i
The
casebook
Thermodynamics at work
ii
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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
v
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
xiii
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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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
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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< j
+ ΣΣΣΣ x i x j x k x l Lijkl + . . .
(I.2.30)
i < j < k
Figure I.2.9, Fig. I.2.10 and Fig. I.2.11 show the different contributions given in equations (I.2.26)–(I.2.29). It should be noted, however, that the excess function has been chosen to be negative for clarity. This is not generally the case. In particular, systems exhibiting miscibility gaps must show positive excess Gibbs energies. Otherwise it is not possible that a Gibbs energy curve has two minima. At this point it is worth mentioning that there are several so-called geometrical models by which one can predict the excess Gibbs energy of a ternary solution as a kind of weighted average of the values for the binary solutions. Such methods have the advantage that they can even be applied to
Total Gibbs energy [J]
–70000
G ref G ref + G id G ref + G id + G ex
G ref
G id
G ex
–90000 0
1 Mole fraction of A (liquid)
I.2.9 All contributions to the Gibbs energy of a binary mixture phase for 1 mol of the liquid phase.
Models and data
27
0
Total Gibbs energy [J]
G id
G ex
G id G id + G ex –20 000 0
1 Mole fraction of A (liquid)
I.2.10 Contribution of ideal and excess term of a binary mixture phase for 1 mol of the liquid phase.
0
Total Gibbs energy [J]
G ex
G ex –20000
0
1 Mole fraction of A (liquid)
I.2.11 Contribution of excess Gibbs energy of mixing of a binary mixture phase for 1 mol of the liquid phase.
hand-drawn curves and without assessing the complete binaries. For computer calculations it is of course necessary that at least all binary subsystems be described by mathematical expressions covering the complete composition range. The most well-known methods are those by Kohler and Toop [60Koh]. A method suggested by Muggianu turns out to give the same mathematical
28
The SGTE casebook
expression as the Redlich–Kister equation used for a ternary. Pelton [001Pel] has recently given a comprehensive summary of these methods. In the above model the L parameters may be given a temperature dependence and thus they can be used to represent deviations from ideal entropy, although in a rough way only. There are more advanced theories considering deviations from random mixing of the atoms which are termed short-range order. In general, they do not yield explicit expressions for the Gibbs energy and are thus difficult to use except by numerical methods. Among the various models treating short-range order, one should mention the simplest, namely the classical quasichemical model [52Gug], later expanded by Pelton and Blander [86Pel], and a very advanced model, namely the cluster variation model (CVM), developed by Kikuchi [74Kik, 85Kik].
I.2.5.2
Partial properties for the substitutional solution
From the expression for the molar Gibbs energy of a substitutional solution, one can calculate the chemical potential for any component element i by the equation (I.2.31) µ i = Gi = ∂ G with G = Σ ni Gm ∂ni One obtains for substitutional solutions, i.e. in cases when the mole fractions are used as composition variables,
Gi = G m +
∂Gm ∂Gm – Σ xj ∂x i ∂x j
(I.2.32)
where all xj are treated as independent variables in the derivatives. It should be noted that the partial properties thus obtained do naturally obey the Gibbs– Duhem equation. Also note that the appropriate derivatives of this equation with respect to temperature yield the partial values for enthalpy, entropy and heat capacity.
I.2.5.3
The sublattice model
Now consider a solid solution where there is more than one kind of lattice site. A simple case with two sublattices and two elements on each site would be represented by the formula (A, B)a(C, D)c where a and c give the numbers of different sites per formula unit. The simplest model for such a solution would be obtained by assuming random mixing of the atoms within each sublattice. It is then convenient to define mole fractions for each sublattice. They are called site fractions and are denoted ys where the superscript s identifies the sublattice. The site fractions are used to define the frame of reference for the Gibbs energy of the solution phase. The Gibbs energy equation for a phase with two sublattices would thus be
Models and data
GmΦ = ΣΣ y i1 y 2j ° GijΦ + aRT Σ y i1 ln y i1 + cRT Σ y 2j ln y 2j
29
(I.2.33)
This is the so-called ‘compound-energy’ model [81Sun] and °GIJϕ represents the molar Gibbs energy for the compound iajc in the structure ϕ. Deviations from the ideal solution are primarily described with terms representing interactions between atoms within a sublattice. Recognising that such an interaction may be influenced by what atoms occupy the other sublattice, the excess Gibbs energy is represented by an expression GmE ,Φ = Σ y i1 ΣΣ y 2j y k2 Σ Lνi : j, k ( y 2j – y k2 ) ν j
+ Σ y i2 ΣΣ y 1j y 1k Σ Lνi , j : k ( y 1j – y 1k ) ν j
(I.2.34)
In addition, one may add so-called ‘reciprocal’ terms of the type y i1 y 1j y k2 y l2 Li,j :k,l . It was suggested that this term can give an approximate account of the effect of short-range order. It would then be negative and proportional to T (see below). It should be noted that the subscripts of the interaction parameters are constructed by separating elements on the same sublattice by a comma and elements on different sublattice by a colon. Thus i,j:k,l represents the index for interaction of components i and j on sublattice 1 with simultaneous interaction of components k and l on sublattice 2.
I.2.5.4
Partial properties for the sublattice model
The molar Gibbs energy for the (two) sublattice models discussed above is given per mole of formula units of the phase (A, B)a(C, D)c. The chemical potential of any compound that can be derived by filling the sublattices with one sublattice component each can be calculated from an equation similar to that for the simple substitutional solution. This time, however, the site fractions must be used in the partial derivatives. One obtains for example for the compound AaCc GA a C c = Gm +
∂Gm ∂Gm ∂Gm ∂Gm + – Σ y i1 – c Σ y 2j ∂y 1A ∂y C2 ∂y i1 ∂y 2j
(I.2.35)
There is no way of calculating the chemical potential of the elements from the model in this form. This is consistent with the fact that chemical potentials of elements are not defined for such a phase. They will not be defined until there is equilibrium with other phases. However, as for the substitutional solution, the temperature derivatives of the above equation yield the appropriate values for the enthalpy, the entropy and the heat capacity of the compounds. Because the compounds AaCc are major building blocks of the phase, the above method is often called the
30
The SGTE casebook
compound-energy formalism. Further applications of this formalism are briefly described below.
I.2.5.5
Interstitial solutions
In an interstitial solution an alloying element does not substitute for a host element but dissolves in interstitial sites. That kind of solution is described by considering two sublattices, the main (substitutional) sublattice and the interstitial sublattice (Fig. I.2.12). It should then be realised that the interstitial sublattice is mainly occupied by vacancies, which must be introduced as a special element. As an example, the solution of Mn and C in fcc Fe is represented by the formula (Fe, Mn)1(Va, C)1. The compound energies °GFe,Va and °GMn,Va represent the Gibbs energy of pure fcc Fe and fcc Mn, but °GFe,C and °GMn,C represent the Gibbs energy of hypothetical carbides, the values of which must be evaluated to fit the experimental information on the solution which may or may not extend to very high C contents. For cases such as TiC or NbC the situtation may actually be inverse; at equilibrium the second sublattice is mainly occupied by carbon. A small number of vacancies on the carbon site can then be used to treat the nonstoichiometry of MC1–x. From the compound-energy model, one can primarily calculate the chemical potential for compounds as shown before. For the above example, one would thus obtain
° C G Me
° C G Me
2
1
° G Me
C
° G Me
1
2
Gm
Me1
Me2
I.2.12 Schematic drawing of the Gibbs energy surface of (Me1, Me2) (Va, C).
Models and data
GFe = GFe,Va = Gm + 2 – y Va
31
∂Gm ∂Gm ∂G ∂G – – y 1Mn 1m – y 1Fe 1m 1 2 ∂y Fe ∂y Va ∂y Mn ∂y Fe
∂Gm ∂Gm – y C2 2 ∂y Va ∂y C2
(I.2.36)
but the chemical potential of the interstitial element would be obtained from the difference
GC = GFe,C – GFe,Va = GMn,C – GMn,Va =
I.2.5.6
∂G ∂Gm – 2m ∂y C2 ∂y Va
(I.2.37)
Lattice defects
The multisublattice approach can also be used for the case of phases with lattice defects, e.g. antisite atoms and vacancies. Thus the homogeneity range of a compound ‘A2B’ could be treated as (A, B)2(B, Va). The formula indicates that one has assumed antisite atoms on the A site and vacancies on the B site. The compound energies for the end members A2B (ideal compound), as well as A2Va, B2B and B2Va, would be needed. The Gibbs energies of the latter three end members can be used as adjustable parameters of the model. It should, however, be borne in mind that these energies are indirectly related to the formation energies of the defects. On the basis of the concept outlined in the previous paragraph it is obvious how interstitial defects could be integrated into this concept.
I.2.5.7
Ionic solid solutions
Ionic phases usually have different sublattices for cations and anions. There can be several cation or anion sites. If an ion of a different valency is dissolved in an ionic compound, it is necessary to maintain electroneutrality by introducing either vacancies or interstitials. The former case may be illustrated by the solution of SrCl2 in KCl which is represented by the formula (K+, Sr2+, Va0)1 (Cl–)1, where the vacant sites are assumed to be without a charge. In order to handle ionic solutions the compound-energy model needs to be combined with the condition of electroneutrality. It gives restrictions on what values are permitted for the site fractions yi. In the present case, one obtains 1(yK + 2ySr) – 1yCl = 0
(I.2.38)
A further application of the crystalline electric field (CEF) for ionic solid solutions is given by its use for the spinel phase. This is a solid oxide solution the main feature of which is the simultaneous occurrence of divalent and trivalent cations together with divalent oxygen anions. A simple formula for
32
The SGTE casebook
this phase would be Me′OMe″2O3 or (Me′)1(Me″)2O4. In this simple form a group of several divalent cations, e.g. Ni and Mg, can be accommodated on the first sublattice while trivalent cations, e.g. Cr and Al, occupy at the same time the sites of the second sublattice and oxygen occupies the third sublattice. This simple approach can even handle a case in which one element can occur in two different oxidation states, e.g. Fe2+ and Fe3+. FeO.Fe2O3, i.e. Fe3O4, would result as a pure stoichiometric compound. However, it is well known that Fe3O4 has a small but not negligible homogeneity range with respect to oxygen. This can only be incorporated into the model if defects are introduced. Even that is not a problem in the framework of the CEF. In fact, a full spinel model incorporating a good many of the elements at present known to dissolve in this phase is (Co2+, Cr2+, Fe2+, Mg2+, Ni2+, Zn2+, Al3+, Co3+, Cr3+, Fe3+)1 [Al3+, Co3+, Cr3+, Fe3+, Co2+, Fe2+, Mg2+, Ni2+, Zn2+, Va0]2 (O2–)4. One can recognise quite clearly the antisite atoms (ions), e.g. Al3+ on the first or Mg2+ on the second sublattice, and there are also vacant sites (Va) on the second sublattice which compensate for the excess charge caused by the anti-site atoms. A total of 10 × 10 × 1 = 100 compounds results from this phase formula, the majority of which are not real substances because they are composed of antisite combinations of the ions. An example is (Al3+)1(Mg2+)2(O2–)4 which carries in total one negative charge. As indicated above, the Gibbs energies of such compounds are the major adjustable parameters of the model while those of the ‘real’ compounds, e.g. (Mg2+)1(Al3+)2(O2–)4, are often already contained in the databases for pure substances.
I.2.5.8
The ideal gas
The gas phase is very often treated by an ideal model, which, however, takes into account the amounts of all species that actually exist in the gas. That is to say, there can be phase internal equilibria between some of the species. Usually, a minimisation of the Gibbs energy must be performed in order to find the equilibrium fractions of all species. The Gibbs energy is given for 1 mol of the final mixture of species and it should be noticed that the number of moles is not known until after the calculation has been performed. The expression for the Gibbs energy is Gm = Σ y i °Gi + RT Σ y i ln y i + RT ln P
(I.2.39)
and yi is used here to express the mole fractions among the species, and not among the elements. As an example, Fig. I.2.13 shows the Gibbs energy versus the composition for the H–O system at 1 bar. The variable on the composition axis was here taken as x O 2 = n O 2 /( n H 2 + n O 2 ) and for consistency the Gibbs energy was not given per mole of actual species but per 2 mol of total H and O, i.e. 2nH + 2nO. It is interesting to note that the curve has a V shape with a very sharp minimum at the position of H2O. Because of the high stability of H2O molecules
Models and data
33
Relative Gibbs energy (J mol–1)
0
∆GH° 2O
–140 000
0
1
xO2
I.2.13 Gibbs energy of the gas phase relative to pure H2 and O2 (T = 1000 K). 1
Equilibrium activities of species
H2 H2O O2
0
0
1
xO2
I.2.14 Activities (partial pressures at 1 bar total pressure) of the species in the system H2–O2 (T = 1000 K).
at T = 1000 K, there are very few other molecules at that composition. To the left of that composition there is hardly any O2 and to the right hardly any H2. This fact is further illustrated by Fig. I.2.14 showing the activities of H2, H2O and O2, where the activities are defined from the chemical potential by
µi = °Gi + RT ln ai and °Gi is the molar Gibbs energy for pure species i at 1 bar.
(I.2.40)
34
I.2.5.9
The SGTE casebook
Liquid solutions
Many models are used to represent the Gibbs energy of liquid solutions. Assuming that all the atoms mix at random with each other, one can directly apply the substitutional model. Assuming that an element with small atoms dissolves between the other atoms, one can apply the interstitial model. There are many cases where a liquid solution shows a strong deviation from random mixing. Such cases can sometimes be described by assuming that molecularlike associates form. By treating an associate as a new compound, one can again apply the mathematics of the substitutional solution model. For instance, suppose the compound A2B shows a high stability in the solid state of the A–B system. Then one may assume that A2B molecules exist in the liquid and one may apply the following expression: Gm = y A0 GA + y B0 GB + y A0 2 B GA 2 B + RT Σ y i ln y i + Gmex
(I.2.41)
where yi is used to define the mole fractions among A, B and A2B similar to those of a set of gas species. The important parameter in this ‘associated solution model’ is
∆ 0 GA 2 B = 0 GA 2 B – 2 0 GA – 0 GB
(I.2.42)
The Gm expression for this model is very similar to Gm for ideal gas. When ∆ 0 GA 2 B has a large negative value, then Gm versus composition would be very similar to the H–O case illustrated in Fig. I.2.13 and Fig. I.2.14. The associate solution approach treats short-range order in this way. Ionic liquids are composed of cations and anions. These cannot mix randomly because of local electroneutrality. Thus it is reasonable to assume that anions are surrounded by cations and vice versa. According to Temkin [45Tem], this situation can be described by introduction of two sublattices, one for the cations and another for the anions. The relative number of sites on the two sublattice now depends upon the valencies which can be illustrated by considering the liquid solution of SrCl2 in KCl. It would be represented by the formula (K+, Sr2+)P(Cl–)Q, (Fig. I.2.15). It is evident that Q/P varies from 1 for pure KCl to 2 for pure SrCl2. In the solid state there would be two quite different phases, the KCl rich solution (K+, Sr2+,Va0)1(Cl–)1 and the SrCl2 rich solution (Sr2+, K+)1(Cl–,Va0)2, while in the liquid there must be continuous transition from KCl to the SrCl2. Vacancies will in general not be required in the liquid state. (For an exception see below.) It has been proposed [85Hil] that a convenient way of defining the values of P and Q is to identify them with the average value of the ions on the other sublattice. For the liquid solution (A+, B2+)P (C–,D2–)Q, one thus obtains
– P = – 1 y C2 – 2 y D2 Q = + 1 y 1A – 2 y 1B
(I.2.43)
Models and data Va0Cl–
Cl
or
K
35
Sr
SrCl2
K+Cl–
Sr2+Cl–
I.2.15 Composition space of an electroneutral crystal.
A liquid solution between a metal and a non-metal is sometimes represented by the above ionic two-sublattice model. For the liquid Fe–O solution, one has thus used a model defined by the formula (Fe2+, Fe3+)P(Vaq–, O2–, O0)Q
(I.2.44)
The neutral O0 atoms have here been introduced in order to allow the solution to extend beyond the O content of Fe2O3. The vacancies have been introduced in order to allow the model to extend all the way to pure Fe. This is accomplished by assuming that these vacancies are not neutral but have an induced charge equal to minus the average of the charges of the cations, here
– q Va = – 2 y 1Fe 2+ + 3 y 1Fe 3+ = Q
(I.2.45)
The ionic two-sublattice model can also be applied to solutions between metals showing a strong deviation from random mixing. For instance, the liquid phase in the Mg–Sb system, which has a very stable solid phase Mg3Sb2, can be described by assuming that Mg is divalent and Sb is trivalent, (Mg2+)P(Sb3–, Vaq–, Sb0)Q. Surprisingly, the result of this model is identical with the result of the associated solution model if the associates are assumed to contain one atom of the more electronegative element, in this case Mg1.5Sb1. Molten oxides, in particular silicates, have been described with the associate solution model using neutral associates such as CaO, Ca2SiO4 and SiO2. This is equivalent to using an ionic two-sublattice model with (Ca2+)P(SiO44–,O2–, SiO20)Q. So far no particular comments have been made to the contributions of the excess function in the liquid. In general, this can be handled in the form of Equation (I.2.34) above. The reciprocal terms for systems of the type (A, B)a(C, D)b have already mentioned above. This case becomes particularly interesting for ionic liquid systems, e.g. the simple case (Na+, K+)P(Cl–, F–)Q. One can see immediately that P and Q will both be equal to one, and that the compounds are the simple salts NaCl, KCl, NaF and KF. This is, however, not a four-
36
The SGTE casebook
component system but has only three independent components. The reason for this is the exchange reaction NaCl + KF = NaF + KCl
(I.2.46)
This, however, provides not only a mass-balance constraint and thus reduces the number of free components to three but also has an effect because of its Gibbs energy change. In the present case, ∆G for the reaction is negative. The salts NaF and KCl are said to form the stable pair. In fact, if ∆G is sufficiently large, it can lead to a miscibility gap that is centred close to the stable diagonal in the reciprocal phase diagram. Blander [64Bla] proposed to derive from ∆G a value for the reciprocal term in the excess function which would be in the present case G ex =
– y Na y K y Cl y F ( ∆G ) 2 ZRT
(I.2.47)
In this equation, Z is the first-nearest neighbour co-ordination number. It is obvious that such a parameter can be derived from the pure substance data of the phase constituents involved. Even if no experimental data for the reciprocal system are available, this term can be given a value and therefore needs to be introduced. Two more liquid models should be mentioned which have also been used to describe silicate melts. One was given by Kapoor and Frohberg [71Kap] and was modified by Gaye and Welfringer [84Gay]. It treats shortrange order with a so-called cell model. The energy of formation of the cells and the energy of interaction between the cells are the adjustable parameters in this model. The other model is the quasichemical model that was first introduced by Guggenheim [35Gug]. The basic concept of this model is to use the bond energies between nearest neighbours for the derivation of a compositiondependent total Gibbs energy. In a binary system A–B there can be A–A and B–B bonds and one could envisage a formation reaction of two A–B bonds from one ‘pure’ pair each: (A–A) + (B–B) = 2(A–B)
(I.2.48)
If one now assumes two different coordination numbers ZA and ZB, one can write mass balance equations ZAXA = 2nAA + nAB
(I.2.49)
ZBXB = 2nBB + nAB where XA and XB are the overall mole fractions of A and B and nIJ is the number of bonds in one mole of solution. Introducing the mole fractions XAA, XBB and XAB for the bond pairs, one can
Models and data
37
write down a equilibrium constant for the above reaction between the bond pairs. One obtains 2 X12 – (ω – ηT ) = 4 exp X11 X12 RT
(I.2.50)
This is the reason why this model is called the quasichemical model. The basis is a chemical reaction but not for a stoichiometric substance, instead the bond pairs are the reactants and products. The XIJ can be retraced to the overall mole fractions XI by way of coordination equivalent fractions
YA = 1 – YB =
Z A XA ZA XA + ZB XB
(I.2.51)
It should be noted that the energy expression ω – ηT can be composition dependent using the overall mole fractions XI of the components. Pelton and Blander [86Pel] originally used a general coordination number Z which could not be adjusted in multicomponent systems. They emphasised that this model gives reasonable values for small deviations from random mixing only, whereas there are very large deviations in silicate melts. Nevertheless, they found that the fixed value of Z = 2 removes this shortcoming of the model and have used that value with considerable success. However, more recent developments by Pelton et al. [000Pel] have overcome the difficulty of having to fix Z. They introduced not only next-nearest-neighbour but also secondnearest-neighbour interactions and arrived at a so-called quadruplet model which is equally applicable to molten salts and oxides. In fact, it was successfully applied to modelling the molten salt phase in the cryolite–alumina system (NaF– AlF3–Al2O3) [002Cha] which implies a continuous transition from salt to oxide. This new model is much more flexible than the old model since it permits the coordination numbers to be composition dependent and the energy parameters ω – ηT to be polynomials in the bond fractions rather than the overall mole fractions.
I.2.5.10
Magnetic effects in solution phases
The general equation used for the calculation of the explicit magnetic Gibbs energy contribution has already been given for pure magnetic substances. It contains the critical temperature Tc and the magnetic moment β. For solution phases both are of course functions of the composition of the phase. This is the only difference between pure magnetic substances and solution phases. The critical temperature separates the range of either ferromagnetism or antiferromagnetism from the range of paramagnetism. That is why it is termed critical temperature rather than the Curie or Néel temperature. However, as ferromagnetism and antiferromagnetism are mutually exclusive, it is possible
38
The SGTE casebook
to treat them in the same formalism. This is illustrated qualitatively in Fig. I.2.16 for the case for the bcc A2 phase in the Fe–Cr system. The continuous part shows the ‘real’ critical temperature as a function of xCr. Note that Tc must always be greater than or equal to zero! The zero point separates the ferromagnetic from the antiferromagnetic range. However, formally it is possible to calculate one continuous and steady curve by introducing ‘negative’ critical temperatures for the antiferromagnetic range. These have to be taken as their absolute values to obtain the real Néel temperature. This procedure also holds for the magnetic contributions to the Gibbs energy of the hcp phase. However, for the fcc A1 phase the situation is slightly more complex. The change from ferromagnetism to antiferromagnetism depends upon the number of nearest neighbours. For the bcc A2 phase this has no effect since the coordination number does not change upon change in the spin direction (ferromagnetic > antiferromagnetic), because bcc A2 is a symmetrical structure. For fcc A1 the situation is different since only one third of the changed spins will contribute in the antiferromagnetic range; the remaining two thirds cancel each other. Therefore a correction by a factor of one third before mirror imaging into the positive range is required. The structuredependent factor is thus defined to be 1 for bcc and hcp structures and one third for fcc structures. For a solution between a ferromagnetic and an antiferromagnetic element, Tc and β must go through zero at the same composition in order for the model to yield reasonable results. In general, the composition dependence of Tc and β may be described with the Redlich–Kister type of polynomial series. It should be noted that the magnetic phase transitions are of second order. Thus the two phase fields are normally separated by a
Curie on neel temperature (K)
1200
TC
TN 0
–400
0
1 Mole fraction of Cr
I.2.16 Curie on Néel temperature for 1 mol of the bcc A2 phase versus mole fraction of Cr in the system Fe–Cr (SGTE data used).
Models and data
39
single line, the line of the critical temperature. However, under certain conditions the magnetic contributions to the Gibbs energy lead to a normal two-phase field, i.e. first-order phase equilibria, which has the shape of a horn. This horn is named after Nishizawa who discovered this effect first (see the paper by Nishizawa et al. [79Nis]). A phase diagram exhibiting a Nishizawa horn with a continuous transition into a second-order phase boundary is the Fe–Ni diagram which is shown in Fig. I.2.17 as calculated from the SGTE solution database. A further effect that leads to second-order phase transitions results from chemical long-range ordering. An order parameter may be defined which decreases to zero at the transformation temperature and composition. Unlike first-order phase transitions which involve a change of state (solid, liquid or gas) or crystal structure (fcc, bcc, etc.) and also involve diffusion over distances large compared with atomistic dimensions, order–disorder transformations occur by atomic rearrangement over distances of the order of atomic dimensions. In order to handle such a situation in the framework of the Gibbs energy relationships already discussed above Ansara, et al. [97Ans] suggested a method similar to the method outlined above for magnetism. An explicit Gibbs energy term which describes the transition from disordered to ordered is added to the Gibbs energy of the disordered reference state.
Gm = Gmdis ( x i ) + ∆Gmord ( y is )
1800
(I.2.52)
Liquid Bcc A2
T (K)
1500
Fcc A1
1200
Tc
900 Bcc A2
Fcc L12
600
300
0
0.2
0.4 0.6 Mole fraction of Ni
0.8
I.2.17 The Fe–Ni phase diagram with magnetic phase boundaries. Note the Nishizawa horn near the centre of the diagram (chemical ordering).
1
40
The SGTE casebook 2000
1800
Liquid
T (K)
1600
1400
Fcc A1
Fe2Si
Bcc A2
Fe5Si3
1200 Bcc B2 1000
FeSi 800 0
0.1
0.2 0.3 Mole fraction of Si
0.4
0.5
I.2.18 The Fe-rich end of the Fe–Si system showing the second-order phase boundary between the bcc A2 and the bcc B2 phases.
The disordered part Gmdis is in general a substitutional solution description on the basis of the mole fractions xi of the system components using a Redlich– Kister polynomial. The transformation energy from disordered to ordered ∆Gmord may be treated with the same mathematics as outlined above for the general multisublattice formalism. However, special conditions hold between the energy parameters for the second part such that the value of ∆Gmord equals zero for the state in which the site fractions y is equal the mole fractions xi, i.e. for the disordered state. A detailed analysis of these relationships has been given by Ansara et al. [97Ans]. Successful applications of this technique have been incorporated in the SGTE solution database, e.g. the transition from bcc A2 to bcc B2 in the Fe–Al and Fe–Si systems (Fig. I.2.18) as well as the transition from fcc A1 to fcc L12 in the Fe–Ni or Al–Ni system. The Fe–Ni phase diagram has already been shown above in the section on magnetic ordering. A further phase diagram exhibiting a case of fcc ordering (Au–Cu) is shown in Chapter I.3 in the section on the phase rule. For more details on Gibbs energy modelling the reader is referred to scientific journals such as Metallurgical Transactions, Chemica Scripta, Calphad, Journal of Physics and Chemistry of Solids, Journal of Alloys and Compounds, Geochimica et Cosmochimica Acta and Physics and Chemistry of Minerals as well as to the textbooks CALPHAD Calculation of Phase Diagrams – A Comprehensive Guide by Saunders and Miodownik [98Sau] and Computational Thermodynamics: the CALPHAD Method by Lukas et al. [007Luk].
Models and data
I.2.6 07Ein 12Deb 13Fen 35Gug 44Mur 45Tem 48Red 49Kel 52Gug 60Koh 62Ken 64Bla 66Ost 67Coh 69Ger 71Kap 73Bar 74Kik 76Ind1 76Ind2 76Jac 77Bar 78Gra 78Rob
79Nis 81Sun 84Gay
85Cha
85Hil 85Kik 86Pel 91Din 94Sax 97Ans
41
References A. EINSTEIN: Ann. Physik 22, 1907, 180. P. DEBYE: Ann. Physik 39, 1912, 787. C.L. FENNER: Am. J. Sci. 36, 1913, 331–384. E.A. GUGGENHEIM: Proc. R. Soc. A 148, 1935, 304. F.D. MURNAGHAN: Proc. Natl Acad. Sci. USA 30, 1944, 244. M. TEMKIN: Acta Phys. Chim. USSR 20, 1945, 411. O. REDLICH and A.T. KISTER: Ind. Eng. Chem. 40, 1948, 345–348. K.K. KELLEY: US Bur. Mines, Bull. No. 476, 1949. E.A. GUGGENHEIM: Mixtures, Clarendon, Oxford, 1952. F. KOHLER: Monatsh. Chem. 91, 1973, 738. G.C. KENNEDY, G.J. WASSERBURG, H.C. HERD and R.C. NEWTON: Am. J. Sci. 260, 1962, 501–521. M. BLANDER: Molten Salt Chemistry, Interscience, New York: 1964, Chapter 3. I.A. OSTROVSKY: Geol. J. 5, 1966, 127–134. L.H. COHEN and W.KLEMENT JR: J. Geophys. Res. 72, 1967, 4245–4251. C. GERTHESEN and H.O. KNESER: Physik, Springer, Berlin, 1969. M.L. KAPOOR and G.M. FROHBERG: Proc. Symp. Chemical Metallurgy of Iron and Steel, Sheffield, UK, 1971, pp. 17–22. I. BARIN and O. KNACHE: Thermochemical Properties of Inorganic Substances, Springer, Berlin, 1973. R. KIKUCHI and C.M. VAN BAAL: Scripta Metall. 8, 1974, 425. G. INDEN: Proc. Calphad V, Düsseldorf, 1976, III.4 , pp. 1–13. G. INDEN: Proc. Calphad V, Düsseldorf, 1976, IV-1, pp. 1–33. L. JACKSON: Phys. Earth Planet. Inter. , J3, 1976, 218–231. I. BARIN, O. KNACKE and O. KUBASCHEWSKI: Thermochemical Properties of Inorganic Substances, Supplement, Springer, Berlin, 1977. P.E. GRATTAN-BELLEW: Exp. Miner. 11, 1978, 128–139. R.A. ROBIE, B.S. HEMINGWAY and J.R. FISHER: ‘Thermodynamic properties of minerals and related substances at 298.15 K and 1 bar (105 Pascals) pressure and at high temperatures’, Geol. Surv. Bull., US Government Printing Office, Washington, DC, 1978. T. NISHIZAWA, M. HASEBE and M. KO: Acta Metall. 27(5), 1979, 817–828. B. SUNDMAN and J. ÅGREN: J. Phys. Chem. Solids 42, 1981, 297–301. H. GAYE and J. WELFRINGER: Proc. 2nd Int. Symp. Metallurgical. Slags and Fluxes (Eds H.A. Fine and D.R. Gaskell), Metallurgical Society of AIME, New York, 1984, pp. 357–375. M.W. CHASE JR, C.A. DAVIES, J.R. DOWNEY JR, D.J. FRURIP, R.A. MCDONALD and A.N. SYVERUD: JANAF Thermochemical Tables, 3rd edition, J. Phys. Chem. Ref. Data 14, Suppl. 1, 1985. M. HILLERT, B. JANNSON, B. SUNDMAN and J. ÅGREN: Metall. Trans. A 16, 1985, 261–266. R. KIKUCHI and J.L. MURRAY: Calphad 9, 1985, 311. A.D. PELTON and M. BLANDER: Metall. Trans. B 17, 1986, 805–815. A.T. DINSDALE: Calphad 15, 1991, 317–425. S. SAXENA: J. Geophys. Res. 99, 1994, 11 787–11 794. I. ANSARA, N. DUPIN, H.L. LUKAS and B. SUNDMAN: J. Alloys Compounds, 247, 1997, 20–30.
42 98Hil
The SGTE casebook
M. HILLERT: Phase Equilibria, Phase Diagrams and Phase Transformations – Their Thermodynamic Basis, Cambridge University Press, Cambridge, 1998. 98Sau N. SAUNDERS and A.P. MIODOWNIK: CALPHAD (Calculation of Phase Diagrams): A Comprehensive Guide (Ed. R.W. Cahn), Materials Series, Vol. 1, Pergamon, Oxford, 1998. 000Pel A.D. PELTON, S.A. DEGTEROV, G. ERIKSSON, G. ROBELIN and C. DESSUREAULT: Metall Mater. Trans. B 31, 2000, 651–659. 001Pel A.D. PELTON: in Phase Transformations in Materials (Ed. G. Kostorz), Wiley– VCH, Weinheim, 2001, Chapter 1. 002Cha P. CHARTRAND and A.D. PELTON: Light Metals 2002, Minerals, Metals & Materials Society, Warrendale, Pennsylvania, 2002, pp. 245–252. 007Luk H. LUKAS, S.G. FRIES and B. SUNDMAN: Computational thermodynamics: the CALPHAD Method, Cambridge University Press, Cambridge, 2007.
I.3 Phase diagrams KLAUS HACK
I.3.1
Introduction: types of phase diagrams
Hillert [85Hil] has pointed out that what is usually called a phase diagram is derivable from a projection of a so-called property diagram. The Gibbs energy as the property is plotted along, for example, the z-axis as a function of two other variables. From the minimum condition for the equilibrium, one can derive the phase diagram of the system as a projection on to the x–y plane. This method is illustrated by Fig. I.3.1, Fig. I.3.2 and Fig. I.3.3: for a one-component system as a P–T diagram (Fig. I.3.1), for a two-component system as a T–x diagram (Fig. I.3.2), and for a three-component system as a x1–x2 diagram (Fig. I.3.3). Note that Fig. I.3.3 uses equilateral triangular coordinates for convenience only. One may draw any part of the triangular figure with rectangular coordinates.
γ α β mA
β+γ
γ
β
β
α+β
γ α+
T
γ
α α
T P
P
I.3.1 Unary system, projection from the µ–T–P diagram.
43
44
The SGTE casebook 1.0 0.5 0.0 G –0.5 –1.0 300 400 500 T 600 1.0 Ni
0.8
0.6
0.4 xNi
0.2
700 0.0 Cu
I.3.2 Binary system, projection from the G–T–x diagram.
α
°GMe
α
°GC
α
GMe α °GFe
GMeaCb °GFeaCb
α
GFe
GFeaCb α
GC Fe
b
a
I.3.3 Ternary system, projection from the G–x1–x2 diagram.
C
Phase diagrams
45
Pelton and Schmalzried [73Pel] have shown how the Gibbs–Duhem relation governs the topology of phase diagrams. The Gibbs–Duhem equation including considering all work terms is: S dT – V dP + ∑ ni dµi + ∑njzj F dΦj + … = 0
(I.3.1)
The generalisation of the different terms in the above equation leads to
Σ QiΦ dΦi = 0
(I.3.2)
One uses generalised potentials Φi which are the temperature T, total pressure p, chemical potential µi, electrical potential Φj, surface tension ∑k, etc., and their generalised conjugate extensive properties QiΦ which are the entropy S, volume V, mole number ni, generalised charge zjFnj (z is the charge number, and F is the Faraday constant), surface area Ak, etc. For each phase Φ present at equilibrium, one specific Gibbs–Duhem equation holds, since the values of the respective potentials hold for the system as a whole but their conjugate extensive properties have phase-related values. For the classification of twodimensional phase diagrams there are now only three different combinations of variable pairs available. – – –
Type 1 or potential diagrams: two potentials, Φi versus Φj. Type 2 or mixed diagrams: one potential and one ratio of conjugate extensive properties, Φi versus Q Φj / QkΦ . Type 3 or extensive property diagrams: two ratios of conjugate extensive properties, QiΦ / QkΦ versus Q Φj / QkΦ .
It is obvious that any higher-order dimensionality is governed by the same reasoning. However, representations with two dimensions are the only ones that will give quantitative results directly readable from the figure. The easiest to read are those two-dimensional diagrams which are sections of multidimensional diagrams, i.e. those for which sufficiently many parameters have been set constant. If two-dimensional projections are used, the readability usually suffers. It is worth noting that the three different types of diagram applied to a given system result in a family of figures which are interrelated by axis transformations. Each of these shows aspects that the others cannot show. Thus it may be necessary in certain cases to draw several diagrams to cover all aspects. The following examples will demonstrate this. It should, however, be emphasised that, for the generation of such a family of diagrams, only one series of calculations is required. The diagrams are generated by selecting the required axis variables from the result tables of the equilibrium calculations. For the unary system pure iron the family of diagrams is shown in Fig. I.3.4, Fig. I.3.5 and Fig. I.3.6. The potentials and their conjugate extensive properties are:
46
The SGTE casebook 2700 Liquid
2400
Temperature (K)
2100
Fcc
1800
Bcc
1500 1200 900
Hcp
600
Bcc
300 0
5
10
15
20 25 30 35 Pressure (GPa)
40
45
50
I.3.4 Type 1 diagram for the unary system Fe. 2700 2400
Liquid
Temperature (K)
2100 1800
Fcc Bcc
1500 1200 900
Negative pressures
600 300 –8.0
Hcp Bcc
–7.5
–7.0 –6.5 Molar volume (cm3)
–6.0
I.3.5 Type 2 diagram for the unary system Fe.
Φ1 and Q1,
Φ2 and Q2,
Φ3 and Q3,
T and S,
–P and V,
µ and n
The first diagram, Fig. I.3.4, is a potential or type 1 diagram using T and –P with the value of µ unspecified. It should be noted that Gibbs used this type of graphical representation in the derivation of the phase rule. The geometrical dimensionality of the phase fields (2), the phase boundary lines (1) and the points (0) is equal to the number of degrees of freedom of the respective equilibrium states. The second diagram, Fig. I.3.5, a mixed or type 2 diagram, is derived from the first by transformation of one axis. The transformation of the x-axis variable P is carried out by using the ratio of VΦ,
Phase diagrams
47
110 Liquid
Entropy (J K–1 mol–1)
100 90
Bcc
80
Fcc
70 60
Negative pressures
50
Bcc Hcp
40 30 –8.0
–7.5
–7.0 –6.5 Molar volume (cm3)
–6.0
I.3.6 Type 3 diagram for the unary system Fe.
the conjugate extensive variable for the total pressure, and n, the conjugate extensive variable of the (so far unspecified) chemical potential. One obtains VΦ/n = vΦ, the molar volume of the respective phase. In Fig. I.3.5 the resulting temperature versus molar volume diagram is shown. It has the topology of the common T–x diagram for a binary system! The triple points of Fig. I.3.4 split into eutectic- and peritectic-like horizontals and the phase boundary lines split into areas filled with tie lines parallel to the x-axis. Now the second axis, T, can also be transformed. One needs to divide the conjugate extensive property SΦ by nΦ to obtain the molar entropy S for the respective phase. This is shown in Fig. I.3.6. The three-phase equilibria split into tie triangles and the tie lines within the two phase fields can have any angle as required by the equilibrium values. From Fig. I.3.4, Fig. I.3.5 and Fig. I.3.6 it is obvious that categorising phase diagrams as unary, binary, ternary, etc., is not really of prime importance. The categories discussed here are not controlled by the number system components but are instead given by the choice of the axis variables. Two more examples will illustrate this fact in more detail. Furthermore, these examples show that it may sometimes be convenient to transform the Gibbs–Duhem relationship to obtain a set of possible axis variables which are related to experimentally more accessible quantities. The first example shows the family of phase diagrams for the binary system Fe–C after transformation of the Gibbs–Duhem equation to contain molar enthalpy and mole fractions instead of entropy and absolute mole numbers [93Hil]. One obtains
( )
µ VΦ H mΦ d 1 + m dP – Σ x iΦ d i = 0 T Τ T
(I.3.3)
48
The SGTE casebook
The resulting phase diagrams are given in Fig. I.3.7. In this series the interrelated quantities are enthalpies (of transformation), temperatures and concentrations (the total pressure is kept constant). The well-known T–x binary phase diagram is of course one diagram of the series. The second example shows the family of diagrams for a ternary system (Fig. I.3.8). The salt–water system KBr–RbBr–H2O rather than a metallic ternary has been chosen to demonstrate the generality of the method. The transformed Gibbs–Duhem equation for constant temperature (298.15 K) and total pressure (1 bar) now is [91Koe] (1 – x aq ) d(ln a kbr ) + x aq d(ln a RbBr ) +
1
M H 2 O Σm
d(ln a H 2 O ) = 0
(I.3.4) The possible relationships for the two-phase solid salt solution–water equilibria involve the chemical potentials, i.e. the activities, of the two salts
7
Liquid
Hm – HFe298.15 J mol–1
Hm – Href298.15 (J mol–1)
8
6 5 4
γ
3 2
1 E4 0 0
0.05 0.10 0.15 0.20 0.25 Mole fraction of C
10 9 8 7 6
5 δ Cementite 4 3 2 1 0 –10 –8 –6 –4 –2 0 µC–°HC (298.15) (J mol–1)
(a)
1500 1400 1300 1200 1100 1000
(b)
δ Liquid γ
900 α 800 700 600 0 1
Cem
2
3 4 5 C (wt%) (c)
6
7
8
Temperature (°C)
Temperature (°C)
1600
Liquid
1600 δ 1500 1400 1300 1200 1100 γ 1000 900 800 α 700 600 0 0.5
Liquid
Cementite
1.0 1.5 Activity of C (d)
I.3.7 Phase diagram family for the binary system Fe–C.
1
Phase diagrams –0.21
49
–0.21
–0.22
–0.22
ln a(H2O)
ln a(H2O)
aq –0.23 –0.24
–0.23 –0.24
S –0.25
–0.25
–0.26 –2
–1.5
–1 –0.5 ln a(RbBr)
0
–0.26 0
0.2
0.8
1
0.8
1
II
10
10
8
8
1/(M(H2O) Σ m)
1/(M(H2O) Σ m)
I
0.4 0.6 x(RbBr)
6 4 2
6 4 2
0 –2
–1.5
–1 –0.5 ln a(RbBr) II
0
0
0
0.2
0.4 0.6 x(RbBr) III
I.3.8 Phase diagram family for the ternary system KBr–RbBr–H2O.
and water as well as the concentrations. For the latter the most convenient units have been chosen, i.e. the mole fractions in the binary solid salt solution and the molality for the aqueous phase. Königsberger and Gamsjäger [91Koe] have pointed out that the type 3 diagram of this series, 1/ M H 2 O Σm versus xRbBr, i.e. the typical ternary diagram with tie lines at variable angles with respect to the diagram’s axes, is equivalent to what has been called a Jänecke [06Jän] diagram among aqueous chemists for decades. The above three series of phase diagram families show that the thermodynamic equilibrium rules permit many possible representations. It is thus up to the user to decide from which type of plot most information can be gained. A unary extensive property (type 3) diagram is certainly more useful when the general principle has to be explained and is hardly related to any experiment or process, whereas the same type of figure for a fivecomponent alloy can give very useful answers to real-world engineering problems. For systems with more than three components it is, however, necessary to generalise the topological rules even further.
50
The SGTE casebook
In most cases, multicomponent phase relationships can best be represented by type 2 or type 3 diagrams. However, the restriction to two dimensions requires sections through (or projections of) a multidimensional space to be drawn. Figure I.3.14 (see below) shows such a section for the system Fe–Cr– C–W for T = 1123 K, P = 1 bar and 0.2 wt% C with the weight percentage of Cr and W as axis variables. It is obvious that such a section cannot be read with the rules applicable to the extensive property (type 3) diagrams discussed so far. It is highly unlikely that any tie line or tie triangle between the possible phase lies within the chosen two-dimensional space. Some new aspects have to be introduced to make proper use of diagrams such as Fig. I.3.14 below. These will be discussed in the following section.
I.3.2
Zero-phase-fraction lines
Masing [44Mas] stated in 1944 that ‘a state space can ordinarily be bounded by another state space only if the number of phases in the second space is one less or one greater than that in the first space considered’. Palatnik and Landau [64Pal] have added in the 1950s a series of papers about the dimensionality of phase boundaries from which comes the law of adjoining phase regions: R1 = R – D – – D + ≥ 0
(I.3.5)
In this, R1 is the dimension of the phase boundary, R is the dimension of the phase diagram (this may also be a section, isothermal or isopleth) and D– and D+ represent the number of disappearing and appearing phases respectively when crossing the phase boundary. For two-dimensional diagrams, as discussed here, R = 2. For the boundaries, one will therefore obtain either a dimension of 1, i.e. a line, or a dimension of 0, i.e. a point, because R1 ≤ R – 1, as was shown by Prince [66Pri]. These are of course all topological elements possible in a two-dimensional figure. Recently Morral and co-workers [84Bra, 84Mor, 86Gup] have published a series of papers in which they introduced the concept of zero-phase-fraction (ZPF) lines and thus add to the lines and points discussed so far one very useful aspect. This will be illustrated by the following series of figures: Fig. I.3.9, Fig. I.3.10 and Fig. I.3.11. Consider an isothermal section of a ternary phase diagram for the system A–B–C with the three terminal solution phases α, β and γ (Fig. I.3.9). What one usually does is to draw the lines around the phase regions of the pure phases, and to add the tie lines between the phase boundaries as well as the tie triangle around the three-phase region. From the tie lines, one can obtain information which is usually considered of importance: the amount of the phase present in the two- or three-phase equilibrium. This is given by the lever rule and can be read directly from Fig.
Phase diagrams
51
C
γ
β
α A
B
I.3.9 Conventional representation of an isothermal section for a ternary system (extensive property (type 3) diagram). C
γ
β=0
. γ = α = 50 mol%
β° = γ = 50 mol%
γ=0 β
α A
B . β° = α = 50 mol%
α=0
I.3.10 The lines of 50 and 0 mol% of the respective phases.
I.3.3. Drawing the lines for 0 and 50 mol% the next figure, Fig. I.3.10, is obtained. The most interesting lines are those for the zero phase amount (or fraction). These lines show for each phase, one by one, the composition range in which they can exist, notwithstanding whether alone or together with others. The major point here is that they do not exclude the other phases such as the lines marked in Fig. I.3.9 but include them as they may occur. Seen from the other side, the lines indicate the range in the diagram in which a particular phase does not exist. For the α phase for example, one obtains the ZPF line shown in Fig. I.3.11.
52
The SGTE casebook C
No α
α + others A
B
I.3.11 The ZPF line for the α phase. Outside
fα = 0
Others Inside α + others
I.3.12 The inside–outside property of a ZPF line. fα = 0
β + others
α + β + others
Others
α + others
fβ = 0
I.3.13 The intersection of two ZPF lines.
This aspect was not explicitly included in the considerations of Masing or of Palatnik and Landau or in Prince’s interpretation thereof but helps considerably to understand a diagram such as Fig. I.3.14 given below, as the inside–outside property relates the lines and phases in a diagram in pairs (Fig. I.3.12) When two such lines meet, they simply intersect, i.e. to follow the line of a particular phase means to go straight through the crossing. At the crossing, the law of adjoining phase regions holds for the four lines and the point of intersection. This is shown in Fig. I.3.13.
Phase diagrams
53
18 16
α + M23C6 + λ
α + M6C + λ 14
W (wt%)
12
α + M6C
10 8
α = γ + M6C α + M6C + M23C6
6 4
γ + M6C
α + M23C6
2 0 0
γ + M23C6
γ 3
6
9 Cr (Wt%)
12
15
I.3.14 Marked ZPF lines in the Fe–W–Cr–C system with 0.2 wt% C and T = 1123 K.
One further property of ZPF lines can be understood from Fig. I.3.11: ZPF lines begin and end on the diagram axes or else form a closed loop within the diagram. Knowing these properties one can now draw Fig. I.3.14, considering all lines as ZPF lines.
I.3.2.1
Special cases of zero-phase-fraction intersections
The more complex a system is, i.e. the more components and phases it has, the more likely it is that the rules outlined above are sufficient to understand the calculated two-dimensional phase diagrams. However, there are some exceptions which occur and which have to be discussed to prevent misunderstandings. Luckily, the exceptions occur in well-understood systems. The general case is that four lines meet at a point and that the law of adjoining phase regions holds. Both the rules seem to be violated in a simple binary eutectic system, as shown in Fig. I.3.15. At the points A and C, only three lines meet and, at point B, going from the liquid one-phase region into the solid two-phase region, the number of phases changes from one to two instead of three. This result stems from the fact that the three-phase field A–(B)–C is what Prince calls a degenerate phase region. Instead of one line there are really three, A–B, B–C and A–C, separating the three phase region α–liquid–β from the respective two-phase
54
The SGTE casebook
Liquid
T
α
A
B
C
β
x
I.3.15 Simple binary eutectic system.
Liquid α + liquid
Liquid + β
α + liquid + β
T α
α+β
β
x
I.3.16 Exaggerated plot of the three-phase α–liquid–β region in a binary eutectic system.
regions. An exaggerated drawing with a finite angle between the lines shows this very clearly (Fig. I.3.16). Now there are only points at which four lines meet and, at point B, going down from the one-phase region, one obtains three phases in accordance with the law of adjoining phase regions. Palatnik and Landau [64Pal] have already pointed out that there are some other exceptions which they called nodal plexi. These may result in six or eight lines meeting at a point. The classification of these nodal plexi has been given by Palatnik and Landau. For further details the reader is referred to the monograph Alloy Phase Equilibria by Prince [66Pri].
Phase diagrams
I.3.2.2
55
Conclusions on zero-phase-fraction lines
From the above, one may conclude that the concept of ZPF lines provides a useful basis in several ways: 1 2
3
For the programmer, because she or he can use the features of ZPF lines to derive algorithms for what is called two-dimensional phase mapping. For the reader of computer-generated multicomponent two-dimensional diagrams, because they provide a ‘read thread’ for each phase contained in the figure. For the experimentalist, because he or she can use the properties of ZPF lines to derive a first diagram from his or her experimental phase equilibria before a computer calculation is possible [84Mor].
I.3.3
Beyond classical phase diagrams
In this section it will be shown how very familar diagrams show new and extended features when regenerated using the latest calculational techniques for phase diagrams. Some cases will be discussed now. Almost every ferrous metallurgist knows what is called the Baur–Glässner diagram [03 Bau]. In this diagram the stability ranges of the various phases in the Fe–O system are shown with the temperature on one axis and the molar ratio of CO and CO2 or the appropriate mole fraction on the other. The major condition in this approach is that the indeed very small oxygen potential can be expressed by the more ‘tangible’ values of CO and CO2 because of the homogeneous gas equilibrium CO + 12 O2 = CO2. In fact, the Boudouard reaction (C + CO2 = 2CO), which is important too, is also taken account of by plotting the appropriate equilibrium line as an overlay into the diagram. In Fig. I.3.17, Fig. I.3.18 and Fig. I.3.19, diagrams are shown which relate to the classical situation described above. They have, however, been generated using the technique of ZPF line calculations. In Fig. I.3.17, all Fe–O phase boundaries are shown disrespecting the Boudouard equilibrium and using pure substance data for solid and liquid Fe as well as for solid and liquid oxides. The well-known hay-fork topology between the metallic phases on the one side and the oxides Fe3O4 and FeO is clearly recognisable in the diagram. It must, however, be noted that this diagram can only be generated by suppressing the graphite phase from all equilibria. Also phase boundaries with the gas phase have been ignored. In other words, the phase diagram contains many metastable phase boundaries. Figure I.3.18 shows the phase diagram for the case when graphite and gas both are permitted to become stable phases in the system. Note, however, that all condensed phases are still treated as stoichiometric pure substances. In the range of lower temperatures’ one can now see how the formation of stable graphite overrides the Fe–Fe3O4 phase boundary shown in Fig. I.3.17.
56
The SGTE casebook 2000 Fe3O4(l) + gas
Fe(l) + gas FeO(l) + gas Gas + Fe(s)
1680
1360
Gas + FeO(s)
T (K)
Gas + Fe(s2)
Gas + Fe3O4(s2)
1040
Gas + Fe(s)
720
Gas + Fe3O4(s)
400 0
0.2
0.4 0.6 Mole fraction CO2 /(CO + CO)
0.8
1
I.3.17 Phase diagram for P = 20 atm and mole Fe/(CO+CO2) = 0.000 01. 2000 Fe3O4 (l) + gas Gas 1680
FeO(l) + gas
Gas + Fe(bcc)
Gas + Fe(bcc) 1360
T (K)
Gas + C + Fe(fcc)
Gas + FeO(s)
Gas + Fe(bcc) 1040
Gas + C + Fe(bcc) Gas + C + FeO Gas + C + Fe3O4(s2)
Gas + Fe3O4(s2)
Gas + Fe3O4(s1)
720 Gas + C + Fe(s1) 400
0
0.2
0.4 0.6 Mole fraction CO2/(CO + CO)
0.8
1
I.3.18 Phase diagram for P = 3 atm and mole Fe/(CO+CO2) = 0.000 01.
It must be noted that in order to obtain Fig. I.3.18 it is necessary to fix the total pressure which was given an arbitrary value of 3 atm. Thus the position of the Boudouard curve has also been fixed arbitrarily. It would shift on
Phase diagrams
57
change in the total pressure. Likewise the phase boundaries between gas and Fe(bcc) as well as gas and FeO(l) will shift with total pressure. Note furthermore that the Boudouard line is not an overlay as in the classical Baur–Gläsner diagrams. It is indeed a true phase boundary in this figure. An additional interesting side effect resulting from the present setting for the pressure is that no range of stability of liquid Fe is found. For the lower temperature range, Fig. I.3.18 represents the real equilibria quite well since the solubility of carbon in Fe is almost negligible. Also Fe3O4 may be treated as stoichiometric for these temperatures. However, when the temperature exceeds that of the α–γ transition of Fe the situation changes considerably since the solubility of C in γ–Fe is not negligible. Also liquid Fe can dissolve large amounts of C, not to mention the lowering of the stability temperature of the melt by the dissolution of the carbon. Furthermore wustite is not a stoichiometric compound and liquid iron oxide (slag) has a wide composition range. Thus, for a proper Baur–Glässner diagram, solid and liquid solutions need to be considered in the calculation. The resulting phase diagram is shown in Fig. I.3.19. It can be seen immediately that there are considerable changes in the phase boundaries on the CO-rich side of the phase diagram. The solubility of C in the condensed Fe phases now leads to phase boundaries that are dependent upon both the temperature and the mole fraction of CO2. Furthermore, the size and shapes of the phase fields of both the wustite and the liquid oxide (slag) are different from the respective phase fields in Fig. I.3.18 because of the wide homogeneity ranges of these phases. Figure I.3.19 should also be used to give a better understanding of the calculations as such. For all phase fields above the Boudouard line the mole fraction of CO2 can be read from the x-axis. Because above this line C does not precipitate from the gas, only the reaction CO + 12 O2 = CO2 is relevant. Since the amount of Fe that was chosen as an input parameter for the calculation is 10–5 mol per mole of gas, there is no significant loss of oxygen from the gas phase on formation of the iron oxides, and thus the mole fraction of CO2 is a direct measure for the oxygen potential at a given point in the diagram. The small amount of Fe also leads to only a small loss of C from the gas phase in the stability ranges of the Fe-based condensed solution. Thus the CO2 mole fraction read from the x-axis is identical with that in the gas phase for these ranges too. As an illustration of this behaviour of the system, two equilibrium tables are given. Table I.3.1 shows the equilibrium at the position of the label in the phase field for wustite. The overall amounts of CO and CO2 respectively shown in the top part of the table are identical with those in the calculated equilibrium gas phase. In Table I.3.2 is shown an equivalent calculation for the position of the label in the spinel (magnetite) plus carbon (+ gas) phase field. Comparison of the data for CO and CO2 with the values for the gas phase shows clearly that these are different from each other.
58
The SGTE casebook 2000 Gas Liquid + Fe + gas Bcc + gas
1680
1360
Slag–liquid + gas
Monoxide + gas
T (K)
Gas + fcc Gas + C + fcc Monoxide + gas
1040
Spinel + gas
720 Spinel + gas + C 400
0
0.2
0.4 0.6 Mole fraction CO2/(CO + CO2)
0.8
1
I.3.19 Phase diagram for P = 3 atm and mole Fe/(CO+CO2) = 0.000 01.
Because of the finite amount of carbon formed in this phase field the CO2 mole fraction in the equilibrium state is not equal to that read from the x-axis. Now let us consider the case of an aqueous system. Classically such systems are depicted in Pourbaix diagrams, i.e. predominance area diagrams with the potential Eh of the half-cell H+(aq) + e– = 12 H2 plotted as the y-axis while on the x-axis the pH value of the system is represented. This type of diagram has been used successfully in Pourbaix’s atlas [63Pou] for the comparison of very many metal–aqueous systems. Compared with one another these diagrams give very useful information on the relative stabilities of metals in aqueous environments. For practical purposes, however, they are not so easy to use since, on the one hand, fixing the Eh value of a system can only be achieved indirectly by controlling either the O2 or the H2 partial pressure in the gas phase of the system and, on the other hand, controlling the pH of the system can also not be performed directly, i.e. by adding H+ ions, but only by adding an acid or a base, i.e. real substances, to the system. Thus a metal–water system is in practice investigated by setting up a metal– water–acid–base–H2 (or O2) system. For the purpose of this discussion the system Cu–H2O–HCl–NaOH–H2 was chosen. The phase diagram shown in Fig. I.3.20 was calculated using the following conditions. 1
A total of 1 mol of acid–base mixture with a variable mole fraction is entered for definition of the x-axis.
Phase diagrams
59
Table I.3.1 Equilibrium table for composition and temperature given by the position of the label “monoxide + gas” in Fig. 3.19 T(K) = 587.7, mole CO2/(CO+CO2) = 0.4486 T = 587.69 K P = 3.00000E+00 atm V = 1.16460E+01 dm3 STREAM CONSTITUENTS Fe CO CO2
AMOUNT/mol 9.9999E-06 5.5138E-01 4.4861E-01
PHASE: gas_ideal CO2 CO C3O2 Fe(CO)5 Fe O2 TOTAL:
EQUIL AMOUNT mol 7.2410E-01 3.7629E-04 9.0366E-25 7.1654E-34 6.1201E-34 5.4116E-36 7.2448E-01
MOLE FRACTION 9.9948E-01 5.1940E-04 1.2473E-24 9.8904E-34 8.4475E-34 7.4696E-36 1.0000E+00
FUGACITY atm 2.9984E+00 1.5582E-03 3.7420E-24 2.9671E-33 2.5343E-33 2.2409E-35 1.0000E+00
PHASE: FCC_A1 Fe:C Fe:Va TOTAL:
mol 0.0000E+00 0.0000E+00 0.0000E+00
MOLE FRACTION 1.1003E-03 9.9890E-01 1.0000E+00
ACTIVITY 4.0133E-11 4.3277E-05 4.3325E-05
PHASE: BCC_A2 Fe:C Fe:Va TOTAL:
mol 0.0000E+00 0.0000E+00 0.0000E+00
MOLE FRACTION 2.8154E-08 1.0000E+00 1.0000E+00
ACTIVITY 2.0980E-37 7.8983E-05 7.8983E-05
PHASE: Fe-LIQUID Fe C O TOTAL:
mol 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00
MOLE FRACTION 6.7414E-01 3.2586E-01 2.4778E-06 1.0000E+00
ACTIVITY 7.0465E-06 7.1598E-10 2.8077E-08 9.1665E-05
PHASE: Slag-liq FeO Fe2O3 TOTAL:
mol 0.0000E+00 0.0000E+00 0.0000E+00
MOLE FRACTION 3.5358E-01 6.4642E-01 1.0000E+00
ACTIVITY 6.3908E-04 2.4595E-06 2.7156E-03
PHASE: Spinel Fe3O4 Fe3O4[1-] Fe3O4[1+] Fe3O4[2-] Fe1O4[5-] Fe1O4[6-] TOTAL:
mol 5.3091E-08 1.5652E-06 1.6651E-06 4.9908E-08 4.6572E-22 1.4849E-23 3.3333E-06
MOLE FRACTION 1.5927E-02 4.6958E-01 4.9952E-01 1.4973E-02 1.3972E-16 4.4549E-18 1.0000E+00
ACTIVITY 3.7409E-02 2.3802E-02 2.5681E-01 4.2537E-05 2.8122E-24 5.1596E-27 1.0000E+00
60
The SGTE casebook
Table I.3.1 (Continued) PHASE: Monoxide FeO Fe2O3 TOTAL:
mol 0.0000E+00 0.0000E+00 0.0000E+00
C_graphite
mol 2.7551E-01
Cp_EQUIL J.K-1 3.86600E+01
H_EQUIL J -2.75038E+05
MOLE FRACTION 6.8987E-01 3.1013E-01 1.0000E+00
S_EQUIL J.K-1 1.72852E+02
ACTIVITY 3.7926E-02 3.3535E-06 1.2759E-01 ACTIVITY 1.0000E+00
G_EQUIL J -3.76621E+05
V_EQUIL dm3 1.16460E+01
Mole fraction of sublattice constituents in FCC_A1: Fe 1.0000E+00 Stoichiometry = 1.0000E+00 C 1.1003E-03 Stoichiometry = 1.0000E+00 Va 9.9890E-01 Magnetic properties for FCC_A1: Neel temperature = 67.00 K Average magnetic moment/atom = 7.0000E-01 Mole fraction of sublattice constituents in BCC_A2: Fe 1.0000E+00 Stoichiometry = 1.0000E+00 C 2.8154E-08 Stoichiometry = 3.0000E+00 Va 1.0000E+00 Magnetic properties for BCC_A2: Curie temperature = 1043.00 K Average magnetic moment/atom = 2.2200E+00 Mole fraction of sublattice constituents in Spinel: 3.0900E-02 Stoichiometry = 1.0000E+00 Fe[2+] Fe[3+] 9.6910E-01 4.8455E-01 Stoichiometry = 2.0000E+00 Fe[2+] Fe[3+] 5.1545E-01 Va[0] 1.4417E-16 Magnetic properties for Spinel: Curie temperature = 848.00 K Average magnetic moment/atom = 4.4540E+01
2 3 4 5 6
The partial pressure H2 is used for the y axis. 55.508 mole of water, i.e. 1 kg, is set as a constant. A constant molar amount relative to the given water amount is entered as the molality of the metallic component. The temperature is set to 25 °C. The total pressure is set equal to 1 atm.
Phase diagrams
61
Table I.3.2 Equilibrium table for composition and temperature given by the position of the label “spinel + gas + C” in Fig. 3.19 T(K) = 1342, mole CO2/(CO+CO2) = 0.5569 T = 1341.54 K P = 3.00000E+00 atm V = 3.66942E+01 dm3 STREAM CONSTITUENTS Fe CO CO2
AMOUNT/mol 9.9999E-06 4.4305E-01 5.5694E-01
PHASE: gas_ideal CO2 CO Fe FeO O2 TOTAL:
EQUIL AMOUNT mol 5.5693E-01 4.4306E-01 3.3516E-10 1.2855E-12 5.8846E-14 9.9999E-01
MOLE FRACTION 5.5693E-01 4.4307E-01 3.3516E-10 1.2855E-12 5.8847E-14 1.0000E+00
FUGACITY atm 1.6708E+00 1.3292E+00 1.0055E-09 3.8566E-12 1.7654E-13 1.0000E+00
PHASE: FCC_A1 Fe:C Fe:Va TOTAL:
mol 0.0000E+00 0.0000E+00 0.0000E+00
MOLE FRACTION 5.1884E-04 9.9948E-01 1.0000E+00
ACTIVITY 5.8284E-06 2.5050E-01 2.5063E-01
PHASE: BCC_A2 Fe:C Fe:Va TOTAL:
mol 0.0000E+00 0.0000E+00 0.0000E+00
MOLE FRACTION 2.2610E-05 9.9998E-01 1.0000E+00
ACTIVITY 3.4281E-25 2.4889E-01 2.4891E-01
PHASE: Fe-LIQUID Fe C O TOTAL:
mol 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00
MOLE FRACTION 9.8101E-01 1.3363E-02 5.6292E-03 1.0000E+00
ACTIVITY 1.8266E-01 1.8206E-06 1.1212E-03 1.8635E-01
PHASE: Slag-liq FeO Fe2O3 TOTAL:
mol 0.0000E+00 0.0000E+00 0.0000E+00
MOLE FRACTION 8.9264E-01 1.0736E-01 1.0000E+00
ACTIVITY 4.9991E-01 2.5805E-03 5.1776E-01
PHASE: Spinel Fe3O4 Fe3O4[1-] Fe3O4[1+] Fe3O4[2-] Fe1O4[5-] Fe1O4[6-] TOTAL:
mol 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00
MOLE FRACTION 1.7712E-01 2.6112E-01 4.6154E-01 1.0021E-01 5.1600E-08 1.9802E-08 1.0000E+00
ACTIVITY 5.3624E-02 3.7564E-02 8.5771E-02 5.6105E-03 2.0080E-15 3.0005E-16 3.2761E-01
62
The SGTE casebook
Table I.3.2 (Continued) PHASE: Monoxide FeO Fe2O3 TOTAL: C_graphite Cp_EQUIL J.K-1 4.73776E+01
mol 8.1193E-06 9.4015E-07 9.0594E-06 mol 0.0000E+00
H_EQUIL J -2.24128E+05
MOLE FRACTION 8.9622E-01 1.0378E-01 1.0000E+00
S_EQUIL J.K-1 2.63995E+02
G_EQUIL J -5.78287E+05
ACTIVITY 8.2982E-01 2.3861E-04 1.0000E+00 ACTIVITY 3.4955E-03 V_EQUIL dm3 3.66942E+01
Mole fraction of sublattice constituents in FCC_A1: Fe 1.0000E+00 Stoichiometry = 1.0000E+00 C 5.1884E-04 Stoichiometry = 1.0000E+00 Va 9.9948E-01 Magnetic properties for FCC_A1: Neel temperature = 67.00 K Average magnetic moment/atom = 7.0000E-01 Mole fraction of sublattice constituents in BCC_A2: Fe 1.0000E+00 Stoichiometry = 1.0000E+00 C 2.2610E-05 Stoichiometry = 3.0000E+00 Va 9.9998E-01 Magnetic properties for BCC_A2: Curie temperature = 1043.00 K Average magnetic moment/atom = 2.2200E+00 Mole fraction of sublattice constituents in Spinel: Fe[2+] 2.7734E-01 Stoichiometry = 1.0000E+00 Fe[3+] 7.2266E-01 3.6133E-01 Stoichiometry = 2.0000E+00 Fe[2+] Fe[3+] 6.3867E-01 Va[0] 7.1402E-08 Magnetic properties for Spinel: Curie temperature Average magnetic moment/atom
= =
848.00 K 4.4540E+01
It should be noted that an increasing partial pressure of H2 on the y-axis is inverse to the classical Eh value while an increase in the mole fraction of the basic component of the acid–base pair defines the same direction along the x axis as an increase in pH. It must also be noted that the resulting diagram is a true phase diagram, i.e. the stability fields relate to the stable phases at equilibrium and not to particular species, be they stable solids or species in
Phase diagrams
63
0
Cu(s) + aqueous –12
2
log10 [p(H ) (bar)]
–6
Aqueous
–18
–24
–30
Cu2O(s) + aqueous
Cu (OH)2 (s) + aqueous
0
0.2
0.4 0.6 Mole fraction NaOH/(HCl + NaOH
0.8
1
I.3.20 Phase diagram for T = 25 °C, mole H2O/(HCl+NaOH) = 55.508 and mole Cu/(HCl+NaOH) = 0.001.
the aqueous phase. The diagram therefore shows on the lower left (high acidity and low H2 potential) the field of stability of the single-phase water equilibrium while at higher mole fractions of NaOH, i.e. high basicity and thus high pH, the precipitations of copper hydroxide, copper oxide and copper metal occur with increasing H2, i.e. decreasing Eh. This is all in accord with the classical Pourbaix diagram for copper. However, on increase in the molality of Cu as shown in Fig. I.3.21, additional reactions occur in this system. When increasing the molality from 0.001 to 0.1 the phase diagram shows that the use of a real acid, here HCl, instead of an assumed pH value, leads to additional phase fields because CuCl can coprecipitate with copper hydroxide, copper oxide or copper metal or can even precipitate as a single solid from the aqueous phase. Such effects can never be realised in Pourbaix diagrams since the anion of the acid and the cation of the base are not entered explicitly into the method of calculation. On the other hand, the method described above can immediately be used to generate a diagram with the same axes as a Pourbaix diagram. For any point in diagrams such as Fig. I.3.20 or Fig. I.3.21 the value of Eh and pH is known. Therefore a simple axis transformation, i.e. different choice of x- and y-axis variables for the plot, will lead to a diagram as shown in Fig. I.3.22. This diagram is the Pourbaix-axes equivalent to Fig. I.3.20. It is clear from Fig. I.3.20 that this new diagram will also only show true phase boundaries; there are no equilibrium lines with single species in the one, and so there cannot be such lines in the other.
64
The SGTE casebook 0
Cu(s) + aqueous
log10 [(pH2) (bar)]
–6
Cu(s) + CuCl(s) + aqueous
–12
CuCl(s) + aqueous Cu2O(s) + CuCl(s) + aqueous
–18
Cu2O(s) + aqueous
Cu(OH)2(s) + CuCl(s) + aqueous –24
–30
Aqueous
0
0.2
Cu (OH)2 (s) + aqueous
0.4 0.6 Mole fraction NaOH/(HCl + NaOH)
0.8
1
I.3.21 Phase diagram for T = 300 K, mole H2O/(HCl+NaOH) = 55.508 and mole Cu/(HCl+NaOH) = 0.1.
0.65 0.55 Cu2+ 0.45 aqueous 0.35 0.25
Eh
Cu(OH)2 + aqueous
Cu 2 O( s)
0.15 0.05
Cu
2O
–0.05
+a
que
ous
–0.15 –0.25 Cu + aqueous –0.35 –0.45 –0.55 0
1
2
3
4
5
6
7 pH
8
9
10
11
12
13
I.3.22 Eh versus pH plot derived from the phase diagram; the figure from the Eh–pH module is overlayed.
14
Phase diagrams
65
For comparison the predominance fields of a classical Pourbaix diagram are overlayed in this figure (thin lines). It is obvious that the line of ‘equilibrium’ between the Cu2+ aqueous ion and the metallic copper is not a true phase boundary, and also parts of the Cu–Cu2O and Cu2O–Cu(OH)2 coexistence lines of the Pourbaix diagram are not true phase boundaries. All these lines extrapolate into the region of stability of the aqueous phase when the phase boundaries are calculated considering all aqueous species and the stoichiometric solids simultaneously. Finally two phase diagrams for combustion systems with variable C–H–O ratios will be considered. A Gibbs triangular plot for an isothermal representation of the equilibria is used. In Fig. I.3.23, only the elements relevant for combustion are considered. The only important phase field that can be depicted in this diagram is that for the precipitation of C from the gas, i.e. the soot formation range. Different types of fuel, depicted by their specific range of fuel-to-oxygen ratios can be marked in the diagram, thus showing their tendency to soot formation. However, if the system C–H– O is considered together with one or more metallic components, then it is possible to use this kind of diagram in order to predict ranges for the corrosion of the metal both with respect to O and C. Figure I.3.24 shows such a diagram for a Fe–20% Cr stainless steel. The ratio of molar amount of alloy to gas has been chosen as 0.0001 in order to make sure that the amounts of
0.8
+ 0.8 H +
cti fra
0.6
ole M 0.5 0.2
0.7
0.3
0.6
0.4
Gas + C
) +O +H (C 0.5 H/ on cti 0.4 fra 3 0. ole M 0.2
on 0.7 C(C
0.1
O 0.9 )
C
O
0.9
0.1
Gas
0.9
0.8
0.7 0.6 0.5 0.4 0.3 Mole fraction O/(C + H + O)
0.2
0.1
H
I.3.23 Phase diagram for T = 1200 K, mole Fe/(C+H+O) = 0 and mole Cr/(C+H+O) = 0.
66
The SGTE casebook
O) 0.8
H
le Mo
0.5
0.4 0.3
)
0.8
+ 0.7 C3 M7 0.6 te + C nti e+ 0.5 eme entit C cem + O3 M2 + C c + fc
Sp ine l+ Sp fcc ine l+ mo no xid e
0.2
+O
0.4
Spinel + fcc + C
M 2O 3
cti o 0.6 n C/ (C +H 0.7 +
C+ C/( on cti 0.3 fra 0.2
fra
le Mo 0.1
0.9
C
M
Sp
ine
l
O
0.9
0.8
0.7
0.6 0.5 0.4 0.3 Mole fraction O/(C + H + O)
2O 3
0.2
0.9
0.1
C
M2O3
+ fc
c
0.1
H
I.3.24 Phase diagram for T = 1200 K, P = 1 atm, mole Fe/(C+H+O) = 0.000 08 and mole Cr/(C+H+O) = 0.000 02.
C and O transferred from the gas to the condensed state on formation of carbides or oxides is negligible. Thus the mole fractions of C, H and O in the diagram can be read as those pertaining to the gas phase, except for those phase fields in which pure C is precipiated. In these ranges the amount of carbon transferred from the gas to the condensed state can lead to significant changes in the mole fractions of C, H and O in the gas phase. This behaviour is similar to that of the phase ranges of the Baur–Glässner diagram below the Boudouard curves (Fig. I.3.19).
I.3.4
The phase rule
Perhaps one of the most puzzling questions for students in thermochemistry is the derivation of the number c of components, to be inserted in the equation of the phase rule f=c+2–Φ
(I.3.6)
It is easy to count the number Φ of phases involved in the equilibrium but, when we consider the definition of c, even Gibbs [887Gib] explanations are not very stringent. In fact, he uses ‘actual’ components, which have amounts
Phase diagrams
67
that can be increased or diminished (p. 117), ‘possible’ components, which may be combined with but cannot be subtracted from the homogeneous mass in question (p. 117) and also ‘ultimate’ components (p. 134), which are the independent substances in the system (?). However, at a later stage (p. 155) Gibbs explains that an equilibrium between two phases (labelled with a prime(′) and a double prime (″) in an ncomponent system has only one degree of freedom (p = p(T) or vice versa) if n – 1 ratios of m′1 to m″n, the mole numbers of the components in the phase labelled with a prime, are equal to n – 1 ratios of m″1 to m″n, the equivalent mole numbers in the phase labelled with a double prime. This latter explanation helps to develop a concept from which it is easier to derive the value of c, which according to Rao [87Rao] should be called the number of Gibbsian components. Let us first consider the case as stated by Gibbs for a real system, the azeotropic point between the solid gold solution and the liquid as well as the two points of congrunet solid–solid transition in the system Au–Cu as marked in Fig. I.3.25. All equilibria are two-phase equilibria (Φ = 2) between two solution phases of a system with two elemental, components (n = 2). Yet there is one additional constraint, namely
1500
1300 Liquid
T (K)
1100
Fcc Al
900
700
500
AuCu
AuCu
Au3Cu 300 0
0.1
0.2
0.3
0.4
0.5 x (Cu)
0.6
0.7
0.8
0.9
I.3.25 System Au–Cu (SGTE Solution Database). 䉭, 䉮 points with equal composition in both phases.
1
68
The SGTE casebook liq fcc m Cu m Cu = liq fcc m Au m Au
(I.3.7)
Note that it is not important that the value of the ratio is known; only the equality is important. Thus f=c+2–Φ=1+2–2
(I.3.8)
It is obvious that c in this equation is not equal to the number of system components or elements but must be reduced by the number of compositional constraints given by the extra condition(s) imposed on the system. Thus c=ε–s
(I.3.9)
where ε is the number of elements in the system and s is the number of compositional constraints within the solution phases which are in equilibrium with each other. At first sight, it seems of course as though the problem has only been shifted from finding the value for c to finding that for s, but at least counting the elements should be easy. The following discussion of some standard textbook cases will demonstrate that it is also not too complicated to find s, the number of compositional constraints. Before going into details, it is, however, necessary to emphasise that phases can be distinguished into two classes. These are stoichiometric pure substances and solutions, i.e. phases with intrinsically fixed ratios and those with variable or partially variable composition. Note that the gas phase always belongs to the latter class. This distinction may seem too strict in some respect but, when it comes to equilibrium calculations using computer programs, one has to decide beforehand what homogeneity range a particular phase will be given. Thus the following two equations have to be considered together when applying the phase rule: f = c + 2 – Φ and c = ε – s
(I.3.10)
The reason for the compositional constraints is twofold. Firstly, one has to investigate what are the smallest stoichiometric units which determine the exchange of matter between the phases. These are not always the elements! Secondly, one has to take into consideration the initial compositional situation of the system. Often both of the above apply at the same time. The following cases will be used to discuss these aspects and how they help to find the values to be used in the above equations. – The ammonium chloride–gas equilibrium, system Cl–H–N. – The water–phosphoric acid equilibrium, system H–P–O. – The 2CaO · SiO2–3CaO · MgO · 2SiO2–CaO · MgO · SiO2 equilibrium.
Phase diagrams
I.3.4.1
69
The ammonium chloride–gas equilibrium
This equilibrium has been discussed in many books that treat the phase rule [62Pri, 82Atk, 89Bar]. Usually one finds the stoichiometric reaction NH4Cl = NH3(g) + HCl(g) as the centrepiece of discussion. This approach can be quite misleading as it does not make use of the fact that the gas phase is capable of attaining any composition based on the elements H, N and Cl. If this was done, one could treat this case along the lines of the previous example. For the gas alone, one obtains
ε=3
N, H, Cl
Φ=1
gas = {N2, H2, Cl2, HCl, NH3, ...}
s=0 c=ε–s=3
no compositional constraint and
f=c+2–Φ=4
T and P as well as two mole fractions of the three elemental components can be varied independently. With this gas we can now require the salt (NH4Cl) to coexist. One obtains
ε=3
N, H, Cl
Φ=2
gas = {... as above ...}, NH4Cl
s=0 c=ε–s=3
no compositional constraint and
f=c+2–Φ=3
T, P and one mole fraction of the elemental components in the gas can be varied. By beginning with a mixture of ammonia and hydrochloric acid (xNH3(g) + yHCl(g) with x + y = 1) all one does is to impose a compositional constraint on the gas phase, thus leaving one compositional degree of freedom: y. One obtains
ε=3 Φ=2 s=1 c = ε – s = 2 and
N, H, Cl gas = (... as above ...), NH4Cl nH : (3nN + nCl) = 1 f=c+2–Φ=2
Now for example the pressure is a function of T and y. What if one starts with solid ammonium chloride? NH4Cl will dissociate into the gas species NH3(g) and HCl(g) in equal proportions. Thus one obtains
70
The SGTE casebook
ε=3 Φ=2 s=2
N, H, Cl gas = (... as above ...), NH4Cl mN(g) : mH(g) = 1:4 and mN(g) : mCl(g) = 1:4
c = ε – s = 1 and
f=c+2–Φ=1
The total pressure P is a function of T or vice versa, the system behaves like a one-component system as discussed by Gibbs. This case shows very clearly how the initial choice of the substances from which the system is made up, together with the fact that the exchange of matter between the phases is restricted by the fixed molar ratios between N and H as well as between N and Cl, limits the degrees of freedom step by step.
I.3.4.2
The water–phosphoric acid equilibrium
When adding phosphoric acid (H3PO4) to water (H2O), one deals with a single solution phase (Φ = 1) with internal phase equilibria between the different aqueous species. The usual way to find c, the only missing parameter for the phase rule, is to list all aqueous species, to search for all independent stoichiometric reactions between them and to subtract these two numbers. A far more straightforward way is given by the method at present employed. One does not need to know about the internal phase equilibria. Instead, the number ε of elements and the number s, of compositional constraints, given by the initial substances is sufficient. A similar example was given by Planck for the system H2O–H2SO4 [30Pla]:
ε=3 Φ=1 s=1 c=ε–1=2
H, P, O water with all possible species nH : (nO – 5nP) = 1 and
f=c+2–Φ=3
One can vary T, p and the mole fraction of H3PO4. The latter fact is obvious as, from xH2O + yH3PO4 (with x + y = 1), one will always obtain the concentration of hydrogen, oxygen and phosphorus from the initial mole fraction of H3PO4. It is, however, not necessary to know which aqueous species will form.
I.3.4.3
The 2CaO · SiO2–3CaO · MgO · 2SiO2– CaO · MgO · SiO2 equilibrium
When considering equilibria in oxide (or salt) systems it is often the case that each of the metallic elements in the system is present in only one oxidation
Phase diagrams
71
state. Thus the possible composition space of the system based on the elements is reduced by one dimension as knowledge of the concentrations of the metallic components is sufficient to work out the value for the oxygen (or the salt-forming element). The four component system Ca–Mg–Si–O can therefore be considered as the CaO–MgO–SiO2 system if no pure metal phase or gas is involved, i.e. at sufficiently high oxygen potentials. Thus the possible composition space is not a three-dimensional tetrahedron but only a two-dimensional Gibbs triangle. However, the particular phases taking part in the equilibrium discussed here underlie yet another compositional constraint. This is best recognised when the phase formulae are rewritten using the mole fractions of the basic oxides. One obtains →
2CaO · SiO2
2 1 3 CaO ⋅ 3 SiO 2
3CaO · MgO · 2SiO2 →
1 1 1 2 CaO ⋅ 6 MgO ⋅ 3 SiO 2
→
1 1 1 3 CaO ⋅ 3 MgO ⋅ 3 SiO 2
CaO · MgO · SiO2
All phases have the same concentration of SiO2, thus reducing the number of components even further. They are all on the so-called orthosilicate section. Thus one could use 2CaO · SiO2 and 2MgO · SiO2 to rewrite all phase formulae! Thus:
ε=4 Φ=3 s=2
Ca, Mg, Si, O 2CaO · SiO2, 3CaO · MgO · 2SiO2, CaO · MgO · SiO2 mCa + mMg + 2mSi = mO and xSiO2 = constant
c = ε – s = 2 and
f=c+2–Φ=1
This is the same result as for a three-phase equilibrium of a real binary system, i.e. we have shown that the equilibrium lies in a quasibinary section.
I.3.5
Conclusions
From the set of examples given above which can all be interpreted fairly straightforwardly using c = ε – s and f = c + 2 – Φ, two consecutive steps can be derived which finally lead to the application of the phase rule on a particular phase equilibrium. 1
2
Define the system by searching the databank based on the list of elements in that system. Extract a list of the phases and their phase components which are to be used in the equilibrium calculations. Transfer the system data to the equilibrium program and define the system composition by a set of substances contained in the phase list.
72
The SGTE casebook
Define the temperature and pressure. Check the resulting equilibrium for compositional constraints. Usually these concern compositional degrees of freedom in the solution phases involved but, as shown by the example for the Ca–Mg–Si–O system, equilibria between purely stoichiometric phases can also be governed by such constraints. In these cases it is possible to redefine the set of independent components as shown above. It should be noted that the more qualitative approach used in the discussion of the above examples of course has its mathematical counterpart. As was shown by Rao [87Rao] and Hillert [93Hil], it is possible by means of linear algebra to determine the number of stoichiometric constraints. Essentially this is achieved by calculating the rank of a stoichiometry matrix, the columns of which are constituted by the elemental components and the lines by the species from which the phases are constituted.
I.3.6
References
887Gib J.W. GIBBS: ‘On the equilibrium of heterogeneous substances’, Trans. Conn. Acad. 3, 1887. 03Bau E. BAUR and A. GLÄSSNER: Equilibrium between iron oxides and carbon monoxide and dioxide, Z. Phys. Chem. 43, 1903, 354–368. 06Jän E. JÄNECKE: Z. Anorg. Chem. 51, 1906, 132. 30Pla M. PLANCK: Thermodynamik, 9. Auflage, Walter de Gruyter, Berlin, 1930. 44Mas G. MASING: Ternary Systems, Reinhold, New York, 1944. 62Pri I. PRIGOGINE and R. DEFAY: Chemische Thermodynamik, VEB Deutscher Verlag für Grundstoffindustrie, Leipzig, 1962. 63Pou M. POURBAIX: Atlas d’Équilibres Électrochimiques, Publication du Centre Blege d’Étude de la Corrosion ‘CEBELCOR’, Gauthier-Villars, Paris, 1963. 64Pal L.S. PALATNIK and A.I.LANDAU: Phase Equilibria in Multicomponent Systems, Holt, Rinehart and Winston, New York, 1964. 66Pri A. PRINCE: Alloy Phase Equilibria, Elsevier, Amsterdam, 1966. 73Pel A.D. PELTON and H. SCHMALZRIED: Metall. Trans 4, 1973, 1395. 82Atk P.W. ATKINS: Physical Chemistry, 2nd edition, Oxford University Press, Oxford, 1982. 84Bra T.R. BRAMBLETT and J.E. MORRAL: Bull. Alloy Phase Diagrams 5, 1984, 433– 436. 84Mor J.E. MORRAL: Scripta Metall. 18, 1984, 407–410. 85Hil M. HILLERT: Int. Metals Rev. 30(2), 1985, 45. 86Gup H. GUPTA, J.E. MORRAL and H. NOWOTNY: Scripta Metall. 20, 1986, 889–894. 87Rao Y.K. RAO: Metall. Trans. A 18, 1987, 327. 89Bar I. BARIN: Thermochemical Data of Pure Substances, VCH Verlagsgesellschaft, Weinheim, 1989. 91Koe E. KÖNIGSBERGER and H. GAMSJÄGER: Bunsenges. Phys. Chem. 95(6), 1991, 734. 93Hil M. HILLERT: J. Phase Equilibria 14(4), 1993, 418–424.
I.4 Summarising mathematical relationships between the Gibbs energy and other thermodynamic information KLAUS HACK
The preceding chapters have demonstrated how the different mathematical relationships of thermodynamics can be applied to a great variety of different tasks, from the simple calculation of a thermodynamic property of a pure substance to the phase diagram of a complex multicomponent multiphase system. Below is given a schematic summary in the form of a tree (Fig. I.4.1) showing which mathematical tools relate the Gibbs energy of a system to the various other items. Using G as a function of T, P and ni, Legendre transformations yield all other thermodynamic potentials with their respective natural variables. This has already been pointed out by Gibbs but is now well known under the term Maxwell relations. Based on G a minimisation leads to equilibrium conditions. In order to obtain a complete description of the respective system (including equilibrium amounts and not only potentials) the method of ‘extent of reaction’ can be applied for equilibria that can be described with simple stoichiometric reactions; for multicomponent, multiphase systems, one usually applies the Phase diagram
µ
,
i
cp
Gibbs–Duhem
,
H
()
i
l el w ,… x a F M U, H,
i
(),
Ai
,
vi
x
Minimisation
T
P
e n dr tio en ma g Le sfor n r ta
(),
i
Equilibria
P ith ar t re ial sp de ec ri t t va o tiv , es or
S
w
Gibbs energy
I.4.1 Schematic summary tree.
73
74
The SGTE casebook S
U
H
V
F F
G
T
I.4.2 The full arrow shows the positive value of the partial derivative.
method of Lagrangian undetermined multipliers. It should be noted that both ways require numerical procedures, i.e. a computer program. From single-equilibrium calculations the method of phase mapping based on the Gibbs–Duhem equation leads to phase diagrams of three different but interrelated kinds [73Pel]. To obtain quantitative relationships in multicomponent systems, two-dimensional sections can be generated, again using appropriate computer programs. Using the appropriate partial derivatives of G with respect to T, P and ni, all integral and partial values of the Gibbs energy, entropy and heat content of a phase can be calculated (Fig. I.4.2). This can be done by programming the derivatives explicitly for all Gibbs energy models which are based on explicit mathematical expressions of T, P and the concentration variables. However, it should be noted that some models, such as the cell model or the quasichemical model for ionic liquids, require a phase internal minimisation with respect to composition and subsequent numerical differentiation (with all the inherent disadvantages).
I.4.1 73Pel
Reference A.D. PELTON and H. SCHMALZRIED: Metall. Trans. 4, 1973, 1395.
Part II Applications in material science and processes
75
76
II.1 Hot salt corrosion of superalloys T O M I. B A R R Y and A L A N T. D I N S D A L E
II.1.1
Introduction
Under normal circumstances a nickel-based superalloy is protected from corrosive attack by an oxide layer on the surface, which is typically Cr2O3. However, in marine environments, turbine blades made from superalloys are particularly susceptible to corrosive attack by hot or molten salts. Under these conditions, NaCl can be swept into the turbine in the air stream to react with any sulphur-containing combustion products in the fuel to form sodium sulphate which condenses on to the oxide layer. This salt droplet may dissolve the protective layer and expose the alloy to corrosive attack from the oxidising and sulphidising atmosphere. In this chapter the range of conditions which might lead to this so-called ‘hot salt corrosion’ are explored in an attempt to provide a basis for understanding the detailed mechanisms involved. The driving force for corrosion is thermodynamic and hence it is attractive to analyse corrosion by reference to thermodynamic criteria. On the other hand, corrosion involves a series of processes that are linked together in a complex way. For the case of hot salt corrosion the mechanism by which chromium is removed from the surface of the turbine blade is particularly complex and may involve some sort of fluxing whereby the Cr2O3 is reprecipitated from the salt phase at its boundary with the gas phase. Traditionally corrosion problems have been analysed thermodynamically in terms of so-called phase stability diagrams. Such diagrams are conceptually simple but are really limited to systems where the material is a pure element. In view of the progress made in calculating phase equilibria involving many components and a wide range of phases it is appealing to analyse the corrosion phenomenon as a multicomponent system.
II.1.2
Data used for the calculations
In order to obtain some thermodynamic insight into this process, data for the C–H–O–N–Cl–S–Cr–Na system from the SGTE pure substance database 77
78
The SGTE casebook
were augmented by data assessed for the liquid phase and various solid salt phases thought to be of importance. Of the phases involved in hot corrosion, only the gas phase approximates closely to thermodynamically ideal behaviour. The phases present in the alloy, corrosion products and adherent salts are all non-ideal solutions. The chemical properties of the deposited salt are usually analysed in terms of its oxide activity. However, since NaOH is much more stable than Na2O in the presence of the water in the gas stream, it was decided to analyse the data required for calculations in terms of the quaternary system NaCl–NaOH–Na2CrO4–Na2SO4 Five condensed solution phases have been considered in this treatment: the liquid, the orthorhombic and hexagonal structures of Na2SO4, the halite structure of NaCl and the monoclinic (high-temperature) structure of NaOH. Mixing of all components was considered for the liquid but only pairs of components in the crystalline phases, as shown in Table II.1.1. None of these salt phases requires a large number of terms to model the solution behaviour and some approach ideality. The assessments for the systems NaCl–Na2SO4, NaCl–Na2CrO4 and Na2SO4–Na2CrO4 are based on the work of Liang et al. [80Lia1, 80Lia2] with some modifications. Different data are used for the pure components, and solution data are added for the orthorhombic sodium sulphate phase. The NaOH–Na2SO4 system has been assessed by Bale and Pelton [82Bal] but, since in that particular case equivalent rather than mole fractions were used, a complete reassessment was necessary. New assessments have also been made for the NaCl–NaOH and the NaOH– Na2CrO4 systems [87Bar]. The most important salt binary system is the Na2CrO4–Na2SO4 system in which the components are completely soluble in each other in both the liquid phase and in the hexagonal and orthorhombic Na2SO4-based solid solution phases. Figure II.1.1 shows the calculated phase diagram for this system. Figures II.1.2(a), Fig. II.1.2(b) and Fig. II.1.2(c) show the diagrams for three of the four ternary systems at 1023.15 K calculated from the critically assessed binary data.
Table II.1.1 List of condensed salt phases considered and their constituents Liquid Halite Monoclinic NaOH Hexagonal Na2SO4 Orthorhombic Na2SO4
NaCl, NaOH, Na2CrO4, Na2SO4 NaCl, NaOH NaCl, NaOH Na2CrO4, Na2SO4 Na2CrO4, Na2SO4
Hot salt corrosion of superalloys
79
1200 Liquid
Temperature (K)
1000
Hexagonal
800
600
Orthorhombic 400 300 0.0 Na2CrO4
0.2
0.4
0.6
xNa SO 2
4
0.8
1.0 Na2SO4
II.1.1 Calculated phase diagram for the Na2CrO4–Na2SO4 binary system.
II.1.3
The gas–salt equilibrium
The following calculations relate to the corrosion in a gas turbine operating at a fuel-to-air ratio of 50 to 1 by weight, a pressure of 15 bar (1.5 MPa) and a temperature of 750 °C. The fuel composition was taken to have the approximate formula CH1.8. With these assumptions the main combustion products have the following partial pressures: CO2, 0.61 bar; H2O, 0.55 bar; N2, 11.6 bar; O2, 2.216 bar On this basis, 1% sulphur in the fuel would cause the sum of the pressures of SO2 plus SO3 to be 0.0026 bar. The presence of 1 ppm of sodium chloride by weight of gas would result in a pressure of HCl of 0.0046 bar. The sodium will accumulate mainly as the sulphate on any surfaces. Hence the amount of sodium that can be used in the calculation is arbitrary. On the other hand, for calculations relating to the gas–salt interface, the amount of chlorine should reflect the levels expected in the gas atmosphere. For a given temperature, pressure and relative amounts or activities of components it is possible to calculate the stable assemblage of phases and their compositions by minimising the Gibbs energy. If necessary, the amounts of minor species are then calculated by equalisation of chemical potentials
80
The SGTE casebook NaOH
0.8
0.2
0.4 xNa2SO4
0.6
xNaOH
Liquid
0.4
0.6
0.2
0.8
NaCl
0.8
0.6
0.4 xNaCl
0.2
Na2SO4
(a)
Na2CrO4
0.8
0.2
0.6
xNa CrO 2
0.4
xNa SO 2
4
4
0.4
0.6 Liquid
0.2
NaCl
0.8
0.8
0.6
0.4 xNaCl (b)
0.2
Na2SO4
II.1.2 Phase diagram for 1023.15 K for ternary systems (a) NaCl– NaOH–Na2SO4, (b) NaCl–Na2SO4–Na2CrO4 and (c) NaOH–Na2SO4– Na2CrO4.
Hot salt corrosion of superalloys
81
Na2CrO4
0.8
0.2
0.4 xNa2SO4
0.6
xNa CrO 2
4
0.4
0.6 Liquid
0.2
NaOH
0.8
0.8
0.6
0.4 xNaOH
0.2
Na2SO4
(c)
II.1.2 (Continued)
followed by re-evaluation of the mass balance. Table II.1.2 shows the results of a calculation for the multicomponent system discussed below. The substances are grouped by phase. A number of less important gas-phase species and condensed stoichiometric substances have been omitted from the table. The values of chemical potential printed towards the end of the table are expressed in accordance with SGTE practice [85Bar]. It is possible to use the chemical potential to test whether the calculated equilibrium mixture would react with materials not considered in the calculation, e.g. for possible corrosion of the container. In accordance with the requirements of equilibrium the chemical potential of a species such as NaCl that is present in more than one phase are the same in all phases. The calculation shows that Cr2O3 does indeed dissolve in the salt phase, potentially exposing the turbine blade to corrosive attack from the oxidising atmosphere. Even under modest pressures of SO3, most of the sodium chloride can be expected to be converted to sodium sulphate. At the temperature of 1023.15 K, sodium sulphate itself is solid, whereas experimental evidence suggests that the corrosive agent at this temperature is a sodium-sulphate-rich liquid phase. However, Fig. II.1.2(a) shows that a liquid can form containing about 28 mol% NaCl, a little NaOH and the remainder Na2SO4 but this would depend on there being a sufficiently high partial pressure of HCl. The pressure of HCl required to cause the liquid to form depends on the partial pressure
82
The SGTE casebook
of SO3 which in turn depends on the partial pressure of oxygen. Figure II.1.3 shows the coexistence line between the liquid and crystalline sodium sulphate and also the less important line for the coexistence of liquid and the NaCl– NaOH of the halite structure. The equilibrium between the liquid and crystalline sodium sulphate is governed by the chemical reaction Na2SO4 + 2HCl = 2NaCl + SO3 + H2O
(II.1.1)
The proportion of NaOH is greatest under conditions of low pressures of HCl and SO3, as can easily be anticipated. The straightness of the line in Fig. II.1.3 arises because the concentration of NaOH is small and the proportions of NaCl and Na2SO4 in the liquid are insensitive to the relative pressures of HCl and SO3, provided that the liquid and crystalline sulphate coexist. However, as indicated by Reaction (II.1.1), the position of the line and hence the conditions for liquid-phase formation are dependent on the partial pressure of water. Thus the rates of diffusion of water in the crystalline and liquid phases may be important parameters in hot corrosion since these rates will influence liquid formation in microenvironments.
II.1.4
The interaction of gas and salt with Cr2O3
For systems of more than three components it is not possible to represent the results of calculations on a conventional phase diagram. Table II.1.2 shows the results of an individual calculation in the eight-component system 0
log pHCl
–1
Halite solid solution (NaCl–NaOH)
–2 Liquid –3 Hexagonal –4
–5 –10
–8
–6 –4 log pSO
–2
0
3
II.1.3 Diagram showing the conditions for the coexistence of the three-component liquid phase NaCl–NaOH–Na2SO4 with the twocomponent halite phase and hexagonal Na2SO4 at 1023.15 K (p H 2O = 0.55 bar).
Hot salt corrosion of superalloys
83
Table II.1.2 Edited output data from a single calculation of equilibrium in the system C–Cl–Cr–H–N–Na–O–S Temperature = 1023.1500 K Fixed gas volume = 8.395 500 × 10–2 m3 Calculated gas pressure = 1.515 009 × 106 Pa Phase Species
Gas phases CClO(g) CCl2O(g) CHNO(g) CO(g) CO2(g) Cl(g) Cl2(g) Cl2CrO2(g) ClH(g) ClHO(g) ClNO(g) ClNO2(g) ClNa(g) Cl2Na2(g) ClO(g) ClO2(g) Cl2O(g) Cl2OS(g) Cl2O2S(g) CrO2(g) CrO3(g) H(g) H2(g) H2N(g) H3N(g) HNO(g) HNO2(g) HNO3(g) HNaO(g) H2Na2O2(g) HO(g) HO2(g) H2O(g) H2O2(g) H2O4S(g) N(g) N2(g) N3(g) NO(g) NO2(g) NO3(g)
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
18 19 37 51 52 58 59 60 61 62 63 64 65 66 67 68 69 70 71 79 80 81 82 84 86 88 89 90 92 93 94 95 96 97 98 102 103 104 105 106 107
Amount (mol)
6.463 5.120 1.836 5.406 6.137 3.072 3.209 4.670 6.422 1.645 2.024 4.534 3.134 3.689 2.671 4.223 1.636 3.322 5.796 5.076 3.446 3.292 6.348 5.320 8.794 1.275 3.443 1.057 4.931 1.246 7.833 6.286 5.500 2.981 1.127 3.661 1.159 2.147 5.570 8.112 3.127
891 377 733 494 000 932 431 889 286 855 754 128 771 358 773 573 331 069 471 084 742 599 202 189 008 204 450 176 999 164 063 854 873 711 982 168 968 879 745 675 762
Mole fraction Partial pressure Notional activity × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×
10–18 10–17 10–22 10–11 10–1 10–6 10–5 10–6 10–3 10–6 10–7 10–10 10–5 10–6 10–7 10–11 10–12 10–19 10–17 10–17 10–11 10–14 10–11 10–22 10–19 10–12 10–7 10–8 10–10 10–16 10–7 10–8 10–1 10–9 10–9 10–21 10 10–23 10–4 10–5 10–11
6.464 5.120 1.836 5.406 6.137 3.073 3.209 4.671 6.422 1.645 2.024 4.534 3.134 3.689 2.671 4.223 1.636 3.322 5.796 5.076 3.446 3.292 6.348 5.320 8.794 1.275 3.443 1.057 4.932 1.246 7.833 6.287 5.501 2.981 1.128 3.661 1.160 2.147 5.570 8.112 3.127
081 528 787 654 181 023 526 027 476 903 814 262 864 467 852 697 379 167 641 233 843 696 389 345 267 241 551 207 144 201 294 039 035 798 015 276 002 942 909 914 854
× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×
10–18 10–18 10–22 10–11 10–1 10–6 10–5 10–6 10–3 10–6 10–7 10–10 10–5 10–6 10–7 10–11 10–12 10–19 10–17 10–17 10–11 10–14 10–11 10–22 10–19 10–12 10–7 10–8 10–10 10–16 10–7 10–8 10–1 10–9 10–9 10–21 10 10–23 10–4 10–5 10–11
84
The SGTE casebook
Table II.1.2 (Continued) Phase Species
N2O(g) N2O3(g) N2O4(g) Na(g) NaO(g) Na2O4S(g) O(g) O2(g) O3(g) OS(g) O2S(g) O3S(g) Phase total
Amount (mol) 10–7 10–13 10–15 10–16 10–16 10–10 10–10
× × × × ×
10–11 10–19 10–7 10–7 10
1.450 2.759 6.611 7.544 7.365 6.088 4.591 2.180 4.169 8.564 2.919 6.014 1.495
437 441 200 342 154 846 051 992 788 745 770 804 198
× × × × × × ×
10–7 10–13 10–14 10–16 10–16 10–10 10–10
× × × × ×
10–11 10–19 10–7 10–7 10
1.721 0.000 0.000 0.000 1.189 4.369 1.261 1.000 8.957 4.230 8.602
567 000 000 000 317 075 701 000 969 024 759
× 10–21
2.479 8.663 1.452 6.654 1.000
055 033 467 496 000
× × × ×
108 109 110 112 114 115 116 117 118 119 121 122
1 3 4 5 7 8 9 11 12 25 27
1 1 1 1 1 1 1 1 1 1 1
28 28 28 28
1 2 3 4
Halite phases ClNa(halite) HNaO(halite)
29 29
1 2
2.872 939 × 10–1 5.161 327 × 10–7
Monoclinic phases ClNa(monoclinic) HNaO(monoclinic)
30 30
1 2
1.632 411 × 10–1 4.444 794 × 10–6
31 31
1 2
32
1
Liquid phases ClNa(l) CrNa2O4(l) HNaO(l) Na2O4S(l) Phase total
Hexagonal phases CrNa2O4(hexagonal) Na2O4S(hexagonal) Phase total Orthorhombic Na2SO4 phases CrNa2O4(ortho Na2SO4)
395 359 005 120 937 666 916 927 665 493 684 627 153
× × × × × × ×
2 2 2 2 2 2 2 2 2 2 2 2
Stoichiometric phases C C2Cr3 C3Cr7 C6Cr23 CNa2O3 Cr CrN Cr2O3 Cr2O12S3 CrH2Na4O6 HNa3O5S
1.450 2.759 6.611 7.544 7.364 6.088 4.590 2.180 4.169 8.564 2.919 6.014 1.495
Mole fraction Partial pressure Notional activity
3.210 702 × 10–2
3.460 1.209 2.027 9.288 1.395
268 187 355 352 801
× × × × ×
10–3 10–3 10–7 10–3 10–2
3.457 211 × 10–2 4.531 808 × 10–1 4.877 529 × 10–1
× 10–6 × 10–23 × 10–20 × 10–19 × 10–11 × 10–6 10–1 10–2 10–5 10–1
7.088 038 × 10–2 9.291 196 × 10–1 1.000 000
6.232 106 × 10–2
Hot salt corrosion of superalloys
85
Table II.1.2 (Continued) Phase Species
Amount (mole)
Mole fraction Partial pressure Notional activity
Na2O4S(ortho Na2SO4)
32
2
2.551 695 × 10–1
Orthorhombic NaOH phase HNaO(ortho NaOH)
33
1
2.396 193 × 10–6
Na2SO4 phase Na2O4S(Na2SO4)
34
1
2.458 958 × 10–1
Component
Phase chemical potential
C Cl Cr H N Na O S Total
–4.199 –1.676 –4.761 –1.745 –9.551 –3.567 –1.098 –4.833
367 515 963 120 675 809 866 949
× × × × × × × ×
Activity
105 105 105 105 104 105 105 105
3.644 2.761 4.891 1.232 1.329 6.106 2.455 2.098
258 495 850 839 623 673 424 815
Amount (mol) × × × × × × × ×
10–22 10–9 10–25 10–9 10–5 10–19 10–6 10–25
0.613 0.010 0.100 1.106 23.200 1.000 8.229 0.462 34.722
70 00 00 60 00 00 40 47 17
Helmholtz energy = –4.327 944 809 8 × 106 J Amount of Phase component 3.1067 × 10 1.6054 × 10–1 8.0404 × 10–2 3.4143
Mole fraction of following components within the phase (mol) C
Cl
Cr
H
N
Na
O
S
Gas
0.019 75 0.000 21 0.000 00 0.035 62 0.746 77 0.000 00 0.197 64 0.000 00
Cr2O3
0.000 00 0.000 00 0.400 00 0.000 00 0.000 00 0.000 00 0.600 00 0.000 00
Liquid 0.000 00 0.043 04 0.015 04 0.000 00 0.000 00 0.304 16 0.522 24 0.115 52 Hexagonal 0.000 00 0.000 00 0.010 13 0.000 00 0.000 00 0.285 71 0.571 43 0.132 73
C–Cl–Cr–H–N–Na–O–S The pressures of N2, CO2, H2O and initially O2 were set to the values expected in the gas turbine environment. The amounts of sodium and chromium were fixed and the oxygen and sulphur proportions were adjusted to explore the range of conditions that might be found within the salt mixture at its contact with Cr2O3. The method allows the user of the system to calculate the partition of chromium between the four phases in which it is involved, namely Cr2O3, the crystalline solid solution of Na2CrO4 with Na2SO4, the liquid and the gas. The effect of the reducing conditions that might be expected within the molten salt deposit was explored by varying the oxygen potential.
86
The SGTE casebook
Because the binary system Na2CrO4–Na2SO4 behaves almost ideally, both for the solid phase as well as for the liquid solution, an appreciation of the equilibrium between Cr2O3 and these phases can be obtained by considering the chemical reaction Na2SO4 + 0.5Cr2O3 + 0.25O2 = Na2CrO4 + SO2
(II.1.2)
The reaction suggests that the equilibrium between Cr2O3 and Na2CrO4 at a particular mole fraction in the crystalline phase can be plotted on a conventional phase stability diagram, as shown in Fig. II.1.4. The slope of the lines in Fig. II.1.4 is 14 , in agreement with Reaction (II.1.2). Moreover the lines are nearly equally separated in accordance with the fact that the activity of Na2SO4 and the activity coefficient of Na2CrO4 in the solid solution are essentially constant. Because of this behaviour it is possible to combine the lines of Fig. II.1.4 into a single relationship by plotting the logarithms of the concentration of sodium chromate in the solid solution as a function of log pSO 2 – 0.25 log pO 2 . The result is shown in Fig. II.1.5. The same procedure can be followed for the solution of sodium chromate in the liquid phase. Despite the fact that the proportion of NaOH, although small, varies substantially and that of NaCl slightly with the imposed pressures of SO2, O2 and HCl, very little deviation from the two lines drawn in Fig. II.1.5 was calculated to occur. Liquid does not form under all conditions of
–2 –3
Corundum (Cr2O3)
xNa CrO 2
–4
4
0.0001 log pSO2
–5 0.001
–6
0.01
–7
0.1 –8 –9 –10 –12
–10
–8
–6 log pO
–4
–2
0
2
II.1.4 The solubility at 1023.15 K of Cr2O3 expressed as the mole fraction of Na2CrO4 dissolved in the hexagonal Na2SO4 phase as a function of log p O 2 and log p SO 2 .
Hot salt corrosion of superalloys
87
–1.0 –1.5
Liquid
–2.5
2
log xNa CrO
4
–2.0
Hexagonal
–3.0 –3.5 –4.0 –4.5 –5.0 –8
–7
–6 –5 –4 log pSO – 0.25 log pO 2
–3
–2
2
II.1.5 The solubility at 1023.15 K of Cr2O3 expressed as the mole fraction of Na2CrO4 in the liquid and hexagonal Na2SO4 phases. The conditions for existence of the liquid phase are indicated by Fig. II.1.3.
oxygen and sulphur dioxide pressure. However, the conditions for liquid formation are not very much altered by the proportion of Na2CrO4 in solution and hence can be predicted by reference to Fig. II.1.3. In the chosen model of the liquid phase the alkalinity, often expressed as oxide activity, is represented instead by the activity of NaOH. In principle the solution of Cr2O3 as Na2CrO4 might be represented by any of the following reactions: Na2SO4 + 0.5Cr2O3 + 0.25O2 = Na2CrO4 + SO2
(II.1.3)
2NaOH + 0.5Cr2O3 + 0.75O2 = Na2CrO4 + H2O
(II.1.4)
Na2O + 0.5Cr2O3 + 0.75O2 = Na2CrO4
(II.1.5)
In conditions of chemical equilibrium these reactions will all give consistent results. On the other hand, in the non-equilibrium conditions obtaining at the interface between the salt and the oxide, material needs to be transported to and from the salt–gas interface. Reaction (II.1.3) would provide a good summary of the process if sulphur dioxide were able to diffuse rapidly through the salt or Reaction (II.1.4) if diffusion of NaOH and H2O were more rapid. The rates of diffusion would depend not only on the diffusion coefficients but also on the activity gradients. The concentration of Na2O is unlikely to be sufficient to act as a vehicle for maintaining the alkalinity of the melt by diffusion from the salt–gas to the salt–oxide surface, as required by Reaction (II.1.5).
88
The SGTE casebook
In principle, it is not necessary to express the problem in terms of specific stoichiometric reactions; indeed this carries dangers of misrepresentation. The reactions have been used here merely to indicate the need to include data for all the constituents that are active in diffusion processes.
II.1.5
Limitations of the data and calculated results
The calculated phase equilibria are of course totally dependent on the data. The authors are reasonably satisfied with the data for the pure substances and solutions considered but it must be stressed that other substances or solution species need to be included. Thus, under conditions of low oxygen potential, sodium carbonate may become a significant component of the liquid. At high pressures of SO3, sodium pyrosulphate (Na2S2O7) may form and chromium may dissolve as the sulphate Cr2(SO4)3. Furthermore it cannot be excluded that sodium chromite (NaCrO2) may dissolve in the liquid phase. There is nothing in principle to stop the inclusion of these constituents except the limited amount of data available. The conditions for liquid-phase formation would in practice be greatly extended by the presence of nickel and cobalt in the alloy [80Gup, 80Lut] and potassium in the gaseous and liquid environments. The calculations discussed above, therefore, are intended mainly as an indication of the power of recent thermodynamic methods.
II.1.6
Extension to higher-order systems
The alloys commonly used in gas turbines contain the components nickel, cobalt and aluminium as well as chromium and operate in an environment containing many others including nitrogen, oxygen, carbon, hydrogen, sulphur, chlorine, sodium, potassium, calcium, magnesium, phosphorus and vanadium. The oxides, sulphides and salts formed should be considered as solution phases as should the metallic and carbide phases which develop in subsurface regions of the alloy. Coatings containing the additional elements silicon, zirconium, titanium and yttrium may also be applied and these will participate in all the phases already mentioned, including the alloy phases and the gas. It will be appreciated that the potential products of reactions are very numerous and a substantial volume of data is required. However, many of the same phases and systems that are important in gas turbine operation are also of interest in aqueous corrosion, pyrometallurgy, hydrometallurgy, electronic materials, ceramics, catalysis and other applications of alloys. There is therefore the prospect that data can be accumulated from many sources, provided that the assessments are based on consistent criteria. In recognition of this fact there is a drive towards standardisation of the models and of key data values. Thus in many respects the methods and
Hot salt corrosion of superalloys
89
organisation required to provide the thermodynamic data needed for analysis of hot corrosion are already in place.
II.1.7
Future developments
To extend the calculations to the kinetics of chemical processes it is usual to assume that there is local achievement of equilibrium. With this assumption, various workers have given analytical descriptions relevant to the oxidation of binary alloys with considerable success [83Laq, 83Nis]. However, for multicomponent alloys carrying multiphase corrosion products an analytical description is not feasible. Indeed, even individual calculations of chemical equilibrium use numerical and not analytical methods. The obvious method of modelling the kinetics of corrosion is therefore through the use of boundary element analysis. For this to be feasible, data will be required in the form of coefficients of equations for the diffusion of species under the gradients of chemical (and perhaps electrical) potential present in the corrosion layers and in the depleted alloy. These coefficients will be temperature and composition dependent and will be required for each phase and for grain boundary diffusion. If, for example, H2O(g) is the agent that transfers oxygen along a crack to cause further oxidation, the calculation requires knowledge of the interdiffusion coefficients of H2O with all the other gases present. Provided that all the data are available and provided that the models and software for determination of chemical equilibria allow the chemical potentials and concentrations of diffusing species to be calculated, the user of the kinetic model does not need to state explicitly the processes involved in terms of chemical reactions. Thus it should not be necessary to decide in advance whether the solution of, say, nickel oxide in sodium sulphate occurs by alkaline or acid dissolution. The correct result will be automatically generated provided that data for both models are incorporated into the database. Further work will be required before a full thermodynamic analysis of the problem can be attempted. This could include the addition of some or all of magnesium, nickel, chromium, aluminium, potassium, vanadium, carbonate, chromite, pyrosulphate and oxide ions to the salt phase. Such ions could have the effect of lowering the melting point of the solid phase or providing a mechanism for the migration of chromium away from the surface of the turbine blade.
II.1.8
Acknowledgements
The authors gratefully acknowledge discussions with Dr S. R. J. Saunders, concerning hot-corrosion phenomena, and with Mr R. H. Davies and Mrs S. M. Martin concerning various aspects of the data and calculation methods. The calculations for this case study have been performed using MTDATA.
90
II.1.9 80Gup 80Lia1 80Lia2 80Lut 82Bal 83Laq
83Nis 85Bar 87Bar
The SGTE casebook
References D.K. GUPTA and R.A. RAPP: J. Electrochem. Soc. 127, 1980, 2194–2202. W.W. LIANG, P.L. LIN and A.D. PELTON: High. Temp. Sci. 12, 1980, 41–50. W.W. LIANG, P.L. LIN and A.D. PELTON: High. Temp. Sci. 12, 1980, 71–88. K.L. LUTHRA and D.A. SHORES: J. Electrochem. Soc. 127, 1980, 2202–2210. C.W. BALE and A. PELTON: Calphad 6,1982, 255–278. W. LAQUA and H. SCHMALZRIED: in High Temperature Corrosion (NACE-6) (Ed. R.A. Rapp), National Association of Corrosion Engineers, Houston, Texas, USA, 1983, pp. 115–120. K. NISHIDA: in High Temperature Corrosion (NACE-6) (Ed. R.A. Rapp), National Association of Corrosion Engineers, Houston, Texas, USA, 1983, pp. 184–191. T.I. BARRY: in Chemical Thermodynamics in Industry: Models and Computation (Ed. T.I. Barry), Blackwell Scientific, Oxford, 1985, pp. 1–39. T.I. BARRY and A.T. DINSDALE: Mater. Sci. Technol. 3, 1987, 501–511.
II.2 Computer-assisted development of high-speed steels P E R G U S TA F S O N
II.2.1
Introduction
The optimisation of the composition of high-speed steel and the recommendation for heat treatment are usually based upon a large amount of experimental work, because an adequate set of phase diagrams is not normally available for such a high-order system. However, by assessing the lowerorder systems thermodynamically it is possible to obtain a data set from which the state of equilibrium can be calculated for any composition and temperature. Such a data set has now been produced for the Fe–Cr–Mo–W–C system and it has been tested by comparing predictions with the results of some experiments. As a result, we can now trust the data set and use it for compositions and temperatures different from the experimental values. The experimental part of the development work can thus be reduced considerably.
II.2.2
Background
The earliest known high-speed steels were tungsten based, 18 wt% W–4 wt% Cr–1 wt% V steel being the best known. Different applications have resulted in the development of several variations on this composition, with a W content ranging from 22 wt% for increased red hardness and a V content up to 5 wt% for greater resistance to abrasive wear. V dissolves in the MC1–x carbide and will stabilise this otherwise rather unstable carbide. All these steels contain about 4 wt% Cr which is important in reducing scaling and in the hardening reaction (it increases the hardenability). More recently, Mo has entered this field. It can, owing to the similarity between the two, replace W atom for atom and induce very similar properties.
II.2.3
Calculation
The following calculations will show a number of examples on how the description of the Fe–Cr–Mo–W–C system can be used to attain a better 91
92
The SGTE casebook
understanding of the properties of the high-speed steel type of alloys. At the present time, the databank contains a description of the Fe–Cr–Mo–W–C system. This description will in the future be extended to include other elements of importance for the properties of many real high-speed steels. The influence of Co and V was thus ignored in the calculations presented in this paper.
II.2.4
Discussion
Figure II.2.1 shows a calculated vertical section in the C–Cr–Fe–Mo–W system at 6 wt% W, 6 wt% Mo and 4 wt% Cr. In the past, various sections in the ternary C–Fe–W and quaternary C–Cr–Fe–W systems have been used as a basis for discussions on the properties of high-speed steels. As an example the vertical dashed line shows the composition of typical high-speed steel, containing 0.6 wt% C. On cooling, this alloy will initially solidify to the α phase, but that reaction will not go to completion because γ will start to form. The γ phase grows peritectically and all α will be consumed before the last liquid disappears and before the first M6C starts to form. At about 876 °C some M23C6 will form and finally at about 820 °C all γ will disappear owing to the formation of α. The calculated section alone does not give all information required. It may be more interesting to know how the amount of the various phases varies with temperature. This is better illustrated in Fig. II.2.2. This diagram 1600 1500 1400
Liquid Liquid + α Liquid + α + γ
Liquid + γ
Temperature (°C)
1300 1200 1100
α + γ +M2C
Liquid + M2C γ + M2C
γ + M2C + M7C2
1000 900
γ + M2C + M22C9
γ + M2C + M23C8 + MC 800 α + λ α + M2C + M22C6 +M2C α + M23C9 + MC 700 α + M6C + M20C9 + MC 600 0 0.5 1.0 1.5 2.0 C (wt%)
II.2.1 Vertical section of the C–Cr–Fe–Mo–W system at 4 wt% Cr, 6 wt% Mo and 6 wt%.
Computer-assisted development of high-speed steels
93
1.0 0.9 α
0.8
Weight fraction
0.7
γ
α
Liquid
0.6 0.5 0.4 0.3 0.2 0.1 M23C6 0 600 800
M6C 1000 1200 Temperature (°C)
1400
1600
II.2.2 The weight fraction of the phases forming in a high-speed steel containing 0.6 wt% C, 4 wt% Cr, 6 wt% Mo and 6 wt% W as a function of temperature. The weight fraction of a phase is represented by the fraction of the vertical axis covered by the phase.
0.20 0.18 0.16
Activity of C
0.14 0.12 0.10 0.08 0.06 0.04 0.02 0 600
800
1000 1200 Temperature (°C)
1400
II.2.3 The activity of C as a function of temperature for a steel of the composition given in Fig. II.2.2.
shows the variation with temperature in the amount of the various phases that exist in a steel containing 0.6 wt% C. At each temperature the weight fraction of a phase is represented by the fraction of the vertical axis covered by the phase. Figure II.2.3 shows how the activity of carbon changes with
94
The SGTE casebook
temperature in the same steel. It may also be of interest to know how a change in the composition influences the amount of the different phases at a given temperature. This is illustrated in Fig. II.2.4. This diagram shows how the amount of the various phases varies along the horizontal deshed line in Fig. II.2.1. The experimental data included in Fig. II.2.4 are taken from an unpublished work on high-speed steels by Wisell [81Wis]. He also determined the carbon activity and the composition of the γ phase in the same alloys. Figure II.2.5 shows the variation in the carbon activity along the same line in comparison with Wisell’s data. Figure II.2.6 shows the variation in composition of the γ phase along the line. The agreement between Wisell’s experimental data and the present calculation is very good. This is encouraging especially since this information was not included in the evaluation of the thermodynamic properties of the C–Cr–Fe–Mo–W system. The type of information given in the diagrams presented above can for example be of help to explain the effect of a given heat treatment or change in composition on the structure and mechanical properties of high-speed steel. The calculations presented here are merely a few examples on what can be done with the data available in the SGTE databank. The calculations for this case study have been performed using ThermoCalc. 1.0 0.9 0.8
Mole fraction
0.7 0.6
γ
0.5 0.4 α 0.3 0.2 Liquid 0.1 0
0
0.5
1.0 C (wt%)
1.5
2.0
II.2.4 The mole fraction of the phases forming in a high-speed steel containing 4 wt% Cr, 6 wt% Mo and 6 wt% W at 1200 °C as a function of the total carbon content: | , experimental data from unpublished work by Wisell [81Wis]. The weight fraction of a phase is represented by the mole of the vertical axis covered by the phase.
Computer-assisted development of high-speed steels
95
0.45 0.40
Activity of C
0.35 0.30 0.25 0.20 0.15 0.10 0.05 0
0
0.5
1.0 C (wt%)
1.5
2.0
II.2.5 The activity of C as a function of the total carbon content, for the steel given in Fig. II.2.4: | , experimental data from unpublished work by Wisell [81Wis]. 2.0 1.8 1.6
C in γ phase (wt%)
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 0
0.5
1.0 C (wt%)
1.5
2.0
(a)
II.2.6 The composition of the γ phase as a function of the total carbon content for the steel given in Fig. II.2.4 showing (a) the carbon content in γ as a function of the total carbon content, (b) the tungsten content in γ as a function of the total carbon content, (c) the molybdenum content in γ as a function of the total carbon content and (d) the chromium content in γ as a function of the total carbon content: | , | , , , experimental data for W: Mo: Cr = 6:6:4 from unpublished work by Wisell [81Wis]. |
The SGTE casebook 5.0 4.5
W in γ phase (wt%)
4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0
0
0.5
1.0 C (wt%)
1.5
2.0
1.5
2.0
(b)
5.0 4.5 4.0
Mo in γ phase (wt%)
96
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0
0
0.5
1.0 C (wt%) (c)
II.2.6 (Continued)
Computer-assisted development of high-speed steels
97
5.0 4.5
Cr in γ phase (wt%)
4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0
0
0.5
1.0 C (wt%)
1.5
2.0
(d)
II.2.6 (Continued)
II.2.5 81Wis
Reference H. WISELL: Laboratory Report ASP-B 3/81, Kloster Speed Steel AB, Sweden.
II.3 Using calculated phase diagrams in the selection of the composition of cemented WC tools with a Co–Fe–Ni binder phase ARMANDO FERNANDEZ GUILLERMET
II.3.1
Introduction: background to the problem
It has long been known [54Gur, 70Mos] that the transverse rupture strength of Co-bonded WC tools is critically dependent upon the carbon content, and that the most favourable results are obtained when only the face-centred cubic (fcc) phase and WC are formed on cooling from the sintering temperature. For Co-bonded WC tools this is achieved if the W-to-C ratio is close to the value corresponding to the stoichiometric composition WC. An excess of carbon leads to the formation of graphite and a deficiency of carbon to the formation of M6C. In this last case, the decrease in the rupture strength is particularly drastic. When replacing Co by an Fe–Ni alloy the compositions leading to fcc + WC do not only necessarily coincide with the stoichiometric ratio. As a consequence, early attempts to produce (Fe–Ni) bonded WC tools along the lines suggested by the experience with the Co-based systems gave unsatisfactory results [60Sch], probably because of M6C formation. More recent work [82Thu] on the effect of carbon upon the martensitic transformation in (Co– Fe–Ni) binder phases further highlights the need for information about the interesting range of carbon contents for WC tools. It is shown here that this type of information can be obtained from calculated phase diagrams based on consistent thermodynamic data sets for all phases involved. The underlying assessment work has been described elsewhere [89Fer1].
II.3.2
The region of favourable carbon contents
Figure II.3.1 shows a vertical section of the Co–W–C phase diagram calculated for a constant Co content of 10 wt%, a so-called isopleth diagram. It is evident from this graph that alloys with a carbon content falling between the compositions indicated by a and b will exhibit immediately after equilibrium solidification the two phases fcc and WC only. Of course, consideration of 98
Composition of cemented WC tools with a Co–Fe–Ni binder Liquid + M6C + WC
1450
99
Liquid + WC
1400 a
Temperature (°C)
1350 Fcc + M6C + WC
1300 1250
b Liquid + graphite + WC
M6C + WC
1200 Fcc + WC 1150 Fcc + M12C + WC 1100 Fcc + graphite + WC 1050 1000 4.0
4.5
5.0 C (wt%)
5.5
6.0
II.3.1 Calculated vertical section (isopleth) of the Co–W–C system, with constant Co content of 10 wt%. See the text for the explanation of the full circle and points a and b.
the precipitation of M12C or pure graphite, which takes place at lower temperatures [89Fer2], will narrow the favourable range even further. However, neglecting these effects in the present discussion the range of carbon contents indicated by a and b will be referred to as ‘the favourable region’. A full circle on the horizontal axis of Fig. II.3.1 and all the following figures indicate the composition of the respective system corresponding to contents of W and C in stoichiometric proportion to form WC. For the system shown in Fig. II.3.1 this composition falls within the favourable region.
II.3.3
Effects of replacing Co by Fe and Ni
Figure II.3.2 and Fig. II.3.3 show calculated vertical sections (isopleths) of the Ni–W–C and Fe–W–C phase diagrams respectively. Comparison with Fig. II.3.1 elucidates the various effects of a full substitution of Co by Ni and by Fe respectively. Ni moves the favourable region towards lower carbon contents compared with the stoichiometric composition without decreasing its width. Fe has the opposite effect, moving the favourable region towards higher carbon contents and decreasing its width. Figure II.3.2 shows also that the solid–liquid temperatures in the Ni–W–C system are higher than in the Co–W–C system. However, the calculations predict that appreciable amounts of Co may be replaced by Ni without a drastic increase in the solid– liquid temperatures. This is demonstrated by Fig. II.3.4 and Fig. II.3.5 which
100
The SGTE casebook 1500 Liquid + WC 1450
Liquid + M6C + WC
a
1400 Fcc + M6C + WC
b
Temperature (°C)
1350
Liquid + graphite + WC
1300 1250 1200
Fcc + graphite + WC 1150 Fcc + WC 1100 1050 1000 4.0
4.5
5.0 C (wt%)
5.5
6.0
II.3.2 Calculated isopleth of the Ni–W–C system, with constant Ni content of 10 wt%. 1500 1450
Liquid + WC
Liquid + M6C + WC
1400 Liquid + graphite + WC
Temperature (°C)
1350 1300
a Fcc + M6C + WC
1250 1200 1150
M6C + WC
Fcc + WC b
1100
Fcc + graphite + WC
1050 1000 4.0
4.5
5.0 C (wt%)
5.5
6.0
II.3.3 Calculated isopleth of the Fe–W–C system, with constant Fe content of 10 wt%.
show sections (isopleths) of the Co–Ni–W–C system calculated for 10 wt% (Co + Ni) and Co–to–Ni weight percentage) ratios of 4 to 1 (Fig. II.3.4) and 1 to 1 (Fig. II.3.5).
Composition of cemented WC tools with a Co–Fe–Ni binder
101
1500 Liquid WC 1450 1400 Liquid + graphite + WC Temperature (°C)
1350 Liquid + M6C + WC
a
1300 1250 1200
Fcc + M6C + WC b
M6C + WC
Fcc + WC
1150 1100 Fcc + graphite + WC
1050 1000 4.0
4.5
5.0 C (wt%)
5.5
6.0
II.3.4 Calculated isopleth of the Co–Ni–W–C system, with constant Co + Ni content of 10 wt% and Co-to-Ni weight percentage ratio of 4 to 1.
1500 Liquid + WC 1450 1400 Liquid + graphite + WC
Temperature (°C)
1350 1300
Liquid + M6C + WC
1250 M6C + WC 1200
Fcc + M6C + WC
1150
Fcc + WC
1100 Fcc + graphite + WC 1050 1000 4.0
4.5
5.0 C (wt%)
5.5
6.0
II.3.5 Calculated isopleth of the Co–Ni–W–C system, with constant Co + Ni content of 10 wt% and Co-to-Ni weight percentage ratio of 1 to 1.
102
The SGTE casebook
A question of practical interest is the replacement of part of the Co + Ni content of the system by Fe. Figure II.3.6 and Fig. II.3.7 illustrate the effect of replacing half the content of Co + Ni (in weight percentage as given in Fig. II.3.4 and Fig. II.3.5) by Fe. Compared with Fig. II.3.4 and Fig. II.3.5 the calculated sections of the quinary Co–Fe–Ni–W–C system suggest relatively small effects of the replacement of Co + Ni by Fe. The favourable region is as expected somewhat narrower than in the Fe-free alloys with the same Coto-Ni weight percentage ratio and this unfavourable effect is more pronounced for the alloys with the lowest Co-to-Ni weight percentage ratio. Also, the substitution by Fe has moved the favourable region towards lower carbon contents. In Fig. II.3.6 and Fig. II.3.7 this region is roughly centred around the stoichiometric composition.
II.3.4
Favourable carbon contents of a family of alloys
As an alternative to the use of single isopleths to determine the limits of the favourable region (i.e. the position of the points a and b in all previous graphs) of a set of specific alloys one can also study a family of alloys as was demonstrated for the Fe–Ni–W–C system by Fernandez Guillermet [87Ferl, 87Fer2]. He showed that the lines defined by the displacement of the points 1500 Liquid + WC 1450 1400 Liquid + graphite + WC
Temperature (°C)
1350 1300
Liquid + M6C + WC
1250 1200
a
Fcc + M6C + WC M6C + WC b Fcc + WC
1150 1100
Fcc + graphite + WC 1050 1000 4.0
4.5
5.0 C (wt%)
5.5
6.0
II.3.6 Calculated isopleth of the Co–Fe–Ni–W–C system, with constant Co + Fe + Ni content of 10 wt%, constant Fe content of 5 wt% and Co-to-Ni weight percentage ratio of 4 to 1.
Composition of cemented WC tools with a Co–Fe–Ni binder
Liquid + WC
1450 1400
Liquid + M6C + WC Liquid + graphite + WC
1350
Temperature (°C)
103
a
1300 Fcc + M6C + WC 1250
M6C + WC
1200
b Fcc + WC
1150 1100
Fcc + graphite + WC 1050 1000 4.0
4.5
5.0 C (wt%)
5.5
6.0
II.3.7 Calculated isopleth of the Co–Fe–Ni–W–C system, with constant (Co + Fe + Ni) content of 10 wt%, constant Fe content of 5 wt% and Co-to-Ni weight percentage ratio of 1 to 1.
a and b in the temperature composition space can be directly obtained by calculation. The results of such calculation for alloys of the Fe–Ni–W–C system with 20 wt% (Fe + Ni) is demonstrated in Fig. II.3.8 which shows the temperature projections of the lines as a function of the Ni/(Fe + Ni) weight percentage ratio. The region between the lines in Fig. II.3.8 defines the favourable region of this family of alloys. If the Fe + Ni content is decreased, the favourable region becomes narrower. This is illustrated by Fig. II.3.9 which shows the result of the calculation for 10 wt% (Fe + Ni).
II.3.5
Conclusions
The present case study demonstrates the possibilities of obtaining information of direct practical use in connection with (Co–Fe–Ni)-bonded cemented WC tools by use of complex equilibrium calculations in multicomponent multiphase systems. By carrying out calculations based on the same set of thermodynamic data for other compositions, ranges outside those covered by the present discussion can easily be covered. The calculations for this case study have been performed using ThermoCalc.
The SGTE casebook 1.0 0.9 1400 °C
0.8
1300 °C
Ni/(Fe + Ni) (wt%)
0.7 M6C formation
Graphite formation
0.6 1250 °C
0.5 1350 °C
0.4
1200 °C
0.3 0.2
1300 °C
1150 °C
0.1 0 4.0
4.5
5.0 C (wt%)
5.5
6.0
II.3.8. Temperature projection of a section of the Fe–Ni–W–C system with 20 wt% (Fe + Ni). The lines describe the composition of a mixture of WC and liquid in equilibrium with fcc and M6C (to the left), and WC and fcc in equilibrium with liquid and graphite (to the right). The asterisk on the horizontal axis represents the favourable region in the system. 1.0 0.9 0.8
1400 °C 1300 °C
0.7 Ni/(Ni + Fe) (wt%)
104
0.6
0.4 0.3 0.2
1350 °C
1200 °C
M6C formation 1300 °C
0.1 0 5.0
Graphite formation
1250 °C
0.5
5.5
1150 °C
6.0 C (wt%)
6.5
7.0
II.3.9 Temperature projection of a section of the Fe–Ni–W–C system with 10 wt% (Fe + Ni). For explanation of the lines see Fig. II.3.8.
Composition of cemented WC tools with a Co–Fe–Ni binder
II.3.6 54Gur 60Sch 70Mos 82Thu 87Ferl 87Fer2 89Ferl 89Fer2
105
References J. GURLAND: Trans. AIME 200, 1954, 285–290. P. SCHWARTZKOPF and R. KIEFFER: Cemented Carbides, Macmillan, London, 1960, pp. 188–190. D. MOSKOWITZ, M.J. FORD and M. HUMENIK JR: Int. J. Powder Metallurgy 6, 1970, 55–64 (and references therein). F. THUMMLER, H. HOLLECK and L. PRAKASH: High Temp–High Press. 14, 1982, 129–141. A. FERNANDEZ GUILLERMET: Z. Metallkunde 78, 1987, 165–171. A. FERNANDEZ GUILLERMET: Int. J. Refractory Hard Metals 6, 1987, 24–27. A. FERNANDEZ GUILLERMET: Z. Metallkunde 80,1989, 83–94. A. FERNANDEZ GUILLERMET: Metall. Trans. A 20, 1989, 935–956.
II.4 Prediction of loss of corrosion resistance in austenitic stainless steels M AT S H I L L E R T and C A I A N Q I U
II.4.1
Introduction
Austenitic stainless steels contain a combination of Cr and Ni which makes them fully austenitic under most practical conditions. The Cr content is usually about 18 wt% or higher which is well above the critical limit for corrosion resistance, about 12 wt% Cr. It is difficult to produce commercial steels without any C and some decades ago it was usual to have around 0.1 wt% C in the austenitic stainless steels. This caused troubles in welded constructions because a Cr-rich carbide may precipitate in the heat-affected zone. The carbide forms at the grain boundaries by C diffusing there from the bulk material. Owing to the slow diffusion of Cr, this element is only taken from a thin layer along the grain boundaries and that layer may thus be drastically depleted of Cr and may lose its corrosion resistance. Clearly, the risk of losing the corrosion resistance would be lower if the C content were lower, and today it is possible to produce austenitic stainless steels with a C content as low as 0.02 wt%. However, for any C content there is a critical temperature below which a heat treatment can produce a depleted zone with less than 12 wt% Cr. It would be of considerable practical value to be able to predict the critical temperature of any composition, i.e. for any combination of Cr, Ni and C contents. Such predictions will now be presented.
II.4.2
Theory
An accurate calculation of the formation of the depleted zone would require a detailed consideration of the diffusion-controlled growth of the C-rich carbide. However, a rough estimate can be made already from the thermodynamic information on the Fe–Cr–Ni–C system, using the following method. For chosen values of the temperature and Ni content, the C activity in the reaction zone along the grain boundaries can be calculated from the carbide– 106
Loss of corrosion resistance in austenitic stainless steels
107
austenite equilibrium assuming that the C content of the austenite will be at the critical value of 12 wt%. In view of the very large difference between the diffusivities of C and Cr, it may be assumed that the rate of precipitation is controlled by the rate of Cr diffusion and that C has sufficient time to diffuse over long distances and eliminate all differences in C activity. One may thus assume that the calculated C activity is also valid in the austenite far away from the grain boundaries and at an early stage of the reaction it will be equal to the initial C activity. One may then calculate the critical value of the initial C content of a steel, which will give a depletion of Cr down to 12 wt%. In the above calculation it is necessary to make some assumption regarding the Ni content in the depleted zone of austenite. The Cr-rich carbide will normally contain less Ni than the austenite and the Cr-depleted zone will thus be enriched in Ni. A quantitative calculation of this enrichment requires a consideration of the rate of diffusion of Ni relative to Cr and is outside the scope of the present work. Stawstrom and Hillert [69Sta] chose to neglect this effect and simply assumed that the initial Ni content is also valid in the Cr-depleted zone. This approximation will be accepted in the present work and the calculations of Stawstrom and Hillert will now be repeated but using an improved description of the thermodynamic properties of the Fe–Cr–Ni– C system.
II.4.3
Results
Figure II.4.1 shows the critical C content in a steel with 8 wt% Ni as a function of the Cr content at four temperatures. For instance, in a steel with 18 wt% Cr (and 8 wt% Ni) a C content of just over 0.05 wt% will give the critical condition at 800 °C. Below that temperature the Cr content in the depleted zone will be less than 12 wt%. These curves were calculated as described in the preceding section. The slopes of the curves in Fig. II.4.1 show that more C can be tolerated if the Cr content is higher. This conclusion holds in spite of the fact that C and Cr both increase the tendency of forming the Cr-rich carbide. This apparent contradiction is demonstrated for 900 °C in Fig. II.4.2. The explanation is that Cr has a strong decreasing effect on the activity coefficient of C in austenite. The relative positions in Fig. II.4.1 demonstrate a very strong effect of temperature. One must go to exceedingly low C contents in order to decrease the critical temperature below 600 °C. Figure II.4.3 shows the critical C content in a steel with 18 wt% Cr as a function of Ni content at four temperatures. The slopes of these curves show that less C can be tolerated if the Ni content is higher. An increase from 8 to 24 wt% Ni decreases the tolerable C content to half.
108
The SGTE casebook 600 °C
24
700 °C
23
Cr (wt%)
22 800 °C
21 20 19
900 °C
18 17 16
0
0.05
0.10 0.15 C (wt%)
0.20
0.25
II.4.I. Conditions for the formation of a depleted zone with a Cr content of 12 wt% in steels with 8 wt% Ni. 25 24
Equilibrium conditions at 900 °C
23
Cr (wt%)
22 21
Critical conditions at 900 °C
20 19 18 17 16 0
0.05
0.10 0.15 C (wt%)
0.20
0.25
II.4.2 Comparison between equilibrium conditions and conditions, where the Cr content in the depleted zone goes down to 12 wt%. The Ni content is 8 wt%.
Figure II.4.4, Fig. II.4.5 and Fig. II.4.6 show similar diagrams for higher Cr contents. The set of diagrams in Fig. II.4.3, Fig. II.4.4, II.4.5 and Fig. II.4.6 covers the most common range of compositions in austenitic stainless steels.
Loss of corrosion resistance in austenitic stainless steels
109
24 22
Ni (wt%)
20 18 900 °C
16 14 12
800 °C
10
700 °C
8 600 °C 0
0.05
0.10 0.15 C (wt%)
0.20
0.25
II.4.3. Conditions for the formation of a depleted zone with a Cr content of 12 wt% in steels with 18 wt% Cr.
24 22
Ni (wt%)
20 18 900 °C 16 14 800 °C
12 700 °C
10
600 °C
8 0
0.05
0.10 0.15 C (wt%)
0.20
0.25
II.4.4 Conditions for the formation of a depleted zone with a Cr content of 12 wt% in steels with 21 wt% Cr.
Figure II.4.7 shows the same kind of information plotted in a different way. Here the critical temperature is plotted as a function of the carbon content for a number of Cr contents and 8 wt% Ni. It is again demonstrated that very low C contents are required in order to decrease the critical temperature to low values. For comparison, the equilibrium temperature is plotted in Fig.
110
The SGTE casebook 24 22
Ni (wt%)
20 18
900 °C
16 14 800 °C 12 10
700 °C
8 600 °C 0
0.05
0.10 0.15 C (wt%)
0.20
0.25
II.4.5 Conditions for the formation of a depleted zone with a Cr content of 12 wt% in steels with 24 wt% Cr.
24 22
Ni (wt%)
20 18
900 °C
16 14 800 °C
12 10 8 0
700 °C 600 °C 0.05
0.10 0.15 C (wt%)
0.20
0.25
II.4.6 Conditions for the formation of a depleted zone with a Cr content of 12 wt% in steels with 27 wt% Cr.
II.4.4. For 0.10 wt% C and 18 wt% Cr the difference is 130 K. For 27 wt% Cr it is twice as high because the higher Cr content increases the equilibrium temperature but decreases the critical temperature. Of course, the curves in Fig. II.4.8 also show how high the temperature is that must be reached in order to dissolve carbide which may have precipitated during a previous heat treatment.
Loss of corrosion resistance in austenitic stainless steels
111
1000 950 900 18% Cr
Temperature (°C)
850 21% Cr 24% Cr
800 750
27% Cr
700 650 600 550 500 0
0.02
0.04 0.06 C (wt%)
0.08
0.1
II.4.7. The variation in the critical temperature, where the C content of the depleted zone goes down to 12 wt%, with the C content of steels with 8 wt% Ni and 18–27 wt% Cr. 1000 950
27% Cr 24% Cr
900 Temperature (°C)
18% Cr 21% Cr
850 800 750 700 650 600 550 500 0
0.02
0.04 0.06 C (wt%) in fcc
0.08
0.1
II.4.8 The variation in the temperature, below which carbide starts to precipitate, with the C content of the steels considered in Fig. II.4.7.
II.4.4
Discussion
In the present work it has been assumed that the critical limit for the Cr content is 12 wt%. It should be realised that this critical limit most probably depends upon the composition of the steel. For instance, it seems to be well
112
The SGTE casebook
established that Mo may also contribute to the corrosion resistance and it seems most likely that Ni will also have an effect, maybe a negative one. As a consequence, the predictions presented in the present work should only be used as a guideline. The predicted effects of changing the Cr, Ni and C contents are probably more reliable and can be used if one wants to predict how a change in composition will affect the tendency of a steel to lose its corrosion resistance. In an attempt to make the predictions more accurate, one may try to estimate how much higher the Ni content will be in the Cr-depleted zone than in the bulk. A very rough consideration of the rates of diffusion of Cr and Ni showed that the local Ni content in the depleted zone may be higher by a factor of 1.3. In order to demonstrate the effect of such an enrichment, a series of calculations were carried out for a bulk content of 8 wt% Ni. See Fig. II.4.9 which should be compared with Fig. II.4.1. It is evident that this effect is almost negligible.
II.4.5
Method of plotting diagrams
The present calculations were carried out using the Thermo-Calc databank [85Sun]. It has a special feature for automatic calculation of diagrams involving two different but related equilibria in each step [90Hal] which can be illustrated by reference to Fig. II.4.3. In this case, each calculation is carried out in two steps. In the first step, one defines the conditions of equilibrium between the carbide and austenite, 25 24
600 °C 700 °C
23 800 °C
Cr (wt%)
22 21
900 °C
20 19 18 17 16 0
0.05
0.10 0.15 C (wt%)
0.20
0.25
II.4.9 Conditions for the formation of a depleted zone with a C content of 12 wt% in steels with 10.4 wt% Ni.
Loss of corrosion resistance in austenitic stainless steels
113
by first giving the pressure (1 bar), temperature, Ni content (a value between 7 and 25 wt%) and Cr content (12 wt%) of the austenite and requiring that the carbide will be present but without giving its composition. The C content of the austenite required by the two-phase equilibrium is then calculated automatically and its C content is stored. In the second step a one-phase system of austenite is considered under the same pressure, temperature and Ni content, with the Cr content of the steel (18 wt%) and with the C activity obtained from the first step. Its C content is then calculated and is tabulated together with the Ni content used. The procedure is repeated with another Ni content until the range of Ni contents has been covered. Then the whole calculation is repeated for a new temperature. Finally, the content of the table is plotted. The other diagrams could be calculated similarly because it is possible to define the conditions for each equilibrium calculation in any way and to transfer any condition or result of the first calculation into the conditions for the second calculation.
II.4.6
Database
The present calculations were carried out with a database for the Fe–Cr–Ni– C system which has recently been completed [90Hil] and included in the Thermo-Calc databank. For the present calculations, information on austenite and M23C6 were used.
II.4.7 69Sta 85Sun 90Hal 90Hil
References C.O. STAWSTROM and M. HILLERT: J. Iron Steel Inst. 207, 1969, 77–85. B. SUNDMAN, B. JANSSON and J.-O. ANDERSSON: Calphad 9, 1985, 153–90. B. HALLSTEDT and L. HOGLUND: Unpublished work, Royal Institute of Technology, Stockholm, Sweden, 1990. M. HILLERT and C. QUI: ‘A thermodynamic assessment of the Fe–Cr–Ni–C system’, Technical Report TRITA-MAC 420, Royal Institute of Technology, Stockholm, Sweden, 1990.
II.5 Prediction of a quasiternary section of a quaternary phase diagram M AT S H I L L E R T and S T E F A N J O N S S O N
II.5.1
Introduction
In order to visualise the phase diagram of a quaternary system under constant temperature and pressure, one would need three dimensions. However, in some cases, much information can be presented in a quasiternary section. This is the case for the Si–Al–O–N system where all the condensed phases fall in or very close to the section defined by the corners Si3N4, SiO2, Al2O3 and AlN as if all the phases were ionic and composed of Si4+, Al3+, O2– and N3–, which is not quite true. It has been possible to model the properties of the condensed phases using models which restrict their existence to the section mentioned. The modelling will now be described briefly and it will be demonstrated that it was possible to extrapolate the properties of the ternary side systems and to use the meagre information available from the quaternary system in order to predict the phase relations [91Hil2].
II.5.2
Solid phases
There are two solid Si3N4 modifications, α and β, but their relative stabilities are not well known and it is difficult to separate information on their individual properties. In the present work the α phase was completely neglected. The βSi3N4 phase extends far into the quaternary Si–Al–O–N system and, from a practical point of view, it is the most important phase in this system. The Si2N2O phase also extends into the system but has found less practical use. Inside the quaternary system these phases are often denoted by β′ and O′ respectively. In addition, there is a quaternary phase denoted by X. Its composition is not well known but will be treated as a stoichiometric phase with the composition Si12Al18O39N8. Finally, there is a whole series of socalled polytype phases related to AlN and situated close to the AlN corner. Their stabilities and properties are not well known and in the present work they were simply represented by one of them, the so-called 27R phase, which was treated as a stoichiometric phase on the AlN–Al2O3 side. 114
Quasiternary section of a quaternary phase diagram
II.5.3
115
Modelling
It was assumed that the β′ phase can dissolve Al in the Si sublattice and O in the N sublattice. Accepting the normal valencies the formula was thus written as (Si4+, Al3+)3(N3–, O2–)4. A general type of model for phases with mixtures of elements on two different sublattices, called the compound energy model [70Hil, 86And], has been developed and programmed for computer calculations [85Sun]. In the present case a special procedure must be used in order to ascertain that the composition stays inside the quasiternary section. It is simply based upon the condition of electroneutrality. With this model the β′ phase extends along a straight line from the Si3N4 corner to the AlN · Al2O3 point on the AlN–Al2O3 side. The O′ phase was treated in the same way using the formula (Si4+, Al3+)2(N3–, O2–)2(O2–)1. With this model the phase extends along a straight line from the Si2N2O point to the Al2O3 corner. The modelling of the liquid phase is most difficult. In an attempt to describe the covalent character of the bonds and the tendency to form a network, the SiO2–Si3N4 liquid was described simply by using hypothetical species SiO2 and SiN4/3 [91Hil1]. However, when analysing the properties of Al2O3 in various systems it was found that they could be represented with an ionic model, mainly using Al3+ and O2– [90Hal]. The liquid phase in the quaternary system was finally described with the formula (Al3+)P(N3–, O2–, SiN4/30, SiO20, SiO44–)Q using the ionic two-sublattice model [85Hil]. P and Q are coefficients to be adjusted to satisfy the condition of electroneutrality.
II.5.4
Results
The melting behaviour is particularly interesting in connection with the production and use of materials in this system. Figure II.5.1 shows a projection of the calculated liquidus surfaces but it should not be trusted close to the AlN corner where a number of polytype phases may form. In the other regions of the system the diagram may be regarded as a reasonable prediction based upon extrapolation of the ternary properties from the four side systems and on the meagre information available inside the system. A vertical section through the straight line of existence for the β′ phase is also of considerable interest. Figure II.5.2 shows when β′ starts to melt (see dashed line) and also the phases that it may be in equilibrium with in the case of a small excess of SiO2 (in parentheses). Phase fields, where the β′ phase of a given composition decomposes into β′ of another composition and another phase upon heating, are given without parentheses. It is possible to combine the calculated properties of the solid phases inside the quaternary section with the known properties of the gas phase,
The SGTE casebook Al2O3
SiO2 1873 100
1973
2073
2173
2073
Equivalent O (%)
70 60
Spinel
Al
X
80
2273
2O 3
Mullite 90
2173 O′
2273
50
2373
2373
β′
40
27R
2473
30
73 24 73 25 73 26 773 3 2 287 AIN 73 29
20 2573
10 0 0 Si3N4
20
40
60
80
2973
100 AIN
Equivalent Al (%)
II.5.1 Predicted projection of the liquidus surfaces with isotherms. The values give the liquidus temperatures in kelvin. 2700 2600
+S
P+
2300
LS
SP
β′ + (LS) 2200
β′ + (M) β′ + (O′)
β′+(SP)
β′ +
2100
IN
0.6 xAl3NO3
)
0.4
3
0.2
l 2O
1800
+A
1900
(A β′+
β′ + (X)
SP
2000
1700 0 Si3N4
S
β′
+L
S
2400
SP + β′ + 27R
+L
SP + LS + 27R
LS
β′
2500
Temperature (K)
116
0.8
1.0 Al3NO3
II.5.2 Predicted section through the Si3N4–Al3NO3 line showing equilibria with β-(Si–Al–O–N) (β ′). Two-phase fields with the other phase in parentheses indicate that the amount of the other phase is zero in the section. All regions below the dashed line actually represent 100% β-(Si–Al–O–N). The mullite and spinel phases are denoted by M and SP respectively.
Quasiternary section of a quaternary phase diagram Al2O3 0.1
3
0.2
90
Mullite
X
80
Equivalent O (%)
Spinel
0.4 0.8 1 2
70 60
O′
4
0 20 00 4
50 40
Al 2O
SiO2 100
8
β′
10
800
30
20
1000
27R
40
20
20 40 AIN 80
80
10 0
117
100 0
20
40
60
Si3N4
80
100 AIN
Equivalent Al (%)
II.5.3 Predicted projection of the liquidus surfaces with curves giving the vapour pressures in bars.
which in general falls outside the section. In this way it was possible to produce a diagram (Fig. II.5.3) showing what pressure is required in order to prevent boiling before complete melting. Even if the pressure is high enough to prevent boiling, there is a considerable risk of losing material by evaporation and thus changing the composition. The composition may thus move away from the quasiternary section which is in principle possible by the formation of metallic droplets. The calculations for this case study have been performed using Thermo-Calc.
II.5.5 70Hil 85Hil 85Sun 86And 90Hal 91Hil1 91Hil2
References M. HILLERT and L.-I. STAFFANSSON: Acta Chem. Scand. 24, 1970, 3618–3626. M. HILLERT, B. JANSSON, B. SUNDMAN and J. ÅGREN: Metall. Trans. A 16, 1985,261– 266. B. Sundman, B. Jansson and J.-O. Andersson: Calphad 9, 1985, 153–190. J.-O. Andersson, A. FernAndez Guillermet, M. Hillert, B. Jansson and B. SUNDMAN: Acta Metall. 34, 1986, 437–445. B. HALLSTEDT: J. Am. Ceram. Soc. 73, 1990, 15–23. M. HILLERT and S.JONSSON: Technical Report TRITA-MAC 465, Royal Institute of Technology, Stockholm, Sweden, 1991. M. Hillert and S. Jonsson: Technical Report TRITA-MAC 470, Royal Institute of Technology, Stockholm, Sweden, 1991.
II.6 Hot isostatic pressing of Al–Ni alloys KLAUS HACK
II.6.1
Introduction
Although there are a number of processes with phase transitions taking place under elevated total pressure, such as in squeeze forging or hot isostatic pressing (HIP), metallurgists tend to neglect the influence of pressure on chemical equilibria since, in most cases, the pressure level is near or even below atmospheric pressure. While geologists have to treat the total pressure as one of their major variables and, therefore, have developed explicit expressions [44Mur] and consistent databases [94Sax], calculations for alloy systems still have to be treated on an approximative level. Often, it is even difficult to find the values of the molar volume of an alloy phase, let alone the thermal expansion, the compressibility or the pressure derivative of the bulk modulus. The following example demonstrates how such a ‘thin’ database can still be applied to provide a reasonable first estimate of the pressure dependence of alloy phase equilibria. However, it should be noted that the Clausius– Clapeyron equation presented in all standard textbooks for the case of a onecomponent system first has to be generalised. In the example, the pressure dependence of the peritectic temperature of the reaction γ + γ ′ = liquid in nickel-based superalloys is calculated. The complex superalloy is approximated by the Ni–Al system, the phase diagram of which is given in Fig. II.6.1. The diagram shows that the phase boundaries between the two solid phases run almost vertically. Since during HIP the γ ′ phase is supposed to dissolve completely in the γ phase, the highest possible temperature is chosen. Theoretically this temperature is the peritectic temperature. On the other hand, the formation of liquid is to be avoided, because, on solidification, microporosity may occur, which is harmful to the mechanical properties. Thus the task is to find a value for the shift of the peritectic temperature on application of a pressure of 2000 bar at which the HIP process is operated. 118
Hot isostatic pressing of Al–Ni alloys
119
1650
1648
Temperature (K)
Liquid 1646
1644 γ 1642 γ′ 1640 0.70
0.72
0.74 0.76 Mole fraction of Ni
0.78
0.80
II.6.1 Phase diagram of the Ni–Al system.
II.6.2
Generalised Clausius–Clapeyron equation
Pelton and Schmalzried [73Pel] have shown that the phase boundaries in a phase diagram are mathematically governed by a Gibbs–Duhem equation with generalised potentials and conjugate extensive properties:
ΣQiϕ dΦi = 0
(II.6.1)
where ϕ is the number of equilibrium phase. With Φi = T, –P or µi as the potential, S, V, and the mole numbers ni must be used as the conjugate extensive properties. If the pressure dependence of equilibrium has to be calculated, only the first two terms (Sϕ dT – Vϕ dP) apply, as all changes in chemical potentials are zero at equilibrium. For a two-phase equilibrium in a one-component case the Clausius–Clapeyron equation is derived easily from 1 2 S 1 dT – V 1 dP = S 2 dT – V 2 dP[= 0] → dT = V 1 – V2 (II.6.2) dP S –S Note that this equation applies not only to a phase transformation of an element but also to all two-phase equilibria with one Gibbsian component, i.e. those cases in which both phases have the same composition with respect to all elementary components as stated by Gibbs [878Gib]. Examples are Zn(s) = Zn(l), but also H2O(s) = H2O(l), CaO · A12O3 · 2SiO2 = liquid, or azeotropic points in organic systems. It is also worth mentioning that the absolute values of entropy and volume are used, i.e. the conjugate extensive
120
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properties to temperature and pressure respectively although, because of the ratio, the result is the same if molar quantities are inserted. For a three-phase equilibrium such as a eutectic or peritectic in a binary system (binary in the sense of the phase rule), three equations of the above type are obtained. In the general case, all three phases are solution phases and thus the composition in each phase may vary with pressure. Thus it becomes more obvious that the slope dT/dP is calculated for a certain fixed value P°. One obtains 3 2 1 1 3 2 2 1 3 dT = ( x – x )V + ( x – x )V + ( x – x )V 3 2 1 1 3 2 2 1 3 dP ( x – x )S + ( x – x )S + ( x – x )S
(II.6.3)
with xϕ the mole fraction of component two in phase ϕ, and Vϕ and Sϕ the integral molar volume and entropy respectively of phase ϕ. The numerator can easily be interpreted as the volume change ∆V(1 + 2 → 3) (eutectic) or ∆V(l→2 + 3) (peritectic) of the respective reaction with the appropriate interpretation of the entropy changes in the denominator. A more general treatment of pressure dependence of all types of phase boundary has been given by Hillert [82Hil].
II.6.3
Application to the Ni–Al equilibrium
The data required for the application of the above equation to the Ni-rich equilibria in the Ni–Al system are summarised in Table II.6.1. Thermochemical data have been given by Murray [89Mur] who found eutectic behaviour between the liquid, γ and γ ′, while Sundman and Ansara [90Sun] in a more recent assessment concluded a peritectic reaction. The majority of the volume data had to be based on assumptions (e.g. the additivity of molar volumes of the component elements for the solution phases), interpolations or comparisons with similar metals. Different databases had to be used for the solid and the liquid state [75Tou, 90Lan]. It became obvious during the search for these data that there is a striking lack of precise data with respect to the pressure dependence for alloy calculations. From the data in Table II.6.1 the shift of the melting point of nickel (Ni(s) = Ni(l)) is calculated to be dT/dP = ∆V/∆S = 3.223 × 10–8 K Pa–1. For the eutectic reaction according to Murray, one obtains dT/dP = 4.92 × 10–8 K Pa–1, while the peritectic reaction according to Sundman and Ansara leads to dT/dP = 5.07 × 10–8 K Pa–1. Thus, for an HIP pressure of 2000 bar = 0.2 GPa, the melting temperature of nickel is raised by 6.446 K, and the three-phase equilibrium by 9.84 K or 10.14 K respectively. Consequently, under HIP conditions, the maximum operating temperature may be chosen to be about 10 K higher than the annealing temperature at ambient pressure.
Hot isostatic pressing of Al–Ni alloys
121
Table II.6.1 Univariant phase equilibrium in the system Ni–Al (a) Pure nickel: Ni(s) = Ni(l), Tm = 1728.3 K
∆V* (m3 mol–1) ∆S† (J mol–1 K–1)
3.26 × 10–7 10.1139
(b) Eutectic reaction [89Mur]: γ ′ + γ = liquid, Teut = 1658 K
XNi V × 106* [ m3 mol–1] S‡ [J mol–1 K–1]
γ′
γ
Liquid
0.740 8.126 2.168
0.798 7.891 4.184
0.750 8.71 15.209
(c) Peritectic reaction [90Sun]: liquid + γ = γ ′, Tper = 1643.4 K
XNi V x 106* [ m3 mol–1] S§ [J mol–1 K–1]
Liquid
γ
γ′
0.756 8.658 6.3774
0.787 7.928 –3.2833
0.760 8.037 –5.2587
*Based on the work by Touloukian et al. [75Tou] and by Lang [90Lan]. †Based on the work by Dinsdale [91Din]. ‡Relative to solid Al and Ni. § Relative to liquid Al and face-centred cubic Al Ni.
II.6.4
Conclusions
The generalised Clausius–Clapeyron equation yields a precise mathematical expression for the pressure derivative of univariant temperatures in binary (or higher-order) systems. The type of the univariant equilibrium (peritectic or eutectic) and data for the entropy change of the reaction can be calculated for a good number of systems from different thermochemical assessments, while volume data are still very scarce. Nevertheless, for the Ni-rich part of the Ni–Al system the resulting pressure derivatives of the univariant temperatures can be used for reasonable estimates of temperature changes on application of technically feasible pressures.
II.6.5
Acknowledgement
The author wishes to thank Professor D. Neuschütz, LTH, RWTH Aachen, for drawing attention to this subject matter.
122
II.6.6
The SGTE casebook
References
878Gib J.W. GIBBS: ‘On the equilibria of heterogeneous substances’, Trans. Conn. Acad. 3, 1878. 44Mur F.D. MURNAGHAN: Proc. Natl Acad. Sci. 30, 1944, 244. 73Pel A. PELTON and H.S. CHMALZRIED: Metall. Trans. 4, 1973, 1395. 75Tou Y.S. TOULOUKIAN et al.: in Thermophysical Properties of Matter. Vol. II, (Ed. Y.S. Touloukian), OFI–Plenum, New York, 1975. 82Hil M. HILLERT: Bull. Alloy Phase Diagrams, 3(1), 1982, 4. 89Mur J. MURRAY: Private communication. 90Lan G. LANG: in Handbook of Chemistry and Physics (Ed. D.R. Lide), CRC Press, Boca Raton, Florida, 1990. 90Sun B. SUNDMAN and I. ANSARA: COST507 Database, 1990. 91Din A.T. DINSDALE: Calphad 15, 1991, 317–425. 94Sax S. SAXENA, N. CHATTERJI, Y. FEI and G. SHEN: Thermodynamic Data on Oxides and Silicates, Springer, Berlin, 1994.
II.7 Thermodynamics in microelectronics A L E X A N D E R P I S C H and C L A U D E B E R N A R D
II.7.1
Introduction
The ongoing size reduction of the individual transistor and the growing number of active and passive devices within an integrated circuit are the current challenges in the microelectronics industry. The higher integration of devices allows higher clock frequencies and improves speed and performance but, at the same time, serious technological problems have to be solved. Some classical materials, such as insulating SiO2 as the dielectric and conducting Al for interconnects, which were used in the last 50 years, have instrinsic properties that are no longer good enough on a reduced scale. They have to be replaced by materials with improved electrical, mechanical and chemical properties. As examples for these new materials, one can cite HfO2 and SrTiO3 as promising new dielectrics. Metallic Cu is the choice for interconnects, but its high chemical reactivity requires the use of diffusion barriers such as TaN or W. However, these barriers are often insulating and therefore other materials such as conducting amorphous Ru–Si–O or Ir–Si–O are also under investigation. Integrated circuits have a multilayered structure of thin films with various thicknesses which are deposited by different techniques and complex process steps. Chemical stability is therefore a critical issue and thermodynamic calculations can provide valuable information. Two main fields are concerned. 1
2
Phase stability. Is the material stable with its environment or is it likely that interface reactions occur during thermal treatment of the complete structure? Is it possible to etch the material in order to obtain the desired geometric structure? If the answer is yes, by which technique and with which reactants? Materials synthesis. By which technique can the new material be deposited (evaporation, sputtering or chemical vapour deposition) and which are the optimum experimental conditions (temperature, total pressure and precursors)? 123
124
The SGTE casebook
Performing a thermodynamic modelling can provide at least qualitative answers to these questions. The thermodynamic approach will be illustrated on four examples: SrTiO3 film deposition by using organometallic precursors and interface stability of the thin film with the Si substrate, reactive ion etching of HfO2 thin films and thermal stability of Ru–Si–O and Ir–Si–O amorphous thin films upon annealing.
II.7.2
Thin-film deposition of SrTiO3 and interface stability with Si
Chemical vapour deposition of complex oxides using organometallic precursors is a process that is now widely used in microelectronics industry. The advantages of this process are process compatibility, good control of the deposition parameters, coverage of non-planar surfaces and selectivity depending on the nature of the substrate. In the case of SrTiO3, organometallic precursors such as Sr(thd)2 (thd = C11H19O2) and Ti(i-Opr)2(thd)2 (i-Opr = C3H7O2) are dissolved with a given stoichiometry in a suitable liquid organic solvent. In this study, monoglyme (1,2 diethoxymethane) (C4H10O2) was used. The dissolution of the highly reactive organometallic precursors in the solvent has the advantage of stabilising them chemically. Small quantities of the liquid are injected into the reaction chamber in a repetitive manner for reaction with O2 to form the oxide layer. The use of calibrated injection pumps increases considerably the reproducibility of this process. This liquid injection CVD process has still a large number of experimental parameters which have to be controlled. – – – – –
Total pressure in the reactor. Deposition temperature. Precursor concentration in the liquid solution. Injection frequency. Ar + O2 carrier gas flow.
The main goal of a thermodynamic modelling of the deposition process is to establish the parameter intervals for which a pure SrTiO3 thin film can be obtained [003Ran]. A two-dimensional plot is generally chosen to present the results of these calculations. The two axes correspond to two independent experimental parameters, all other conditions remain constant. It is important to mention that kinetic effects are not taken into account. It is supposed that the reactions at the surface are fast enough to obtain a local equilibrium. This is in general the case for oxide deposition from organometallic precursors. The SGTE Substance Database [006SGT] is used for thermodynamic calculations. No thermodynamic data are available for the organometallic compounds; therefore their complete decomposition at the surface was
Thermodynamics in microelectronics
125
supposed. As an example for the results of this thermodynamic modelling, the precursor ratio of Sr(thd)2 to Ti(i-Opr)2(thd)2 as a function of Ar+O2 carrier gas flow is simulated at 700 °C and a total pressure of 6.6 mbar and the results are reproduced in Fig. II.7.1. The precursor concentration is fixed at 0.02 M and the injection speed is 2.5 × 10–5 cm3 s–1. Pure SrTiO3 can only be obtained for an equimolar ratio of the two precursors and log(Ar+O2) partial pressures greater than –3. For lower carrier gas flow rates, carbon co-deposition is expected, because the oxygen potential is too low to transform the carbon and hydrogen of the solution into CO(g) or CO2(g) and H2O(g). If the precursor ratio deviates from unity, secondary phases such as TiO2 or Sr4Ti3O10 may occur. Experiments confirm the validity of this simulation approach. Thin-film deposition in the vicinity of the full code in Fig. II.7.1 revealed that it is difficult to obtain single-phase SrTiO3 because of process fluctuations during deposition. Secondary TiO2 and Sr4Ti3O10 are identified by X-ray diffraction and transmission electron microscopy [004Lho], in excellent agreement with the modelling. It is important to mention that the experimental precursor ratio to obtain near single-phase SrTiO3 differs from unity. This disagreement arises because transport phenomena and gas-phase diffusion are neglected. The qualitative agreement is good and the real diagram is only shifted when taking into account the transport and the diffusion in the modelling. The thermodynamic calculations serve therefore as a starting point for further simulations. In order to simulate quantitatively the deposition behaviour in a specific type of chemical vapour deposition reactor, a coupling of homogeneous and heterogeneous thermodynamics, kinetics, diffusion and heat and mass transport phenomena is necessary [002Ber]. After having demonstrated that it is possible to deposit near single-phase SrTiO3, the interface compatibility with the Si substrate is the main issue. The interface quality is the key parameter for the electrical properties of the material. In order to investigate the stability of SrTiO3 on Si, the Sr–Ti–O–Si quaternary phase diagram was calculated at 700 °C. The isothermal tetrahedron is presented in a reduced form in Fig. II.7.2. Nine four-phase equilibria including SrTiO3 were identified and are plotted separately in Fig. II.7.2 for better readability. It can be easily seen from these results that an SrTiO3–Si interface is not in thermodynamic equilibrium. There are, however, three possible threephase equilibria with SiO2: SrTiO3–SiO2–TiO2–Ti4O7, SrTiO3–SiO2–Ti4O7– Ti3O5 and SrTiO3–SiO2–SiSrO3–Ti3O5. As a consequence, the occurrence of secondary phases at the interface during film formation or annealing of SrTiO3 on Si are expected. An experimental study has shown that this interface reaction layer actually forms during thin film deposition. This layer has two distinct components: SiO2 and an amorphous or nanocrystalline phase with
PT i
log10(PSr)/log10(PTi) 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
126
PSr
1.4
–1
SrCO3
TiO2
SrTiO3 + Sr4Ti3O10
–2.5
–3
–3.5
–4
–4.5
SrO + C
log10 [flow rate(O2 + Ar)]
–2
SrTiO3 + TiO2
SrCO3 + Sr4Ti3O10
SrO + C + Sr4Ti3O10
SrTiO3 + TiO2 + C
TiO2 + C
SrTiO3 + Sr4Ti3O10 + C
–5
II.7.1 Deposited phases for a precursor ratio of Sr(thd)2 to Ti(i-Opr)2(thd)2 as a function of Ar + O2 carrier gas flow at 700 °C and 6.6 mbar. The precursor concentration is 0.02 M and the injection speed is 2.5 × 10–5 cm3 s–1.
The SGTE casebook
–1.5
Thermodynamics in microelectronics Si3Ti5
SrTiO3
TiO Si4Ti5
TiO
Si3Ti5
SiSrO3
SiSr2O4 SrTiO3
SrTiO3
TiO SiO2
SiO2
Si2Ti
SrTiO3
Ti3O5 Ti2O3
TiO
SiO2
SiTi
TiO2
SiSrO2
Ti4O7
SiS2O4
Ti3O5
SrTiO3 TiO2 SrO
Sr
SrTiO3
Ti2O3
SrTiO3
SiSrO3
SrTiO3
Si4Ti5
Si
TiO
Sr4Ti3O10
SiSr2O4
SiSrO3
127
Sr4Ti3O40
Si4Ti5
SiSrO3
Si3Ti5 SiTi3
SrTrO3 Ti3O5
TiO Ti
II.7.2 Isothermal tetrahedron of the Sr–Ti–O–Si system at 700 °C. The nine four-phase equilibria including SrTiO3 are plotted separately.
a silicate signature in X-ray photoelectron spectroscopy analysis [004Lho]. From the equilibrium diagram, one can conclude that this silicate may correspond to SrSiO3.
II.7.3
Reactive ion etching of HfO2 dielectric films
Insulating HfO2 is a promising material for use as dielectric in complementary metal–oxide-semiconductor devices. For process integration of this new highk material, a process-compatible etch step has to be developed. Different process integration issues must be addressed: etch rate, etch uniformity and etch selectivity with respect to the SiO2 interface layer without damaging the underlying Si substrate. Plasma etching is considered to be the most viable technique to etch thin films of HfO2.To determine the most promising chemistry among the usually adopted halogen-based reactive gases, thermodynamic equilibrium calculations are performed. The goal is to select the most suitable gas or gas mixture with optimised process conditions, which means a maximum amount of volatile etch by-products for a given temperature and overall pressure. Although the process takes place at a low temperature, the use of a plasma assures the activation of the etch gases and thermodynamic equilibrium can be expected at the sample surface. The SGTE Substance Database [006SGT] is used for the simulation of the etch process. The first results show that fluorine-based gases are not suitable for this etch process, because the vapour pressure of HfF4(s) is too low.
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Therefore, the main focus is on chlorine-based chemistry. HfCl4(s) has a low melting point and exhibits high vapour pressures even at low temperatures. Different mixtures are investigated and the results of this study are summarised in Table II.7.1. Pure chlorine gas alone or a Cl2 + O2 mixture does not etch the HfO2 thin films even at temperatures as high as 700 K. One has to add a carboncontaining gas species to reduce the oxide and to transport the oxygen via the gas phase. CCl4 works theoretically but was not investigated experimentally. A Cl2 + CO gas mixture results in an HfCl4(g) + CO2(g) gas phase with considerable etching at low temperatures. The etch rate can be calculated from the Hertz–Knudsen relation [001Dum] by using the results from the thermodynamic equilibrium calculations [006Hel].
II.7.4
Annealing of amorphous Ru–Si–O and Ir–Si–O thin films
RuO2 and IrO2 are promising materials because of their high electrical conductivity, which is a rather unusual property among oxides. By simultaneous deposition with Si, one can obtain ternary amorphous layers with a nominal composition of Ru0.2Si0.15O0.65 and Ir0.2Si0.15O0.65. The films are obtained by reactive sputtering in an Ar + O2 plasma [99Gas, 001Che]. These materials can be used as a gate in GaAs metal–semiconductor field effect transistors or as a diffusion barrier against Al, Au, Ag and Cu [000Nic]. The thickness ranges from 100 to 200 nm. In order to relax mechanical constraints in the films due to the low-temperature deposition technique, a high-temperature anneal must be performed. This process step also increases the electrical conductivity. The annealing behaviours of the two different materials are completely different and the results are summarised in Table II.7. 2. Thermodynamic equilibrium calculations were performed in the Ru–Si– O and Ir–Si–O ternary systems using the SGTE Substance Database [006SGT]. No Gibbs energy data are available for iridium silicides. Their thermodynamic functions are estimated using the experimental enthalpy of formation from Table II.7.1 Thermodynamic and plasma etching studies for HfO2 at 5 × 10–6 bar Thermodynamic calculations Gas mixture
Etching
Gas species
Plasma etching at 423 K
Cl2 Cl2+O2 CCl4 Cl2+CO HCl
No No Yes, from 400 K Yes, from 400 K No
Cl2, Cl Cl2, O2, Cl HfCl4, CO2, CO, Cl2 HfCl4, CO2, CO, Cl2 HCl
No No Not performed Yes No
Thermodynamics in microelectronics
129
Table II.7.2 Annealing behaviour of Ir–Si–O films Atmosphere
Ru–Si–O
Ir–Si–O
Vacuum (≈10–9 bar)
800 °C for 5 h : amorphous 1000 °C for 30 min, nanocrystalline
1000 °C, metallic Ir
1 bar O2
700 °C for 5 min, crystalline + Ru loss + reaction with Si substrate
700 °C for 1 h, amorphous
the work of Meschel and Kleppa [98Mes1, 98Mes2]. A complete modelling of the Ru–Si and Ir–Si binary systems was not necessary because the process temperature is low enough to avoid the formation of a liquid phase and the oxygen chemical potential is high enough to avoid the reduction of SiO2 to Si. The amorphous phase was not modelled because the relative stability of this phase upon anneal seems to indicate that its Gibbs energy is close to that corresponding to a mixture of crystalline phases. The first step in the thermodynamic analysis of the annealing behaviour is the calculation of the ternary phase diagrams at the corresponding annealing temperatures [001Pis]. The isothermal section of Ru–Si–O and Ir–Si–O at 700 °C are presented in Fig. II.7.3(a) and Fig. II.7.3(b). Both diagrams show similar topologies with tie lines from metallic Ru or Ir to SiO2 and a threephase equilibrium Ru–RuO2–SiO2 or Ir–IrO2–SiO2. The main difference is the oxygen chemical potential inside these three-phase equilibria. While in the case of Ru this value is 2 × 10–8 bar and therefore close to the secondary vacuum pressure, it is 2 × 10–4 bar in the case of Ir. The consequence is that the amorphous phase in the Ru–Si–O system has no driving force to decompose, because the imposed pressure is close to the equilibrium value. This is not the case for Ir, where the equilibrium pressure is four orders of magnitude higher. The amorphous phase is reduced and metallic Ir is formed, which is observed experimentally. The equilibrium pressures of the gas species are low in vacuum. In the case of Ru–Si–O, RuO3(g) has a partial pressure of 5 × 10–11 bar and RuO4(g) a partial pressure of 10–13 bar. For Ir–Si–O, the partial pressures are 5 × 10–9 bar for IrO3(g) and 8 × 10–14 bar for IrO2(g). It is important to mention that the species IrO4(g) does not exist. The partial pressures change dramatically if the anneal is performed in oxygen at 1 bar. In this case, the partial pressure of RuO4(g) reaches a value of 4 × 10–6 bar and that of RuO3(g) a value of 3 × 10–7 bar. The sum of both values is five orders of magnitude higher than in the case of the vacuum anneal. The observed loss of Ru in the layer can therefore easily be explained by the vaporisation of Ru as RuO4(g). At the same time, the vaporisation seems to favour recrystallisation in the film. In
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The SGTE casebook O 1.0
0.8 SiO2 0.6
RuO2
0.4
Mo
le f
rac
tio
no
fO
2 × 10–8 bar
0.2
0 0 Si
0.4 RuSi 0.6
0.2
0.8
Ru4Si3
Ru2Si3
1.0 Ru
Mole fraction of Ru (a)
1.0
O
0.8
2 × 10–4 bar
IrO2
Mo le fra cti on of O
SiO2 0.6
0.4
0.2
0 0 Si
0.2
0.4 IrSi 0.6
IrSi3 IrSi2
0.8
Ir2Si Ir3Si
1.0 Ir
Mole fraction of Ir (b)
II.7.3 Calculated isothermal sections of (a) the Ru–Si–O ternary system and (b) the Ir–Si–O system, both at 700 °C.
the case of Ir–Si–O, the partial pressures remain small, even under oxygen at 1 bar and the amorphous layer is stable.
II.7.5
Conclusions
Thermodynamic equilibrium calculations using the calculation-of-phasediagrams approach and the SGTE databases can give valuable insight into all
Thermodynamics in microelectronics
131
fields in the microelectronics industry. The benefits range from the definition of the chemical stabilities of new materials with improved properties to the determination of optimised process conditions. Thermodynamic process simulations can therefore save time and money in the development of new integrated-circuit devices with improved performance.
II.7.6 98Mes1 98Mes2 99Gas 000Nic 001Che 001Dum 001Pis 002Ber
003Ran
004Lho 006Hel 006SGT
References S.V. MESCHEL and O.J. KLEPPA: J. Alloys Compounds 274, 1998, 193. S.V. MESCHEL and O.J. KLEPPA: J. Alloys Compounds 280, 1998, 231. S.M. GASSER, E. KOLAWA and M.-A. NICOLET: J. Appl. Phys. 86(4), 1974. M.-A. NICOLET: Vacuum 59, 2000, 716. H.B. CHERRY, P.H. GIAUQUE and M.-A. NICOLET: Microelectron. Eng. 55, 2001, 403. L. DUMAS, C. CHATILLON and E. QUESNEL: J. Cryst. Growth 222, 2001, 215. A. PISCH and C. BERNARD: Calphad 25(4), 2001, 639–644. C. BERNARD: in Fundamental Gas-phase and Surface Chemistry of Vapor Phase Deposition II/Process Control, Diagnostics and Modeling in Semiconductor Manufacturing IV, Electrochemical Society Proceedings, Vol. 2001-13, Electrochemical Society, Pennington, New Jersey, 2001, p. 245. E. RANGEL-SALINAS, A. PISCH, C. CHATILLON and C. BERNARD: in Chemical Vapour Deposition XVI and EUROCVD 14, Electrochemical Society Proceedings, Vol. 2003-08, Electrochemical Society, Pennington, New Jersey, 2003, pp. 243– 248. S. LHOSTIS: PhD Thesis, Institut National Polytechnique de Grenoble, France, 2004. M. HELOT, T. CHEVOLLEAU, L. VALLIER, O. JOUBERT, E. BLANQUET, A. PISCH, P. MANGIAGALLI and T. LILL: J. Vac. Sci. Technol. A 24(1), 2006, 30–40. SGTE: SGTE Substance Database, V.11, 2006.
II.8 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 O L G A F A B R I C H N A YA
II.8.1
Introduction
According to geophysical data the Earth consists of a core (solid inner core and liquid outer core), a solid mantle and a crust. The borders between these parts of the Earth are indicated by seismic wave discontinuities. Figure II.8.1 presents the distribution of seismic waves VP and VS and density ρ in the Earth’s interior derived from seismological data. The mantle consists of a homogeneous lower mantle, where there are no pronounced seismic discontinuities, transition zones with two main discontinuities at 400 and 650 km depth accompanied by increases in the seismic wave velocities
Velocity (km–1 s) or density (g cm–3)
VP 12
r
10
VP
8
VS Core 6
r
VS
4
2 Mantle 0
2000
4000 Depth (km)
II.8.1 Seismic profile of the Earth.
132
6000
Phase diagrams of the MgO–FeO–Al2O3–SiO2 system
133
and density, and an upper mantle where the partial melting zones are indicated by decreases in the seismic wave velocities. Chemical analyses of rocks originating from deepest parts of upper mantle are available (i.e. peridotites, eclogites and lherzolites) and the possible composition of the whole mantle could be assumed on the basis of these data. The mantle composition up to about 95% is described by the MgO–FeO–Al2O3–SiO2 system. Ringwood [75Rin] assumed that the transition zone and lower mantle are formed by ‘pyrolite’ rock consisting of olivine (Mg, Fe)2SiO4, pyroxene (Mg, Fe)SiO3 and garnet (Mg, Fe)3Al2Si3O4 minerals. Experiments at high pressures and temperatures demonstrated that silicate minerals transform to denser structures at pressures corresponding to the depth of seismic discontinuities [83Jea]. Therefore, if data for the Gibbs energy of mantle-forming minerals are available, it is possible to calculate mineral assemblages stable at pressures and temperatures of the mantle transition zone and to attribute seismic discontinuities to changes in mineral assemblage. To calculate the Gibbs energy of the stoichiometric compound or end member of solid solution at high pressures, equation-of-state (EOS) parameters are necessary additionally to thermochemical information (enthalpy, entropy and heat capacity). One of the simplest equations of state is the Murnaghan [44Mur] equation. To calculate the volume of a compound at a high pressure and temperature, information on the volume in the standard state, the isobaric bulk modulus, its pressure derivative and the thermal equation is necessary. This kind of information for pyrolite-forming minerals and their high-pressure modification are partially available from experimental measurements. These data were combined with the results of modelling based on the theory of lattice vibrations, Raman and infrared spectra of compounds by Saxena et al. [93Sax]. Thus a consistent set of EOS parameters was derived. The calculationof-phase-diagrams (CALPHAD) method was applied to derive the thermodynamic database for minerals in the system FeO–MgO–Al2O3–SiO2 using thermochemical data and EOS parameters together with experimental phase equilibria at high pressures [004Fab]. The Gibbs energy of solidsolution minerals were described by compound energy formalism except for pyroxene, which is described by substitutional model. To verify the reliability of the thermodynamic description, many types of phase diagram for subsystems were calculated and compared with experimental data. The pressure distribution in the Earth’s interior is quite well determined from geophysical data, while the temperature estimates are less precise (±200 K in the transition zone). The calculated P–T diagram for pyrolite composition allows us to relate the seismic discontinuities observed in the mantle transition zone to chemical transformations in minerals. Based on the obtained results a density profile could be calculated [92Ita]. These data combined with the shear modulus make it possible to calculate the sound velocity profile [89Duf]. A comparison of the calculated profiles with seismic data allows us to verify
134
The SGTE casebook
the chemical composition of the Earth’s mantle [92Ita]. The thermodynamic data for minerals are also important for modelling convection in the mantle [88Tru].
II.8.2
Phases and models
Most mantle minerals are solid solutions and their Gibbs energies depend on the composition. The mixing parameters L depend on the temperature and pressure. The P–T dependence of the Gibbs energy for end members of solid solutions is described below. The Gibbs energy G of a pure phase and the end members of solid solutions at a certain P and T is expressed as
∫
° + G ( P , T ) = ∆ f H 298
T
298
+
∫
° C P dT – T S298 +
∫
T
298
CP dT T
p
(II.8.1)
V dP
1
° is the entropy, ∆ f H 298 ° is the enthalpy of formation from the where S298 elements and CP is the heat capacity given by CP = a + bT + cT–2 + dT2 + eT–3 + f T – 0.5 + gT –1
(II.8.2)
The molar volume as a function of pressure and temperature is calculated using the Murnaghan equation
K′ P V ( P , T ) = V (1, T ) 1 + P KT
–1/ K P′
(II.8.3)
where KT is the isothermal bulk modulus expressed as KT =
1 β 0 + β 1 T + βT 2 + β 3 T 3
(II.8.4)
and K′P is the pressure derivative of bulk modulus which in some cases has a temperature dependence: K P′ = K PT ′ r + K PT ′ ( T – Tr ) ln T Tr
(II.8.5)
K PT ′ r is the pressure derivative of bulk modulus at Tr = 298.15 K, and K′PT is its temperature derivative. The molar volume at 1 bar is expressed as a function of temperature by
V(1, T ) = V1,0Tr exp
∫
T
Tr
α ( T ) dT
(II.8.6)
Phase diagrams of the MgO–FeO–Al2O3–SiO2 system
135
where V1,0Tr is the molar volume at 1 bar and Tr=298.15 K, and α(T) is the thermal expansion depending on temperature and given by
α(T) = α 0 + α 1T + α 2T
–1
+ α 3T
–2
(II.8.7)
The parameters are related to each other by C P = C V + α 2 V TK T T
γ=
α KT V CV
(II.8.8) (II.8.9)
where γ is the Grüneisen parameter; also
KS =
K T CP CV
(II.8.10)
where KS is the adiabatic bulk modulus. Compressibility could be derived from different experimental data (static compression, shock-wave data and Brillouin scattering). CP is measured by calorimetric techniques (differential scanning calorimetry at high temperatures and adiabatic calorimetry at low temperatures). The volume and thermal expansion are determined from X-ray diffraction data, dilatometry or picnometry. If information on Raman and infrared spectra is available, the heat capacity CV at constant volume could be estimated using the Kieffer [79Kie] model. A review of experimentally determined EOS parameters and thermochemical values for stoichiometric minerals and end members of solid solution has been given by Fabrichnaya et al. [004Fab]. A list of the phases and models of solid solutions are presented in Table II.8.1.
II.8.3
Phase equilibria in subsystems
The development of high-pressure apparatus (piston–cylinder, Bridgman anvils, diamond anvil cell and multianvil split-sphere apparatus) with simultaneous heating made it possible to reach the pressures and temperatures of the Earth’s mantle [87Aki]. It was discovered that quartz (SiO2) transforms to denser phases of coesite and stishovite (rutile structure) under pressure increase. Ringwood and Major [70Rin] were the first to discover that olivine transforms at 14 GPa to the structure of distorted spinel (β) and to normal spinel (γ) at 18 GPa. Later olivine–β–γ phase transformations in the Fe2SiO4–Mg2SiO4 system were studied in detail by Akimoto [72Aki]. Liu [75Liu] was the first to find the post-spinel transformation of γ -spinel to a mixture of perovskite and magnesiowustite and the formation of phases with ilmente structure and perovskite at pressures of 20–25 GPa in the MgSiO3 composition [76Liu]. The phase diagram of FeSiO3–MgSiO3 system was studied in detail by Ito
136
The SGTE casebook
Table II.8.1 Phases and models in the system MgO–FeO–Al2O3–SiO2 Phase (abbreviation)
Formula
Olivine (Ol) β-spinel γ-spinel Orthopyroxene (Opx) Protopyroxene (Ppx) HP-clinopyroxene (Hpcpx) Ilmenite (Ilm, Il1) Perovskite (Pv) Garnet (Gar) Magnesiowustite (Mw) Spinel (Sp) Magnesioferrite (Mf) α-quartz (α-Q) β-quartz (β-Q) Tridymite (Tr) Cristobalite (Cr) Coesite (Coes) Stishovite (St) Hamatite (Hem)
(Fe2+,Mg2+) 2SiO4 (Fe2+,Mg2+) 2SiO4 (Fe2+,Mg2+) 2SiO4 Al2O3–FeSiO3–MgSiO3 (Fe2+,Mg2+)SiO3 (Fe2+,Mg2+)SiO3 (Al3+,Fe2+,Mg2+)(Al3+,Si4+)(O2–)3 (Al3+,Fe2+,Mg2+)(Al3+,Si4+)(O2–)3 (Fe2+,Mg2+)3(Al3+,Fe2+,Mg2+)(Al3+,Si4+)(Si4+) 3O12 (Fe2+,Fe3+,Mg2+,Va)(O2–) (Fe2+,Mg2+)(Al3+)2(O2–)4 (Fe2+,Mg2+)(Fe3+)2(O2–)4 SiO2 SiO2 SiO2 SiO2 SiO2 SiO2 Fe2O3
and Yamada [82Ito] at 1373 K using the multianvil split-sphere apparatus. Phase relations in the Fe2SiO4–Mg2SiO4 system were studied by Yagi et al. [79Yag] in a laser-heated diamond anvil cell at 1273 K and pressures up to 30 GPa. Experimental investigations of the Mg4Si4O12–Mg3Al2Si3O12 system indicated that the Mg4Si4O12 solubility in the garnet phase increases with increasing pressure, attaining a maximum at about 15 GPa (see the papers by Akaogi and Akimoto [77Aka] and Kanzaki [87Kan]). At high temperatures and high pressures, garnet and ilmenite structures form solid solutions in the whole composition range of the Mg4Si4O12–Mg3Al2Si3O12 system. According to the experimental data of Irifune et al. [96Iri] the solubility of the Al2O3 in perovskite structure is limited. A review of the experimental data for different subsystems in the FeO–MgO–Al2O3–SiO2 system has been presented by Fabrichnaya et al. [004Fab]. A review of experimental thermodynamic data for mantle minerals has also been given by Fabrichnaya et al. [004Fab]. Navrotsky [94Nav] derived the enthalpy of formation for several high-pressure phases by high-temperature oxide melt solution calorimetry. The temperature dependence of CP was determined by Watanabe [82Wat] using the technique of differential scanning calorimetry for several mantle minerals. Missing values were derived using the CALPHAD technique by Fabrichnaya et al. [004Fab].
Phase diagrams of the MgO–FeO–Al2O3–SiO2 system
II.8.4
137
Phase diagrams for selected subsystems of the FeO–MgO–Al2O3–SiO2 system
The P–T phase diagrams for the systems of olivine (Mg2SiO4) and pyroxene (MgSiO3) compositions are presented in Fig. II.8.2(a) and Fig. II.8.2(b). The calculated phase diagrams were compared with experimental data by Fabrichnaya et al. [004Fab]. It was shown that calculations were in agreement with available experimental phase equilibria within uncertainty limits except 26 Pv + MgO 24 γ
P (GPa)
22 20 18
β
16 14
Ol 12 1000
1500
2000
2500
T (K) (a) 26 Pv 24
P (GPa)
22 Ilm 20 Gar
γ + St 18 β + St 16 14 1000
Hpcpx
1500
2000
2500
T (K) (b)
II.8.2 Phase diagrams of (a) the Mg2SiO4 system and (b) the MgSiO3 system. For the meanings of the abbreviated phases, see Table II.8.1.
138
The SGTE casebook
for the data obtained by Irifune et al. [98Iri], indicating that the pressure of γ-spinel decomposition is 2 GPa lower than other studies. The calculated phase diagram of the Mg2SiO4–Fe2SiO4 system at pressures up to 30 GPa is presented in Fig. II.8.3. The phase relations in the pyroxene compositions (MgSiO3–FeSiO3) at 1373 K and 2073 K are shown in Fig. II.8.4(a) and Fig. II.8.4(b) respectively. The calculated phase diagrams for both systems were compared with available experimental data obtained by Fabrichnaya et al. [004Fab]. It was indicated that they agree within experimental uncertainty. It should be mentioned that iron partition between perovskite and magnesiowustite was extensively studied experimentally because of its importance for understanding the mineral composition of the lower mantle. However, there is still uncertainty in the partition coefficient obtained by different experimental techniques. The calculated partition coefficient [004Fab] is in agreement with data obtained by Ito and Yamada [82Ito] and by Fei et al. [91Fei]. The calculated phase diagrams for the Mg4Si4O12–Mg3Al2Si3O12 system at 1273 K and 1773 K at pressures up to 30 GPa are presented in Fig. II.8.5(a) and Fig. II.8.5(b) respectively. These calculations clearly demonstrate that the solubility of the Mg4Si4O12 in garnet in equilibrium with pyroxene increases with increasing pressure. The solubility of the Mg4Si4O12 component in garnet reaches a maximum in invariant equilibrium with pyroxene, stishovite and β-spinel. At higher 30 Pv + Mw
Pv + Mw + St Mw + St
25
P (GPa)
20
γ + Mw + St
γ β
15
Ol + γ 10 Ol 5 0 Mg2SiO4
0.2
0.4
0.6 x(Fe SiO ) 2
4
0.8
1.0 Fe2SiO4
II.8.3 Phase diagram of the Mg2SiO4–Fe2SiO4 system at 1873 K. For the meanings of the abbreviated phases, see Table II.8.1.
Phase diagrams of the MgO–FeO–Al2O3–SiO2 system
139
26 24 22
Pv
Pv + Mw + St
llm
P (GPa)
20 18
γ + Mw + St
γ + St
Mw + St
Gar
16 14 Hpcpx
12 10 0 MgSiO3
0.2
0.4
0.6
0.8
x(FeSiO ) 3
1.0 FeSiO3
(a) 30 Pv
Pv + Mw + St Mw + St
25 llm
γ + St + Mw
P (GPa)
20 γ + St β + St 15 Hpcpx + γ + St 10 Hpcpx 5 0 MgSiO3
0.2
0.4
0.6
x(FeSiO ) 3
0.8
1.0 FeSiO3
(b)
II.8.4 Phase diagram of the MgSiO3–FeSiO3 system at (a) 1373 K and (b) 2073 K. For the meanings of the abbreviated phases, see Table II.8.1.
140
The SGTE casebook 30 28 26
Pv Pv + ll1 Pv + llm
llm
24
P (GPa)
22
llm + Gar
20 γ = St + Gar
18
β = St + Gar
16
Gar
14 Hpcpx + Gar
12 10
0 Mg4Si4O12
0.2
0.4 0.6 xMg3Al2Si3O12
0.8
1.0 Mg3Al2Si3O12
(a) 30 28
Pv + II1
Pv
26
P (GPa)
22
Gar + II1
Pv + Gar
24 llm llm + Gar 20 18 γ + St + Gar
Gar
16 14 Hpcpx + Gar 12 10 0 Mg4Si4O12
0.2
0.4 0.6 xMg3Al2Si3O12
0.8
1.0 Mg3Al2Si3O12
(b)
II.8.5 Phase diagrams of the Mg4Si4O12–Mg3Al2Si3O12 system at (a) 1273 K and (b) 1773 K. For the meanings of the abbreviated phases, see Table II.8.1.
pressures, garnet is in equilibrium with ilmenite or β/γ + stishovite and its stability field decreases. The solubility of the Mg4Si4O12 in garnet also increases with increasing temperature. The calculations indicated the formation of continuous solid solutions in the whole composition range of
Phase diagrams of the MgO–FeO–Al2O3–SiO2 system
141
Mg4Si4O12–Mg3Al2Si3O12 at temperatures above 2000 K. The sequence of phase transformations depends on temperature, at 1273 K, the garnet phase directly transforms to the ilmenite structure while, at 1773 K, garnet decomposes to a mixture of perovskite and an Al2O3-rich ilmenite structure. It should be mentioned that the ilmenite phase has a miscibility gap at 1273 K, decomposing to MgSiO3-rich and Al2O3-rich phases.
II.8.5
Phase diagram of the mantle composition at pressures up to 30 GPa
The phase equilibria in samples modelling mantle compositions at the temperatures and pressures of mantle transition zone have been experimentally investigated in several studies [87Iri1, 87Iri2, 94Iri]. The phase compositions calculated at the same conditions as in experiments (bulk chemical composition, pressure and temperatures) have been compared with experimental data by Fabrichnaya et al. [004Fab] to verify the derived thermodynamic database. The phase diagram of the FeO–MgO–Al2O3–SiO2 system for the pyrolite composition ( x SiO 2 = 0.4013, xMgO = 0.5146, x Al 2 O 3 = 0.0216 and xFeO = 0.0624) at the pressures and temperatures of the mantle transition zone is presented in Fig. II.8.6. Two possible geotherms are shown by dashed lines. 32
853.9 809.2
30 Pv + Mw 28
764.4 Pv + Gar + Mw γ + Pv + Gar
P (GPa)
24 22
719.2 673.8
γ + llm + Gar
20
624.0
γ + Gar
5
H (km)
26
573.6
γ + St + Gar
2
18
522.4 4
16 3 14
470.3
β + Gar β + Gar + Hpcpx
6
417.3
Ol + Gar + Hpcpx 12 1500
1800
2100
361.5 2400
T (K)
II.8.6 P–T diagram of the FeO–MgO–Al2O3–SiO2 system for pyrolite composition at pressures up to 30 GPa. For the meanings of the abbreviated phase, see Table II.8.1. Two possible geotherms are shown by dashed lines. The following phase assemblages are indicated as follows: 1, Ol + β + Gar + Hpcpx; 2, β + γ + Gar; 3, β + St + Gar; 4, β + γ + St + Gar; 5, γ + St + Ilm + Gar; 6, Ol + β + Gar.
142
The SGTE casebook
This figure demonstrates that the sequence of phase assemblages depends on the temperature distribution in the Earth’s mantle. Between the two main seismic discontinuities at 400–420 km and 650–670 km attributed to the transformations of olivine to β-spinel and of γ-spinel to a mixture of perovskite and magnesiowustite respectively, there are other transformations indicating the fine mineralogical structure of the mantle. According to calculations, the garnet phase disappears completely at a depth of 730–750 km depending on the temperature. The lower mantle is composed of coexisting perovskite and magnesiowustite, in agreement with experiments by O’Neill and Jeanloz [90One].
II.8.6
Acknowledgements
This study was performed in Uppsala University (Sweden) under the financial support of CAMPADA project. The author is grateful to S.K. Saxena and B. Sundman for close cooperation.
II.8.7 44Mur 70Rin 72Aki 75Liu 75Rin 76Liu 77Aka 79Kie 79Yag 82Ito
82Wat 83Jea 87Aki 87Iri1 87Iri2 87Kan 88Tru 89Duf 90One 91Fei 92Ita
References F.D. MURNAGHAN: Proc. Natl Acad. Sci. 30, 1944, 244. A.E. RINGWOOD and A. MAJOR: Phys. Earth Planet. Inter. 3, 1970, 89–108. S. AKIMOTO: Tectonophysics 13, 1972, 161–188. L. LIU: Nature 258, 1975, 510–512. A.E. RINGWOOD: Composition and Petrology of the Earth’s Mantle, McGrawHill, New York, 1975. L. LIU: Earth Planet. Sci. Lett. 31, 1976, 200–208. M. AKAOGI and S. AKIMOTO: Phys. Earth Planet. Inter. 15, 1977, 90–106. S. KIEFFER: Rev. Geophys. Space Phys. 17, 1979, 1–59. T. YAGI, P. BELL and H.K. MAO: Carnegie Inst. Washington Yearbook 78, 1979, 614–618. E. ITO and H. YAMADA: in High-Pressure Research in Geophysics (Eds S. Akimoto and M.H. Manghnani), Center for Academic Publishing, Tokyo, 1982, pp. 405– 419. H. WATANABE: in High-Pressure Research in Geophysics (Eds. S. Akimoto and M.H. Manghnani), Center for Academic Publishing, Tokyo, 1982, pp. 441–464. R. JEANLOZ and A.B. THOMPSON: Rev. Geophys. 21, 1983, 51–74. S. AKIMOTO: in High Pressure Research in Mineral Physics (Eds M.H. Manghnani and Y. Syono), Terra, Tokyo, 1987, pp. 1–13. T. IRIFUNE: Phys. Earth Planet Inter. 45, 1987, 324–336. T. IRIFUNE and A.E. RINGWOOD: Earth Planet. Sci. Lett. 86, 1987, 365–376. M. KANZAKI: Phys. Earth Planet Inter. 49, 1987, 169–175. L.M. TRUSKINIVSKY: Izv. Akad. Nauk SSSR, Fiz. Zemli 9, 1988, 3–14. T.S. DUFFY and D.L. ANDERSON: J. Geophys. Res. 94, 1989, 1895–1912. B. O’NEILL and R. JEANLOZ: Geophys. Res. Lett. 17, 1990, 1477–1480. Y. FEI, H.K. MAO and B.O. MYSEN: J. Geophys. Res. 96, 1991, 2157–2169. J. ITA and L. STIXRUDE: J. Geophys. Res. – Solid Earth 97B, 1992, 6849–6866.
Phase diagrams of the MgO–FeO–Al2O3–SiO2 system 93Sax
143
S.K. SAXENA, N. CHATTERJEE, Y. FEI and G. CHEN: Thermodynamic Data on Oxides and Silicates, Springer, New York, 1993. 94Iri T. Irifune: Nature 370, 1994, 131–133. 94Nav A. NAVROTSKY: Physics and Chemistry of Earth Minerals, Cambridge University Press, Cambridge, 1994. 96Iri T. IRIFUNE, T. KOIZUMI and J. ANDO: Phys. Earth Planet. Inter. 96, 1996, 147– 157. 98Iri T. IRIFUNE, N. NISHIYAMA, K. KURODA, T. INOUE, M. ISSHIKI, W. UTSUMI, K. FUNAKOSHI, S. URAKAWA, T. UCHIDA, T. KATSURA and O. OHTAKA: Science 279, 1998, 1698–1700. 004Fab O.B. FABRICHNAYA, S.K. SAXENA, P. RICHET and E.F. WESTRUM: Thermodynamic Data, Models and Phase Diagrams in Multicomponent Oxide Systems, Springer, Berlin, 2004.
II.9 Calculation of the concentration of iron and copper ions in aqueous sulphuric acid solutions as functions of the electrode potential J Ü R G E N K O R B and K L A U S H A C K
II.9.1
Introduction
When reprocessing metal containing aqueous solutions by electrolysis, it is very useful to know the behaviour and theoretically possible concentrations of the metal ions in the area near the electrodes as a function of the prevailing electrode potential and pH value. The present example concerns a sulphuric acid solution with variable contents of iron and copper, and an acid concentration of 100 g of free H2SO4 per litre. In the first part of this discussion the system is treated with the usual reaction thermochemistry for which the following simplifications are made. –
– – –
2– Because of kinetic inhibitions it is not possible to reduce SO 2– 4 or HSO 4 – 2– ions to elementary sulphur, or HS or S ions to H2S at the cathode. The – system will, therefore, be described with SO 2– 4 and HSO 4 as the stable sulphur containing species. The total metal contents for the elements Fe and Cu are calculated as the sum of Fe2+ and Fe3+ and the sum of Cu+ and Cu2+ respectively. Because of the lack of higher order interaction terms ideal aqueous behaviour is assumed. Neither cathode nor anode materials take part in the reactions.
In the second part, the system is investigated with complex equilibrium methods. The differences from the stoichiometric reaction approach are pointed out. Since all reactions take place in an aqueous solution, it is necessary to take into account the stability range of water when choosing the electrode potential Eh. For O2 partial pressures of 1 bar the upper limit of stability is described by 2H2O(l) ↔ O2(g) + 4H+(aq) + 4e–(aq)
(II.9.1)
with Eh = 1.23–0.0592 pH The lower limit for variable partial pressure of H2 is described by 144
Concentration of iron and copper ions in acid solutions
H2(g) ↔ 2H+ (aq) + 2e–(aq)
145
(II.9.2)
with Eh = –0.0296 log p H 2 – 0.0592 pH. However, under practical conditions a wider range of stability of water is found. Excess voltages of more than 0.5 V are generally necessary for a dissociation of the water molecules [65Gar]. For the excess voltage η, which is found for H2 separation, an empirical equation was given by Tafel [83Paw]:
η = a + 0.117 log i
(II.9.3)
where a is a material constant related to the electrode material and i is the current density. For copper, excess voltages of 0.6–0.8 V are found for cathodic current densities of 0.01–0.1 A cm–2. With decreasing current density the value of the excess voltage also drops. In the production of metals by electrolysis of aqueous metal salt solutions, lead anodes are often used. For these, one finds, for comparable current densities, excess voltages of 0.7 V for the oxygen separation. Considering the above empirical aspects, the range of stabilities of water is assumed to be between –0.8 V and +2.0 V for the following discussion.
II.9.2
Fe–H2SO4–H2O subsystem
The electrochemical equilibrium between Fe2+ and Fe3+ ions in a sulphuric acid solution is discussed for total contents of 1 g l–1 (1.7906 × 10–2 mol l–1) and 5 g l–1 (8.9503 × 10–2 mol l–1), respectively. Referred to the standard hydrogen electrode, one obtains for the standard electrode potential E° of the half-cell Fe2+ (aq) ↔ Fe3+ (aq) + e– (aq) a value of E° = 0.771 V. Under the assumptions outlined in the introduction, the Nernst equation can be employed for the calculation of the concentrations of Fe2+ (aq) and Fe3+ (aq). One obtains E h = E ° + RT ln F
c Fe 3+ (aq) c 2+ Fe (aq)
c Fe total = c Fe 2+ (aq) + c Fe 3+ (aq) E h – 0.771 0.0592 c Fe total c Fe 2+ (aq) = mol l –1 1 + 10 E1 c Fe total c Fe 2+ (aq) = mol l –1 1 + 10 – E1 E1 =
(II.9.4) (II.9.5) (II.9.6) (II.9.7) (II.9.8)
The value of 55.847 g mol–1 for the molar mass of iron was used for the conversion of moles per litre to grams per litre.
146
The SGTE casebook Table II.9.1 Concentrations of Fe2+ and Fe3+ as functions of the electrode potential for T = 25 °C and cFe total = 1 g l–1 Potential Eh (V) +2.000 +1.500 +1.000 +0.900 +0.800 +0.700 +0.600 +0.500 +0.400 +0.300 +0.200 +0.100 +0.000 –0.100 –0.200 –0.300 –0.400 –0.500 –0.600 –0.700 –0.800
Concentration (g l–1) Fe2+ 1.7373 4.8508 1.3542 6.5778 2.4454 9.4056 9.9871 9.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Fe3+ × × × × × × × ×
10–21 10–13 10–4 10–3 10–1 10–1 10–1 10–1
1.0000 1.0000 9.9986 9.9342 7.5546 5.9438 1.2910 2.6441 5.4089 1.1064 2.2632 4.6296 9.4700 1.9371 3.9625 8.1056 1.6580 3.3916 6.9377 1.4191 2.9029
× × × × × × × × × × × × × × × × × × ×
10–1 10–1 10–1 10–2 10–3 10–5 10–7 10–8 10–10 10–12 10–14 10–15 10–17 10–19 10–20 10–22 10–24 10–25 10–27
Table II.9.1 and Table II.9.2 contain the calculated concentrations for total iron contents of 1 g l–1 and 5 g l-1 respectively and for a voltage range from +2.0 V to –0.8 V. The concentration curves given in Fig. II.9.1 and Fig. II.9.2 intersect at the equiconcentration points of Fe2+ (aq) and Fe3+ (aq). From the Nernst equation, a potential of 0.771 V independent of the total iron content is obtained for the chosen conditions and with Eh = E°.
II.9.3
Cu–H2SO4–H2O subsystem
The electrochemical equilibrium between Cu+ and Cu2+ ions in a sulphuric acid solution will be discussed for total contents of 1 g l–1 (1.5738 × 10–2 mol l–1) and 5 g l–1 (7.8691 × 10–2 mol l–1). Referred to the standard hydrogen electrode, one obtains for the standard electrode potential E° of the half-cell Cu+ (aq) ↔ Cu2+ (aq) + e– (aq) a value of E° = 0.167 V. Again, accepting the assumptions outlined in the introduction the Nernst equation can be employed for the calculation of the concentrations of Cu+ (aq) and Cu2+ (aq). One obtains
Concentration of iron and copper ions in acid solutions
147
Table II.9.2 Concentrations of Fe2+ and Fe3+ as functions of the electrode potential for T = 25 °C and cFe total = 5 g l–1 Concentration (g l–1)
Potential Eh (V)
Fe2+
+2.000 +1.500 +1.000 +0.900 +0.800 +0.700 +0.600 +0.500 +0.400 +0.300 +0.200 +0.100 +0.000 –0.100 –0.200 –0.300 –0.400 –0.500 –0.600 –0.700 –0.800
Concentration (g l–1)
1.0
8.6863 2.4254 6.7712 3.2889 1.2227 4.7028 4.9935 4.9999 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000
Fe3+ × × × ×
10–21 10–12 10–4 10–2
5.0000 5.0000 4.9993 4.9671 3.7773 2.9719 6.4550 1.3221 2.7044 5.5321 1.1316 2.3148 4.7350 9.6857 1.9813 4.0528 8.2902 1.6958 3.4688 7.0957 1.4515
× × × × × × × × × × × × × × × ×
10–1 10–3 10–4 10–6 10–8 10–9 10–11 10–13 10–15 10–16 10–18 10–20 10–21 10–23 10–25 10–26
Fe2+
0.8 0.6 0.4 0.2 0.0
Fe3+ –0.6 –0.3 0.0
0.3
0.6 0.9 1.2 Eh (V)
1.5
1.8
II.9.1 Fe2+ and Fe3+ concentrations as functions of Eh at 25 °C and 1 g I–1 total ferrous content in a sulphuric acid solution.
c Cu 2+ (aq) E h = E ° + RT ln F c Cu + (aq)
c Cu total = c Cu + (aq) + c Cu 2+ (aq)
(II.9.9) (II.9.10)
148
The SGTE casebook 5.0
Concentration (g l–1)
Fe2+ 4.0 3.0 2.0 1.0 Fe3+ 0.0
–0.6 –0.3 0.0
0.3
0.6 0.9 Eh (V)
1.2
1.5
1.8
II.9.2 Fe2+ and Fe3+ concentrations as functions of Eh at 25 °C and 5 g I–1 total ferrous content in a sulphuric acid solution.
E2 =
E h – 0.167 0.0592
(II.9.11)
c Cu + (aq) =
c Cu total mol l –1 1 + 10 E2
(II.9.12)
c Cu 2+ (aq) =
c Cu total mol l –1 1 + 10 – E2
(II.9.13)
The value of 63.540 g mol–1 for the molar mass of copper was used for the conversion of moles per litre to grams per litre. Table II.9.3 and Table II.9.4 contain the calculated concentrations for total copper contents of 1 g l–1 and 5 g l–1 respectively and for a voltage range from +2.0 V to –0.8 V. Figure II.9.3 and Fig. II.9.4 show the appropriate graphical representations. The E° values for iron and copper have been taken from the tables given by Rauscher et al. [65Rau].
II.9.4
The complete system Cu–Fe–H2SO4–H2O
In contrast with the ‘classical’ stoichiometric reaction approach described above, the complete system Cu–Fe–H2SO4–H2O has been analysed by complex equilibrium computations using the program ChemSage [90Eri]. A system with the phases and species listed in Table II.9.5 has been composed from different databases [82Wag, 92SGT]. It should be noted that the last mentioned group of substances in the table is only listed to indicate the completeness of the data set. None of the solid phases is involved in any of the equilibria discussed in this article.
Concentration of iron and copper ions in acid solutions
149
Table II.9.3 Concentrations of Cu+ and Cu2+ as functions of the electrode potential for T = 25 °C and cCu totai = 1 g l–1 Concentration (g l–1)
Potential Eh (V)
Cu+
+2.000 +1.500 +1.000 +0.900 +0.800 +0.700 +0.600 +0.500 +0.400 +0.300 +0.200 +0.100 +0.000 –0.100 –0.200 –0.300 –0.400 –0.500 –0.600 –0.700 –0.800
1.0893 3.0416 8.4929 4.1519 2.0297 9.9225 4.8508 2.3714 1.1591 5.6354 2.1695 9.3124 9.9849 9.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Cu2+ × × × × × × × × × × × × × ×
10–31 10–23 10–15 10–13 10–11 10–10 10–8 10–6 10–4 10–3 10–1 10–1 10–1 10–1
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 9.9988 9.9436 7.8305 6.8755 1.5080 3.0892 6.3194 1.2927 2.6442 5.4089 1.1064 2.2632 4.6296
× × × × × × × × × × × × ×
10–1 10–1 10–1 10–2 10–3 10–5 10–2 10–8 10–10 10–12 10–13 10–15 10–17
Table II.9.4 Concentrations of Cu+ and Cu2+ as functions of the electrode potential for T = 25 °C and cCu total = 5 g l–1 Concentration (g l–1)
Potential Eh (V)
Cu+
+2.000 +1.500 +1.000 +0.900 +0.800 +0.700 +0.600 +0.500 +0.400 +0.300 +0.200 +0.100 +0.000 –0.100 –0.200 –0.300 –0.400 –0.500 –0.600 –0.700 –0.800
5.4467 1.5208 4.2464 2.0759 1.0149 4.9613 2.4254 1.1857 5.7957 2.8177 1.0847 4.6562 4.9925 4.9998 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000
Cu2+ × × × × × × × × × ×
10–31 10–22 10–14 10–12 10–10 10–9 10–7 10–5 10–4 10–2
5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 4.9994 4.9718 3.9153 3.4378 7.5400 1.5446 3.1597 6.4633 1.3221 2.7044 5.5321 1.1316 2.3148
× 10–1 × 10–3 × 10–4 × × × × × ×
10–8 10–9 10–11 10–13 10–14 10–16
150
The SGTE casebook
Concentration (g l–1)
1.0
Cu+
0.8 0.6 0.4 0.2 Cu2+ –0.6 –0.3 0.0
0.3
0.6 0.9 Eh (V)
1.2
1.5
1.8
II.9.3 Cu+ and Cu2+ concentrations as functions of Eh at 25 °C and 1 g I–1 total copper content in a sulphuric acid solution.
5.0
Concentration (g l)
Cu+ 4.0 3.0 2.0 1.0 Cu2+ 0.0
–0.6 –0.3 0.0
0.3
0.6 0.9 Eh (V)
1.2
1.5
1.8
II.9.4 Cu+ and Cu2+ concentrations as functions of Eh at 25 °C and 5 g I–1 total copper content in a sulphuric acid solution.
For the aqueous species, ideal Debye–Hückel behaviour is assumed and the temperature of the system is fixed at 25 °C because of a lack of data on activities, i.e. parameters for the non-ideal Pitzer model, and temperature dependence. The composition of the system is defined by the values in Table II.9.6 (reference 1 kg = 1 l of water). The total concentrations of Cu and Fe respectively are thus 1 g l–1 each. The redox potential of all part reactions in the system has the same value at equilibrium. By influencing an arbitrarily chosen pair, e.g. H2(g)–H+ (aq), the computational procedure can find the equilibrium as a function of the potential Eh. Figure II.9.5 shows the calculated results. Of the copper species considered, only Cu + (aq), Cu 2+(aq) and CuSO 3– (aq) are of importance up to a molality of 1 × 10–14. As expected from the calculations discussed above
Concentration of iron and copper ions in acid solutions
151
Table II.9.5 Composition of the system Cu–Fe–H2SO4–H2O Gas (g) (four major species) H2
O2
SO2 SO3
Aqueous solution (aq) (35 species including H2O) H+ OH– SO2 SO 2– 3 SO 2– 4 S 2 O 2– 3 S 2 O 2– 4 S 2 O 2– 8 S 4 O 2– 6 HSO –3 HSO –4 H2SO3 HS 2 O –4 H 2S2O4
Cu+ Cu2+ CuO 2– 2 HCuO –2 CuSO –3 Cu(SO 3 ) 3– 2 Cu(SO 3 ) 5– 2
Fe2+ Fe3+ FeO 2– 2 FeOH+ FeOH2+ HFeO –2 Fe(OH) +2 Fe(OH)3 Fe(OH) –3 Fe(OH) 2– 4 Fe 2 (OH) 4+ 2 FeSO +4 – Fe(SO 4 ) 2
Solid precipitates (s) (25 species (=phases) in total) Cu CuO Cu2O CuS Cu2S CuSO4 CuSO4 · H2O CuSO4 · 3H2O CuSO4 · 5H2O Cu2SO4 CuSO4 · 2Cu(OH)2 CuSO4 · 3Cu(OH)2 CuSO4 · 3Cu(OH)2 · H2O
Fe Fe0.947O (Wuestite) Fe2O3 Fe3O4 Fe(OH)2 Fe(OH)3 FeS(α) FeS2(pyrite) Fe7S8 FeSO4 FeSO4 · 7H2O
S(rhombic)
Table II.9.6 Quantitative composition of the system Species
Amount (g l–1)
Amount (mol l–1)
H2SO4 CuSO4 FeSO4
100 2.52 2.72
1.0196 1.5738 × 10–2 1.7906 × 10–2
152
The SGTE casebook
Molality of phase component as log y
0
Cu+ Cu2+ CuSO3–
–14
0
1.2 Redox potential Eh (V)
II.9.5 Concentrations of Cu species as functions of Eh at 25 °C and 1 g I–1 total copper content in a sulphuric acid solution.
(Cu–H2SO4–H2O subsystem), the concentration curves for Cu+(aq) and Cu2+(aq) intersect at 0.167 V and 7.869 × 10–3 mol total Cu l–1, the logarithm of which is –2.1242. These results are equivalent to those calculated from the Nernst equation (Cu–H2SO4–H2O subsystem). For iron, however, the major species at equilibrium are Fe2+(aq), Fe3+(aq) and Fe(SO4)2–(aq). This indicates a major change in the ions that influence the potential in comparison with the choice taken for the calculations with the Nernst equation (Fe–H2SO4–H2O subsystem). Figure II.9.6 shows a decrease in species containing divalent iron (as Fe2+ and FeOH+) and an increase in species containing trivalent iron. As can be expected, the sulphur species HSO 4– (aq) and SO 2– 4 (aq) dominate the aqueous solution from a potential of +0.3 V (Fig. II.9.7). The formation of H2S, solid sulphur (S) and copper or iron sulphides does not occur because of strong kinetic inhibitions. Therefore, the hydrogen which is added for the fixation of the potential leads to a reduction of sulphate and formation of S 2 O 32– (aq) and S 4 O 62– (aq) ions at potentials less than 0.3 V.
II.9.5
Conclusions and further developments
The calculation results permit an estimation and judgement of the conditions that have to be chosen in potential aqueous metal recovery processes. However, it is evident that the application of the ‘classical’ method, i.e. the exclusive use of the Nernst equation for a preselected stoichiometric reaction, permits no comprehensive understanding of the equilibrium state of a complex multicomponent system. For copper, the application of the simple method
Concentration of iron and copper ions in acid solutions FeSO4 Fe2+
Molality of phase component as log y
0
153
Fe3+ FeOH+ FeOH2+ +
Fe2(OH)2
4+
Fe2(OH)2 +
FeSO4
–
Fe(SO4)2
–14 0
1.2 Redox potential Eh (V)
II.9.6 Concentrations of Fe species as functions of Eh at 25 °C and 1 g I–1 total ferrous content in a sulphuric acid solution. 0
SO2 (aq)
Molality of phase component as log y
SO42– 2–
S2 O3
S4O62– HSO3– HSO4– H2SO3 (aq)
–14 0
1.2 Redox potential Eh (V)
II.9.7 Concentrations of S species as functions of Eh at 25 °C and 1 g I–1 total copper and ferrous content in a sulphuric acid solution.
was successful, because the assumed reaction was correct. For iron, it was shown that the initial choice, reasonable although it seemed, was not good. The formation of complexes of multivalent elements can lead to a considerable shift in the reactions that actually define the potential. Complex equilibrium computation has been used successfully to overcome this problem.
154
The SGTE casebook
It should, however, be noted that more advanced calculations would have to include non-ideal interactions between the aqueous species since the compositional range covered in the present calculations exceeds the validity range of the Debye–Hückel limiting law. Furthermore, the present results pertain only to 25 °C. If higher temperatures are of interest, CP functions of the aqueous species have to be introduced. Indeed, in some cases, it is even necessary to derive values for H298 and S298, since only ∆G298 values are at present available for many aqueous species. Experience shows that extrapolation of such simplified room temperature data beyond 35 °C can lead to unacceptably large deviations. The calculations for this case study have been performed using dedicated software and the program ChemSage.
II.9.6 65Gar 65Rau
82Wag
83Paw 90Eri 92SGT
References R.M. GARRELS and CH. L. CHRIST: Solutions, Minerals and Equilibria, Harper & Row, New York; Evanston, London; John Weatherhill, Tokyo, 1965. K. RAUSCHER, J. VOIGT, I. WILKE and K.-TH. WILKE: Chemische Tabellen und Rechentafeln fur die analytische Praxis, VEB Deutscher Verlag für Grundstoffindustrie, Leipzig, 1965. D.D. WAGMAN, W.H. EVANS, V.B. PARKER, R.H. SCHUMM, I. HALOW, S.M. BAILEY, K.L. CHURNEY and R.I. NATTALL: ‘The NBS tables of chemical thermodynamics properties, selected values for inorganic and C1 and C2 organic substances in SI units, American Chemical Society and the American Institute of Physics for National Bureau of Standards’, J. Phys. Chem. Ref. Data 11, 1982, Suppl. 2. F. PAWLEK: Metallhuttenkunde, Walter de Gruyter, Berlin, 1983. G. ERIKSSON and K. HACK: Metall. Trans. B 21, 1990, 1013–1024. SGTE: SGTE Pure Substance Database, STGE, Grenoble, 1992.
II.10 Thermochemical conditions for the production of low-carbon stainless steels KLAUS HACK
II.10.1
Introduction
Austenitic stainless steels classically contain the elements iron, chromium and nickel; chromium (typically 18 wt%) to give corrosion resistance and nickel (typically 8 wt%) to improve the ductility of the Fe–Cr alloys. However, during the production of these alloys, the presence of carbon can never be avoided. Unfortunately, higher carbon contents lead to the formation of M23C6 mainly containing chromium, thereby reducing the corrosion resistance. It is, therefore, necessary to establish a production process that yields Fe–Cr–Ni–C alloys with low carbon and high chromium contents. In the converter, oxygen is blown through the liquid Fe bath, thus removing carbon as CO gas. However, chromium also has a high affinity for oxygen, and oxidation of chromium from the melt is quite probable, especially if the activity of chromium oxide is reduced because it is dissolved in the slag. Thus, the thermochemical problem is one of a six-component system, the components being Fe, C, Cr, Ni, O and possibly Ar (see below) The major phases to be considered are the metallic melt, the gas phase and a slag (of unknown composition). The temperature and pressure conditions are as yet undefined. This situation is summarised in Table II.10.1. Table II.10.1 Summary of the thermochemical aspects of the production of low-carbon stainless steel Global conditions
8 wt% Ni
Elements
Fe, C, Cr, Ni, O, (Ar)
Phases
Liquid metal, gas, slag
155
156
II.10.2
The SGTE casebook
The mass action law approach
As carbon and chromium both compete for the oxygen, the following stoichiometric exchange reaction can be used to describe the decisive equilibrium: 2Cr + 3CO(g) = 3C + Cr2O3
(II.10.1)
However, the reactants are not pure substances. Instead, chromium and carbon will be dissolved species in an iron-based melt, Cr2O3 can be dissolved in a slag or precipitate as pure solid oxide and CO is a species in the gas phase. Thus, none of the activities of the reactants or products is equal to unity. The temperature and total pressure in the process are variables, which in practice can be modified over a wide range. The mass action law provides a relationship between all the parameters to decide which of them is the most important and should, therefore, be controlled in practice. One obtains from the equilibrium condition
log a C = 2 log a Cr + 1 (log K – log a Cr2 O 3 + 3 log p CO ) (II.10.2) 3 3 For the activity of chromium, a value near 0.2 can be assumed since the iron–chromium system is nearly ideal and an 18/8 steel contains about 20 at.% Cr. The temperature and the partial pressure of CO can be controlled in a converter and, to a lesser extent, the activity, i.e. the contents, of Cr2O3 in the slag. In Fig. II.10.1 the phase diagram of the slag system CaO–SiO2– Cr2O3 is shown. The slag path in a converter is overlaid, showing that the Cr2O3 amounts, and thus the activities, have essentially very low values. Only in the intermediate stage of the process might precipitation of Cr2O3 (i.e. a Cr2 O 3 = 1) be reached. The losses of chromium to the slag are usually recovered, as can be seen from last section of the slag path. For a quick check at a reasonable value of the Cr2O3 activity ( a Cr2 O 3 = 0.1) a matrix (Table II.10.2) of the carbon activity as a function of temperature and CO partial pressure can be obtained. The trends in the lines and columns show that both parameter changes (increase in temperature and decrease in pCO) result in a decrease in the carbon activity, i.e. the carbon content in the iron bath. However, a change in pCO has a far greater effect than a change in temperature.
II.10.3
The complex equilibrium approach
If confirmation is required that a certain level of carbon content (e.g. 0.01 wt%) is reached, only a series of proper complex equilibrium calculations will provide the desired result [90Spe]. Figure II.10.2 shows the weight percentage of C versus the weight percentage of Cr for different temperatures between 1600 and 1900 °C. The concentration
Thermochemical conditions for the production of low-carbon steels
157
Table II.10.2 Carbon activity aC as a function of temperature and CO partial pressure for a Cr2 O 3 = 0.1 and aCr = 0.2 pCO(bar)
aC for the following temperatures
1 0.1 0.01
1800 K
1900 K
2000 K
2.9 × 10–2 2.9 × 10–3 2.9 × 10–4
1.2 × 10–2 1.2 × 10–3 1.2 × 10–4
5.2 × 10–2 5.2 × 10–3 5.2 × 10–4
40
60 SiO2 0 100
70
20
)
1800
(m O
141 3
2151 CS 1500 C3S2 1407
60
1900 2000
2100 Cr2O3
60 C2S
) s% as (m
Ca
2 Liquid
2 SiO
as
s%
40
C : CaO K: Cr2O3 S: SiO2
80
SiO2
1430 1568
40
C3KS3
C 3S
80
100 0 CaO
Ca2Cr2Si2O15 Ca5Cr2SiO12
20
1450 1060 1061
20
40 Ca Cr O 60 3 2 8 Ca3(CrO4)2
CaCrO4
CK
80
0 100 Cr2O3
Cr2O3 (mass %)
II.10.1 Phase diagram of the slag system CaO–SiO2–Cr2O3 [83Gei, 92Kow]. The inset shows a typical slag path.
of Ni is set to 8 wt%, the value for pCO is set to 0.1 atm, and the activity of Cr2O3 is fixed at 0.1. All values are representative of practical conditions. From the diagram, it is obvious that increasing the temperature reduces the concentration of carbon, but with detrimental consequences for the refractories in the converter and at high energy costs.
158
The SGTE casebook 0.5
0.1 C (wt%)
T = 1600 °C T = 1700 °C T = 1800 °C
0.01
T = 1900 °C
0.001 5.0
10.0
20.0 18 wt% Cr
Cr (wt%)
II.10.2 Weight percentage of C versus weight percentage of Cr for different temperatures between 1600 and 1900 °C (8 wt% Ni, rest Fe; pCO = 0.1; a Cr 2O 3 = 0.1; 2Cr + 3CO = 3C + Cr2O3). 0.5
0.1
C (wt%)
aCr2O3 = .01 aCr2O3 = .1 aCr2O3 = 1
0.01
0.001 5.0
10.0
20.0 18 wt% Cr
Cr (wt%)
II.10.3 Weight percentage of C versus weight percentage of Cr for different values of the activity of Cr2O3 (8 wt% Ni, rest Fe; pCo = 0.1; T = 1700 °C; 2Cr + 3CO = 3C + Cr2O3).
Figure II.10.3 shows the same relationship for different values of the activity of Cr2O3. The temperature is fixed to an intermediate value of 1700 °C, the partial pressure of CO and the Ni concentration are kept as in Fig. II.10.2 (0.1 atm and 8 wt% respectively). Clearly, only increasing the activity of Cr2O3 to a value of one will reduce the concentration of carbon to the desired level, resulting in the transfer of chromium from the melt to the slag (see Fig. II.10.1).
Thermochemical conditions for the production of low-carbon steels
159
Figure II.10.4 shows the influence of the variation in pco at a fixed activity of Cr2O3 (0.1), fixed concentration of Ni (8 wt%) and again for T = 1700 °C. This diagram shows the strongest trend of the curves with a variation in the parameter.
II.10.4
Engineering conclusions
It is obvious that, of all the parameters varied, the partial pressure of CO is the most important. The lower its value (with all other parameters fixed), the lower is the carbon concentration in the iron bath. The engineering solution process can now be either reduction of the total gas pressure level, a process that is called vacuum oxygen decarburisation, or reduction of the partial pressure of CO by strong dilution in the gas phase, called argon oxygen decarburisation [87Lin]. It should be noted that using calculations based on the stoichiometric reaction, e.g. the law of mass action including the Gibbs energy data for pure substances, the assumptions about the possible values of the equilibrium activities already enable these conclusions to be made. In principle, a thermochemist of the 1930s would have been able to perform these calculations since all the Gibbs energy data needed were available at that time. Although the majority of this discussion is qualitative in nature, the results show that the conclusion as to which is the major parameter in the process is stringent. However, in the calculations above, it was possible to tackle the quantitative aspects of the redox reaction only because the Gibbs energy models and data available today permit complex equilibrium calculations to take into proper 0.5
pCO = 1
C (wt%)
0.1
pCO = 0.1 0.01
pCO = 0.01 0.001 5.0
10.0
20.0 18 wt% Cr
Cr (wt%)
II.10.4 Weight percentage of C versus weight percentage of Cr for different values of pCO (8 wt% Ni; rest Te; a Cr 2O 3 = 0.1; T = 1700 °C; 2Cr + 3CO = 3C + Cr2O3).
160
The SGTE casebook
account the concentrations of the different components in the different solution phases. For example, it is now possible to determine whether a defined pressure level is sufficient to keep the concentration of C below 0.01 wt%, the limit of formation of M23C6. The transformation of carbon and chromium activities into concentrations can be carried out for liquid steels taking into account the presence of 8 wt% Ni. With thermodynamic data available for the Gibbs energy of mixing of a multicomponent non-ideal metallic Fe melt as well as for a multicomponent oxidic slag, it is possible to produce the above series of diagrams. Thus, the question concerning the low carbon concentration can also be answered, but only using complex equilibrium calculations.
II.10.5
Acknowledgements
The author wishes to thank Professor D. Neuschutz, LTH, RWTH Aachen, for his helpful discussions. The calculations for this case study have been performed using ChemSage.
II.10.6 83Gei 87Lin 90Spe 92Kow
References J. GEISELER, K. GRADE and P. VALENTIN: Stahl Eisen 103, 1983, 1013–1017. H.-U. LINDENBERG, K.-H. SCHUBERT and Z. ZORCHER: Stahl Eisen 107, 1987, 1197–1204. P.J. SPENCER and K. HACK: Swiss Mater. 2 (3a), 1990, 69–73. M. KOWALSKI, P.J. SPENCER and D. NEUSCHUTZ: ‘Evaluation and critical compilation of thermochemical data and physical property values of slag for iron and steelmaking, phase diagrams, Part 3’, ECSC Research Report 7210-CF/107, 1992.
II.11 Interpretation of complex thermochemical phenomena in severe nuclear accidents using a thermodynamic approach P I E R R E - Y. C H E VA L I E R, B E R T R A N D C H E Y N E T and E V E LY N E F I S C H E R
II.11.1
Introduction
In the framework of the nuclear reactor safety, the analysis of all possible scenarios of hypothetical accidents is needed. In the first step of an accident of loss of coolant in a pressurised-water reactor (PWR), control rods may melt and interact with the steel structures. As the temperature increases in the core, the Zircaloy cladding may be oxidised and liquefied. Then, core degradation occurs, with the interaction of the fuel with the Zircaloy cladding. The partially melted complex mixture of all these materials, including the lower internal structures of the vessel, is called the ‘corium’. In the ultimate steps of a severe accident, the corium may melt through the vessel and slump into the concrete reactor cavity; the phenomenon is called molten corium– concrete interaction (MCCI). The associated inauspicious phenomena are mainly the ablation of all the encountered solid materials by the very corroding high-temperature corium, the continuation of the process due to the residual power produced by irradiated fuel and non-volatile fission products, and the release of dangerous radioactive elements in the outside environment. The general objective of the presented work was thus to provide useful information on the key corium interactions and thermochemical or thermophysical properties which affect significantly the corium behaviour progression during a variety of severe PWR scenarios. Simultaneously, new experiments were proposed in areas characterised by poor knowledge or the lack of knowledge. In this aim, the development and assembling of a global, consistent and reliable nuclear thermodynamic database (corium, concrete and non-volatile fission products) was initiated in 1989, from the thermodynamic modelling of the multicomponent system including selected elements representative of major components, linkable to a thermochemical equilibrium calculation software, for in-vessel and ex-vessel applications. 161
162
The SGTE casebook
This tool would allow the user to calculate the thermochemical equilibrium state of corium or corium–concrete mixtures at any step of a potential severe accident and to validate this approach with existing experimental values. In a general way, the equilibrium state consists of the proportion and composition of condensed phases and partial pressure of gaseous species for any given initial conditions, composition, temperature, pressure or volume. The predicted results could be used to obtain evidence about the unknown domains for which new experiments are needed in either basic or multicomponent systems, or to feed other thermal–hydraulic safety codes with directly calculated thermochemical properties (phase equilibria) or modelled thermophysical properties (viscosity, etc.).
II.11.2
The nuclear thermodynamic database
II.11.2.1 Pure elements and oxide components A schematic diagram of a PWR is shown in Fig. II.11.1.
Water input
................................... Vessel (bottom) Steel 14550 kg Fe 14400 kg Ni 150 kg ................................................
Water output ................................ Core Inconel 950 kg Steel 304 3040 kg 79600 kg UO2 Zr 18150 kg Fe 2300 kg Cr 800 kg Ni 900 kg Ag 1900 kg In 360 kg Cd 120 kg ................................ Lower internal structures Steel 304: 25900 kg Fe 18300 kg Cr 5400 kg Ni 3200 kg ..........................................
II.11.1 Schematic diagram of a PWR.
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The main identified PWR materials are as follows: UO2 (fuel), Zr (Zircaloy), Fe–Cr–Ni (steel), Ag–In–Cd (control rods) and H2O (water). In some other reactor types (light-water reactor), control rods are composed of boron carbide (B4C). Four elements, Ba, La, Ru and Sr, have been selected as representative of different families of presumably non-volatile fission products. In all reactors, the different types of concrete used for the reactor cavity (siliceous, limestone and limestone–common sand) may be represented by the quaternary oxide system Al2O3–CaO–MgO–SiO2 consisting of the major components. In some scenarios of accidents the system may become open and air is then present, allowing a variation in the oxygen (O2) potential. Thus, the selected elements of the multicomponent system are O–U–Zr–Fe–Cr–Ni–Ag–In–B–C–Ba–La– Ru–Sr–Al–Ca–Mg–Si + Ar–H. Cd was not considered because it is highly volatile and thus the early thermochemistry of the gas phase may be treated separately. Hydrogen was not introduced into the condensed solution phases, but only added in the gas phase and stoichiometric compounds. The thermodynamic database covers the entire field from metal to oxide domains, including in particular the oxide system UO2–ZrO2–FeO–Fe2O3– Cr2O3–NiO–In2O3–B2O3–BaO–La2O3–SrO–Al2O3–CaO–MgO–SiO2. Specific simplified subsets may be automatically extracted: – – –
In-vessel (without concrete components). O–U–Zr–Fe (quaternary system with key elements). O–U–Zr–Fe–Al–Ca–Si (major core–concrete elements).
II.11.2.2 History Historically, the work on the nuclear thermodynamic database began with condensed and gaseous species in 1988. At this time, oversimplified assumptions as ideal behaviour were too often used for modelling condensed solutions. For that reason, a more realistic modelling of the oxide phases was initiated by THERMODATA in 1989, with the impulsion of backing of L’Institut de Radioprotection et de Sûreté Nucléaire [91Rel]. Because of the importance of the task, the modelling of UO2–ZrO2–BaO– La2O3–SrO–Al2O3–CaO–MgO–SiO2 was continued in collaboration with the Atomic Energy Agency-Technology (AEA-T) between 1990 and 1992 [92Che2], [93Bal]. The database was extended to include FeO–Fe2O3 in 1994. The metallic and metal–oxygen systems were then added in 1995 and 1996. The work was then pursued from 1997 to 2000 under European contracts separately by THERMODATA for in-vessel applications (‘Corium interactions and thermochemistry’ (CIT) between 1997 and 1999 [99Adr]) and by AEAT for ex-vessel applications (‘Thermodynamic modelling and data’ (THMO) between 1997 and 1999 [99Cor]). Common work was initiated with European
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support between 2001 and 2003 to merge the two existing thermodynamic databases and to include two new elements, boron and carbon (ENTHALPY, between 2001 and 2003 [003Bre]). The actual version of the current thermodynamic database for nuclear applications called NUCLEA [003Che], commonly developed by THERMODATA–Institute National Polytechnique de Grenoble – Centre National de la Recherche Scientifique and AEA-T, will be continuously updated and improved. In particular, the validation work will continue with the improvement of the Fe–O–U–Zr and Ca–Fe–O–Si key quaternary systems in the coming years.
II.11.2.3 Thermodynamic modelling of substance and solution phases To calculate the equilibrium state in a multicomponent system, the minimisation of the whole Gibbs energy with regards to the composition variables and under mass balance constraints needs preliminary knowledge of the analytical description of the Gibbs energy of all possible substance and solution phases as a function of composition and temperature, referred to a given reference state: ∆G(xi, T)i, = 1, ..., N. The model of a phase may require internal variables yj linked to the composition by mass balance relations, which are in this case supplementary constraints. The thermodynamic modelling of substance and solution phases has been well described in a general way by [85Hil]. It is essentially based on the use of international standards defined by the SGTE [87Ans, 91Din, 96Hac]. In the present work, the liquid phase has been described with an associate model [000Kru], with the following formula: (Ag, Al, AlO1.5, Al1.333O4Si, AlCa0.5O4Si, B, B2BaO4, B2Ba3O6, B4CaO7, B2Ca2O5, BLaO3, BLa3O6, BO1.5, B2O4Sr, Ba, BaO, C, Ca, CaO, CaO3Si, Ca2O4Si, Cr, CrO1.5, Fe, FeO, FeO1.5, In, InO1.5, La, LaO1.5, Mg, MgO, Mg2O4Si, Ni, NiO, O, O2Si, OSr, O2U, O2Zr, Ru, Si, Sr, U, Zr)(l). This model allows one to describe the liquid phase in one single set from the metallic domain to the oxidic domain, and also miscibility gaps between metallic liquids, between oxide liquids or between metallic and oxide liquids. It needs a full description of binary interaction parameters between elements, between oxides or between elements and oxides. The introduction of ternary and quaternary associate oxides allows one to describe interactions due to chemical short-range order. The solid-solution phases (Table II.11.1) have been described by using the general compound energy formalism [81Sun]. For example, most of the metallic, intermetallic and carbides were represented with either a simple substitutional model (orthogonal A20, tetragonal metal, etc.), a two-sublattice model (face-centred cubic (fcc) Al, body-centred cubic (bcc) A2, hexagonal close-packed (hcp A3, MxCy, etc.) or a three-sublattice model (M23C6, σ, etc.).
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Table II.11.1 List of modelled solution phases in the database NUCLEA (Tmax is the approximate maximal stability temperature). The underlined elements are the major constituents of the solution N 1
Name as listed in NUCLEA
Elements
Miscibility gap index
LIQUID
Ag, Al, B, Ba, C, Ca, Cr, Fe, In, La, Mg, Ni, O, Ru, Si, Sr, U, Zr O, U, Zr, Ba, Ca, In, La, Mg, Sr O, U, Zr, In, Mg O, Ba, Sr, U, Zr O, Ba, Sr, U C, U, Zr, O C, O, U B, C Fe, Cr, Ni, C, Ru, In, U, Zr U, Zr, O, Ag, C, Fe, In, Ru, Cr, Ni Fe, Cr, Ni, Ru, In, Al, Zr, C Ru, Zr, O, C, Ag, Al, In, Fe, Cr, Mg, Ni, U Cr, Fe, U, Zr Fe, O, Sr, Ca, Mg, Ni Mg, Ni, O, Cr, Fe, Ca Ba, Ca, Sr, O, Fe, Mg, Ni Al, Cr, Fe, O Al, Cr, Fe, O, Mg, Ni La, O, Ba, Ca, Sr, Zr O, U, Zr, Si C, Cr, Fe, Ni C, Cr, Fe, Ni B, C, Cr, Fe, Ni C, La C, La B, Si Si, B, Ru Ag, Al, In, Mg In, Ni In, Ni B, C, Fe La, In, Ag, C, Ca, Mg La, In, Ag, Ba, C, Ca, Mg Ba, Ca, Sr Cr, Fe, Ni AL, Al, In, Mg U, Ru, Zr, Cr, Fe, Al, Si Ag, Al, In U, Zr, Fe U, Zr Sr, Ba, Ca, La Al, Mg Ag, Al In, Mg In, Mg In, Mg
1
2 3 4 5 6 7 8 9 10 11 12
FCC C1 TET(OXIDE) PEROVSKIT1 PEROVSKIT2 FCC B1(4) BCT B 4C BCC A2(1) BCC A2(2) FCC A1(1) HCP A3(1)
13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
LAVES FCC B1(1) FCC B1(2) FCC B1(3) RHO SPINEL CC TCHERNO M 7C 3 M23C6(1) M 3C C2La(H T) C3La2(S) BETA B DIA A4 FCC A1(2) HCP A3(3) BCC A2(5) M23C6(2) BCC A2(4) FCC A1(4) BCC A2(3) SIGMA BCC A2(6) TET(METAL) HCP A3(2) ORT A20 DELTA FCC A1(3) BCC A12 CUB A13 FCC L10 FCC L12 TET A6
TMax
1 0 0 0 1 0 0 1 1 0 1
3120 2650 3023 3200 2100 2800 2180 2250 1800 2607
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1947 1650 3100 3172 2603 2500 2586 1910 2100 1900 1600 2800 1700 2000 1687 1235 1210 1200 1200 1193 1134 1115 1088 1052 1049 1008 942 900 820 734 732 630 611 440
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The three-sublattice formula for the (U, Zr)O2+x solid solution, fcc C1, (U, Zr)(O, 䊐)2 (O, 䊐) (where 䊐 represents a vacancy on the sublattice) used by [004Che], was modified by using a simplified associate model in order to include a limited solubility of iron, barium, lanthanum and strontium in that phase: (BaO, CaO, FeO, InO1.5, LaO1.5, MgO, O, OSr, O2U, O2Zr, U, Zr)(fcc C1). A similar simplification has been made for most of the oxide solid solutions (fcc B1, spinel, corundum, etc.). Different polynomial expressions have been developed depending on the order of the subsystem and the number of sublattices; in our work, the binary interactions are described by Redlich–Kister [48Red] polynomials and the ternary interactions by symmetrical terms using a Margules [895Mar] type of expression. In some specific cases (iron–nickel phases), a magnetic contribution has to be added.
II.11.2.4 Critical assessment of binary and ternary subsystems (calculation-of-phase-diagrams method) The initial step is therefore to analyse all binary and the most important ternary subsystems, in order to identify all possible phases. Firstly, all substances (stoichiometric compounds) and binary terminal or intermediate solutions (non-stoichiometric phases) were inventoried and, secondly, their crystallographic structures were analysed. Phases with identical structures formed multicomponent solution phases. Then, a thermodynamic model was chosen for each phase, and all Gibbs energy parameters involved were set. Some parts of the Gibbs energy were directly calculated from the precise thermodynamic knowledge and basic data, such as the ‘lattice stability’ of pure elements [91Din], the magnetic parameters (Tc, β and f(τ) for the bcc A2, fcc A1 and hcp A3 structures) [85Hil], the reference and ideal terms, and the fundamental thermodynamic properties of substances [88Che]. The rest of the contribution to the Gibbs energy represents the deviation from ideal behaviour for the selected model. The parameters of the excess terms (binary and ternary interactions) or nonideal magnetic or ordering contributions have to be evaluated from the experimental information. Such work is called ‘critical assessment’ or optimisation. Thus, the building of a multicomponent thermodynamic database needs the critical assessment of all the binary and quasibinary (metallic, metal–oxygen and oxide) and of the most important ternary or higher-order systems. The optimisation is usually made by using a sophisticated procedure [77Luk, 002And], which takes into account all the available experimental information (phase diagram and thermodynamic properties, e.g. enthalpy, entropy, heat capacity and activity), in order to establish the best self-consistency
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between calculated and experimental values, on the basis of the international standards defined above. Practically, the critical assessment of any basic subsystem is a considerable task, tedious and time consuming. It consists of several consecutive steps: an update of bibliographic references, a check of the set of experimental data, inclusion of new experimental results, estimation of heat capacity and selection of fundamental thermodynamic properties of substances (∆H298, S298, CP), optimisation of binary or ternary interaction parameters, numerical and graphical comparison of calculation and experiments, production of a new set of Gibbs energy parameters, identification of the lack of experimental knowledge, and proposition of a quality criterion. The process of modelling, extrapolating, comparing with experiments and re-evaluating parameters is continued until satisfactory self-consistency is obtained. It is internationally known as the calculation-of-phase-diagrams (CALPHAD) method.
II.11.2.5 Content, assessed subsystems, solution and substance phases All the binary systems of the multicomponent system O–U–Zr–Fe–Cr–Ni– Ag–In–B–C–Ba–La–Ru–Sr–Al–Ca–Mg–Si have been critically assessed (18 × 17/2 = 153), as well as all the pseudobinary systems of the oxide system UO2–ZrO2–FeO–Fe2O3–Cr2O3–NiO–In2O3–B2O3–BaO–La2O3–SrO–Al2O3– CaO–MgO–SiO2 (15 × 14/2 = 105). Only selected ternary systems among the most important have been modelled, B–C–Fe, B–C–U, B–C–Zr, B–Fe–U, B–Fe–Zr, C–Cr–Fe, C–Cr– Ni, C–Fe–Ni, C–O–U, C–O–Zr, C–U–Zr, Cr–Fe–Ni, Cr–Fe–O, Cr–Fe–Zr, Cr–Ni–O, Fe–Ni–O, Fe–O–U, Fe–O–Zr, Fe–U–Zr and O–U–Zr (20), as well as selected pseudoternary oxide systems, such as Al2O3–CaO–O2Si [90Gis], which contains major components of concrete. Many ternary oxide systems have been modelled directly from the binaries, without adding ternary interactions: Al2O3–CaO–FeO, Al2O3–CaO–Fe2O3, Al2O3–SiO2–UO2, Al2O3– SiO2–ZrO2, SiO2–UO2–ZrO2, Al2O3–FeO–Fe2O3, Al2O3–FeO–SiO2, Al2O3– Fe2O3–SiO2, CaO–FeO–Fe2O3, CaO–FeO–SiO2, CaO–Fe2O3–SiO2, FeO– Fe2O3–SiO2, Al2O3–B2O3–CaO, Al2O3–B2O3–SiO2, B2O3–CaO–SiO2, B2O3– FeO–Fe2O3, Al2O3–B2O3–MgO, B2O3–CaO–MgO and B2O3–CaO–SiO2 (20). For some of these systems, the agreement with experimental data is satisfactory while, for some others, supplementary work has been undertaken owing to their importance for practical applications, e.g. the quaternary system Ca–Fe–O–Si and selected sub-systems (Ca–Fe–O, Fe–O–Si, CaO–FeO–Fe2O3– SiO2). For yet others, international programmes are continuously providing for new experimental points and consequently allow continuous improvement in specific systems, such as Fe–O–U–Zr.
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For each critically assessed system, the source and a quality criterion are indicated. To illustrate this huge work, some highly relevant systems for the nuclear safety field have been more extensively studied, e.g. the O–U binary and O–U–Zr ternary systems [002Che2, 004Che], which are part of the key Fe–O–U–Zr quaternary system. Much new experimental work has been taken into account in the optimisation process [96Mau, 98Gue, 001Bai, 001Lab, 002Bai]. It must be pointed out that these diagrams may still be improved owing to the continuation of the validation process with newly available experimental points, in both oxidising conditions ((U, Zr)O2+x) and reducing conditions ((U, Zr)O2–x). For example, new experiments provided by current international programmes will be used to improve the oxide part of the Fe–O–U (FeO/Fe2O3–UO2) and Fe–O–Zr (FeO/Fe2O3–ZrO2) ternary systems. The C–U binary system [001Che1] was also identified as very important when boron carbide control rods were considered. It contains refractory nonstoichiometric carbides which may include O and Zr. Many systems including boron oxide (B2O3) have been developed by AEA-T [001Mas, 002Mas]. At this time, the assembled nuclear thermodynamic database NUCLEA [003Che], contains 46 solution phases enumerated in Table II.11.1: some of these present miscibility gaps, and their modelling results in 271 reference substances. The nomenclature used for the names aimed only to give a concise information on the structure of the corresponding phase, but not the entire crystallographic designation used in the existing classifications. It includes also 420 condensed substances consistent with the phase diagram information, 95 complementary stoichiometric compounds and 203 gaseous species directly extracted from the substance database [88Che], with some more recent updates [001Che2].
II.11.3
Equilibrium calculation software
In a multicomponent system (N elements), as the Gibbs energy of all possible phases (Φ substances and solutions) are known versus temperature and composition, the thermodynamic equilibrium state, i.e. the number, the name, the composition and the proportion of phases at equilibrium, as well as the activity or chemical potential of elements, and the partial pressure of gaseous species, are determined from the minimisation of the global Gibbs energy at constant temperature T and pressure P, for the global composition of the system, with the constraints of the mass balance in and between phases. All the thermodynamic properties (∆G, ∆H, ∆S, ∆Cp, ∆Gi and ai) may be calculated at equilibrium. Much equilibrium calculation software is commercially available, such as Factsage [002Bal, 002Che1, 002Dav] Thermo-Calc [002And] and ThermoSuite
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[002Che3]. They have all their own specific features, among which are phase diagrams, complex equilibria, rapidity or robustness. The NUCLEA thermodynamic database linked to the GEMINI equilibrium calculation software [92Che3] integrated in ThermoSuite was used for the interpretation of thermochemical phenomena in severe nuclear accidents described by several examples in the following.
II.11.4
Complex thermochemical phenomena in severe nuclear accidents
In the nuclear field, practical applications of equilibrium state calculations for users are as follows. – – – –
– – –
Phase diagrams (binary, ternary and higher-order). Phase transitions (liquidus, solidus, ...). Proportion of solid and liquid phases (impacting the viscosity of partially melted mixtures). Composition of condensed phases (trapping non-volatile fission products for estimating the residual power and fission products release in the atmosphere). Calculating the miscibility gap between oxide and metallic liquids (in order to estimate the heat fluxes through the vessel). Corium–concrete equilibria (to design the reactor containment). Two-dimensional diagrams (influence of key parameters playing a major role in the observed phenomena, such as oxygen potential or iron content).
The capabilities of the thermodynamic approach can be used in two different ways. – –
To drive and understand global experiments, by predicting unknown domains or limiting the experimental investigated field. To feed other thermal–hydraulic safety codes with calculated thermochemical properties or ideally by a direct link.
In the following, all results are presented as qualitative examples but must not be considered as quantitative, because they were calculated with available versions of the nuclear thermodynamic database at a given time. Thus, they do not take always into account the most recent improvements in the actual nuclear thermodynamic database.
II.11.4.1 In-vessel applications A major in-vessel application is the calculation of the first interaction of UO2 with Zircaloy (O–U–Zr) followed by the interaction with the steel, leading to
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three main solution phases: (U, Zr)O2–x or fcc C1, liquid (oxide) and liquid (metal). The influence of two parameters, namely the oxygen potential and the iron content (steel), on the repartition of the main identified phases, was studied graphically during the thermochemical interaction of the core (UO2, Zr, Fe–Cr–Ni and Ag–In–Cd) with the lower internal structures and the spherical bottom of the vessel under either reducing or oxidising conditions, with prototypical (the PWR Framatome, the boiling-water reactor in Finland and the PWR 900, France) corium compositions resulting in safety code simulations. An example of graphical output obtained from NUCLEA– GEMINI2 is presented in Fig. II.11.2, which gives the proportion of phases versus temperature for the PWR Framatome corium composition in reducing conditions. One important point of this calculation is the coexistence of two liquid phases in the high-temperature range. The impact of this liquid miscibility gap in the key ‘Fe–O–U–Zr’ quaternary system on the reactor safety was recently discussed [004Bar]; in particular, the stratification of the layers has a strong influence on the heat fluxes and the possible failure of the lower head of the vessel. Another in-vessel application was the calculation of the release of some selected non-volatile fission products (Ba, La, Ru, Sr, U and Zr) in Vercors experiments ([99Fro, 000Sei]. 1.0 0.9
Liquid (2)
FCC C1
0.8 0.7
x
0.6 0.5 0.4 0.3 0.2
ZrO2 monoclinic Tet(Oxide)
Liquid (1) Gas
0.1 0
1000
1500
2000
2500
3000
3500
T (K) HCP A3(1)
FCC C1
ZrO2 (monoclinic)
(ZrH2(s))
(Zr2Fe(s))
Liquid (1)
Gas
tetragonal oxide
Liquid (2)
II.11.2 Proportion of phases versus temperature for a typical PWR Framatome corium composition.
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II.11.4.2 Ex-vessel applications The first important ex-vessel application [92Che1] was the calculation of liquidus and solidus temperatures of selected core (UO2–ZrO2)–concrete (Al2O3–CaO–SiO2) mixtures, with different types of concrete (siliceous, limestone–sand and lime). As shown in Fig. II.11.3, the first type of mixture presents a tendency to liquid immiscibility and was validated by experiments [93Roc]; the two other types present a eutectic behaviour. An extension of that application is the determination of the proportion of liquid (liquid/(liquid + solid)) for the various mixtures between the solidus and liquidus temperatures obtained in an antiproton cell experiment (ACE) as shown in Fig. II.11.4. That ratio directly impacts on the viscosity estimation (Fig. II.11.5 [000Sei]). The second major ex-vessel application [92Cen] was the calculation of selected fission products (Ba, La, Ru, Sr, U and Zr) released during ACEs simulating an interaction between corium and concrete. An example is shown in Fig. II.11.6 for the ACE L7.
5
3000
Temperature (K)
4 2500 1 2 5 2000
3
1
4 3 2
1500
0
20
Corium (UO2–ZrO2)
40 60 80 100 Concrete (wt%) Concrete (Al2O3–CaO–SiO2)
II.11.3 Liquidus-solidus of typical core–concrete mixtures; curve 1, SiO2; curve 2, siliceous; curve 3, limestone–sand; curve 4, limestone; curve 5, lime.
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2800
2600
2400
1
T (K)
2200 0.8 2000
0.7 0.6
1800 0.5 1600 0 1400 0
0.1
0.2
0.3
0.4
20
40 60 80 100 Concrete (wt%) included in the mixture Corium (85% UO2, 15% ZrO2) Concrete (4.6% Al2O3, 15.6% CaO, 79.8% SiO2
II.11.4 The proportion of liquid in typical core–concrete mixtures versus temperature (ACE L6). The values on the curves are the liquid/ (liquid + solid) ratios. Melt Composition Temperature Shear rate
Gemini
Liquid phase Composition Emulsion (?)
Urbain model Taylor model
Solid phase Solid volume fraction (Fvs)
Einstein model (low Fvs) Thomas model (spheres) Stedman model (dendrites)
Apparent viscosity
II.11.5 Deterministic methodology for calculating the viscosity of solid–liquid corium–concrete mixtures.
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Gemini Test L7
Release fraction
1e-2
1e-3
1e-4
1e-5
1e-6
Ba
La
Sr
U
Zr
II.11.6 Calculated and experimental fission products released during the ACE L7.
II.11.5
Conclusions
Thirty years ago, only substance databases existed. Nowadays, at the beginning of the third millenium, binary solution databases have been developed and will increase rapidly in the next few years. However, the number of critically assessed ternary systems is still very limited. International standards are extensively used for unary substances and ‘lattice stabilities’, and thermodynamic models for solid solutions. However, models for oxide and metal–oxygen liquids are still a subject of scientific discussion. The progress made in the development of a consistent nuclear thermodynamic database allowed users to interpret complex thermochemical phenomena in both inVessel and ex-vessel applications, and to obtain satisfactory agreement with observed experimental results, on a qualitative and quantitative aspects. We can hope that in the future there will be overall agreement on the models to be used for extrapolation to multicomponent systems and at high temperature, especially in the field of oxides and metal–oxygen systems. The development of a ternary solution database at a similar rate to the binary database in the past should be undertaken. The validation of the internal experimental database on thermochemical properties and phase diagrams could be made. These expected improvements will be possible only if users themselves are convinced of the interest in the global thermodynamic approach for solving practical problems and limiting the field of possible experiments.
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Specialists have to ensure the recruitment of students and research workers in thermodynamic assessment, but users must understand that this work, even if it is less expensive that experimental research, also needs financial support and is time consuming. This approach increases scientific patrimony and must be continued in cooperation with common standards, to increase the number of components and systems, as well as the validation state.
II.11.6
References
895Mar M. MARGULES: ‘Uber die zusammensetzung der gesättigten dämpfe von mishungen’, Sitzungsber. Akad. Wiss. Wien., Math. Naturwiss. K1. 11a(104) (1895) 1243–1278. 48Red O. REDLICH and A.T. KISTER: ‘Algebraic representation of thermodynamic properties and the classification of solutions’, Ind. Eng. Chem. 40(2), 1948, 345. 77Luk H.L. LUKAS, E. TH. HENIG and B. ZIMMERMANN: ‘Optimisation of phase diagrams by a least square method using simultaneously different types of data’, Calphad 1(3), 1977, 225. 81Sun B. SUNDMAN and J. ÅGREN: ‘A regular solution model for phases with several components and sublattices, suitable for computer applications’, J. Phys. Chem. Solids 42, 1981, 297. 85Hil M. HILLERT: ‘Thermodynamic modeling and phase diagrams … A call for increased generality’, in Proc. Symp. Computer Modeling of Phase Diagrams, Fall Meeting of the Metallurgical Society (Ed. L.H. Bennett), Toronto, Canada, 13–17 October, 1985, Metallurgy Division, National Bureau of Standards, Gaithersburg, Maryland, 1985, 17 pp. 87Ans I. ANSARA and B. SUNDMAN: ‘The Scientific Group Thermodata Europe’, in Computer Handling and Dissemination of Data (Ed. P.S. Glaeser), Elsevier, Amsterdam, 1987, pp. 154–158. 88Che B. CHEYNET: ‘THERMODATA, on-line integrated information system for inorganic and metallurgical thermodynamics’, in Proc. Int. Conf. Computerized Metallurgical Databases, Fall Meet. of American Socity of Metals and American Institute of Mechanical Engineers (Eds J.R. Cuthill, N.A. Gokcen and J.E. Morral) Cincinnati, Ohio, USA, 10–16 October 1987, Metallurgical Society of AIME, Warrendale, Pennsylvania, 1988, pp. 28–40. 990Gis J.A. GISBY: ‘Assessment of thermodynamic data for the CaO–Al2O3–SiO2 system’, NPL report DMM(D)15, March 1990. 91Din A.T. DINSDALE: ‘SGTE data for pure elements’, Calphad 15(4), 1991, 317. 91Rel O. RELAVE, P.Y. CHEVALIER, B. CHEYNET and G. CENERINO: ‘Thermodynamical calculation of phase equilibria in a quinary oxide system of first interest in nuclear energy field: UO2–ZrO2–SiO2–CaO–Al2O3’, in Proc. Int. Conf. User Aspects of Phase Diagrams (Ed. F.H. Hayes), Petten, The Netherlands, 25–27 June 1990, Institute of Metals, London, 1991, pp. 55–63. 92Cen G. CENERINO, P.Y. CHEVALIER and E. FISCHER, ‘Thermodynamical calculation of phase equilibria in oxide complex systems: prediction of some selected fission products (BaO, SrO, La2O3) release’, in Proc. 2nd OECD (NEA) CSNI Specialist Meet. Molten Core Debris Concrete Interactions (Ed. H. Alsmeyer), Karlsruhe, Germany, 1–3 April 1992, Publ. KFK 5108, NEA/CSNI/R(92)10, Institüt für
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Angewandte Thermo and Fluiddynamik Projekt Nukleare Sicherheitsforschung, Kernforschungszentrum, Karlsruhe, November 1992, pp. 271–278. 92Che1 P.Y. CHEVALIER: ‘Thermodynamical calculation of phase equilibria in a quinary oxide system of first interest in nuclear energy field: liquidus and solidus temperatures of some selected core (UO2–ZrO2)–concrete (Al2O3–CaO–SiO2) mixtures’, J. Nucl. Mater., 186, 1992, 212. 92Che2 P.Y. CHEVALIER and G. CENERINO: ‘Thermodynamical databases and calculation code adapted to the modelling of molten core–concrete interaction (MCCI) developed jointly by THERMODATA and the Institut de Protection et de Sûreté Nucléaire (France)’, in Proc. 2nd OECD (NEA) CSNI Specialist Meet. Molten Core Debris Concrete Interactions (Ed. H. Alsmeyer), Karlsruhe, Germany, 1– 3 April, 1992, Publ. KFK 5108, NEA/CSNI/R(92)10, Institüt für Angewandte Thermo and Fluiddynamik Projekt Nukleare Sicherheitsforschung, Kernforschungszentrum, Karlsruhe, November 1992, pp. 279–286. 92Che3 B. CHEYNET, J.N. BARBIER, P.Y. CHEVALIER, A. RIVET and E. FISCHER: ‘GEMINI: Gibbs Energy MINImizer codes for complex equilibria determination’, in Calphad XXI, Jerusalem, Israel, 14–19 June 1992, in Calphad 16(4), 1992, 339. 93Bal 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. 93Roc M.F. ROCHE, L. LEIBOWITZ, J.K. FINK and L. BAKER JR: ‘Solidus and liquidus temperatures of core–concrete mixtures’, Report NUREG/CR-6032, ANL-93/ 9, Argonne National Laboratory, 1993. 96Hac K. HACK (Ed.): The SGTE Casebook – Thermodynamics at Work, Institute of Materials, London, 1996. 96Mau A. MAURISI, C. GUENEAU, P. PERODEAUD, B. SCHNEIDER, O. DUGNE, F. VALIN and G. BORDIER: ‘Experimental determination of the liquidus L/(U, Zr)O2–x in the (U, Zr, O) system at T = 2273 K’, Proc. Int. Conf. EUROMAT, October 1996. 98Gue C. GUENEAU, V. DAUVOIS, P. PERODEAUD, C. GONELLA and O. DUGNE: ‘Liquid immiscibility in a (O, U, Zr) model corium’, J. Nucl. Mater. 254, 1998, 158. 99Adr B. ADROGUER, M. BARRACHIN, F. JACQ, F. DEFOORT, K. FROMENT, P. MASON, M. MIGNANELLI, P.Y. CHEVALIER, B. CHEYNET, F. FUNKE, S. HELLMANN, C. JOURNEAU, P. PILUSO, S. MARGUET, Z. HOZER, V. VRILKOVA, L. BELOVSKY, L. SANNEN, M. VERWERFT, P.H. DUVIGNEAUD, K. MWAMBA and C. RONNEAU: ‘Corium interactions and thermochemistry (CIT project), in-vessel core degradation and coolability (INV cluster), in and ex-vessel corium properties and thermochemistry’, in Euratom Framework Programme for Community Research and Training Activities 1994 to 1998/ Nuclear Fission Safety (Fisa-99) (Eds G.V. Goethem, G. Keinhorst, J. M. Bermejo, A. Zurita, H. Bischoff and G. Keinhorst), 29 November–2 December 1999, Publ. EUR 19532EN, 1999, pp. 202–210. 99Cor E.H.P. CORDFUNKE, M.E. HUNTELAAR, F. FUNKE, M.K. KOCH, CH. KORTZ, P.K. MASON and M.A. MIGNANELLI: ‘Thermochemical modelling and data (THMO project), ex-vessel corium behaviour and coolability (EXV cluster), in and exvessel corium properties and thermochemistry’, in Euratom Framework Programme for Community Research and Training Activities 1994 to 1998/ Nuclear Fission Safety (Fisa-99) (Eds G.V. Goethem, G. Keinhorst, J. M. Bermejo, A. Zurita, H. Bischoff and G. Keinhorst), 29 November–2 December 1999, Publ. EUR 19532EN, 1999, pp. 285–293.
176
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99Fro
K. FROMENT, F. DEFOORT, M. BAICHI and G. DUCROS: ‘Thermodynamical calculations applied to the fission products release from a degraded fuel in pin in Vercors V and VI experiments and to residual power distribution’, in Calphad XXVIII, Grenoble, France, 2–7 May 1999. H.-G. KRULL, R.N. SINGH and F. SOMMER: ‘Generalised association model’, Z. Metallkunde. 91(5) 2000, 356–365. J.M. SEILER and K. FROMENT: ‘Material effects on multiphase phenomena in late phases of severe accidents of nuclear reactors’, Multiphase Sci. Technol. 12(2), 2000, 17–257. M. BAICHI, C. CHATILLON, C. GUENEAU and S. CHATAIN: ‘Mass spectrometry study of UO2–ZrO2 pseudo-binary system’, J. Nucl. Mater. 294, 2001, 84. P.Y. CHEVALIER and E. FISCHER: ‘Thermodynamic modelling of the B–U and C– U binary systems’, J. Nuclear Mater. 288, 2001, 100. B. CHEYNET and P. CHAUD: ‘Pressions de vapeur et points d’ébullition, Cd, Cr, Pb, U, Zn, Zr’, J. Phys. Paris, IV 11, 2001, Pr10-165–Pr10-174. P.K. MASON and M.A. MIGNANELLI: ‘Extension of the oxide components of the nuclear thermodynamic database for in-vessel applications’, Report SAMENTHA(01)P007, AEAT/R/NS/0493, November 2001. J.O. ANDERSSON, T. HELANDER, L. HOGLUND, P. SHI and B. SUNDMAN: ‘ThermoCalc and DICTRA, computational tools for material science’, Calphad 26(2), 2002, 273–312. M. BAICHI, C. CHATILLON, C. GUENEAU and J. LE NY: J. Nucl. Mater. 303, 2002, 196. C.W. BALE, P. CHARTRAND, S.A. DEGTEROV, G. ERIKSSON, K. HACK, R. BEN MAHFOUD, J. MELANCON, A.D. PELTON and S. PETERSEN: ‘Factsage thermochemical software and databases’, Calphad 26(2), 2002, 189–228. S.L. CHEN, S. DANIEL, F. ZHANG, Y.A. CHANG, X.Y. YAN, F.Y. XIE, R. SCHMIDTFETZER and W.A. OATES: ‘The Pandat software package and its applications’, Calphad 26(2), 2002, 175–188. P.Y. CHEVALIER, E. FISCHER and B. CHEYNET: ‘Progress in the thermodynamic modelling of the O–U binary system’, J. Nucl. Mater. 303 2002, 1. B. CHEYNET, P.-Y. CHEVALIER and E. FISCHER: ‘ThermoSuite’, Calphad 26(2), 2002, 167–174. R.H. DAVIES, A.T. DINSDALE, J.A. GISBY, J.A.J. ROBINSON and S.M. MARTIN: ‘Mtdata–thermodynamic and phase equilibrium software from the national physical laboratory’, Calphad 26(2), 2002, 229–271. D. LABROCHE, O. DUGNE and C. CHATILLON: J. Nucl. Mater. 312, 2003, 21, 50. P.K. MASON and M.A. MIGNANELLI: ‘Extension of the oxide components of the nuclear thermodynamic database for ex-vessel applications’, Report SAMENTHA(02)P013, AEAT/R/NS/0573, July 2002. A. DE BREMAECKER, M. BARRACHIN, M. COQUERELLE, D. BOTTOMLEY, P. HOFMAN, M. STEINBRÜCK, P.Y. CHEVALIER, B. CHEYNET, M. FISCHER, S. HELLMANN, J.M. SEILER, G. COGNET, M. BELLON, F. TOCI, K. FORCEY, P. PACENTI, R. NANNICINI, S. MARGET and G. AZARIAN: ‘European nuclear thermodynamic database validated and applicable in severe accident codes (ENTHALPY project)’, in Symp. EU Research on Severe Accidents (Fisa-2003), 10–13 November 2003. B. CHEYNET, P. CHAUD, P.Y. CHEVALIER, E. FISCHER, P. MASON and M. MIGNANELLI: ‘Nuclea, Propriétés thermodynamiques et équilibres de phases dans les systèmes d’intérêt nucléaire’, J. Phys. Paris, IV 113, 2004, 61–64.
000Kru 000Sei
001Bai 001Che1 001Che2 001Mas
002And
002Bai 002Bal
002Che1
002Che2 002Che3 002Dav
002Lab 002Mas
003Bre
003Che
Complex thermochemical phenomena in severe nuclear accidents 004Bar
177
M. BARRACHIN, F. FICHOT and M. SALAY: ‘Thermodynamical and thermal–hydraulic behaviour of a molten pool during a LWR severe accident’, in Proc. Symp. Thermodynamics of Nuclear Materials, Karlsruhe, Germany, September 2004. 004Che P.Y. CHEVALIER, E. FISCHER and B. CHEYNET: ‘Progress in the thermodynamic modelling of the O–U–Zr ternary system’, Calphad, 28(1) 2004, 15–40.
II.12 Nuclide distribution between steelmaking phases upon melting of sealed radioactive sources hidden in scrap K L A U S H A C K , J Ü R G E N K O R B and DIETER NEUSCHÜTZ
II.12.1
Introduction
Sealed radiation sources are widely used in industry, medicine and research. Although there are regulations in many states on the management of disused radioactive sources [003Bun], the steel industry is well aware of the possibility that disused sources are lost to regulatory control [99Wac]. Since most sealed sources have an outer containment made of steel, a certain fraction of these ‘orphan’ sources may eventually be discarded as plain steel scrap and be sold to the steel industry. In most steelworks, incoming scrap is nowadays subjected to a strict γradiation survey at the works gates. Some works have additional monitors installed at the scrap bins and around the scrap buckets. Sealed sources with strongly γ-radiating nuclides can thus be detected with high probability before being melted. A matter of greater concern is the group of sources containing nuclides with weak γ- or with β- and α-emission. In a recent study, the number of γ-emitting sealed sources in use within the European Union (EU) has been estimated to be 110 000, while about 30 000 disused sources are held in storage at the users’ premises [000Ang]. Since steel scrap is not only traded inside the EU but also shipped to the EU from many non-European countries, the case that a ‘lost’ radioactive source enters the scrap, passes the gate monitors and ends up in the basic oxygen furnace (BOF) or electric are furnace (EAF) meltshops has become rare [003Bun] but cannot be totally disregarded. According to a report published in 2002 by the (US) National Council on Radiation Protection and Measurements (NCRP) [002NCR], 71 inadvertent melting events had been confirmed from the early 1980s to the year 2000 in 23 countries, with 33 of these events occurring in the USA. Over 20 of the US events were encountered in the steel industry. The most frequently detected radionuclides after inadvertent melting were 137Cs (41%), 60Co (18.5%), 226 Ra (7%) and 241Am (6.5%) [002NCR]. 178
Nuclide distribution upon melting of sealed radioactive sources
179
With respect to such cases, a study has been carried out as a European Coal & Steel Research steel research project [96Spe] to predict the resulting distribution of nuclides between steel melt, slag, dust and off-gas. The study was to set up a list of relevant nuclides actually or formerly used as radiation sources in medicine, industry and research, and to give recommendations as to the instalment of suitable monitors for quick nuclide detection. This paper reports the thermochemical calculations related to nuclide partitioning between the steelmaking phases according to equilibrium calculations. Further information on realistic nuclide distribution ratios additionally based on operational experiences and on campaigns of controlled melting of slightly contaminated steel scrap from the nuclear cycle can be found elsewhere [003Neu].
II.12.2
List of relevant nuclides
All nuclides that are or have been used as sealed radiation sources in industry, research and medicine have been taken into account. The relevant nuclides are listed in Table II.12.1. Among them, plutonium and radium are no longer or only rarely used, and nickel and promethium have only limited applications. The others are frequently used in medicine, as thickness and level gauges, for weld tests and crack inspection, as smoke detectors or as radionuclide batteries. Their half-lives vary between 0.2 years (192Ir) and 24.110 years (239Pu). According to the type of radiation, the nuclides under consideration can be classified into three groups. 1 2 3
Strong γ emitters (60Co, 137Cs and 192Ir). Weak γ emitters (226Ra and 241Am). β and α emitters (63Ni, 90Sr, 147Pm, 238Pu,
II.12.3
239
Pu and
244
Cm).
Preparation of a suitable set of thermochemical data
The aim was to determine partition ratios by thermodynamic calculations taking into account the steel melt, the slag phase and the vapour phase. For Kr and Ir, no calculations were made because Kr as a noble gas can be expected to be completely in the gas phase and finally in the cleaned off-gas, and Ir as a noble metal to be totally dissolved in the steel melt. For several nuclides, thermodynamic data as needed for complex equilibrium calculations were not available. They have therefore been substituted by chemically related elements with better known thermodynamic data. Promethium as a lanthanide was replaced by lanthanum, and the actinides radium, americium, plutonium and curium were as a whole group substituted by the more thoroughly investigated actinide uranium.
180
Table II.12.1 Nuclides at present or formerly used in sealed radioactive sources for medicine, industry or research Name
Emitter
Half-life (years)
Form that it is used in
Applications
60
Co
Cobalt
γ (+β)
5.3
Metal
Thickness and level gauges; medical applications; weld tests
63
Ni
Nickel
β
96.0
Metal
Electron capture detectors (e.g. in gas chromatography)
85
Kr
Krypton
γ (+β)
10.7
Gas
Thickness gauging (plastics)
90
Sr
Strontium
β
28.7
Carbonate
Thickness gauges (paper, plastics, metal foils); medical applications
Caesium
γ (+β)
30.0
Chloride
Thickness gauges, crack inspection; medical applications
137
Cs
147
Pm
Promethium
β
2.6
Oxide
Medical applications nuclear-fuelled generators
192
Ir
Iridium
γ
0.2
Metal
Radiographic weld tests; medical applications
226
Ra
Radium
α (+γ)
1600
Oxide
Formerly moisture gauges, medical applications; now dose calibration
238
Pu,
Plutonium
α
87.7 (238Pu) 24 110 (239Pu)
Oxide
Smoke detectors (obsolete)
241
Am
Americium
α (+γ)
432
Oxide
Thickness, moisture and level gauges; smoke detectors
244
Cm
Curium
α
18.1
Oxide
Thickness gauges; radionuclide batteries
239
Pu
The SGTE casebook
Nuclide
Nuclide distribution upon melting of sealed radioactive sources
181
As a common basis for the calculations, a steel melt with 0.07 wt% C, balance Fe, and a slag with 49 wt% CaO, 43 wt% Al2O3 and 8 wt% SiO2 plus variable FeO contents up to 10 wt% with a slag-to-metal ratio of 133 to 1000 were assumed. The partition ratios of the nuclides or of their respective substitutes were calculated for a content of 3 wt% of these nuclides in the scrap. The equilibrium calculations were carried out with the Equilib module of the program package FactSage which is designed to take into account complex systems involving phases with non-ideal mixing properties [002Bal]. The thermodynamic models applied for the various phases were the Wagner unified interaction parameter formalism for dilute solutions (for the steel melt), the Blander–Pelton quasichemical formalism for ionic liquids (for the slag), and the ideal mixing model for the gas phase. To predict the partition ratios of the elements under consideration (Co, Ni, Sr, Cs, La and U), a case-specific thermodynamic database had to be set up using as basic data sources the SGTE Pure Substance Database [000SGT] and the FACT database [001FAC] which cover the regular steelmaking components in the gas, the liquid metal and the liquid slag phases. For the present purpose, additional thermodynamic information had to be collected or assessed to calculate the partition ratios in question with appropriate accuracy. 1
2
The liquid metal phase. Data for the elements Sr and Cs in the liquid iron phase have been added, taking them as pure liquids and estimating the interactions using the Miedema method for ∆H∞ together with Kubaschewski’s rule of experience: ∆H∞ = 3400 ∆S ∞. These estimates result in strong positive interactions which is in accordance with the phase diagrams of Fe–Sr and Fe–Cs. Uranium has been added to the metal phase using information on the Fe–U system available from the thermodynamic database for nuclear chemistry at Thermodata, Grenoble [000The]. The data for the substitutional model used there have been adjusted to fit the requirements of the dilute solution approach in the present database. The slag phase. CoO, SrO and Cs2O have been added to the slag phase for the present work using the data for the respective pure liquid oxides from the SGTE Pure Substance Database. Interaction data for CoO have been included according to the availability of phase diagrams. Mainly CoO–MxOy systems (CoO–SiO2, CoO–Al2O3, CoO–FexO and CoO–CaO) were available. SrO interactions had to be estimated since no phase diagram information was available for the derivation of appropriate data. It was assumed that, because of the chemical similarity between Sr and Ca, using interactions from the relevant CaO–MxOy systems would be a reasonable starting point. On similar grounds, Cs2O interactions have been taken to be equal to equivalent data for K2O stored in the FACT
182
The SGTE casebook
database. La2O3 could be moved from a separate data set available within the FACT Database which covers interactions with Al2O3, SiO2 and CaO. All other interactions of La2O3 in the slag were taken to be ideal. Information on UO2 in the liquid oxide phase was available in the thermodynamic database for nuclear chemistry (Thermodata). Again only the phase diagrams calculated with the database could be used as information since the quasichemical model for the liquid slag used in the present work is not directly compatible with the associated solution model used by Thermodata. CsCl has been added to the slag phase using the dilute solution feature that is available for the quasichemical interaction model.
II.12.4
Calculated partition ratios
The results of the equilibrium calculations have been plotted as partition ratios of the respective components between liquid metal, liquid slag and gas phase for 1500, 1600 and 1700°C as a function of the FeO content in the slag in Fig. II.12.1 to Fig. II.12.6, thus indicating possible differences between reducing and oxidising conditions. Nickel and cobalt clearly are contained to the steel melt (Fig. II.12.1 and Fig. II.12.2). While the slag may contain up to 0.01% of the total Ni, the Co content in the slag may reach up to 0.1% in strongly oxidising conditions. Neither Ni nor Co is transferred to the gas phase. Strontium (Fig. II.12.3) is exclusively found in the slag, as expected because of its similarity to calcium. The maximum partition ratio in the metal is 0.001%, and in the gas phase 0.008%. In steel
Co partition ratio (%)
100.0
99.9
, , ,
99.8
T = 1973 K T = 1873 K T = 1773 K
In slag
0.1 0.0 In gas phase 0
2
4 6 8 FeO content in slag (wt%)
10
II.12.1 Partitioning between steel, slag and the gas phase as a function of temperature and FeO content in the slag (starting Co content in scrap, 3 wt%).
Nuclide distribution upon melting of sealed radioactive sources
183
100.000
Ni partition ratio (%)
In steel 99.995 99.990
, , ,
88.985
T = 1973 K T = 1873 K T = 1773 K
In slag
0.010 0.005 In gas phase 0.000 0
2
4 6 8 FeO content in slag (wt%)
10
II.12.2 Nickel partitioning as in Fig. II.12.1 (starting Ni content in scrap, 3 wt%).
100.000
Sr partition ratio (%)
99.998 99.996 In slag
99.994 99.992
, , ,
T = 1973 K T = 1873 K T = 1773 K
In gas phase
0.008 0.006 0.004
In steel
0.002 0.000 0
2
4 6 8 FeO content in slag (wt%)
10
II.12.3 Strontium partitioning as in Fig. II.12.1 (starting Sr content in scrap, 3 wt%).
Caesium (Fig. II.12.4) is the only nuclide of Table II.12.1 to be present almost completely in the gas phase assuming CsCl as the starting compound. Since the chloride is mostly used in Cs sources, this case appears to be the most realistic. With other Cs compounds (such as sulphates or hydroxides) the partition ratio between slag and vapour phase is closer to 1. The steel melt is always free of Cs. The partition calculations for lanthanum (which substitutes for Pm) shows clearly that La is completely in the slag. The amounts expected to be transferred to metal and gas phase are in the range of a few parts per million (Fig. II.12.5).
184
The SGTE casebook 100.000 In gas phase
Cs partition ratio (%)
99.998 99.996 99.994 0.005
, , ,
T = 1973 K T = 1873 K T = 1773 K In slag
0.004 0.003 0.002
In steel
0.001 0.000 2
4 6 8 FeO content in slag (wt%)
10
II.12.4 Caesium partitioning as in Fig. II.12.1 (starting CsCl content in scrap, 3 wt%).
La partition ratio (%)
100.000 , , ,
99.999 In slag
T = 1973 K T = 1873 K T = 1773 K
99.998 In gas phase 0.0005 0.0000 In steel 0
2
4 6 8 FeO content in slag (wt%)
10
II.12.5 Lanthanum partitioning as in Fig. II.12.1 (starting La2O3 content in scrap, 3 wt%). La was chosen as substitute for Pm because of its similar chemical behaviour.
The results obtained for uranium (Fig. II.12.6) are very similar to those shown in Fig. II.12.5. Again, this element is predicted to be completely in the slag; only traces are to be found in the metal and the vapour phases. The U results are taken as representative for the nuclides Ra, Pu, Am and Cm. With respect to the partitioning of the nuclides under consideration, the thermodynamic calculations have led to the result that the respective nuclide should definitely exist in one phase only and not dissipate among the phases to any noticeable degree. Co and Ni should be in the metal phase, Cs in the
Nuclide distribution upon melting of sealed radioactive sources
185
100.0000
U partition ratio (%)
99.9998 In slag
99.9996 99.9994 99.9992
In steel
0.0003
, , ,
T = 1973 K T = 1873 K T = 1773 K
0.0002 0.0001 0.0000 –0.0001
In gas phase 2
4 6 8 FeO content in slag (wt%)
10
II.12.6 Uranium partitioning as in Fig. II.12.1 (starting UO7 content in scrap, 3 wt%). U was chosen as a substitute for Ra, Pu, Am and Cm because of its similar chemical behaviour.
vapour phase (and after cooling to the dust), while all other nuclides (Sr, Pm, Ra, Am, Pu, Cm) should exclusively be in the slag phase.
II.12.5
Realistic distribution ratios
According to industrial studies discussed by Neuschütz et al. [003Neu], in practice phase separation is incomplete. This results in a small amount of liquid metal typically remaining suspended in the slag when the slag is removed from the furnace. Although this amount varies with the slag-tometal ratio, with the slag viscosity and its temperature during deslagging, an average of 1% appears to be a realistic value for the metal to be found as granules in the slag. A greater part of this metal is often recovered during subsequent slag beneficiation. Another cause for deviations in the realistic element distribution from the equilibrium partition ratios is the interaction of gas bubbles rising to the surface of a melt. The effect called ‘bubble bursting’ produces extremely small droplets which are easily carried up with the gas flow, forming very fine off-gas dust both in BOF and in EAF steelmaking [001Gri]. Again, about 1% of the metal and the slag are typically transferred from the steel and slag melt to the dust. Combining the equilibrium partition ratios with these process-specific effects leads to realistic distribution ratios as they are expected to be observed in operating steelworks, in the case when one of the nuclides under consideration happens to be melted down in an EAF or a BOF (Table II.12.2). According to Table II.12.2, the steel melt is expected to contain about 98% of the
186
The SGTE casebook
Table II.12.2 Realistic distribution ratios expected for the nuclides listed in Table II.12.1 upon inadvertent meltdown in the EAF or BOF Nuclide 60
Co Ni 85 Kr 90 Sr 137 Cs 147 Pm 192 Ir 226 Ra 238 Pu, 239Pu 241 Am 244 Cm 63
Radiation γ (+β) β γ (+β) β γ (+β) β γ α (+γ) α α (+γ) α
Realistic distribution (%) Melt Slag*
Dust†
98 98 – – – – 98 – – – –
1 1 100 gas 1 > 99‡ 1 1 1 1 1 1
1 1 – 99 <1 99 1 99 99 99 99
* About 1% of the metal is lost to the slag as suspended granules. † About 1% of metal + slag ends up in the dust as fine droplets via bubble bursting. ‡ When in the source as CsC.
radioactive Co, Ni and Ir, while the dust will contain almost all the Cs and about 1% of all the other nuclides expect for Kr which should completely go to the off-gas. The slag is expected to contain about 99% of the nuclides Sr, Pm, Ra, Am, Ru and Cm, and about 1% of Co, Ni and Ir dissolved in the suspended metal granules.
II.12.6
Conclusions
The occasional inadvertent meltdown of a disused radioactive source hidden in purchased scrap cannot completely be excluded by thorough scrap monitoring at the steelworks gates. For the safety of their workforce and their products, steelmakers are well advised to monitor the steel samples, the dust and the off-gas of their meltshops with respect to strong γ emission [002Uni]. With respect to weak γ emitters and to the α and β emitters, online detection is technically not possible at present. It should be kept in mind that the primary responsibility for the safe handling of radioactive sources throughout their complete life cycle is not with the steelmakers but with the producers and the users of these sources. In the NCPP report [002NCR], enhancements needed in the regulatory control of orphan sources are discussed and preventive and corrective measures suggested. Global regulations and control managements [002Uni, 003Bun] should become sufficiently strict and efficient that ‘lost’ sources cannot find their way into steel meltshops any more.
Nuclide distribution upon melting of sealed radioactive sources
II.12.7
187
References
96Spe P.J. SPENCER: Z. Metallkunde 87, 1996, 35. 99Wac H.-J. VON WACHTENDONK: Stahl Eisen, 119(1), 1999, 61–67. 000Ang M.J. ANGUS, C. CRUMPTON, G. MCHUGH A.D. MORETON and P.T. ROBERTS: ‘Management and disposal of disused sealed radioactive sources in the European Union’, eur18186.pdf.pdf, Safeguard International Ltd, Abingdon, Oxfordshire, 146 pp. 000SGT SGTE: SGTE Pure Substance Database, SGTE, Grenoble 2000. 000The TDBCR, Thermodynamic Database for Nuclear Chemistry, Thermodata, Grenoble, 2000. 001FAC FACT Database, Centre de Recherche en Calcul Thermochimique, Ecole Polytechnique, Montreal, 2001. 001Gri A. GRITZAN and D. NEUSCHÜTZ: Steel Res. 71, 2001 324–330. 002Bal 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. 002NCR ‘Managing potentially radioactive scrap metal’, NCRP Report 141, National Council on Radiation Protection and Measurements, Bethesda, Maryland, November 2002. 002Uni ‘Report on the improvement of the management of radiation protection aspects in the recycling of metal scrap’, UN. Publ. E. 01. II. E.22, United Nations Economic Commission for Europe, European Commission, and International Atomic Energy Agency, New York, 2002. 003Bun ‘Joint Convention on the Safety of Spent Fuel Management and on the Safety of Radioactive Waste Management’, Report of the Federal Republic of Germany for the First Review Meeting in November 2003, Section J: Disused Sealed Sources, Bundesministerium für Umwelt, Naturschutz und Reaktorsicherheit, 5 May 2003. 003Neu D. NEUSCHÜTZ, D. SPIRIN, K. HACK, U. QUADE and J. METER-KORTWIG: Steel Res. 74(11–12), 2003, 762–767.
II.13 Pyrometallurgy of copper–nickel–iron sulphide ores: the calculation of the distribution of components between matte, slag, alloy and gas phases T O M I. B A R R Y, A L A N T. D I N S D A L E, S U S A N M. M A R T I N and J E F F R. T A Y L O R
II.13.1
Introduction
A basic principle of many processes for extracting copper and nickel from sulphide ores is to overlay the molten ore or matte with a lime–silica–iron oxide slag and blow with air or oxygen. Some of the iron present in the ore oxidises and, as a result, partitions to the slag phase, while some of the sulphur is carried off as sulphur dioxide. In this example of the application of computerassisted thermochemistry a series of calculations is described which model the phase equilibria and partition of elements between the matte, slag, alloy and gas phases at various stages of the pyrometallurgical process. The results agree well with practical experience and also provide an insight into the detailed phase relationships to be expected during such processes. The subsequent processing depends very much on the composition of the ore, including the amounts of impurities, some of which may themselves be of value. The following discussion does not relate to any particular process and is intended only to illustrate the calculation procedures. Not all factors have been taken into account in the thermodynamic modelling. For example, the solution of Cu, Ni and S in the slag is not considered. Three types of process are considered, which have been more fully described by Dinsdale et al. [88Din2]. – – –
Partition of iron to the slag phase and sulphur to the gas phase while blowing the matte. The effect of sulphur content on phase separation in the liquid mattes especially those with low nickel content. Crystal transformation during solidification of a matte containing reduced amounts of sulphur and iron, which can be used to effect a further partition of the components.
II.13.2
Blowing the matte
For these calculations, data are required for the matte, slag and gas phases. The plant operator needs to process ores of a range of compositions and to 188
Pyrometallurgy of copper–nickel–iron sulphide ores
189
achieve the desired result, the temperature, the composition of the slag and the amount of air or oxygen used in the blow can be varied. Using these variables, some degree of independent control over the amounts of iron and sulphur left in the matte can be achieved. The partial pressure of sulphur dioxide in the off-gas can be used as a monitor of the process. The initial compositions of the mattes and slags used in the calculations below are given in Table II.13.1. The mass of the slag is made three times that of the matte so that its composition does not change markedly. In practice the slag would be replenished either from time to time or continuously. In continuous processing, the slag could be made to flow countercurrent to the matte. Two different matte compositions are used to illustrate different aspects of processing. Figure II.13.1 shows the calculated effect of blowing a matte (matte A) which originally contained equimolar proportions of nickel, iron and copper. Subsequently the oxygen potential and partial pressure of sulphur dioxide rise and the amount of sulphur remaining in the matte falls. In agreement with practical experience, calculations show that initially the added oxygen is consumed mainly in the process of oxidation of iron, which is transferred to the slag. As a result the proportion of sulphur in the matte actually increases. In Fig. II.13.1, oxygen is added progressively until a substantial pressure of sulphur dioxide develops. Most of the iron is removed from the matte during this phase. In the second phase of the process the partial pressure of sulphur dioxide is considered to be maintained at 0.1 atm. Figure II.13.2 shows the calculated relationship between the amounts of iron and sulphur in the matte under these chosen conditions. The relationship between these two variables is a function of the composition of the slag and the relative proportions of nickel and copper in the matte, the temperature and the partial pressure of sulphur dioxide. Knowledge of this relationship is important to the control of the blowing process in order to prepare the matte for subsequent processing. Good agreement between calculated and experimental results has been demonstrated elsewhere [88Dinl, 90Tayl]. The strongly non-ideal behaviour is reflected in both the calculated and the experimental results. Table II.13.1 Initial compositions of the matte and slag phases Component
Ni Fe Cu S Total mass (g)
Matte A (wt%) 26.7 25.4 28.9 18.9 219.7
Matte A (mol) 1.0 1.0 1.0 1.3
Matte B (wt%) 7.9 25.0 48.4 18.6 223.0
Matte B Component (mol) 0.3 1.0 1.7 1.3
CaO ‘FeO’ SiO2
Slag (wt%)
Slag (mol)
10.0 60.0 30.0
1.18 5.52 3.33
662.9
The SGTE casebook 0.40 Cu 0.35 Ni
Mass fraction
0.30 0.25
S
0.20 0.15
Fe 0.10 0.05 0.00
13.5
14.0 14.5 Oxygen (mol)
15.0
II.13.1 Composition of matte A as a function of oxygen content of the system at 1573 K. The starting amount of oxygen before blowing is 13.36 mol.
0.25
0.20
0.15
WS(l)
190
0.10
0.05
0.00 0.00
0.01
0.02 WFe(l)
0.03
0.04
II.13.2 Interdependence of the weight fraction of iron and sulphur in the liquid for a particular nickel-to-copper ratio, slag composition and sulphur dioxide pressure.
Pyrometallurgy of copper–nickel–iron sulphide ores
II.13.3
191
Phase separation in the matte
There is a considerable tendency for mattes, particularly those of low nickel content, to unmix into sulphur-rich and sulphur-poor liquids. This is exploited in the Noranda process in which copper-rich mattes are blown down to a sulphur level at which a sulphur-poor liquid enriched in nickel separates. A feature of copper-rich nickel-poor mattes is that they tend to develop liquid–liquid immiscibility. Thus, at 1573 K, matte B exists as two liquids. As the amount of iron is reduced, the two liquids become miscible but immiscibility again develops as the amount of sulphur declines and a sulphurpoor phase separates. The proportions of the two phases and the distribution of components between them in this second regime are illustrated in Fig. II.13.3. The proportion of iron in the matte has been reduced to low levels and thus the composition lies nearly within the Cu–Ni–S ternary system of Fig. II.13.4. In this diagram, the position of the miscibility gap, which is not perfectly in accord with the experimental data reviewed by Chang et al. [79Cha], suggests that, for the compositions of Fig. II.13.3, phase separation should begin at about 30 wt% S. Because the tie lines in Fig. II.13.4 run at an angle to lines of constant copper-to-nickel ratio, enrichment of nickel occurs in the sulphur-poor phase. If a differential flow is established between the phases, it is possible to reduce the level of nickel in the sulphur-rich phase (which will follow the upper edge of the miscibility gap) to low levels. This possibility is exploited in the Noranda process, the thermodynamics of which has been explored by Nagamori and Mackey [78Nag]. It is instructive to examine the results of a single step in the calculations displayed graphically in Fig. II.13.3. This is given in Table II.13.2.
II.13.4
Solidification and recrystallisation
Further separation of the components occurs during crystallisation of the matte on cooling. It may sometimes be advantageous not to reduce the iron to very low levels; among other things this avoids loss of values to the slag. Figure II.13.5(a) shows the predicted sequence of phases formed during cooling of a matte of original composition A in which the amount of iron has been reduced to 10 wt% and the sulphur to 22.3 wt%. The compositions of the face-centred cubic (fcc) phase are given in Fig. II.13.5(b) and the distribution of components between the phases is given in Fig. II.13.6. Solidification sets in at 1050 K with the formation of the β and metallic fcc phases. Nickel is strongly segregated to the β phase, which approximates to Ni3S2 in composition. Plots similar to Fig. II.13.5(b) show that the iron maintains its concentration in the diminishing amount of liquid, whilst the concentration of copper is enhanced. At 950 K, digenite precipitates, depleting
192
The SGTE casebook –1.0 Cu
log10 [mass of species (kg)]
–1.5
S
–2.0 Ni –2.5 –3.0 –3.5 Fe –4.0
–4.5
0.240
0.250 0.260 Mass of O (kg) (a)
0.270
–1.0 Cu log10 [mass of species (kg)]
–1.5 Ni
–2.0 S
–2.5 –3.0 –3.5 –4.0
Fe
–4.5 0.240
0.250 0.260 Mass of O (kg) (b)
0.270
II.13.3 Two-liquid formation in matte B as a function of oxygen content of the system at 1573 K, showing the compositions of (a) the sulphur-rich liquid and (b) the sulphur-poor liquid.
the liquid in copper and sulphur and leading to the disappearance of the liquid at 900 K. At this temperature, iron is partitioned to the β and transient pyrrhotite phase, which has only a small temperature range of stability before
Pyrometallurgy of copper–nickel–iron sulphide ores 0.4
0.6
xNi
Liquid
xS
0.2
193
Liquid + liquid
0.8 Liquid + fcc
Cu
0.8
0.6
0.4
0.2
Ni
Fcc
xCu
II.13.4 Calculated phase diagram for the Cu–Ni–S system at 1573 K.
the emergence of pentlandite. Not obvious in these plots but evident in the copper–nickel phase diagram is a tendency to immiscibility in the fcc phase at low temperatures. This contributes to a further transformation at 750 K, in which the amount of copper combined with sulphur as digenite sharply declines, whereas that of nickel and iron combined with sulphur in pentlandite increases. The calculations relate only to a single composition. Nevertheless they demonstrate that the crystallisation processes in a multiphase matte can be very complex and that calculations based on thermodynamic data can contribute greatly to the selection and optimisation of processing methods.
II.13.5
Thermodynamic models and data
The phases and thermodynamic models are given in Table II.13.3. The data for the sulphide phases were derived exclusively from the assessments of Dinsdale and co-workers [82Din, 84Din] and Fernandez Guillermet et al. [81Fer]. The metal–matte liquid is described by a two-sublattice model with variable site fractions similar to that given by Hillert et al. [85Hil]. Both binary and ternary experimental data were taken into consideration in the assessment. Most of the several crystalline phases are modelled as solid solutions. The data for the slag phase used for this work are based with some modification on the work of Gaye and Welfringer [84Gay] and Taylor and Dinsdale [90Tay2], who used the cellular model of Kapoor and Frohberg [71Kap]. Other models used elsewhere for slags include the sublattice model, the quasichemical model of Pelton and Blander [86Pel] and the associated solution model [93Bar]. Data for the gaseous species were obtained from the SGTE Pure Substance Database [85Bar, 87Ans]. All the data were converted to the G-Hser formalism, in which the Gibbs energy data for all substances are referred to the properties of the pure elements in their stable states at 298.15 K and 101 325 Pa, rather than to the elements in a specified phase at current temperature.
194
The SGTE casebook
Table II.13.2 Edited results showing the composition of the slag and liquid phases for a single set of conditions (temperature, 1573.00 K; fixed gas pressure, 1.013 250 × 105 Pa; calculated gas volume, 7.172 848 × 10–1 m3). Ferric iron in the slag is identified as Fe2/3O Number of phases
Number of species
Amount (mol)
Mole fraction
Liquid phases Ni(l) Fe(l) Cu(l) S(l) Phase total
4 4 4 4
1 2 3 4
0.19876 0.00095 0.61835 0.13979 0.95786
0.20751 0.00100 0.64556 0.14594 1.00000
Liquid phases Ni(l) Fe(l) Cu(l) S(l) Phase total
4 4 4 4
1 2 3 4
0.10124 0.00323 1.08153 0.60326 1.78926
0.05658 0.00181 0.60446 0.33719 1.00000
Slag phases Si0.5O Fe2/3O CaO FeO Si0.5O.Fe2/3O Si0.5O.CaO Si0.5O.FeO Fe2/3O.CaO Fe2/3O.FeO CaO.FeO Phase total
11 11 11 11 11 11 11 11 11 11
1 2 3 4 5 6 7 8 9 10
1.44802 0.04811 0.00017 1.90744 0.20923 1.10093 3.84182 0.02883 0.31542 0.05007 8.95004
0.19179 0.00538 0.00002 0.21312 0.02338 0.12301 0.42925 0.00322 0.03524 0.00559 1.00000
Gas phases Cu(g) N2(g) SO(g) SO2(g) S2(g) Phase total
12 12 12 12 12
1 4 8 10 12
0.00012 4.99999 0.00114 0.55571 0.00003 5.55703
0.00002 0.89976 0.00021 0.10000 0.00001 1.00000
Component
Chemical potential
Activity
Amount (mol)
Ni Fe Cu S O Si Ca N
–1.023565E+05 –1.566855E+05 –9.278681E+04 –2.181046E+05 –2.814964E+05 –5.128231E+05 –5.594041E+05 –1.719695E+05
3.991462E–04 6.267199E–06 8.296679E–04 5.722415E–08 4.493439E–10 9.356250E–18 2.656610E–19 1.993019E–06
3.000000E–01 6.520000E+00 1.700000E+00 1.300000E+00 1.560896E+01 3.300000E+00 1.180000E+00 1.000000E+01
Pyrometallurgy of copper–nickel–iron sulphide ores
195
0.0 Liquid –0.2
log10 massphase (kg)
Digenite –0.4
β
–0.6 –0.8 Pentlandite –1.0
Fcc
–1.2 Pyrrhotite
–1.4 –1.6 700
800
900 1000 Temperature (K) (a)
1100
0.0 Cu
Ni
log10 [wspecies (kg)]
–0.5
–1.0
Fe Ni Cu
–1.5
–2.0 Fe –2.5
–3.0 700
800
900 1000 Temperature (K) (b)
1100
II.13.5 Phase equilibria as a function of temperature in matte A depleted in sulphur and iron (the composition is 0.325 Ni, 0.1 Fe, 0.352 Cu and 0.223 S parts by weight): (a) mass of phases (cf. Table II.13.3); (b) composition of the fcc phase.
II.13.6
Acknowledgements
The authors gratefully acknowledge the help of T. G. Chart for his advice and help in the critical assessment of data and R. H. Davies and J. A. Gisby
196
The SGTE casebook
0.0
log10 (fraction of Ni in system)
β –0.5 Fcc Liquid
–1.0 Pentlandite
–1.5
–2.0
Pyrrhotite
700
800
900 1000 Temperature (K) (a)
1100
0.0
log10 (fraction of Fe in system)
Pentlandite β –0.5
Liquid Pyrrhotite
Fcc –1.0
–1.5
–2.0
–2.5 700
800
900 1000 Temperature (K) (b)
1100
II.13.6. Distribution of the components of the matte ((a) Ni; (b) Fe; (c) Cu; (d) S) between the phases as listed in Table II.13.3. The total mass of each phase is given in Fig. II.13.5(a).
for assistance with the calculation of the phase equilibria. This work was sponsored in part by Matthey Rustenberg Refiners. The calculations for this case study have been performed using MTDATA.
Pyrometallurgy of copper–nickel–iron sulphide ores 0.0
log10 (fraction of Cu in system)
Digenite
Liquid
–0.5 –1.0 Fcc –1.5 –2.0 –2.5 –3.0 β
–3.5 700
800
900 1000 Temperature (K) (c)
1100
0.2
log10 (fraction of S in system)
0.0
Liquid
–0.2
β
–0.4
Digenite
–0.6 –0.8
Pentlandite
–1.0 Pyrrhotite
–1.2 –1.4 –1.6 700
800
II.13.6 (Continued)
900 1000 Temperature (K) (d)
1100
197
198
The SGTE casebook Table II.13.3 Phases and thermodynamic models
Gas Slag Liquid Fcc Bcc β(‘Ni3S2’) Pyrrhotite Pentlandite Digenite Heazlewoodite Millerite Ni7S6
II.13.7 71Kap
78Nag
79Cha
81Fer
82Din
84Din 84Gay
85Bar
85Hil
86Pel
Phases
Thermodynamic model
Ni–Fe–Cu–O–S–N CaO–FeO–Fe2O3–SiO2 Ni–Fe–Cu–S Ni–Fe–Cu Ni–Fe–Cu Ni–Fe–Cu–S (Ni, Fe)S (Ni, Fe)S0.889 Cu2S Ni3S2 NiS Ni7S6
Ideal gas Cellular model Variable sublattice Redlich–Kister Redlich–Kister Redlich–Kister Redlich–Kister Redlich–Kister Stoichiometric Stoichiometric Stoichiometric Stoichiometric
References M.L. KAPOOR and G.M. FROHBERG: ‘Theoretical treatment of activities in silicate melts’, Proc. Symp. Chemical Metallurgy of Iron and Steel, Sheffield, UK, 1971, Iron and Steel Institute, London, pp. 17–22. N.A. NAGAMORI and P.J. MACKEY: ‘Thermodynamics of copper matte converting, Part 1: fundamentals of the Noranda process’, Metall. Trans. B 9, 1978, 255– 265. Y.A. C HANG , J.P. N EUMANN and U.V. C HOUDRAY : ‘Phase diagrams and thermodynamic properties of ternary copper–sulphur–metal systems’, in The Metallurgy of Copper, INCRA Monograph VII, INCRA, New York, 1979. A. FERNANDEZ GUILLERMET, M. HILLERT, B. JANSSON and B. SUNDMAN: ‘An assessment of the Fe–S system using a two-sublattice model for the liquid phase’, Metall. Trans. B 12, 1981, 745–754. A.T. DINSDALE, T.G. CHART, T.I. BARRY and J.R. TAYLOR: ‘Phase equilibria and thermodynamic data for the Cu–S system’, High Temp. – High Pressures 14, 1982, 633–640. A.T. DINSDALE: ‘The generation and application of thermodynamic data’, Thesis, Brunei University, 1984. H. GAYE and J. WELFRINGER: ‘Modelling of the thermodynamic properties of complex metallurgical slags’, in Proc. 2nd Int. Symp. Metal Slags and Fluxes (Eds H.A. Fine and D.R. Gaskell), Lake Tahoe, California, USA 1984, Metallurgical Society of AIME, New York, 1984, pp. 357–375. T.I. BARRY: ‘High temperature inorganic chemistry and metallurgy’, in Chemical Thermodynamics in Industry: Models and Computation (Ed. T.I. Barry, Blackwell Scientific, Oxford, 1985, Chapter 1, pp. 1–39. M. HILLERT, B. JANSSON, B. SUNDMAN and J. ÅGREN: ‘A two-sublattice model for molten solutions with different tendency to ionization’, Metall Trans. A 19, 1985, 261–266. A.D. PELTON and M. BLANDER: ‘Thermodynamic analysis of ordered liquid solutions by a modified quasichemical approach – application to silicate slags’, Metall Trans. B 17, 1986, 805–815.
Pyrometallurgy of copper–nickel–iron sulphide ores 87Ans
199
I. ANSARA and B. SUNDMAN: ‘The Scientific Group Thermodata Europe’, in Computer Handling and Dissemination of Data (Ed. P.S. Glaeser), Elsevier, Amsterdam, 1987, pp. 154–158. 88Dinl A.T. DINSDALE, S.M. HODSON and J.R. TAYLOR: ‘Application of MTDATA to the modelling of slag, matte, metal, gas phase equilibria’, Proc. 3rd Int. Conf. Molten Slags and Fluxes, Strathclyde, UK, 1988, Institute of Metals, London, 1988. 88Din2 A.T. DINSDALE, S.M. HODSON, T.I. BARRY and J.R. TAYLOR: ‘Computations using MTDATA of metal-matte-slag-gas equilibria’, Proc. 27th Annual Conf. of Metallurgists, Canadian Institute of Mining, Metallurgy and Petroleum, Montreal, 1988. 90Tayl J.R. TAYLOR and A.T. DINSDALE: ‘Application of the calculation of phase equilibria to the pyrometallurgical extraction from sulphide ores’, Proc. Conf. User Aspects of Phase Diagrams, Petten, The Netherlands, 25–27 June 1990, Institute of Metals, London, 1990. 90Tay2 J.R. TAYLOR and A.T. DINSDALE: ‘Thermodynamic and phase diagram data for the system CaO–SiO2 system’, Calphad 14(1), 1990, 71–88. 93Bar T.I. BARRY, A.T. DINSDALE and J.A. GISBY: ‘Predictive thermochemistry and phase equilibria of slags’, JOM, 45(4), 1993, 32–38.
II.14 High-temperature corrosion of SiC in hydrogen–oxygen environments K L A U S G. N I C K E L, H A N S L. L U K A S and GÜNTER PETZOW
II.14.1
Introduction
SiC is used commercially for high-temperature applications such as heating elements because of its resistance to oxidation corrosion to at least 1600 °C. It is also one of the very few materials that have the capability to retain strength at high temperatures. Thus it is a prime candidate for currently developed applications such as heat engine linings or heat exchangers. In such applications, gas temperatures may exceed 3000 °C and imply extremely fast gas flow (e.g. in turbines of super-fast planes). The prevailing atmosphere in such engines depends on the fuel used. Hydrogen–oxygen is known as a very efficient fuel system in rockets. Heat exchangers may also be exposed to various atmosphere compositions including steam at high temperatures. The resistivity of SiC against hydrogen–oxygen corrosion is thus a limiting factor for the applicability of this material. We have simulated three basic types of gaseous corrosion situation by computing models for static, moderate and extremely high flow rates of the atmosphere. The importance of the analysis of the application environment is demonstrated. The validity of the use of simple models is evaluated by comparing the predicted damage with those of experimental data. It is obvious from the modelling procedure that the prediction of the corrosion kinetics is based on thermodynamic analysis and in particular on the calculation of partial pressure of gas species. Hence we present the main points of a thermodynamic analysis of parts of the systems Si–C–H and Si– C–O–H, which are of interest for the problem of steam, ‘dry’ and ‘wet’ hydrogen corrosion of SiC. The partial pressures over SiC + C are a model system for pressureless sintered SiC (‘SSiC’), which usually contains some free carbon. Other types of commercially free available SiC (‘SiSiC’) contain free silicon. Hence partial pressures over Si + SiC are relevant for this type of material. Thermodynamically, equilibria which are not invariant at constant pressure and temperature can be calculated but are of little general significance because 200
High-temperature corrosion of SiC in hydrogen–oxygen
201
of their compositional dependence. However, some bulk compositions are of technical interest, because they refer to an existing material. Such a case is pure SiC, which may be purchased as SiC from chemical vapour deposition production or will be produced in a dynamic process such as the removal of carbon from SSiC. Hence we have included some calculations for this type of material.
II.14.2
Models for the corrosion process
Corrosion processes, which are controlled by the emission of gaseous particles from a material, are strongly dependent on the boundary conditions of the system. We may distinguish three basic types of corrosion situation: static, flowing and extremely fast flowing atmosphere: n = PV RT
(II.14.1)
A static atmosphere with a defined available gas volume has a defined maximum loss of material to the atmosphere, because at equilibrium the equation of state for gases defines the number of moles of the individual species that may enter the available gas chamber volume. Because this type of equilibrium is rarely overstepped, the value is a maximum, which will, however, be in good agreement with reality at high temperatures. We may thus calculate the amount of any element in the atmosphere from the thermodynamic analysis of this relation by adding the contribution of each gas species. From here we may calculate further the amount of elements from SiC now present in gas form. While this procedure may be performed by a computer, in practice this is hardly necessary, because the gas phase is usually strongly dominated by a single species at a given temperature; the sum of all minor species will thus only add up to a minor or negligible fraction of the total amount of species containing elements of the material to be corroded. From the dominating gas species we can immediately deduce the main reaction. If, for example the only condensed species is SiC and the main gas species is monatomic Si, it follows that SiC → SiT + C is the dominating reaction. A list of important reactions in the Si–C–H(–O) systems is given in Table II.14.1. Knowing the dominating reaction we have also defined the amount ∆m of SiC destroyed, because the reaction equation relates the moles of evaporating and destroyed species: ∆m = M
n gas u
(II.14.2)
where M is the molecular weight of SiC and u the factor relating the moles
202
The SGTE casebook Table II.14.1 Reaction equations Si ↔ Si ↑
(R1)
1 H + Si ↔ SiH ↑ 2 2 2H2 + Si ↔ SiH4 ↑
(R2)
1 H + 2C ↔ C H ↑ 2 2 H2 + 2C ↔ C2H2 2H2 + C ↔ CH4 ↑ SiC ↔ Si ↑ + C 2SiC ↔ SiC2 ↑ + Si 2SiC ↔ Si2C ↑ + C 3SiC ↔ SiC2 ↑ + Si2C
(R4)
1 H + SiC ↔ SiH ↑ + C 2 2 H2 + 2SiC ↔ C2H2 ↑ +2Si H2 + 2SiC ↔ C2H2 ↑ + 2Si ↑ 2H2 + SiC ↔ CH4 ↑ + Si 2H2 + SiC ↔ SiH4 ↑ + C H2O + Si ↔ SiO ↑ + H2 H2O + C ↔ CO ↑ +H2 2H2O + Si ↔ SiO2 ↑ + 2H2 2H2O + C ↔ CO2 ↑ + 2H2 2H2O + SiC ↔ SiO ↑ + CO ↑ +2H2 H2O + H2 + SiC ↔ CH4 ↑ + SiO↑
(R11)
(R3)
(R5) (R6) (R7) (R8) (R9) (R10)
(R12) (R13) (R14) (R15) (R16) (R17) (R18) (R19) (R20) (R21)
of evaporating species and destroyed SiC in the reaction equation. We thus have ∆m =
MPi V uRT
(II.14.3)
or, if we know the exposed surface a, the corrosion depth via the density relation
Xl =
MPi V a ρuRT
(II.14.4)
The corrosion depth in Equation (II.14.4) is mainly dependent on V and Pz and cannot be large unless the partial pressure or the volume of the gas chamber is huge. The introduction of a factor u implies that the corrosion depth is defined as the depth to which the material is destroyed and not by the width of shape change, because the corrosion may leave a (non-protective) layer of residue, e.g. of carbon, if the main reaction is Reaction (R7) in Table II.14.1. A corrosion rate (corrosion depth per time unit) follows from Equation (II.14.4) directly, if we assume a regular exchange of the atmosphere (i.e. a regular removal of the species containing Si or C) with time. We obtain the
High-temperature corrosion of SiC in hydrogen–oxygen
203
total amount of corrosion by multiplying by the exchange rate v (the number of exchanges per time unit): MPi V X˙ l = v a ρuRT
Since the flow rate of an atmosphere is defined by V˙ = V v
(II.14.5)
(II.14.6)
we also have
MPi V˙ X˙ l = a ρuRT
(II.14.7)
or, in terms of a mass flux J, MPi V˙ J = X˙ l v = a ρuRT
(II.14.8)
We have formulated the mass flux from equilibrium assumptions. Hence the calculation will give precise values if equilibrium is obtained and kinetic hindrances are negligible. If these conditions are not met, we need to determine experimentally an effective flow rate, which allows us to keep the form of Equations (II.14.7) and (II.14.8) and to give good corrosion predictions. At extremely high gas flow rates the physics of the corrosion process is poorly known and the assumption of obtained equilibrium is certainly not valid. As the most extreme case of gaseous decomposition we consider the removal of any gas species particle as soon as it is formed without being slowed down by any physical or chemical interaction. In this case the speed of removal is solely controlled by the speed of the gas particles, which is known from v=
8 kT πm
1/2
(II.14.9)
Since the molecular flux of gas is
Φ= 1N v 4V
(II.14.10)
and Equation (II.14.1) is still valid, we may calculate the maximum possible flux as J=
Pi (2 πRTM )1/2
(II.14.11)
It should be emphasised that this is a worst-case calculation, which will only be approached in extreme conditions such as ultrahigh vacuum. We wanted
204
The SGTE casebook
to evaluate conditions of extreme cases, which are not easily accessible to experimental testing.
II.14.3
Thermodynamic analysis
Thermodynamic analysis was performed using the programs PMLFKT [82Luk] and SOLGASMIX [75Eri]. Using the SGTE data set [87Ans] as input parameters we considered the species listed in Table II.14.2. In the corrosion models we refer to reaction equations. The relevant reaction equation is defined (see above) by those gas species that have the highest partial pressures. Hence in a plot of partial pressures of gas species versus temperature we need to show only these species. The resulting plot is thus a ‘maximum partial pressure surface’ (MPPS).
II.14.4
Si–C–H system
The equilibria Si–C gas and Si–SiC gas are invariant at constant temperature and pressure. Therefore they can be calculated at constant total pressure for each temperature. All calculations in this work are for a total pressure of 1 bar. Figure II.14.1(a) and Fig. II.14.1(b) show the MPPSs for Si–SiC gas and SiC–C gas equilibria respectively in the Si–C–H system. All stable twophase equilibria between SiC and gas lie between these three-phase equilibria. In the SiC–C gas equilibria up to about 2600 °C, there is more C than Si in the gas phase (Fig. II.14.1(a)). In the Si–SiC gas equilibria below 1620 °C, C also dominates over Si in the gas phase (Fig. II.14.1(b)). Any corrosion of SiC by dry H2 below 1620 °C therefore is expected to remove free carbon or to generate free Si. There are T–X regions that have SiC as the only condensed phase present. In this two-phase field SiC + gas, a bulk composition has to be defined besides temperature and pressure. An example is shown in Fig. II.14.1(c) for the bulk composition SiC:H2 = 1:1. At temperatures above 1620 °C, C- and Si-bearing species have to balance each other and corrosion here leads to direct removal of SiC. Table II.14.2 Species considered in the thermodynamic calculations Gas species
C, C+, C–, C –2 , C3, C4, C5, H, H–, H+, H2, H +2 , H –2 , Si, Si+, Si2, Si3, C2H, CH, CH+, C2H2, C2H4, C2H6, C3H8, CH3, CH4, SiC, SiC2, Si2C, SiH, SiH+, SiH4, Si2H6, O, O+, O–, O2, O +2 , O –2 , O3, C2O, CO, CO2, CO –2 , C3O2, OH, OH+, OH–, HO2, H2O, H3O+, HCO, HCO+, H2CO, H4CO, SiO, SiO2
Condensed species
C(graphite), Si, Si(l), α-SiC, β-Sic, SiO2(quartz), SiO2(tridymite), SiO2(cristobalite), SiO2(l)
High-temperature corrosion of SiC in hydrogen–oxygen
205
log [partial pressures (bar)]
0 –1 –2
CH4
C2H2
–3
Si
–4 –5 –6
SiH
–7 –8 500
SiH4 1000
1500 2000 2500 Temperature (°C) (a)
3000
log [partial pressures (bar)]
0 –1
Si
–2 –3 –4
C2H2 CH4
–5 –6
SiH SiH4
–7 –8 500
1000
1500 2000 2500 Temperature (°C) (b)
3000
log [partial pressures (bar)]
0 –1
Condensed phases
–2
Si + SiC
SiC
Si2C + SiC2 Si
–3 –4
C2H2
CH4
–5 SiH –6 –7 –8 500
SiH4
1000
1500 2000 2500 Temperature (°C) (c)
3000
II.14.1 MPPSs in the Si–C–H system (a) for SiC–C gas equilibrium, (b) for Si–SiC gas equilibrium and (c) for a bulk composition SiC:H2 = l:l.
206
II.14.5
The SGTE casebook
Si–C–O–H system
In the problem of steam or ‘wet’ hydrogen corrosion, one of the modifications of SiO2 (e.g. tridymite, cristobalite or an ionic melt) may appear as an additional condensed phase. The possible equilibria of interest, which are invariant under isothermal isobaric conditions, are SiC–SiO2–C gas and SiC– SiO2–Si gas. The invariant equilibria involving SiO2 cannot be used for the simulation of a corrosion with the models outlined above because they imply the formation of a SiO2 layer, which may be protective (‘passivation’). Outside the four-phase equilibria we have to specify the bulk compositions to analyse the situation thermodynamically. Because all commercial SiC materials have bulk compositions close to pure SiC we have calculated the MPPS for pure SiC with H2 H2O ratios ranging from pure steam to l ppm H2O in H2. Examples of this type of calculation are shown in Fig. II.14.2. From Fig. II.14.2 it is obvious that a temperature exists, above which SiO2 becomes unstable. This temperature may be regarded as the theoretical active– passive oxidation boundary. In three-phase equilibria in the Si–C–O–H system the transition temperature is dependent on composition. For pure SiC this transition temperature is plotted as a function of the initial H2:H2O ratio of the corroding gas in Fig. II.14.3.
II.14.6
Discussion
To evaluate the corrosion of SiC by hydrogen we have computed a model case, where the gas flow has a value of 1 m3 min–1 through a SiC tube of 200 mm inner diameter and 320 mm length, corresponding to an exposed surface of approximately 2000 cm2 and a gas velocity of –2 km h–1. Figure II.14.4 shows the predicted losses from Equation (14.7) for SiC in Ar, pure H2 and various H2–H2O mixtures. SiC in Ar would suffer losses of the order of millimetres per annum only at temperatures above 1700 °C, and pure Si in H2 already at about 1500 °C. The peculiar shape of the loss– temperature curve for SiC in H2 is a direct reflection of the appropriate MPPS (Fig. II.14.1), which is dominated by methane (CH4) up to temperatures of 1500–1600 °C. CH4 has a low enthalpy of formation and is thus less stable at higher temperatures. Therefore the partial pressures of CH4 over C and SiC and the predicted losses decrease with increasing temperature, as long as CH4 is the dominating and hence rate-controlling gas species. Predicted losses within the model of moderate gas flow in H2–H2O environments (Fig. II.14.4) indicate that above the passive–active transition the presence of H2O at more than l ppm levels increases the corrosion rate strongly. Figure II.14.5 shows the calculated influence of gas speed within this
High-temperature corrosion of SiC in hydrogen–oxygen
207
log [partial pressures (bar)]
1
SiO2 SiC SiC, Si SiC
SiO2, SiC, C
0
CH4
–1
Si
euou
2
Si
CO
–2 –3 –4 –5
SiO
–6 –7 –8 500
1000
1500 2000 Temperature (°C) (a)
2500
3000
log [partial pressures (bar)]
0 –1 –2
SiO2 SiO2 SiC SiC Si
Si
–3 –4
Si + C
SiC
CH4
C2H2
CO
–5 –6
SiO
–7 –8 500
1000
1500 2000 Temperature (°C) (b)
2500
3000
II.14.2 MPPSs in the Si–C–O–H system for bulk compositions: (a) SiC:H2O=l:l; (b) SiC:H2:H2O=l:l:10–4. The stable condensed phases are indicated at the top.
model on the corrosion rates; increasing the velocity from 2 km h–1 to sonic speed means a reduction in tolerable temperatures by several hundred kelvins. The calculation of the ‘worst case’ gives a safe temperature limit nearly 1000 °C lower. Figure II.14.2 illustrates that already quite small water additions to a system may cause the formation of an SiO2 layer with protective potential. Furthermore it is known that SiC reacts already at room temperature with oxygen–water, yielding a thin SiO2 layer. Heating of SiC in an atmosphere containing any amount of oxygen will thus almost inevitably lead to the formation of some SiO2. If the temperature chosen is above the active– passive transition, it depends on the kinetics of the SiO2 removal (e.g. SiO2
208
The SGTE casebook
1800
Temperature (°C)
1600 Active 1400 1200
Passive
1000 800 600 100 104 H2 : H2O ratio (mol)
1
106
II.14.3. Theoretical boundary between active and passive oxidation of pure SiC defined by the instability of SiO2 in the phase assembled.
Temperature (°C)
2500
200 mm
3000
1 m3 min–1
Si–C–H–O
Gas flow 2 km h–1 320 mm
2000 1 1500
0
Ar 104 H2:H2O = 106
1000 H2 500 –3
–2
–1
0 1 2 log [loss (mm a–1)]
Passivation 3
4
5
II.14.4 Predicted losses of pure SiC in different environments for the model situation described by the inset drawing.
→ SiO↑+ 12 O2 and SiO2 + H2 → SiO↑ + H2O↑) if and when corrosion occurs. The kinetic formulations used in this paper are under the assumption of complete SiO2 removal. There are other limitations to the methods described: the kinetics of gas removal from a surface at low temperature are complex; the driving forces
High-temperature corrosion of SiC in hydrogen–oxygen
Temperature (°C)
2500
200 mm
3000
209
Gas flow
2 km m–1
320 mm
Sonic speed
2000
Maximum loss
1500
1000 mm a–1 500 –1
0
mm h–1 1
2 3 4 log [loss (mm a–1)]
5
6
7
II.14.5 Dependence of the corrosion rate on the velocity using the models for moderate (for 2 km h–1 and sonic speed) and extremely fast flow.
are strongly influenced by absorption or absorption phenomena. Hence we do not expect Equation (II.14.7) to give good predictions at low temperatures. The transition temperatures, above which theoretical calculation and reality are in good agreement, have to be determined empirically. Hallum and Herbell [88Hal] have presented data for the behaviour of SiC under ‘wet’ hydrogen (H2 with 25 ppm H2O). They reported grain boundary attack at temperatures below 1300 °C followed by simultaneous SiC and grain boundary attack to 1600 °C. The investigated material was a commercial SSiC material and contained randomly distributed free carbon (probably both in pockets and in grain boundaries). From their boundary conditions (475 cm3 min–1 and 4.86 cm2 exposed geometrical surface) we have calculated the losses from the MPPS of the bulk composition. Hallum and Herbell [88Hal] gave estimations of partial pressures using SOLGASMIX [75Eri] type of calculations. However, they did not take into account the free carbon present, which influences the equilibrium values profoundly (compare Fig. II.14.1(a), Fig. II.14(b) and Fig. II.14(c)). We do not know the exact amount of free C in their samples and estimated it to be 0.3 wt%. This crude estimate gives for a random distribution of C an exposed surface of approximately 0.02 cm2. The form of Equations (II.14.7) and (II.14.8) show that a higher corrosion depth X or a higher flux J is needed for a small exposed area to maintain the equilibrium pressure Pi. In the MPPSs for C-bearing SiC, quite high partial pressures for CH4 are indicated, which would effectively remove free C from pockets. However, it is known that the low-temperature kinetics of CH4 formation from C involve several steps including a transposition of CH2
210
The SGTE casebook 1600 C(CO)
Temperature (°C)
1500
C(CH4)
1400 [88 Hal]
1300 1200 1100 SiC
CH4 + SiO
1000 900 0
20
40 60 80 Loss after 50 h (mg)
100
120
II.14.6 Comparison of losses predicted from Equation (II.14.7) for Reactions (R6), (R17) and (R21) of Table II.14.1 and the data of Hallum and Herbell [88Hal].
molecules, which may be slow up to temperatures of approximately 1400 °C [68Gme]. The removal of C via CO is a much more effective process in the temperature range considered in the experimental study. Because the two phases of the material are very different in their exposed surface, we cannot use the bulk flux quoted by Hallum and Herbell [88Hal] for an analysis. We have rather calculated the total loss after 50 h from their data and the MPPS (Fig. II.14.6). The corrosion depth for C removal is large; hence grain boundary corrosion is predicted. We interpret their data not as an exponential increase in loss with increasing temperature but as a three-step feature: below 1100 °C, very little corrosion takes place, which is compatible with the removal of C as CO; at T > 1100 °C, SiC corrosion starts via reaction (R21) in Table II.14.1; At T > 1300 °C, additional CH4 production due to increased kinetics is favoured. This interpretation is consistent with the observed corrosion style. Thus the calculations presented in this work may be used for the following. – – – –
To assess corrosion processes chemically (main reaction determination). To allow extrapolation of corrosion kinetics under steady-state or moderate flow conditions. To estimate safe temperature limits for extreme conditions. To predict corrosion type (active or passive) and to predict corrosion style (grain boundary or bulk).
High-temperature corrosion of SiC in hydrogen–oxygen
II.14.7 68Gme 75Eri 82Luk 87Ans 88Hal
211
References Gmelins Handbuch de-anorganischen Chemie. Kohlenstoff, 8th edition B3, Verlag Chemie, Weinheim, 1968, p. 795. G. ERIKSSON: ‘Thermodynamic studies of high-temperature equilibria’, Chem. Scripta 8, 1975, 100–103. H.L. LUKAS, J. WEISS and E.T. HENIG: ‘Strategies for the calculation of phase diagrams’, Calphad 6, 1982, 229–251. I. ANSARA and B. SUNDMAN: ‘Computer handling and dissemination of data, CODATA Report, 1987, pp. 154–158. G.W. HALLUM and T.P. HERBELL: ‘Effect of high-temperature hydrogen exposure on sintered α-SiC, Advd. Ceram. Mater. 3, 1988, 171–75.
II.15 The carbon potential during the heat treatment of steel T O R S T E N H O L M and J O H N Å G R E N
II.15.1
Introduction
The properties of all steels depend strongly on their carbon content. In metals, carbon diffuses interstitially and is quite mobile even at comparatively low temperatures. The carbon exchange between the furnace atmosphere and the steel will thus have a large impact on the properties of the steel. For example, in case hardening, one applies an active atmosphere, i.e. carbon is transferred from the atmosphere of the steel surface, resulting in carburising. When heat treating tool steels, one rather wants an inactive or inert atmosphere, i.e. there should be no transfer between the atmosphere and the steel. In other cases, one wants the steel to be decarburised and the atmosphere must then pick up carbon from the steel surface. The transfer of carbon between the atmosphere and the steel surface depends on two factors. The first factor, the driving force for the transfer, is the difference between the carbon activities of the atmosphere and the surface. The second factor is the kinetics of the surface reactions. An inert atmosphere can thus be achieved in two different ways. Either way can make sure that the driving force for carbon transfer is sufficiently small or can inhibit the surface reactions. From the practical point of view it is thus important to be able to predict and control the carbon activity of the furnace atmosphere as well as to know the carbon activity of a particular steel.
II.15.2
The carbon potential
The carbon activity of the atmosphere or the steel can be predicted by equilibrium calculations. In practical heat treatment it is common to apply the so-called carbon potential rather than the carbon activity. It is defined as the carbon content, expressed in weight per cent, that an initially pure iron specimen would have if carbon is equilibrated between the atmosphere or the alloy under consideration. 212
The carbon potential during the heat treatment of steel
213
The relation between carbon activity and carbon potential is easily calculated from the thermodynamic description of the binary Fe–C system [85Gus]. A series of such calculations have been performed on the Thermo-Calc system [85Sun] and the result is summarised in Fig. II.15.1. First the relation between the carbon activity and the carbon content of γ-iron has been calculated and plotted, one curve for each temperature. Thereafter the Fe–C phase diagram has been calculated and plotted. However, rather than using the temperature as one axis variable the carbon activity has been chosen. A carbon activity of 1 corresponds to equilibrium with graphite. This freedom in the choice of axis variables is built into the POLY post-processor in Thermo-Calc.
II.15.3
The carbon activity in industrial furnace atmospheres
A typical furnace atmosphere for carburising consists of a mixture of CO and N2. It has been quite common to produce this type of atmosphere by incomplete combustion of some fuel, for example propane (C3H8), in a so-called generator and then to lead the products into the furnace. Such a reaction is endothermic and the atmosphere produced by this method is often called endogas. It may now be of practical interest to see how the conditions of the generator affects the constitution and more specifically the carbon potential of the resulting gas. A series of equilibrium calculations was thus performed using the ThermoCalc program and the SGTE Pure Substance Database [87Ans]. The calculations were made as follows. 1.5
°C
γ
90 0
Carbon potential (wt% C)
1.2
γ + Fe3C
120 0° C 11 00 °C 10 00 °C
γ + liquid
0
0.9
80
°C
0.6 α + Fe3C
0.3 α+γ 0 0
0.5 1.0 Carbon activity
1.5
II.15.1 Relation between the carbon potential and carbon activity for various temperatures.
214
The SGTE casebook
First a temperature of 930 °C and a pressure of 1 atm (0.101 325 MPa) were fixed. The overall composition given as the numbers of moles of the different elements was subsequently fixed. For a mixture of 1 mol of propane and x mol of air we can thus write the scheme of overall contents (assuming that the volume fraction of oxygen in air is 0.209) shown in Table II.15.1. The above overall contents can be directly imposed as conditions in the equilibrium calculation program and with the temperature and pressure fixed we shall have a well-posed problem with a unique solution. For example, if we consider the mixture of air and propane that yields equal amounts of C and O, we shall have x=
3 = 7.18 2.0.209
For this particular case, Table II.15.1 yields the following overall contents of the elements to be fixed as conditions in the Thermo-Calc program: C H N O
3 mol 8 mol 11.358 76 mol 3.001 24 mol
The result of such a calculation is shown in Table II.15.2. The table corresponds to the output data generated by the program when giving the command ‘listequilibria’. The conditions specified by the user are first listed. The overall composition corresponds to 1 mol of propane. The next set of information concerns the temperature, the pressure and the elements. It should be noted that the activities listed here are given relative to the so-called stable element reference (SER), which is the reference state used in the databank, is used. The activity relative to any other reference state is readily obtained in the program by a special command. In this case the activity of carbon relative to graphite is 1.337, i.e. there is a tendency for soot or graphite formation. Table II.15.3 shows the same type of calculation performed with 7.23 mol of air instead of 7.18 mol. In this case the carbon activity is 0.91 and there is no tendency for formation of soot or graphite. Table II.15.1 Content of elements in endogas from C3H8 and air Element
C H N O
Content of elements in the following 1 mol of C3H8
x mol of air (0.209 O2, N2 balancing)
3 8 – –
– – 2 × x (l – 0.209) 2 × x × 0.209
Table II.15.2 Equilibrium in endogas: edited output data from POLY-3 in Thermo-Calc Output from POLY-3, equilibrium number = 1
N(C) = 3
N(H) = 8
N(N) = 11.358 76
N(O) = 3.001 24
Potential
Reference state
Temperature = 1203.00, Pressure = 1.013 250 × 10–5 Number of moles of components = 2.356 00, Mass = 2.512 13 × 10–1 Total Gibbs energy = –3.409 40 × 106, Enthalpy = 1.540 50 × 104, Volume = 1.246 57 Component
Amount (mol)
Fraction
C H N O
3.0000 8.0000 1.1359 × 10 3.0012
1.4344 3.2097 6.3332 1.9114
Gas 1 Number of moles = 2.3560, Driving force = 0.0000 N 6.333 22 × 10–1 O 1.911 45 × 10–1 4.497 3.113 2.347 2.007 1.386 7.575 3.820 2.244
18 04 65 20 72 18 08 01
× × × × × × × ×
10–1 10–1 10–1 10–3 10–3 10–4 10–5 10–5
C 3 H6 C 3 H8 C3 O2 C1 N1 C3H6(1) N1O1 C 2 H1 C 1 H2
10–1 10–2 10–1 10–1
2.2045 6.9937 2.1009 7.8788
× × × ×
10–1 10–5 10–6 10–17
–1.5124 –9.5702 –1.3076 –3.7089
× × × ×
104 104 105 105
SER SER SER SER
Status ENTERED C 1.434 36 × 10–1 2.979 1.665 1.018 2.337 2.258 2.177 1.230 4.582
39 45 17 24 78 81 44 64
× × × × × × × ×
10–12 10–13 10–13 10–14 10–14 10–14 10–14 10–15
H 3.209 71 × 10–2 O1 N2O1 C4 N2 C1 N2 C 1 H1 O2 H 2O2 N3
1.587 9.945 2.450 8.309 8.837 3.577 2.024 2.554
24 84 47 67 21 94 56 13
× × × × × × × ×
10–18 10–19 10–19 10–20 10–21 10–21 10–21 10–22
215
Constitution 6N2 H2 C1O1 C1H4 H2O1 C1 O2 C1H1N1 H 3N1
× × × ×
Activity
The carbon potential during the heat treatment of steel
Conditions T = 1203 P = 101 325 Degrees of freedom = 0
216
C 2 H4 H1 C1H2O1 C 2 H2 C 1 H3 C 1 H 1 N 1O 1 C 2 H6 C 4 H8 C1H1O1 C1H4O1 H 2N1 C2 N2 H 1O1 C3H4(l)
2.263 49 × 10–7 1.157 84 × 10–7 7.668 74 × 10–8 7.195 45 × 10–8 5..692 02 × 10–8 1.258 78 × 10–8 1.214 10 × 10–8 1.253 93 × 10–9 1.744 95 × 10–10 8.155 84 × 10–11 1.327 26 × 10–11 5.185 94 × 10–12 4.334 22 × 10–12 3.946 04 × 10–12
C 4 H4 C 4 H2 C4H6(1) C1N1CC1 H 1N1 H 2N2 C2 O1 C4H8(1) C2H4O1 H 4N2 C2 N1 H 1N 1 O 1 C4H1O(1) N1
3.278 4.390 3.755 2.185 1.859 1.116 8.983 7.034 2.964 5.072 4.567 3.882 3.165 3.151
09 20 23 38 25 94 24 12 57 36 18 15 81 36
× × × × × × × × × × × × × ×
10–15 10–16 10–16 10–16 10–16 10–16 10–17 10–17 10–17 10–18 10–18 10–18 10–18 10–18
C1 H 1O2 H 1N 1 O 2 C3 N1O2 C2 H 1N 1 O 3 C5 C4 N1O3 N2O3 N2O4 O3
1.503 7.025 1.907 1.334 4.608 1.016 1.000 1.000 1.000 1.000 1.000 1.000 1.000
60 05 64 09 48 85 00 00 00 00 00 00 00
× × × × × × × × × × × × ×
10–23 10–24 10–24 10–24 10–26 10–26 10–30 10–30 10–30 10–30 10–30 10–30 10–30
The SGTE casebook
Table II.15.2 (Continued)
Table II.15.3 Equilibrium in endogas: edited output data from POLY-3 in Thermo-Calc Output from POLY-3, equilibrium number = 1
Component
Amount (mol)
Fraction
C H N O
3.0000 8.0000 1.1438 × 10 3.0221
1.4262 3.1914 6.3409 1.9138
10–1 10–2 10–1 10–1
1.5007 6.9854 2.1036 7.1538
× × × ×
10–1 10–5 10–6 10–16
N(O) = 3.02214
Potential
Reference state
–1.8971 –9.5714 –1.3075 –3.6707
× × × ×
104 104 105 105
SER SER SER SER
Status ENTERED C 1.426 17 × 10–1 9.332 6.887 5.204 3.193 1.593 7.075 5.695 3.112 3.428
12 63 27 25 05 00 09 19 85
× × × × × × × × ×
10–13 10–14 10–14 10–14 10–14 10–15 10–15 10–15 10–16
H 3.191 38 × 10–2 N2O1 C4H1O(1) C1 N2 C4 N2 O2 C 1 H1 H 2O2 N3 H 1O2
1.460 6.718 5.671 5.275 7.672 6.008 4.331 2.563 1.504
17 36 03 41 81 68 37 87 73
× × × × × × × × ×
10–18 10–19 10–20 10–20 10–21 10–21 10–21 10–22 10–23
217
Gas 1 Number of moles = 25.460, Driving force = 0.0000 N 6.340 92 × 10–1 O 1.913 77 × 10–1 Constitution N2 4.508 60 × 10–1 C 3 H6 H2 3.105 70 × 10–1 C3 O2 C1 O1 2.340 30 × 10–1 C 3 H8 H 2O1 2.025 93 × 10–3 N1O1 C 1 H4 1.359 93 × 10–3 C1 N1 C1 O2 1.105 84 × 10–3 C3H6(1) C1H1N1 2.600 68 × 10–5 C 2 H1 H 3N1 2.238 92 × 10–5 C 1 H2 H1 1.156 47 × 10–7 C 4 H4
× × × ×
Activity
N(N) = 11.43186
The carbon potential during the heat treatment of steel
Conditions T = 1203 P = 101 325 N(C) = 3 N(H) = 8 Degrees of freedom = 0 Temperature = 1203.00, Pressure = 1.013 250 × 10–5 Number of moles of components = 25.46 00, Mass = 2.526 55 × 10–1 Total Gibbs energy = –3.427 45 × 106, Enthalpy = 1.332 32 × 104, Volume = 1.252 09
218
C 2 H4 C1H2O1 C 2 H2 C1H1NO1 C 1 H3 C 2 H6 C 4 H8 C1H1O2 C1H4O1 H 2N1 H 1O1 C2 N2 C3H4(l)
1.043 7.626 3.326 1.254 1.147 5.586 2.667 1.737 8.091 1.325 6.339 2.409 1.238
95 68 48 95 74 37 32 43 98 81 56 25 91
× × × × × × × × × × × × ×
10–7 10–8 10–8 10–8 10–8 10–9 10–10 10–10 10–11 10–11 10–12 10–12 10–12
C 1 N1 O 1 H 1N1 H 2N2 C 4 H2 C4H6(1) C2 O1 C4H8(1) C2H4O1 H 1N 1 O 1 H 4N2 N1 O1 C2 N1
2.181 1.859 1.117 9.405 8.006 6.096 4.637 2.002 5.685 5.061 3.155 2.324 2.119
30 42 13 12 93 02 03 28 53 28 36 36 10
× × × × × × × × × × × × ×
10–16 10–16 10–16 10–17 10–I7 10–17 10–17 10–17 10–18 10–18 10–18 10–18 10–18
C1 H 1N 1 O 2 C3 N1O2 C2 C5 H 1N 1 O 3 C4 N1O3 N2O3 N2O4 ON3
1.023 4.091 4.208 9.895 4.712 1.000 1.000 1.000 1.000 1.000 1.000 1.000
55 25 38 31 07 00 00 00 00 00 00 00
× × × × × × × × × × × ×
10–23 10–24 10–25 10–26 10–27 10–30 10–30 10–30 10–30 10–30 10–30 10–30
The SGTE casebook
Table II.15.3 (Continued)
The carbon potential during the heat treatment of steel
219
The above procedure can be extended in order to take into account the influence of various factors on the constitution of a furnace atmosphere. For example, if we want to study the so-called synthetic atmospheres obtained from mixtures of cracked methanol CH3OH and N2 we can apply the same scheme as for the endogas and obtain for a mixture of cracked methanol and N2 the overall contents shown in Table II.15.4. This atmosphere will thus contain 100x/(3 + x) wt% N2. The conditions are given in the same way as in the previous type of calculations. The results of a series of calculations are given in Fig. II.15.2.
II.15.4
The carbon activity of multicomponent steels
As mentioned, the driving force for carbon transfer between a steel and a furnace atmosphere is given by the difference between their carbon activities. Table II.15.4 Content of elements in mixtures of cracked methanol and N2 Content of elements in the following Element
CO + 2H2
XN2
C H N O
1 4 – 1
– – 2x –
Carbon activity
1.00
Cracked CH3OH T = 930 °C
0.50
0
0
50 N2 (%)
II.15.2 Calculated carbon activity.
100
220
The SGTE casebook
It is thus important to know the carbon activity of a given steel at the temperature under consideration. For a binary Fe–C steel this information can be found from Fig. II.15.1. For a multicomponent steel the information is conveniently accessible by means of computerised thermodynamic calculations. Let us assume for example that we want to know the carbon activity of the stainless steel SS/AISI2310/02 at 1000 °C. The composition of this steel is shown in Table II.15.5. The calculation is now performed by specifying the temperature (1273 K) and the pressure (0.101 325 MPa) and the overall alloy content in weight fraction. In addition the size of the system has to be fixed. This choice is arbitrary in the present case and we chose to consider 1 mol of atoms. The result of the calculation is shown in Table II.15.6. As in the previous case the activities are given relative to the SER. The carbon activity relative to graphite is 0.122. From Fig. II.15.1 we can see that this corresponds to a carbon potential of around 0.30 wt%, i.e. a rather low carbon potential despite the high carbon content of 1.5 wt% for this steel.
II.15.5
Summary
The Thermo-Calc program has been applied in order to investigate the behaviour of furnace atmospheres for heat treatment of steel. Particular attention is paid to the so-called carbon potential. In order to control the process, the carbon potential of the atmosphere must be adjusted in accordance with the carbon potential of the steel. A higher carbon potential in the atmosphere will result in carburising of the steel and a lower carbon potential in decarburising. Table II.15.5 Composition of stainless steel SS/AISI2310/02 Element Amount
C 1.5
Si 0.30
Mn 0.45
Cr 12.00
Mo 0.80
V 0.90
Table II.15.6 Equilibrium in stainless steel SS/AISI 2310/02: edited output data from POLY-3 in Thermo-Calc Output from POLY-3, equilibrium number = 1
Temperature = 1273.00, Pressure = 1.013 250 × 105 Number of moles of components = 1. 000 00, Mass 5.247 81 × 10–2 Total Gibbs energy = –6.526 33 × 104, Enthalpy = 3.488 81 × 104, Volume = 0.000 00 Component
Amount (mol)
Va C Cr Fe Mn Mo Si V
0.0000 6.5538 1.2111 7.8980 4.2985 4.3759 5.6055 9.2715
× × × × × × ×
Fraction
10–2 10–1 10–1 10–3 10–3 10–3 10–3
0.0000 1.5000 1.2000 8.4050 4.5000 8.0000 3.0000 9.0000
× × × × × × ×
Activity
10–2 10–1 10–1 10–3 10–3 10–3 10–3
1.0000 1.8280 7.9024 2.4900 5.2795 9.1238 1.9837 6.2957
Fcc A1 1 Status ENTERED Number of moles = 8.5332 × 10–1, Driving force = 0.0000 Fe 9.123 18 × 10–1 Mo 6.428 98 × 10–3 Cr 6.360 60 × 10–2 C 5.341 01 × 10–3
× × × × × × ×
Potential
10–2 10–4 10–3 10–6 10–5 10–8 10–6
V Mn
0.0000 –4.2358 –7.5606 –6.3458 –1.2862 –9.8456 –1.8772 –1.2675
4.469 17 × 10–3 4.449 82 × 10–3
× × × × × × ×
Reference state SER? SER? SER? SER? SER? SER? SER? SER?
104 104 104 105 104 105 105
Si
3.387 43 × 10–3
The carbon potential during the heat treatment of steel
Conditions T = 1273, P = 101 325, N = 1, W(C) = 1.5 × 10–2, W(Si) = 3 × 10–3, W(Mn) = 4.5 × 10–3, W(Cr) = 1.2 × 10–1, W(Mo) = 8 × 10–3, W(V) = 9 × 10–3 Degrees of freedom = 0
221
222 The SGTE casebook
Table II.15.6 (Continued) M7C3 1 Status ENTERED Number of moles = 1.3803 × 10–1, Driving force = 0.000 Cr 5.866 05 × 10–1 C Fe 2.996 77 × 10–1 Mo
8.731 82 × 10–2 2.124 90 × 10–2
V3C2 1 Status ENTERED Number of moles = 8.6568 × 10–3, Driving force = 0.0000 V 8.641 65 × 10–1 Si 0.000 00 C 1.358 35 × 10–1 Mo 0.000 00
Mn Si
5.151 37 × 10–3 0.000 00
V
0.000 00
Mn Fe
0.000 00 0.000 00
Cr
0.000 00
The carbon potential during the heat treatment of steel
II.15.6 85Gus 85Sun 87Ans
223
References P. GUSTAFSON: Scand. J Metall. 14, 1985, 259–267. B. SUNDMAN, B. JANSSON and J.-O. ANDERSSON: Calphad, 9, 1985, 153–190. I. ANSARA and B. SUNDMAN: ‘Computer handling and dissemination of data’, CODATA Report, 1987, 154–158.
II.16 Preventing clogging in a continuous casting process BO SUNDMAN
II.16.1
Introduction
The problem presented here was caused by an attempt to modify an alloy produced by a continuous casting process. This process worked well for a stainless steel with 20 wt% Cr and the manufacturer now wanted to use the same process for steel with 25 wt% Cr. However, he then obtained problems with clogging by solid oxide formation which prevented the flow of liquid steel. The oxide formed at the outlet was found to consist mainly of Cr2O3. The manufacturer thus faced an expensive and time-consuming experimental scheme in order to find out how to prevent the formation of this Cr2O3. As an alternative route he tried to use the Thermo-Calc thermodynamic databank in order to simulate the process on the computer in order to find a remedy. Such a simulation can usually be made in less than a day if the necessary thermodynamic data are available. The problem arises of course because the partial pressure of oxygen, or equivalently the oxygen activity, in the liquid steel is high enough to precipitate Cr2O3 at the higher chromium content. The stability of Cr2O3 is determined by the product of the oxygen and chromium activities raised to their respective powers. Therefore the solution must be to decrease the oxygen activity in the liquid steel.
II.16.2
Setting up the calculation
On solving a problem of this type with a thermodynamic databank, one must first try to reproduce the original process. This requires fairly good knowledge of the conditions under which the process works. Some simplifications may be necessary if the conditions are uncertain or if there is a lack of thermodynamic data. Then one may make changes to the process by varying the conditions and in this way try to find the simplest and cheapest solution to the problem. Finally this solution should be tested by experiment before being put to use. If it works, one has usually saved several weeks of experimental testing of 224
Preventing clogging in a continuous casting process
225
various alternative solutions. If it does not work, it means one has overlooked one or more of the important factors in modelling the process. The original oxygen activity was not known and so a preliminary calculation was made in order to obtain an almost stable Cr2O3 at the original 20 wt% Cr content. The database useful for this slag + metal liquid problem is rather small and so only the contents of Si, Mn and Ca in the alloy and slag were taken into account together with the Cr content. The Thermo-Calc software has more flexible ways to specify the conditions for the calculation than most other software for thermodynamic calculation. Thus one may specify the composition of an individual phase in the system as conditions and not only the overall amounts or fractions of the components. This was used to set the contents of Cr, Mn and Si in the liquid metal to the values that they have in the steel. The amount of CaO was set according to what had been found in the slag inclusions in the ordinary steel. In this way the oxygen activity will be fixed by the metal + slag equilibrium. The calculation with 20 wt% Cr did not give any Cr2O3, which is correct according to the original alloy. The oxygen activity relative to a pure O2 gas was 6.059 × 10–13 according to the calculation. In this case the activity is the same as the partial pressure of O2. The alloy content of Cr was then changed to 25 wt% and a new calculation was made. The result from the second calculation showed that solid Cr2O3 was stable. Thus it was evident that the calculations could indeed reproduce the precipitation problems.
II.16.3
Solution
After satisfying oneself that the calculations are close enough to what is found in practice, one may try to predict what will happen if one changes some conditions. In this case the interest was to decrease the oxygen activity that would make the Cr2O3 oxide less stable. All components have an effect on the oxygen activity but the question is which would have the largest effect with the smallest change. Thermo-Calc here offers a possibility to show, without any new calculation, the ‘rate of change’ in or partial derivative of any state variable, e.g. the oxygen activity, with respect to any of the conditions that has been set for the calculation. This is done interactively by using a simple ‘dot’ notation as shown here: command>show 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.
234
The SGTE casebook
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.
236
The SGTE casebook
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
The SGTE casebook
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.
244
The SGTE casebook
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) <0.2 (500 °C)
Al12Mn TiAl3
? ?
36 42
Cubic A13 Hcp
Co–Cr Cr–Co Cu–Fe
3.5 (600 °C) 4.9 (600 °C) 0.3 (RT)
σ σ Bcc
? ? Bcc
65 35 50
Bcc Hcp Bcc
Fe–Cu Fe–Ni Ni–Fe Ta–Al Ti–Al
<0.5 (RT) 2 (200 °C) 3.8 (200 °C) <1 (600 °C) 11 (500 °C)
Fcc Fcc FeNi3 σ Ti3Al
18.5 3, 10.4, 12, 15–36, 20, 25, 35, 36 40 30, 40 20, 30, 40, 50, 55, 60, 65 15, 20, 25, 30, 40 36, 40 40 33, 50 10, 16, 25, 28, >30, 33, 50, 60
Fcc Fcc Bcc ? ?
50 41 59 55 58
Fcc Fcc Bcc Fcc Fcc
The SGTE casebook
Table II.20.1 Comparison of predicted and observed solid solubilities (RT, room temperature). Equilibrium solubility values and precipitated phases are taken from the Pauling File [002Vil] and solubilities after mechanical alloying from the work by Suryanarayana [001Sur]
Prediction of metastable phase ranges
259
1400
1200 Liquid
T (°C)
1000
Bcc Cu9Si2
800 Hcp 600
fcc A1 Cu33Si7 Cu15Si4
400
Cu19Si6 200 0
0.05
0.10
0.15 0.20 0.25 Mole fraction of Si
0.30
0.35
0.40
II.20.9 Calculated Cu-rich range of the Cu–Si system using assessed thermodynamic parameters [002SGT].
200 °C, presented in Fig. II.20.10. In this plot, no contribution to the Gibbs energies of the pure components has been made. It can be seen that the Gibbs energies of the fcc and hcp phases have very similar values and that, if no compound phase precipitates to form a two-phase structure in the alloyed material, then the hcp phase can be expected to form at compositions with xSi ⱜ 0.085 up to a composition with xSi ≈ 0.175, when the bcc phase becomes stable. If the Gibbs energies of the phases are now plotted for a temperature of 500 °C (Fig. II.20.11), the range of the hcp phase is seen to be more restricted (xSi = 0.1–0.17). At the still higher temperature of 800 °C, the points of intersection of the hcp and bcc curves with the fcc curve are close to being superimposed (Fig. II.20.12). This is consistent with the temperature of the fcc–hcp–bcc threephase equilibrium in Fig. II.20.9. The phase formed from the fcc phase at 800 °C could be hcp or bcc, depending on just very small energy changes. All the above figures, and in particular Fig. II.20.10, Fig. II.20.11 and II.20.12, clearly demonstrate that the amount of energy imparted by the alloying process, as well as the temperature achieved during milling of the powdered components, can have a significant influence on the Gibbs energies of the potentially forming phases. The relative values of these energies will, in turn, determine which of the phases form under given conditions.
260
The SGTE casebook
4000
2000
∆G (J)
0
–2000 Fcc –4000 Hcp Bcc
–6000
–8000 0
0.10
0.20 Mole fraction of Si
0.30
0.40
II.20.10 Gibbs energies of formation of the fcc, hcp and bcc phases of the Cu–Si system at 200 °C (intermetallic compound phases are omitted).
4000
2000
∆G (J)
0
–2000 Fcc –4000
–6000
Hcp Bcc
–8000 0
0.10
0.20 Mole fraction of Si
0.30
0.40
II.20.11 Gibbs energies of formation of the fcc, hcp and bcc phases of the Cu–Si system at 500 °C (intermetallic compound phases are omitted).
Prediction of metastable phase ranges
261
5000
3000
∆G (J)
1000
–1000
–3000 Fcc –5000 Hcp –7000 Bcc –9000 0
0.10
0.20 Mole fraction of Si
0.30
0.40
II.20.12 Gibbs energies of formation of the fcc, hcp and bcc phases of the Cu–Si system at 800 °C (intermetallic compound phases are omitted).
II.20.4
Summary and conclusions
These first attempts to investigate the potential of thermodynamic calculations, using critically assessed data, for the prediction of metastable phase ranges produced by mechanical alloying, appear to provide encouraging results. Clearly, an accurate knowledge and ‘calibration’ of the parameters of the mechanical alloying process, in terms of their contribution to the energy imparted to the metal powders, are needed to allow appropriate amendment of Gibbs energy parameters in thermodynamic prediction of solubility ranges in selected systems to be made. Systematic calorimetric studies of the energy due to, for example, the time of ball milling, milling speed, initial particle size and temperature are needed. The results obtained could be directly used, as outlined above, in thermodynamic calculations of metastable phase ranges, thereby providing a much clearer picture of the extent of agreement between observed and calculated solubility limits. Of major importance also for the calculations is experimental information with regard to the phases forming from solution when the solubility limit is reached in each case and, in particular, whether these are the expected phases as presented in the equilibrium phase diagram or other metastable phases with different stability ranges. Such information is vital to achieve a correct thermodynamic modelling and interpretation of the phase formation resulting from the mechanical alloying process.
262
The SGTE casebook
In summary, while more detailed experimental investigations incorporating both systematic measurement of the energy imparted to powder components during milling and metallographic determination of the nature of the phases precipitating from the solvent solution phase are needed, these first thermodynamic predictions of metastable solubility limits suggest that the present approach might be quite fruitful.
II.20.5 77Cal
References
Z.-K. LIU (Ed.): Calphad, Computer Coupling of Phase Diagrams and Thermochemistry, Elsevier, Amsterdam, 1977 onwards. 86Sau N. SAUNDERS and A.P. MIODOWNIK: J. Mater. Res. 1, 1986, 38. 90Spe P.J. SPENCER and H. HOLLECK: High Temp. Sci. 27, 1990, 295. 91Uen K. UENISHI, K.F. KOBAYASHI, K.N. ISHIHARA and P.H. SHINGU: Mater. Sci. Eng. A134, 1991, 1342. 92Eck J. ECKERT, J.C. HOLZER, C.E. KRILL and W.L. JOHNSON: J. Mater. Res. 7, 1992, 1751. 96Hac K. HACK (Ed.): The SGTE Casebook, Institute of Materials, London, 1996. 97Bat L. BATTEZZATI: Mater. Sci. Forum 235–238, 1997, 317. 98Spe P.J. SPENCER, G. ERIKSSON and A. VON RICHTHOFEN: Proc. CODATA’94, Chambery, France, September 1994, Springer, Berlin, 1998. 000Spe P.J. SPENCER: Ringberg Workshop on Computational Thermodynamics, Group 5, ‘New applications of thermodynamic calculations’, Calphad 24, 2000, 72. 001Spe P.J. SPENCER: Z. Metallkunde 92, 2001, 1145. 001Sur C. SURYANARAYANA: Prog. Mater. Sci. 46, 2001, 1–184. 002Vil P. VILLARS (Editor-in-chief): Pauling File, Inorganic Materials Database and Design System, Binaries edition, JST, Tokyo; MPDS, Vitznau; 2002. 002SGT SGTE: Thermodynamic Properties of Inorganic Materials, Landolt–Börnstein New Series, Group IV, Vol. 19, Subvol. B, Binary Systems, Springer, Berlin, 2002 onwards.
II.21 Adiabatic and quasi-adiabatic transformations M A L I N S E L L E B Y and M AT S H I L L E R T
II.21.1
Introduction
By rapid cooling of droplets, one can delay the start of solidification to temperatures well below the melting point. Solidification of a pure metal will then proceed very quickly and the temperature will rise drastically in spite of the rapid heat extraction. That temperature increase can be used for analysing the solidification process. It is a manifestation of the difference between the enthalpies of the solid and liquid. It could thus be used for testing the method of extrapolating the properties of the liquid well below the melting point or whether the properties of the solid phase formed at high undercoolings are similar to the equilibrium properties. As a background to such an analysis it may be convenient to make a simulation of the solidification under ideal conditions.
II.21.2
Theory
It is theoretically possible that a phase transformation at a large enough undercooling or superheating can occur adiabatically, i.e. without any heat exchange between the growing phase and the parent phase. Under isobaric conditions the molar enthalpies must then be the same in both phases, evaluated for each at its own temperature if they differ. Except for a very unlikely case, that implies that the new phase grows with a different temperature, which would normally result in heat flow. A transformation can thus be adiabatic only if it proceeds at an extremely high rate, high enough to prevent heat conduction. The only possible case seems to be some martensitic transformations that advance with a speed close to that of sound. Normally, there will be a heat exchange between the two phases but it is never discussed in isothermal experiments. However, it will be important in rapid transformations where the new phase will grow behind a thermal spike in the parent phase. The spike will be pushed forwards and, if it widens, then it will eventually affect all the remaining parent phase. On the other hand, the spike 263
264
The SGTE casebook
does not need to widen if the new phase at its temperature has the same molar enthalpy as the bulk of the parent phase at the temperature where the transformation was initiated. Except for an initial transient, the width of the spike would then be constant and depend on the growth rate, and a very high growth rate is required in order for the spike not to cover a major part of the parent phase. Such rates are found only in diffusionless transformations and at high temperatures where the mobility of the interface is high. It is difficult to imagine this kind of transformation except in solidification. It has been described as quasi-adiabatic [58Hil]. In the ideal case of quasi-adiabatic transformation there is local equilibrium between the two phases at the top of the thermal spike. The local temperature must there be the normal melting temperature. The whole of the growing phase must be at the same temperature and the question is how to find the temperature where the bulk of the parent phase has the same enthalpy as the new phase has at the melting temperature. That would be the temperature to which the parent phase must be undercooled or superheated. It could be found by examining thermodynamic tables with properties of the phases extrapolated into the metastable ranges. However, it would be more illustrative to examine the curves for the two phases in an Sm–Hm diagram. The reason is partly that a true adiabatic transformation under ideal conditions, i.e. without any internal production of entropy due to friction in the migrating interface, can be obtained directly from the intersection of the two curves because a reversible adiabatic transformation is also isentropic [98Hil].
II.21.3
Numerical calculations
With modern databases and computer programs for thermodynamic calculations it is easy to evaluate the conditions for the two kinds of adiabatic transformation in any given system. The purpose of the present chapter is just to demonstrate such a calculation and to illustrate the results graphically. In particular, the Sm–Hm diagram will be applied to illustrate the final situation if a system is first undercooled to a temperature not low enough for quasi-adiabatic growth. It will be assumed that the system is thermally insulated from the moment when the chosen temperature has been reached. The numerical calculations were carried out with the Thermo-Calc databank [002And]. Figure II.21.1 presents the relation between molar entropy and enthalpy of liquid and bcc tungsten. The temperature scales for the two phases are inserted below the diagram. They are displaced with respect to each other because Hm has different values in the two phases at any one temperature. The melting point, 3695 K, will thus fall at Hm = 117 kJ mol–1 for the body-centred cubic (bcc) phase and at 169 kJ mol for the liquid. See the points marked m(bcc) and m(l).
Adiabatic and quasi-adiabatic transformations
265
125 m(l)
Entropy Sm (J mol–1 K–1)
120
q-a(bcc) Partial solidification
115 Adiabatic transformation 110
m(bcc)
c
Bc
q-a(l)
Liq
ui
d
105
100 100
110
120
130 140 150 160 Enthalpy Hm (kJ mol–1)
q-a(l)
ad(l)
170
180
m(l)
Tl 2000
2500
3000
m(bcc)
3500 ad(bcc)
q-a(bcc)
T bcc 3500
4000
4500
II.21.1 Entropy–enthalpy property diagram for pure tungsten.
The liquid can solidify quasi-adiabatically if cooled down to the point q-a(l). There it can transform to the bcc phase of the same enthalpy but at a higher temperature, that of the melting point.
II.21.4
Discussion
The point of intersection represents the true adiabatic conditions for solidification on cooling as well as melting on heating because it is the only situation where the entropy will not change when the system transforms between the two phases. Theoretically, the liquid could solidify there instantaneously, not being dependent on the rate of heat conduction. In practice the heat of solidification will heat the remaining liquid more quickly than the phase interface can migrate. The conditions for adiabatic solidification will thus be destroyed. If one manages to cool the liquid to a point marked q-a(l), then the system could in principle solidify to the bcc phase of the same enthalpy but a higher
266
The SGTE casebook
entropy. That entropy increase is produced internally by heat flow in the thermal spike. This is thus the quasi-adiabatic solidification and its enthalpy is obtained from the bcc phase at the melting point because full local equilibrium between the phases was assumed. If the mobility of the interface is limited, then there must be a driving force acting on the interface and that can be accomplished by a deviation from local equilibrium. That requires that the solidification does not start until below the point marked q-a(l). The entropy increase will then be larger and some of it will now be produced by friction in the migrating interface. The temperature of the bcc phase will now be below the melting point. If the solidification starts before the point q-a(l) has been reached, then the rate will be controlled by heat conduction. It could still be fairly rapid and almost come to a stop when the whole system has reached the melting temperature. This reaction could thus be regarded as approximately adiabatic with respect to the whole system but not with regard to its interior. It has not been described with its own term. The solidification will not be complete without further heat extraction and the two-phase mixture at the melting temperature can be evaluated from the lever rule applied to the tie line drawn between the points representing the phases at the melting temperature, m(bcc) and m(l). Solidification will be completed when there is time for further heat extraction from the system.
II.21.5 58Hil 98Hil
02And
References M. HILLERT: Acta Metall. 6, 1958, 122–124. M HILLERT: Phase Equilibria, Phase Diagrams and Phase Transformations – Their Thermodynamic Basis, Cambridge University Press, Cambridge, 1998, pp. 349–353. J.-O. ANDERSSON, T. HELANDER, L. HÖGLUND, P. SHI and B. SUNDMAN: Calphad 26, 2002, 273–312.
II.22 Inclusion cleanness in calcium-treated steel grades J E A N L E H M A N N and R A Y M O N D M E I L L A N D
II.22.1
Introduction
The aim of calcium treatment performed during secondary metallurgy operations on aluminium-killed steels is to avoid nozzle clogging problems. The success of such a treatment can be evaluated from the ratio of the calcium content to the total oxygen content of the grade. To ensure good castability, the Ca content must be high enough to assure complete transformation of solid alumina oxides into completely liquid calcium aluminates, but low enough to avoid precipitation of calcium sulphides at high temperature (Fig. II.22.1). This upper limit depends on the sulphur content of the steel grade. From an industrial point of view, the adjustment of a Ca treatment may be problematic. The difficulty arises from several factors. One of these factors is the Ca addition yield. This yield is always lower than 30% and depends on the addition process and on the metal composition. Therefore, the adjustment of a calcium treatment process on a given industrial installation requires 50
Total Ca (ppm)
40 CaS + liquid oxides 30 Liquid oxides
20
Partially or completely solid oxides
10 0 0
5
10 15 Total O (ppm)
20
25
II.22.1 Nature of oxide and sulphide inclusions present at 1550 °C as functions of the total calcium and oxygen content in a steel grade with 1.5% Mn, 0.2% Si, 0.04% Al and 0.005% S.
267
268
The SGTE casebook
specific industrial trials where the treatment is qualified by the nature of the obtained oxides at the casting temperature. This study shows that the type and, more specifically, the size of the samples taken in the tundish to qualify the Ca treatment are very important. Indeed, oxide inclusion composition may evolve during cooling by reaction with elements that remain dissolved in the steel or with other precipitated phases such as sulphides. This evolution depends on the cooling rate, and the cooling rate is in turn directly linked to the size of the sample.
II.22.2
Choice of the most adapted sample to qualify the calcium treatment
A specific industrial trial was conducted in order to select the most appropriate type of liquid metal samples to characterise the Ca treatment. Five heats of the steel grade given in Table II.22.1 have been sampled and their inclusion population has been analysed by scanning electron microscopy (SEM). Two types of sample have been taken in the tundish. – –
Flushed-sucked lollipop samples. LUS samples taken at the same time as the lollipop samples.
These results have been compared with the inclusion population present on hot-rolled product. As mentioned before, the observed differences can be attributed to the size and, as a consequence, to the cooling rate of the various samples. We shall see that this effect is clearly shown by SEM results and can be explained by simple thermodynamic considerations.
II.22.2.1 Nature and composition of the inclusions obtained by scanning electron microscopy In the studied steel grade, the inclusions are composed of two phases that are, most of the time, spatially associated: an oxide and a sulphide phase (Fig. II.22.2). Because of the usually small size of these inclusions, it appeared impossible to analyse separately the two phases. Therefore, the quantitative 5 µm size of the observed inclusions, as given by SEM investigations, has been conveniently represented in two different ternary diagrams.
Table II.22.1 Composition of the studied steel grade Element Amount (wt%)
C 0.05
Mn 1.5
Si 0.2
Al 0.04
S 0.005
Ca 0.002
N 0.004
O 0.002
Inclusion cleanness in calcium-treated steel grades
269
5 µm
II.22.2 Example of inclusion observed on the studied steel grade (Table II.22.1). %CaO
Liquid oxide (C12A7)
Lollipop LUS Coil
CA CA6 CA6 0 10 %CaS+%MnS
20
30
40
50
60
70
80
90
100 %Al2O3
II.22.3 Composition of the oxide phase of the inclusions in the different samples (SEM results), where A stands for aluminate. We can observe a decrease in the lime content in the oxide phase between the lollipop, LUS and coil.
–
–
The Al2O3–CaO–(CaS+MnS) diagram (Fig. II.22.3) that represents precisely the composition of the oxide phase. Pure oxide inclusions have a composition situated on the binary Al2O3–CaO, the composition of pure sulphides are positioned on the %CaS + %MnS apex and associated sulphide–oxide inclusions are located in-between. The (Al2O3+CaO)–CaS–MnS diagram (Fig. II.22.4) that gives the precise composition of the sulphide phase. Figure II.22.3 and Fig. II.22.4 show
270
The SGTE casebook %MnS
10% CaS
Lollipop LUS Coil
40% CaS
10
20
30
40
50
60
70
%CaS
80
90
100 %Al2O3 + %CaO
II.22.4 Composition of the sulphide phase of the inclusions in the different samples (SEM results). We can observe an increase in the MnS content in the sulphide phase between the lollipop, LUS and coil.
–
–
–
that the compositions of the oxide and sulphide phases depend on the sample type even if those samples have been taken at the same time. For the lollipop samples, the oxide phase composition is close to C12A7 (about 50 wt% CaO and about 50 wt% Al2O3). This is the oxide composition aimed at when performing a calcium treatment. The sulphide phase is nearly pure CaS. For the LUS samples, the oxides contain more alumina than in the lollipop samples; their compositions vary between CA (about 65 wt% Al2O3) and CA2 (about 78 wt% Al2O3) or even CA6 (about 90 wt% Al2O3). The sulphide composition is roughly 90 wt% CaS–10 wt% MnS. For the hot-rolled sample, the oxide phase is composed mainly of CA6. The sulphide phase contains mainly MnS with a CaS content between 10 and 40 wt%.
II.22.2.2 Thermodynamic modelling The fact that these analytical differences can be explained by the different cooling rates of the samples is substantiated by the thermodynamic calculations carried out with CEQCSI software [92Gat]. These calculations show that, in lollipop samples, the inclusions are quenched as they are in liquid metal or
Inclusion cleanness in calcium-treated steel grades
271
at the beginning of the solidification whereas, in coil samples, their composition is as it is at the end of the solidification. The LUS sample is an intermediate case. The details of these calculations are represented in Fig. II.22.5. The sequence of precipitation is as follows when equilibrium is assumed to be reached at each temperature. – – –
Liquid Ca aluminates precipitate in liquid metal. Then, CaS starts to precipitate at the beginning of the solidification. Sulphides became increasingly enriched in MnS with decreasing temperature. At the end of the solidification, nearly pure MnS precipitates.
With the assumption of a re-equilibration at each temperature, the composition of the oxide phase changes with decreasing temperature. Because of their reaction with dissolved sulphur and aluminium given by 2Al + 3 CaO + S → Al2O3 + 3 CaS
(II.22.1)
they become increasingly enriched in Al2O3 and, as a consequence, they are transformed into solid aluminates: first CA, then CA2 and finally CA6. Such a situation is more or less encountered for the coil sample that cools quite slowly. On the contrary, for lollipop samples that cool very rapidly, the situation, as it existed at the early stages of the solidification, has been quenched and the oxide phase is much richer in CaO. The loss of CaO of the oxides can continue for a slab sample during reheating before hot rolling. For example, it has been shown that, on
60
1
0.8
40 MnS
0.6
30 0.4 20
CaS 0.2
10 CA6
Liquid oxides 0 1550
Liquid metal fraction
Precipitate O, S (ppm)
50
1525
1500
1475
0 1450
CA CA2 Temperature (°C)
II.22.5 Calculated evolution of the composition of the oxides during solidification of the metal (0.05% C, 1.5% Mn, 0.2% Si, 0.04% Al, 22 ppm Ca, 50 ppm S, 10 ppm O and 45 ppm N).
272
The SGTE casebook
resulphurised steel grades [95Gat], manganese sulphides and oxides could react with dissolved aluminium to form oxides richer in Al2O3 and sulphides richer in CaS according to the reaction 3CaO +3MnS + 2Al → 3CaS + 3Mn + Al2O3
II.22.3
(II.22.2)
Conclusion
SEM observations coupled with thermodynamic calculations have shown that lollipop samples appear to be the most adapted type of sample to qualify for a Ca treatment. This type of samples cools quickly enough to ‘quench’ the different phases with the composition that they have in the tundish. LUS samples will be more appropriate if one wants to obtain an idea of the inclusion state in products. This problem of sample choice is less critical for Al-killed ULC steel grades or even for Ca-treated steels with low S contents (typically less than 10 ppm). In ‘conventional’ Al-killed steel, no reaction occurs between alumina and sulphides which, in this case, are MnS precipitating at low temperatures. In Ca-treated steels with low S contents, the quantity of CaS will be very limited and, therefore, the oxide compositions will not drastically change with temperature. On the contrary, in Si–Mn semikilled steels, where oxide precipitation occurs all along the solidification, the problem of choice of the sample type is again quite important, depending on which fraction of the inclusions population it is necessary to characterise.
II.22.4 92Gat 95Gat 97Pub 99Mei
References C. GATELLIER, H. GAYE, J. LEHMANN and Y. ZBACZYNIAK: Rev. Métall.-CIT, October 1992, 887–888. C. GATELLIER, H. GAYE and J. LEHMANN: Rev. Métall.-CIT, April 1995, 541–553. F. RUBY-MEYER and G. WILLAY: Rev. Métall.-CIT, March 1997, 367–378. R. MEILLAND, H. HOCQUAUX, C. LOUIS, L. POLLINO and F. HOFFERT: Rev. Métall.CIT, January 1999, 89–97.
II.23 Heat balances and CP calculations K L A U S H A C K and M I C H A E L H. G. J A C O B S
II.23.1
Introduction
Values for CP are often used by engineers in practical heat calculations. This is done on the basis of the definition that CP is the amount of heat needed for a temperature change of 1 K. However, what happens if a phase transition occurs in this 1 K interval? Searching in the early literature on this subject, one finds that for example Max Planck has already addressed this problem. In the ninth edition of his book Thermodynamik (1930) he states (§53 in Section 1, Chapter 3, translation from German): ‘While in general the heat capacity changes steadily with temperature, there are for each substance under a given outside pressure certain singular temperature points at which with other properties also the heat capacity becomes discontinuous. At these points an amount of heat added from outside will not go into the body as a whole but only partially and it will not serve to raise the temperature but change the state of aggregation, this being melting, evaporation or sublimation, depending upon whether the substance goes from solid to liquid, or from liquid to gaseous, or from solid to gaseous. Only after the whole body at the given temperature has become homogeneous again in the new state of aggregation, the temperature will rise again on continued heating and a heat capacity can be defined again.’ He then introduces the term ‘latent heat’ for all enthalpies of transformation. Finally he also introduces ‘similar to changes in the state of aggregation’ processes of mixing or dissolution as well as chemical processes which are ‘in general accompanied by larger or smaller heat evolvements depending upon the external conditions’. It is obvious that he refers to additional terms that need to be considered in proper heat balance calculations. However, the decisive sentence above is: ‘Only after the whole body at the given temperature has become homogeneous again in the new state of aggregation ... a heat capacity can be defined again.’ Planck himself argues mainly about ‘pure substances’, but for a complex multicomponent chemical 273
274
The SGTE casebook
system all his arguments hold too. The ‘whole body’ can of course be composed of many chemical components which occur in several phases. The ‘state of aggregation’ is then a particular phase assemblage. The expression for the total enthalpy of this phase assemblage becomes H = Σ nΦ H Φ
(II.23.1)
Φ
In turn, one can derive CP as Φ ∂n C P = Σ nΦ ∂H + Σ H Φ Φ ∂T P Φ Φ ∂T P
∂n = Σ n Φ C PΦ + Σ H Φ Φ Φ Φ ∂T P
(II.23.2)
It is important to note that there is not only the contribution of the stoichiometric sum of the CP values of all phases (first sum) but also a contribution of the change in phase amounts within the phase assemblage (second sum). The following three figures for the phase diagram of the Ag–Cu system, the enthalpy versus temperature curve for a composition with 80 at.% Cu and the resulting CP curve illustrate the above relationships. Figure II.23.1 shows the simple eutectic phase diagram of the system with the composition line for 80 at.% Cu marked. At lower temperatures the alloy is a two-phase solid because of the face-centred cubic (fcc)–fcc miscibility gap. Then after the transition through the eutectic temperature a Cu-rich fcc solid is in equilibrium with the liquid, and finally, after transition through the liquidus line, the alloy is all liquid. 1500 Liquid
Temperature (K)
1300
1100
900
700 fcc + fcc 500 300 0.0 Ag
0.2
0.4
0.6
xCu
0.8
1.0 Cu
II.23.1 The Ag–Cu phase diagram calculated from the SGTE Database.
Heat balances and CP calculations
275
In Fig. II.23.2 the enthalpy shows a pronounced jump when the alloy goes through the eutectic temperature, comparable with the melting of a pure substance. On the other hand, the transformation of the remaining Cu-rich fcc alloy into liquid is drawn-out from the eutectic to the liquidus temperature. This is reflected in a drawn out increase in enthalpy which contains contributions of the melting as well as the mixing behaviour in the fcc solid and the liquid phase. At the temperature of the liquidus line there is a cusp in the curve indicating the transition into the one-phase range. In Fig. II.23.3 finally the CP curve is shown. An almost constant CP in the miscibility gap range with an increase when approaching the eutectic temperature is followed by a sharp jump at the eutectic. In turn, this is followed by a strong peak-like increase in the melting range, while the pure liquid has an almost constant CP again. The effect of the contributions from the second sum in Equation (II.23.2) is clearly visible. As soon as the phase boundaries become flatter, e.g. when the miscibility gap approaches the eutectic, and all the way through the liquid–fcc two-phase field, the contribution of the second sum in Equation (II.23.2) has a strong influence on the total CP. This is not always realised properly in standard software. The conclusions from the above are as follows: A unique value for CP cannot be defined under all circumstances, neither for pure substances nor for multiphase systems (see comments on Planck’s work and the jumps in Figure II.23.2). It is therefore not advisable to use values of CP for heat balance calculations unless it is clear that no phase
50 45
Liquid
Ag–Cu, xCu = 0.8
40
Enthalpy (kJ mol–1)
1
Liquid + fcc
35 30 25 20
fcc + fcc
15 10 5 0 400
600
800 1000 Temperature (K)
1200
II.23.2 Enthalpy of the 20% Ag–80% Cu alloy.
1400
276
The SGTE casebook 180 160
Ag–Cu, xCu = 0.8
CP (J K–1 mol–1)
140 120 Liquid + fcc
100 80 60 fcc + fcc
Liquid
40 20 400
600
800 1000 Temperature (K)
1200
1400
II.23.3 CP curve for the 20% Ag–80%Cu alloy. The dashed curve indicates the contribution of the first sum in Equation (II.23.2).
2
transitions occur in the temperature interval under consideration. However, even then it is useful to check how steep the phase boundaries are in this region in order to have an idea of how much the second sum in Equation (II.23.2) contributes to the total CP. Engineers who carry out ‘heat calculations’ need to keep in mind that CP does not contain contributions from phase transitions and/or mixing (dissolution). If they want to be sure that all contributions to the enthalpy are included in their calculations, they need to calculate proper enthalpy balances, i.e. to calculate one equilibrium state with its corresponding enthalpy, and then the other with the second enthalpy. The difference between the two is the desired result (see also Part I.1.5).
The above method is ‘safe and sound’ and will not overlook any contributions from phase transitions or mixing effects. The ∆H values coming from such calculations will automatically give the proper differences, i.e. balances, for the enthalpy (heat). It should be added here for completeness that the above holds for all extensive property balances, i.e. also for entropies or Gibbs energies. All of these are path independent and therefore only their values for the initial state and the final state of the system need to be known.
II.23.2
Practical calculations
The following paragraphs give a more practical view on the use of heat balance calculations and the way that these may be executed with inclusion
Heat balances and CP calculations
277
of effects from non-ideal mixing of the phases even if complete databases are not available. It was customary to state in the introductory chapters of classical printed data compilations (see, for example, the books by Kubaschewski and Alcock [51Kub], by Barin and Knacke [73Bar], by Barin et al. [77Bar], by Glushko and Gurvich [79Glu] and by Kubaschewski et al. [93Kub] and also the JANAF Thermochemical Tables [65Jan]) how to apply the data in the framework of what is called ‘stoichiometric reaction thermochemistry’. In all such applications it is assumed that a stoichiometric balance (reaction) formula can be provided such as H 2 + 12 O 2 = H 2 O . The major purpose of such a calculation would be to gain insight into the equilibrium between the reactants and products. This can only be done on the basis of isothermal conditions. However, often the mass balance (reaction) equation that is considered is not at all related to an equilibrium. On the contrary, it holds for a complete transformation of the reactants at some initial temperature and pressure into the products at some final temperature and pressure. Such a case is given for example by the thermite process which is represented by the classical aluminothermic reaction Fe2O3 + 2Al = Al2O3 + 2Fe. In this redox reaction the two reactants Fe2O3 and Al are completely consumed and the final equilibrium state consists only of Al2O3 and Fe. For such reactions the lawof-mass-action approach mentioned above is not applicable since the process is not isothermal. An equilibrium constant cannot be defined. However, it is possible to calculate non-isothermal changes for the extensive properties mentioned above. Assuming that the reactants are initially at room temperature and the final products end up at a temperature of 2500 K, one can calculate from the tabulated values for the four substances involved the total enthalpy change of the reaction from ∆H = ∆ H = H Al 2 O 3 (2500) + 2HFe(2500) – H Fe 2 O 3 (298.15) – 2HAl(298.15). One finds that ∆H = –235.72 kJ, i.e. at 2500 K there is still quite some enthalpy left to heat the products (or to be dissipated into the surroundings). However, what if the reactants and products are not pure substances, i.e. compounds with fixed stoichiometry? In a meltshop the aim usually is to produce a mixture phase with a given overall composition, e.g. a liquid steel G-X20Cr14, from different ‘raw materials’, e.g. pure iron, pure graphite and pre-alloyed ferrochrome. One could of course write down a mass balance (reaction) equation of the kind xFe + yC + z ferrochrome = G-X20Cr14. The data for the pure substances iron and carbon in the initial states Fe(25 °C, 1 bar, ferrite) and C(25 °C, 1 bar, graphite) can be found in the above-mentioned data compilations. For the preheated reactant ferrochrome (800 °C, 1 bar, solid) and the product G-X20Cr14(1600 °C, 1 bar, liquid), however, there are no tabulations available. Thus the balance calculation turns into a rather tedious, if not impossible, job since additional data have to be searched for in the literature.
278
The SGTE casebook
H (kJ/100 g)
Often it is possible to make assumptions which permit the use of pure substance data nevertheless. For example, for ferrochrome at room temperature, one can work with pure Fe and pure Cr with good approximation, since the Fe–Cr system exhibits a large miscibility gap for body-centred alloys at low temperatures (see Table II.23.3). For temperatures upwards from 713 °C, however, ferrochrome is a solid solution phase (also see Table II.23.3), such that the related enthalpy of mixing needs to be considered in the calculations. For liquid G-X20Cr14 a simplification is not possible at all, since the enthalpy of the melt is not given by the weighted sum of the elementary liquid components only but also contains a contribution from the enthalpy of mixing. The calculations below are for solution phase systems at particular initial (reactant) and final (product) temperatures. Therefore the data which are needed for the different materials are represented in tabular form similar to the classical pure substance tables. The reader may execute calculations for other initial or final temperature by way of interpolation of the tabulated data. The general calculational method will be explained using the tables and also Fig. II.23.4 as applied to the production of the above-mentioned steel GX20Cr14. In Fig. II.23.4 the enthalpy curves according to Table II.23.1, Table II.23.2, Table II.23.3 and Table II.23.4 are given. These relate to the three reactants pure Fe, pure C, ferrochrome (35wt% Fe–65 wt% Cr) and the product GX20Cr14 (85.8 wt% Fe, 0.2wt% C and 14 wt% Cr) respectively. In all curves
150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 –10
C (graphite) Fe G–X 20 Cr14
Ferrochrome
0
200
400
600
800 1000 T (°C)
1200
1400
1600
II.23.4 The enthalpy curves of the various substances according to Tables 23.2.1–23.2.4.
Heat balances and CP calculations
279
the phase transformations of the respective material can be recognised from the enthalpy shifts at the respective temperatures. It must be mentioned especially that the steel G-X20Cr14 has of course no ‘melting point’ like a pure substance. It melts through a temperature interval. The same holds even if not so pronounced for the other phase transitions of G-X20Cr14. For these transitions also see Table II.23.4. Furthermore, it needs to be mentioned that in the enthalpy of the solid (fcc and bcc) as well as the liquid solutions for G-X20Cr14 the contributions of the enthalpy of mixing are automatically contained. For the calculation of the enthalpy balance of the formation reaction the enthalpy values of the respective substances at the respective temperatures only need to be read either from Tables II.23.1 to Table II.23.4 or from the
Table II.23.1 Pure iron T (°C)
H (kJ/100 g)
Phase
25.00 200.00 400.00 600.00 800.00 911.66 911.66 1000.00 1200.00 1394.32 1394.32 1537.80 1537.80 1600.00 1700.00
0.00 8.40 19.60 33.29 51.30 60.12 61.93 67.36 80.10 93.07 94.55 105.03 129.76 134.88 143.12
Bcc Bcc Bcc Bcc Bcc Bcc Fcc Fcc Fcc Fcc Bcc Bcc Liquid Liquid Liquid
Table II.23.2 Pure carbon (graphite) T (°C)
H (kJ/100 g)
Phase
25.00 200.00 400.00 600.00 800.00 1000.00 1200.00 1400.00 1600.00 1700.00
0.00 16.47 43.49 75.86 111.39 148.98 188.02 228.14 269.12 289.88
Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite
280
The SGTE casebook Table II.23.3 Ferrochrome (35 wt% Fe–65 wt% Cr) T (°C)
H (kJ/100 g)
Phase(s)
25.00 200.00 400.00 511.77 511.77 600.00 713.14 800.00 1000.00 1200.00 1400.00 1600.00 1700.00
0.007 8.77 22.27 32.59 35.76 41.54 50.29 55.60 68.55 82.68 98.17 115.24 124.47
Bcc+bcc Bcc+bcc Bcc+bcc Bcc+bcc Bcc+σ Bcc+σ Bcc+σ Bcc Bcc Bcc Bcc Bcc Bcc
Table II.23.4 G-X20Cr14 (85.8 wt% Fe, 0.02 wt% C and 14 wt% Cr) T (°C)
H (kJ/100 g)
Phase(s)
25.00 200.00 400.00 434.92 600.00 800.00 843.06 885.65 976.06 1050.00 1126.51 1200.00 1406.60 1410.25 1496.55 1600.00 1700.00
0.00 8.05 21.98 24.99 36.73 54.36 57.73 62.47 68.90 73.61 78.59 83.77 99.80 101.27 125.61 134.20 142.58
Bcc+bcc+M23C6 Bcc+bcc+M23C6 Bcc+bcc+M23C6 Bcc+bcc+M23C6 Bcc+M23C6 Bcc+M23C6 Bcc+M23C6 Bcc+M23C6+fcc Bcc+M23C6+fcc Fcc Fcc Fcc+bcc Fcc+bcc Fcc+bcc+liquid Bcc+liquid Liquid Liquid
graph. Then they must be multiplied by the substance amounts and added to the balance with the appropriate sign (+ for products and – for reactants). If necessary, linear interpolations between the temperatures given in the tables can be applied to obtain heat balances for temperatures other than those listed. It should be noted that the tables and curves give values for 100 g of the respective materials. The balances can thus be easily executed for 100 g of the product if the fractions (factors) of the reactants all sum up to 1.
Heat balances and CP calculations
II.23.3
281
Calculational example
The reaction–mass balance equation (substance amounts are in weight fractions; data for 100 g of substance) is 0.7826Fe (25 °C,1 bar, ferrite) + 0.0020C (25 °C,1 bar, graphite) + 0.251 54 ferrochrome (800 °C,1 bar, stable state) = G-X20Cr14 (1600 °C,1 bar, liquid) The heat balance equation (in kilojoules per 100 g of alloy) is ∆H = –0.7826 × 0 – 0.0020 × 0 – 0.2154 × 55.60 + 134.20 = 122.23 kJ/100 g Note that pure ferritic iron and pure graphite at room temperature and 1 bar total pressure are the reference states for Fe and C respectively and therefore have by definition the enthalpy value 0. On preheating the reactants, one obtains of course finite positive values, as can be seen from the value used for ferrochrome above and also from Table II.23.1 to Table II.23.4 and the curves in Fig. II.23.4. The value for the total heat balance will then become smaller accordingly.
II.23.4 51Kub 65Jan 73Bar 77Bar 79Glu
93Kub
References O. KUBASCHEWSKI and C.B. ALCOCK: Metallurgical Thermochemistry, 1st to 5th editions, Pergamon, Oxford, 1951 to 1979. JANAF Thermochemical Tables, US Department of Commerce, National Bureau of Standards, Washington, DC, 1965 to 1968. I. BARIN and O. KNACKE: Thermochemical Properties of Inorganic Substances, Berlin, 1973. I. BARIN, O. KNACKE and O. KUBASCHEWSKI: Thermochemical Properties of Inorganic Substances, Supplement, Springer, Berlin, 1977. V.P. G LUSHKO and L.V. G URVICH : IVTAN, Termodinamicheskie Svoistva Individualnikh Veshchestv, Vols I–IV, Akademiya Nauk SSSR, Moscow, since 1979. O. KUBASCHEWSKI, C.B. ALCOCK and P.J. SPENCER, Materials Thermochemistry, Pergamon, Oxford, 1993.
II.24 The industrial glass-melting process REINHARD CONRADT
II.24.1
Introduction to some fundamentals of industrial glass melting
The present chapter gives an example of the application of thermodynamic data to a quite complex technical process, i.e. the industrial melting of glass. The glass-melting process starts from a granular mixture of natural and synthetic raw materials (the so-called batch) and yields a thermally and chemically homogeneous melt made available at a well-defined temperature level. The consecutive steps of glass fabrication (forming, annealing, etc.) are not within the scope of interest of this chapter. Figure II.24.1 illustrates how a continuously working glass-melting furnace functions in principle. Figure II.24.2 presents in very general terms a heat balance of the glassmelting process. It consists of amounts of heat related to the flow of matter Heat recovery system
Heat loss Hot stream through combustion space
Tad, hot gas from fuel (T0) and air (Tre)
T0, batch
Toff, off-gas Heat transfer
Cold stream through basin
Tex, melt
Heat loss
II.24.1 Sketch of the principal function of a continuously working glass-melting tank furnace: T0, ambient temperature; Tre, temperature of preheated air; Tad, adiabatic combustion temperature; Toff, offgas temperature; Tex, pull temperature of the glass melt.
282
The industrial glass-melting process Heat exchanger system
283
Glass-melting tank furnace
in 4000 re
2100 sf fire ht
wx 200
ex 1690
off 3100 exch stack
wu
800
700 wo 540
II.24.2 Heat balance of a continuously working glass-melting tank furnace. Typical values of the individual heat terms given in kilojoules per kilogram of molten glass. The abbreviations denoting individual heat terms are as follows: in, heat input by fuel and electricity; sf, heat set free in the furnace; fire, heat transferred to the furnace body; ht, heat transferred to the basin; ex, exploited heat; off, heat stored in the off-gas leaving the heat exchanger; stack, heat stored in the off-gas leaving the heat exchanger; exch, heat transferred to the body of the heat exchanger; re, heat recovered by the air passing through the heat exchanger; wu, wall losses through the basin, wo, wall losses through the lining of the combustion space; wx, wall losses of the heat exchanger.
led through the combustion space, amounts of heat related to the flow of matter led through the melting basin, and heat losses through the boundaries of the system. For simplicity, all quantities are referred to the standard temperature level T0 = 298 K of the environment and to isobaric conditions at p = 1 bar. The quantity of specific interest in this paper is the so-called exploited heat Hex. It consists of both the heat required to bring about the batch-to-melt conversion and the heat stored in the homogeneous melt leaving the system boundary at a temperature Tex. For the quantification of the heat balance in Figure II.24.2, it is crucial to have an accurate account of Hex. It is true that the contributions Hin, Hoff, Hstack and Hre related to the combustion space can be calculated in a most accurate way from measured process temperatures and from the known amounts of fuel and air used. However, for the wall losses Hwo and Hwu through the boundaries of the combustion space
284
The SGTE casebook
and the basin respectively, only rough estimates are available. This is due to the complicated shape of a real glass furnace, to the presence of a considerable number of openings in the furnace walls and (most of all, after some years of continuous furnace operation) to the unknown residual thickness of the refractory bricks of the furnace lining. Thus, if Hex is assessed at a high accuracy, then the balance in Figure II.24.2 can be completed, and even an accurate account of the wall losses may be obtained. As stated above, the exploited heat Hex consists of both the heat required for the batch-to-melt conversion and the heat physically stored in the homogeneous melt at T = Tex. Irrespective of the actual reaction path leading from the batch at T0 to the melt at Tex, Hex may be presented as ° H ex = ∆H chem + ∆H melt ( Tex )
(II.24.1)
° denotes the enthalpy difference at T = 298 K between the where ∆H chem batch on one side, and the glass plus the gases released from the batch on the other side according to
batch (298 K) → glass (298 K) + batch gases (298 K)
(II.24.2)
and ∆Hmelt (Tex) denotes the enthalpy difference between the glass at 298 K and the glass melt at T = Tex. This approach to Hex poses a dual challenge: we need an accurate approach to the thermodynamics of multicomponent glasses at room temperature, and to multicomponent melts.
II.24.2
Description frame for the thermodynamic properties of industrial glass-forming systems
II.24.2.1 Description frame for one-component glasses and glass melts The thermodynamic state of a one-component system in its stable liquid, metastable undercooled, glassy and crystalline state at an ambient pressure of p = 1 bar is described by the following seven quantities in a comprehensive way: H° = standard enthalpy at 298 K, for the crystalline solid, stable at T = Tg S° = standard entropy at 298 K, for the crystalline solid, stable at T = Tg Hfus = enthalpy of fusion Tliq = liquidus temperature cP(T) = the heat capacity of the crystalline solid as a function of
The industrial glass-melting process
285
temperature as, for example, represented by the polynomial cP(T) = A + BT + C/T2 Hvit = vitrification enthalpy Svit = vitrification entropy (0 K entropy of the glass) ∆cP = jump of the heat capacity at the glass transition temperature Tg = glass transition temperature In principle, all quantities referring to the glassy state depend on the cooling rate at which this state is reached. With the cooling rate defined, they assume unambiguous values. The details are not elaborated here. The set of quantities Hfus, Sfus, Tliq, Hvit, Svit, ∆cP and Tg is redundant. It is linked by the relations given in the following equations, where Hc(T) and Sc(T) denote the configurational enthalpy and entropy respectively: fus S fus = H Tliq
(II.24.3)
H c ( T ) = H vit +
∫
T
∫
Tliq
( c P , liq – c P ,cryst ) dT
Tg
= H fus –
( c P , liq – c P ,cryst ) dT
(II.24.4a)
T
≈ Hvit + ∆cP · (T – Tg) ≈ Hfus – ∆cP · (Tliq – T) Hvit ≈ Hfus – ∆cP · (Tliq – Tg) Sc ( T ) = S vit +
∫
T
∫
Tliq
Tg
= S fus –
T
(II.24.4b) (II.24.4c)
c P ,liq – c P ,cryst dT T c P ,liq – c P ,cryst dT T
(II.24.5a)
Tg ≈ S vit – ∆c P ⋅ ln T Tliq ≈ S fus – ∆c P ⋅ ln T
(II.24.5b)
Tliq S vit ≈ S fus – ∆c P ⋅ ln Tg
(II.24.5c)
The knowledge of any four of the above redundant set of quantities is sufficient to derive the rest. As shown by a large number of calorimetric experiments
286
The SGTE casebook
[95Ric], the error introduced by the approximation of the real shape of the ∆cP jump by a constant value is calorimetrically insignificant. Thus Hc(T) and Sc(T) may be calculated as suggested by Equation (II.24.4a–c) and Equation (II.24.5a–c).
II.24.2.2 Description frame for multicomponent glasses and glass melts The thermodynamic properties of multicomponent systems have been approached by different elaborate models, among which are the (modified) quasichemical model [86Bla], the cell model [84Col] and the model of ideal mixing of complex components [90Bon, 94Sha]. However, even for the elaborate computer codes and databases used in computational thermochemistry [90Eri, 004GTT], the generation of reliable data for multicomponent systems is still a major problem. The present author’s own approach [99Con, 001Con] outlined below is especially well suited to the multicomponent systems typical of industrial glasses. The rigid glass as well as the glass melt are described by their energy and entropy difference from a normative state of mineral phases k which would form and coexist at the glass transition temperature Tg under equilibrium conditions. This state has been termed the ‘crystalline reference system’ (CRS). In the temperature interval from absolute zero to Tg, the rigid glass differs from the CRS by an enthalpy Hvit of vitrification and entropy Svit of vitrification. In the same way, the melt at liquidus temperature Tliq differs by an enthalpy Hfus of fusion and entropy Sfus of fusion. The glass and the melt are regarded as a mixture of glassy and melted compounds k respectively. Heats (enthalpies) and entropies of mixing, which usually make very large contributions in silicate systems if referred to the oxide components j, become negligibly small if referred to the CRS compounds k. The crucial step is the identification of the appropriate set of compounds k. An adequate strategy is developed by exploiting two fundamental principles found to be valid in the mineral world. These are as follows. 1
2
The principle of majority partition. By experience, even complicated multicomponent systems, such as magmatic and igneous rock melts, metallurgical slags and commercial glasses, can be represented by a predominant quaternary typically consisting of more than 85–95% of the oxides on a molar basis. The principle of parsimony. The very large number of combinatorial possibilities of compound formation is not exploited by nature. Rather, quite a limited set of binary and ternary compounds is found. The constitutional relations in a given multicomponent system are therefore approximated in the following way. First, the minority oxides are allotted
The industrial glass-melting process
287
to a set of normative phases as suggested by the CIPW norm calculation (see, for example, the book by Philpotts [90Phi]). The remaining four majority oxides are allotted to the respective constitutional subrange in the predominant quaternary identified and reconstructed by the evaluation of existing phase diagrams. The details have been described by Conradt [004Con]. According to Gibbs phase rule, the number j of oxides in a glass composition is identical with the number k of compounds in the corresponding CRS; the molar amounts n or masses m of j and k (given in kilomoles or kilograms respectively per 100 kg of glass) are thus related by a linear equation system nj = (vjk) nk ⇒ nk = (Bkj) n
(II.24.6a)
mj = (µjk) mk ⇒ mk = (Akj)mj
(II.24.6b)
(Akj) = (Vjk)–1, (Bkj) = µjk)–1
(II.24.6c)
Here, vjk is the matrix element stating how many moles of oxide j are found in compound k; µjk states how many kilograms of oxide j are contained in 1 kg of compound k. Akj and Bkj are the elements of the inverted matrices (vjk) and (µjk) respectively. As an example, Table II.24.1 presents the main oxides j and compounds k, and the matrix elements Bkj used for E glass compositions. The composition of an E glass depicted as an example and reference [003Sew] is shown in Table II.24.2 in terms of both oxides j and compounds k. The thermodynamic quantities of a glass or its melt are obtained from the following set of equations: ° H glass = Σ n k ( H k° + H kvit )
(II.24.7a)
° ° H1673,melt = Σ n k H1673,melt, k
(II.24.7b)
° Sglass = Σ n k ( Sk° + Skvit )
(II.24.7c)
° ° S1673,melt = Σ n k S1673,melt, k
(II.24.7d)
c P ,melt = Σ n k c P ,melt, k
(II.24.7e)
° H T ,melt = H1673,melt + c P ,melt ( T – 1673)
(II.24.7f)
k
k
k
k
k
° ST ,melt = S1673,melt + c P ,melt ln
T ( 1673 )
(II.24.7g)
° is the standard enthalpy (heat) of the rigid glass (at 25 °C and 1 bar), H glass ° H1673,melt is the heat of the melt at 1400 °C (= 1673.15 K); HT,melt is the heat of the melt at arbitrary temperature T; the entropies S have analogous meanings; cP,melt is the heat capacity of the melt above Tliq. The quantities of the individual
288
Compound k
SiO2 CaO · TiO2 CaO · Al2O3 · 2SiO2 B2 O3 FeO · Fe2O3 CaO · MgO · 2SiO2 CaO · SiO2 Na2O · Al2O3 · 6SiO2 K2O · Al2O3 · 6SiO2
(µ jk) for the following oxides j SiO2
TiO2
Al2O3
B2 O3
Fe2O3
MgO
CaO
Na2O
K2 O
1.000 — — — — — — — —
0.752 1.702 — — — — –1.454 — —
–0.589 — 2.729 — — — –1.139 — —
— — — 1.000 — — — — —
— — — — 1.000 — — — —
–1.491 — — — — 5.372 –2.882 — —
–1.071 — — — — — 2.071 — —
–4.847 — –4.489 — — — 1.874 8.462 —
–3.189 — –2.953 — — — 1.233 — 5.909
The SGTE casebook
Table II.24.1 Matrix (µjk) for the calculation of the normative compounds k of E glasses from their oxide composition given by the amounts mj of oxides j in kilograms per kilogram of glass; the calculation proceeds like m(k = SiO2) = 1.000m(SiO2) + 0.752m(TiO)2 – 0.589m(Al2O3) –1.491m(MgO) –1.071m(CaO) –4.847m(Na2O) –3.189m(K2O); m(k = CaO · TiO2) = 1.702m(TiO2), etc.
Table II.24.2 Compositions of a reference E glass [003SEW] given in terms of both oxides j and normative compounds k; M is the molar mass in kilograms per kilomole; m is the mass in kilograms per 100 kg of glass; n is the molar amount in kilomoles per 100 kg of glass Mj
SiO2 TiO2 Al2O3 B2 O3 Fe2O3 FeO MgO CaO Na2O Sum
60.084 79.898 101.961 69.619 159.691 71.846 40.311 56.079 61.979
mj 55.15 0.57 14.42 6.86 0.44 4.22 17.73 0.61 100.00
nj
Compound k
Mk
mk
nk
0.9179 0.0071 0.1414 0.0985 0.0055 0.0055 0.1047 0.3162 0.0098
SiO2 CaO · TiO2 CaO · Al2O3 · 2SiO2 B2 O3 FeO · Fe2O3 FeO · SiO2 CaO · MgO · 2SiO2 CaO · SiO2 Na2O · Al2O3 · 6SiO2
60.084 135.977 278.208 69.619 231.537 131.930 216.558 116.163 524.444
18.77 0.97 36.61 6.86 0.34 0.15 22.67 8.45 5.16 99.98
0.3124 0.0071 0.1316 0.0985 0.0015 0.0011 0.1047 0.0728 0.0098
The industrial glass-melting process
Oxide
289
290
The SGTE casebook
compounds k used in Equations (II.24.7a)–(II.24.7g) are compiled in Table II.24.3. This table allows us to calculate the properties not only of E glasses, but also of A-fibre, C-fibre, stone and slag wool, crystal, low-expansion, container and float glasses. For an appropriate determination of the CRSs of the different types of industrial glass, see the paper by Conradt [004Con]. The following data give an impression of the accuracy of the approach: the Gibbs energies of formation were calculated for four different mineral wool glasses. The values were checked by calorimetry by an independent laboratory [004Ric], yielding the following experimental versus calculated values for the standard Gibbs energies of formation from the elements (in kilojoules per mole of oxides): –852.0 versus –849.6; –865.0 versus –867.2; –880.8 versus –881.8; –855.4 versus –852.7. The standard Gibbs energies of formation from the oxides are as follows: –12.9 versus –10.6; –35.4 versus –37.7; –34.0 versus –35.0; –44.1 versus –41.4.
II.24.2.3 Heat content of glass melts For our reference E glass (see Table II.24.2), the following results are obtained: H° = –15,112 kJ kg–1 = –4197.8 kW h t–1 Hvit = 291 kJ kg–1 = 80.9 kW h t–1 ° H glass = –14 821 kJ kg–1 = –4116.9 kW h t–1
H1673,liq = –13 039 kJ kg = –3621.8 kW h t–1 cP,liq = 1454 J kg–1 K–1 = 404.0 W h t–1K–1 Svit = 135 J kg–1K–1 = 37.6 W h t–1K–1 All quantities are given in the SI units J, kg and K. In order to allow an easy comparison with electrical energy, the quantities are also given in k W h t–1 and W h t–1 k–1 for heats and entropies respectively; 1 t = 1000 kg. From the above data, a number of data with high practical importance are derived. As an immediate example, the heat content of a given glass melt (relative to 25 °C) at arbitrary temperature T is given by ° ∆H T ,melt = H T ,melt – H glass
(II.24.8)
For our reference E glass, the following results are obtained: ∆HT,liq = 1782 kJ kg–1 = 495.0 kW h t–1 for 25–1400 °C = 1636 kJ kg–1 = 454.6 kW h t–1 for 25–1300 °C For a glass-melting process with a pull temperature Tex = 1300 °C, the last line represents the value ∆Hmelt(Tex) in Equation (II.24.1). Thus, an essential
Table II.24.3 Thermodynamic data of compounds k employed to represent the CRSs of industrial glasses. The superscripts have the following meanings: degree(°), standard state at 298.15 K and 1 bar; vit, vitrification. The subscripts have the following meanings: melt, liquid state; 1673, 1673.15 K. (Origin: multiple sources) S° (J mol–1K–1)
H vit (kJ mol–1)
S vit (kJ mol–1 K–1)
–H 1673,melt (kJ mol–1)
S 1673,melt (kJ mol–1 K–1)
cP, melt (kJ mol–1K–1)
P2O5·3CaO P 2O5 Fe2O3 FeO·Fe2O3 FeO·SiO2 2FeO·SiO2 MnO·SiO2 2ZnO·SiO2 ZrO2·SiO2 CaO·TiO2 TiO2 BaO·Al2O3·2SiO2 BaO·2SiO2 BaO·SiO2 Li2O·Al2O3·4SiO2 Li2O·SiO2 K2O·Al2O3·6SiO2 K2O·Al2O3·2SiO2 K2O·4SiO2 K2O·2SiO2 Na2O·Al2O3·6SiO2 Na2O·Al2O3·2SiO2 B2 O3 Na2O·B2O3·4SiO2 Na2O·4B2O3 Na2O·2B2O3
4117.1 1492.0 823.4 1108.8 1196.2 1471.1 1320.9 1643.1 2034.7 1660.6 903.7 4222.1 2553.1 1618.0 6036.7 1648.5 7914.0 4217.1 4315.8 2508.7 7841.2 4163.5 1273.5 5710.9 5902.8 3284.9
236.0 114.4 87.4 151.0 92.8 145.2 102.5 131.4 84.5 93.7 185.4 236.8 154.0 104.6 308.8 79.9 439.3 266.1 265.7 190.6 420.1 248.5 54.0 270.0 276.1 189.5
135.1 18.2 45.2 82.8 36.7 55.2 40.2 82.4 86.6 67.4 40.2 130.5 81.6 56.5 184.1 16.7 106.3 80.4 26.4 12.6 125.0 92.0 18.2 42.7 58.3 48.8
51.5 9.5 17.2 31.4 13.8 20.5 15.1 31.4 32.6 25.5 19.7 95.4 26.8 41.0 12.1 6.3 29.3 22.1 21.3 23.9 28.4 27.9 11.3 21.1 40.1 26.6
3417.1 1138.5 550.2 677.8 962.3 1118.8 1085.3 1261.1 1686.2 1365.7 741.0 3454.3 2171.1 1349.8 5235.4 1416.7 6924.9 3903.7 3697.8 2153.1 6870.1 3614.1 1088.7 4988.0 4986.7 2735.9
898.7 586.6 370.3 579.9 342.7 512.1 345.2 494.5 381.2 360.2 335.6 1198.3 533.5 361.1 1173.2 339.7 1559.4 666.5 983.7 595.4 1512.5 856.9 271.1 1090.2 1275.5 780.3
324.3 181.6 142.3 213.4 139.7 240.6 151.5 174.5 149.4 124.7 87.9 473.2 241.4 146.4 498.7 167.4 765.7 517.6 410.0 275.3 648.1 423.8 129.7 637.6 704.2 444.8
291
–H ° (kJ mol–1)
The industrial glass-melting process
k
292
k
–H ° (kJ mol–1)
S° (J mol–1K–1)
H vit (kJ mol–1)
S vit (kJ mol–1 K–1)
–H 1673,melt (kJ mol–1)
S 1673,melt (kJ mol–1 K–1)
Na2O·B2O3 2MgO·2Al2O3·5SiO2 MgO·SiO2 2MgO·SiO2 CaO·MgO·2SiO2 2CaO·MgO·2SiO2 CaO·Al2O3·2SiO2 2CaO·Al2O3·SiO2 3Al2O3·2SiO2 CaO·SiO2 2CaO·SiO2 Na2O·2SiO2 Na2O·SiO2 3Na2O·8SiO2 * Na2O·3CaO·6SiO2 Na2O·2CaO·3SiO2 2Na2O·CaO·3SiO2 Na2O·CaO·5SiO2 * SiO2
1958.1 9113.2 1548.5 2176.9 3202.4 3876.9 4223.7 3989.4 6820.8 1635.1 2328.4 2473.6 1563.1 9173.0 8363.8 4883.6 4763.0 5934.0 908.3
147.1 407.1 67.8 95.4 143.1 209.2 202.5 198.3 274.9 83.1 120.5 164.4 113.8 597.0 461.9 277.8 309.6 349.0 43.5
43.6 135.8 46.6 61.4 92.3 106.7 103.0 129.9 188.3 49.8 101.3 29.3 37.7 94.2 77.3 57.7 87.0 63.3 6.9
19.5 41.4 13.6 11.0 25.7 32.0 37.7 49.4 71.5 18.8 38.5 13.2 9.8 34.2 20.5 13.4 22.6 30.4 4.0
1585.7 7994.8 1318.0 1876.1 2733.4 3319.2 3628.8 3374.0 5816.2 1382.0 1868.2 2102.5 1288.3 – 7372.6 4240.9 4029.6 – 809.6
538.7 1606.2 296.2 402.9 621.7 775.3 791.2 787.8 1231.8 329.7 509.2 588.7 415.1 – 1555.6 990.4 1107.9 – 157.3
*Only found in highly pure ternary Na2O–CaO–SiO2.
cP, melt (kJ mol–1K–1) 292.9 1031.8 146.4 205.0 355.6 426.8 380.7 299.2 523.4 146.4 174.5 261.1 179.1 – 786.6 470.3 501.2 – 86.2
The SGTE casebook
Table II.24.3 (Continued)
The industrial glass-melting process
293
part in determining Hex has been accomplished. What is left is the determination ° . of the chemical term ∆H chem An additional comment should be made. It is true that the heat content of a melt may also be estimated from existing oxide increment systems [55Zie], [58Moo], [85Gud]. As shown in Table II.24.4 for the example of a mineral fibre glass, however, the direct thermodynamic approach is more accurate. Since the increment systems are based on quite a restricted composition range only, the thermodynamic approach is also more versatile compositionally.
II.24.3
The batch-to-melt conversion
II.24.3.1 Heat demand of the batch-to-melt conversion; simple raw materials Earlier work [53Kro] on the calculation of the heat demand of batch melting yielded considerable success for batches with a small number of chemically pure raw materials. The former calculation strategy was based on the formulation of a gapless sequence of chemical and physical reactions linking the stage of the batch at 25 °C to the stage of the glass melt at a given temperature. For a realistic industrial batch, this is virtually impossible to accomplish. Beyond this, the strategy gives up a noble principle of thermodynamics, i.e. the path independence of the properties of thermodynamic states. With the successful thermodynamic quantification of the states of industrial glasses and glass melts (Sections II.24.1 and II.24.2), we may fully exploit the principle of path independence and present the batch-tomelt conversion by the hypothetical reaction, given in Equation (II.24.2). The energy difference between the right- and left-hand sides of Equation (II.24.2) is the standard heat of formation of glass and batch gases from the ° also termed the chemical heat demand of raw materials, denoted by ∆H chem ° batch melting. ∆H chem is calculated as ° ° ° ° ∆H chem = H glass + H gas – H batch
(II.24.9)
Table II.24.4 Heat content ∆HT,liq of a mineral fibre glass melt with a composition of 58.2 wt% SiO2, 1.1 wt% Al2O3, 3.4 wt% Fe2O3, 9.0 wt% MgO, 23.5 wt% CaO, 4.6 wt% Na2O and 0.2 wt% K2O at different temperatures; calculated and experimental values (inverse drop calorimetry) ∆HT,liq (kW h t–1) for the following T
After Schwiete and Ziegler [55Sch] After Moore and Sharp [58Moo] After Gudovich and Primenko [85Gud] Own model Experimental value (± 21)
1408 °C
1360 °C
1352 °C
448 441 391 465 472
429 424 374 447 445
426 421 371 444 440
294
The SGTE casebook
° ° ° and H batch where H glass is determined from Equation (II.24.7a) and H gas are the weighted sums of standard heats of the individual batch gases and raw materials respectively. For simple batches containing chemically pure raw materials, Equation (II.24.9) may be evaluated in a straightforward way. Table II.24.5 summarises earlier calorimetric results [56Kro] on the standard heats of formation of Na2O–CaO–SiO2 glasses from the pure raw materials quartz, calcite and soda ash. The calculated values are obtained by characterising the glasses as described before. The good agreement between calculation and experiment shows that additional mixing terms can be neglected even in these sodium-rich systems. Note that, for the above scientific rather than technical glass compositions, the true equilibrium phases N3S8 and NCS5 (see [98ACS, No. 5321]) were taken into account as normative phases of the CRS. These phases, however, are identified in very pure systems and under very slow cooling only. Otherwise, the metastable substitutes NS2 and NC3S6 are formed. Therefore, for industrial soda–lime-based mass glasses, the oxides Na2O, CaO and SiO2 are always allotted to the normative phases NS2, NC3S6 and S. Since this important class of glass composition stems from a narrow composition range only, the solution of Equation (II.24.6a)–Equation (II.24.6c) can be presented in a straightforward way by
NAS6 = 5.1440 Al2O3 – 5.5697 K2O KAS6 = 5.9102 K2O Hm = 0.6 Fe2O3 FS = 0.7345 Fe2O3 MS = 2.4907 MgO NC3S6 = 3.5112 CaO NS2 = 2.9386 Na2O + 1.9346 K2O – 1.7867 Al2O3 – 1.0824 CaO S = rest to the total mass All quantities given in mass amounts; neutral redox conditions of the melt assumed. Table II.24.5 Heats ∆H f of formation of N–C–S glasses (N = Na2O; C = CaO; S = SiO2) from pure quartz, calcite and soda ash: calculated and experimental data. For each glass, the corner compounds of the respective constitutional subranges are given ∆Hf (kJ/100 g)
∆H f (kJ/100 g)
SiO2
CaO
Na2O
CRS
Calculated [56Kro]
Experimental
74.1 75.3 71.0
10.1 11.7 13.8
15.8 13.0 15.3
N3S8–NCS5–S NC3S6–NCS5–S NS2–NC3S6–NCS5
50.5 47.5 53.2
51.3 ± 0.8 49.3 ± 1.0 55.7 ± 1.0
The industrial glass-melting process
295
Until now, we have been approximating the raw materials by pure chemical substances. This may look like an acceptable simplification, but the contrary is the case; the formation data of some natural raw materials deviate from those of their chemically pure counterparts in a considerable way. In principle, each of such raw materials represents an individual multicomponent minerals system of its own and has to be treated this way. For different feldspar and sand qualities, the standard enthalpies in units of kilojoules per gram are directly found from H° = 15.174 SiO2 + 17.100 Al2O3 + 4.633 Fe2O3 + 15.030 MgO + 12.035 CaO + 10.52 Na2O + 7.639 K2O which is again an easy and straightforward solution of Equation (II.24.6a)– Equation (II.24.6c) for the narrow compositional range typical of these minerals. The oxide amounts have to be inserted in grams per 100 g of the mineral. Other natural raw materials require more attention. In the following section, glass-grade dolomites and limestones are treated as examples.
II.24.3.2 Dolomite and limestone as examples of complex raw materials Dolomites and limestones are the minerals typically used as carriers of MgO and CaO. They are added to the batch as carbonates or, alternatively, in their partially or fully calcined form as ‘dolime’ MgO + CaCO3 or as burned dolomite MgO + CaO and burnt lime CaO respectively. The decomposition of alkaline-earth carbonates makes the largest contribution to the chemical heat demand of the batch-to-melt conversion. Thus, from the point of view of on-site production efficiency, it may be advisable to use partially or fully calcined products in the batch. In order to assess an accurate value for ° , reliable data on the energy situation of limestone and dolomite are ∆H chem required. When, however, inspecting literature data, there is a striking uncertainty, especially with respect to the heats of formation of dolomite. This issue deserves a closer look. In Table II.24.6, the heats of formation of dolomite from the elements as taken from several renowned databases are contrasted. The uncertainty amounts to 30 kJ mol–1, which is equivalent to 163 kJ per kilogram of dolomite or 311 kJ per kilogram of MgO + CaO equivalent. From the point of view of mineralogy, natural dolomite and limestone are not pure phases, but rather minerals from the system CaCO3–MgCO3–FeCO3–MnCO3, accompanied by minor amounts of quartz, olivine and feldspatic minerals. They display a most complex polycrystalline microstructure of coexisting carbonates, even in the individual grains, ranging form coarse to fine and cryptocrystalline phases. Figure II.24.3 illustrates the phase relations in the ternary subsystem
296
The SGTE casebook Table II.24.6 Compilation of literature data on the standard heat H ° of formation of dolomite from the elements H ° kJ (mol CaMg(CO3)2)–1
Kind
Source
–2315.0 ± 5.0 –2329.9 –2331.7 –2324.5 –2314.2 ± 0.5 –2300.2 ± 0.6 –2325.7 –2326.3 –2317.6 –2329.9
Unspecified Unspecified Unspecified Unspecified Disordered Ordered Unspecified ‘CaCO3 · MgCO3’ Disordered Ordered
[93Kub] [90Phi] [85Bab] [78Rob] [87Nav, 95Cha, 96Cha] [87Nav, 95Cha, 96Cha] [93Sax, 004GTT] [000Roi] [000Roi] [000Roi]
CaCO3 One phase: Ca-rich ss
Two phases: Ca-rich ss, dolomite ss
One phase: disordered or cation-ordered dolomite
CaMg(CO3)2
Two phases: Ca-rich ss, Ca-poor ss
Three phases: Ca-rich ss, Ca-poor ss, dolomite ss
Two phases: Ca-poor ss, dolomite ss
CaFe(CO3)2
II.24.3. Phase diagram of the system CaCO3–CaMg(CO3)2–CaFe(CO3)2 showing the stability fields of one-, two- and three-phase equilibria; ss, solid solution.
CaCO3–CaMg(CO3)2–CaFe(CO3)2, redesigned after data from [98ACS, Nos. 2753 and 4664]. According to Fig. II.24.3, calcite may dissolve considerable amounts of Mg. By contrast, dolomite may dissolve much Fe (such a dolomite would not be used in glass industry), but hardly any excess Ca. Thus, a natural glass-grade dolomite always contains at least two kinds of phase, i.e. Mg-saturated limestone and Ca-saturated dolomite.
The industrial glass-melting process
297
A very careful study [87Nav, 95Cha, 96Cha] may help to resolve the discrepancies found in Table II.24.6. As can be expected from Fig. II.24.3, even small amounts of Ca excess in dolomite considerably shift the resulting heat of formation of the mineral. The Fe versus Mg substitution yields less strong effects. The smallest shift is observed for Mg excess in pure limestone. Let us consider the heat of formation of Ca1+xFe(1–x)yMg(1–x)(1–y)(CO3)2. For x = 1, the formula denotes pure limestone, given as Ca2(CO3)2. For x = 0, it is CaFeyMg1–y(CO3)2 with pure dolomite and pure ankerite as end members (y = 0 and y = 1 respectively). Then, based on the results given by Navrotsky and Capobianco [87Nav], Chai et al. [95Cha] and Chai and Navrotsky [96Cha], the standard enthalpies of formation, given in units of kilojoules per mole of formula unit, are calculated from the chemical composition as * H dolo = – 2314.2 + 129.4 x + 74.0 y
(II.24.9)
for the one-phase dolomite solid solution and * H lime = – 2413.8 – 9.97(1 – x )
(II.24.10)
for the one-phase limestone solid solution. The stoichiometric coefficients x and y are derived from the analytically determined mass ratio u of MgO to CaO and mass ratio v FeO to MgO as
x = 0.7188 – u 0.7188 + u
(II.24.11a)
v 1.7832 + v
(II.24.11b)
y=
with the values 0.7188 and 1.7832 representing the molar mass ratios of MgO to CaO and of FeO to MgO respectively. Thus, the standard enthalpy of a natural dolomite and limestone can be swiftly calculated from analytical data.
II.24.3.3 Modelling the batch-to-melt conversion With the principles explained in Sections II.24.3.1 and II.24.3.2, we may ° of the exploited now complete our task and determine the chemical part ∆H chem heat Hex, (Equation (II.24.1)). In Table II.24.7, the calculation procedure is demonstrated for two different batches, namely batch 1 using dolomite and limestone, and batch 2 using fully burned dolomite and lime, yielding a glass ° identical with our reference E glass (see Table II.24.2). The values of ∆H chem for both batches differ considerably. When glass cullets are added to the batch (which is standard industrial ° , practice), then the chemical contribution to Hex is reduced to (1 – yC) ∆H chem where yC denotes the mass fraction of cullets referred to the total mass of
298
Sand Al2O3 3H2O · B2O3 Na2O · 2B2O3 · 5H2O Dolomite Burned dolomite Limestone Burned lime I: Sum of batch CO2 H 2O II: Sum of gases III: Glass ° = I + II + III ∆H chem
M
H°
(g mol–1)
(kW h kg–1)
60.084 101.961 123.664 291.292 184.410 96.390 100.089 56.079
–4.2112 –4.5652 –4.9152 –4.5676 –3.4859 –3.5634 –3.3495 –3.1449
44.010 18.015
–2.4837 –3.7284 –4.1065
° ∆H chem
Batch 1 (kg) –562.00 –144.00 –97.30 –28.90 –192.80 — –211.50 — –1236.50 185.04 51.46 236.50 1000.00
Batch 1 (kW h) 2366.7 657.4 478.2 132.0 672.1 0.0 708.4 0.0 5014.8 –459.6 –191.9 –651.4 –4106.5 256.9
Batch 2 (kg)
Batch 2 (kW h)
–562.00 –144.00 –97.30 –28.90 — –100.80 — –118.50 –1051.50 0.00 51.50 51.50 1000.00
2366.7 657.4 478.2 132.0 0.0 359.2 0.0 372.7 4366.2 0.0 –192.0 –192.0 –4106.5 67.7
The SGTE casebook
° Table II.24.7 Calculation of the chemical heat demand ∆H chem of two different batches given in amounts per 1000 kg of glass, both yielding the reference E glass (see Table II.24.2). M is the molar mass, and H° is the standard enthalpy
The industrial glass-melting process
299
produced glass. With yC = 0.2 and a value of 454.6 kW h t–1 for ∆Hmelt at Tex = 1300 °C, the exploited heat for the two batches in Table II.24.7 amounts to 660 kWh per tonne of produced glass and 509 kW h per tonne of produced glass respectively. Hex is an integral quantity referring to the entire melting process. The principles elaborated before can also be used to give an approximate image of the reaction path itself. This is demonstrated in Table II.24.8 for a simple soda–lime–silicate glass batch. The path starts from the batch at 25 °C and passes through milestone states reached at arbitrarily selected temperatures. These are the melting temperature of the soda ash (860 °C) as the temperature of primary melt formation, the decomposition temperature of the limestone (900 °C), the liquidus temperature of the glass melt (960 °C), plus two process temperatures (the maximum temperature in the basin (1400 °C), and the pull temperature Tex = 1200 °C). The heat balance is given in terms of the enthalpy difference ∆H from the initial state. Thus, the ∆H value given in the ° . As a special last column is identical with the chemical heat demand ∆H chem feature, the viscosity of the melt is calculated for every state. It is interesting to note that, in spite of a steady increase in temperature from 860 to 1400 °C, the viscosity does not decrease steadily but rather passes through minima and maxima. Simultaneously, the mass fraction ysolid of solid matter in the melt decreases steadily. In soda–lime glass batches, silica is usually the phase that dissolves last. Finally, it is demonstrated what additional information may be gained by employing a sophisticated commercial program and database such as Fact. Sage [004GTT]. Surprisingly enough, some of these programs do not make any use of the well-established experimental experience of phase coexistence at 298 K but rather have the ambition to calculate such phase coexistence relations at 298 K from the fundamental thermodynamic data of individual phases. This makes it especially difficult to describe multicomponent frozenin phases (i.e. glasses) at 298 K. On the other hand, such concepts enfold their strengths in presenting partially crystalline equilibrium stages in the range from the appearance of the first liquid phase towards complete melting. Figure II.24.4 gives an example, again referring to the E glass composition given in Table II.24.2 and to batch 1 in Table II.24.7. The technologically most relevant liquidus temperature of the E glass is determined as 1245 °C; the corresponding primary phase is anorthite. Another most significant temperature level is the temperature at which solid silica disappears (1150 °C). Thus, for this E glass batch, the sand grains may dissolve via chemical driving forces even before the liquidus of the system is reached. At this point, the melt has a composition of 56.9 wt% SiO2, 11.9 wt% Al2O3, 7.7 wt% B2O3, 0.49 wt% Fe2O3, 4.74 wt% MgO, 17.56 wt% CaO and 0.69 wt% Na2O and a viscosity of 103.27 dPa s. Let us also discuss what happens when such a batch is heated under industrial non-equilibrium conditions. The hydrous
Solid phases Quartz l/h Limestone Soda ash Cristobalite Total solids Liquid or glassy components Soda ash Na2O·SiO2 Na2O·2SiO2 Na2O·3CaO·6SiO2 SiO2 Total melt Batch gases CO2
25 °C
860 °C
900 °C
960 °C
1200 °C*
740.0 178.5 273.6 — 1192.1
740.0 178.5 — — 918.5
— 178.5 — 480.8 659.3
— — — 287.0 287.0
— — — — — —
273.6 — — — — 273.6
— 103.6 315.6 — — 419.2
—
—
113.6
Heat ∆H; content ysolid of dispersed solids; viscosity η of liquid phase ∆H (kW h/1000 kg of glass) 0 350 431 ysolid 1.00 0.77 0.66 log [η(melt) (dPa s)] — –2.0 1.9
1400 °C
1200 °C†
— — — 287.0 287.0
— — — —
— — — —
— — — —
— — 361.9 351.1 — 713.0
— — 361.9 351.1 — 713.0
— — 361.9 351.1 287.0 1000.0
— — 361.9 351.1 287.0 1000.0
— — 361.9 351.1 287.0 1000.0
192.1
192.1
192.1
192.1
192.1
471 0.29 4.3
585 0.29 1.8
681 0 2.1
589 0 2.8
134 0 —
25 °C
* Under equilibrium conditions, the amount of solid phases at T > Tliq is zero; in the industrial process, however, the residual silica dissolves at a considerable rate only if log η < 2–3. † Pull temperature Tex.
The SGTE casebook
Amount (kg/1000 kg of glass) for the following temperatures
300
Table II.24.8 Reaction path of a simple soda–lime–silicate glass batch yielding a glass of 74 wt% SiO2, 10 wt% CaO and 16 wt% Na2O; the reaction path is given as a function of temperature in amounts of solid phases, CO2 gas and melt (Tliq = 960 °C) and reaches from the batch at 25 °C (second column) to the glass at 25 °C (last column). ∆H is the enthalpy difference from the cold batch, ysolid is the mass fraction solid/(solid + melt) and η is the viscosity
The industrial glass-melting process
301
100
Mass (g/100 g of glass)
80
60
Melt The batch: Sand
CAS2
40
S Limestone 20
0 0
Rest Dolomite CMS2 H3BO3 Melt Borax 5H2O
200 TEU of B2O3–SiO2
400
600 800 Temperature (°C)
Tliq CS 1000
1200
1400
Tliq of B2O3
II.24.4 Phase coexistence in the E glass batch 1 shown in Table II.24.7 as a function of temperature; the hydrous boron carriers dehydrate well below 200 °C; a primary molten phase occurs between 350 and 450 °C; under industrial non-equilibrium conditions, this melt is absorbed by the limestone and dolomite to form solid calcium– magnesium borates, and the batch may remain ‘dry’ even up to Tliq.
boron carriers dehydrate well below 200 °C, which is only slightly above the equilibrium decomposition temperatures. A primary molten phase in the batch occurs at the melting point of B2O3 (450 °C) at the latest. In the presence of limestone and dolomite, however, this melt may be resorbed to form solid magnesium–calcium borates. So the batch may remain a granular bulk solid even up to the liquidus temperature of the system. This unfavourable behaviour is well known for E glass batches. As a consequence, information on a reaction path cannot be obtained from equilibrium data without hesitation. Nevertheless, the determination of the liquidus temperature alone makes it worthwhile to perform such calculations.
II.24.4
Conclusions
The present author hopes that this chapter will encourage glass technologists to make increasing use of thermodynamic calculations to optimise their processes. In this chapter, only one (however, important) issue was elaborated. This is the accurate determination of the heat involved in the glass-melting process. Many other useful tasks can also be accomplished by using thermodynamic calculations, e.g. the quantitative description and potential optimisation of the following:
302
– – –
The SGTE casebook
The fining and refining process. Evaporation processes from the melt. Corrosion processes between melt and refractories, or between vapours above the melt and refractories.
These processes altogether are undoubtedly of high relevance to every glass technologist.
II.24.5 53Kro
55Sch
56Kro 58Moo 78Rob
84Col
85Bab 85Gud 86Bla
87Nav 90Bon 90Eri 90Phi 93Kub 93Sax 94Sha
References C. KRÖGER: ‘Theoretischer Wärmebedarf der Glasschmelzprozesse’ (‘Theoretical heat demand of the glass melting processes’), Glastech. Ber. 26, 1953, 202– 214. H.E. SCHWIETE and G. ZIEGLER: ‘Beitrag zur spezifischen Wärme der Gläser’, (‘Contribution to specific heat of glasses’), Glastech. Ber. 28, 1955, 137– 146. C. KRÖGER and G. KREITLOW: ‘Heats of solution and formation of silicates of sodium and calcium’, Glastech. Ber. 29, 1956, 393–400. J. MOORE and D.E. SHARP: ‘Note on calculation of effect of temperature and composition on specific heat of glass’, J. Am. Ceram. Soc. 41, 1958, 461–463. R.A. ROBIE, B.S. HEMINGWAY and J.R FISHER: ‘Thermodynamic properties of minerals and related substances at 298.15 K and 1 bar (105 Pascals) pressure and at high temperatures’, Geol. Survey Bull., US Government Printing Office, Washington, DC, 1978. H. GAYE, ‘Donneés thermochimiques et cinétiques relatives à certains matériaux sidérurgiques’, in Commission de la Communautés Européennes Convention CEEC 7210-CF/301 TCM-RE 1064 (Eds H. Gaye and D. Colombet), Brussels, Belgium, 1984. V.I. B ABUSHKIN , G.M. M ATVEYEV and O.P. M CHEDLOV -P ETROSSYAN : Thermodynamics of Silicates, Springer, Berlin, 1985. O.D. GUDOVICH and V.I. PRIMENKO: ‘Calculation of the thermal capacity of silicate glasses and melts’, Soc. J. Glass Phys. Chem. 11, 1985, 206–211. A.D. PELTON and M. BLANDER: ‘Thermodynamic analysis of ordered liquid solutions by a modified quasi-chemical approach—application to silicate slags’, Metall. Trans. B 17, 1986, 805–815. A. NAVROTSKY and C. CAPOBIANCO: ‘Enthalpies of formation of dolomite and magnesia calcites’, Am. Mineralogist 72, 1987, 782–787. D.W. BONNEL and J.W. HASTIE: ‘Ideal mixing of complex components’, High Temp. Sci. 26, 1990, 313–334. G. ERIKSSON and K. HACK: ‘ChemSage – a computer program for the calculation of complex chemical equilibria’, Metall. Trans. B 21, 1990, 1013. A.T. PHILPOTTS: Principles of Igneous and Metamorphic Petrology, PrenticeHall, Englewood Cliffs, New Jersey, 1990. O. KUBASCHWESKI, C.B. ALCOCK and P.J. SPENCER: Materials Thermochemistry, Pergamon, Oxford, 1993. S.K. SAXENA, N. CHATTERJEE, Y. FEI and G. SHEN: Thermodynamic Data on Oxides and Silicates, Springer, Berlin, 1993. B.A. SHAKHMATKIN, N.M. VEDISHCHEVA, M.M. SCHULTZ and A.C. WRIGHT: ‘The
The industrial glass-melting process
95Cha 95Ric 96Cha
98ACS 99Con
000Roi 001Con 003Sew
004Con
004GTT 004Ric
303
thermodynamic properties of oxide glasses and glass forming liquids and their chemical structure’, J. Non-Crystalline Solids 177, 1994, 249–256. L. CHAI, A. NAVROTSKY and R.J. REEDER: ‘Energetics of calcium-rich dolomite’, Geochim. Cosmochim. Acta 59, 1995, 939–944. P. RICHET and Y. BOTTINGA: ‘Rheology and configurational entropy of silicate melts’, Rev. Mineralogy 32, 1995, 6593. L. CHAI and A. NAVROTSKY: ‘Synthesis, characterization, and energetics of solid solution along the dolomite–ankerite join, an implication for the stability of ordered CaFe(CO3)2’, Am. Mineralogist. 81, 1996, 1141–1147. AMERICAN CERAMIC SOCIETY: Phase Equilibria Diagrams, CD-ROM Database. Version 2.1, American Ceramic Society, Westerville, Ohio, 1998. R. CONRADT: ‘Thermochemistry and structure of oxide glasses’, in Analysis of the Composition and Structure of Glass and Glass Ceramics (Eds H. Bach and D. Krause), Springer, Berlin 1999, pp. 232–254. A. ROINE: HSC Chemistry Software Version 3.0., Outokumpu Research Oy, Pori, Finland, 2000. R. CONRADT: ‘Modeling of the thermochemical properties of multicomponent oxide melts’, Z. Metallkunde 92, 2001, 1158–1162. T.P. SEWARD (principal investigator): ‘Modeling of glass making processes for improved efficiency’, High Temperature Glass Melt Property Database for Modeling, work conducted under US Department of Energy Grant DE-FG0796EE41262, Alfred University, Alfred, New York, 2003. R. CONRADT: ‘Chemical structure, medium range order, and crystalline reference state of multicomponent liquids and glasses’, J. Non-Crystalline Solids 345– 346, 2004, 16–23. GTT-Technologies: FactSage Software Version 5.2, Thermfact Montreal and GTT-Technologies Aachen, 2004. P. RICHET: Private communication.
II.25 Relevance of thermodynamic key data for the development of high-temperature gas discharge light sources T O R S T E N M A R K U S and U L R I C H N I E M A N N
II.25.1
Introduction
Electrical lighting became an important part of human life since the development of incandescent lamps at the end of the nineteenth century. Today, electrical light sources based on incandescent, gas discharge and solid-state lighting are used in various applications [72Ele, 97Coa].The role of lamp research is an essential part with respect to further improvements of lamp performance such as light quality and high efficiency [005Bor]. In 2000 the worldwide lighting market consisted of about 14 billion lamps, as shown in Fig. II.25.1. By far the largest contribution arises from halogen and incandescent lamps followed by fluorescent lamps. Recently, energysaving fluorescent lamps, also known as compact fluorescent lamps, have been introduced to the market. A third block is represented by high-pressure CFLs HID lamps (150) (300) FLs (2500)
Incandescent on halogen lamps (11000)
II.25.1 World market in 2000 for electrical light sources in a million pieces: incandescent on halogen lamps high-intensity discharge (HID) lamps, fluorescent lamps (FLs) and compact fluorescent (CFls).
304
Thermodynamic data for gas discharge light sources
305
discharge lamps with various applications for general lighting, e.g. shop lighting, office lighting, city beautification and road lighting. Another market segment includes lamps for special lighting purposes, such as projection lamps for beamers or automotive headlight lamps. The lighting market has a volume of about 25 billion Euros (advertising lighting excluded) and is growing annually by some per cent. Major players are Philips, Osram and General Electric, each having a market share of some 25–30% followed by a number of other manufacturers such as Ushio, Toshiba, Matsushita and Stanley. In the near future, highly efficient white or coloured light-emitting diodes are expected to be applied substantially in existing and new lighting applications competing with incandescent lamps, for example. However, in this chapter we restrict ourselves to gas discharge lamps. A major aspect of lamps is to convert electrical power efficiently into (visible) light. Worldwide about 1% of the primary energy, i.e. about 10% of the electrical energy, is used for electrical lighting. This value corresponds to energy of about 8 × 1011 kW h year–1. For example, this energy would be delivered by about 90 power plants each having a power of 1 GW. Thus, energysaving aspects of electrical light sources has a significant environmental impact.
II.25.2
Operation principle of high-intensity discharge lamps
In this chapter we would like to focus on high-intensity discharge (HID) lamps. Therefore here solely this type of high-temperature discharge lamp is described concerning the functionality. For a description of other lamp types, the reader should refer to the book by Born and Markus [005Bor] and the articles cited there. In high-pressure discharge lamps, also known as HID lamps, the operating pressure is about several 10 bar or even up to several hundred bars. Under these conditions, electron and heavy-particle temperatures are close to each other, typically in the range between 4000 and 10 000 K. In high-pressure Hg lamps the luminous efficiency is limited up to 60 lm W–1, owing to a lack of transitions in the visible spectrum. In addition, thermal and infrared losses are significant [001Hil]. High-pressure mercury lamps are widely applied, e.g. for industrial and special lighting applications. High-pressure Na lamps emit a more white light than the low-pressure lamps do because of spectral broadening of the Na D lines. Because of corrosion of the hot Na vapour with quartz, polycrystalline alumina (Al2O3) is used as a wall material. Application areas are outdoor or horticultural lighting, for example. An important variant of high-pressure mercury lamps are the so-called metal halide lamps. Here, Hg is still used as a buffer gas in order to adjust sufficiently large electrical field strengths. The radiation is emitted from various other metal atoms and molecules with low excitation energies. In multiline radiators,
306
The SGTE casebook
rare earths and associated elements such as Dy, Ho, Tm or Sc and Na are added to the lamp filling. Argon or xenon is used as a starting gas. In order to establish sufficiently large vapour pressures, elements are dosed as metal halides, e.g. NaI or DyI3, which are much more volatile than the pure metals. As a result of the large number of transitions in the visible spectrum, such plasmas emit multiline radiation with high luminous efficiencies of up to 100 lm W–1. Another class, the so-called three-band radiators, emit line radiation from Na I, Tl I and In I in the yellow, green and blue region, respectively. Metal halide lamps are used for a variety of indoor and outdoor applications, such as shop lighting or office lighting. At powers ranging between 35 and 1000 W they offer luminous efficiencies up to 100 lm W–1 with excellent colour quality and long lamp life of about 20 000 h. Discharge vessels of advanced light sources are made of translucent polycrystalline alumina. Higher operating temperatures are possible because of the use of polycrystalline alumina instead of quartz as discharge vessel material. These tubes contain salt mixtures which essentially consist of metal halides. The salt mixture vaporises partly under operating conditions. The condensed phase is present as a melt at the coldest spot of the vessel. The temperatures of the coldest spot range between 1300 and 1310 K depending on the lamp performance. The maximum wall temperatures range up to 1380 K and for special purposes up to 1700 K. Important constituents of the melt are alkali halides AX (A = Na or Cs; X= Br or I) and rare-earth metal halides LnX3 (Ln = Dy, Ho or Tm) because they are efficient radiators in the visible range of the emission spectrum. The vapour species of the metal halides enter the arc column and are used for the light emission. During prolonged burning of metal halide lamps, corrosion phenomena due to the reaction between the filling and the wall can be observed. Al2O3 is transported from hotter parts of the discharge vessel to colder parts on the wall. Figure II.25.2 shows a schematic representation of the polycrystalline alumina discharge vessel of a metal halide lamp with temperature distribution and the different corrosion processes. It shows the depletion of the Al2O3 at the hot parts of the vessel and its deposition at places of the wall with reduced temperatures. The aim of our work is to elucidate the corrosion and transport mechanisms of the alumina wall material and their influence on the gas phase chemistry. Studies on the high-temperature lamp chemistry have been reported by Hilpert and Niemann [97Hil]. The results of investigations of the corrosion attack of discharge vessels made of quartz glass have been given by Hilpert et al. [001Hil] and van Erk [000Erk].
II.25.3
Thermochemical modelling
Thermochemical modelling leads to a detailed insight of the chemical interactions. These interactions originate from evaporation of condensed
Thermodynamic data for gas discharge light sources
307
Interior with gaseous species e.g. Na, I, Hg, Tl, Dy, Ar and Xe
Wall material Al2O3 1340 K
1380 K
5000 K
1340 K
6000 K
Tungsten electrode 3800 K
3000 K
Wall corrosion 1310 K
1300 K Alumina deposition
Metal halide melt e.g. Nal/Tll/Dyl3
1300 K Tungsten deposition
II.25.2 Schematic representation of a discharge vessel made of polycrystalline alumina with the temperature distribution and corrosion processes.
phases as well as from corrosion reactions, such as from electrode and wall materials. Also, interaction of the liquid phases of the lamp filling with gaseous species must be taken into account. The simulation of the chemical equilibrium, which is assumed in HID lamps, is performed using the FactSage code [002Bal]. This tool is used for thermodynamic calculations based upon a minimisation of the Gibbs free energy. Complex equilibrium simulations are possible for systems with many components over a broad range of temperatures and pressures. In this section a short introduction into thermochemical basics with application to lamp research is given. In general, the basis of all thermodynamic simulations is to minimise the Gibbs free energy of the system under consideration. Hence, the first step is to express the Gibbs free energy for a chemical reaction as
Σ ν i Ai ↔ Σ ν f A f i
(II.25.1)
i
with Ai denoting the educts, Af the products formed, and νi and νf the stoichiometric coefficients. The Gibbs free energy is a function of pressure, temperature and stoichiometric coefficients: G = G(p, T, ν1, … ν n)
(II.25.2)
The derivative of Equation (II.25.2) can be written in its differential form dG = V dp – S dT + Σ µ i d ν i i
(II.25.3)
308
The SGTE casebook
At a minimum of G its derivative vanishes: dG = 0. In chemical equilibrium we assume that dp = 0 and dT = 0 so that G remains as a function of the chemical potentials µi. The chemical potentials are calculated as a function of temperature and pressure according to p µ i ( T , pi ) = µ i0 ( T ) + R T ln i p0
(II.25.4)
with pi for the partial pressures of species i, p0 = 105 Pa for the reference pressure and R = 8.314 J mol–1 K–1 for the gas constant. µ i0 is the chemical potential at reference pressure. Commercially available thermodynamic simulation tools are based upon different numerical procedures for minimising G assuming chemical equilibrium. As an example, the software package FactSage consists, among other parts of a database, of a numerical simulation tool and a data assessment module. User-defined databases may be implemented. Typically, HID lamp modelling involves a large number of various species and substances resulting in complex partial pressure distributions. Such data are used as input parameters for physical modelling of lamps. The key for the accomplishment of computer-based model calculations is the availability of reliable thermodynamic databases, where the basic thermodynamic data for the pure components and the interaction parameters for mixtures are stored. Reliable thermodynamic databases for lighting applications are rarely available. Building up private databases from literature data and one’s own experimental investigations is often the only possibility. However, this is a challenging time-consuming process. Details on demands for thermodynamic databases for lighting applications and experimental facilities to build up custom-made databases have been given in detail by Davies et al. [007Dav]. In Fig. II.25.3 an example of such a calculation is given for a rare-earth metal halide lamp filling. The composition of the salt mixture serves as an input parameter assuming an equimolar dose between NaI(s) and DyI3(s). FactSage computes the partial pressures of gaseous species, which are in equilibrium with the condensed phases as a function of indicated temperatures. Each value corresponds to an independent simulation run. Parameters of species formed during vaporisation are taken from the implemented custombuilt database. The heterocomplex NaDyI4 is formed during evaporation of NaI(s) and DyI3(s) according to the reaction NaI(g) + DyI3(g) ↔ NaDyI4(g)
(II.25.5)
Obviously, this heterocomplex is dominating with respect to partial pressure. Consequently, Dy and Na plasma densities are enhanced compared with those of the pure phases (DyI3 and NaI) by formation of NaDyI4 [97Hil]. The
Thermodynamic data for gas discharge light sources
309
101 I I2
100
Nal Na2l2
pi (bar)
10–1
Dyl2 Dyl3
10–2
Dy2l6 NaDyl4
10–3
Na2Dyl5
1 × 10–4 1 × 10–5 10–6 800
1000
1200 T (K)
1400
1600
II.25.3 Calculation of the partial pressures of species in a 35 W lamp burner filled with DyI3 and NaI.
formation of heterocomplexes leads to a significant enhancement of the sodium concentration in the high-temperature arc column of metal halide lamps with ceramic envelopes, thereby decreasing the colour temperature. The result is an increase in spectral radiation power at improved colour properties. Calculations on the corrosion behaviour is a very important issue for HID lamps since the market demands stability in light output over a maximum time period. These kinds of calculation complement experimental investigations and consequently can help in order to reduce experimental effort. An example for a corrosion-related calculation is given in Fig. II.25.4. Here the calculation of the equilibrium partial pressures in the system NaI–DyI3 together with Al2O3 is shown. The vapour pressure of AlI3 is dominant. Also the vapour pressure of the heterocomplex NaAlI4 is remarkable. The AlI3 partial pressures formed as gaseous corrosion products are advantageous for the enhancement since it causes in turn high NaAlI4(g) partial pressures as a result of the mass action law. From the corrosion point of view it is important to know the amount of the partial pressures of these products since, on the one hand, they may influence the colour properties of the lamp in a non-predictable way and, on the other hand, they may cause a severe corrosion attack as the wall material (polycrystalline alumina) is transported chemically from hot parts of the lamp to cooler parts. Details of those mechanisms have been given by Markus and Hilpert [003Mar] and by Markus et al. [005Mar].
310
The SGTE casebook 101 I I2
100
Nal Na2l2
pi (bar)
10–1
All All2
10–2
All3 NaAll4
10–3
Dyl2 Dyl3
1×10–4
Dy2l6 NaDyl4
1 ×10–5
Na2Dyl5
10–6 800
1000
1200 T (K)
1400
1600
II.25.4 Calculation of the partial pressures of species in a 35 W lamp burner filled with DyI3 and NaI with corrosion attack.
II.25.4
Conclusions
Thermochemical modelling of the gas-phase composition in HID lamps under operating conditions is a very important issue in order to gain an insight into the complex phenomena that are observed in those lamps. These are, for example, reactions between salt filling components in the liquid phase as well as in the gas phase and reactions between those species and the wall material of the burner vessel which can cause severe corrosion attack. Corrosion attack of the wall limits the lifetime of a lamp and also influences the stability of the light technical properties during the lifetime. Model calculations allow the development and improvement of HID light sources which can lead to higher luminous efficiency and can contribute to energy-saving light sources that have a certain demand on the lighting market.
II.25.5 72Ele 97Coa 97Hil 000Erk 001Hil
References W. ELENBAAS: Light Sources, Crane, Russek & Company, New York, 1972. J.R. COATON and A.M. MARSDEN: Lamps and Lighting, Arnold, London, 1997. K. HILPERT and U. NIEMANN: Thermochim. Acta 299, 1997, 49–57. W. VAN E RK : in High Temperature Materials Chemistry, Schriften des Forschungszentrum Jülich GmbH, Central Library, Jülich, 2000, pp. 267–276. K. HILPERT, T. KARWATH, T. MARKUS, U. NIEMANN and L. SINGHEISER: Proc. 6th Int. Conf. Molten Salt Chemistry and Technology (MS-6), Shanghai, People’s Republic of China, 2001, pp. 140–145.
Thermodynamic data for gas discharge light sources 002Bal
003Mar
005Bor
005Mar 007Dav
311
C.W. BALE, P. CHARTRAND, S.A. DEGTEROV, G. ERIKSSON, K. HACK, R. BEN MAHFERED, J. MELANCON, A.D. PELTON and S. PETERSEN: Calphad 26, 2002, 189– 228. T. MARKUS and K. HILPERT: in High Temperature Corrosion and Materials Chemistry IV (Eds E. Opila, P. Hou, T. Maruyama, B. Pieraggi, M. McNallan, D. Shifler and E. Wuchina), Electrochemical Society, Pennington, NJ, 2003. M. BORN and T. MARKUS: ‘Research on modern gas discharge light sources’, in Confinement, Transport and Collective Effects, Lecture Notes in Physics, vol. 670 (Eds A. Dinklage, T. Klinger, G. Marx and L. Schweikhard), Springer, Berlin, 2005, pp. 399–423. T. MARKUS, U. NIEMANN and K. HILPERT: J. Phys. Chem. Solids 66, 2005, 372– 375. R.H. DAVIES, A.T. DINSDALE, T. MARKUS, S.A. MUCKLEJOHN and G. ZISSIS: Requirements for thermochemical data in the lighting community, Z. Naturforsch., to be published.
II.26 The prediction of mercury vapour pressures above amalgams for use in fluorescent lamps A L A N T. D I N S D A L E, G R A H A M M. F O R S D Y K E and S T E WA R T A. M U C K L E J O H N
II.26.1
Introduction
Fluorescent lamps are now widely used as energy-efficient alternatives to incandescent lamps. Not only do they have long lives but also they can now be sold in a wide variety of shapes and sizes to match conventional light fittings. One of the most important factors influencing the luminous flux from low-pressure fluorescent lamps is the mercury vapour pressure. Successful lamp design relies on reproducible mercury pressures stable over the operating temperature of the lamp and over the product lifetime. In this chapter it will be shown how computational thermochemistry can be used as a tool for the lamp engineer to select the composition of amalgams in order to generate the desired mercury pressure for the required temperature ranges.
II.26.2
Use of amalgams in compact fluorescent lamps
For each lamp geometry there is an optimum mercury vapour pressure. This can be expressed easily through a quantity called the ‘relative light output’. Figure II.26.1 shows how this varies with the Hg pressure for a conventional linear fluorescent tube. In this case the relative light output reaches a maximum at an optimum mercury vapour pressure of about 0.8 Pa or 6 mTorr. From Fig. II.26.2, which shows the calculated partial pressure of pure mercury as a function of temperature, it can be seen that this optimum mercury pressure can be achieved for lamps which use a source of pure mercury, such as many conventional linear fluorescent tubes, at temperatures in the region of 45–50 °C. Compact fluorescent lamps are designed as replacements for incandescent lamps in standard light fittings for domestic use. Here the source of the mercury is closer to the heat source and this leads to a somewhat higher operating temperature, possibly in the region of 90–120 °C. The vapour pressure of mercury above pure liquid mercury at these temperatures (see 312
Prediction of mercury vapour pressure in fluorescent lamps
313
100
Relative light output
80
60
40
20
0 0
5
10
15 20 p(Hg)(mTorr)
25
30
35
II.26.1 Variation in the relative light output with mercury pressure for a conventional linear fluorescent lamp. 3.5
3.0
log [p(Hg) (mTorr)
2.5
2.0
1.5
1.0
0.5
0.0 –0.3 0
50
100 Temperature (°C)
150
II.26.2 Calculated partial pressure of mercury above pure liquid mercury.
314
The SGTE casebook
Fig. II.26.2) is much too high for efficient operation of a fluorescent tube. However, forming an amalgam with one or more low-melting-point metals such as bismuth, indium, tin, lead or zinc can reduce the vapour pressure. Moreover, use of amalgams can offer a means to control the variation in mercury pressure with temperature, to extend the range of operating temperatures and to choose the appropriate mercury concentration to give the maximum relative light output. However, it would be prohibitively expensive and time consuming to carry out the necessary experimental work to determine the mercury vapour pressure for a wide range of candidate amalgams although a limited number of compositions have been studied experimentally and have demonstrated the potential for use of amalgams [71Fra, 77Blo, 78Blo]. It is therefore appealing to use the techniques developed for the calculation of phase equilibria to obtain a more quantitative description of the variation in vapour pressure of mercury over amalgams with temperature and composition and to aid in the prediction of the luminous flux. This subject has been covered already in some detail by Dinsdale et al. [97Din].
II.26.3
Calculation of phase equilibria for amalgam systems
The In–Hg system has been examined as a potential system for use in compact fluorescent lamps and shows a typical variation in the vapour pressure of mercury. The data for this system were taken from an assessment by Hansen [98Han]. The assessed data had been derived by careful analysis of all the experimental thermodynamic and phase diagram data for the system. The phase diagram calculated using these data is shown in Fig. II.26.3. The vapour pressure of mercury above alloys with different mercury contents can be calculated as a function of temperature as shown in Fig. II.26.4 for mercury concentrations of 3 mol% and this is in very good agreement with the experimental data [77Blo, 78Blo]. The curve shows a kink at a temperature of about 148 °C, corresponding to the liquidus temperature as can be seen from the phase diagram (Fig. II.26.3). Similar curves can be obtained for other compositions as shown in Fig. II.26.5. At first sight the behaviour shown in Fig. II.26.4 is rather strange with an increase in mercury vapour pressure with increasing temperature between 100 and 135 °C and above 148 °C, in line with expectation, but with a decrease in mercury vapour pressure between 135 and 148 °C, which certainly is not expected. This strange effect can be understood by noting that between 127.5 and 148 °C the overall composition of the amalgam lies in the twophase region between the tetragonal_A6 phase and the liquid phase (see Fig. II.26.3). Below 127.5 °C the amalgam lies in the tetragonal_A6 phase field. While the amalgam is in the two-phase liquid + tetragonal_A6 region the
160
100 150
140
50
Temperature (°C)
Temperature (°C)
Liquid
Tetragonal_A6
0
Fcc A1
130
120
InHg Rhombotedal A10 –50
InHg4
InHg + HgIn2 0.0 In
0.2
110
0.4
0.6
xHg
1.0 Hg
100 0.00 In
0.02
0.04
0.06
0.08 xHg
0.10
0.12
0.14
(b)
II.26.3 Calculated In–Hg phase diagram: (a) complete diagram; (b) expanded section for the indium-rich composition in higher temperature region. In both diagrams the dashed line represents a composition containing 3% Hg.
315
(a)
0.8
Prediction of mercury vapour pressure in fluorescent lamps
150
316
The SGTE casebook 2.0
log10 [p(Hg)(mTorr)]
1.5
1.0
0.4
0.0 100
120
140
160
180
200
T (°C)
II.26.4 Calculated partial pressure of mercury for Hg–In alloy containing 3% Hg.
mercury vapour pressure is determined by the vapour pressure of mercury corresponding to compositions on the liquidus curve. Here there are two competing effects. As the temperature increases, this would normally lead to an increase in the mercury vapour pressure. This is offset, however, by a decrease in the mercury concentration which in itself tends to reduce the mercury vapour pressure. This can best be understood by studying Fig. II.26.6. The dotted line in Fig. II.26.6 represents the behaviour of the mercury vapour pressure as the amalgam containing 3% Hg is cooled. As the temperature falls below the liquidus temperature, the mercury pressure becomes dependent on the composition of the liquid phase in equilibrium with the indium-based solid solution. This now follows the solid curve in Fig. II.26.6 towards increasing mercury concentration. In this region the mercury pressure increases as the mercury concentration increases until a composition is reached when the effect of decreasing temperatures becomes predominant. At 127.5 °C the last amount of liquid is converted to the tetragonal_A6 indium-based solid solution. For other mercury concentrations the behaviour is similar but less marked. Examination of Fig. II.16.5 shows that the vapour pressure of mercury for an amalgam containing 3% Hg is stable over a relatively wide temperature
Prediction of mercury vapour pressure in fluorescent lamps 4
3
6% Hg
log10 [p(Hg)(mTorr)]
2
12% Hg
Hg
3% Hg
1
0
Along liquidus curve
–1
–2
–3
0
50
100 T (°C)
150
200
II.26.5 Calculated mercury partial pressure for various In–Hg amalgams. 1.5 1.0
log10 [p(Hg)(mTorr)]
0.0
–1.0
–2.0
–3.0 –3.5 0.0
0.1
0.2 xHg
0.3
0.4
II.26.6 Calculated partial pressure of mercury along the liquidus curve.
317
318
The SGTE casebook
window. This effect gives lamp manufacturers some tolerance in lamp design. This can be compared with the vapour pressure for a 12% Hg amalgam where this effect is shown only for a narrow temperature range. In this case the 3% Hg amalgam appears to be a good candidate for use in lamps as the plateau is close to the optimum mercury vapour pressure of 0.8 Pa. The discussion thus far has been concerned with binary amalgams and in this case the choice of an appropriate composition and temperatures regime for use in compact fluorescent lamps could be made simply on the basis of experimental work. Use of ternary amalgams or amalgams with even more components offers even greater potential for control of the mercury vapour pressure but requires a different approach in order to limit the amount of costly experimental work. The possibilities offered by computational thermodynamics are ideal for this. One of the ternary amalgam systems of most interest to manufacturers of compact fluorescent lamps is the Bi–In–Hg system. Figure II.26.7 shows the ternary isothermal section of the phase diagram for 300 K (26.85 °C) calculated ln
0.8 Liquid tetragonal_A6 BiIn2
0.2
Liquid BiIn2 Bi3In5
0.6
0.4
xHg
Liquid BiIn Bi3In5
xln 0.4
Liquid
0.6
Liquid rhombohedral_A7 BiIn 0.2
Bi
0.8
Liquid rhombohedral_A7
0.8
0.6
0.4
0.2
Hg
xBi
II.26.7 Calculated ternary isothermal section of the Bi–In–Hg system at 300 K (26.85 °C) and 1.013 25 × 105 Pa.
Prediction of mercury vapour pressure in fluorescent lamps
319
using a critical assessment of data for the Bi–In–Hg system from the work of Mucklejohn and Dinsdale [001Muc] which incorporates data for the Bi–In system from Boa and Ansara [98boa] and for the In–Hg system from Hansen [98Han]. By studying the calculated phase diagram it is possible to match the desired lamp operating temperature to particular relative amounts of bismuth and indium and then to vary the mercury content until the optimum mercury vapour pressure is obtained. One of the compositions of interest is based on 53% Bi–47% In with the addition of 3% Hg expressed in molar terms. At 26.85 °C this composition lies in the three-phase liquid + rhombohedral-A7 + BiIn phase field. The phase equilibrium for this composition can be calculated over a range of temperatures as shown in Fig. II.26.8 which shows that the three-phase region persists up to close to the liquidus temperature. The form of the calculated mercury vapour pressure (Fig. II.26.9) is similar to that shown in Fig. II.26.4 for the In–Hg amalgam. When this is converted into a plot of the relative light output (Fig. II.26.10), it shows a wide range of temperatures (from approximately 60 to 120 °C) for which the light output is sufficiently stable to give lamp engineers considerable flexibility over its operating conditions. 0.18 0.16 BiNi
0.14
Mass phase (kg)
0.12
0.10
0.08 0.06
0.04
Liquid Rhombohedral + A7
0.02 0.00 0
20
40
60
80 T (°C)
100
120
140
II.26.8 Calculated phase fraction as a function of temperature for a amalgam based on 53% Bi–47% In with the addition of 3% Hg.
The SGTE casebook 2.0 1.8 1.6 1.4
log10 [p(Hg) (mTorr)]
1.2 1.0 0.8 0.6 0.4 0.2 0.0 –0.2 –0.4 –0.6 –0.8 –1.0
0
20
40
60
80 T (°C)
100
120
140
II.26.9 Calculated mercury vapour pressure as a function of temperature for an amalgam based on 53% Bi–47% In with the addition of 3% Hg. 105 100 95 90 Relative light output
320
85 80 75 70 65 60 55 50 40
60
80
100 T (°C)
120
140
II.26.10. Calculated relative light output as a function of temperature for an amalgam based on 53% Bi–47% In with the addition of 3% Hg.
Prediction of mercury vapour pressure in fluorescent lamps
321
A range of alloy systems involving bismuth, indium, tin, lead and silver have been studied in this way in order to determine amalgams suitable for use in fluorescent lamps. Results of the calculations have been validated experimentally in lamp tests and have formed the basis of a patent.
II.26.4
Conclusions
Computational thermochemistry has been used successfully to model the vapour pressure of mercury above candidate amalgams of potential use in fluorescent lamps. The results have been validated experimentally. A database for amalgams and other low-melting-point alloys has been developed in order to extend the range of systems covered.
II.26.5 71Fra 77Blo
References
G. FRANCK: Z. Naturforsch. 26A, 1971, 150. J. BLOEM, A. BOUWKNEGT and G.A. WESSELINK: J. Illum. Eng. Soc. 6, 1977, 141– 147. 78Blo J. BLOEM, A. BOUWKNEGT and G.A. WESSELINK: Philips Tech. Rev. 38(3), 1978– 1979, 83. 97Din A.T. DINSDALE, G.M. FORSDYKE and S.A. MUCKLEJOHN: ‘Use of MTDATA to predict mercury vapour pressures above amalgams for fluorescent lamps’, in Proc. 9th Int. Conf. High Temperature Materials Chemistry, State College, Pennsylvania, USA, 19–23 May 1997. 98Han S.C. HANSEN: Calphad 22(3), 1998, 359–374. 98Boa D. BOA and I. ANSARA: Thermochim. Acta 314, 1998, 79–86. 001Muc S.A. MUCKLEJOHN and A.T. DINSDALE: Unpublished work, 2001.
II.27 Modelling cements in an aqueous environment at elevated temperatures J O H N A. G I S B Y, H U G H D AV I E S, A L A N T. D I N S D A L E, M A R K T Y R E R and C O L I N W A L K E R
II.27.1
Introduction
The engineered component of a low- and intermediate-level nuclear waste repository in the UK will almost certainly consist primarily of ordinary Portland cement (OPC)-based materials. OPC provides a chemical barrier by its ability to buffer the pH > 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
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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
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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
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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.
358
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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.
362
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.
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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 (<100> 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,
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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.
428
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 <
m˙ Fe,hot spot M Fe
(IV.8.7)
between the bath and the metal–slag zone, mass transfer of Fe droplets takes place. During a large part of the process, the decarburisation reaction is limited by the oxygen supply. After reaching the critical point, it is limited by the carbon transport to the reaction zone. Therefore at the critical point, ˙ ˙ 0 = m Fe,metal–slag c 0 2 n˙ O 2 < NV c 0 = NVc ρ Fe t′
(IV.8.8)
These mass flows are a function of the oxygen blow rate and conditions as well as the gas production, which are constant over the main process time. Towards the end of the process, gas formation depends on carbon concentration and therefore the mass transfer conditions change. As the kinetic and transport limitations of the process are modelled by the interlink of the reaction zones, the zones themselves can be treated assuming thermochemical equilibrium. Although derived with regard to the decarburisation reaction, the present model is at the same time also valid for other chemical components with similar behaviour present in hot metal and slag. Processes determined by other kinetic phenomena, such as the melting of scrap and the dissolution of lime, need to be modelled separately. The dissolution of fluxes, especially lime, is limited by the formation of a dicalcium silicate layer around the particles for a major part of the process. This behaviour is mainly dependent on the composition and temperature of the slag and mixing between metal and slag [73 Bap]. A dissolution function was implemented in the model to regulate lime participation in reactions leading to the formation of slag. With this function, the heating of the lime
Non-equilibrium modelling for the LD converter
429
prior to its dissolution and the withdrawal of energy from bath and slag owing to this heating are treated as well. A similar problem exists for the addition of steel scrap. Scrap melting depends not only on its temperature but also on its carbon concentration. Diffusive carbon transport to the scrap–metal bath interface is therefore an important factor for the melting behaviour. A function was implemented which treats the melting as well as the heating of the scrap prior to melting. Not only are material streams to and from ideal reaction zones defined in the model, but so are energy streams. Energy losses are calculated and appropriate amounts of enthalpy are withdrawn from the ideal reaction zones throughout the simulation. Thus radiation losses from the converter mouth as well as radiation and convection losses through the converter walls are regarded. Furthermore, enthalpy losses due to the hot off-gas stream are directly taken into account in the reaction zone model as the hot off-gas stream is removed throughout the process. A temporal discretisation of the described model is needed for simulation purposes. According to a chosen time increment ∆t and the determined mass flows between the ideal reaction zones, in every time step the mass exchange between the reaction zones and the reaction progress is calculated. Hence, a step-by-step transport and conversion of the different mass streams results.
IV.8.3
Modelling tool
The process modelling tool SimuSage is a combination of the thermochemical programmers’ library ChemApp and a library of additional graphical components for Borland’s programming environment Delphi. Initially developed by Mannesmann Demag Metallurgy as an in-house tool under the name ProMoSys, it is now co-developed by GTT-Technologies and SMSDEMAG under the new name SimuSage. It consists of flow sheeting components, which add to Borland Delphi’s powerful programming language the capability to generate easily stand-alone flow-sheeting models with a sound thermodynamic basis. The elements of the reaction zone model shown in Fig. IV.8.2, e.g. material streams and equilibrium reactors, are readily available in SimuSage as graphic components. This has made it possible to set up the converter model and to test modifications and additions to the model very quickly. As indicated above, the ChemApp programmers’ library (in the form of a Dynamic Link Library, DLL) has been used to form the thermodynamic backbone of SimuSage. The general concept is represented in Fig. IV.8.3. The ChemApp library permits the easy use of full complex equilibrium calculations within software by way of a set of interface routines. These interface routines are used to define the conditions for a ‘local’ equilibrium calculation, to execute the equilibrium calculation and to extract information
430
The SGTE casebook SimuSage program
Process parameters
Call ChemApp
Results
Interface
ChemApp library Data handling and phase equilibrium calculation module
IV.8.3 Integration of ChemApp into SimuSage.
from the calculated equilibrium state (such as phase amounts or phase internal concentrations) that is needed for the process model to proceed. The equilibrium calculations in ChemApp are performed by the same Gibbs energy minimisation code as in the well-known interactive software ChemSage, now FactSage5, and are thus of proven reliability. The thermodynamic data used in the present modelling have been taken from the extensive file store of GTT-Technologies. 109 phases and altogether 202 species have been included in the calculations. The gas phase (60 species) has been treated as ideal, while the liquid Fe phase (dilute solution approach, 14 species) and the liquid slag (Gaye–Kapoor–Frohberg model, eight species) have been treated as non-ideal chemical solutions.
IV.8.4
Simulation results
For the simulation calculations, the following process conditions are assumed according to Asai and Muchi [70Asa]: – – – –
150 t converter. oxygen blow rate 10.87 kg O2/s for 0 to 15th min., from 15th min. 7.99 kg O2/s 5 t scrap input 10 t CaO input
Non-equilibrium modelling for the LD converter
431
From the experimental results [70Asa] shown in Fig. IV.8.4 and the corresponding process conditions stated above, model parameters can be calculated. The lower limit for the exchange ratio between the bath and hotspot reaction zone defined as m˙ hog spot / m˙ total is calculated from Equation (IV.8.7); it amounts to 0.023. The exact value for this exchange ratio is derived by comparing measured and calculated decarburisation results. Other reactions, e.g. the reactions of Si, and the temperature development also have to be considered. From the comparison of experimental and calculated results, the exchange ratio between the bath and hot-spot reaction zone is found to be 0.04. In Fig. IV.8.4 the critical point, where the change from a constant decarburisation rate to a rate controlled by carbon concentration takes place, is found to be at 13.5 min at a carbon concentration of 0.006. From this, the exchange ratio between bath and metal–slag reaction zone defined as m˙ metal–slag / m˙ total is calculated to be 0.7. The results of the calculation with these parameters are shown in Fig. IV.8.5. The decarburisation reaction, the critical point and carbon content are modelled correctly as can be seen from the calculation of the decarburisation rate. With the calculated exchange ratio between the bath and metal–slag reaction zone, a critical carbon content of 0.006 results, as expected. As discussed above, Equation (IV.8.8) is valid for the process conditions up to the critical point. The decrease in decarburisation rate from the critical point onwards results in a decrease in CO production and therefore less stirring is caused by the gas bubbles. Hence mass transfer in form of metal droplets in the slag decreases. In Fig. IV.8.6, this effect is taken into account and modelled by a decrease in the exchange ratio between the bath and the metal–slag reaction zone 0.045
C content in hot metal
0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0
2
4
6
8 10 Time (min)
IV.8.4 Carbon content in hot metal.
12
14
16
18
432
The SGTE casebook
Decarburisation rate (min–1)
0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 Ckrit 0 0
0.005
0.01
0.015 0.02 0.025 0.03 C content in hot metal
0.035
0.04
0.045
IV.8.5 Decarburisation rate. 0.000 5
0.045 C content, measured [70 Asa] Si content, measured [76 Hor) C content, calculated Si content, calculated Mn content, calculated O content, calculated
C, Si, Mn content
0.035 0.03
0.000 45 0.000 4 0.000 35 0.000 3
0.025
0.000 25 0.02
0.000 2
0.015
0.000 15
0.01
0.000 1
0.005
0.000 05
0
O content
0.04
0
2
4
6
8 10 Time (min)
12
14
16
18
0
IV.8.6 Components in hot metal.
from the critical point onwards. After reaching the critical point, the exchange ratio is set to 0.35. Obviously, the decrease in decarburisation rate can be modelled quite well with this set of parameters. The calculated carbon content of the metal phase shows good agreement with measured values [70Asa]. At the beginning of the process, the oxidation of Si from the hot spot is limited by the transport of Si to the phase boundary. Hence decarburisation starts although, normally, Si is oxidised preferentially to carbon. The limitation of transport to the phase boundary is modelled by a limited mass flow through the hot-spot reaction zone and therefore limited amount of Si in this reaction zone. The resulting excess oxygen reacts to form CO and FeO-rich slag. This slag facilitates the dissolution of lime and forms the main reaction zone for
Non-equilibrium modelling for the LD converter
433
further reactions. The oxidation of Si therefore takes place not only in the hot spot but also in the metal–slag reaction zone. The amount of Si calculated in the metal phase shows good agreement with the measured values. Additionally, the contents of Mn and O in the metal phase were calculated. Both components show qualitatively good agreement with measured values. Mn is oxidised towards the end of the process; the oxygen content of the metal phase increases. Figure IV.8.7 shows the composition of the slag phase, which demonstrates qualitatively good agreement with measured values. The FeO content rises towards the end of the process, resulting in iron losses. The calculated gas production is shown in Fig. IV.8.8. The shift in the main reaction zone from the hot spot to the metal–slag dispersion in the 25 000 FeO, calculated CaO, calculated
Component (kg)
20 000
SiO2, calculated MnO, calculated Total slag, calculated
15 000
10 000
5 000
0 0
5
10 Time (min)
15
IV.8.7 Components in the slag phase: simulation results. 1600 1400
Amount (kg)
1200 1000 800 600 400 200 0
0
1.5
3
4.5
6
Hot spot, calculated
IV.8.8 Gas production.
7.5 9 10.5 Time (min)
12
13.5 15
16.5 18
Metal–slag reaction zone, calculated
434
The SGTE casebook
course of the process is modelled well, as can be seen from the fact that the gas production decreases in the hot spot while it increases in the metal–slag reaction zone. The decreasing gas production at the end of the process is caused by the change in the oxygen blow rate and the slower decarburisation. As discussed above, the ideal reaction zones in the cell model are coupled not only by defined mass transfer but also by heat transfer. Energy losses from the converter are calculated in every time step and enthalpy is withdrawn from the various reaction zones, mainly from the metal–slag reaction zone. In Fig. IV.8.9, the evolution of the temperature in the hot spot and of the mean temperature of the hot metal bath are shown. The temperature in the hot spot is well described. Measurements give values from 2300 to 2400 °C for the first 8 min of the process [76Koc]. The mean temperature shows a very good quantitative agreement with the experiments [70Asa].
IV.8.5
Conclusions
The aim of the research presented in this chapter is the evaluation of a technique for modelling metallurgical processes. In the long term, the process models developed by use of this technique will be used to improve process design. Special focus is given to modelling non-equilibrium phenomena which are caused, for example, by transport limitations or dissolution processes. The objective is to model the process with a relatively simple model structure and only a few model-specific parameters.
2600
Temperature (°C)
2400 2200
Thot spot, calculated Tbath calculated Tbath [70 Asa]
2000 1800 1600 1400 1200 0
2
4
6
8 10 Time (min)
IV.8.9 Temperature evolution.
12
14
16
18
Non-equilibrium modelling for the LD converter
435
The LD converter process was modelled according to this intention. Based on experimental results for the decarburisation reaction and possible reaction routes, a simple cell model was developed which describes well the transport limitations of the decarburisation. Simple relationships were deduced which define the mass transfer between the cells (reaction zones) and thus the model parameters. Only a little further adjustment is needed. Energy losses from the process are calculated throughout the simulation. The resulting temperatures of the hot spot show good agreement with real process data. Regarding the reactions, the model describes not only the decarburisation but also the behaviour of other elements dissolved in metal and slag. The temperature of the metal bath could also be reproduced by the model. Altogether, these facts show that, although only a simple model structure was employed, the physical and chemical conditions in the converter process are handled correctly.
IV.8.6
List of symbols
A ci cX c0 kX KX,eff m˙ bath m˙ hot spot m˙ metal–slag m˙ total
area of the phase boundary concentration of component i concentration of component X critical carbon concentration reaction rate constant transport coefficient mass flow rate through the bath mass flow rate through the hot spot mass flow rate through the metal–slag reaction zone total mass flow rate through the converter = m˙ hot spot + m˙ metal–slag + m˙ bath mass flow rate of component X molar mass of component X molar flow rate of component X number of droplets droplet flow rate reaction rate droplet residence time in the slag volume of a droplet time increment density dissolved in liquid iron dissolved in slag in the gas phase
m˙ X MX n˙ X N N˙ RX t′ V ∆t ρ [] () {}
436
IV.8.7 70Asa 73Bap 76Hor
76Koc 87Bar
The SGTE casebook
References S. ASAI and I. MUCHI, Trans. Iron Steel Inst. Japan 10, 1970, 250–263. V.I. BAPTIZMANSKII, V. KULIKOV, B. BORCHENKO and E. TRETYAKOV: Steel USSR, 8, 1973, 634–638. A.I. VAN HOORN, J.T. VAN KONYENBURG and P.J. KREYGER: Proc. Symp. Role of Slag in Basic Oxygen Steelmaking (Ed. W.-K. Lu), McMaster University Press, Hamilton, Ontario, 1976. K. KOCH, W. FIX and P. VALENTIN: Arch. Eisenhüttenwes. 47(10), 1976, 583– 588. I. BARIN, M. MODIGELL and F. Sauert: Metall. Trans. B 18, 1987, 347–354.
IV.9 Modelling TiO2 production by explicit use of reaction kinetics P E RT 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
IV.9.1
Introduction
Titanium dioxide (TiO2) is a bulk commodity, which is used as a white pigment, e.g. for paints, plastics, paper and rubber. TiO2 pigment is produced by two major industrial routes from its ore, which is either ilmenite (ferrous titanate) or either natural or synthetic rutile. Ilmenite-based raw materials are used for the wet sulphuric acid process, while rutile is the major input for the high-temperature chloride process. After years of development, both manufacturing processes remain in extensive use and represent both economically effective and environmentally sound industrial practice. In the following examples, the combined reaction rate–multiphase calculation has been applied to two characteristic unit processes of TiO2 production.
IV.9.2
Anatase–rutile transformation – a simple example of the constrained Gibbs energy method
The first example presents the formation of titanium dioxide in a calciner, which is one of the final stages of the sulphate process. To simplify the treatment here, the feed is assumed to consist of titanium oxyhydrate slurry without sulphuric acid or sulphate residues. The chemical composition of the oxyhydrate is approximated as TiO(OH)2 · nH2O. During calcination, the slurry is dried and finally the hydrate decomposes, leaving the product titanium dioxide in the bed. From the oxyhydrate, at relatively low temperatures (about 200 °C) the crystalline form anatase, TiO2(An), is formed first, and only in the high-temperature zone of the furnace end does the thermodynamically stable rutile form TiO2(Ru) appear as the desired product. The reactions are TiO(OH)2 · nH2O ↔ TiO2(An) + (n+1)H2O(gas)
(IV.9.1)
↔ TiO2(Ru)
(IV.9.2) 437
TiO2(An)
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As rutile is the more stable form of the two titanium dioxide species, Gibbs free energy minimisation would at all temperatures result in rutile and water. This would lead to a 100% rutilisation of the titania already at temperatures where the titanium oxyhydrate is calcined by Reaction (IV.9.1). It is, however, well known from practical experience that the rutilisation reaction (Reaction (IV.9.2)) is slow and only takes place with a finite rate at elevated temperatures (above 850 °C). The simulation of the calcination process must take this feature into account. In Table IV.9.1, the equilibrium system is presented incorporating additional virtual components and species as outlined in Chapter III.5. The additional system component R represents the kinetically constrained rutile content, the amount of which is set by using R+ or R– inputs. These two additional constituents are used for input amounts; yet they are not allowed to form in the calculated (restricted) equilibrium. As R+ or R– are defined as pseudoconstituents in the multicomponent system, it is necessary to define an input value for their chemical potentials. It is assumed that they do not affect the energy or entropy balances of the system, and thus the essential input condition is µ 0R+ = µ 0R– = 0 (see Chapter III.5). Here again, the reaction constraint must not affect the mass balance of the system, and the molecular mass of the system component R is also set to zero (MR = 0). With these input settings, the Gibbsian method can again be used to solve for both the equilibrium conditions and the condition for a kinetically constrained multicomponent system. As explained in Chapter III.3 the conditions for the chemical potentials of anatase and rutile are obtained for the kinetically constrained system: µ4 – 2πO – πTi = 0
(IV.9.3)
µ5 – 2πO – πTi – πR = 0
(IV.9.4)
The chemical potential of a constituent species is denoted as µk and the chemical potential of a system component as πj; subscripts 4 and 5 are used for anatase and rutile respectively. The kinetic condition for the anatase– rutile transformation then becomes Table IV.9.1 Stoichiometric matrix for TiO(OH)2 calcination with kinetically constrained rutilisation Index k
Species
O
H
Ti
R
1 2 3 4 5 6 7
O2(g) H2O(g) TiO(OH)2 TiO2(anatase) TiO2(rutile) R+ R–
2 1 3 2 2 0 0
0 2 2 0 0 0 0
0 0 1 1 1 0 0
0 0 0 0 1 1 –1
Modelling TiO2 production by reaction kinetics
µ 5 – µ 4 = πR
439
(IV.9.5)
which gives the affinity of Reaction (IV.9.2) in terms of the component potential πR. At equilibrium, this affinity is zero, as can be seen if Equation (IV.9.3) is applied to the equilibrium system (last four columns of Table IV.9.1). For reaction (IV.9.1), both the last four columns of Table IV.9.1 and the extended matrix give the zero-affinity condition. The additional constraint in the calculation follows the deviation of the affinity of anatase–rutile transformation from zero and can be used as a control for the extent of this reaction. In practice, this is achieved algorithmically through the addition of the pseudoconstituent R+ in each calculation step. For unrestricted equilibrium, πR = 0, so that with µ 0R+ = µ 0R– = 0 the chemical potentials of R+ and R– also become zero. This is equal to the condition that the activities of these two ‘phases’, as defined by Eriksson [75Eri], are equal to unity. This kind of equilibrium condition is then directly applicable to calculations using standard Gibbs energy minimisers. The reaction rate can be presented in the form of the rutile fraction x, which is measured as a function of time at constant temperature and pressure. For example, the following formula has been presented for rutile fraction in an anatase–rutile calcination system without additives [75McK]: x = 1 – (1 – kt)3
(IV.9.6)
Here, x is the fraction of rutile in the reaction mixture, t is time (h) and k is the reaction rate constant (h–1). This formula was used in the multicomponent system to calculate the respective molar amount for R+, which determines the kinetic conversion rate from anatase to rutile. The input data of the calculation system have been collected in Table IV.8.2. The parameters for the rate constant are given in terms of the Arrhenius equation k = A exp(–Ea/RT), where A is the frequency factor and Ea is the activation energy. The thermodynamic (Gibbs energy) data for the species are obtained from standard sources [001FAC] (see also [97Ket]). The calculation is then performed in 60 steps with 10 min intervals to cover the experimental data of McKenzie [75McK], covering in total 10 h at 995 °C. For each step, the Gibbs energy of the system is minimised and as a result the composition and the Gibbs energy of the system are calculated. Table IV.9.2 Input data for the Ti(OH)2 calcination model O2 (mol)
TiO(OH)2 (mol)
R+ (mol)
Temperature (°C)
Pressure (atm)
A (h–1)
Ea (KJ mol–1)
0.5
1.0
Equation (IV.8.6)
995–1045
1
1.8E17
442
440
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Figure IV.9.1(a) shows the anatase–rutile transformation curves for the TiO(OH)2–O2 system for three different temperatures. The curves represent the degree of rutilisation (increasing from 0 to 1) and the total decreasing Gibbs energy of the system during the course of the reaction. At each –1 734 000
1.2
–1 736 000 1.0 –1 738 000
Measured, 995 °C Measured, 1020 °C
0.6
–1 742 000
Measured, 1045 °C Gibbs energy 995 °C
–1 744 000
0.4
Gibbs energy (J)
–1 740 000
Model curves
–1 746 000 0.2 –1 748 000 0 0
100
Rutilisation exotherm (J mol–1 min–1)
Rutile fraction
0.8
200
300 400 Time (min) (a)
500
–1 750 000 700
600
300 995 °C 1020 °C 1045 °C
250 200 150 100 50 0 0
50
100
150 Time(min) (b)
200
250
300
IV.9.1 Anatase–rutile transformation kinetics in a Gibbs free energy model at 995–1045 °C. The kinetic data are reproduced from the work of McKenzie [75McK]. (a) The descending Gibbs energy of the system at 995 °C is also shown. (b) The exotherms of the rutilisation reaction as obtained from the thermodynamic Gibbs energy model.
Modelling TiO2 production by reaction kinetics
441
temperature, the Gibbs energy is a monotonically descending curve as a function of time, reaching its minimum value at equilibrium. As the Gibbs energy model inherently calculates other thermodynamic properties, such as heat capacities, enthalpies and entropies, it is usually beneficial to use Gibbs energy-based modelling in process calculations. As an example, the exothermal heat effect of the rutilisation reaction has been deduced from the Gibbs energy model at the three reaction temperatures and is presented in Fig. IV.9.1(b). As for non-isothermal systems, the entropy production could be used to assess the thermodynamic consistency of coupled kinetic– thermodynamic models [006Kou]. The presented method can include more complex mechanisms in a multicomponent calculation [006Kou]. The matrix must then be extended with one column and two rows for each linearly independent kinetically restricted reaction. The method of the extended matrix is now directly applicable in the commonly used Gibbs free energy routines, such as ChemApp. In fact, a particular data-assembling program is available from the developers of the ChemApp program library, so that the appropriate Gibbs energy data files can be adapted to solve problems including kinetic constraints.
IV.9.3
Model for the TiCl4 burner: comparison with the image component technique
An early approach to combining reaction rates with the multiphase Gibbsian calculations was made by Koukkari [93Kou] while modelling the production of TiO2 pigment with the chloride process. The main reaction is burning TiCl4 in a plug flow reactor. The original work was based on the use of the image component technique discussed briefly in Chapter III.5; the calculations have been performed here both with the constrained Gibbs energy method and with the image technique using the computation of the entropy of mixing. The particular reactor modelled in this work is the titanium(IV) chloride oxidation burner, which is well known for its use in high-temperature TiO2 pigment production. In this process, rutile (either natural or synthetic) is used as raw material. In a reducing fluidised bed the ore is chlorinated at about 900 °C. The ore consists of 94–96% TiO2, the rest being oxides of other metals. The impurity chlorides from these other cations become separated in a series of condensers and the pure TiCl4 is then gasified, heated and fed to a tubular burner together with a flow of superheated oxygen. A vigorous oxidation reaction then follows between the ‘tetrachloride’ and oxygen, the overall reaction being TiCl4(g) + O2(g) ↔ TiO2(s) + 2Cl2(g)
(IV.9.7)
The two gaseous reactants produce an aerosol, with titanium dioxide as the
442
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condensed phase in a gas stream consisting of the reactants, chlorine and gaseous intermediates. The activation energy of Reaction (IV.9.7) is quite high (average, 88 kJ mol–1), and thus the preheating of the reactant streams is essential. In an industrial burner, ignition takes place at 850–900 °C (mixed reactant temperature). Furthermore, the oxidation is an exothermic reaction with ∆RH = –177 kJ mol–1 (1000 °C). The Gibbs energy change is –102 kJ mol–1 at 1000 °C and –54 kJ mol–1 at 1500 °C. Thus, the increase in temperature in the reaction mixture may become a limiting factor for the overall conversion of titanium(IV) chloride to TiO2. To avoid excessive heating of the mixture, the reactor is provided with a cooling jacket with water circulation. Thermodynamic studies, e.g. that performed by Karlemo et al. [96Kar], showed that gaseous chlorine dissociates significantly at temperatures exceeding 1100 °C. This dissociation is an endothermic process and thus is an additional factor affecting the temperature of the process. The reaction rate parameters for the consumption of TiCl4 in the oxidation were determined by Pratsinis et al. [90Pra]. The overall reaction rate equation was given as d[TiCl 4 ] = – {k + k ′ ([O 2 ])1/2} [TiCl 4 ] dt
(IV.9.8)
k = A exp – Ea RT
(IV.9.9)
E′ k′ = A′ exp a RT
(IV.9.10)
where k and k′ are given in the usual Arrhenius form. The numerical values for the frequency factors A (= 8.26 × 104 s–1) and A′ (= 1.4 × 105 dm3/2mol–1/2s–1) and the activation energy Ea (= Ea′ = 88 kJ mol–1), average) were taken from the work of Pratsinis et al. One may conclude that, at close to stoichiometric compositions, k′[O2]1/2 Ⰶ k and the reaction rate is almost first order in terms of the TiCl4 concentration. As there are no measured data of the reassociation of chlorine to Cl2 in the burner conditions, it is a viable assumption that chlorine atoms and molecular chlorine coexist in mutual equilibrium according to the reaction Cl2(g) ↔ 2Cl(g)
(IV.9.11)
With the overall reaction rate of Reaction (IV.8.7), by using the rate parameters measured by Pratsinis et al. and by using the partial equilibrium assumption for chlorine dissociation, a thermodynamic model controlled by reaction kinetics can be presented for the TiCl4 oxidation. To simulate the operation of the burner, the extent of reactions and the enthalpies of reaction must be combined with the heat transfer from the
Modelling TiO2 production by reaction kinetics
443 T
TiCl4 (g), T1
Q = Q (x)
ξr
T = T (x), P = P (x) X v = v (x)
O2(g), T2 G = G (T, P, nj)
Cooling media flow
Q
IV.9.2 The schematic presentation of the one-dimensional TiCl4 burner model. 1 Simulated Measured
0.9 0.8
Qmax / Q
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Cone
RT1
RT2
CT1 CT2–3 Section
CT4–5
CT6–7
CT8
IV.9.3 The calculated and measured heat transfer of the reactor for predefined segments along the reactor’s longitudinal axis.
reaction mixture to the surrounding cooling media. A schematic representation of the one-dimensional reactor model is given in Fig. IV.9.2. In the recent modelling work performed by Koukkari et al. [000Kou], different values of the activation energy Ea in the range 71–102 kJ mol–1 were tested. The best agreement with the measured heat transfer and temperature data from the process could actually be achieved with low Ea values, and thus Ea = 71 kJ mol–1 was adopted for the actual model (Fig. IV.9.3 and Fig. IV.9.4). The measured gas temperatures (published earlier by Koukkari and Niemelä [96Kou]) were also in satisfactory agreement with the results of the model calculations. The TiCl4 burner model is a practical example of the usefulness of the image component technique. The reactant TiCl4 is introduced to the model as an image (unreactive) constituent, which is then algorithmically transferred to its reactive counterpart. The latter is stoichiometrically connected with the
444
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1600 100
1200
Gas
80
1000 60 800 600
40
400
20
Yield (%)
Temperature (°C)
1400
Inner wall of reactor Outer wall of reactor Reactor coolant TiO2 yield
200 0 0
20 Position (m)
40
0
IV.9.4 The calculated temperature and TiO2 yield profiles. The O2-toTiCl4 mole ratio was 1.1 and the initial mixing temperature about 950 °C. For measured gas temperatures see the paper by Koukkari and Niemelä [96Kou]. The reaction is slightly retarded owing to the thermodynamic conversion limit at high temperatures. Additional heat transfer completes the reaction. The exothermal behaviour can also be utilised for secondary TiCl4 injection.
rest of the system in the stoichiometric matrix of the system. As all the reactive TiCl4 is readily consumed to form TiO2, Cl2, Cl and (minor amounts of) oxychlorides in the minimisation calculation, the image reactant TiCl4, with its thermophysical properties equal to those of the actual reactant provides for the necessary upkeep of the heat and material balances in the sequential calculation. When the temperature is raised to about 1300 °C, however, part of the reactive TiCl4 remains thermodynamically stable in the reactant stream, and a situation is encountered where both the active and the image constituent can be present as species in the calculated gas phase. Their combined amount equals the physical amount of TiCl4 at those temperatures, and thus heat and material balances are not violated. However, there is a deviation in the entropy of mixing owing to the ‘double presence’ of a single constituent in this high-temperature range. The ‘entropy-of-mixing’ effect of the image component method in the TiCl4 burner model is illustrated in Fig. IV.9.5. When the image component is used, its mole fraction in the reactive mixture can be denoted xI whereas its reactive counterpart has a mole fraction xR. The image component is transferred to its reactive counterpart with discrete increments ∆nR = –∆nI (expressed in molar amounts). When all the ∆nR is consumed in the subsequent reaction, it follows that xR ≈ 0, and only the terms due to xI contribute to the entropy of mixing at each stage of the calculation. This gives a result for ∆Smix, which is physically justified. If, however, the thermodynamic procedure leaves some of the ∆nR unreacted in the mixture, both terms: the term involving xR and the term involving xI, contribute to the value of ∆Smix as if they were
0.4
1600
0.35
1400
0.3
1200
0.25
1000
0.2
800
0.15
600 400
0.1 0.05 0 0
445
Temperature (°C)
Smix (TiCl4)/R
Modelling TiO2 production by reaction kinetics
Image component method Exact result T °C 10
20 30 Length (arbitrary units)
200 0 40
IV.9.5 The deviation in the entropy of mixing in the TiCl4 burner model when the image model is compared with the single-reactant approach.
different constituents of the mixture. The error in the calculated ∆Smix is thus due to the unreacted amount of the reactive species at such stages of the calculation, where ∆nR is not entirely consumed. In such cases, which prevail in the intermediate stages of the TiCl4-reactor, the entropy using the image component is calculated too high (see Fig. IV.9.5). When, on the other hand, the constrained Gibbs energy method is used, the changing chemical state can be described entirely with well-defined functions and the erroneous entropy effect is eliminated. The image component method remains applicable for cases either where the driving force of the consumption of the reactive species is sufficiently large to prevent any remnants of this constituent from appearing in the equilibrium mixture or where the reactive species and its image component are stoichiometric condensed substances. For practical process modelling purposes, the image approach is often advantageous, as it provides an easy method to maintain the correct mass and heat balances for the reactor.
IV.9.4
References
75Eri G. ERIKSSON: Chem. Scripta 8, 1975, 100. 75McK K.J.D. MCKENZIE: Trans. J. Br. Ceram. Soc. 74, 1975, 77. 90Pra S. E. PRATSINIS, H. BAI, M. FRENKLOCH and S.V.R. MASTRANGELO: J. Am. Ceram. Soc. 73, 1990, 2158. 93Kou P. KOUKKARI: Computers Chem. Eng. 17(12), 1993, 1157.
446
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96Kar
B. KARLEMO, P. KOUKKARI and J. PALONIEMI,: Plasma Chem. Plasma Proc. 16, 1996, 59. P. KOUKKARI and J. NIEMELÄ: Computers Chem. Eng. 21(3), 1996, 245. M. KETONEN, P. KOUKKARI and K. PENTTILÄ: Simulation Studies of the Calcination Kiln Process, European Control Conf. (ECC 97), Brussels, Belgium, 1–4 July 1997. P. KOUKKARI, M. KEEGEL and K. PENTTILÄ, in High Temperature Materials Chemistry, Schriften des Forschungszentrum Julich, Vol. 15, Part I (Eds K. Hilpert, F.W. Froben and L. Singheiser), 2000, p. 253. FACT Database, 2001, in FactSage 5.0, CRCT, Montreal, Canada, and GTT. Technologies, Herzogenrath, Germany, 2001. P. KOUKKARI and S. LIUKKONEN: Ind. Eng. Chem. Res. 41(12), 2002, 2931. P. KOUKKARI and R. PAJARRE: Computers Chem. Eng., 30, 2006, 1189.
96Kou 97Ket
000Kou
001FAC 002Kou 006Kou
Index
adiabatic transformations 263–6 Al-Cu-Mg system 413–14 Al-Cu-Mg-Zn system 375 quaternary system 414–15 Al-Cu-Zn system 414 Al-Mg-Zn system 408–13 Al-Ni alloys 118–21 alkali metals 239, 242 alloys aluminium alloys 375–84 calculation of metastable phase ranges 249–51 casting alloys 410 in corrosive atmospheres 441–2 magnesium alloys 375–84 melting of alloys 392–6 metastable phases 248–62 milling process 248 partial equilibrium calculation 378–80 solid solubilities 248 solidification path 344–5, 372–4 aluminium alloys 375–84 aluminium-killed steels 267 amalgam systems 312–21, 314–21 phase equilibria calculation 314–21 ternary amalgam systems 318–19 ammonium chloride-gas equilibrium 69–70 anatase-rutile transformation 437–41 annealing of thin films 128–30 aqueous environments, cement modelling 322–35 aqueous sulphuric acid 144–54 austenitic stainless steels 106–13
ball-milling 249–51 batch-to-melt conversion 293–5 modelling 297–301 Baur-Glässner diagram 55–7 binary and ternary subsystems 166–7 blades of turbines 77 blowing the matte 188–9 Boudouard equilibrium 55 Bragg-Williams approach 25 C-S-H solubility at higher temperature 331–3 at room temperature 328–31 Ca-C-O system 231–8 caesium 182 calcination process 437 calcium-treated steel grades 267–72 carbon potential during heat treatment of steel 212–22 carbon activity and carbon potential relation 213 carbon activity in furnace atmospheres 213–19 carbon activity of multicomponent steels 219–20 carbon steel, partial equilibrium calculation 378–80 casting alloys 410 cell model 36, 74 cement modelling 322–35 C-S-H solubility at higher temperature 331–3 at room temperature 328–31 compound energy model 325–6
447
448
Index
conventional models 323 free-energy-based models 323 leaching simulation 333–4 MTDATA 323–5 Chem App 422 chemical long-range ordering 39–40 chemical stability in microelectronics 123 chemical vapour deposition 124 Clausius-Clapeyron equation 118, 119–20 clogging in continuous casting 224–7 Co-bonded WC tools 98–104 favourable region of carbon content 98–103 Fe-Ni alloy replacement 98, 99–102 transverse rupture strength 98 W-to-C ratio 98 coal-fired combined cycle power systems 239 complex thermochemical phenomena 169–73 compound energy model 29, 30–1, 325–6 compound steel 386–91 computational phase studies 375–84 Al-Cu-Mg system 413–14 Al-Cu-Mg-Zn system 375 quaternary Al-Cu-Mg-Zn system 414–15 Al-Cu-Zn system 414 Al-Mg-Zn system 408–13 Cu-Mg-Zn system 413 ternary subsystems 375–83 continuous casting processes 224–7 control of material flows 355–6 conventional models 323 copper extraction from sulphide ores 188–98 copper ions 144–54 Corium-concrete interaction (MCCI) 161 corrosion gas phase corrosion 442–5 of HID lamps 309 high temperature corrosion of SiC 200–210 hot corrosion in CPFBCs 241–4 hot corrosion in gas turbines 239–46 hot corrosion of heat exchanger components 398–403 hot corrosion in PPCCs 244–5
hot salt corrosion 77–89 resistance loss 106–13 wet hydrogen corrosion 206, 209 countercurrent reactors 352, 355, 400–6 cracked methanol 219 Crank-Nicholson algorithm 443 critical temperature 37–8 crystalline electric field (CEF) 31–2 crystalline reference system (CRS) 286 crystallisation processes 343 Cu-Mg-Zn system 413 Curie temperature 18 data see models and data Debye function 16 Debye-Hückel behaviour 150, 154 DICTRA simulations 382 dielectrics 123, 127–8 diffusion calculations in compound steel 386–91 in multicomponent phases 347–50 dolomite 296–7 Duhem equation see Gibbs-Duhem equation dynamic processes 351–7 Earth’s mantle transition zone 132–42 electric arc furnaces 415–24 electric lighting 304–5 HID (high-intensity discharge) 305–6, 309 mercury vapour pressure 312–21 electrochemistry 3 electrode potential 144–54 EMF evaluation 228–30 endogas 213 engine linings 200 enthalpy 3, 12 entropy-of-mixing effect 437 equilibrium calculations amalgam systems 314–21 ammonium chloride-gas equilibrium 69–70 carbon steel and alloy steel 378–80 gas-salt equilibrium 79–82 kinetic controls 359–67 Ni-Al equilibrium 120 and nuclear accidents 168–9
Index oxide systems 70–1 salt solution-water equilibria 48–9 water-phosphoric acid equilibrium 70 equilibrium reactors 352 equilibrium thermochemistry 3–4, 7 eutectic phase diagrams 274 ex-vessel calculations 171–3 extensive property balances 12–13 extent of reaction method 73–4 Fe-Ni alloys 98, 99–102 Fick’s law 347 first order phase transitions 18 five-component Ni-base model 405–14 fluorescent lamps 304–5 mercury vapour pressure 312–21 relative light output 312 free energy for a chemical reaction 307–8 free-energy-based models 323 furnace atmospheres carbon activity 213–19 electric arc furnaces 415–24 gas discharge light sources 304–10 gas phase 32–3 corrosion 442–5 gas and salt interactions 82–8 gas turbines 239–46 gas-salt equilibrium 79–82 geometric models 26–8 Gibbs energy free energy for a chemical reaction 307–8 geometric models 26–8 integral molar 8–9 of liquid solutions 34–7 magnetic Gibbs energy 37–40 of mantle-forming minerals 133, 134–5 natural variables 3 relative Gibbs energies 22 disadvantages 24 solution phases 9–12, 22–40 standard element reference (SER) state 16, 24, 25 stoichiometric substances data 16–21 sublattice model 28–30 thermodynamic databank 14–16
449
Gibbs-Duhem equation 3, 45, 47–8, 119 Gibbs-Helmholtz relation 16 glass-melting process 282–302 batch-to-melt conversion 293–5 modelling 297–301 crystalline reference system (CRS) 286 dolomite addition 296–7 glass cullets added to 297 heat content of glass melts 290–3 limestone addition 296–7 majority partition principle 286 multicomponent glasses 286–90 neutral redox conditions 296 one-component glasses 284–6 parsimony principle 286–7 thermodynamic properties 284–93 grain boundary distribution 443 Guggenheim model 36–7 Gulliver-Scheil method 341, 343–6, 373–4, 410 heat balance calculations 273–81 heat content of glass melts 290–3 heat exchangers 200, 398–403 heat treatment of steel 212–22 Helmholtz energy 3 Gibbs-Helmholtz relation 16 Hertz-Knudsen relation 128 HID (high-intensity discharge) lamps 305–6, 309 high temperature corrosion of SiC 200–210 high-speed steels 91–7 tungsten based 91 high-temperature gas discharge light sources 304–10 thermochemical modelling 306–10 hot corrosion in CPFBCs 241–4 hot corrosion in gas turbines 239–46 thermodynamic modelling 240–1 hot corrosion of heat exchanger components 398–403 hot corrosion in PPCCs 244–5 hot isostatic pressing 118–21 Clausius-Clapeyron equation 118, 119–20 Ni-Al equilibrium 120
450
Index
hot salt corrosion 77–89 data limitations 88 data requirements 77–9 extensions to higher-order systems 88–9 future developments 89 gas and salt interactions 82–8 gas-salt equilibrium 79–82 sodium-sulphate-rich liquid phase 81–2 hydrogen-oxygen environments, SiC corrosion 200–210 ideal gas model 32–3 image component technique 434–8 in-vessel calculations 169–70 inclusion cleanness 267–72 InCorr 398, 442 industrial furnace atmospheres see furnace atmospheres industrial glass-melting process 282–302 integral molar 8–9 integrated circuits 123 interface stability in microelectronics 124–7 internal energy 3, 12 interstitial solutions 30–1 ionic liquids 34 ionic phases 31–2 ionic solid solutions 31–2 ionic two-sublattice model 35 iron ions 144–54 isostatic pressing 118–21 kinetic controls 359–67 kinetics 341 lanthanum 182 latent heat 273 lattice defects 31 leaching simulation 333–4 Legendre transformation 3–4 lighting see electric lighting limestone 296–7 Linz-Donawitz converter process 425–35 modelling tool 422–3 process model development 419–22 simulation results 423–7
liquid injection CVD process 124 liquid solutions, Gibbs energy 34–7 low-carbon stainless steels 155–60 complex equilibrium 156–9 mass action law 156 magnesium alloys 375–84 magnetic Gibbs energy 37–40 majority partition principle 286 mantle-forming minerals 133, 134–5 mass action law 4–5, 156 mass balance equation 405 materials synthesis 123 matte phase of copper and nickel extraction 188–98 Maxwell relations 3, 4 mechanical alloying 248–62 calculation of metastable phase ranges 249–51 milling process 248 solid solubilities 248 melting of alloys 392–6 melting of a tool steel 392–6 mercury vapour pressure 312–21 metallurgical-grade silicon 415–24 complex equilibrium approach 398–9 countercurrent reactor approach 400–6 stoichiometric approach 398 metastable phases 9–12, 248–62 microelectronics see thermodynamics in microelectronics microstructure simulation 390–4 milling process 248 models and data 14–40 cell model 36, 74 cement modelling 322–35 compound energy model 29, 30–1 first order phase transitions 18 geometric models 26–8 Guggenheim model 36–7 ideal gas model 32–3 ionic solid solutions 31–2 ionic two-sublattice model 35 lattice defects 31 Ni-base model 405–14 nuclear thermodynamic database 162–8 phase field models 390
Index quasichemical models 36–7, 74 relative data 21–2 second order phase transitions 18 solution phase systems 22–40 stoichiometric substances data 16–21 sublattice model 28–30 testing predictions 15–16 thermodynamic databanks 14–16, 226–7 molten Corium-concrete interaction (MCCI) 161 MTDATA 323–5 multicomponent diffusion in compound steel 386–91 multicomponent glasses 286–90 multicomponent steels 219–20 multisublattice approach 31 Murnaghan equation 19, 133 Néel temperature 37, 38 Nernst equation 146 neutral redox conditions 296 Ni-Al equilibrium 120 Ni-base model 405–14 nickel extraction from sulphide ores 188–98 nickel-based superalloys 77 non-equilibrium modelling 425–35 nuclear accidents 161–74 binary and ternary subsystems 166–7 complex thermochemical phenomena 169–73 critical assessment 166 equilibrium calculation software 168–9 ex-vessel calculations 171–3 in-vessel calculations 169–70 molten Corium-concrete interaction (MCCI) 161 thermodynamic modelling 164–6 nuclear thermodynamic database 162–8 nuclear waste repositories 322 nuclide distribution in scrap steel 178–86 distribution ratios 185–6 liquid metal phase 181 partition ratios 182–5 relevant nuclides 179 slag phase 181
451
thermochemical data preparation 179–82 Ohm’s law 348 one-component glasses 284–6 OPTICORR 398, 442 oxide systems equilibrium 70–1 Parrot program 378 parsimony principle 286–7 partition ratios 182–5 pentlandite 193 phase diagrams 43–72, 74 alloys in corrosive atmospheres 441–2 amalgam systems 314–21 Baur-Glässner diagram 55–7 Earth’s mantle transition zone 132–42 eutectic phase diagrams 274 phase rule 66–71, 231–8 Porbaix diagram 58, 63–6 potential phase diagrams 228–30 quaternary phase diagrams 114–21, 228–30 solidification paths 371 two-dimensional diagrams 45 types of 43–50 zero-phase fraction lines 50–5 phase field models 390 phase rule 66–71, 231–8 phase stability in microelectronics 123 phenomenological treatment 347–9 Planck, Max 273 plasma etching 127 polytype phases 114, 115 Porbaix diagram 58, 63–6 Portland cement see cement modelling potential phase diagrams 228–30 pressurised pulverised coal combustion (PPCC) 244–5 pressurised-water reactors 161 pyrometallurgy of copper-nickel-iron sulphide ores 188–98 blowing the matte 188–9 phase separation in the matte 191 recrystallisation 191–3 solidification 191–3 thermodynamic models and data 193–5
452
Index
quasi-adiabatic transformations 263–6 quasichemical models 36–7, 74 quaternary Al-Cu-Mg-Zn system 414–15 quaternary phase diagrams 114–21, 228–30 modelling 115 polytype phases 114, 115 solid phases 114 radiation sources 178–86 reactive ion etching 127–8 reactor modelling 351–5 Redlich-Kister series 25, 26, 38 relative data 21–2 relative Gibbs energies 22 disadvantages 24 relative light output 312 resistance loss 106–13 salt binary systems 78 salt corrosion see hot salt corrosion salt solution-water equilibria 48–9 Scheil method see Gulliver-Scheil method scrap steel 178–86 sealed radiation sources 178–86 second order phase transitions 18 SER (standard element reference) state 16, 24, 25 Si-C-H system 204–5 Si-C-O-H system 206 SiC corrosion in hydrogen-oxygen environments 200–210 models for the corrosion process 201–4 thermodynamic analysis 204 silicon arc furnaces 353 silicon production 415–24 simulation of dynamic and steady-state processes 351–7 control of material flows 355–6 reactor modelling 351–5 SimuSage 351, 422 soda ash 299 sodium-sulphate-rich liquid phase 81–2 solid solubilities 248 solidification paths 343–6, 344–5, 371–4 solidification of pure metals 263
solution phase systems 22–40 gas phase 32–3 interstitial solutions 30–1 ionic phases 31–2 liquid solutions 34–7 magnetic effects 37–40 multisublattice approach 31 sublattice model 28–30 partial properties 29–30 substitutional solutions 25–8 partial properties 28 solution phases, Gibbs energy 9–12, 22–40 squeeze forging 118 stable phases 9–12 stainless steels austenitic 106–13 diffusion calculation 386–91 low-carbon 155–60 standard element reference (SER) state 16, 24, 25 steady-state processes 351–7 stoichiometric reactions 4–7, 16–21, 398 strontium 182 sublattice model 28–30 partial properties 29–30 substitutional solutions 25–8 partial properties 28 sulphide ores 188–98 sulphuric acid solutions 144–54 superalloys 77, 405 tempering steel 386–9 ternary amalgam systems 318–19 ternary subsystems 166–7, 375–83 testing predictions 15–16 thermochemical relationships 3–13 complex systems 7–9 equilibrium thermochemistry 3–4, 7 extensive property balances 12–13 Gibbs-Duhem equation 3, 45, 47–8, 119 solution phases 9–12 stoichiometric reactions 4–7 thermodynamic databank 14–16, 226–7 thermodynamics in microelectronics 123–31 annealing of thin films 128–30
Index chemical stability 123 interface stability 124–7 materials synthesis 123 phase stability 123 plasma etching 127 reactive ion etching 127–8 thin film deposition 124–7 thermostatics 341 thin film deposition 124–7 titanium dioxide modelling 437–45 anatase-rutile transformation 437–41 calcination process 437 entropy-of-mixing effect 437 image component technique 434–8 titanium(IV) chloride oxidation burner 434–8
453
tool steel 392–6 transverse rupture strength 98 tungsten based high-speed steels 91 turbines blades 77 hot corrosion 239–46 two-dimensional diagrams 45 uranium 182 W-to-C ratio 98 water-phosphoric acid equilibrium 70 wet hydrogen corrosion 206, 209 zero-phase fraction lines 50–5 Zircaloy cladding 161
454
Index
