]. The thermodynamic description of the Al–Mg–Zn system [98Lia2] was used to simulate the solidification of the magnesium alloy AZ91. The analysis of alloy solidification is crucial to understand the microstructure development of casting light alloys. The magnesium alloy AZ91 consists of 9 wt% Al (8.2 mol% Al) and 1 wt% Zn (0.38 mol% Zn) and it can be expected from calculation that (Mg) solid solution is the primary crystallisation product during the cooling of a liquid with this composition [98Lia1]. The solidus and solvus in the Mg-rich corner are projected in Fig IV.2.2 with the composition of AZ91 indicated in mole percent. A vertical section through the composition of AZ91 alloy and pure magnesium is shown in Fig IV.2.3. The composition line of this vertical section is shown as a dashed line in Fig IV.2.2. The dashed line in Fig IV.2.3 indicates the composition of the AZ91 alloy. On cooling a melt of the AZ91 alloy, (Mg) solid solution precipitates as primary phase when crossing the liquidus at a temperature of 600 °C. If solidification continues to follow equilibrium, further cooling results in crossing of the solidus at a temperature of 456 °C, where the liquid phase disappears and the alloy becomes single phase (Mg). At a temperature of 377 °C the solvus is reached and the γ phase (Mg17Al12) precipitates from the magnesium solid solution. Quantitative information on weight fractions of the different phases at specific temperatures can be derived from the calculated phase fraction diagram for equilibrium solidification, as shown in Fig IV.2.4(a). From these calculations it is expected that the γ phase precipitates directly from the (Mg) primary crystals at temperatures lower than 377 °C.

25

MgZn 40 0

20

45

15

50

Temperatures are given in °C

τ

Solvus Solidus Univariant lines

φ 0 35

0 0 30

0 25 0

10 550

γ

0

20

Mole fraction of Zn (× 10–3)

30

15 0

5

60

AZ91

0

0 0 Mg

0.02

0.04

0.06

0.08

0.10

0.12

Mole fraction of Al

IV.2.2 Solvus and solidus of (Mg) solid solutions of the Al–Mg–Zn systems. The composition of the magnesium alloy AZ91 is indicated as well as the concentration line shown in Fig. IV.2.3.

378

The SGTE casebook 700

AZ91

Liquid

Temperature (°C)

600

500 Liquid + (Mg) 400

(Mg)

300 (Mg) + γ 200 100 0 Mg

0.1

0.2 0.3 Mole fraction of Al

0.4

IV.2.3 Calculated vertical section along the concentration line shown in Fig. IV.2.2; the AZ91 magnesium alloy composition is indicated.

However, casting alloys show non-equilibrium microstructures resulting from restricted diffusion mainly in the solid phases. The microstructures of as-cast AZ91 alloys consist of (Mg) solid solution plus interdendritic γ phase [96Nei]. To simulate such casting microstructures the Scheil–Gulliver model can be used. Calculation modules for this model are implemented in modern software. It is simulated that no diffusion occurs in the solid phases but infinitely fast diffusion in the liquid phase. The resulting calculations are illustrated in Fig. IV.2.4(b). This non-equilibrium phase fraction diagram shows that the liquid phase can be found in AZ91 casting alloys down to a temperature of 338 °C which is 118 K lower than the equilibrium solidus temperature (T = 456 °C (Fig. IV.2.4(a))). As in the equilibrium case, (Mg) is the primary crystal but more magnesium rich. Owing to the non-equilibrium conditions the liquid becomes enriched in aluminium and zinc, and at a temperature of 430 °C the γ phase precipitates from the liquid phase and segregates at the grain boundaries and interdendritically. This temperature corresponds to the temperature of the univariant equilibrium liquid+(Mg)+ γ (as shown in the vertical section (Fig. IV.2.3)). Further cooling results in the crystallisation of small amounts of Φ phase at a temperature of 366 °C. This corresponds to the transition reaction liquid L + γ = (Mg) + Φ at 366 °C (see Table 8 in the book by Saunders and Miodownik [98Sau]; for further details see the paper by Seifert [99Sei]). Using this calculation the microstructure of the magnesium casting alloy AZ91 can be explained, containing interdendritic γ phase and a lamellar eutectic microstructure (owing to the simultaneous precipitation of (Mg) and γ phase along the eutectic monovariant equilibrium). Additionally, the chemical composition of the individual phases can be calculated. For example, Fig. IV.2.5(a) and Fig. IV.2.5(b) show the variations in composition,

Computational phase studies in commercial Al and Mg alloys

379

600

Temperature (°C)

550 500

Liquid

(Mg)

450 (Mg) 400

350

γ

(Mg)

300 0

0.2

0.4 0.6 Weight fraction (a)

0.8

1.0

600

550

500

Liquid

(Mg)

450 (Mg) + γ 400 (Mg) + γ + Φ

350

300 0

0.2

0.4 0.6 Weight fraction (b)

0.8

1.0

IV.2.4 Phase fraction diagrams for the AZ91 magnesium alloy; (a) equilibrium solidification; (b) solidified according to the Scheil– Gulliver model.

as functions of temperature, for the liquid phase and for the precipitating magnesium solid solution respectively. The calculated latent heat evolution during the Scheil solidification is shown in Fig. IV.2.6. The reaction paths related to the equilibrium and Scheil solidifications respectively mark limiting cases. In reality, some diffusion does occur in the solid phases, and diffusion in the liquid phase may be incomplete. Furthermore, microstructural development of AZ alloys is influenced by additional alloying

380

The SGTE casebook 0.9

Weight fraction of elements

0.8 0.7

Mg

0.6 0.5 0.4 Al

0.3 0.2 0.1 0 350

Zn 400

450 500 Temperature (°C) (a)

550

600

1.0

Weight fraction of elements

0.9 Mg

0.8 0.7 0.6 0.5 0.4 0.3 0.2

Al 0.1 0 350

Zn 400

450 500 Temperature (°C) (b)

550

600

IV.2.5 Variation in composition for (a) the liquid phase and (b) the precipitating magnesium solid solution during Scheil–Gulliver solidification of AZ91 magnesium alloy.

elements such as manganese or impurities (e.g. iron, copper or nickel). More work taking into account these effects is in progress. However, comparison between the present Scheil calculations and experimental results already indicates very good agreement. The development of continuous precipitation of the γ phase observed by

Computational phase studies in commercial Al and Mg alloys

381

Latent heat evolution/(J g–1)

0

–50

–100

(Mg)

–150

–200

(Mg) + γ + φ

–250

–300 350

(Mg) + γ

400

450 500 Temperature (°C)

550

600

IV.2.6 The calculated latent heat evolution during the Scheil–Gulliver solidification of AZ91 magnesium alloy.

transmission electron microscopy upon ageing the AZ91 cast alloy between 70 and 300 °C [00Cel] is explained by referring to the phase diagram shown in Fig. IV.2.3 and the description which is given above.

IV.2.2.2

Cu–Mg–Zn system

Experimental investigations by energy-dispersive X-ray analysis were carried out on ternary Cu–Mg–Zn alloys to provide missing data on the copper solubilities in the Mg–Zn phases [98Lia2]. Ternary solubilities are described in the literature for only the Laves phases C15 (MgCu2), C14 (MgZn2) and C36 (Mg2CuZn3) along the quasibinary section MgCu2–MgZn2. These phases were modelled by Cu–Zn exchange, Mg(Cu1–xZnx). The weak tendency for antistructure atom formation (copper and zinc on the magnesium sublattice and magnesium on the Cu–Zn sublattice) was interpolated between the binary boundary systems. The Gibbs free energies of the three Laves phases were optimised along the quasibinary join MgCu2–MgZn2 using the published liquidus, solidus and enthalpy of mixing data. The binary intermetallic phases MgZn, Mg2Zn3 and Mg2Zn11 were modelled to have Cu–Zn exchange on one sublattice. The corresponding parameters were adjusted to reproduce the copper solubility measurements obtaned by three of three of the present authors and coworkers [98Lia2].

IV.2.2.3

Al–Cu–Mg system

The thermodynamic description of the Al–Cu–Mg system is mainly based on the work by Buhler et al. [98Buh] with some modifications regarding the description

382

The SGTE casebook

of the Al–Mg system according to Liang et al. [98Lia3]. The T phase in the Al– Cu–Mg system is isotypic with the T phase in the Al–Mg–Zn system, and these phases form continuous solid solutions in the quaternary Al–Cu–Mg–Zn system. Therefore, the same model was used for this phase in both ternary subsystems, (Mg)26(Mg, Al)6(Al, Zn, Cu, Mg)48(Al)1, and the binary Al–Mg end–members are the same in these descriptions [98Lia1, 98Buh]. The ternary phases Q (Al7Cu3Mg6), S (Al2CuMg) and V (Mg2Cu6 Al5) were modelled as stoichiometric phases. Experimental work for further refinement of the Al–Cu–Mg thermodynamic data set is investigated [98Fau, 99Fau].

IV.2.2.4

Al–Cu–Zn system

The thermodynamic description of the Al–Cu–Zn system is based on the work by Liang and Chang [98Lia4] and by Chen et al. [97Che]. The body-centred cubic (bcc) A2 phase shows a continuous series of solid solutions from Cu3 Al to Cu Zn. For its B2 (CsCl type) ordering at lower temperature the binary Cu– Zn description was taken from the work of Ansara [95Ans], and for Al–Cu and Al–Zn ordering the parameters were set to approximately zero. For the γ phase the description as a disordered solid solution by Liang and Chang [98Lia4] and by Chen et al. [97Che] was adopted as a first approach. The ternary τ phase in the Al–Cu–Zn system splits into two separate ranges of homogeneity, τ and τ′, at lower temperatures. The hR9 (rhombohedral) structure of τ′ is a superstructure of the CsCl type of the τ phase with ordered vacancies. The τ phase was modelled as (Al, Cu)1 Al4 Cu4 Zn1, covering both ranges.

IV.2.2.5

Quaternary Al–Cu–Mg–Zn system

All the phases known to exist in the Al–Cu–Mg–Zn quaternary system are already present in the binary or ternary subsystems, but several of them have large homogeneity ranges inside the quaternary system, ranging from one ternary boundary system to another. Compatible models for the solution phases of the different boundary systems were used to allow them to merge in the higher-order system. In most of these phases there is a large homogeneity range along a substitution of zinc atoms by an equimolar mixture of aluminium and copper atoms. The T phase appears in the Al–Mg–Zn system as (Al, Zn)49Mg32 and, with a smaller homogeneity range, in the Al–Cu–Mg system as (Al, Cu)49Mg32. It extends into the quaternary Al–Cu–Mg–Zn system, forming a continuous series of solid solutions. According to the crystal structure [57Ber] and the extension of the solubility range, this phase was modelled in the compound energy formalism as (Mg)26(Mg, Al)6(Al, Zn, Cu, Mg)48(Al)1, which includes the descriptions used above for the T phase in the Al–Mg–Zn and Al–Cu–Mg systems. For the T phase, two quaternary interaction parameters had to be introduced to reproduce the quaternary homogeneity range.

Computational phase studies in commercial Al and Mg alloys

383

The Laves phases C14, C15 and C36 exist in the Al–Cu–Mg and Cu–Mg–Zn systems. Between these ternary phases, quaternary homogeneity ranges extend approximately along lines of constant valence electron concentration, i.e. substitution of Zn by Cu + Al in the ratio 1:1. The Laves phases C14, C15 and C36 were described by the compound energy formalism with Cu ↔ Zn ↔ Al exchange on one sublattice, Mg on the other sublattice and a weak tendency for antistructure atom formation: (Mg, Al, Cu, Zn)(Cu, Zn, Al, Mg)2. The V phase appears as the binary phase Mg2Zn11, with some solubilities for Al and Cu and as the ternary, nearly stoichiometric phase Mg2Cu6Al5 in the boundary systems. It extends through the quaternary system from Mg2Cu6Al5 until Mg2Zn11 with the same crystal structure. In the Al–Cu–Mg system, aluminium and copper occupy crystallographically different sites. The solubilities of aluminium or copper in Mg2Zn11 are relatively small, but the combined substitution of zinc by both aluminium and copper leads to a continuous series of quaternary solid solutions. For the quaternary system this phase was modelled as Mg2(Cu, Zn)6(Al, Zn)5, which is the simplest model covering the ternary descriptions, but which needs slight modification of the published Al–Mg–Zn and Cu–Mg–Zn descriptions. A continuous range of solid solutions could be reproduced. The other ternary phases, S (Al2 CuMg), Q (Al7 Cu3Mg6) and Φ (approximately (Al, Zn)5Mg6), have only limited extensions of solubility into the quaternary system [47Str]. The quaternary thermodynamic parameters derived from the discussed modelling and optimised thermodynamic descriptions have been given by Liang et al. [99Lia]. Using this thermodynamic data set, arbitrary sections can be calculated in the quaternary Al–Cu–Mg–Zn system. As an example, the calculated isothermal section at 673 K (400 °C) and 40 at.% Mg is shown in Fig IV.2.7. It illustrates the continuous solubility of the T phase in the composition range between the Al–Mg–Zn and Al–Cu–Mg ternary systems. The quaternary system modelled to date has already given important insights into the phase relations within this system. A further refinement of the quaternary data set taking into account new experimental investigations is in progress.

IV.2.3

Conclusions

The development of aluminium and magnesium alloys can be supported by taking into account possible equilibrium and non-equilibrium phase reactions. From the results presented here it can be concluded that the combination of CALPHAD-type calculations and well-selected experiments offers an efficient way for the treatment of these materials. This approach supports the understanding of materials microstructure development and moreover can be used to predict and understand the solidification of aluminium and magnesium casting alloys.

384

The SGTE casebook 0.6

0.5 C15 + Mg2Cu

Cu

0.4

0.3 C36 + Mg

Mo

le

fra c

tio n

of

C15 + Mg

C14 + Mg

0.2 C14 + liquid C14 + T + 0.1

Q+T

C36 + liquid

C14 + T T

0

β+T

0 Al60Mg40

0.1

0.2

0.3 0.4 Mole fraction of Zn

0.5

0.6

IV.2.7 Isothermal section of the Al–Cu–Mg–Zn system at 673 K (400 °C) and 40 at.% Mg.

IV.2.4

Acknowledgement

Financial support by the ‘Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie’ (Contract 03K7208 0) within the framework of the European Community Project COST 507 is gratefully acknowledged.

IV.2.5 47Str 57Ber 77Luk 85Sun 92Luk 93Jan

References D. I. STRAWBRIDGE, W. HUME-ROTHERY and A. T. LITTLE: J. Inst. Metals 74, 1947, 191–225. G. BERGMAN, L. T. WAUGH, and L. PAULING: Acta Crystallogr., 10, 1957, 254– 259. H. L. LUKAS, E.-TH. HENIG and B. ZIMMERMANN: Calphad, 1, 1977, 225–236. B. SUNDMAN, B. JANSSON and J.-O. ANDERSSON: Calphad, 9, 1985, 153–190. H. L. LUKAS and S. G. FRIES: J. Phase Equilibria 13, 1992, 532–541. B. JANSSON, M. SCHALIN, M. SELLEBY and B. SUNDMAN: in Computer Software in Chemical and Extractive Metallurgy (Eds C. W. Bale and G. A. Irons), Metallurgical Society of the Canadian Institute of Mining, Metallurgy and Petroleum, Montreal, Quebec, 1993, p.57.

Computational phase studies in commercial Al and Mg alloys 95Ans

385

I. Ansara (Ed.): COST 507, Thermochemical Database for Light Metal Alloys, Action on Materials Sciences, European Commission, DG XII, Science Research and Development, L-2920 Luxembourg, 1995. 96Nei G. NEITE, K. KUBOTA, K. HIGASHI and F. HEHMANN: in Magnesium-based Alloys (Eds R. W. Cahn, P. Haasen and E. J. Kramer), Materials Science and Technology, Vol. 8 (Ed. K. H. Matucha), Structure and Properties of Nonferrous Alloys, VCH, Weinheim, 1996, pp. 113–212. 97Che S.-L. CHEN, Y. ZUO, H. LIANG and Y. A. CHANG: Metall. Mater. Trans. A 28, 1997, 435–446. 97Don P. DONNADIEU, A. QUIVY, T. TARFA, P. OCHIN, A. DEZELLUS, M. G. HARMELIN, P. LIANG, H. L. LUKAS, H. J. SEIFERT, F. ALDINGER and G. EFFENBERG: Z. Metallkunde, 88, 1997, 911. 98Buh T. Buhler, S. G. Fries, P. J. Spencer and H. L. Lukas: J. Phase Equilibria 19, 1998, 317–333. 98Fau F. FAUDOT, P. OCHIN, M. G. HARMELIN, S. G. FRIES, T. JANTZEN, P. J. SPENCER, P. LIANG and H. J. SEIFERT: Comptes Rendus, Journées d´Etude des Equilibres Entre Phases (XXIV JEEP) (Eds F. A. Kuhnast and J.J. Kuntz), Nancy, France, April 1998, pp. 173–176. 98Fri S. G. FRIES, I. HURTADO, T. JANTZEN, P. J. SPENCER. K. C. HARI KUMAR, F. ALDINGER, P. LIANG, H. L. LUKAS and H. J. SEIFERT: J. Alloys Compounds. 267, 1998, 90– 99. 98Hur I. HURTADO: Private communication, 1998. 98Lia1 P. LIANG, T. TARFA, J. A. ROBINSON, S. WAGNER, P. OCHIN, M. G. HARMELIN, H. J. SEIFERT, H. L. LUKAS and F. ALDINGER: Thermochim. Acta 314, 1998, 87–110. 98Lia2 P. LIANG, H. J. SEIFERT, H. L. LUKAS, G. GHOSH, G. EFFENBERG and F. ALDINGER: Calphad, 22, 1998, 527–544. 98Lia3 P. LIANG, H.-L. SU, P. DONNADIEU, M. G. HARMELIN, A. QUIVY, P. OCHIN, G. EFFENBERG, H. J. SEIFERT, H. L. LUKAS and F. ALDINGER: Z. Metallkunde, 89, 1998, 536–540. 98Lia4 H. LIANG and Y. A. CHANG: J. Phase Equilibria, 19, 1998, 25–37. 98Sau N. SAUNDERS and A.P. MIODOWNIK: in Calphad (Calculation of Phase Diagrams): A Comprehensive Guide, Materials Series, Vol. 1. (Ed. R.W. Cahn), Pergamon, Oxford, 1998. 99Fau F. FAUDOT, M. G. HARMELIN, S. G. FRIES, T. JANTZEN, P. LIANG, H. J. SEIFERT, and F. ALDINGER: Comptes Rendus, Journées d´Etude des Equilibres Entre Phases (XXV JEEP), (Eds J. L. Jorda, M. Lomello-Tafin and C. Opagiste), Annecy, France, March 1999, 1999, pp. 199–202. 99Lia P. LIANG, H. J. SEIFERT, H. L. LUKAS and F. ALDINGER: Proc. Conf. Werkstoffwoche ´98, Vol. 6, Metalle/Simulation (Eds R. Kopp, K. Herfurth, D. Böhme, R. Bormann, E. Arzt and H. Riedel), München, Germany, October 1998, Wiley–VCH, Weinheim, 1999, pp. 463–468. 99Sei H. J. SEIFERT: Z. Metallkunde, 90, 1999, 1016–1024. 00Cel S. CELOTTO: Acta Mater. 48, 2000, 1775–1787. 07Luk H. L. LUKAS, S. G. FRIES, and B. SUNDMAN: Computational Thermodynamics: The Calphad Method, Cambridge University Press, 2007.

IV.3 Multicomponent diffusion in compound steel JOHN ÅGREN

IV.3.1

Introduction

Compound materials are increasingly being used as structural materials. For example, one can combine a corrosion-resistant steel with an inexpensive low-alloy high-strength steel and thereby lower the costs. However, if the different materials are chemically incompatible, the heat treatment of such a compound material will present some difficulties. There may be strong driving forces for the transfer of atoms by means of diffusion between the two component materials. This transfer may result in a drastic change in the properties close to the interface between the two materials. In some cases this may be beneficial but in most cases it is not, and the amount of transfer must be minimised. Over the last decade a general package of computer programs for multicomponent diffusional transformations (see, for example, the papers by Ågren [82Agr1, 82 Agr2]) has been developed and the purpose of the present chapter is to describe its application to some problems of practical interest.

IV.3.2

Numerical calculation of diffusion between a stainless steel and a tempering steel

In this section we consider the heat treatment of a compound material which is composed of a 16Cr–10Ni austenitic stainless steel (A) and a 1Cr–4Ni tempering steel (B). The material is heat treated in two steps: 2 h at 1250 °C and 30 min at 900 °C. The important question now is the extent to which the diffusional reactions occur. Further, we want to know whether the reactions can be inhibited by introducing a thin sheet of pure nickel between the steels. The complete heat treatment cycle was now simulated under various conditions with Ågren’s [82Agr1, 82 Agr2] program package. The full chemical compositions given in Table IV.3.1 were entered as initial conditions in the two parts of the compound material and the program calculated the subsequent development of the concentration profiles. 386

Multicomponent diffusion in compound steel

387

Table IV.3.1 Alloy composition in weight per cent Alloy

A B C D

Amount (wt%) C

Cr

Mn

Mo

Ni

Si

V

W

0.025 0.31 0.40 0.95

16.5 1.12 — 4.00

0.65 0.43 0.30 0.30

2.08 0.05 — 5.00

10.3 4.21 — —

0.66 0.26 0.30 0.30

— — — 2.00

— — — 6.30

0.50

C (wt%)

0.40

0.30

0.20 B

A

0.10

0.00 –1.0

–0.5

0 Distance (cm)

0.5

1.0

IV.3.1 Calculated carbon concentration profile in compound steel after 2 h at 1250 °C and 30 min at 900 °C.

The thermodynamic data compiled by Uhrenius [78Uhr] and the diffusivities compiled by Fridberg et al. [69Fri] were applied. It was assumed that both steels are one-phase austenitic during the heat treatment. The results plotted by the computer are shown in Fig. IV.3.1 to Fig. IV.3.4 and will now be discussed. The stainless steel denoted A is on the left-hand side in all diagrams. Figure IV.3.1 shows the carbon concentration profile at the end of the heat treatment. As can be seen, there is a considerable exchange of carbon between the two steels. The affected zone ranges over several millimetres and there is a very drastic change close to the interface. The profiles of all the components except for Ni are shown in Fig. IV.3.2 with a finer scale on the x-axis. The profiles of Cr, Mo, Mn and Ni (not seen in the diagram) have more or less a step behaviour whereas the Si profile shows a complex variation similar to that of carbon. The behaviour is caused by the fact that C diffuses several

388

The SGTE casebook 2.0 B

A

Concentration (wt%)

1.6

1.2

Cr

Mo

0.8

Mn 0.4 Si

–0.10

C

–0.06

–0.02 0.02 Distance (cm)

0.06

0.10

IV.3.2 Calculated concentration profiles of Cr, Mn, Ni, Si and C in compound steel after 2 h at 1250 °C and 30 min at 900 °C. 1.00

0.80

Si (wt%)

B 0.60

0.40 A 0.20

0.00 –0.10

–0.06

–0.02 0.02 Distance (cm)

0.06

0.10

IV.3.3 Calculated Si concentration profile in compound steel after 2 h at 1250 °C and 30 min at 900 °C.

orders of magnitude more rapidly than the other elements. The Si profile is plotted separately in Fig. IV.3.3. The effect of introducing a 100 µm layer of pure Ni between the two steels was now investigated by the same procedure. The resulting carbon concentration profile is shown in Fig. IV.3.4 (solid curve). As a comparison the profile without the Ni layer, i.e. Fig. IV.3.1, has been superimposed

Multicomponent diffusion in compound steel

389

0.50

C (wt%)

0.40

0.30

0.20 B

A

0.10

0.00 –1.0

–0.5

0 Distance (cm)

0.5

1.0

IV.3.4 Calculated carbon concentration profile in compound steel with l00 µm Ni layer after 2 h at 1250 °C and 30 min at 900 °C (solid curve). The corresponding carbon concentration profile without the Ni layer (see Fig. IV.3.1) is superimposed (dashed curve).

(dashed curve). As can be seen, the carbon redistribution is inhibited to some degree by the Ni layer.

IV.3.3

Calculation of partial equilibrium between a carbon steel and an alloy steel

In this section we shall consider the partial equilibrium with respect to carbon between a carbon steel and an alloy tool steel. The full compositions are given in Table IV.3.1, steels C and D. The calculation should be a reasonable approximation if the temperature is low enough to neglect the redistribution of substitutional elements between the steels but high enough to allow complete equilibration of carbon. The calculation is performed by defining two separate equilibria having the same carbon activities. The two equilibria are connected by the condition that the total number of carbon atoms must be equal to the sum of the initial carbon contents. The calculation is performed by means of the Parrot program developed by Jansson [84Jan]. Two equilibria are defined. Both equilibria are defined by fixing a common temperature, pressure and carbon activity. In addition, the appropriate alloy contents of the individual steels are fixed. However, the carbon activity is not known but must be chosen in such a way that the sum of the carbon contents must equal a fixed value. In principle, this can be done by trial and error but the Parrot program allows this condition to be added as an extra constraint and the unknown carbon activity to be obtained from an optimisation.

390

The SGTE casebook

Table IV.3.2 Listing from the Parrot program for equilibrium in the tool steel at 600 °C (873 K). Output from POLY-3, Equilibrium number = 2 Conditions P = P0, T = T0, AC(c) = AC1, B(Si) = 0.3, B(Mn) = 0.3, B(Cr) = 4, B(Mo) = 5, B(V) = 2, B(W) = 6.3, B(Fe) = 81.25 Degrees of freedom = 0, Mass = 1.004 99 × 10–1 kg, Volume = 0.000 00, Pressure = 1.0132 50 × 10–5 Pa, Temperature = 873.00 K, Enthalpy = 1.706 32 × 104 J, Total Gibbs energy = –1.324 65 × 104 J, Number of moles of components = 1.785 93 Component

Amount (mol)

Fraction

Activity

Va C Cr Cu Fe Mn Mo Ni Si V W

0.0000 1.12 × 103 7.69 × 102 0.0000 1.45 × 104 5.46 × 10 5.21 × 102 0.0000 1.07 × 102 3.93 × 102 3.43 × 102

0.0000 1.34 × 102 3,98 × 102 0.0000 8,08 × 103 2.99 × 10 4.98 × 102 0.0000 2.99 × 10 1.99 × 102 6.27 × 102

1.00 2.86 5,74 1.00 8.61 2.21 8.38 1.00 1.44 5.34 5.19

× × × × × × × × × × ×

Ferrite 1 Status Entered Number of moles = 1.3579, Driving force = 0.0000 Fe 9.871 17 × 10–1 Mn 1.075 26 × 10–3 Ni 0.000 00 Cr 6.367 45 × 10–3 C 6.245 85 × 10–6 Si 3.970 66 × 10–3 Cu 0.000 00

104 103 102 104 103 10 10 104 105 10–3 10

Potential 0.00000 –9.0759 × –2.0738 × 0.0000 –1.0840 × –4.4392 × –3.4712 × 0.0000 –1.4778 × –1.0484 × –3.8182 ×

V Mo W

M23C6 1 Status Entered Number of moles = 1.2839 × 10–1, Driving force = 0.0000 Fe 4.387 25 × 10–1 C 5.069 97 × 10–2 Ni Si 0.000 00 Cr 3.194 41 × 10–1 Mn V 0.000 00 Mo 1.421 16 × 10–1 W Cu 0.000 00 M6C 1 Status Entered Number of moles = 1.5889 × 10–1, Driving force 0.0000 W 4.244 52 C 2.059 29 × 10–2 Mn Si 0.000 00 Fe 2.720 52 × 10–1 Cr V 0.000 00 Mo 2.649 43 × 10–1 Ni CU 0.000 00 M7C3 1 Status Entered Number of moles 3.6774 Cr 7.028 34 × 10–1 W 0.000 00 V 0.000 00 Cu 0.000 00

× 10–2, Mn Fe C

Driving force = 0.0000 3.113 39 × 10–3 Ni 2.052 61 × 10–1 Si 8.879 13 × 10–2 Mo

Reference state 103 104 103 104 104

SER SER SER SER SER SER SER SER

105 105 104

1.073 62 × 10–5 8.902 77 × 10–4 5.625 29 × 10–4

0.000 00 3.397 01 × 10–2 1.504 74 × 10–2

0.000 00 1.796 03 × 10–2 0.000 00

0.000 00 0.000 03 0.000 00

? ? ? ? ? ? ? ?

Multicomponent diffusion in compound steel

391

Table IV.3.2 (Continued) MF fcc carbide 1 Status ENTERED Number of moles = 1.0402 × 10–1, Driving force = 0.0000 V 5.099 32 × 10–1 Mo 1.353 73 × 10–1 Fe Ni 0.000 00 C 1.593 45 × 10–1 Cr Si 0.000 00 W 1.385 80 × 10–1 Mn Cu 0.000 00

4.367 46 × 10–5 5.663 87 × 10–2 8.785 42 × 10–5

A calculation was now performed for a compound material consisting of equal weights of a carbon steel and a tool steel of type M2. The calculation was performed for T = 600 °C. The initial carbon contents are 0.4 wt% C in the carbon steel and 0.95 wt% C in the tool steel. The corresponding carbon activities calculated from the data obrained by Uhrenius [78Uhr] at 600 °C are 2.05 and 0.02 respectively. Despite the much lower carbon content there is thus a strong tendency for carbon to diffuse from steel C to steel D. The final result yields the common carbon activity 0.29 and almost all the carbon redistributed to the tool steel. Table IV.3.2 shows the final equilibrium of steel D as listed in the Parrot program. As can be seen, four different carbides and a ferritic matrix coexist at equilibrium.

IV.3.4

Summary

It has been shown that diffusion calculations and modified equilibrium calculations can give valuable information for the heat treating of compound materials. As input the diffusivities of the various components and a thermodynamic description are required. A strong redistribution of carbon is predicted. In order to design a proper heat treatment of such a compound material it is absolutely necessary to take this carbon redistribution into account.

IV.3.5 69Fri 78Uhr

References

J.FRIDBERG, L.-E.TÖRNDAHL and M.HILLERT: Jernkont. Ann. 153, 1969, 263. B.UHRENIUS: in Hardenability Concepts with Applications to Steel (Eds D.V.Doane and J.S.Kirkaldy), Metallurgical Society of AIME, Warrendale, Pennsylvania, 1978, p. 28. 82Agr1 J. ÅGREN: J.Phys.Chem.Solids 43,1982, 385. 82Agr2 J. ÅGREN: Acta Metall. 30, 1982, 841. 84Jan B. JANSSON: Technical Report TRITA-MAC-0234, Division of Physical Metallurgy, Royal Institute of Technology, Stockholm, Sweden, 1984.

IV.4 Melting of a tool steel BENGT HALLSTEDT

IV.4.1

Introduction

Here we use DICTRA [002And] to simulate the melting behaviour of the tool steel X210CrW12 (Fe–2%C–12%Cr–0.8%W). This is a ledeburitic cold work steel containing about 40% austenite + carbide (M7C3) eutectic structure after solidification. This steel has been identified as a very suitable material for semisolid processing [006Uhl]. After semisolid forming from about 1270 °C the austenite can be retained down to room temperature, producing a relatively soft material. After proper heat treatment, at least the same hardness and wear resistance as for conventionally produced material can be achieved. A key parameter for semisolid processing is the amount of liquid phase as a function of temperature (and possibly time). The original purpose of this work was to study the influence of the heating rate, which can be quite high using induction heating, on the amount of liquid phase. In the end it turned out that the heating rate is quite unimportant, but other factors (in particular, the segregation state) are important. In contrast with solidification, melting of alloys has rarely been studied in any detail. Most investigations have dealt with incipient melting in connection with homogenisation treatments, hot working or welding [98Sam, 006Zhu] where the formation of even a small amount of liquid can have catastrophic consequences. In these cases, incipient melting occurs below or well below the solidus temperature. The incipient melting temperature is the temperature where liquid starts to form in an actual sample and the solidus temperature is the temperature where liquid starts to form at equilibrium. Here we shall see an example where incipient melting can occur above the solidus temperature. Another area where the melting behaviour is important is in the design and interpretation of differential thermal analysis experiments. This has been dealt with by Boettinger and Kattner [002Boe] for homogeneous single-phase alloys.

392

Melting of a tool steel

IV.4.2

393

Calculation

DICTRA simulations of the melting were performed for two extreme cases. In the first case the solidification was first simulated (at a cooling rate of 20 K min–1) in order to produce concentration profiles representing maximum segregation. Concentration profiles were taken after cooling to 1100 °C and used as the start condition for the melting simulation. In the actual simulation, both cooling and heating were performed in a single simulation run. In the second case the melting of a sample completely homogenised (i.e. at full equilibrium) at 1100 °C was simulated. For the conditions used here it takes about 50 h to reduce the segregation levels by 90% at 1100 °C. This time is determined by Cr, and not, as might have been expected, by W. The actual feedstock for semisolid processing has typically been hot rolled, leading to segregation levels somewhere between these two extremes. In the semisolid state the microstructure consists of globules of austenite in a liquid matrix. Below about 40% liquid there is in addition coarse M7C3 present. At melting, the first liquid forms at the interface between austenite and coarse M7C3, which is located in regions of maximum segregation. It is expected that the size of the globules in the semisolid state correlates with the distance between the coarse carbides and the distance between the segregation peaks in the solid state. The cell size in DICTRA was set to 20 µm, which is about half the distance between the centre of two globules as determined from samples quenched from the semisolid state. A region of austenite was placed at the left-hand side of the cell and a region of M7C3 at the right-hand side. At the austenite–M7C3 interface an inactive liquid region was placed. For simplicity a linear cell geometry was used. A heating rate of 240 K min–1 was used. This heating rate was typically achieved by induction heating. The precise composition used was 2.18 mass% C, 11.64 mass% Cr and 0.8 mass% W. In the homogenised case the austenite size was 17.30 µm with a composition of 1.12 mass% C, 6.05 mass% Cr and 0.60 mass% W and the M7C3 size was 2.70 µm with a composition of 8.61 mass% C, 45.46 mass% Cr and 2.00 mass% W. In the as-solidified case there was considerably less carbide (about 1.7 µm) at the start of heating. Thermodynamic data were taken from the TC-Fe 2000 Steels/Alloys Database [99The1], which is equivalent to the SGTE Solution Database for this system, and mobility data were taken from the MOB2 Database [99The2]. Diffusion was considered in austenite and liquid, but not in M7C3. The calculated solidus and liquidus temperatures are 1245 °C and 1367 °C respectively.

IV.4.3

Discussion

The amount of liquid as a function of temperature is shown in Fig. IV.4.1. Above the knee, austenite and liquid are present and, below the knee, austenite,

394

The SGTE casebook 1.0 0.9

Mole fraction of liquid

0.8

Equilibrium Solidification Reheating from 1100 °C Heating after homogenisation at 1100 °C

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1200

1250

1300 Temperature (°C)

1350

1400

IV.4.1 Amount of liquid phase as a function of temperature.

M7C3 and liquid are present. The solid curve shows the equilibrium amount of liquid. The dotted curve shows the amount of liquid during solidification. The dashed curve shows the amount of liquid during heating directly after solidification and cooling to 1100 °C. This curve is very similar to the solidification curve; only the lower part is shifted to somewhat higher temperatures, as a result of diffusion in the austenite. The dash–dotted curve shows the amount of liquid during heating in the completely homogenised case. The most striking feature here is that melting starts about 15 K above the solidus temperature. Also the rest of the curve is rather different from the others. It is certainly not very close to the equilibrium curve, except at complete melting where all curves are very close to the liquidus temperature. A detail of Fig. IV.4.1 is shown in Fig. IV.4.2. In Fig. IV.4.3 the composition profiles after solidification and cooling to 1100 °C are shown. W shows a rather strong positive segregation and Cr shows a positive segregation before M7C3 forms and a strong negative segregation after it forms. The maximum in the Cr curve corresponds to the start of M7C3 formation. The geometry of the DICTRA cell is not quite realistic after M7C3 starts to form, since the structure is expected to be eutectic at solidification. Nevertheless, the very high Cr content in the primary austenite compared with the eutectic austenite is realistic and is observed

Melting of a tool steel

395

0.6

0.5

Equilibrium Solidification Reheating from 1100 °C

Mole fraction of liquid

Heating after homogenisation at 1100 °C 0.4

0.3

0.2

0.1 Solidus 0 1220

1230

1240 1250 1260 Temperature (°C)

1270

1280

IV.4.2 Detail of Fig. IV.3.1.

experimentally. For the melting simulations, however, the geometry should be quite reasonable also when M7C3 is present. In M7C3, there is a very strong W gradient. This gradient would probably be considerably less strong if diffusion in M7C3 would be considered. The reason for the large differences in incipient melting temperature (Fig. IV.4.2) is that, depending on the initial state at the start of the heating, the operating tie line between austenite and M7C3 will be quite different when heating through the solidus temperature. The liquid may then hit the operating tie line considerably below or above the (equilibrium) solidus temperature. The lower the homogenisation (equilibrium) temperature is chosen, the higher the incipient melting temperature will be for this material. The other feature worth noting is that all curves in Fig. IV.4.1 are very close to the liquidus temperature at complete melting. This is a consequence of the fact that the composition gradient in the liquid is very small at the conditions considered. This also causes the amount of liquid to become insensitive towards the heating rate. Heating rates between 5 K min–1 and 500 K min–1 gave very similar results. The melting curves only shifted slightly towards higher temperatures below the knee with increasing heating rate. However, if the heating rate is increased considerably more or a considerably coarser structure is sampled, so that the composition gradient in the liquid phase is no longer

396

The SGTE casebook 0.10 0.09

Cr

0.08

Mass fraction

0.07 0.06 0.05 0.04 0.03 0.02

C

0.01 0 0

W 2

4

6

8 10 12 14 Distance (µm)

16

18

20

IV.4.3 Simulated composition profiles at 1100 °C after solidification and cooling.

negligible, then the whole melting curve becomes less steep and complete melting will occur clearly above the liquidus temperature.

IV.4.4

Conclusions

A number of conclusions can be drawn. 1

2

3 4

5

The most striking finding is that incipient melting can occur also above the (equilibrium) solidus temperature. In principle, this could be possible for any ternary (or higher) alloy with at least two phases below the solidus. A well-homogenised single-phase alloy, however, should always show incipient melting at the solidus temperature. Complete melting occurs very close to the liquidus within a wide range of conditions. This is true as long as the composition gradients in the liquid can be neglected. The heating rate (in the conditions studied) has only a small influence on the melting behaviour. The initial state (segregation state, coarseness of the microstructure, etc.) of the solid alloy is very important in order to determine its melting behaviour. This will also determine whether it is possible to form a globular microstructure (and its coarseness) on partial melting, which is a necessary requirement for semisolid processing. The solidus of this alloy cannot be determined using thermal analysis. This is probably the general case for ternary (or higher) alloys with two phases or more.

Melting of a tool steel

IV.4.5

397

Acknowledgement

The author gratefully acknowledges financial support from the Deutsche Forschungsgemeinschaft within the Collaborative Research Centre (SFB) 289 ‘Forming of metals in the semisolid state and their properties’.

IV.4.6

References

98Sam F.H. SAMUEL: J. Mater. Sci., 33, 1998, 2283–2297. 99The1 Thermo-Calc: TC-Fe 2000 Steels/Alloys Database, Thermo-Calc AB, Stockholm, Sweden, 1999. 99The2 Thermo-Calc: MOB2 Database, Thermo-Calc AB, Stockholm, Sweden, 1999. 002And J.-O. ANDERSSON, T. HELANDER, L. HÖGLUND, P. SHI and B. SUNDMAN: Calphad 26, 2002, 273–312. 002Boe W.J. BOETTINGER and U.R. KATTNER: Metall. Mater. Trans. A 33, 2002, 1779– 1794. 006Uhl D.I. UHLENHAUT, J. KRADOLFER, W. PÜTTGEN, J.F. LÖFFLER and P.J. UGGOWITZER: Acta Mater. 54, 2006, 2727–2734. 006Zhu T. ZHU, Z.W. CHEN and W. GAO: Mater. Sci. Eng. A 416, 2006, 246–252.

IV.5 Thermodynamic modelling of processes during hot corrosion of heat exchanger components U L R I C H K R U P P, V I C E N T E B R A Z D E T R I N D A D E F I L H O and K L A U S H A C K

IV.5.1

Introduction

Corrosion is a process that takes place in space and time. Although classical equilibrium thermochemistry can already help to investigate which phases may occur during such a process, as will be shown below using standard phase diagrams, it is ultimately necessary to apply a coupled approach of diffusion and local equilibria. In the context of a European project on the corrosion of heat exchanger materials called OPTICORR [005Bax] a method and code called InCorr was developed in which diffusion treated along the lines outlined in Chapter III.3 using a two-dimensional finite-difference method was coupled with a code that provides phase equilibria, ChemApp, for the compositional conditions generated in the diffusional steps [005Bax]. Results from classical phase diagrams for a stainless steel represented by an Fe–Cr alloy with a given content of Cr were used as a first means for understanding the possible phase formation in an atmosphere containing a given oxygen potential. Figure IV.5.1 gives a view of the situation of corrosion of the metal under a gas with a defined oxygen potential. Diffusion of oxygen into the material and diffusion of metallic components towards the surface interact with each other and lead to formation of an outside and an inside oxide layer. The inward flow of the oxygen is governed by both bulk and grain boundary diffusion in the alloy. Experimental results by Krupp et al. [004Kru] showed that the grain size in the metal has a strong influence in the propagation of the layer growth.

IV.5.2

Database work

For the calculations in the project extensive databases have been created, extracts of which were employed to execute the calculations discussed in the presentation. They covered Fe-based alloys (Al–C–Ce–Co–Cr–Cu–Fe–Mn– Mo–Nb–Ni–S–Si–V), Ni-based alloys (C–Cr–Fe–Mo–Ni–Si–Ti–W), and an 398

Thermodynamic modelling during hot corrosion of heat exchanger Outer scale (Fe3O4)

pO Fe O

Inner scale (Fe3O4, FeCr2O4)

pO FeCr O

2

3

2

4

2

p(O )Cr O 2

Intercrystalline oxidation (Cr2O3)

2

399

4

3

Moving interface

Deff Dx Dy DGB

Steel B (550 °C, 72 h, air) 5 µm

IV.5.1 Schematic view of the gas-phase corrosion process, including the two paths along the grain boundaries and across the grains of the metal, also showing a micrograph of a real sample.

oxide, sulphate and sulphide database (MeS, MeO, MeSO4, where Me = Al, C, Ce, Co, Cr, Cu, Fe, Mn, Mo, Nb, Ni, Si, Ti, V or W).

IV.5.3

Calculational results

Before complex thermodynamic models including explicit kinetics (here the diffusional transport of reactive elementary or molecular species) are employed, it is most useful to obtain with the aid of classical thermochemical calculations a picture of the momentary situation, i.e. of the frozen-in state at a certain moment in time. For that purpose the phase diagram module of the integrated themodynamic databank system FactSage was employed. The results of the kinetic simulations, i.e. the time-dependent gas and salt corrosion respectively, are given separately further below.

IV.5.3.1

Two-dimensional mappings (phase diagrams) for alloys in corrosive atmospheres

Best suited to the present purpose are phase diagrams of type 1 (here T versus log pi) for Fe–Cr alloys. Figure IV.5.2 shows a phase diagram in which for variable temperature and variable oxygen potential (log pO 2 ) the phase fields are shown for an Fe–Cr alloy with 20 wt% Cr. Two curves (curves 1 and 2) are remarkable in this diagram. Curve 1 depicts the outer line of stability of the pure metallic state. One can see that the ferritic state of the alloy changes to austenite because of the loss of Cr by formation of Me2O3 solid solution which is under these conditions almost pure Cr2O3. Curve 2 depicts the outer curve of existence of metal. Beyond that curve, all metal will be consumed by the formation of oxide solid solutions. At lower

400

The SGTE casebook

1700 Me2O3 + Liquid Fe spinel + liquid

Liquid 1500

Bcc A2 + Fe spinel Bcc A2

Temperature (°C)

1300

1 cA

1100

O3 e2 M

+

fc

Fe

+ el in p s

c Fc

1 A Fe

l+ ne i sp

Me2O3 + Fe spinel

e tit us w

Fe spinel

1

900

el

pin

700

Bcc

A2

+

s Fe

2

500 Me2O3

Me2O3 + Fe spinel 300 –30

–25

–20

–15 log10 pO

–10

–5

0

2

IV.5.2 The Fe–Cr–O2 phase diagram for 20 wt% Cr.

temperatures and very low oxygen potentials (partial pressures) the primary phase will be Fe spinel, i.e. the solid solution between (FeO)(Fe2O3) and (FeO)(Cr2O3), replacing trivalent Fe by Cr. At higher temperatures and intermediate oxygen potentials there is the additional formation of wustite solid solution. It should be noted that for high oxygen potentials the solid oxide is a solid solution (corundum) between Fe2O3 and Cr2O3 with the initial 80–20 composition of the two metals. It should be emphasised here that the phase diagram calculation is based on the assumption that the Fe-toCr ratio is the same at each point in the diagram. In reality this condition has of course not to be satisfied at each point in space since a difference in diffusion velocity between the different metallic components of an alloy can lead to strong gradients in composition. For the system Fe–Cr this is indeed decisive.

IV.5.3.2

Model calculations for gas-phase corrosion

The software InCorr permits a general approach to the phase formation under the conditions of internal oxidation, nitridation, carburisation and sulphidation. For the project OPTICORR it has especially been employed to investigate the behaviour of steels and nickel-based alloys under gases with oxygen potentials as known from heat exchanger atmospheres. The model is capable of simulating multiphase internal corrosion processes controlled by solid-state diffusion into the bulk metal as well as intergranular corrosion

Thermodynamic modelling during hot corrosion of heat exchanger

401

occurring in polycrystalline alloys owing to the fast inward transport of the corrosive species along the grain boundaries of the material. A treatment of internal corrosion problems that involve more than one precipitating phase, compounds of moderate stability, high diffusivities of the metallic elements or time-dependent changes in the (test) conditions, e.g. temperature or interface concentrations, is not possible on the basis of Wagner’s classical theory of internal corrosion. To simulate such systems the application of numerical methods to the differential equation for the diffusion and to the thermochemistry of the system is required. Such an approach has been taken here, which leads to a finite-difference method that is solved by a Crank– Nicholson algorithm. The distribution and structure of grain boundaries play important roles in the kinetics of many high-temperature degradation processes since the transport of matter along interfaces is orders of magnitude faster than throughout the bulk. Therefore, reducing the grain size, i.e. increasing the fraction of fast diffusion paths, may have a detrimental effect, as is known for the creep behaviour of metals and alloys [95Sut]. On the other hand, the high-temperature oxidation resistance of Cr–Ni 18-8-type stainless steels, which are widely used for superheater tubes in power plants, can benefit from smaller grain sizes. As reported by Teranishi et al. [89Ter] and Trindade et al. [005Tri], the formation of protective Cr-rich oxide scales (FeCr2O4 and/or Cr2O3) is promoted by the fast outward flux of Cr along the substrate grain boundaries. A similar effect can be used by providing nanocrystalline surface layers on Ni-based superalloys. Wang and Young [97Wan] have shown that an increase in the fraction of grain boundaries can decrease the critical Al concentration required for the establishment of a superficial Al2O3 scale on a material that usually forms a Cr2O3 scale. It has been shown [005Tri] that in the case of low-Cr steels, typically used for cooling applications in power generation up to temperatures of approximately 550 °C, the beneficial effect of grain refinement disappears. Here, the grain boundaries seem to act as fast-diffusion paths for the oxygen transport into the substrate. The parabolic rate growth obviously decrease as the alloy grain size increases. Furthermore, the oxidation kinetics decrease as the Cr content increases for alloys with similar grain sizes. Results of the model calculations with InCorr are from inert-gold-marker experiments at 550 °C (Fig. IV.5.3(a)); one knows that oxide scale growth occurs by both outward Fe diffusion, leading to the formation of haematite (Fe2O3, outermost) and magnetite (Fe3O4), and inward O transport, leading to (Fe, Cr)3O4 formation. As a consequence of the Cr content in the substrate, a gradient in the Cr concentrations is established, reaching from the outer– inner scale interface (cCr = 0) to the inner-scale–substrate interface, where the Cr concentration corresponds to the sole formation of the spinel phase FeCr2O4. This is in agreement with the thermodynamic prediction using a

402

The SGTE casebook

Fe3O4 (at.%) 50 0 0

0 0.5 × 10–5

1

1 1.5

y (m)

2

2

(a)

× 10

–5

x (m)

Grain boundaries

FeCr2O4 (at.%)

2 1 0 0

0 0.5 × 10–5

1

1 1.5

y (m)

2

2

× 10–5

x (m)

(b)

IV.5.3 The oxides in (a) the outer layer and (b) the inner layer of an Fe–1.5 wt% Cr alloy.

specific data set developed for these kinds of alloy [004Hei]. The inward oxide growth itself is governed by an intergranular oxidation mechanism (see Fig. IV.5.1) that can be described as follows: oxygen atoms that have reached the scale–substrate interface by short-circuit diffusion through cracks or pores [003Che,003Sch] or by O anion transport penetrate into the substrate along the grain boundaries, leading to the formation of Cr2O3 and, consequently, FeCr2O4. Progress of the scale–substrate interface occurs as soon as the bulk of the grains are oxidised completely, as shown in Fig. IV.5.3 and Fig. IV.5.4. Figure IV.5.3(a) and Fig. IV.5.3(b) show the simulated lateral concentration profiles of the oxide phases Fe3O4 and FeO·Cr2O4 (chrome spinel) while Fig. IV.5.4(a) shows the Cr distribution and Fig. IV.5.4(b) shows the Cr2O3 formed during exposure of the low-alloy steel X60 (1.44 wt% Cr) with a grain size of 30 µm for T = 550 °C under air. Note that y = 0 corresponds to the original inner-scale–metal interface at t = 0 s. It is obvious that the outer layer consists of Fe3O4 as found in the experimental investigations (see Fig. IV.5.1), while the inner layer dependent upon the penetration depth consists of chrome spinel and pure Cr2O3, the Cr2O3 being the first phase that is formed in the intergranular region. This behaviour is in full agreement with the phase diagrams calculated for the Fe–Cr–O system. These too (see, for example, Fig. IV.5.2) show the sequence of phases to be Cr2O3, chrome spinel and Fe3O4 from low to high oxygen potentials. Finally Fig. IV.5.5 shows the clear influence of the grain size on the

Thermodynamic modelling during hot corrosion of heat exchanger

403

Cr in bcc Fe (at.%)

1 0 0 0

× 10–5

1 0.5 × 10–5

1

1.5

x (m)

2

2

y (m) (a) Cr2O3 (at.%) 0 × 10–5

0.5 0 0

1 0.5 × 10–5

x (m)

1 y (m)

1.5

2

2

(b)

IV.5.4 (a) Chromium content and (b) intergranular Cr2O3 content in the surface zone of an Fe–1.5 wt% Cr alloy.

corrosion behaviour. The smaller the grain size, the higher is the corrosive loss of material. Note how the experimental data for the 30 µm grain size have been used for the calibration of the data.

IV.5.4

Conclusions

Special databases have been assembled dedicated to the field of corrosion of heat exchanger materials (steels and nickel based alloys) under liquid salt layers and by direct gas diffusion through the bulk and along grain boundaries in the metal. These databases expand the scope of the standard databases for alloys (SGTE [002SGT, 004SGT]) and also for salts and oxides (FACT [003FAC]; see the paper by Bale et al. [002Bal]). The new databases have been successfully applied in the generation of classical thermodynamic one- and two-dimensional mappings but also in the kinetic models developed in the OPTICORR project. The results are very well suited to understanding the processes of corrosion, leading to deterioration of heat exchanger materials under experimental conditions that simulate the real situation in power plants. However, for the description of the full situation in a power plant the databases need further extension, especially with respect to the high complexity of real-world salt deposits and their interaction with silicate deposits which will also form. A full-scale model of the combustion chamber and the resulting gas–aerosol flow from the combustion chamber into and through the exchanger is now needed.

404

The SGTE casebook

Thickness of the inner oxide scale (µm)

25 Experiment Simulated (grain size, 10 µm Simulated (grain size, 30 µm) Simulated (grain size, 100 µm)

20

15

10

5

0

0

10

20

30

40 Time (h)

50

60

70

80

IV.5.5 Comparison of the simulated inner-oxide growth kinetics for the low-alloy steel X60 (cCr = 1.44 wt%) with three different grain sizes and with the experimentally measured inner-oxide thickness for specimens having a grain size d = 10 µm.

IV.5.5 89Ter 95Sut 97Wan 002Bal 002SGT 003Che 003Fac 003Sch 004Kru 004Hei

004SGT 005Bax 005Tri

References H. TERANISHI, Y. SAWARAGI and M. KUBOTA: Sumitomo Res. 38, 1989, 63. A.P. SUTTON and R.W. BALLUFFI: Interfaces in Crystalline Materials, Oxford University Press, Oxford, 1995. F. WANG and D.J. YOUNG: Oxidation Metals 48, 1997, 497. C.W. BALE, P. CHARTRAND, S.A. DEGTEROV, G. ERIKSSON, K. HACK, R. BEN MAHFOUD, J. MELANCON, A.D. PELTON and S. PETERSEN: Calphad 26(2), 2002, 189–228. SGTE: SGTE Pure Substance Database, 2002. R.Y. CHEN and W.Y.D. YUEN: Oxidation Mater. 59, 2003, 433. FACT Database, 2002. M. SCHÜTZE: J. Corros. Sci. Eng. 2003, 6. U. KRUPP, V.B. TRINDADE, B.Z. HANJARI, H.-J. CHRIST, U. BUSCHMANN and W. WIECHERT Mater. Sci. Forum 461–464, 2004, 571–578. L. HEIKINHEIMO, K. HACK, D. J. BAXTER, M. SPIEGEL, U. KRUPP, M. HÄMÄLÄINEN and M. ARPONEN: Proc. 6th Int. Symp. High Temperature Corrosion and Protection of Materials, Les Embiez, France, Vol 461-464, 2004, 473–480. SGTE: SGTE Solution Database, 2004. D. BAXTER (Ed.): OPTICORR GuideBook, VTT Research Notes 2309, VTT, Espoo, 2005. V.B. TRINDADE, U. KRUPP, PH. E.-G. WAGENHUBER and H.-J. CHRIST: Mater. Corros. 56(11), 2005, 785–790.

IV.6 Microstructure of a five-component Ni-base superalloy: experiments and simulation NILS WARNKEN, BERND BÖTTGER, D E X I N M A, S U Z A N A G . F R I E S , N AT H A L I E D U P I N and B O S U N D M A N

IV.6.1

Introduction

The ever-increasing demand for high-performance alloys for high-temperature applications has led to the development of the present Ni-base superalloys. These alloys are nowadays routinely used as turbine blades in gas turbine engines [97Dur, 000Kar]. Their development, e.g. alloying, casting, heat treatments and homogenisation, was essentially based on experimental observation and metallurgical intuition. Numerical simulations providing indication of trends, correlation between measurable quantities, determination of detrimental effects, etc., even when qualitative, are very welcome, allowing time and costs to be saved as well as efficiency to be improved when compared with the usual trial-and-error methods. Many steps in the design and production of these alloys cannot, nowadays, be quantitatively simulated. Solidification simulations, however, were achieved, which enabled a revision of the empirical methods to be made, bringing more understanding and thereby implying an improvement in the entire alloy development process. The work presented here was performed within the frame of the Collaborative Research Center SFB 370 ‘Integrated modeling of materials’ which aimed at understanding, controlling and optimising the microstructure evolution during solidification, heat treatment, coating and operational service [006Her]. In order to reduce the number of variables to be controlled during the process, the high-alloyed (usually more than ten components) secondgeneration material was substituted by a model alloy that was complex enough to reproduce the main features of the commercial alloy, but with fewer components: Ni–13 at.% Al–10 at.% Cr–2.7 at.% Ta-3 at.% W. To simulate realistic microstructure evolution in multicomponent alloys, coupling of the phase-field code to a thermodynamic database is mandatory. Figure IV.6.1 schematically shows the coupling as it is implemented in the ACCESS phase field code, as used throughout this work [96Ste, 97Tia, 006Eik]. Much attention is given to a close control of all the tools used in the project; therefore a realistic thermodynamic database is prepared by one of 405

406

The SGTE casebook Thermodynamic description T

CALPHAD software Programming interface

φ 1–φ

α α+β mβα

β

Relinearisation

0, k c c 0, k cβα αβ

mαβ c (k)

Thermodynamic database

Time loop

∆G, c α, c β PF solver Temp. solver Conc. solver Output 50µm

Phase field software

Results

IV.6.1 Schematic representation of the approach for coupling the phase field method with thermodynamic calculations.

the project partners, the LTH (now called MCh) at the RWTH Aachen, by means of the calculation-of-phase-diagrams (CALPHAD) method [98Sau,97Kat] which quite successfully describes phase equilibria for multicomponent systems with an accuracy sufficient for technological applications. The phases of interest within this framework are basically the liquid phase and the face-centred cubic (fcc) phases: the ordered L12, called γ ′ and the disorderd A1, the γ phase. Details about the thermodynamic database constructed specifically for the model alloy are not given in this chapter; however, some equilibrium experimental data are presented here that can be used to validate the thermodynamic database. The modelling of a similar Nibase database has been described by Dupin and Sundman [01Dup].

IV.6.2

Experimental work

Experimental work was performed on a model alloy with a nominal composition of Ni–13 at.% Al–10 at.% Cr–2.7 at.% Ta–3 at.% W, produced by Doncasters in Bochum. Differential thermal analysis (DTA) was employed to determine the liquidus and solidus temperatures. Several samples were subjected to isothermal heat treatment at different temperatures within the solidification interval followed by quenching, allowing for the measurement of the equilibrium partition coefficients. This information can be used to validate the solid–liquid tie lines calculated using the thermodynamic database of the quinternary system Ni–Al–Cr–Ta–W. Finally, samples were unidirectionally solidified, to obtain morphological and element distribution patterns that can be compared with those obtained by numerical simulation of the solidification microstructure.

Microstructure of a five-component Ni-base superalloy

407

The actual composition of the alloy measured by the manufacturer is shown in Table IV.6.1. All calculations presented in this work are made using this composition. Our own analysis of one of the samples revealed a slight variation in the overall composition. For determination of the liquidus and solidus temperatures of the alloy, DTA runs were performed. Several samples were heated to 1723 K at 5 K min–1 and subsequently cooled at the same rate, which is believed to be small enough to provide data close to thermodynamic equilibrium. Figure IV.6.2 shows the thermal effects measured by DTA compared with the evolution of phase fractions with temperature calculated with the first version of the thermodynamic database provided by LTH, RWTH Aachen. The comparison indicates that the calculated liquidus temperature differs from the experimental value by less than 20 K which is rather good as a first approximation. It should be noted that the database was constructed only by extrapolation from ternary subsystems and the true composition of the analysed alloy might slightly differ from that given by the specifications of manufacturer, which is used for the calculations. The experimental and calculated solidus temperatures are in slightly better agreement when referring to the heating curve of the DTA measurement. Good agreement between the calculations and measurements was also found for the partitioning alloying elements. This was measured from samples prepared by isothermal holding and quenching as described by Sung and Poirier [99Sun]. It was found that the database provides a good description of the equilibrium solidification behaviour of our alloy. The dendritic structure of the primary phase can clearly be seen on micrograph obtained in cross-sections of unidirectional solidified samples (Fig. IV.6.3). Energy-dispersive X-ray spectroscopy and electron backscatter diffraction measurements have confirmed the dendritic phase as being γ and the elliptically shaped phases as being γ ′. Recently it was shown that, during solidification, γ ′ nucleates from the γ dendrites, i.e. on the solid side of the existing solid–liquid interface [005War]. This was proved by showing that Table IV.6.1 Composition of the model alloy as determined by the manufacturer’s and our own analysis Element

Al Cr Ta W Ni

Manufacturer’s analysis

Our analysis

Amount (at.%)

Amount (at.%)

Amount (at.%)

Amount (at.%)

13.06 10.49 2.67 2.92 Balance

5.80 8.98 7.94 8.84 Balance

13.08 11.22 2.74 2.98 Balance

5.80 9.50 8.15 9.00 Balance

408

The SGTE casebook 1.5 1.0 0.5 0

Cooling

–0.5 Heating

–1.0 –1.5 –2.0 1100

1200

1300

1400

1500

1.0 Liquid

0.9 0.8

Phase amount

0.7

γ

γ

0.6 0.5 0.4 0.3

γ′

0.2 0.1 0 1100

1200

1300 T (°C)

1400

1500

IV.6.2 DTA experimental curves obtained at a heating and cooling rate of 5 K min–1 compared with the calculated equilibrium phase evolution; reasonable agreements of the experimental and calculated liquidus and solidus temperatures are observed.

the precipitated crystals have the same crystallographic orientation as the surrounding dendrites. Metallographic sections of a unidirectionally solidified sample were analysed by microprobe (wavelength-dispersive X-ray spectroscopy (WDXS) to map element distributions within the area of a primary dendrite and the interdendritic surrounding.

IV.6.3

Microstructure simulation

Simulations were made using a multiphase multicomponent phase field code [98Ste, 97Tia, 006Eik]. Phase field models are known to be quite successful for describing moving phase boundaries during microstructure formation.

Microstructure of a five-component Ni-base superalloy

409

50 µm

IV.6.3 Vertical sections of a directionally solidified sample (G = 20 K mm–1; v = 0.8 mm min–1), exhibiting γ dendrites and interdendritic γ ′.

For multicomponent systems, like the model alloy investigated in this chapter, the demands for the simulation tool are that not only do multiphase and multicomponent differential equations need to be handled but also that thermodynamics have to be properly described. This is achieved by using thermodynamic databases. For this purpose, online thermodynamic calculations have been incorporated via a Fortran programming interface to a CALPHAD code. The thermodynamic calculations provide the driving force for the phase transformation and the equilibrium composition of the phases. The coupling has been described in more detail by Eiken et al. [006Eik]. Directional solidification was simulated using a unit-cell model for the isothermal dendritic cross-section, which is chosen under the assumption of fourfold symmetry of the dendrite and the surrounding dendritic array [000Ma]. The size of the unit cell corresponds to the primary dendrite spacing. A constant cooling rate was derived from the experimental process parameters (temperature gradient and solidification velocity) which correspond to the steady-state conditions in the Bridgman furnace. Secondary phases are allowed to nucleate, when the driving force for the formation of a new phase overcomes the nucleation barrier. This driving force is calculated from thermodynamic data. The following section presents calculated the microstructures of the model alloy (Table IV.6.1), obtained from simulations made with the coupled-phasefield model. Figure IV.6.4 presents distribution maps for tantalum (Ta) and

The SGTE casebook

2.5

3.0 3.5

4.0

4.5

5.0

410

60 s

620 s

Ta (at.%)

5s

60 s

620 s

W (at.%)

1.5

2.0

2.5

3.0

3.5

5s

IV.6.4 Calculated distribution maps for Ta and W within the isothermal cut through the mushy zone for directional solidification (cooling rate, 0.25 K s–1).

tungsten (W) within isothermal sections through the mushy zone. The time labels correspond to the cooling time, i.e. the time since the solidification started. The upper row shows the Ta distribution while the lower row shows the W distribution. The first two images of each row show the growth of the primary phase (γ) into the liquid; the last images exhibit the final microstructure after solidification. Because of the large difference in the solubilities of the two elements Ta and W in the γ ′ phase, the interdendritic γ ′ phase can easily be identified as light spots in the Ta, and dark spots in the W maps. The morphology of the primary γ phase has a strong influence on the size and distribution of the secondary γ′ phase. Distribution maps for Ta and W were calculated and measured by WDXS (Fig. IV.6.5). In each row the grey scale covers the same value range. The γ phase can clearly be distinguished as the Ta-poor and W-rich region with fourfold symmetry. The solubility is reversed in the γ ′ phase and thus can be seen as light spots in the Ta mapping and dark spots in the W mapping. The results of the solidification simulations were used as starting points to simulate the homogenisation treatment. The concentration-dependent diffusion matrix was obtained from a kinetic database [002Cam]. Figure IV.6.6 shows the calculated concentration profiles along the dendrite arms ( direction) in the as-cast, 1 h homogenised and 4 h homogenised states. All plots start in the centre of the primary phase (dendrite). As Al and Cr diffuse significantly more rapidly than Ta and W, these profiles flatten

0.03

0.04

0.04 0.03

50 µm

411 0.05

0.05

Microstructure of a five-component Ni-base superalloy

50 µm

xTa

0.02

0.02

0.03

0.03

xTa

50 µm

50 µm

xW

xW

IV.6.5 Measured and calculated distribution maps, scaled to the same range, for W from 1.5 to 3.5 at.% and Ta from 2.1 to 5.3 at.%. 13

Concentration (at.%)

16 Al

Cr

1h 4h As cast

15 14

12

13

11

12 10

11

4 Concentration (at.%)

Ta

W

4 3 3 2 2 0

25

Dendrite core

50 75 Distance (µm)

100 Interdendritic

1 0

25

50 75 Distance (µm)

100

IV.6.6 Concentration profiles along the dendrite arms for the as-cast state and after solutioning heat treatment at 1558 K for 1 h and for 4 h respectively.

412

The SGTE casebook

much more quickly in the beginning. Owing to cross-diffusion effects, the overall homogenisation kinetics, however, are controlled by Ta and especially by W. Figure IV.6.7 shows the volume fraction of interdendritic γ′ as a function of time for different holding temperatures. For all temperatures the homogeneous alloy would consist only of γ; thus the appearance of interdendritic γ ′ is related to the appearance of microsegregation. The symbols with error bars show the results of experiments, performed under the same conditions as for the simulated data. In all cases the heat treatment leads to a reduction in interdendritic γ ′ and an increase in the overall dissolution kinetics with increase in the temperature. For 1275 and 1285 °C, γ ′ is stabilised by the surrounding matrix. This occurs owing to microsegregation, which has to be diminished in order to destabilise the interdendritic γ ′. This incubation period is shorter or even absent for higher temperatures. The experiments show very similar kinetics.

IV.6.4

Discussion

Although the results shown here look very promising, a few comments need to be made regarding the comparison between real three-dimensional structures and simulated two-dimensional structures. The measured element distribution patterns at present relate to an arbitrary transverse cut in an arbitrarily selected dendrite of a dendrite array. A more statistical approach could help to reveal

2

1285 °C 1295 °C

Fraction of γ ′ (%)

1.75 1.5 1275 °C

1.25 1 0.75 1285 °C

0.5 1295 °C

0.25 1305 °C 0

0

1

2

3

4

5 6 Time [h]

7

8

9

10

11

IV.6.7 Dissolution of interdendritic γ ′ as a function of time, for different temperatures. The curves are obtained from simulations and the symbols from experiments.

Microstructure of a five-component Ni-base superalloy

413

the topology of an even more representative segregation pattern. This would provide better benchmark data for the simulation. Simulating microstructure evolution in three dimensions will be addressed in future simulations, using a three-dimensional unit-cell approach, but calculation times, especially with the phase field code being coupled to the thermodynamic database, will be much higher. A further aspect is the fact that simulated dendrites of the γ phase develop a more ‘compact’ morphology than experimental dendrites. Current theories [87Lan, 91Mue] have shown that the interfacial properties (interfacial energy and attachment kinetics) and their anisotropy have significant influences on the dendrite morphology. Interfacial properties are important parameters and future phase field simulations will have to focus on a closer matching of real and simulated morphologies. Despite these limitations, the qualitative agreement between experimental and simulated results is already quite good. The main factors determining a realistic microstructure evolution thus are already integrated in our approach. Future work will concentrate on a quantitative comparison between experiments and simulations.

IV.6.5

Conclusions

A five-component model alloy representative for single-crystal superalloy application, Ni–13 at.% Al–10 at.% Cr–2.7 at.% Ta–3 at/% W was produced and experiments were performed regarding the measurement of data relevant to thermodynamic equilibrium, such as the liquidus and solidus temperatures, and the partition coefficients. Unidirectional solidification experiments have generated dendritic off-equilibrium microstructures, which are needed for comparison with numerical simulation obtained by a phase field model coupled to the thermodynamic database of the Ni–Al–Cr–Ta–W system. A multicomponent multiphase field code coupled to the thermodynamic database was applied to simulate the formation and evolution of microstructures. Realistic microstructures which reproduce the influence of the primary-phase morphology on the formation and distribution of the secondary phase were simulated. The results show that the segregation of species is in good qualitative agreement with experimental results. Future work will focus on increasing the quantitative agreement between simulation and experiment. We have shown that the model can be applied successfully not only to solidification but also to homogenisation treatment. The results obtained so far look very promising and with some fine tuning a comprehensive model to describe microstructure evolution in directional solidified superalloys and homogenisation heat treatment may be expected.

414

IV.6.6

The SGTE casebook

Acknowledgement

The authors gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft within the Collaborative Research Center 370 ‘Integrated modeling of materials’.

IV.6.7 87Lan

91Mue

96Ste

97Dur 97Kat 97Tia

98Sau

99Sun 000Ma

000Kar

001Dup 002Cam 005War 006Eik 006Her

References J. S. LANGER: ‘Lectures in the theory of pattern formation’, in Chance and Matter, Proc. Les Houches Summer School, Session XLVI, Elsevier, New York, 1987. H. MUELLER-KRUMBHAAR and W. KURZ: ‘Solidification’, in Materials Science and Technology, Vol. 5, Phase Transformations in Materials (Ed P. Haasen), VCH Weinheim, 1991. I. STEINBACH, F. PEZZOLLA, B. NESTLER, M. SEEBELBERG, R. PRICLER, G. J. SCHMITS and J. L.REZENDE ‘A phase field concept for multiphase systems, Physica D 94, 1996, 135–147. M. DURAND-CHARRE: The Microstructure of Superalloys, Gordon and Breach, New York, 1997. U. R. KATTNER: ‘Thermodynamic modeling of multicomponent phase equilibria’, JOM 49(12), 1997, 14–19. J. TIADEN, B. NESTLER, H. J. DIEPERS and I. STEINBACH: ‘The multiphase-field model with an integrated concept for modelling solute diffusion’, Physica D, 115, 1998, 73–86. N. SAUNDERS and A. P. MIODOWNIK: CALPHAD (CALculation of PHase Diagrams) A Comprehensive Guide, Pergamon Materials Series, Vol. 1 (Ed. R. W. Cahn), Elsevier, Oxford, 1998. P. K. SUNG and D.R. POIRIER: ‘Liquid–solid partition ratios in nickel-base alloys’, Metall. Mater. Trans. A and 30, 1999, 2174–2181. D. MA and U. GRAFE: ‘Dendrite and microsegregation during directional solidification: an analytical model and experimental studies on the superalloy CMSX-4’, Int. J. Cast. Metals Res. 13, 2000, 85–92. M.S.A. KARUNARATNE, D.C. COS, P. CARTER and R.C. REED: ‘Modelling of the microsegregation in CMSX-4 superalloy and its homogenisation during heat treatment’, Proc. 9th Int. Symp. Superalloys (Eds K.A. Green, T.M. Pollock and R.D. Kissinger Champion, Pennsylvania, USA, 17–21 September 2000, Minerals, Metals and Materials Society, Warrendale, Pennsylvania, 2000. N. DUPIN and B. SUNDMAN: ‘A thermodynamic database for Ni-base superalloys’, Scand. J. Metall. 30(3), 2001, 184–192. C.E. CAMPBELL, W.J. BOETTINGER and U.R. KATTNER: Acta Mater. 50, 2002, 775. N. WARNKEN, D. MA, M. MATHES and I. STEINBACH: ‘Investigation of eutectic island formation in SX-Superalloys’, Mater. Sci. Eng. A, 413, 2005 267–271. J. EIKEN, B. BÖTTGER and I. STEINBACH: Phys. Rev. E 73, 2006, 066 122. R. HERZOG, N. WARNKEN, I. STEINBACH, B. HALLSTEDT, C. WALTER, J. MÜLLER, D. HAJAS, E. MÜNSTERMANN, J. M. SCHNEIDER, R. NICKEL, D. PARKOT, K. BOBZIN, E. LUGSCHEIDER, P. BEDNARZ, O. TRUNOVA and L. SINGHEISER: Advd. Eng. Mater. 8, 2006, 535–562.

IV.7 Production of metallurgical-grade silicon in an electric arc furnace G U N N A R E R I K S S O N and K L A U S H A C K

IV.7.1

Introduction

Metallurgical-grade silicon is produced in an electric arc furnace. The process is shown schematically in Fig. IV.7.1. Quartz sand and carbon are fed in appropriate proportions through the top, and liquid silicon is extracted at the bottom. The temperature in the production zone is approximately 2200 K. This is achieved through an electric arc burning between a graphite electrode and the metal bath. Hot gases are produced in the bottom zone of the reactor during the formation of silicon under the input of energy from the electric arc. These gases flow upwards as a convective flux. On their way up, heat exchange with condensed matter falling downwards takes place. To what extent can this process be understood on the grounds of equilibrium thermodynamics?

SiO2, C (s)

Graphite electrode

CO, SiO (g)

Electric arc

Si (l)

IV.7.1 A schematic drawing of the silicon arc furnace.

415

416

The SGTE casebook

IV.7.2

The stoichiometric reaction approach

It is often claimed that the production of silicon is governed by the simple stoichiometric reaction SiO2 + 2C = Si + 2CO (g)

(IV.7.1)

At equilibrium, the Gibbs energy change of the reaction must be zero. As all four phases can be considered as stoichiometric pure substances, the equilibrium constant is equal to unity. (The process takes place at atmospheric pressure, and CO is assumed to be the only gas species involved.) Thus the standard Gibbs energy change of the reaction must be zero too: K = 1 → ∆G° = 0

(IV.7.2)

where K is the equilibrium constant. Figure IV.7.2 shows ∆G° as a function of temperature. Indeed, there is a value of T for which the curve changes sign: T ⬇ 1940 K. However, this value is far below the one known from the process. Such a difference cannot be explained by deviations from unit activities or errors in the thermodynamic data. There must be other reasons.

IV.7.3

The complex equilibrium approach

If quartz is permitted to react freely with carbon in a system at a given total pressure and temperature, a different type of calculation must be carried out. All phases possible must be considered for set values of temperature, total

∆G° (J mol–1)

15 000

0

–15 000 1900

1920

1940 1960 Temperature (K)

1980

IV.7.2 ∆G° for the reaction SiO2 + 2C = Si + 2CO(g) as a function of T.

Production of metallurgical-grade silicon in an electric arc furnace

417

pressure and system composition. In particular, all possible gas species have to be introduced into the calculation. A databank search reveals the following list of phases and phase components for the system Si–O–C. Gas: Si, Si2, Si3, SiO, SiO2, C, O, O2, O3, CO, CO2 Stoichiometric condensed: C (graphite), SiC, SiO2 (quartz), SiO2 (tridymite), SiO2 (cristobalite), SiO2 (liquid), Si(s), Si(l). Assuming stoichiometric behaviour of the reaction, 1 mol of SiO2 and 2 mol of carbon are needed as input, together with the values for the total pressure (equal to 1 bar) and the temperature. The complex equilibrium calculation will describe a reaction SiO2(quartz) + 2C → ηgas + η SiO 2 ( x ) SiO2(x) + ηSiCSiC + ηSiSi

(IV.7.3)

If the temperature is varied through an interval from below the value of equilibrium for the simple stoichiometric reaction (Equation (IV.7.1)) to a value high enough to obtain liquid silicon, the resulting yield factors η can be plotted as in Fig. IV.7.3. From this figure it is obvious that the gas phase in this system contains SiO as an essential species. Thus, the simple stoichiometric reaction initially assumed cannot be correct. On the other hand, the assumption that the process can be described as a single although complex equilibrium state is also disproved. The temperature at which silicon would be produced is near 2900 K and the yield is not more than 50%. This is not in agreement with values known from the real process.

2 Equilibrium amount of species (mol)

CO(g) Si(g) Si2(g) SiO(g) Si(l) SiC SiO2 (cristobalite) SiO2(l)

0 1600

Temperature (°C)

3000

IV.7.3 One-dimensional phase map for 1 mol SiO2 + 2 mol C as a function of T.

418

IV.7.4

The SGTE casebook

The countercurrent reactor approach

In order to simulate the arc furnace as a whole, it is necessary to take into account the fact that the substances taking part in the process move in a temperature field while reacting. Cold condensed matter is fed through the top of the furnace, falling downwards, and hot gases flow rapidly upwards. On their way, they meet and exchange heat or even react with each other. Thus, the local mass balances need not be identical with the overall mass balance of the process. Additionally, the temperatures at different levels of the furnace are not controlled from outside but are mainly determined by the heat exchange and the reactions taking place. Such a complex situation can only be simulated by a thermodynamic equilibrium approach if several separate zones of local equilibrium that are interconnected by materials and heat exchange are assumed. For the silicon arc furnace, it was found that a reactor with four stages which are controlled by the internal heat balance is well suited for the modelling. The flow scheme, the values for the heat balances in each stage, the input substances, their amounts and initial temperatures as well as the distribution coefficients for the non-ideal flow between the stages are given in Fig. IV.7.4. It should be emphasised that all values are the result of a series of parameter optimizations. This set of parameters represents best the process data obtained from the silicon arc furnace of KemaNord at Ljungaverk, Sweden. Gas *∆H (kJ) = 0.000E + 00 Stage 1

T (K) P (bar)

Chemical inputs

= 1784.12 = 1.0000E + 00

C SiO2 (quartz)

*∆H (kJ) = 0.000E + 00 Stage 2

T (K) P (bar)

= 2058.82 = 1.000E + 00

*∆H (kJ) = 0.000E + 00 Stage 3

T (K) P (bar)

= 2080.17 = 1.0000E + 00

*∆H (kJ) = 8.750E + 02 Stage 4

T (K) P (bar)

= 2245.88 = 1.0000E + 00

* Regulated quantity

Pure phases

IV.7.4 Flow diagram for the silicon arc furnace.

Production of metallurgical-grade silicon in an electric arc furnace

419

Table IV.7.1 Converged solution of the reactor simulation, stage 1 STAGE 1 (ITERATION 1) *T = 1784.12 K ; P = 1.00000E+00 bar; V = 2.3002E+02 dm3

C O Si

INPUT AMOUNT mol 2.8378E+00 3.8187E+00 1.7810E+00

GAS FLOW mol 1.8000E+00 2.0000E+00 1.9988E-01

CONDENSED FLOW Mol 1.2948E+00 2.2679E+00 1.7733E+00

INPUT AMOUNT GAS mol CO 1.0377E+00 SiO 7.8081E-01 CO2 9.1223E-05 TOTAL: 1.5507E+00

EQUIL AMOUNT mol 1.5428E+00 7.7238E-03 1.2212E-04 1.0000E+00

MOLEFRACTION mol 9.9494E-01 4.9810E-03 7.8751E-05 1.0000E+00

FUGACITY

FLOW

bar 9.9494E-01 4.9810E-03 7.8751E-05 1.0000E+00

mol 1.7999E+00 1.9985E-01 1.4525E-04

SiO2 (cristobali) C SiC SiO2 (tridymite) SiO2 (liquid) SiO2 (quartz) T Si(l)

mol 0.0000E+00 1.8000E+00 0.0000E+0O 0.000OE+0O 0.0000E+00 1.0000E+00 0.0000E+00

mol 1.1339E+00 6.5550E-01 6.3932E-01 0.0000E+00 0.0000E+00 0.0O0OE+0O 0.0000E+00

ACTIVITY 1.0000E+00 1.0000E+00 1.0000E+00 9.9978E-01 9.3374E-01 9.1635E-01 2.0937E-02

mol 1.1339E+00 6.5550E-01 6.3932E-01 O.O0O0E+O0 0.0000E+00 O.OOOOE+00 0.0000E+00

ENTHALPY OF REACTION = 0.0000E+00 J ENTROPY OF REACTION = 1.1262E+02 J.K-1 THE CUTOFF LIMIT HAS BEEN SPECIFIED TO 1.000E-05 Data on 1 species identified with “T” have been extrapolated *Was calculated from enthalpy change

Using the above input data for stage I and the flow data for the streams between the stages, the converged solution of the reactor simulation is given by the set of Tables IV.7.1 to IV.7.4. Each of the tables represents one stage of the reactor, stage 1 (Table IV.7.1) being the top stage, and stage 4 (Table IV.7.4) the bottom stage. The tables contain information on the local equilibrium state (T, P and the phase amounts and compositions), the flow of matter between the stages (inflowing and outflowing substance amounts as well as elementary flows), and the heat balance conditions (adiabatic behaviour (stages 1–3) or constant-enthalpy input (stage 4)). The most important information for the derivation of a materials flow diagram of the reactor (Fig. IV.7.5) is given in the column named Flow. If an ordinate is chosen,

420

The SGTE casebook

Table IV.7.2 Converged solution of the reactor simulation, stage 2 STAGE 2 (ITERATION 1) *T = 2058.82 K P = 1.00000E+00 bar V = 3.0467E+02 dm3

C O Si

GAS CO SiO CO2 Si TOTAL:

INPUT AMOUNT mol 2.0703E+00 4.1050E+00 2.8352E+00 INPUT AMOUNT mol 7.7540E-01 1.0616E+00 5.7548E-05 3.1171E-04

SiO2 (liquid) SiC SiO2 (cristobali) SiO2 (tridymite) T SiO2 (quartz) T Si (l) C

GAS FLOW mol 1.2948E+00 2.2679E+00 9.7313E-01

CONDENSED FLOW Mol 9.8154E-01 2.3251E+00 2.1441E+00

EQUIL AMOUNT mol 1.0887E+00 6.9100E-01 9.9487E-05 7.3605E-05 1.7798E+00

MOLEFRACTION mol 3.8824E-01 4.9810E-03 5.5897E-05 4.1355E-05 1.0000E+00

FUGACITY

FLOW

bar 6.1166E-01 3.8824E-01 5.5897E-05 4.1355E-05 1.0000E+00

mol 1.2947E+00 9.7295E-01 1.1436E-04 1.6745E-04

mol 0.0000E+00 6.3932E-01 1.1339E+00 O.O000E+00 0.0000E+00 0.0000E+00 6.5550E-01

mol 1.1626E+00 9.8154E-01 0.0000E+00 O.0OO0E+00 0.0O0OE+0O 0.0000E+00 O.0OO0E+00

ACTIVITY 1.0000E+00 1.0000E+00 9.8254E-01 9.8012E-01 8.7802E-01 5.0165E-01 1.2585E-01

mol 1.1626E+00 9.8154E-01 O.O0O0E+O0 0.0000E+00 O.OOOOE+00 0.0000E+00 0.0000E+00

ENTHALPY OF REACTION = 0.0000E+00 J ENTROPY OF REACTION = 1.3954E+01 J.K-1 THE CUTOFF LIMIT HAS BEEN SPECIFIED TO 1.000E-05 Data on 2 species identified with “T” have been extrapolated *Was calculated from enthalpy change.

which shows in arbitrary units the width and sequence of the four stages, the calculated materials flow can be marked for the upflowing gases on the upper (left) abscissa in each segment, and for the downflowing condensed phases on the lower (right) abscissa in each segment. Thus, continuous curves are obtained which represent the total materials flow through the reactor relative to 1 mol of silica and 1.8 mol of carbon in the feed. The most important result, the silicon yield, can be read on the lowest abscissa to be 0.8 mol per mole of silica. The total elementary mass balance of the reactor (Table IV.7.5) is readily obtained from the values on the outermost abscissae.

Production of metallurgical-grade silicon in an electric arc furnace

421

Table IV.7.3 Converged solution of the reactor simulation, stage 3 STAGE 3 (ITERATION 1) *T = 2080.17 K P = 1.00000E+00 bar V = 3.2259E+02 dm3 INPUT AMOUNT mol C 1.762E+00 O 4.1654E+00 Si 3.2049E+00

GAS SiO CO Si CO2 TOTAL:

INPUT AMOUNT mol 1.0596E+00 7.8063E-01 1.0190E-03 3.0834E-05

SiO2 (liquid) SiC SiO2 (cristobali) SiO2 (tridymite) T SiO2 (quartz) T Si (l) C

GAS FLOW mol 9.8155E-01 2.3251E+00 1.3440E+00

CONDENSED FLOW Mol 9.7589E-01 2.3003E+00 2.1261E+00

EQUIL AMOUNT mol 1.0786E+00 7.8630E-01 1.5081E-04 6.4708E-05 1.8652E+00

MOLEFRACTION mol 5.7831E-01 4.2157E-01 8.0856E-05 3.4693E-05 1.0000E+00

FUGACITY

FLOW

bar 5.7831E-01 4.2157E-01 8.0856E-05 3.4693E-05 1.0000E+00

mol 1.3435E+00 9.8145E-01 4.0555E-04 7.2417E-05

mol 1.1626E+00 9.8154E-01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E-00

mol 1.1502E+00 9.7589E-01 0.0000E+00 0.0000E+00 0.0000E+0O 0.0000E+00 0.0000E+00

ACTIVITY 1.0000E+00 1.0000E+00 9.7691E-01 9.87388-01 8.7109E-01 7.7554E-01 8.7603E-02

mol 1.1502E+00 9.7589E-01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

ENTHALPY OF REACTION = 0.0000E+00 J ENTROPY OF REACTION = 2.2424E-01 J.K-1 THE CUTOFF LIMIT HAS BEEN SPECIFIED TO 1.000E-05 Data on 2 species identified with “T” have been extrapolated *Was calculated from enthalpy change.

As a mass balance with respect to the main species of the system one obtains: SiO2 (quartz) + 1.8C → 0.8Si(l) + 1.8CO(g) + 0.2SiO(g) (IV.7.4) Note that this is not an isothermal reaction equation as silica and carbon enter the reactor at room temperature, silicon leaves the reactor at 2200 K and the gas phase leaves at 1874 K. The temperature distribution that results from the energy input in the production zone (∆H = +875 kJ per mole of SiO2) and the assumed adiabatic behaviour (∆H = 0) of the three upper zones is also given in the diagram. The highest temperature is reached in the production zone. Its value (2245 K) is

422

The SGTE casebook

Table IV.7.4 Converged solution of the reactor simulation, stage 4 STAGE 4 (ITERATION 1) *T = 2245.88 K P = 1.00000E+00 bar V = 4.2980E+02 dm3

C O Si

GAS SiO CO Si Si2C CO2 Si2 TOTAL:

INPUT AMOUNT mol 9.7589E-01 2.3003E+00 2.1261E+00 INPUT AMOUNT mol 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Si (l) SiC C SiO2 (liquid) SiO2 (cristobali) SiO2 (tridymite) T SiO2 (quartz) T

GAS FLOW mol 9.7589E-01 2.3003E+00 1.3259E+00

CONDENSED FLOW Mol 0.0000E+00 0.0000E+00 8.0012E-01

EQUIL AMOUNT mol 1.3245E+00 9.7578E-01 1.2737E-03 6.9523E-05 3.8543E-05 2.4640E-05 2.3017E+00

MOLEFRACTION mol 5.7544E-01 4.2395E-01 5.5340E-04 3.0206E-05 1.6746E-05 1.0705E-05 1.0000E+00

FUGACITY

FLOW

bar 5.7544E-01 4.2395E-01 5.5340E-04 3.0206E-05 1.6746E-05 1.0705E-05 1.0000E+00

mol 1.3245E+00 9.7578E-01 1.2737E-03 6.9523E-05 3.8543E-05 2.4640E-05

mol 0.0000E+00 9.7589E-01 0.0000E+00 1.1502E+00 0.0000E+00 0.0000E+00 0.0000E+00

mol 8.0012E-01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+0O 0.0000E+00 0.0000E+00

ACTIVITY 1.0000E+00 8.2082E-01 9.3929E-02 5.9571E-02 5.5865E-02 5.5186E-02 4.8820E-02

mol 8.0012E-01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

ENTHALPY OF REACTION = 8.7500E+05 J ENTROPY OF REACTION = 4.1861E+02 J.K-1 THE CUTOFF LIMIT HAS BEEN SPECIFIED TO 1.000E-05 Data on 2 species identified with “T” have been extrapolated *Was calculated from enthalpy change.

higher than that for which the standard Gibbs energy change of the simple reaction in the first paragraph changes sign (1940 K), but it is also considerably lower than the value for which the single-cell complex equilibrium calculation shows a maximum of the silicon yield (approximately 2900 K). A thermochemical understanding of the different reactions taking place at different levels within the reactor can be obtained from the materials flow between stages 3 and 4 and stages 1, 2 and 3 respectively. The reactor can obviously be split into two distinct zones that are governed by two separate

Production of metallurgical-grade silicon in an electric arc furnace 1784 K

2059 K

2080 K

423

2246 K

Flow (mol)

2 CO C 1

SiO

SiO2

SiC CO SiC Si

Stage 1

Stage 2

Stage 3

Stage 4

Top

Bottom

IV.7.5 The materials flow for the calculated steady state. Table IV.7.5 Total elementary mass balance of the reactor Component

Inpuτ (mol)

Output (mol)

Si C O

1SiO2 1.8C 2SiO2

0.8Si + 0.2SiO(g) 1.8CO(g) 1.8CO(g) + 0.2SiO(g)

processes. In the bottom zone, silicon is produced according to the mass balance equation SiO2 + SiC → Si + SiO(g) + CO(g)

(IV.7.5)

In the top zone, the upflowing silicon monoxide reacts with the carbon according to the mass balance equation C + SiO(g) →

2 3

SiO2 +

1 3 SiC

+

2 3

CO(g)

(IV.7.6)

This reaction is, however, usually incomplete because the amount of incoming carbon is too low to allow all SiO(g) to react. Thus, a loss of silicon (20%) cannot be avoided. It is worth noting that SiC is both formed and consumed within the reactor and, therefore, does not occur in the total mass balance. However, it is an essential phase in the whole process! A more detailed analysis of the process was given by Eriksson and Johansson [78Eri]. They have also employed modifications of the above set of calculational parameters, e.g. to study the influence of the energy supply and the amount of carbon fed into the process. Figure IV.7.6 shows the silicon yield as a function of energy supply for two different amounts of carbon feed. The

424

The SGTE casebook 100

Si yield (%)

2 mol C per mole SiO2

1.8 mol C per mole SiO2

65 865

890 Energy supply (kJ per mol SiO2)

IV.7.6 The silicon yield for two different amounts of carbon feed.

lower curve (1.8 mol of carbon) shows that the optimum value for the silicon yield (80%) is obtained for 875 kJ per mole of silica. This value was used in the calculations discussed above. A higher-energy supply will not raise the silicon yield. For a higher carbon feed (2 mol per mole of silica), the upper curve indicates a possible silicon yield of more than 95%. However, the resulting temperature level in the bottom stage (>2700 K) would make such a combination of process parameters technologically unfeasible. The calculations for this case study have been performed using ChemSage and a new process model generated by using the SimuSage package.

IV.7.5 78Eri

Reference G.ERIKSSON and T. JOHANSSON: Scand. J. Metall. 7, 1978, 264–270.

IV.8 Non-equilibrium modelling for the LD converter M I C H A E L M O D I G E L L, A N K E G Ü T H E N K E, P E T E R M O N H E I M and K L A U S H A C K

IV.8.1

Introduction

In the LD (Linz–Donawitz) converter process, pure oxygen is blown on a molten iron bath for refining purposes. Elements dissolved in the molten iron, e.g. C, Si, Mn and P, but also part of the molten iron itself are oxidised. They form either a slag phase covering the hot metal or, in the case of C, gas bubbles containing CO and CO2. Several reaction zones can be identified in Fig. IV.8.1. In the hot spot, the oxygen directly reacts with iron and dissolved elements. Owing to the impact of the oxygen jet, iron droplets are dispersed in the slag phase as well as slag droplets in the metal bath. The metal–slag dispersion is mixed further by CO and CO2 bubbles and serves as the main reaction zone. A third zone contains the hot metal which is not dispersed in the slag but forms the bath underneath. Droplets from the dispersion fall back into this bath.

Slag Gas bubbles

Metal–slag dispersion Gas–metal reaction zone hot spot

Metal droplets

Metal bath

IV.8.1 LD converter process

425

426

IV.8.2

The SGTE casebook

Process model development

The task of developing a suitable converter model will be discussed on the basis of the basis of the decarburisation reaction of the iron melt. Obviously, the rate of the decarburisation reaction depends on the reaction rate of [C] + (O) = {CO}

(IV.8.1)

and the transport conditions in the converter. Thereby, these are determined by oxygen blowing conditions and CO formation. On the other hand, CO formation is influenced by the decarburisation reactions. Intermediary formation of FeO in the hot spot is the main oxygen source for the decarburisation. FeO is dispersed in the slag and the metal bath as well as Fe–C droplets, which are accelerated by the oxygen jet. The subsequent reaction of FeO with carbon dissolved in iron occurs in both phases. In the slag phase, dissolved FeO reacts with dispersed Fe–C droplets; in the metal bath, FeO droplets form the dispersed phase. For the decarburisation, several reaction routes can be formulated: (FeO) + [C] = [Fe] + {CO}

(IV.8.2)

(FeO) + {CO}= [Fe] + {CO2}

(IV.8.3)

{CO2} + [C] = 2{CO}

(IV.8.4)

Reaction (IV.7.2) is kinetically limited [87 Bar]. The time for the complete reduction of an FeO droplet with a diameter of 1 mm (equalling 4.7 × 10–5 mol) in an Fe–C melt amounts to between 30 and 175 s. The same amount of FeO reacts with pure CO according to Equation (IV.8.3) in only 0.2 s. The consecutive Reaction (IV.8.4) takes place in only 2 × 10–3 s. From these data it can be concluded that the direct reaction between FeO and C contributes little to the decarburisation owing to its kinetic limitation. However, Reaction (IV.8.3) and (IV.8.4) can only occur spontaneously when CO2 bubbles come into contact with Fe droplets in the slag phase or when CO bubbles come into contact with FeO droplets in the metal phase. The probability for such contacts depends on the transport conditions in the slag and metal bath respectively. In principle, the conditions close to the phase boundary of the droplets can be modelled by transport and reaction equations. In a simple two-layermodel, a transport equation according to Fick’s law can be stated for every component X: nX = KX,effA(cX, metal – cX, phase boundary)

(IV.8.5)

The transport coefficients KX,eff are functions of the macroscopic and the local microscopic fluid movement, which is caused by the stirring of the gas bubbles. Additionally, the increase in the diffusion boundary layer during the

Non-equilibrium modelling for the LD converter

427

reaction due to the formation of a pure Fe phase has to be taken into consideration for all three reaction routes Equation (IV.8.2), Equation (IV.8.3) and Equation (IV.8.4). For the reaction rate, Equation (IV.8.6) can be stated –RX = kX ∏ c ini

(IV.8.6)

However, there is no reliable information on transport coefficients, reaction rate constants and concentration ratios that are needed for solving Equation (IV.8.5) and Equation (IV.8.6). Hence, a process model based on these equations is not promising. Instead, the process will be modelled on a less detailed level employing a cell model, which is based on the concept of local equilibrium. As shown in Fig. IV.8.2, according to the main reaction zones, the converter is divided into four sections. These are treated assuming complete (local) equilibrium: the hot spot, where the reaction between oxygen and iron melt

Gas

Slag EQ

Oxygen

Gas

Slag

Slag

Slag Flux

Gas Slag

Hot Spot EQ

Slag Metal slag EQ

Gas Metal

Metal Metal

Gas Metal

Metal Bath EQ

Metal

Metal

Gas Metal Slag

Melt

Metal

Scrap

Metal

All phases

Slag

Input

Gas

Output

Phase or amount splitter

EQ Equilibrium reactor Cut for time step

IV.8.2 Cell model for the LD converter process.

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The SGTE casebook

takes place, the metal–slag zone, where the conversion of the FeO from the hot spot with melt droplets takes place, and the metal bath. The fourth zone, the slag, as shown in Fig. IV.8.2 is needed for the temporal discretisation of the model. It is assumed that the total amount of FeO formed in the hot spot is transported to the metal–slag zone, where it reacts with the carbon-containing iron melt and slag already available. This assumption is reasonable as reactions taking place in the bath zone can be transferred to the metal–slag zone. The individual ideal reaction zones are interlinked by material streams in such a way that a circular flow through these zones results. This flow models the circulation of the material in the converter and the stream of metal droplets through the slag phase as has been observed in experimental studies. With a given oxygen blow rate and lance height, the circulation rate between the bath reaction zone and hot spot reaction zone can be defined. This also determines the mass flow between the assumed reaction zones, for which a boundary condition can be stated. Assuming the experimentally proven full conversion of oxygen in the hot spot leads to 2 n˙ O 2