Fundamentals of metallurgy
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Fundamentals of metallurgy Edited by Seshadri Seetharaman
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 and Maney Publishing Limited on behalf of The Institute of Materials, Minerals & Mining Published by Woodhead Publishing Limited, Abington Hall, Abington, Cambridge CB1 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 2005, Woodhead Publishing Limited and CRC Press LLC ß Woodhead Publishing Limited, 2005 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 Limited ISBN-13: Woodhead Publishing Limited ISBN-10: Woodhead Publishing Limited ISBN-13: Woodhead Publishing Limited ISBN-10: CRC Press ISBN-10: 0-8493-3443-8 CRC Press order number: WP3443
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Contents
Contributor contact details Preface
xi xiii
Part I Understanding the effects of processing on the properties of metals 1
Descriptions of high-temperature metallurgical processes
H Y S O H N , University of Utah and S S R I D H A R , Carnegie Mellon
3
University, USA
1.1 1.2 1.3 1.4 1.5 1.6 1.7
2
Introduction Reactions involving gases and solids Reactions involving liquid phases Casting processes Thermomechanical processes References Appendix: notation
Thermodynamic aspects of metals processing R E A U N E and S S E E T H A R A M A N , Royal Institute of
3 4 17 27 31 34 37
38
Technology, Sweden
2.1 2.2 2.3 2.4 2.5 2.6 2.7
Introduction Basic concepts in thermodynamics Chemical equilibrium Unary and multicomponent equilibria Thermodynamics of solutions Thermodynamics of multicomponent dilute solutions Modelling of metallic systems
38 39 44 49 57 66 70
vi
Contents
2.8 2.9 2.10 2.11 2.12
Thermodynamics of ionic melts Basics of electrochemical thermodynamics Conclusions Further reading References
3
Phase diagrams, phase transformations, and the prediction of metal properties
K M O R I T A , The University of Tokyo and N S A N O , Nippon Steel
72 79 79 80 80
82
Corporation, Japan
3.1 3.2 3.3 3.4 3.5 3.6 3.7
4
Introduction Phase diagrams and potential diagrams Ternary phase diagrams Solidification in ternary systems and four-phase equilibria Examples of solidification behaviour from a phase diagram perspective Conclusions References
Measurement and estimation of physical properties of metals at high temperatures K C M I L L S , Imperial College London, UK
4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11
5 5.1 5.2 5.3
82 83 87 95 102 107 108
109
Introduction Factors affecting physical properties and their measurement Measurements and problems Fluid flow properties Properties related to heat transfer Properties related to mass transfer Estimating metal properties Acknowledgements References Appendix A: calculation of structural parameters NBO/T and optical basicity Appendix B: notation
109 113 120 122 136 146 148 169 169
Transport phenomena and metals properties
178
Introduction Mass transfer Heat transfer
178 178 200
A K L A H I R I , Indian Institute of Science, India
175 176
Contents 5.4 5.5 5.6
6
Fluid flow Further reading References
Interfacial phenomena, metals processing and properties K M U K A I , Kyushu Institute of Technology, Japan
6.1 6.2 6.3 6.4 6.5 6.6
7
7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10
8 8.1 8.2 8.3 8.4 8.5
Introduction Fundamentals of the interface Interfacial properties of a metallurgical melts system Interfacial phenomena in relation to metallurgical processing Further reading References
The kinetics of metallurgical reactions
S S R I D H A R , Carnegie Mellon University, USA and H Y S O H N , University of Utah Introduction Fundamentals of heterogeneous kinetics Solid-state reactions Gas±solid reactions Liquid±liquid reactions Solid±liquid reactions Gas±liquid reactions Comprehensive process modeling References Appendix: notation
Thermoanalytical methods in metals processing O N M O H A N T Y , The Tata Iron and Steel Company, India
Introduction Thermogravimetry (TG) Differential thermal analysis (DTA) and differential scanning calorimetry (DSC) Evolved gas analysis (EGA) and detection (EGD) References
vii 217 235 236
237 237 238 257 260 267 267
270
270 270 278 290 311 313 318 321 341 346
350 350 356 358 363 365
viii
Contents
Part II Improving process and product quality 9
Improving process design in steelmaking
D S I C H E N , Royal Institute of Technology, Sweden
9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8
Introduction Overview of process design Thermodynamics and mass balance Kinetics ± mass transfer and heat transfer Optimization of interfacial reactions Micro-modelling Conclusions References
10
Solidification and steel casting
10.1 10.2 10.3 10.4 10.5 10.6 10.7
Introduction Solidification fundamentals The growth of solids The casting of steels Conclusions Acknowledgements References
11
Analysing metal working processes
11.1 11.2 11.3 11.4 11.5 11.6 11.7
Introduction Work hardening Rate effects Interaction with phase transformations Examples of material behaviour during processing Development trends References
12
Understanding and improving powder metallurgical processes
A W C R A M B , Carnegie Mellon University, USA
G E N G B E R G , SSAB TunnplaÊt AB and MIK Research AB (MIKRAB) and L K A R L S S O N , Dalarna University, Sweden
F L E M O I S S O N and L F R O Y E N , Katholieke Universiteit Leuven,
369 369 369 375 385 387 393 396 396
399 399 400 413 428 449 450 450
453
453 454 457 462 463 468 469
471
Belgium
12.1 12.2 12.3
Introduction Production processes for powders Forming processes towards near-net shape
471 471 486
Contents 12.4 12.5
13 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8
Conclusions References
Improving steelmaking and steel properties
T E M I , Royal Institute of Technology, Sweden
ix 500 500
503
Introduction Developing processes and properties with reference to market, energy, and environment Optimization of processes to meet properties and productivity Economic optimization Environmental optimization Future trends Further reading References
503 506 523 537 546 550 553 553
Index
555
Contributor contact details
(* = main contact) Chapters 1 and 7 Professor H. Y. Sohn* Department of Metallurgical Engineering University of Utah 135 S 1460 E Salt Lake City UT 84112-0114 USA E-mail: [email protected] Professor S. Sridhar Department of Materials Science Carnegie Mellon University Pittsburgh PA 15213-3890 USA E-mail: [email protected] Chapter 2 Dr R. E. Aune* and Professor S. Seetharaman Division of Materials Process Science Department of Materials Science and Engineering Royal Institute of Technology SE-100 44 Stockholm
Sweden E-mail: [email protected] E-mail: [email protected] Chapter 3 Professor K. Morita* Department of Metallurgy The University of Tokyo Bunkyo-ku Tokyo 113-8656 Japan E-mail: [email protected] Professor N. Sano Executive Advisor Nippon Steel Corporation E-mail: [email protected] Chapter 4 Professor K. C. Mills Department of Materials Imperial College of Science, Technology and Medicine Prince Consort Road South Kensington London SW7 2BP UK E-mail: [email protected]
xii
Contributor contact details
Chapter 5 Professor A. K. Lahiri Department of Metallurgy Indian Institute of Science Bangalore 560012 India
Carnegie Mellon University Pittsburgh PA 15213 USA
E-mail: [email protected]
Chapter 11 Dr G. Engberg* MIK Research AB (MIKRAB) Teknikdalen Forskargatan 3 SE-781 27 Borlange Sweden
Chapter 6 Professor Emeritus K. Mukai Department of Materials Science and Engineering Kyushu Institute of Technology Sensui-Cho Tobata-ku Kitakyushu 804 8550 Japan E-mail: [email protected] Chapter 8 Professor O. N. Mohanty Research and Development Services The Tata Iron and Steel Company Ltd 11T Kharagpur-721302 India E-mail: [email protected][email protected] Chapter 9 Professor Du Sichen Department of Materials Science and Engineering Royal Institute of Technology SE-100 44 Stockholm Sweden E-mail: [email protected] Chapter 10 Professor A. W. Cramb Department of Materials Science and Engineering
E-mail: [email protected]
E-mail: [email protected] Dr L. Karlsson Dalarna University Chapter 12 Dr F. Lemoisson* and Professor L. Froyen Physical Metallurgy and Materials Engineering Section Katholieke Universiteit Leuven Kasteelpark Arenberg 44 BE 3001 Heverlee Belgium E-mail: fabienne.lemoisson @mtm.kuleuven.ac.be E-mail: [email protected] Chapter 13 Professor T. Emi Takasu 5-1 B1905 Urayasu-shi Chiba 279-0023 Japan E-mail: [email protected]
Preface
Metallurgy refers to the science and technology of metals. The subject area can be considered as a combination of chemistry, physics and mechanics with special reference to metals. In later years, metallurgy has expanded into materials science and engineering encompassing metallic, ceramic and polymeric materials. Metallurgy is an ancient subject linked to the history of mankind. The development of civilisations from stone age, bronze age and iron age can be thought of as the ages of naturally available ceramic materials, followed by the discovery of copper that can be produced relatively easily and iron that needs higher temperatures to produce. These follow the pattern of the Ellingham diagram known to all metallurgists. Faraday introduced the concept of electrolysis which revolutionised metal production. Today, we are able to produce highly reactive metals by electrolysis. The prime objective to produce metals and alloys is to have materials with optimised properties. These properties are related to structure and thus, physical as well as mechanical properties form essential parts of metallurgy. Properties of metals and alloys enable the choice of materials in production engineering. The book, Fundamentals of Metallurgy is a compilation of various aspects of metallurgy in different chapters, written by the most eminent scientists in the world today. These participants, despite their other commitments, have devoted a great deal of time and energy for their contributions to make this book a success. Their dedication to the subject is admirable. I thank them sincerely for their efforts. I also thank Woodhead Publishing Company for this initiative which brings the subject of metallurgy into limelight. Seshadri Seetharaman Stockholm
Part I
Understanding the effects of processing on the properties of metals
1
Descriptions of high-temperature metallurgical processes H Y S O H N , University of Utah and S S R I D H A R , Carnegie Mellon University, USA
1.1
Introduction
Metallurgical reactions take place either at high temperatures or in aqueous solutions. Reactions take place more rapidly at a higher temperature, and thus large-scale metal production is mostly done through high-temperature processes. Most metallurgical reactions occurring at high temperatures involve an interaction between a gas phase and condensed phases, which may be molten liquids or solids. In some cases, interactions between immiscible molten phases are important. High-temperature metallurgical reactions involving molten phases are often carried out under the conditions of near equilibria among all the phases; other such reactions proceed under the control of interphase mass transfer with equilibria at interphase boundaries. Reactions involving gas±solid contact also often take place under the rate control of mass transfer with chemical equilibrium at the interface, but the chemical kinetics of the heterogeneous reactions are more often important in this case than those involving molten phases. Even in this case, mass transfer becomes increasingly dominant as temperature increases. The solid phases undergo undesirable structural changes, such as fusion, sintering, and excessive reduction of internal porosity and surface area, as temperature becomes too high. Thus, gas±solid interactions are carried out in practice at the highest possible temperatures before these undesirable changes in the solid structure become damaging. In the case of high temperature oxidation, the structure of the product oxide determines the mass transport of gases and ions. The treatment of metals in their molten state, e.g. refining and alloying, involves reactions between the melt and a gas phase or a molten slag. Interfacial reaction kinetics, mass transport in the molten or gaseous phase becomes important. The production of metals and alloys almost always involves solidification, the rate of which is often controlled by the rate of heat transfer through the mold.
4
Fundamentals of metallurgy
1.2
Reactions involving gases and solids
Since metals occur in nature mostly as compounds (minerals), the first step in the utilization of the naturally occurring sources is their chemical separation into elemental forms. More often than not, the first reaction in this chemical separation step involves an interaction between the solid-phase minerals and a reactant gas.
1.2.1 Reduction of metal oxides by carbon monoxide, carbon, or hydrogen Metal oxides are most often reduced by carbon or hydrogen. The reason why reduction by carbon is treated in this section on gas±solid reaction is because the actual reduction is largely effected by carbon monoxide gas generated by the reaction of carbon dioxide with carbon, when carbon is used to reduce metal oxides in solid state. These reactions can in general be expressed as follows: MxO (s) ‡ CO (g) = xM (s) ‡ CO2 (g) yC (s) ‡ yCO2 (g) = 2yCO (g) Overall, MxO (s) ‡ yC (s)
(1.1) (1.2)
= xM ‡ (1 ÿ y)CO2 + (2y ÿ 1)CO
(1.3)
The amount y, which is determined by the pCO2 =pCO ratio in the product gas mixture, depends on the kinetics and thermodynamics of the two gas±solid reactions (1.1) and (1.2). In many systems of practical importance, reaction (1.1) is much faster than reaction (1.2), and thus the pCO2 =pCO ratio approaches the equilibrium value for reaction (1.1). The overall rate of reaction (1.3) is then controlled by the rate of reaction (1.2) taking place under this pCO2 =pCO ratio (Padilla and Sohn, 1979). The reduction of iron oxide is the most important reaction in metal production, because iron is the most widely used metal and it occurs in nature predominantly as hematite (Fe2O3). The production of iron occupies more than 90% of the tonnage of all metals produced. The most important reactor for iron oxide is the blast furnace, in the shaft region of which hematite undergoes sequential reduction reactions by carbon monoxide (Table 1.1). Table 1.1 Reaction 3Fe2O3 ‡ CO ˆ 2Fe3O4 ‡ CO2 Fe3O4 ‡ CO ˆ 3FeO ‡ CO2 FeO ‡ CO ˆ Fe ‡ CO2
Equil. CO/CO2 ratio at 900 ëC
Heat generation (900 ëC)
0 0.25 2.3
10.3 kcal ÿ8.8 kcal 4.0 kcal
(1.4) (1.5) (1.6)
Descriptions of high-temperature metallurgical processes
5
In the blast furnace, the solid charge flows downward, and the tuyere gas with a high CO/CO2 ratio flows upward. Thus, the tuyere gas first comes into contact with wustite (FeO), the reduction of which requires a high CO/CO2 ratio, as seen above. The resulting gas reduces magnetite (Fe3O4) and hematite (Fe2O3) on its way to the exit at the top of the furnace. The equilibrium percentage of CO in a mixture with CO2 is shown in Fig. 1.1 as a function of temperature. Again, it is seen that the equilibrium concentration of CO for the reduction of hematite to magnetite is essentially zero; i.e., CO is completely utilized for the reduction. The equilibrium content of CO for the reduction of Fe3O4 to FeO and that of FeO to Fe depend on temperature. It is also noted that wustite is a non-stoichiometric compound FexO with an average value of x equal to 0.95 in the temperature range (approximately 600 ëC± 1400 ëC) of its stability. The actual value of x and thus oxygen content depend on temperature and CO/CO2 ratio, as illustrated by the curves drawn within the wustite region in Fig. 1.1. Furthermore, the CO/CO2 ratio is limited by the Boudouard reaction given by eqn (1.2) and shown as a sigmoidal curve in Fig. 1.1 (for 1 atm total pressure without any inert gas). Thus, the reduction reactions indicated by the dashed lines to the left of this curve are thermodynamically not feasible. (In practice, however, reduction by CO to the left of the Boudouard curve is possible because the carbon deposition reaction (the decomposition of CO) to produce solid carbon is slow.)
1.1 Equilibrium gas compositions for the reduction of iron oxides by carbon monoxide. (Adapted from Evans and Koo, 1979.)
6
Fundamentals of metallurgy
The reduction of iron oxide by hydrogen is important in the production of direct reduced iron. This method of iron production is gaining increasing significance as an alternative route to the blast furnace technology with the many difficult issues facing the latter, the most important being the problems related to environmental pollution and the shear size of the blast furnace. Direct reduction technology for iron encompasses the processes that convert iron oxides into metallic iron in solid state without going through a molten phase. In this technology, iron-bearing materials are reduced by reacting with reducing substances, mainly natural gas or a coal, at high temperatures but below the melting point of iron. The product, direct reduced iron (DRI), is a porous solid, also known as sponge iron. It consists primarily of metallic iron with some unreduced iron oxides, carbon and gangue. Carbon is present in the range of 1± 4%. The gangue, which is the undesirable material present in the ore, is not removed during reduction as no melting and refining take place during the reduction process. The main usage of DRI products is in the electric arc furnace (EAF). However, due to its superior characteristics, DRI products have found their way into other processes such as blast furnaces, basic oxygen furnaces and foundries. Globally, DRI comprises about 13% of the charge to the EAF (Kopfle et al., 2001). Nowadays, the percentage of crude steel produced by BOF is approximately 63%, while that of EAF is about 33%, and the balance 4% is made up of the open hearth (OH) steel (International Iron and Steel Institute, 2004). However, the contribution of EAF to the world crude steel output is expected to increase to reach 40% in 2010 (Gupta, 1999) and 50% in 2020 (Bates and Muir, 2000). Direct reduction technology has grown considerably during the last decade. The main reasons that make this technology of interest to iron and steel makers are as follows: 1. 2. 3. 4. 5. 6.
Shortage, unpredictability and high price of scrap. The movement of EAF producers into high quality products (flat products). High capital cost of a coke plant for the blast furnace operation. Desire of developing countries to develop small steel industries and capabilities. Availability of ores that are not suitable for blast furnace operation. Necessity for increasing iron production within a shorter time frame.
There have been major developments in direct reduction processes to cope with the increasing demand of DRI. DRI production has increased rapidly from 0.80 million tons per year in 1970 to 18 million tons in 1990, 44 million tons in 2000, and 49 million tons in 2003 (MIDREX, 2004). The worldwide DRI production is expected to increase by 3 Mt/y for the period 2000±2010 (Kopfle et al., 2001). Reduction of iron bearing materials can be achieved with either a solid or gaseous reductant. Hydrogen and carbon monoxide are the main reducing gases used in the `direct reduction' (DR) technology. These gases are largely
Descriptions of high-temperature metallurgical processes
7
generated by the reforming of natural gas or the gasification of coke/coal. In reforming, natural gas is reacted with carbon dioxide and/or steam. The product of reforming is mainly H2 and CO, whereas CO is the main product from coal gasification. Reduction reactions by reducing gases take place at temperatures in the range of 850 ëC±1100 ëC, whereas those by solid carbon occur at relatively higher temperatures of 1300 ëC±1500 ëC. Carburization reactions, on the other hand, take place at relatively lower temperatures below 750 ëC. For reforming reactions, the reformed gas temperature may reach 950 ëC for the stoichiometric reformer, and 780 ëC in the case of a steam reformer. Various reactions important in direct reduction processes are listed below. Reduction reactions Fe2O3 (s) + 3CO (g) = 2Fe (s) + 3CO2 (g) Fe2O3 (s) + 3H2 (g) = 2Fe (s) + 3H2O (g) FeO (s) + CH4 (g) = Fe (s) + 2H2 (g) + CO (g) 3Fe2O3 (s) + 5H2 (g) + 2CH4 (g) = 2Fe3C (s) + 9H2O (g) C (s) + CO2 (g) = 2CO (g)
(1.7) (1.8) (1.9) (1.10) (1.11)
Reforming reactions CH4 (g) + 0.5 O2 (g) = CO (g) + 2H2 (g) CH4 (g) + H2O (g) = CO (g) + 3H2 (g) CH4 (g) + CO2 (g) = 2CO + 2H2 (g) 2CH4 (g) + H2O (g) + 0.5 O2 (g) = 2CO (g) + 5H2 (g) CH4 (g) = C (s) + 2H2 (g) H2O (g) + C (s) = CO (g) + H2 (g) H2O (g) + CO (g) = H2 (g) + CO2 (g)
(1.12) (1.13) (1.14) (1.15) (1.16) (1.17) (1.18)
Carburizing reactions 3Fe (s) + CO (g) + H2 (g) = Fe3C (s) + H2O (g) 3Fe (s) + CH4 (g) = Fe3C (s) + 2H2 (g) 3Fe (s) + 2CO (g) = Fe3C (s) + CO2 (g)
(1.19) (1.20) (1.21)
Direct reduction (DR) processes have been in existence for several decades. The evolution of direct reduction technology to its present status has included more than 100 different DR process concepts, many of which have only been operated experimentally. Most were found to be economically or technically unfavorable and abandoned. However, several were successful and subsequently improved to develop into full-scale commercial operations. In some instances, the best features from different processes were combined to develop improved processes to eventually supplant the older ones. Direct reduction processes may be classified, according to the type of the reducing agent used, to gas-based and coal-based processes. In 2000, DRI produced from the gas-based processes accounted for 93%, while the coal-based
8
Fundamentals of metallurgy
processes produced 7%. Gas-based processes have shaft furnaces for reducing. These furnaces can be either a moving bed or a fluidized bed. The two most dominant gas-based processes are MIDREX and HYL III, which combined to produce approximately 91% of the world's DRI production. Fluid-bed processes, by contrast, have recently received attention, because of its ability to process fine iron ores. These processes are based either on natural gas or coal. A list of the processes together with their relevant characteristics is given in Table 1.2 (MIDREX, 2001). The gaseous (or carbothermic) reduction of nickel oxide, obtained by dead roasting of nickel sulfide matte or concentrate, is an important intermediate step for nickel production. In the Mond process, crude nickel is obtained this way before undergoing refining by carbonylation. Crude nickel has sometimes been cast into anodes and electrolytically refined. Nickel laterite ores are reduced by carbon monoxide before an ammoniacal leach. Nickel oxide reduction reactions are simple one-step reactions, as follows: NiO (s) + H2 (g) or CO (g) = Ni (s) + H2O (g) or CO2 (g)
(1.22)
Both reactions have negative Gibbs free energy values. For hydrogen reduction it is ÿ7.2 and ÿ10.3 kcal/mol, respectively, at 600K and 1000K. For reduction by CO, the free energy values are ÿ11.1 kcal/mol at both 600K and 1000K. The thermodynamic data used here as well as below were obtained from Pankratz et al. (1984). The corresponding equilibrium ratio pH2 =pH2 O is 2.4 0002 10ÿ3 and 5.6 0002 10ÿ3, respectively, at 600K and 1000K, and the equilibrium ratio pCO2 =pCO is 9.5 0002 10ÿ5 and 3.7 0002 10ÿ3, respectively, at the same temperatures. Therefore, the reactant gases are essentially completely consumed at equilibrium in both cases. These reduction reactions are mildly exothermic, the standard enthalpy of reaction (0001Hrë) being ÿ2 to ÿ3 kcal/mol for hydrogen reduction and ÿ11.2 to ÿ11.4 kcal/mol for reduction by carbon monoxide. Zinc occurs in nature predominantly as sphalerite (ZnS). The ZnS concentrate is typically roasted to zinc oxide (ZnO), before the latter is reduced to produce zinc metal by the following reactions (Hong et al., 2003): ZnO (s) + CO (g) = Zn (g) + CO2 (g) C (s) + CO2 (g) = 2CO (g)
(1.23) (1.24)
ZnO (s) + C (s) ! Zn (g) + CO (g)
(1.25)
The overall reaction is essentially irreversible (0001Gë ˆ ÿ12.2 kcal/mol at 1400K) and highly endothermic (0001Hë ˆ ‡84.2 kcal/mol at 1400K) and the gaseous product contains a very small amount of CO2 at temperatures above 1200K (Hong et al., 2003). It is also noted that this reaction is carried out above the boiling point of zinc (1180K), and thus zinc is produced as a vapor mixed with CO and the small amount of CO2 from the reaction. Zinc is recovered by condensation. Zinc vapor is readily oxidized by CO2 or H2O (produced when
Table 1.2 Characteristics of some DR processes Process
Builder
MIDREX HYL FINMET CIRCORED IRON CARBIDE
MIDREX HYLSA VAI Lurgi Nucor Qualitech CIRCOFER Lurgi REDSMELT SMS Demag IRON DYNAMICS Mitsubishi Demag FASTMET MIDREX COMET ITmk3
CRM MIDREX
Charge
Product
Reactor
Fuel/ Reductant
Capacity, kt/y
Lump/pellet Lump/pellet Fines Fines Fines
DRI/HBI DRI/HBI HBI HBI Iron carbide
Natural gas Natural gas Natural gas Natural gas Natural gas
Fines Green pellets Dried pellets
HBI Liquid iron Liquid iron
Shaft furnace Shaft furnace 4 fluidized beds 2 fluidized beds 1 fluidized bed 2 fluidized beds 2 fluidized beds Rotary hearth furnace Rotary hearth furnace
Coal Coal Coal
Up to 1600 Up to 700 2200±2500 500 300 660 5t/d 50±600 520
Dried briquettes and pellets Fines Dried briquettes and pellets
DRI/HBI Rotary hearth furnace Liquid iron Low carbon slab Rotary hearth furnace Iron nuggets Rotary hearth furnace
Coal
150±650
Coal Coal
100 kg/h 100 kg/h
10
Fundamentals of metallurgy
coal is used as the reducing agent) at lower temperatures. Thus, zinc condensation should be done as rapidly as possible, and the CO/CO2 ratio in the product gas must be kept as high as possible by the use of excess carbon in the reactor. Small amounts of copper have been produced by the reduction of oxides that occur naturally, obtained by the dead roasting of sulfide, or produced by precipitation from aqueous solution. The final process in tungsten production is the hydrogen reduction of the intermediate tungsten oxide (WO3 or W4O11) obtained through the various processes for treating tungsten ores. Because of the substantial volatility of the higher oxide, its reduction is carried out at a low temperature to obtain nonvolatile WO2. WO2 is then reduced to tungsten metal at a higher temperature (Habashi, 1986), as indicated below: WO3 (s) + H2 (g) = WO2 (s) + H2O (g); WO2 (s) + 2H2 (g) = W (s) + 2H2O (g);
T = 510 ëC T = 760±920 ëC
(1.26) (1.27)
Molybdenum is also produced by the hydrogen reduction of its oxide MoO3. This process is also carried out in two stages for the same reason as in the case of tungsten oxide reduction: MoO3 (s) + H2 (g) = MoO2 (s) + H2O (g); MoO2 (s) + 2H2 (g) = Mo (s) + 2H2O (g);
T = 600 ëC (1.28) T = 950±1100 ëC (1.29)
Limited amounts of magnesium are produced by the carbothermic reduction of MgO, according to MgO (s) + C (s) = Mg (g) + CO, CO2 (g)
(1.30)
The reduction reaction is carried out at about 2200 ëC and thus magnesium is produced as a vapor (B.P. ˆ 1090 ëC). Much like zinc vapor mentioned earlier, magnesium vapor is susceptible to oxidation and requires similar measures for its condensation and collection. For other aspects of gaseous reduction of metal oxides, including reduction by carbon involving gaseous intermediates, the reader is referred to the literature (Alcock, 1976; Evans and Koo, 1979; Habashi, 1986).
1.2.2 High temperature oxidation High temperature oxidation (tarnishing) is a form of environmental degradation of metals and alloys as a result of the following chemical reaction: Me (s) ‡ 0:5yX2 (g) ! MeXy
…1:31†
Due to the high temperatures involved, these reactions are generally rapid and thus are a concern for high temperature application of structural parts such as turbines, jet propulsion systems and reactors. While the electro-negative gaseous
Descriptions of high-temperature metallurgical processes
11
oxidant (X) could be sulfur, chlorine, etc., the discussion here will mainly be limited to oxidation by oxygen. Thermodynamically, a reaction will be favorable when the free energy is negative. The free energy is decreased by a lower nobility of the metal (or a higher activity of a metallic alloying element), a lower temperature and a higher partial pressure of the oxidizing gas. In the case of alloy oxidation where temperatures are high enough to form oxide slag mixtures, the activities of the oxide species need to be considered too. In general, oxidation occurs according to the following steps: (i) First, oxygen adsorbs on the surface of the metal, which is usually not a rate-limiting step. (ii) Second, oxide forms and covers the surface, forming an oxide scale. (iii) Finally, the oxide scale thickens at the expense of the metal. From an engineering point of view, it is the rate of the third stage that is of interest since this stage constitutes the major loss of metal. Also, from a mechanistic point of view, surface coverage is generally fast and not rate determining. The thickening rate of the oxide scale is known empirically to occur through one or a combination of distinct time dependent behaviors of weight gain vs time, linear, parabolic, logarithmic and cubic. The type of relation observed depends largely on the micro-structural properties of the scale formed. If the resulting oxide layer is porous enough, pore diffusion is very fast and thus access of the oxidizing gas to the metal is easy. Chemical reaction at the interface controls the rate of oxidation. The rate of mass gain (dm/dt) per unit area (A) will then follow first order heterogeneous reaction kinetics, expressed by 1 dm ˆ krxn 0001 ‰aMe Š‰pO2 Š A dt
…1:32†
If the oxide-scale±metal contact area remains constant and the bulk oxygen partial pressure is kept unchanged with time, the above expression becomes dm ˆ k0 dt
…1:33†
and after integrating, this results in a linear dependence of weight gain vs time: 0001m ˆ k 0 0001 0001t
…1:34†
The parabolic time dependency is the most commonly observed type in metal oxidation and occurs when a dense oxide scale does not allow for gaseous diffusion through the product layer, necessitating ionic diffusion through the oxide layer and an electrochemical reaction process. The surface of the oxidescale that is exposed to air serves as a cathode where reduction takes place according to, 0:5yO2 ‡ 2yeÿ ˆ yO2ÿ
…1:35†
The oxide-scale±metal interface serves as an anode where oxidation takes place:
12
Fundamentals of metallurgy Me ˆ Me2y‡ ‡ 2yeÿ
…1:36†
The oxide scale itself serves as both electrolyte and electron lead. In this case, the growth of the oxide will occur either at the surface or at the oxide-scale± metal interface depending on the point defect types that dominate the oxide structure. In the case where the defects allow for a rapid transport of metal ions, due to the abundance of metal ion interstitials or vacancies, the oxide growth will take place on the outer surface of the oxide scale, according to Me2y‡ ‡ yO2ÿ ˆ MeOy
…1:37†
This is the case for Zn oxidation where the product ZnO has an excess of Zn. On the other hand, if defects in the scale promote oxygen transport, i.e. in Zr oxidation where ZrO2 has a high oxygen vacancy concentration, oxide growth will occur internally at the oxide-scale±metal interface. Whichever is the case, we can view the process, as a first approximation, as a steady state diffusion across a growing scale, where the flux of the reactant (through ionic diffusion) controls the rate of oxide growth. dm C1 ÿ C2 …1:38† ˆ ÿkD 0001x dt C corresponds to the ion (metal or oxygen) that is most mobile as a result of oxide-scale defect chemistry. Its concentration at the surface is C1 and that at the interface C2, and the thickness of the oxide layer is 0001x ˆ f …t†. Since the mass is related to the thickness by 0001x…t† 0001 A…001aoxide ÿ 001ametal † ˆ m…t†, we can rewrite the last expression as: dm 1 ˆ k 00 dt m…t† Integrating, we get: p 0001m ˆ k 00 0001 0001t1=2
…1:39†
…1:40†
which results in a parabolic time dependence. A logarithmic rate of oxidation, 0001m ˆ klog 0003 log…at ‡ 1†
…1:41†
where klog and a are constants is observed for thin oxide-scales at low temperatures where electronic conductivity is expected to be low. To a large extent, the resulting oxide film structure determines the rate of oxidation, i.e. whether the oxide forms a dense, compact, continuous scale that acts as a barrier towards gas diffusion or not. In general, the scale offers increased protection with increasing oxide-film adherence, increasing oxide melting point, decreasing oxide vapor pressure, similar thermal expansion of metal and oxide, increasing oxide plasticity and low mobilities of ions and electrons in the oxide. An empirical relation that is called the Pilling±Bedworth ratio (Pilling and Bedworth, 1923) is based on the fact that if the growing oxide
Descriptions of high-temperature metallurgical processes
13
is under compression, it is more likely to be protective. This suggests that when the ratio Oxide Volume …1:42† Metal Volume is moderately greater than 1, the oxide is likely to be protective. Too high ratios are undesired since excessive compressive stress will be detrimental to adherence. As examples of PB values, the Al-Al2O3 system, which is protective, has a ratio of 1.28, whereas the Ca-CaO system, which is non-protective, has a ratio of 0.64. It is noted that a protective oxide-scale that results in a parabolic oxidation rate may transition into a linear rapid oxidation rate when a failure of the protective layer occurs. This can be caused by fracturing of the film or liquefaction due to slag formation. Most commercial metals used in high temperature applications are alloys and the oxidation of alloys is an important topic. The oxidation process in a multicomponent metallic system is extremely complex due to the fact that (i) different oxidation products can be formed and oxidation rates are determined by (ii) thermodynamics of the reaction (1.31) of individual alloying elements and transport properties of alloying elements and ions. A detailed discussion on alloy oxidation is beyond the scope of this chapter, but a few common alloying additions that lead to improved oxidation resistance should be mentioned. Chromium is commonly added to ferrous alloys to form Cr-rich protective scales. Beyond 20 wt% Cr in Fe±Cr alloys, the parabolic rate constant drops drastically (Jones, 1992). Nickel in conjunction with Cr enhances the oxidation resistance, primarily in applications involving thermal cycling. For an in-depth study on oxidation and tarnishing, a text by Kofstad (1988) may be consulted. PB ratio ˆ
1.2.3 Coking Coal is constituted of partially decomposed organic matter in the presence of moisture. The term `coking' is used for a process in which all the volatile constituents in coal are eliminated. This is carried out by heating coal in the absence of air in retort ovens that consist of vertically oriented chambers that are heated from the outside (Rosenqvist, 1983). The chambers are about 5 m in height but relatively narrow in width (0.5 m) in order to allow for appreciable heat flux through thermal conduction from the outside into the chamber. The heat from the evolved gases is used to help heat the retorts. In the case of coke for metallurgical applications, it is carbonized at a high temperature range (between 900 ëC and 1096 ëC) (Wakelin, 1999) and it is important that the product remains structurally stable and does not break into powder during the process. The coking process can be separated into the following three stages
14
Fundamentals of metallurgy
(Wakelin, 1999): first, the primary coal breaks down below 700 ëC and water, oxides of C, H2S, aromatic hydrocarbon compounds, paraffins, olefins, phenolic and nitrogen-containing compounds are released. Second, above 700 ëC, large amounts of hydrogen are released along with aromatic hydrocarbons and methane. Nitrogen-containing compounds react to form ammonia, HCN, pyridine bases and nitrogen. Finally, in the third stage, hydrogen is removed and hard coke is produced. During the process, 20±35% by weight of the original coal is released as volatiles. Inorganic non-volatile constituents in the coal remain in the coke as ash.
1.2.4 Decomposition reactions A number of decomposition reactions are important in metallurgy. Examples include the decomposition of alkali earth carbonates (especially calcium carbonate in limestone and dolomite) to oxides; sulfates to oxides, oxysulfates and metals; carbonyls and iodides in the refining, respectively, of nickel and of titanium and zirconium; and sulfides to lower sulfides. Decomposition reactions are endothermic and thus consume large amounts of energy with the associated cost and environmental implications. The most important decomposition reaction relevant to metal production is the decomposition of limestone to lime according to CaCO3 (s) ˆ CaO (s) ‡ CO2 (g);
0001Hë ˆ 40.3 kcal/mol at 1100K
(1.43)
The thermodynamic decomposition temperature of CaCO3 is 907 ëC. This reaction occurs in the blast furnace and other smelting furnace operations in which limestone is charged to produce lime as a flux. The calcination of limestone is also performed separately for the specific purpose of producing lime to be used as a flux in such processes as oxygen steelmaking as well as a neutralizing agent for acid pickling and leaching solutions. Limestone calcination is carried out in rotary kilns, shaft kilns and rotary hearth reactors. Magnesite (magnesium carbonate) is calcined to obtain magnesia (MgO), which is used to make refractory bricks. The decomposition reaction occurs according to (Hong et al., 2003), MgCO3 (s) ˆ MgO (s) ‡ CO2 (g); 0001Hë ˆ 25.8 kcal/mol at 1200K (1.44) This reaction also occurs in the calcinations of dolomite, which proceeds according to the following reactions: CaCO3 0001 MgCO3 (s) ˆ CaCO3 0001 MgO (s) + CO2 (g) at 700±850 ëC at 850±950 ëC CaCO3 0001 MgO (s) ˆ CaO 0001 MgO (s) + CO2 (g)
(1.45) (1.46)
According to the results of differential thermal analysis (Habashi, 1986), the decomposition of the MgCO3 component in dolomite takes place at 100 ëC higher than pure MgCO3. This is because dolomite contains calcium and
Descriptions of high-temperature metallurgical processes
15
magnesium atoms forming a complex crystal structure, rather than being a physical mixture of CaCO3 and MgCO3. Strontium carbonate is decomposed to SrO according to the following reaction (Arvanitidis et al. 1997), SrCO3 (s) ˆ SrO (s) + CO2 (g)
0001Hë ˆ 49.3 kcal/mol at 1200K (1.47)
Examples of sulfate decomposition are listed below (Habashi, 1986): SnSO4 3MnSO4 2CuSO4 CuO 0001 CuSO4 Ag2SO4
ˆ ˆ ˆ ˆ ˆ
SnO2 ‡ SO2 Mg3O4 ‡ 2SO3 ‡ SO2 CuO 0001 CuSO4 ‡ SO3 2CuO ‡ SO3 2Ag ‡ SO3 ‡ ÝO2
375 ëC 700±800 ëC 650±670 ëC 710 ëC 800±900 ëC
(1.48) (1.49) (1.50) (1.51) (1.52)
Nickel carbonyl, which is obtained by reacting crude nickel with carbon monoxide, is decomposed at about 180 ëC to obtain pure nickel according to Ni(CO)4 (g) ˆ Ni (s) + 4CO (g)
0001Hë ˆ 31 kcal/mol
(1.53)
Some sulfides decompose to lower sulfides upon heating and release sulfur. Examples are: 2FeS2 (s) 4CuS (s) 6NiS (s) 5CuFeS2 (s)
ˆ ˆ ˆ ˆ
2FeS (s) ‡ S2 (g) 2Cu2S (s) ‡ S2 (g) 2Ni3S2 (s) ‡ S2 (g) Cu5FeS4 (s) ‡ 4FeS (s) ‡ S2 (g)
(1.54) (1.55) (1.56) (1.57)
1.2.5 Chemical vapor synthesis of metallic and intermetallic powders Many metals in the form of powder, especially ultrafine powder (UFP), display useful physical properties. With a large specific surface area, they are the raw materials for powder metallurgical processing. They also possess other exceptional properties involving light absorption (Nikklason, 1987), magnetism (Okamoto et al., 1987), and superconductivity (Parr and Feder, 1973). For much the same reasons, the powders of intermetallic compounds are also expected to offer promising possibilities. Several methods have been practiced in the production of metallic and intermetallic powders. Here, we will summarize developments in the synthesis of such powders by the vapor-phase reduction of metal chlorides, with an emphasis on the synthesis of intermetallic powders. A reaction between a metal halide and hydrogen can in general be written as follows: MXn (g) ‡ 0.5nH2 (g) ˆ M (s) ‡ nHX (g)
(1.58)
16
Fundamentals of metallurgy
where M and X represent the metal and the halogen, respectively. The hydrogen reduction of single-metal chlorides for the preparation of metallic UFP has been studied for tungsten and molybdenum (Lamprey and Ripley, 1962), cobalt (Saeki et al., 1978), and nickel, cobalt, and iron (Otsuka et al., 1984). By reducing vaporized FeCl2, CoCl2, and NiCl2 by hydrogen at 1200K to 1300K, Otsuka et al. (1984) were able to prepare corresponding metal particles in the size range 52 to 140 nm with up to 99.7% metal chloride conversion. The synthesis of metal carbide UFP has been practiced by the vapor-phase hydrogen reduction (Zhao et al., 1990; Hojo et al., 1978). Hojo et al. (1978) produced the UFP of tungsten carbide (WC, W2C) of 40 to 110 nm size by vapor-phase reaction of the WCl6-CH4-H2 system at 1000 ëC to 1400 ëC. Magnesium is a much stronger reducing agent for chlorides than hydrogen. Thus, the reduction of titanium and aluminum chlorides by magnesium is feasible, whereas the reduction of these chlorides by hydrogen is not feasible up to 2500K. Titanium sponge is prepared by the chlorination of rutile, followed by the reduction of the resulting titanium chloride by liquid or gaseous magnesium (Barksdale, 1966). A reaction between a metal halide and magnesium vapor can be written as follows: MXn (g) ‡ 0.5nMg (g) ˆ M (s) ‡ 0.5nMgX2 (l,g)
(1.59)
where M and X are metal and halogen, respectively. In recent years, Sohn and co-workers (Sohn and PalDey, 1998a; 1998b; 1998c; 1998d; Sohn et al., 2004) applied the basic concepts of the above chloride reduction methods to the `chemical vapor synthesis' of intermetallic and metal alloy powders. These reactions can in general be written as follows, when hydrogen is used: mMClx (g) ‡ nNCly (g) ‡ 0.5(mx ‡ ny)H2 ˆ MmNn (s) ‡ (mn ‡ ny)HCl (g)
(1.60)
where M and N represent two different metals, with x and y being the valences, and MmNn the intermetallic compound formed. Sohn and PalDey (1998a) synthesized fine powder (100±200 nm) of Ni4Mo at 900 ëC to 1100 ëC using hydrogen as the reducing gas. These authors also prepared a coating of Ni4Mo of 0.7 0016m thickness on a nickel substrate. Sohn and PalDey (1998b) also synthesized nickel aluminide (Ni3Al) particles (50±100 nm) at 900 ëC to 1150 ëC using hydrogen as the reducing agent. The fact that aluminum chloride is reduced by this reaction scheme is very significant, because the reduction of AlCl3 alone by hydrogen is thermodynamically unfavorable at moderate temperatures. The negative free energy of formation of the intermetallic compound makes the overall reaction feasible. Using the same chemical vapor synthesis process, Sohn et al. (2004) prepared ultrafine particles of Fe-CO alloys by the hydrogen reduction of FeCl2-CoCl2 mixtures. Sohn and
Descriptions of high-temperature metallurgical processes
17
PalDey (1998c; 1998d) also synthesized ultrafine powders of the aluminides of titanium and nickel using magnesium as the reducing agent.
1.3
Reactions involving liquid phases
1.3.1 Smelting and converting The term `smelting' has broad and narrow definitions. In the broadest sense, any metal production process that involves a molten stage is called `smelting', the word having its origin in the German word `schmelzen' ± to melt. Thus, aluminum smelting and iron smelting in addition to sulfide smelting would be included in this category. The next level of definition is the overall process of producing primary metals from sulfide minerals by going through a molten stage. The narrowest definition is the first step of the two-step oxidation of sulfur and iron from sulfide minerals, mainly Cu and Ni, i.e., matte smelting or `mattemaking' as opposed to `converting' in which the matte is further oxidized, in the case of coppermaking, to produce metal. Thus, especially in coppermaking, we talk about a `smelting' step and a `converting' step. The reason for doing it in two stages has largely to do with oxygen potentials in the two stages as well as heat production, the former in turn affecting the slag chemistry (magnetite formation, for example) and impurity behavior. If one goes all the way to metal in one step, much more of the impurities go into the metal, rather than the slag, and too much heat is produced. Thus, in the first stage ± the `smelting step', as much iron, sulfur and harmful impurities as possible are removed into the large amount of slag formed in that stage, and the matte is separated and treated in a subsequent step, usually the converting step. Figure 1.2 presents a simplified flowsheet of a typical copper production operation. The copper contents at various stages are indicated in the flowsheet. The major chemical reactions that occur in the smelting and the converting steps are shown in Figs 1.3 and 1.4. In the smelting (mattemaking) step, which takes place in a molten state, large portions of sulfur and iron contained in the copper mineral (typically chalcopyrite, CuFeS2, mixed with some pyrite, FeS2) are oxidized by oxygen supplied in the form of oxygen-enriched air of various oxygen contents. The sulfur dioxide is sent to the acid plant to be fixed as sulfuric acid. The oxidized iron combines with silica, contained in the concentrate and added as a flux, to form a fayalite slag. The remaining metal sulfides Cu2S and FeS, which are mutually soluble, form a copper matte of a certain copper content, which varies from smelter to smelter (50±70%). The matte and the slag form an immiscible phase, enabling their separation, with the lighter slag floating above the matte. Another important aspect of the mattemaking step, in addition to the removal of iron and sulfur, is that large portions of undesirable impurities in the concentrate such as As, Bi, Sb, and Pb are absorbed into the slag and thus removed from copper. Valuable metals such
18
Fundamentals of metallurgy
1.2 A typical copper production operation.
as gold, silver, and other precious metals present in the concentrate largely remain in the copper. The matte, consisting of Cu2S and FeS, is separated from the slag and fed to the converting furnace. In the converting step, the FeS in the matte is first oxidized into FeO and sulfur dioxide. Silica or limestone is added to absorb the FeO by forming a slag. The remaining Cu2S is then further oxidized to form metallic copper, called the blister or crude copper, which still contains small amounts of undesirable impurities as well as valuable minor elements plus residual amounts of sulfur and iron. The sulfur and iron in the blister copper are removed by further oxidation in a fire-refining furnance. To remove iron completely, the oxygen potential must be
Descriptions of high-temperature metallurgical processes
19
1.3 Major chemical reactions in copper smelting.
1.4 Major chemical reactions in copper matte converting.
sufficiently high, and thus some copper is oxidized in this step. After iron oxide is removed, therefore, a reducing agent such as reformed natural gas is used to remove oxygen from the copper. The relevant chemical reactions are as follows: Oxidation period:
S ‡ O2 (g) ˆ SO2 (g) 2Fe ‡ O2 (g) ˆ 2FeO (l) 4Cu (l) ‡ O2 (g) ˆ 2Cu2O (l)
(1.61) (1.62) (1.63)
20
Fundamentals of metallurgy
Reduction period:
Cu2O (l) ‡ H2 (g) ˆ 2Cu (l) ‡ H2O (g) Cu2O (l) ‡ CO (g) ˆ 2Cu (l) ‡ CO2 (g)
(1.64) (1.65)
The fire-refined copper is cast into anodes that go to the electrolytic cell to be refined to 99.99 % pure copper cathodes. Thus, the fire-refining furnace is also called the anode furnace. The equilibrium that is important in the mattemaking (smelting) step as well as for iron removal in the converting step is FeS (l) ‡ Cu2O (l, slag) ˆ FeO (l, slag) ‡ Cu2S (l) 0001Gë ˆ ÿ35,000 ÿ 4.6T, cal/mol (T in K) 0012 0013 ÿ0001G000e aCu2 S 0001 aFeO ˆ K ˆ exp RT aCu2 O 0001 aFeS
(1.66) (1.67) …1:68†
K (1200 ëC) 0019 104, assuming aCu2 S =aFeS 0019 1; aFeO 0019 0:3, and aCu2 O 0019 3 0002 10ÿ5 at 1200 ëC (Biswas and Davenport, 1976). This indicates that FeS will be oxidized long before Cu2S. The thermodynamic relations of coppermaking reactions in the converting step are as follows: Cu2S (l) ‡ 1.5 O2 (g) ˆ Cu2O (s) ‡ SO2 (g), 0001G000e1200000e C ˆ ÿ54,500 cal (1.69) (1.70) Cu2S (l) ‡ 2Cu2O (s) ˆ 6Cu (l) ‡ SO2, 0001G000e1200000e C = ÿ11,500 cal Cu2S (l) ‡ O2 ˆ 2Cu (l) ‡ SO2, 0001G000e1200000e C ˆ ÿ40,200 cal (1.71) These reactions indicate that the oxidation of Cu2S (l) to Cu (l) is highly favorable. The removal of sulfur in the fire-refining step is represented by [S]in
Cu
‡ 2[O]in
Cu
ˆ SO2 (g)
(1.72)
for which K0 ˆ
pSO2
‰%SŠ‰%OŠ2
0019 (90 at 1100 ëC; 20 at 1300 ëC)
(1.73)
The smelting of nickel sulfide concentrates is similar to that of copper sulfide concentrates, except that the converting step in this case produces high-grade, low-iron matte for further treatment, rather than crude nickel metal. The smelting of lead sulfide (galena) concentrate is somewhat different in that lead sulfide is easily oxidized to PbO. Thus, the reactions for producing lead from the sulfide concentrate are as follows: PbS ‡ O2 ˆ PbO (l, slag) ‡ SO2, 1100 ëC±1200 ëC PbO (l, slag) ‡ C ˆ Pb (l) ‡ CO2 In this process, the slag is typically composed of CaO, FeO, and SiO2.
(1.74) (1.75)
Descriptions of high-temperature metallurgical processes
21
1.3.2 Slag refining After producing a metal from its ore or from recycled scrap, it will contain impurities. Some of these may be acceptable but many are not and therefore have to be removed. This depends on the elements constituting the impurity. The impurity elements can be classified as: · Economically valuable elements, e.g. Ag, Au in Cu. · Elements that are harmful for the final metal properties, e.g. P, S, Sb, Sn, H, O, N. During product formation, the metal will be cast into one form or another. This means that the metal undergoes a physical conversion from liquid into solid state. Me (l) ) Me (s)
(1.76)
The amount of an impurity element (or any other element for that matter) that can be contained in the metal lattice is always greater in the liquid metal than in the solid. In other words, the solubility of an element is greater in a liquid than in a solid. Generally the solubility of an impurity in a metal follows the trend shown in Fig. 1.5 (at constant total pressure) during cooling and solidification that occurs in casting. From the solubility change with temperature shown in this figure, it can be seen that a substantial drop in solubility happens at the melting point. Hence during casting, if the impurity level is higher in the liquid than what the solid can incorporate, there is a rejection of the impurity solute from the solidifying front into the remaining liquid. Thus, the remaining liquid continuously gets enriched by the impurity. At some point the amount of impurity in the liquid will be high enough that one of the following happens:
1.5 Solubility change during solidification.
22
Fundamentals of metallurgy
· The impurity forms a gas: e.g. 2H (dissolved in the liquid) ) H2 (g) · The impurity forms a solid or liquid compound: e.g. Mn ‡ S ) MnS or Fe ‡ O ) FeO Both pores and inclusions form in-between the grains and end up at dendrite boundaries since that is where the last liquid existed and where the concentration of the impurities were highest. The gaseous species can be entrapped during casting and result in pores which degrade the fatigue and fracture strength of the metal. The inclusion compounds form second phase inclusions inside the metal that also degrade the fatigue and fracture strength and ductility. Furthermore, they cause a loss in metals since the metal reacts with the impurity to form a second phase. In the steelmaking process the primary method of removing sulfur, phosphorous and oxygen is to separate them into a second phase, namely a slag. Generally, industrial slags and fluxes contain SiO2, MexO (metal oxides) and, depending on the slag, additional compounds like Al2O3, CaF2 and P2O5. The ratio SiO2/MexO is an indication of the degree of polymerization. This is because each MexO is considered to break a bond of the three dimensional .. ÿ network of tetrahedral units of SiO4ÿ 4 or (.Si-O ) by supplying an additional oxygen and charge, compensating the electron at the broken bond with the cation. When the ratio of SiO2/MexO is 2, each tetrahedral unit has one unshared corner and the structure is expected to resemble that of an endless sheet (Richardson, 1974) and at a ratio of 1, endless chains. At higher MexO contents the network breaks down further to form rings and then to discrete units of silica compounds. While P2O5 can easily accommodate itself by substituting P for Si in the silica network (PO4ÿ 4 ), Al2O3 is amphotheric and accommodates itself in the silica network in silica rich melts as AlO5ÿ 4 but acts as a network breaker in melts with low silica contents. Fluorides are generally thought to break the network (Kozakevitch, 1954; Mills and Sridhar, 1999) according to the reaction: . . . . (..Si-O-Si..) ‡ (Fÿ) = (..Si-Oÿ) ‡ (F-Si..). There still is uncertainty, however, concerning (i) the individual effect of the cation, (ii) whether fluorine acts as a . network breaker also at basic compositions and (iii) whether a unit of (F-Si..) is .. equivalent to a unit of (.Si-Oÿ) with respect to physical properties such as viscosity and thermal conductivity. Unlike most other elements, sulfur does not need to form a compound before partitioning to the slag phase. Molten slags are able to absorb sulfur from either molten pig iron or steel through a reduction process:
S ‡ O2ÿ ˆ S2ÿ ‡ 0.5O2
(1.77)
In the equation above, the underline denotes dissolved state in the steel melt. The distribution ratio of sulfur in the slag to that in the metal can then be written as:
Descriptions of high-temperature metallurgical processes …S† CS fS0003 ˆ p ‰SŠ K pCO2
23 …1:78†
where (S) and [S] denote sulfur concentrations in weight percent in the slag and metal, respectively. K is the equilibrium constant for the reaction 1/2S2 ˆ S and CS is the sulfide capacity of the slag, defined as: r pO2 CS ˆ …S† …1:79† p S2 From the equation above, it is clear that de-sulfurization is favored by a high sulfide capacity, an increased sulfur activity coefficient in the metal melt and a low oxygen potential. The activity coefficient of sulfur (fS0003 ) is increased by carbon and silicon, and thus de-sulfurization is best achieved prior to the oxygen steelmaking (de-carburization) process. The removal of silicon, phosphorous and manganese are carried out through oxidation, and are therefore favored during oxygen steelmaking, but since they are all highly exothermic, they are favored at lower temperatures. In the cases of Si and Mn, Si + 2O = SiO2 0001Gë ˆ ÿ594,000 ‡ 230.1 0003 T, J Mn ‡ 2O ˆ MnO (s) 0001Gë ˆ ÿ291,000 ‡ 129.79 0003 T, J
(1.80) (1.81) (1.82) (1.83)
In the case of phosphorous, P ‡ 2.5O ‡ 1.5O2ÿ ˆ PO3ÿ 4 The phosphate capacity of a slag can be defined (Wakelin, 1999): 0012 0013 …%P† 21,740 ÿ5=2 log KPO ˆ log ˆ ‰%OŠ ÿ 9:87 ‡ 0:071 0003 BO ‰%PŠ T
(1.84)
(1.85)
where BO ˆ %CaO ‡ 0.3 0003 (%MgO). Slags and their properties play a crucial role in the removal of non-metallic inclusions during clean steel manufacturing. Non-metallic inclusions are generally removed in the ladle, tundish and continuous casting mold. In all these three vessels, the molten metal is covered by a molten slag in order to provide thermal and chemical protection and, in the case of the caster, also to lubricate the mold±strand interface. Inclusions are removed by (i) transporting the inclusion to the steel±slag interface, (ii) separating across the interface and (iii) dissolving into the slag phase. Among the three steps for inclusion removal, the second step involving separation across the interface is probably the least understood and is strongly influenced by interfacial properties. The thermodynamics of inclusion removal has been studied in a number of papers (Kozakevitch et al., 1968; Kozakevitch and Olette, 1970, 1971; Riboud and Olette, 1982; Cramb and Jimbo, 1988).
24
Fundamentals of metallurgy
Consider an inclusion at the slag±metal interface. For an inclusion to be removed it is necessary for it to travel through the slag±metal interface and on into the slag phase. In terms of interfacial energies, a favorable separation will be achieved when the free energy change given by the following relationship is negative: 0001G ˆ inclusionÿslag ÿ inclusionÿmetal ÿ metalÿslag
…1:86†
While the above-mentioned model for separation of inclusions across a metal± slag interface is based on thermodynamics, it is a rather simplified view and thus its applicability limited since the kinetics may be slow. As a spherical particle approaches the interface, the film between the particle and the other phase must be drained and the hydrodynamic forces determine the speed. At closer distances to the interface, the assumption of a continuous medium is no longer valid and the thin liquid film ahead of the particle is removed slowly. The final rupture of the interface is probably rapid, as this has been found in the case of droplet separation. The residence time of particles at a fluid±fluid interface may thus be long although it is energetically favorable to separate it from one phase to another. The steps of drainage and rupture are schematically shown in Fig. 1.6. The separation time would likely depend strongly on whether the inclusion is solid or liquid. In the case of solid inclusions, it is primarily a hydrodynamic problem. Shannon and Sridhar (2004), Bouris and Bergeles (1998), Nakajima and Okamura (1992) and Cleaver and Yates (1973) have all studied the mechanism of solid particle separation across steel±slag interfaces. First, the existence of a film (that needs to be drained) was contingent upon the Reynolds number, i.e. if Re < 1, no film formation was assumed. The drainage step was computed based on (i) a force balance between the buoyancy on one hand and the drag, gravity and a so-called rebound force on the other and (ii) fluid flow past a sphere. The rebound force resulting from a normal interfacial stress is a function of the steel-melt±slag interfacial energy ( melt±slag). It should be mentioned that the presence of Marangoni forces might delay the drainage due to differences along the droplet surface as explained for the case of drainage around gas bubbles by Lahiri et al. (2002). Upon reaching a critical separation
1.6 Schematics of kinetic/transport issues in inclusion separation across metal±slag interfaces.
Descriptions of high-temperature metallurgical processes
25
from the interface, the interface was assumed to be ruptured and continued separation occurred based on the balance between the drag, buoyancy and the dynamically changing interfacial forces. Here, as long as the inclusions were not wetted by the melt, i.e. inclusion±melt > inclusion±slag, the inclusions would initially be pushed towards the slag. According to calculations (Bouris and Bergeles, 1998), an Al2O3 inclusion of 20 0016m, would separate completely into a SiO2-Al2O3-CaF2-MgO-CaO (mold flux type) slag within roughly 0:5 0002 10ÿ5 seconds. Incomplete separation occurred when the steel wetted the inclusions and TiO2 in the inclusions had a profound effect on this.
1.3.3 Processes for reactive/high requirement alloys Vacuum degassing Impurities can be removed by forcing them to form gases and float out (e.g. C ‡ O ˆ CO in the oxygen steelmaking furnace). For certain applications, extremely low impurity levels must be obtained. This can be achieved through vacuum degassing. The principle is as follows, taking as an example the degassing of hydrogen: 2H ˆ H2 (g)
(1.87)
At equilibrium, assuming Henry's law applies in the dilute solution standard state, we get: 0010 00111=2 0001G000e …1:88† aH 0019 wt%H ˆ pH2 0001 e… RT † By reducing the partial pressure of hydrogen we can lower the thermodynamic limit hydrogen solubility in the liquid metal. By maintaining a vacuum, the partial pressure of the gas, corresponding to the impurity that is to be removed, is minimized and the thermodynamic conditions for refining are improved. The kinetic mechanisms by which impurities are removed are diffusion of the dissolved impurity species though a boundary layer in the melt to the melt surface and evaporation from the surface. If boundary layer diffusion is assumed to be the slower of the two steps (Pehlke, 1973), a simple model can be used to describe the degassing kinetics. Using Fick's first law and assuming steady state diffusion across the liquid-side boundary layer, setting the impurity concentration at the surface to Co, and assuming that the bulk of the melt is stirred enough to maintain a constant concentration, the change in the melt impurity concentration (C) with time is given by: dC A ˆ ÿD …C ÿ Co † …1:89† dt 000eV Here, A is the gas±melt interfacial area, V is the melt volume, and 000e is the boundary layer thickness. Co is the concentration of the impurity at the surface
26
Fundamentals of metallurgy
(established by equilibrium with the partial pressure in the gas). D is the diffusion coefficient of the dissolved impurity element in the melt. When a boundary layer is absent, a model has been developed (Danckwerts, 1951; Machlin, 1960; Darken and Gurry, 1952) that assumes that rigid body elements of fluid move parallel to the melt±gas interface, and during their time of contact with the interface, they get degassed. The streamline flow of the elements is assumed to be free of convective currents and within the elements, semi-infinite diffusion is assumed to be valid. The rate of degassing, is then: dC dM dA 1 ˆ dt dA dt V
…1:90†
Here dA/dt is the rate at which gas±melt interface is created and V is the volume of the melt. dM/dt is the change in solute per unit area of gas±metal interface, and can be expressed (Darken and Gurry, 1952) as: dM ˆ 1:1284…Co ÿ C†…Dt0 †1=2 dA
…1:91†
where t0 is the time interval during which solute is lost from the fluid element. Due to the dependence upon diffusion, vacuum degassing is a very slow and expensive method and is therefore primarily used for applications where high purity is needed. Electro slag remelting In this refining method, an impure ingot is used as an electrode that is immersed into a molten, ion-conducting slag consisting of CaO, Al2O3 and CaF2. A high current is passed through the ingot, whereupon the ingot melts. During this process, inclusions dissolve in the slag. The molten metal, now purified from inclusions, drops through the slag, and resolidifies as an ingot in the bottom in a water-cooled mold. The water-cooled mold allows for a skull to be formed and thus enables the refining of reactive molten alloys that cannot be easily contained. Zone-refining Zone refining is used to produce primarily materials for electronic components such as Si or Ge. The principle is to utilize the fact that most impurities have a higher solubility in the melt than the solid metal (Fig. 1.5) and thus partition into the melt ahead of the solidification front during solidification. During zone refining, an impure bar is passed slowly through a hot zone that melts a small section of the bar as shown in Fig. 1.7. As the melt zone is moved along the bar, impurity successively builds up in the melt as shown in Fig. 1.8. A model developed by Pfann (1958) describes the concentration in a bar, with initially a uniform impurity concentration of Co during a single pass.
Descriptions of high-temperature metallurgical processes
27
1.7 Purification by zone melting (Pfann, 1952).
1.8 Schematic of concentration change during zone melting.
C ˆ 1 ÿ …1 ÿ k†eÿkx=l Co
…1:92†
Here, C is the impurity concentration, k is the partition coefficient of the impurity between the solid and liquid metal, x is the length solidified and l is the length of the molten zone. Several passes are carried out to reach extreme levels of cleanliness.
1.4
Casting processes
In order to convert a liquid metal into a solid it is necessary to remove heat to (i) cool the liquid below its melting point and (ii) remove the latent heat (exothermic enthalpy release) during the transformation from liquid to solid. This is carried out in a container called a mold. In general, the transformation of an element of liquid into solid follows a heat balance given by: 001acp
@T @fs S ÿQ ˆL @t @t V
…1:93†
28
Fundamentals of metallurgy
where 001a, cp and L are the density, specific heat capacity and volumetric latent heat of the metal, respectively. S and V are the surface area and volume of the element, and fs is the fraction solid. Q is the net heat flux to/from the volume element. This equation can be closed by specifying an fs-t-T relationship if Q and fs can be identified. Q is dependent upon the thermal field caused by the structure around the element and the mold characteristics. The fraction solid, fs, depends on the solidification kinetics, which is governed by the rates of nucleation and growth. The specific conditions of the alloy chemistry and mold will govern nucleation and growth rates and thermal fields, and this in turn will determine solidification structure in terms of solid crystal size and shape, micro and macro segregation and porosity. Casting processes are carried out in a wide variety of mold types and processes. They includes continuous processes such as: 1.
2. 3.
Oscillating molds to cast profiles with uniform cross-sections such as billet, blooms, slabs, thin slabs or rails. This is by far the dominant process for producing steel bars and sheets. In this process, molten metal is continuously poured through a submerged entry nozzle from a tundish into an oscillating, water-cooled mold. A mold slag is used to cover the surface of the melt meniscus and also to fill the gap between the mold and the solidifying shell. The roles of the mold flux are to control heat transfer, lubricate the mold± strand interface and to refine and protect the melt from impurities. Moving molds using two large rotating wheels in between which thin sections can be cast have resulted in the strip casting process that reduces to a large extent the need for rolling in the production of sheet steels. Atomization is a process by which metal powder is produced by intercepting melt pouring from a tundish with a high velocity gas. By controlling gas velocity and angle of intercept different shapes and sizes of powders can be produced. Atomized powder is a major source for the powder metallurgy industry. Casting is also carried out in batch processes such as:
1. 2.
Large permanent ingot molds. This is used to caste large parts that are to be later machined into specific shapes and sizes. In the case of steels, this has largely given way to continuous casting. Permanent or expendable molds. This is used to shape-cast various structural parts of complex geometry, e.g. in the automotive and aerospace industries. Expendable molds include those made from sand or ceramics.
As mentioned before, the solidified microstructure depends on the cooling rate. The cooling rate is determined by how fast heat is removed from the melt (heat transfer). In casting processes, the heat transfer is thus the most pertinent variable that needs to be controlled. During solidification, the melt adjacent to the mold wall usually solidifies first since this is where it is coolest. The heat
Descriptions of high-temperature metallurgical processes
29
1.9 Schematic of the heat transfer resistances during solidification.
from the melt must thus be transported through (as shown in Fig. 1.9): (a) (b) (c) (d)
the liquid metal; the solid metal shell that forms at the mold wall; across the interface between the solid shell and the mold; and the mold.
The rate of heat transfer depends on the design of the mold and thus the mold is designed according to the desired microstructure and production rates in various industrial casting processes. Heat is transferred by conduction, convection and radiation. Usually the heat transfer through the liquid metal is fast due to the high degree of convection in the melt. The temperature profile and corresponding heat flux is calculated by simultaneously solving the multi-scale problem of heat transfer and solidification kinetics. Solving a general coupled problem of solidification and heat transfer is complicated and beyond the scope of this chapter. Two simplified cases, each having some practical importance in the casting industry, are discussed below.
1.4.1 Heat conduction in the mold controls the heat transfer: expendable ceramic molds (sand casting, investment casting) used for shape casting Molds made out of ceramics conduct heat so slowly that the entire temperature drop occurs across the mold (i.e. temperature in solid and liquid metal and mold±metal interface are equal and thus at thermal equilibrium). Furthermore, the mold can often be assumed to be infinitely thick in such a batchwise casting process. The situation is schematically shown in Fig. 1.10. The Fourier's law solution to this problem yields the following expression for the heat flux: p kmold 001amold Cp;mold p …Tm ÿ To † (1.94) Heat flux ˆ q ˆ 0019t
30
Fundamentals of metallurgy
1.10 Schematic temperature profile across a sand mold.
The heat generated per unit area during the growth of the solid is given by the latent heat: Released heat ˆ 0001Hm 0001
dx 001ametal 0001 dt MW ;metal
(1.95)
The released heat is the heat that is to be transported in the heat flux and thus we can equate the two: released heat ˆ heat flux. After integrating (with x ˆ 0 at t ˆ 0) we end up with an expression for the solid thickness: xˆ
2…Tm ÿ To † 0001 MW ;metal p p p 0001 kmold 0001 001amold 0001 Cp;mold 0001 t 001ametal 0001 0001Hm 0001 0019
…1:96†
1.4.2 Heat transport across the mold±solid metal controls the heat transfer: strip casting, thin slab casting, wire casting, die casting and atomization This is often the case when casting thin cross-sections in short times using metal molds. Mold thickness is usually large compared to the cross-section of cast metal. This situation is shown in Fig. 1.11. In this case heat transfer in the ingot and mold are so fast that it is the interfacial heat transfer that controls how fast heat is transported. The heat transfer equations yield the following expressions for the heat flux and shell thickness in this case:
Descriptions of high-temperature metallurgical processes
31
1.11 Schematic temperature profile across a metal mold and metal strip.
q ˆ h…Tm ÿ To † xˆ
h…Tm ÿ To † 0001 MW ;metal 0001 t 001ametal 0001 0001Hm
…1:97† …1:98†
In the first equation, h is the interfacial heat transfer coefficient [J/(m2.K)]. Its value depends on both the contact between the solid shell and the mold as well as the surface properties of both and whether or not a gas fills the gap. Due to its complexity, it must usually be determined experimentally for each type of mold.
1.5
Thermomechanical processes
There are two basic goals in thermomechanical processing of metals: 1.
2.
Plastic deformation at ambient or elevated temperature of a cast metal into a desired shape. This is necessary for metals that have not been cast into their final shape such as continuously cast metal or ingots from electroslag refining or vacuum arc re-melting. Alternative methods to achieve this are shape casting (die or investment molds) or powder compaction. Use of deformation and heat treatment to obtain micro-structural changes in order to improve properties. This applies not only to mechanical properties but also to others, e.g. electrical, magnetic and ferroelectric properties.
1.5.1 Deformation processing to obtain shape changes The principle of deformation processing is to apply a mechanical force resulting in stresses that exceed the metal's yield stress. For example, for the case of uniaxial
32
Fundamentals of metallurgy
tension, we can define a yield stress at the end of the Hooke's law regime. The condition for reaching the yield stress is approximated through a suitable yield criterion, e.g. Tresca or Von Mises. The metal flows due to the creation and movement of dislocations. The yield stress increases as the deformation increases, because more dislocations are created as they get entangled more into one another. This causes an increase in the yield stress and is called work hardening: 001b ˆ 001bo (no work hardening) ‡ k* (001ad )1/2
(1.99)
Here, 001ad ˆ dislocation density. Most deformation processes can be described by the following characteristics (shown in Fig. 1.12): · · · ·
The metal has an initial shape and initial properties. It goes through a plastic deformation zone. The metal has a final shape and final properties. The process is affected by the friction condition at the interface.
A few examples of deformation processes are listed below: Rolling Material is passed between two rolls of equal radius. This is used for reducing the thickness of the plates and sheets (see Fig. 1.13). Forging A die is used to apply a force and change the shape of the metal sample. The material is in compression during deformation (see Fig. 1.14). Wire drawing The material is drawn through a die and is in tension during the deformation (see Fig. 1.15).
1.12 A generic schematic of a deformation process.
Descriptions of high-temperature metallurgical processes
1.13 Rolling.
1.14 Forging.
1.15 Wire drawing.
33
34
Fundamentals of metallurgy
1.16 Extrusion.
Extrusion Similar to wire drawing but material is in compression (see Fig. 1.16).
1.6
References
Alcock C.B. (1976) Principles of Pyrometallurgy, London, Academic Press, 180±226. Arvanitidis I., Sichen D., Sohn H.Y. and Seetharaman S. (1997) `The Intrinsic Thermal Decomposition Kinetics of SrCO3 by a Nonisothermal Technique', Metall Mater Trans B, 28B, 1063±68. Barksdale J. (1966) Titanium Its Occurrence, Chemistry and Technology, New York, Ronald Press Co., 213±23, 400±79. Bates P. and Muir A. (2000) `HIsmelt ± Low Cost Iron Making', Gorham Conference, Commercializing New Hot Metal Processes ± Beyond The Blast Furnace, June 2000. Biswas A.K. and Davenport W.G. (1976) Extractive Metallurgy of Copper, Oxford, Pergamon, 81±82. Bouris D. and Bergeles G. (1998) `Investigation of inclusion re-entrainment from the steel±slag interface', Metall Mater Trans B, 29B, 641±649. Cleaver J.W. and Yates B. (1973) `Mechanism of detachment of colloidal particles from a flat substrate in a turbulent flow', J Coll Interface Sci, 44(3), 464±74. Cramb A.W. and Jimbo I. (1988) `Interfacial considerations in continuous casting', W.O. Philbrook memorial symposium, Toronto, Ontario Canada, 17±20 April 1988, Warrendale, PA, USA, ISS, 259±71. Danckwerts P.V. (1951) `Significance of liquid-film coefficient in gas adsorption', Ind Eng Chem, 43, 1460±67. Darken L.S. and Gurry R.W. (1952) Physical Chemistry of Metals, New York, McGrawHill. Evans J.W. and Koo C-H. (1979) `The Reduction of Metal Oxides', in Sohn H.Y. and Wadsworth M.E., Rate Processes of Extractive Metallurgy, New York, Plenum, 285±321. Gupta S. (1999) `Steelmaking-Technological Options', in International Conference on Alternative Routes of Iron and Steelmaking (ICARISM `99), 15±17 September 1999, Perth, WA, Australia, 81±84. Habashi F. (1986) Principles of Extractive Metallurgy, Vol. 3 Pyrometallurgy, New York, Gordon and Beach, 326.
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Hojo J., Oku T. and Kato A. (1978) `Tungsten Carbide Powder Produced by the Vapor Phase Reaction of the WCl6-CH4-H2 System', J Less Common Metals, 59(1), 85± 95. Hong L., Sohn H.Y. and Sano M. (2003) `Kinetics of Carbothermic Reduction of Magnesia and Zinc Oxide by Thermogravimetric Analysis Technique', Scandinavian J. Metallurgy, 32, 171±76. International Iron and Steel Institute (2004) `World Steel in Figures', from website, http:// www.worldsteel.org/wsif.php. Jones D. (1992) Principles and Prevention of Corrosion, 2nd edn, New Jersey, Prentice Hall. Kofstad P. (1988) High Temperature Corrosion, 2nd edn, New York, Elsevier. Kopfle J., Anderson S.H. and Hunter R. (2001) `Scrap, DRI and Pig Iron: Whassup?' ISRI Ferrous Conference, Chicago, IL, 2001; http://www.midrex.com/uploadedfiles/ Whassup.pdf. Kozakevitch P. (1954) `Viscosity of blast-furnace slags', Revue Metall, 51, 569±87. Kozakevitch P., Lucas L. and Louis D. (1968) `Part placed by surface phenomena in the elimination of solid inclusions in a metal bath', Revue Metall, 65, 589±98. Kozakevitch P. and Olette M. (1970) `Role of surface phenomena in the mechanism of removal of solid inclusions', Production and application of clean steel, Intl. Conference, 1970, Balatonfured, Hungary, London, The Iron and Steel Institute, 42±49. Kozakevitch P. and Olette M. (1971) `Role of surface phenomena in the mechanism used for eliminating solid inclusions', Revue Metall, 68, 636-46. Lahiri A.K., Yogambha R., Dayal P. and Seetharaman S. (2002) `Foam in Iron and Steelmaking', in Aune R.E. and Sridhar S., Proceedings of the Mills Symposium, Metals, Slags, Glasses: High Temperature Properties and Phenomena, 22±23 August 2002, London, vol. 2, London, The Institute of Materials. Lamprey H. and Ripley R.L. (1962) `Ultrafine Tungsten and Molybdenum Powders', J. Electrochem Soc, 109(8), 713±15. Machlin E.S. (1960) `Kinetics of vacuum induction refining: theory', Trans AIME, 218, 314±26. MIDREX Technologies Inc. (2001) `2000 World Direct Reduction Statistics', http:// www.midrex.com/info/world.asp. MIDREX Technologies Inc. (2004) `2003 World Direct Reduction Statistics', http:// www.midrex.com/info/world.asp. Mills K.C. and Sridhar S. (1999) `Viscosities of iron and steelmaking slags', Ironmaking Steelmaking, 26, 262±68. Nakajima K. and Okamura K. (1992) `Inclusion transfer behavior across molten steel± slag interfaces', 4th Intl. Conf. on Molten Slags and Fluxes, ISIJ, Sendai, 1992, 505±10. Nikklason G.A. (1987) `Optical Properties of Gas-Evaporated Metal Particles: Effects of a Fractal Structure', J Appl Phys, 62, 258±65. Okamoto Y., Koyano T. and Takasaki A. (1987) `A Magnetic Study of Sintering of Ultrafine Particles', Japan J Appl Phys, 26, 1943±45. Otsuka K-I., Yamamoto H. and Yoshizawa A. (1984) `Preparation of Ultrafine Particles of Nickel, Cobalt and Iron by Hydrogen Reduction of Chloride Vapors', J Chem Soc Japan, 6, 869±78. Padilla R. and Sohn H.Y. (1979) `Reduction of Stannic Oxide with Carbon', Metall Trans
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B, 10B, 109±15. Pankratz L.B., Stuve J.M. and Gokcen N.A. (1984) Thermodynamic Data for Mineral Technology, U.S. Bureau of Mines Bull. 677, Washington DC, US Government Printing Office. Parr H. and Feder J. (1973) `Superconductivity in -Phase Gallium', Phys Rev, 7, 166± 81. Pehlke D. (1973), Unit Processes of Extractive Metallurgy, New York, Elsevier. Pfann W.G. (1958) Zone Melting, New York, Wiley. Pilling N.N. and Bedworth R.E. (1923) `Oxidation of metals at high temperatures', J Inst Met, 29, 529±83. Riboud P.V. and Olette M. (1982) `Mechanisms of some of the reactions involved in secondary refining', Proc 7th International conference on vacuum metallurgy, Tokyo, Japan, 879±89. Richardson, F.D. (1974) Physical Chemistry of Melts in Metallurgy, Volume 1, London, Academic Press, 81. Rosenqvist T. (1983) Principles of Extractive Metallurgy, 2nd edn, New York, McGrawHill, 202. Saeki Y., Zaki R.M., Nishikara H. and Ayoama N. (1978) `Preparation of Cobalt Powder by Hydrogen Reduction of Cobalt Dichloride', Denki Kagaku, 46(12), 613±17. Shannon G.S. and Sridhar S. (2004) `Separation of Al2O3 Inclusions Across Interfaces between Molten Steel and Ladle-, Tundish- and Mold-Slags', METAL SEPARATION TECHNOLOGIES III, Symposium in Honor of Professor Lauri E. Holappa, Copper Mountain Colorado, June 20±24, 2004, Engineering Foundation Conferences. Sohn H.Y. and PalDey S. (1998a) `Synthesis of Ultrafine Particles and Thin Films of Ni4Mo by the Vapor-Phase Hydrogen Coreduction of the Constituent Metal Chlorides', Mater Sci Eng A, 247, 165±72. Sohn H.Y. and PalDey S. (1998b) `Synthesis of Ultrafine Nickel Aluminide Particles by the Hydrogen Reduction of Vapor-Phase Mixtures of NiCl2 and AlCl3', J Mater Res, 13, 3060±69. Sohn H.Y. and PalDey S. (1998c) `Synthesis of Ultrafine Particles of Intermetallic Compounds by the Vapor-Phase Magnesium Reduction of Chloride Mixtures: Part I. Titanium Aluminides', Metall Mater Trans B, 29B, 457±64. Sohn H.Y. and PalDey S. (1998d) `Synthesis of Ultrafine Particles of Intermetallic Compounds by the Vapor-Phase Magnesium Reduction of Chloride Mixtures: Part II. Nickel Aluminides', Metall Mater Trans B, 29B, 465±69. Sohn H.Y., Zhang Z., Deevi S. and PalDey S. (2004) `Chemical Vapor Synthesis of Ultrafine Fe-Co Powder', High Temperature Materials and Processes, 23, 329±33. Wakelin D. (ed.) (1999) The Making, Shaping and Treating of Steel, 11th edn, Pittsburgh, PA, The AISI Steel Foundation, 403±567. Zhao G.Y., Revenkar V.V.S. and Hlavacek V. (1990) `Preparation of Tungsten and Tungsten Carbide Submicron Powders in a Chlorine-Hydrogen Flame by Chemical Vapor Phase Reaction', J Less Common Metals, 163(2), 269±80.
Descriptions of high-temperature metallurgical processes
1.7
37
Appendix: notation
Nomenclature Italics A a C Cp Cs D fi fs G H H k K l m M MW p PB Q R S t T V x
effective area or interface where reaction takes place chemical activity or constant in eqn 1.41 concentration per unit volume heat capacity sulfide capacity diffusion coefficient activity coefficient of specie `i' in the infinite dilution standard state fraction solid Gibbs free energy heat transfer coefficient enthalpy reaction rate constant, partition coefficient or thermal conductivity chemical equilibrium constant melt zone length mass of product formed or reactants consumed solute per unit area molar weight partial pressure Pilling±Bedworth ratio net heat flux ideal gas constant surface area time temperature in K volume distance or thickness
Greek symbols 000e boundary layer thickness in a fluid phase 001a density 001b stress Subscripts o reference state m mold Superscript o starting value 0 interval or defines intermediate constant
2
Thermodynamic aspects of metals processing R E A U N E and S S E E T H A R A M A N
2.1
Introduction
According to Albert Einstein,1 thermodynamics is marked by its simplicity, the different kinds of things it relates to and the wide area of its applicability. Thermodynamics is by definition a subject that describes the link between heat and motion. With the development of physics and chemistry and the application of mathematical principles, the subject area today covers all forms of energy including, thermal, electrical, mechanical energies and the impact of the same in the change of the states of the systems we are interested in. The area of thermodynamics stretches from the atomistics to macro systems including huge metallurgical reactors. The applications of the concepts of thermodynamics to metallurgy reached significant advancements during the past four decades with stalwarts like Wagner from Germany, Darken, Chipman and Elliott from the US, Richardson in UK as well as Hillert from Sweden. The present chapter is intended to cover the areas of thermodynamics that are of relevance to metallurgy, particularly, the processing of metals and the properties of metals and alloys. For further reading, the readers are requested to resort to the classical textbooks in this area presented in the bibliography at the end of this chapter. The present chapter is also to an extent inspired by the course literature in thermodynamics at the Royal Institute of Technology by the present authors and their predecessors. Since thermodynamics is a subject that can be almost philosophical, it is imperative to have clear definitions of the various terms involved in order to apply its concepts in metallurgical applications. As mentioned earlier, we are concerned with the changes of state of a system due to energy impact. Thermodynamics does not provide any information as to the rate of this change.
Thermodynamic aspects of metals processing
2.2
39
Basic concepts in thermodynamics
2.2.1 State and state functions System A system in thermodynamics is a limited but well-defined part of the universe focused on presently. The rest of the universe can be considered as the surrounding. The aim is to examine the interaction between the system and the universe in a simple but well-specified way. An open system can exchange with its surroundings both matter and energy. A closed system on the other hand can exchange only energy with the surroundings, but not matter. An isolated system can neither exchange matter nor energy with the surroundings. A homogeneous system is identical in physical and chemical properties in all parts of the system, as for example, liquid steel at 1600 ëC. A system that has differences in physical and chemical properties within the system is referred to as a heterogeneous system, as for example, water and ice at 0 ëC. A system is composed of different types of molecules. For example, air, as a system consists of nitrogen, oxygen and other minor gas molecules. These are referred to as components of a system. Considering a gas system consisting of three molecular types, H2, O2 and H2O, there is mutual reaction between these molecular types, namely: 2H2 ‡ O2 ˆ 2H2O
(2.1)
The number of these molecular types in the system can be altered by introducing some of these molecules from the surroundings or change, for example, the total pressure of the system. Thus, it is enough to define two of the three types of molecules, viz. the system has two components. A heterogeneous system that may have different physical properties with the same components throughout may have different phases. For example, water at its freezing point contains a solid (ice) and a liquid, and thus, two phases. Within each phase, the molar properties have the same value at every point. Generally, the three phases, solid, liquid and gas are considered in a heterogeneous system. In metallurgy, it is often necessary to consider different allotropic modifications in the solid state, as for example, , or 000e iron with different crystal structures. Water at the triple point will have three phases, viz. ice, water and steam. State The macroscopic definition of `state', which is relevant to metallurgy, is defined by its macroscopic properties like temperature, pressure, volume, vapour pressure, viscosity, surface tension, etc. With a focus on the chemical properties, surface energy is also related to the state of the system. For example, in defining
40
Fundamentals of metallurgy
10 moles of nitrogen, it is important to define the temperature and pressure, as the other properties get defined implicitly. State properties The properties stated above along with others being defined in the following chapters that define the state of the system are the state properties. The properties that are additive are called the extensive properties, as for example mass, volume and energy and, in the case of a homogeneous system, are proportional to the total mass. On the other hand, properties, to which a value could be assigned at each point in the system are intensive properties. Some of the common intensive properties are temperature, pressure, density etc. Since the ratio of two extensive properties is independent of total mass, and may be assigned a value at a point, these ratios fall under intensive properties. Examples of such properties are molar properties like volume per mole. In order to differentiate these from intensive properties like temperature and pressure, the latter are often classified as potentials. It is useful to introduce, at this point, the concept of chemical potential, represented usually as 0016, which is the potential corresponding to the chemical energy in the system.The equality of the potentials and the inequality in molar properties between phases is illustrated in Fig. 2.1. A system is in a state of mechanical equilibrium if the pressure at all points in the system is the same. If the system has no thermal gradients, it is supposed to be in thermal equilibrium. The system is in chemical equilibrium if the chemical potential is uniform throughout the system. The system is in complete thermodynamic equilibrium if it is having mechanical, thermal as well as chemical equilibria. The properties of the system can be varied by interaction between the system and the surrounding. Mass transfer could change the material content of the system while heat transfer could alter the energy content. In order to define the macroscopic state of the system unequivocally, all the properties of the system need be known. On the other hand, due to the interdependency of the properties, it is sufficient to define only a few. For example, in the case of a gas in a container, it is only necessary to define the temperature and pressure. The volume of the system, V in m3.molÿ1 gets
2.1 Two phases in thermodynamic equilibrium.
Thermodynamic aspects of metals processing
41
V
d a c b T P1 P2
T1
T2
P
2.2 The states of a defined amount of gas±schematic diagram.2 The letters `a', `b', `c' and `d' refer to different states of the system. P1 and P2 refer to two different pressures where P1 > P2. T1 and T2 are the two temperatures, T1 < T2.
implicitly defined by the gas law. Thus, in this case, we can define pressure, P, N.mÿ2 and temperature, T, K are the independent variables and volume is the dependent variable. The gas law V ˆ NRT/P
(2.2)
where N is the number of moles in a system is an equation of state. A schematic representation of the various states of a system, where the variables are T, P and V, is presented in Fig. 2.2. In some cases, especially in the case of phase transformations, it is sometimes advantageous to define a new term `inner state variable'3 as, for example, the degree of change due to conditions imposed on the system, as represented by the symbol 0018. Since thermodynamics is concerned with the changes associated with the interactions between the system and the surroundings, the degree of change could be a useful parameter in following the path of a reaction. This is illustrated in Fig. 2.3.
42
Fundamentals of metallurgy 1
z/(xi)
B
A 0
T1
T2
Temperature
2.3 Gradual change of an inner state variable caused by a quick change in temperature.3
2.2.2 The first law of thermodynamics Energy change between the system and its surroundings is defined by the first law of thermodynamics. It is also variously considered as a definition of energy or a law of conservation of energy. dU ˆ 000eQ ‡ 000eW
(2.3)
where dU is the change in the internal energy of the system (without defining the microscopic state), 000eQ is the energy added to the system and 000eW is the work imposed on the system. (Please note that the symbol `d' is used for infinitesimal, defined change in the state property of a system while, `000e' is used for an infinitesimal, undefined amount of energy or work coming in from the surroundings). It is to be noted that the term `energy' includes all forms of energy including thermal, electrical, chemical and other known forms. Similarly, the term `W' covers all forms of work: chemical, mechanical, etc. The essence of the first law of thermodynamics is the law of conservation of energy. Further, the first law of thermodynamics does not contradict the Einstein's mass±energy relationship, viz. E ˆ m.c2
(2.4)
where E is the energy, m is the mass and c is the velocity of light in the sense that mass could be considered a form of energy. Q and W could be expressed in a suitable energy unit, Joule. The above figure illustrates the energy±work relationship considering mechanical energy. In the case of a chemical reaction, a negative value of Q refers to exothermic reaction while a positive Q would mean an endothermic one. One of the direct consequences of the first law of thermodynamics is Hess's law of constant heat summation which states the the heat change in a chemical reaction is the same whether it takes place in one or several steps. In other
Thermodynamic aspects of metals processing
43
2.4 A schematic illustration of the first law of thermodynamics.
words, the heat of a reaction is dependent only on the initial final states of the chemical system. Using the cylinder±piston analogy in Fig. 2.4, it is possible to express the Q and W terms as system variables. With an external pressure, P, acting on the piston, the change in volume of the system, dV will be PdV (the `minus' sign indicating the decrease in volume). Thus, equation (1.3) can be rewritten as: dU ˆ 000eQ ÿ P.dV
(2.5)
Between two states of the system 1 and 2, equation (2.6) can be integrated as Z U2 Z V2 dU ˆ Q ÿ P dV U1
V1
U2 ÿ U1 ˆ Q ÿ P…V2 ÿ V1 †
(2.6)
(U2 ‡ PV2) ÿ (U1 ‡ PV1) ˆ Q
(2.7)
The system properties U and PV terms for the two states can be combined to get an expression for Q, viz. H2 ÿ H1 ˆ Q
(2.8)
The term H, which is a system property, includes the internal energy of the system and the mechanical work and is known as enthalpy. The internal energy and enthalpy terms can be expressed in terms of the molar heat capacities.
44
Fundamentals of metallurgy
The molar heat capacity at constant volume, CV, represents the change in the internal energy between the states 1 and 2 and thus, Z 2 U2 ÿ U2 ˆ CV dT (2.9) 1
Correspondingly, the molar heat capacity at constant pressure CP can be expressed as Z 2 CP dT (2.10) H2 ÿ H1 ˆ 1
In the case of ideal gases, it can be very easily shown that CP ÿ CV ˆ N.R, where N is the Avogadro no. and R is the gas constant.
2.3
Chemical equilibrium
2.3.1 The second law of thermodynamics In thermodynamics, in order to arrive at the maximum work done for a given supply of energy, the concepts of reversible and irreversible processes are often used. The term reversibility is applied to a process which takes the system only infinitesimally away from the equilibrium state. The direction of a reversible process can be changed by an infinitesimal change of the parameters of the process, as, for example, changing the voltage in a galvanic cell. Other processes like the flow of heat from a hotter to a colder body are irreversible. Among the reversible processes, an isothermal process is one where there is an exchange of energy between the system and the surroundings (see definition of `closed systems', page 39) during the process. An adiabatic process is one where the system is insulated from the surroundings (see definition of `isolated systems', page 39) and there is no exchange in energy. In calculating the energy absorbed for isothermal as well as adiabatic expansions, it can be shown as: Isothermal expansion: external energy required ˆ RT ln (V2/V1)
(2.11)
where the subscripts 1 and 2 refer to the initial final states of the system. Similarly, Adiabatic expansion: external energy required ˆ 0
(2.12)
But, in the latter case, the internal energy of the used will be expended for the work done, which is given by the relationship: CV ln (T2/T1) ˆ R ln (V2/V1)
(2.13)
Equations (2.11), (2.12) and (2.13) represent the change of state of the system. Considering a cylinder±piston system, as shown in Fig. 2.4, and a series of reversible processes, viz. adiabatic expansion, isothermal expansion, adiabatic
Thermodynamic aspects of metals processing
45
compression and isothermal compression, an exercise termed Carnot cycle can be performed. By this, it can be shown that 0006(000eQ/T) ˆ 0
(2.14)
for reversible processes. If a state function, entropy can be defined as Q/T, dS ˆ 000eQRev/T and dS > 000eQIrrev/T
(2.15)
is a statement of the second law of thermodynamics. QRev and QIrrev are the energies associated with reversible and irreversible processes.
2.3.2 Concept of entropy An irreversible process in a system together with its surroundings is accompanied by an increase in entropy. If the enthalpy change associated with the process is 0001H, the entropy change of the surroundings is ÿ0001Hsystem/T. The total entropy change of the system plus surroundings is 0001S ˆ 0001SSystem ˆ 0001SSurroundings ˆ 0001SSystem ÿ 0001Hsystem/T
(2.16)
Thus, for a process at constant pressure, 0001H ÿ T.0001S ˆ 0 for reversible processes
(2.17)
0001H ÿ T.0001S < 0 for irreversible processes
(2.18)
2.3.3 Concepts of Gibbs and Helmholtz energies When 0001H ÿ T.0001S > 0, the reaction would be impossible. Thus, the driving force of a chemical reaction can be expressed in terms of 0001H ÿ T.0001S. A new term, Gibbs energy, is defined as G ˆ H ÿ TS
(2.19)
The change in Gibbs energy, 0001G can be expressed as 0001G ˆ 0001H ÿ T.0001S
(2.20)
All spontaneous reactions occur with a decrease in the Gibbs energy, while at equilibrium, the Gibbs energy change is zero. The products and reactants have the same Gibbs energy and the reaction lacks a driving force in any direction. In an analogous way, a corresponding expression for the driving force for a process with constant volume can be derived. The expression, U ÿ T.S is termed Helmholtz energy, A. Thus, at constant volume, for a spontaneous reaction, dA < 0. At equilibrium, dA ˆ 0 and when dA > 0, the reaction is impossible.
46
Fundamentals of metallurgy
2.3.4 Maxwell's relations Combination of first and second laws yields the following expressions: dU ˆ TdS ÿ PdV
(@T/@V)S ˆ ÿ(@P/@S)V
(2.21a)
dH ˆ TdS ‡ VdP
(@T/@P)S ˆ (@V/@S)P
(2.21b)
dG ˆ VdP ÿ SdT
(@S/@V)T ˆ (@P/@T)V
(2.21c)
dA ˆ ÿPdV ÿ SdT
(@V/@T)P ˆ ÿ(@S/@P)T
(2.21d)
These four equations are termed Maxwell's relations.
2.3.5 Gibbs energy variation with pressure and temperature From Maxwell's relations, it is quite easy to derive the pressure and temperature dependencies of Gibbs energy. Equation (2.21c) leads to the relationship: (@G/@P)T ˆ V and (@G/@T)P ˆ ÿS
(2.22)
Further, according to equation (2.20), 0001G ˆ 0001H ÿ T.0001S ˆ 0001H ‡ T.(@0001G/@T)P [@(0001G/T)/@T]P = ÿ0001H/T
2
(2.23) (2.24)
and [@(0001G/T)/@(1/T]P ˆ 0001H
(2.25)
Equations (2.24) and (2.25) are referred to as two different forms of Gibbs± Helmholtz equation.
2.3.6 The third law of thermodynamics Nernst postulated that, for chemical reactions between pure solids, the Gibbs energy and enthalpy functions, [@(0001G/T)/@T]P as well as [@0001H/@T]P approach zero as the temperature approaches absolute zero. This further led to the theorem that, for all reactions involving condensed phases, 0001S is zero at absolute zero. This was further developed by Planck to the third law of thermodynamics in a new form, viz. `the entropy of any homogeneous substance in complete internal equilibrium may be taken as zero at 0 K'. Glasses, solid solutions or other systems (for example, asymmetric molecules like CO) that do not have internal equilibrium will obviously deviate from third law. The same is true for systems with different isotopes.
Thermodynamic aspects of metals processing
47
2.3.7 Entropy and disorder The second law of thermodynamics states that the total entropy is increasing for spontaneous processes. In the atomistic level, entropy is considered to be a degree of disorder arising due to randomness in configuration as well as energy contents. The latter can manifest itself as vibrational, magnetic or rotational disorders. From a statistical mechanics treatment of disorder, Boltzman arrived at the equation for configurational entropy as S ˆ k ln !
(2.26)
where k is the Boltzman constant and ! number of arrangements within the most probable distribution. Richard's and Trouton's rules The degree of disorder increases as a substance melts. As a stronger bonding energy between atoms need a higher temperature to cause disorder, Richard's rule states that 0001Sf ˆ Lf/Tf ˆ ca 8.4 J. Kÿ1
(2.27)
where 0001Sf is the entropy of fusion, Lf is the latent heat of fusion and Tf is the melting point. In reality, Richard's rule, which is empirical, holds approximately for metals, the values tend to be higher for metalloids and salts. Another useful empirical rule for the heat of vaporization at the boiling point is given by Trouton's rule, viz. 0001Sv ˆ Lv/Tv ˆ ca 88 J. Kÿ1
(2.28)
where 0001Sv is the entropy of vaporization, Lv is the latent heat of vaporization and Tv is the boiling point.
2.3.8 Reference states for thermodynamic properties The absolute values of enthalpy are not known; but only changes can be measured. In order to facilitate the computation of enthalpies, especially as a function of temperature, the accepted convention is to assign a value of zero to all pure elements in their stable modifications at 105 Pa and 298.15K, which are the reference points. The enthalpies at other temperatures can be calculated by using the expression Z T22 …@[email protected]†P ˆ CP ) 0001H ˆ CP dT ˆ H…T2 ; P† ÿo HRe f …2:29† T1Re f
The enthalpies of compounds can be calculated by adding the enthalpy of the reaction between the elements. The superscript `o' stands for pure substance.
48
Fundamentals of metallurgy
In the case of entropies, the computation of entropies at temperatures other than the reference temperatures is carried out using the equation: Z T1 o T S ˆ oSREF ‡ CP d ln T …2:30† TREF
Newmann±Kopp rule In the estimation of enthalpies and entropies of compounds from elements, the Newmann±Kopp rule can be applied with reasonable success, even though it is claimed that this rule lacks a theoretical basis. According to this rule CP (compound) ˆ 0006CP (components)
(2.31)
2.3.9 Some thermodynamic compilations It is important that a metallurgist has access to thermodynamic data of the systems of interest at temperatures of relevance. Some of the classical compilations are listed below: Thermochemical tables 1. 2. 3. 4. 5.
Materials Thermochemistry, O. Kubaschewski, C.B. Alcock and P.J. Spencer, Pergamon Press (1993). Thermochemical Properties of Inorganic Substances, I. Barin and O. Knacke, Springer-Verlag, Berlin (1973, Supplement 1977). JANAF Thermochemical Tables, N.B.S., Michigan (1965±68, Supplement 1974±75). Selected Values of Thermochemical Properties of Metals and Alloys, R. Hultgren, R.L. Orr, P.D. Anderson and K.K. Kelley, John Wiley & Sons, NY (1963). Thermochemistry for Steelmaking, Vol. I & II, J.F. Elliot, M. Gleiser, J.F. Elliott, M. Gleiser and V. Ramakrishna, Addison-Wesley, London (1963).
Thermochemical databases 1. 2. 3. 4. 5.
HSC databas ± Finland Thermo-Calc ± Sweden Chemsage/F.A.C.T. ± Canade, Germany, Australia MT Data ± England THERMODATA ± France
Thermodynamic aspects of metals processing
2.4
49
Unary and multicomponent equilibria
2.4.1 Unary systems As stated previously, the condition for chemical equilibrium is that the Gibbs energy of the products and reactants are equal and that there is no net driving force for the reaction in either direction. This can be easily understood in the case of systems consisting of a single component. At the melting point of ice, the equilibrium condition is: S
GH20 ˆ lGH2O
(2.32)
The Gibbs energy of the solid, liquid and gas phases in a G±P±T diagram, where the intersections of the Gibbs energy planes represent equality in the Gibbs energy values at given temperatures and pressures is presented in Fig. 2.5. Solid and liquid planes cross at the melting point, while the liquid and gas phase intersect at the boiling point. The three planes meet at a unique temperature and pressure when all the three phases have the same Gibbs energy. This point is known as the triple point. The projection of the Gibbs energy planes in Fig. 2.5 on to the P±T base plane would provide an understanding of the stability areas of various phases as functions of temperature and pressure. Such a projection is presented in Fig. 2.6.
2.4.2 Clapeyron and Clausius±Clapeyron equations An imaginary line in Fig. 2.5 from the triple point along the intersection of the solid and liquid planes represents the variation of melting point as a function of
2.5 Gibbs energy±temperature±pressure diagram in the case of a onecomponent system.
50
Fundamentals of metallurgy P
S
l
g
T
2.6 Projection of the Gibbs energy planes in Fig. 2.5 on the P-T base plane.
pressure. The mathematical relationship can easily be derived from equation (2.21c) of the Maxwell relationship, viz. dG ˆ VdP ÿ SdT (@S/@V)T ˆ (@P/@T)V
(2.21c)
For a phase transformation reaction, the S and V terms could be replaced by 0001S and 0001V, the entropy and volume changes accompanying the transformations respectively. Equation (2.21c) can then be rewritten as ÿSSdT ‡ VSdP ˆ ÿSldT ‡ VldT
(2.33)
(Sl ÿ SS)dT ˆ (Vl ÿ VS)dP
(2.34)
dP 0001Sf 0001Hf 1 ˆ ˆ 0001 Tf 0001Vf dT 0001Vf
…2:35†
where 0001Sf, 0001Vf and 0001Hf are the entropy, volume and enthalpy changes respectively accompanying melting at the fusion temperature, Tf. The above equation holds for any phase transformation in a single component system and is referred to as the Clapeyron equation. This equation is extremely useful in estimating the variation of the melting point as the ambient pressure changes and finds applications in high pressure synthesis of materials. When the Clapeyron equation is applied to the vaporization of a liquid or solid, the 0001V term reduces to V, which is much larger compared to the condensed phases. Thus equation (2.33) is rewritten, combining with the ideal gas law, as dP 0001HV 0001HV ˆ 0001P ˆ dT T0001VV R 0001 T2 dP 0001HV ˆ P 0001 dT R 0001 T2 d ln P 0001HV ˆ dT R 0001 T2
…2:36† …2:37† …2:38†
Thermodynamic aspects of metals processing
51
where the subscript `V' refers to vaporization. The Clausius±Clapeyron equation is very useful in determining the latent heat of vaporization or the vapour pressure of a substance at a temperature with a knowledge of the same at another temperature and the latent heat.
2.4.3 Multicomponent systems Maxwell's relationship for a single component system is expressed by equation (2.21c). In the case of systems with more than one component, the Gibbs energy of the component should be added to the equation. Hillert3 suggests that the inner driving force, 0018 could be added with great advantage in describing reactions like, for example, nucleation. Thus, the integrated driving force can be rewritten for multicomponent systems as dG ˆ V.dP ÿ S.dT ‡ 00060016i.dni ÿ D.d0018
(2.39)
where 0016i is the chemical potential of component `i' per mole of `i', ni is the number of moles of `i', D is defined as ÿ(@U/@0018)S,V,ni. The common practice is to express equation (2.39) without the last term on the right-hand side. It is important to realize that the different phases in the system have thermal, mechanical and chemical equilibria prevailing. In the case of a binary system, it is illustrative to add a third composition axis to Fig. 2.6 on the basis of equation (2.39) without the last term on the right-hand side. This is shown in Fig. 2.7. While vertical planes corresponding to the pure substances are identical with Fig. 2.6, the region in between has three dimensional regions of stability of single phases as well as those with two phases. The two phase regions are shaped as convex lenses. A section of the diagram in Fig. 2.7 corresponding to 1 bar (105 Pa) plane will yield a diagram as shown in Fig. 2.8. Figure 2.8 is a simple binary phase diagram showing the stabilities of solid and liquid phases as a function of temperature. Among the phase diagram types commonly used in metallurgy, eutectic and peritectic types of diagrams are important. These are presented in Fig. 2.9. In Fig. 2.8, the two components are chemically similar and thus exhibit complete solubilities in solid and liquid phases. The repulsive or attractive forces between the components in a binary system can result in partial solubilities or compound formation in the solid state respectively. In the former case, the liquid formation is favoured at lower temperatures. The contrary is true in the case of compound formation. Figure 2.10 presents some examples of how attractive and repulsive forces influence the phase diagrams for binary systems.
2.4.4 Gibbs phase rule In accordance with the requirements for equilibrium, different phases in a system have the same temperature and pressure. Further, each of the components
52
Fundamentals of metallurgy
g+l
s+l
l
g T
P s
s+g
A
B
XB
2.7 Composition±temperature±pressure diagram for a binary system.4 P = 1 atm
T
1
a+1
a
A
XB
B
2.8 A simple binary phase diagram showing the complete mutual solubilities of the two components A and B in solid as well as liquid state.
Thermodynamic aspects of metals processing
53
T
T
l
l a
a
b
b
A (a)
B
A
XO
B
(b)
2.9 Some common types of binary phase diagrams (a) eutectic diagram (b) peritectic diagram.
2.10 (a) Binary phase diagrams where the repulsive forces between the components are manifested.5 (b) Binary phase diagrams where the attractive forces between the components are manifested.5
54
Fundamentals of metallurgy
has the same chemical potential in all the phases in equilibrium. This led to the derivation of the Gibbs phase rule, which states that FˆCÿP‡2
(2.40)
where `F' is the minimum number of degrees of freedom required to reproduce the system, `C' represents the number of independent components in the system and `P' refers to the number of phases. The Gibbs phase rule is applicable to multicomponent macro systems in determining the number of phases at a given temperature and pressure.
2.4.5 Gas mixtures Maxwell's relationship (equation 2.21c) gets reduced at constant temperature to the form dG ˆ VdP ˆ R 0001 Td ln P Z
P2
P1
Z dG ˆ R 0001 T
P2
P1
d ln P ˆ R 0001 T 0001
…2:41† 0012 0013 P2 P1
ˆ G…P2 ; T† ÿ G…P1 ; T†
…2:42† …2:43†
If P1 = 1 bar (105 Pa), G(P1, T) is the Gibbs energy in the standard state at temperature T and can be denoted as ëG, thus equation (2.43) can be rewritten as G(P2,T) ˆ ëG ‡ R 0001 T 0001 ln(P2/1) ˆ ëG ‡ R 0001 T 0001 ln(P2)
(2.44)
In chemistry literature, it is common to refer to a term `fugasity' in the case of non-ideal gases. At high temperatures as it is common in metallurgy, gases are near ideal and the the term fugasity is considered superfluous. Dalton's law of Partial pressures states that the partial pressure of a gas `A', pA, in a mixture of gases `A', `B', `C', etc., is given by PA ˆ XA 0001 P
(2.45)
where P is the total pressure of the gas mixture (P ˆ pA ‡ pB ‡ pC . . .) and XA is the mole fraction of gas species A.
2.4.6 Ellingham diagrams Consider the reaction between metal M (s) and oxygen to form oxide MO2 (s): M ‡ O2 ˆ MO2
(2.46)
At equilibrium 0001G ˆ (Gproducts ÿ 0006Greactants) ˆ 0
(2.47)
Thermodynamic aspects of metals processing
55
Assuming that the metal and the oxide phases are mutually insoluble and that they are in their standard state, and, further remembering from equation (2.44), that GO2 = ëGO2 + RT ln pO2, equation (2.47) can be rewritten as ëGMO2ÿ (ëGM ‡ ëGO2) ˆ 0001ëG ˆ RT ln pO2
(2.48)
Figure 2.11 shows a plot of 0001ëG as a function of temperature for a number of metal±metal oxide systems. The stability of the oxide increases as we go down
2.11 Elligham diagram for oxides.6
56
Fundamentals of metallurgy
the diagram. The slope of the lines are indications of ÿ.0001ëS for reaction (2.45). The intercept at 0K corresponds to 0001ëH at 0K. The diagram assumes that the enthalpy and entropy changes are constant if there is no phase transformation for the metal or oxide phase. The increase in the 0001ëG with increasing temperature is indicative of the decreasing stability of the oxide. With melting of the metal or the oxide, there are accompanying changes in 0001ëS and correspondingly, in the slope of the lines, marked by sharp break points. The line for CO formation has a negative slope indicating the increase in entropy due to the formation of two gas molecules from one oxygen molecule. As crossing of the lines marks the relative stabilities of the oxides, the diagram indicates the reducibility of a number of oxides by carbon if the temperature is sufficiently raised. Logarithmic scales corresponding to pO2 as well as CO/CO2 and H2/H2O ratios, introduced in the Ellingham diagram by Richardson and Jeffes, enable the direct reading of the equilibrium partial pressures of oxygen or the corresponding CO/CO2 and H2/H2O ratios directly from the diagram. The diagram is extremely useful in metallurgical processes, as, for example, the choice of reductants, the temperature of reduction as well as the partial pressure ratios. Similar diagrams have been worked out for sulphides, chlorides, nitrides and other compounds of interest in high temperature reactions.
2.12 The phase stability diagram for Ni-S-O system at 1000K.7
Thermodynamic aspects of metals processing
57
2.4.7 Phase stability diagrams (predominance area diagrams) Another extremely useful set of diagrams representing the stabilities of pure phases is the phase stability diagram or predominance area diagrams. These are generally isothermal diagrams wherein the chemical potential of one component in a ternary system is plotted as a function of another. These diagrams have been found useful in the case of M-S-O system, where M is a metal like Cu, Ni, Fe or Pb (occurring as sulphide). A typical such diagram used for studying the roasting of nickel sulphide is presented in Fig. 2.12. The usefulness of the diagram is well demonstrated in the case of mixed sulphides, where it is possible to superimpose two stability diagrams.
2.5
Thermodynamics of solutions
The concept of pure substances is mainly of theoretical interest. In reality, the systems that are encountered are often multicomponent systems when the components dissolve in each other forming solutions. Even the so-called ultra pure substances have dissolved impurities, albeit in extremely small amounts. Thus, it is an important part of thermodynamics to deal with solutions. The concept of solution is essentially two components forming a single phase in the macroscopic sense. In the micro level, this refers to an intimate mixing of atoms or molecules. The process of solution is often referred to as `mixing', which is somewhat misleading. Gases `mix' completely. In liquid phase, there are many cases where two liquids do not mix with each other, as, for example, oil and water at room temperature or silver and iron at 1600 ëC. In the case of solids, those of similar crystal structure often form `mixed crystals' or solid solutions, which are of single phase, as can be seen by X-ray diffraction measurements.
2.5.1 Integral and partial molar properties In dealing with extensive thermodynamic properties like enthalpy, entropy or Gibbs energy, it is common to refer to one mole of the substance. Exemplifying this in the case of Gibbs energy, ëGA refers to one mole of substance `A' in pure state. On the other hand, in a solution containing `i' different species, the molar Gibbs energy, Gm is given by Gm ˆ G(total)/(nA ‡ nB ‡ nC . . . ni)
(2.49)
where the `n' terms refer to the number of moles of the different species in solution and Gm is the integral molar Gibbs energy of the solution. If the increment in G, caused by the addition of dnA moles of component A to a very 0016A large amount of the solution is dG, this increment per mole of A, denoted as G will be
58
Fundamentals of metallurgy 0016 A ˆ (@G/@nA)P,T,nB,nC. . . G
(2.50)
0016 A is referred to as the partial molar Gibbs energy of A in the solution of G defined pressure, temperature and composition. This leads to the relationship for the total change in the Gibbs energy due to the addition of the various components as 0016 A 0001 dnA ‡ G 0016 B 0001 dnB . . . dG ˆ G
(2.51)
By the addition and removal of nA ‡ nB ‡ nC . . . ni moles of the components, and considering the Gibbs energy of the solution per mole, it can be shown that 0016 A ‡ XB 0001 G 0016B . . . Gm ˆ XA 0001 G
(2.52)
By considering the Gibbs energy for nA ‡ nB ‡ nC . . . ni moles of solution followed by complete differentiation, it can be shown that 0016 A + X B 0001 dG 0016B . . . ˆ 0 X A 0001 dG
(2.53)
The above expression is referred to as Gibbs±Duhem equation and is used to compute the partial molar quantity of a second species with a knowledge of that of the first one. The above relationships hold for even other thermodynamic 0016 B is identical with properties like enthalpy and entropy. It is to be noted that G the chemical potential of component B in the solution, represented usually as 0016B. From a knowledge of the integral molar property, the partial molar properties can be arrived at. In the case of a binary solution A-B, the relationship is given by 0016 B ˆ Gm ‡ …1 ÿ XB † dGm G dXB
…2:54†
This equation can be used graphically to get the partial molar quantities by drawing a tangent to the integral molar Gibbs energy curve with respect to composition (drawn with composition on the x-axis) at a desired composition and reading of the intersection of the tangent on the y-axes (both sides) corresponding to the pure components. Similar relationships can be derived for ternary and multicomponent systems as well.
2.5.2 Relative integral and relative partial molar properties Except in the case of molar volumes, it is not possible to determine the absolute values of integral molar properties of solutions experimentally. On the other hand, the difference in the integral molar property of the solution and those corresponding to a `mechanical' mixture of components is experimentally determined. This difference, referred to as the relative integral molar Gibbs energy, 0001GM represented by the equation:
Thermodynamic aspects of metals processing 0001GM ˆ Gm ÿ (XA 0001 ëGA ‡ XB 0001 ëGB)
59 (2.55)
Similarly, the relative partial molar Gibbs energy of component A, 0001GM A , can be described as corresponding to one mole of each component in solution. The mathematical relationships between relative integral and partial molar properties are analogous to those of the integral and partial molar properties presented earlier in equations (2.52) to (2.54). The relative partial molar Gibbs energy of A, 0001GM A is related to the partial molar Gibbs energy of A by means of the relationship 0016 0001GM B ˆ GB ÿ ëGB
(2.56) M
The relative integral molar enthalpies, 0001H are negative for solutions with exothermicity while they are positive in the case of endothermic solution formation. On the other hand, the relative integral molar entropies are always positive as the configurational entropy increases by the solution of one component in another. Consequently, the relative integral molar Gibbs energies are always negative in the case of spontaneous solutions as otherwise, the driving force is in the opposite direction. The extent of the negative value is dependent on the relative magnitudes and signs of the enthalpy and entropy terms.
2.5.3 The concept of activity Activity of a component in a solution, introduced by G.N. Lewis in 1907, is often referred to as the concentration corrected for the intercomponent interactions. It is represented mathematically as 0016 0001GM B ˆ …GB ÿ ëGB) ˆ RT ln aB
(2.57)
where aB refers to the activity of component B in solution. Equation (2.57) can be graphically represented in Fig. 2.13 along with the partial and integral molar properties.
2.5.4 Chemical potentials and equilibrium constant For a chemical reaction in chemical equilibrium, the sum of the Gibbs energies of the reactants will be equal to that of products. as shown in equation (2.47). This provides a basis for an expression for equilibrium constant on the basis of Gibbs energies. For example, for a reaction 2Fe (l) ‡ O2 (g) ˆ 2FeO (l)
(2.58)
where underlines signify solutions. Equation (2.47) can be formulated as 0016O ˆ G 0016 FeO 0016 Fe ‡ G G 2
(2.59)
By incorporating equations (2.47) and (2.57), equation (2.59) can be rewritten as
60
Fundamentals of metallurgy
2.13 Graphical representation of the integral and partial molar properties of the system A-B. The line XA 0001 ëGA ‡ XB 0001 ëGB corresponds to the `mechanical' mixture of the components A and B.
2ëGFe ‡ RT ln a2Fe ‡ ëGO2 ‡ RT ln pO2 ˆ 2ëGFeO ‡ RT ln a2FeO (2.60) And this equation can be rewritten as (2ëGFeO ÿ 2ëGFe ÿ ëGO2) ˆ 0001ëG = ÿRT ln (a2FeO/a2Fe 0001 pO2) ˆ ÿRT ln KR (2.61) where `KR' is the equilibrium constant. The temperature coefficient for the equilibrium constant can be derived as 0012 0013 1 @0001GR d ln KR 1 B 0001000e GR @T P C C ˆ B ÿ @ A 2 dT T T R 0
1 0001000e H R …0001000e GR ‡ T0001000e SR † ˆ 2 R 0001 T2 R0001T 0001ëHR corresponding to the enthalpy of the reaction. ˆ
…2:62†
2.5.5 Ideal solutions ± Raoult's law Examining the vapour pressures of components in solution in condensed state, Raoult postulated that
Thermodynamic aspects of metals processing PB ˆ ëpB 0001 XB
61 (2.63)
where PB and ëpB are the partial pressures of B in solution and pure state respectively. This would lead to the relationship for an ideal solution as aB ˆ XB
(2.64)
which is often termed as Raoult's law. In reality, many solutions deviate from equation (2.59). Solutions wherein there is repulsive interaction between the components show a positive deviation (aB > XB) while, in the case of the components exhibiting attractive interactions with each other, the solution would show a negative deviation (aB < XB). For non-ideal solutions, the deviation from Raoult's law is denoted by the ratio between the activity and the mole fraction, aB/XB, referred to as the activity coefficient, B. In the case of ideal solutions, the value of the activity coefficient is unity, while values less than unity are indicative of attractive forces between the components in solution and negative deviation from Raoult's law. Activity coefficient values more than unity mark repulsive forces between the components and positive deviation from Raoult's law. The relative integral molar enthalpy of mixing for ideal solutions is zero as there are no attractive or repulsive interactions between the components. For the same reason, the different component atoms have no preferential sites and the mixing will be random. Thus, 0001SM will have a maximum value. From statistical mechanics considerations, an expression for relative integral molar entropy for ideal solutions could be derived, which is presented in equation (2.65). 0001SM ˆ ÿR 0006Xi ln Xi
(2.65)
The relative molar Gibbs energy for ideal solutions will thus be the combined effect of the enthalpy and entropy terms, viz. 0001GM ˆ RT 0006Xi ln Xi
(2.66)
2.5.6 Excess properties The excess property is defined as the difference between the actual value that that would be expected if the solution were ideal. For example, in the case of Gibbs energy, the integral molar excess Gibbs energy, GXS, can be expressed as GXS ˆ 0001GM ÿ 0001GM,
ideal
(2.67)
The relationship between 0001GM and GXS in the case of a binary solution A-B is presented in Fig. 2.14.
62
Fundamentals of metallurgy
2.14 Excess Gibbs energies in the case of a binary system A-B. I Repulsive interactions between A and B. II Ideal solution. III Attractive interactions between A and B.
2.5.7 Ideality and bond energies The enthalpies and entropies of reactions as well as the concept of ideality can be illustrated by considering the changes in the bond energies involved in the reaction A-A ‡ B-B ˆ 2A-B
(2.68)
The energy change associated with the above reaction can be denoted as 0001E. 0001E ˆ 2EA-B (EA-A ‡ EB-B)
(2.69)
where EA-B, EA-A and EB-B are the energies associated with the atom pairs in the subscript. If the probability of forming a A-B bond is denoted as PA-B, the enthalpy of reaction (2.63) can be derived as 0001HM ˆ PA-B 0001 0001E ˆ PA-B 0001 [2EA-B ÿ (EA-A ‡ EB-B)]
(2.70)
If the energy associated with an A-B bond is the average of the energies of A-A and B-B bonds, the solution process does not involve any net energy change. Consequently, 0001E and 0001HM will be zero. Formation of ideal solutions require that the components involved are chemically similar.
Thermodynamic aspects of metals processing
63
2.5.8 Regular solutions The concept of regular solutions was first proposed by Hildebrand in 1950. According to this, a regular solution can have non-ideal enthalpies, while the entropies of mixing are considered ideal. The entropy term has been restricted, by later workers, to configurational entropy and does not include thermal entropy. Thus, the relative molar entropy of mixing for a regular solution would be given by equation (2.60). 0001HM can be shown to be a symmetrical function with respect to composition in the case of a binary solution A-B and is given by the equation 0001HM = 0001 XA 0001 XB
(2.71)
where is a constant independent of temperature and composition. From bond energy considerations, the constant can be shown to be
ˆ Z 0001 N 0001 0001E
(2.72)
where Z is the coordination number of the atoms in solution, N is the Avogadro's number and 0001E has the same significance as in equation (2.64). Assuming that these terms are constant with respect to temperature and composition, the relative integral molar enthalpy of mixing is given by the relationship 0001HM ˆ (Z 0001 N 0001 0001E) 0001 XA 0001 XB
(2.73)
which is identical with equation (2.66). The concept of regular solutions has an inbuilt inconsistency as solutions with non-zero enthalpies of mixing can not have ideal entropies of mixing. But the concept is found very helpful in the case of high temperature systems.
2.5.9 Henry's and Sievert's laws In the case of dilute solutions, the activity coefficient of the solute is found to vary linearly with composition. This behaviour is referred to as Henry's law which is normally designated as aB/XB ˆ 1 B
(2.74)
where 1 B is referred to as Henry's constant. It is observed that when the solute obeys Henry's law, the solvent obeys Raoult's law in the same concentration range. This could be explained on the basis of the interatomic interactions between the solute and the solvent. In the case of diatomic gases dissolving in metals in low amounts in atomic form, the amount of the gas dissolved was found by Sievert to be proportional to the square root of the partial pressure of the diatomic gas. Thus, the solubility of nitrogen in liquid iron can be represented as
64
Fundamentals of metallurgy XN ˆ k 0001
p pN 2
(2.75)
Sievert's law is a corollary of Henry's law.
2.5.10 Standard states In most of the industrial production of base metals, the choice of pure metal as the standard, viz.
i ! 1 when Xi ! 1
(2.76)
is often impractical as the impurity elements are in low concentrations. Hence, standard states corresponding to Henry's law, viz. fi ! 0 when Xi ! 0
(2.77)
fi ! 0 when atom % i ! 0
(2.78)
fi
(2.79)
(wt %)
! 0 when wt. % i ! 0
where the `f'-terms refer to the activity coefficients corresponding to Henrian standard states.
2.15 Activity of tin and gold in the binary Ni-Cu system at 600 ëC.
Thermodynamic aspects of metals processing
65
2.5.11 Stabilities and excess stabilities From a consideration of the thermodynamics of a number of binary metallic molten systems, Darken8 suggested that the thermodynamic behaviour of the components in the terminal regions of the solution with respect to composition is expressed by simple expressions and that the solvent and solute have slightly differing behaviours. For a solution A-B, in the region where A is the solvent and B, the solute, the activity coefficients are given by the expressions
2.16 Excess stability as a function of composition in the system Mg-Bi at 700 ëC.8
66
Fundamentals of metallurgy ln A ˆ AB (1 ÿ XA)2
and
ln B ˆ AB (1 ÿ XB)2 ‡ I
(2.80)
where AB is a constant independent of composition in the range where A is the solvent and I is an integration constant. Hillert suggests that the difference between the solvent and the solute can be considered as being due to different standard states and the integration constant may be considered as factor for the change of standard state. Darken further evolved the concept of excess stability, which is the second derivative of the excess Gibbs energy. The mathematical expression for excess stability in the case of the binary system A-B, as shown by Darken, is (d2GXS/dXB2) ˆ ÿ2RT[d ln B/d(1 ÿ XB)2]
(2.81)
With the occurrence of strong interactions in the system as, for example, formation of the intermetallic compound Mg2Bi in the Mg-Bi binary, the excess stability function showed a sharp peak corresponding to this composition. This is shown in Fig. 2.16. Darken further showed that the concept of excess stability can be applied, with advantage to even molten oxides as well as aqueous systems.
2.6
Thermodynamics of multicomponent dilute solutions
In industrial production of base metals like iron or copper, often dilute solutions are encountered with a number of solute elements dissolved in the same. An understanding of the thermodynamic behaviour of solutes in such solutions is imperative when optimizing these processes. The thermodynamic behaviour of a solute in a binary solution is described by Henry's law. In ternary solutions with two solutes, it can be assumed that Henry's law holds when the concentrations of the solutes are extremely low, as solute±solute interactions are negligible. With increasing concentrations, the solute±solute interactions affect the thermodynamics of the system and deviations from Henry's law occur. Such deviations in the case of the activity coefficient of oxgen in liquid iron at 1600 ëC for a number of solutes are presented in Fig. 2.17.
2.6.1 Wagner's equation The deviation from Henry's law by solute±solute interactions has been expressed mathematically by means of a MacLaurin type of equation shown below: ln B ˆ ln
1
B ‡ XB 0001 (@ ln B/@XB) ‡ XC 0001 (@ ln B/@XC) . . .
‡ (XB/2)2(@ 2 ln B/@X2B) ‡ (XC/2)2(@ 2 ln B/@X2C) . . . ‡ (XBXC)(@ 2 ln B/@XB 0001 @XC) + (XBXD)(@ 2 ln B/@XB 0001 @XD) . . . (2.82)
Thermodynamic aspects of metals processing
67
2.17 The variation of the activity coefficient of oxygen in liquid iron at 1600 ëC for a number of solutes. Henrian standard state is used for oxygen activity coefficient.9
Wagner suggested that, at low concentrations, the second order terms can be neglected and equation (2.82) may be reduced to ln B ˆ ln
1
B ‡ XB 0001 000f(B)B + XC 0001 000f(C)B . . .
(2.83)
where 000f(C)B ˆ [@ ln B/@XC]XA!1;
XB, XC. . .!0
(Standard state: i ! 1 when Xi ! 1.) If the standard state is changed to fB(wt %) ! 1 when wt %B ! 0, the above expression becomes log10 fB ˆ log10
1
fB ‡ wt %B 0001 e(B)B ‡ wt %C 0001 e(C)B
(2.84)
where e(C)B = [@ log B/@ wt %C]wt %A!1; wt %B, wt %C!0. The `000f' and `e' terms are termed interaction parameters. At higher concentrations of the solute, the second order terms in the MacLaurin expression in equation (2.77) may have to be taken into account. ln B ˆ ln
1
B ‡ XB 0001 000f(B)B ‡ XC 0001 000f(C)B . . . ‡ (XB/2)2001aB(B)
‡ (XC/2)2001aB(C) ‡ (XBXC) 0001 001aB(B,C) ‡ (XBXD) 0001 001aB(B,D)
(2.85)
where the `001a'-terms correspond to the second order terms in the MacLaurin expression in equation (2.77).
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Fundamentals of metallurgy
2.6.2 The central atom description10 The solute±solute interactions can be elegantly described by the central atom model by Lupis and the illustration reproduced from Chemical Thermodynamics of Materials by Lupis10 is presented in Fig. 2.18. The circle filled with lines is solute atom B and the one with dots is solute atom C are solute atoms and blank circles are solvent atoms. Configuration 1 ˆ 000f(C)B, 2 ˆ 000f(B)C, 3, 4 ˆ 001aB(B,C), 5 ˆ 001aC(B) The deviation from Henry's law and the application of the MacLaurin expression is illustrated in Fig. 2.19. Equations (2.78) and (2.80) are commonly used to calculate the interaction parameters. It should be remembered that these are valid only when the solute
2.18 Solute±solute interactions according to the central atom model by Lupis.10
2.19 Application of MacLaurin expression for describing the deviation from Henry's law.11
Thermodynamic aspects of metals processing
69
concentrations approach zero. Darken has shown, by applying the Gibbs±Duhem equation, that, if these equations are used for solutes at finite concentrations, it would lead to erroneous results. Equation (2.78) has been modified by Pelton and Bale as ln B ˆ ln
1
B ‡ ln A ‡ XB 0001 000f(B)B ‡ XC 0001 000f(C)B . . .
(2.86)
This equation is compatible with Gibbs±Duhem equation and can be used for finite concentrations of solutes.
2.6.3 Estimations of interaction parameters Extrapolation of interaction parameters from one temperature to another can be carried out by introducing enthalpy interaction parameter, 0011(C)B and entropy interaction parameter, 001b(C)B.12 These terms are related to the interaction parameter, 000f(C)B by the relationship: RT 000f(C)B 0001 XC ˆ 0011(C)B 0001 XC + T 0001 001b(C)B 0001 XC
(2.87)
In analogy with the Gibbs±Helmholtz equation, it can be written (@000f(C)B/@(1/T)) ˆ 0011(C)B/R
(2.88)
and the ratio between 000f(C)B and 0011(C)B is given by (000f(C)B/0011(C)B) ˆ 1/R ((1/T) ÿ (1/001c)
(2.89)
The term `001c' is a correction term having the unit of temperature. s is related to the enthalpy and entropy interaction parameters by the equation 0011(C)B = 001c 0001 001b(C)B
(2.90)
The value of 001c has been estimated in the case of a number of non-ferrous systems as 1800K and in the case of ferrous systems as circa 2100K. This enables the prediction of 0011(C)B from a known value of 000f(C)B as well as estimation of 000f(C)B at the temperatures using equation (2.85). In the case of systems where there is no data on interaction parameters, the empirical equation proposed by Jacob and Alcock can be used with some success. This is given here: 000f(C)B = ÿn[( B(A)/ B(C))]1/n 0001 C(A)) ÿ 1
(2.91)
where the ` ' terms are binary Henry's coefficients and n and are empirical constants, with the values n ˆ 4 and ˆ 12. While equation (2.87) is reasonably successful in the case of non-ferrous solutes, it has been shown that caution should be exercised in applying the same to transition metal solutes.
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Fundamentals of metallurgy
2.6.4 Interaction parameters and solubility of oxides in metallic melts The concept of interaction parameters has been applied to predict the minima in the solubility curves in a number of metallic melts by St Pierre.13 Successful prediction of the minimum in the case of the precipitation of alumina in the case of Fe-O-Al system is a classical case that has applications in the calculations of precipitation of alumina inclusions from steel melts.
2.7
Modelling of metallic systems
Thermodynamic modelling is essential in the case of systems, where there is lack of experimental data or where experimental measurements are extremely difficult. This is particularly true in the case of multicomponent systems. In modelling the thermodynamic properties, attention has been focused on two properties, viz. the enthalpies and Gibbs energies of mixing. In order to extrapolate thermodynamic data from known to unknown composition and temperature ranges, it is generally felt necessary to express enthalpies and Gibbs energies by means of a suitable mathematical expression. In view of the rapid developments in metallurgy and materials, it is necessary to handle the thermodynamics of systems with many components over a wide range of temperatures, necessitating computerized calculations. Thus, the need for models for computer calculations has been felt during the past three decades. One of the earliest expressions for integral molar excess Gibbs energy was due to Margules.14 Hillert15 has proposed the use of the Redlich±Kister polynomial to express excess Gibbs energy. It is noteworthy that, as Darken suggests, in the case of binary systems, that series expressions may be unnecessary as the thermodynamic behaviour of these systems is fairly simple except in the central composition region. Today, based on various empirical or semi-empirical models, thermodynamic databases have been developed that can perform a variety of operations in the case of muticomponent systems. FACT-Sage, MT-data and Thermo-Calc are a few to be named and which have been widely used. In this chapter, only the sublattice model, which is the basis for the Thermo-Calc software will be presented.
2.7.1 The two sublattice model16 The above model places the substitutional and interstitial sites in a metallic lattice as two distinct sublattices. The entropy of mixing is restricted in each sublattice and the total entropy would be the sum of the entropies of the sublattices. A similar description for ionic melts, with anionic and cationic groupings has been presented by Temkin17 earlier. In the two sublattice model, the composition is described in terms of lattice
Thermodynamic aspects of metals processing
71
fraction, Yi. The basic description of the integral molar Gibbs energy is given by the expression Gm ˆ 0006Yi 0001 ëGi ÿ T 0001 0001SM, ideal + GXS
(2.92)
The model is extremely suitable for multicomponent systems with both substitutional and interstitial elements, especially in the solid state. With suitable modifications, the same type of description can be applied to ionic melts as well. The model is extremely suited for computerized applications for multicomponent systems. The Thermo-Calc system18 is based on this model, with more sublattices when needed. The system has a variety of administrative programmes and is used worldwide.
2.7.2 CALPHAD approach CALPHAD is an abbreviation for CALculation of PHAse Diagrams. In Section 2.4, the link between thermodynamics and phase diagrams has been presented. A
2.20 A schematic representation of Gibbs energy curves corresponding to the phase diagram shown in Fig. 2.8 with complete solubility in solid and liquid states corresponding to temperature T1.
72
Fundamentals of metallurgy
2.21 The Cr-C phase diagram calculated using the evaluated parameters.19 The temperatures of the three-phase equilibrium shown in the figure are those calculated. The values in parentheses show the difference from the selected experimental values.
schematic representation of Gibbs energy curves corresponding to the phase diagram shown in Fig. 2.8 is illustrated in Fig. 2.20. Since phase diagrams are essentially thermodynamic descriptions of the phases in the system, it is necessary that the thermodynamic descriptions are compatible with the phase diagram information available. It should be possible to generate thermodynamic information by combining phase diagram data available with the results of thermodynamic experimental studies. Such efforts have progressed extremely well and the thermodynamic assessments of various systems available are based on CALPHAD approach. The Thermo-Calc system has been extremely valuable in this regard. A typical phase diagram for Cr-C binary system,19 developed recently on the basis of the available thermodynamic and phase diagram information is presented in Fig. 2.21.
2.8
Thermodynamics of ionic melts
Salts as well as oxide melts are ionic in nature. An understanding of the thermodynamics of these materials is of importance in fused salt metals extraction as well as slag practice in pyrometallurgy. The thermodynamic description of ionic liquid is complicated as the entropy of mixing is caused by the mixing of cations among themselves and anions in a
Thermodynamic aspects of metals processing
73
similar fashion. The analogy with the two sublattice model described in Section 2.7.1 is seen clearly here. In addition, in the case of silicate melts, the polymerization as SiO2 content increases the complexity of the system. Pure SiO2 has a three dimensional network of SiO4 tetrahedra which are linked to each other by sharing of the corners, edges and sides. Depolymerization is effected by the addition of basic oxides like CaO, which break the silicate network. This leads to the existence of a variety of silicate polymers as functions of composition and also with respect to temperature, the latter contributing to the entropy by the destabilization of the silicate polymers.
2.8.1 Temkin's17 and Flood et al.'s20 description of ionic melts It was long realized that the Raoultian description of ideal systems was not compatible with the experimental results of ionic melts. Temkin realized that this is due to the entropy of mixing in these melts and proposed that anions and cations should be grouped separately and the entropies of mixing should be calculated separately for each subgrouping. On this basis, Temkin suggested that the activity of a component, MA2, in a salt melt, is given by aMA2 ˆ NM2+ 0001 N2Aÿ
(2.93) 2+
ÿ
where NM2+ is the cation fraction of M ions and A the anion fraction of Aÿ ions. Later on Flood, Fùrland and Grjotheim20 introduced the concept of equivalent ion fractions. For example, in this case of a salt melt NaCl-CaCl2, the activity of NaCl is given by component MA2 in turn given by aNaCl ˆ N0 Na+ 0001 N0 Clÿ
(2.94)
where N0 terms are the equivalent ionic fractions. NNa+ can be defined as N0 Na+ ˆ (nNa+/(nNa+ ‡ nV ‡ nCa2+)) ˆ (nNa+/(nNa+ ‡ 2nCa2+))
(2.95)
The choice of Temkin's or Flood et al.'s ion activity concept, according to Sridhar and Jeffes,21 is to be based on the the extent of deviation of the system from ideality. For low values of 0001HM, Temkin activities can be used while, for systems with high values of 0001HM, it is more appropriate to use to concept of Flood et al. Another word of caution is with respect to ion activities. Since the standard state becomes ambiguous, it is recommended to use Henrian standard states when ion activities are referred to. In the case of systems with common ions, the bond energy descriptions (equation 2.70) can be used to describe the enthalpies of mixing. But the interactions are between the next-nearest neighbouring ions; for example, in the case of the system NaCl-CaCl2, the interactions are between Na+ and Ca2+ ions, while O2ÿ are the nearest neighbours to the cations.
74
Fundamentals of metallurgy
2.8.2 Richardson's theory of ideal mixing of silicates22 Richardson proposed that binary silicates of equal silica molfraction (such as FeSiO3 ‡ CaSiO3) mix ideally with one another. This would mean that the enthalpy of mixing is zero and the entropy of mixing, which is the same as the configurational entropy arising from the mixing of cations only will be ideal. This is given by 0001SM ˆ ÿR[(XMO ln (XMO/(XMO ‡ XYO))) ‡ (XYO ln (XYO/(XMO ‡ XYO)))]
(2.96)
The Gibbs energy of mixing is 0001GM ˆ ÿT 0001 0001SM
(2.97)
The Gibbs energy surface for the formation of solutions MO ‡ YO ‡ SiO2 is presented in Fig. 2.22. The theory of ideal mixing is very useful in estimating the ternary activities from the binary values. The validity of the theory is somewhat uncertain when the cation sizes differ widely.
2.22 The Gibbs energy surface of MO ‡ YO ‡ SiO2 solutions.22
Thermodynamic aspects of metals processing
75
2.8.3 Lumsden's description of silicates5 Lumsden proposed that the silicate melts can be considered as melts consisting of O2ÿ ions and cations like Ca2+, Fe2+ and even Si4+, the latter by considering the SiO44ÿ tetrahedral as dissociating into Si4+ and O2ÿ ions. Lumsden's consideration demanded the visualization of pure covalent liquid silica in contrast to the fully ionized silica in the silicate. Lumsden introduced a change of the standard state in order to account for this. While the Lumsden description is contrary to the structure of silicates, it is found to be a very useful tool in empirical modelling of slags.
2.8.4 Slag models A number of models have been developed for providing an adequate thermodynamic description of slags. These can be classified into structural models and semi-empirical models. Structural models are based on the polymer theory as applied to silicate melts, the pioneering work being that of Masson.23 The calculated weight fractions of the various anionic species by the polymeric model is presented in Fig. 2.23. While these are successful for simple silicates, a
2.23 Calculated weight fractions of the anions in a silicate system. Wg refers to infinite chain.
76
Fundamentals of metallurgy
great deal of further work is necessary to apply to multicomponent slags. Among the semi-empirical models, a few are briefly mentioned here. Kapoor and Frohberg24 presented a statistical thermodynamic treatment of silicate melts where units of one oxygen and two cations are visualized. The model was later expanded to more complex systems and is referred to as the IRSID model. Models based on silicate structure and quasichemical approach25 have been developed by Pelton and his group, which form the basis for silicate descriptions for the Chemsage/F.A.C.T. system. Hillert and co-workers have extended their sublattice model to silicate systems. A simple, empirical description of silicates could be achieved by combining the Temkin description with Lumsden description and describe the complex silicates as consisting of cations including Si4+ and O2- ions. This was developed at the Royal Institute of Technology, Stockholm, and is often referred to as the KTH model.26 The ionic solution can be represented as (A2+, B2+, Si4+. . .)P (O2ÿ)Q
(2.98)
where P and Q are stoichiometric constants. The Gibbs energy of mixing in the melt is expressed as Gm ˆ 0006XCici Oai G000eCici Oai ‡ RTp0006yCi ln yCi ‡ GXS
…2:99†
where CiciOai is the oxide component and `y' terms refer to ion fractions. The excess Gibbs energy could be expressed by Redlich±Kister polynomial and the parameters could be derived from the available experimental data. The isoactivity lines of FeO, computed by the above model in the case of a ladle slag are presented in Fig. 2.24. The computation is enabled by a software, THERMOSLAGÕ developed by the present group. The activities of Al2O3 in the Al2O3-CaO-`FeO'-MgO-SiO2 system at 1873K corresponding to blast furnace (BF), electric arc furnace (EAF) and ladle furnace (LF) processes computed by the KTH model using the THERMOSLAG Õ software is presented in Fig. 2.25. Model developments need accurate experimental data, which in many cases is lacking today.
2.8.5 Slag capacities In refining of metals, it is important to know the ability of a slag to absorb impurities like sulphur or phosphorus. In order to enable this, in the case of sulphur, Fredriksson and Seetharaman27 introduced the sulphide capacities, on the basis of equilibrium between slags and a gas phase with defined oxygen and sulphur potentials. The expression for sulphide capacity, CS, as defined by Fincham and Richardson,28 is
Thermodynamic aspects of metals processing
77
2.24 Calculated and experimentally determined activities of `FeO' in the Al2O3`FeO'-SiO2 system at 1873K.27
2.25 The activity of Al2O3 in the Al2O3-CaO-`FeO'-MgO-SiO2 system at 1873K for the BF, EAF and LF process.27
78
Fundamentals of metallurgy Table 2.1 Standard electrode potentials29 Electrode reaction
000f0H
Acid solutions Li = Li+ + e K = K+ + e Cs = Cs+ + e Ba = Ba++ + 2e Ca = Ca++ + 2e Na = Na+ + e Mg = Mg++ + 2e Hÿ = 12H2 + e Al = Al+++ + 3e Zn = Zn++ + 2e Fe = Fe++ + 2e Cr++ = Cr+++ + e Cd = Cd++ + 2e Sn = Sn++ + 2e Pb = Pb++ + 2e Fe = Fe+++ + 3e D2 = 2D+ + 2e H2 = 2H+ + 2e H2S = S + 2H+ + 2e Sn++ = Sn+4 + 2e Cu+ = Cu++ + e 2S2O3= = S4O6= + 2e Fe(CN)6ÿ4 = Fe(CN)6ÿ3 + e Cu = Cu++ + 2e 2Iÿ = I2 + 2e Fe++ = Fe+++ + e Ag = Ag+ + e Hg = Hg++ + 2e Hg2++ = 2Hg++ + 2e 2Brÿ = Br2 (l) + 2e Mn++ + 2H2O = MnO2 + 4H+ + 2e 2Cr+3 + 7H2O = Cr2O7= + 14H+ + 6e Clÿ = 12Cl2 + e Ce+3 = Ce+4 + e Co++ = Co+++ + e 2SO4= = S2O8= + 2e 2Fÿ = F2 + 2e
3.045 2.925 2.923 2.90 2.87 2.714 2.37 2.25 1.66 0.763 0.440 0.41 0.403 0.136 0.126 0.036 0.0034 0.000 ÿ0.141 ÿ0.15 ÿ0.153 ÿ0.17 ÿ0.36 ÿ0.337 ÿ0.5355 ÿ0.771 ÿ0.7991 ÿ0.854 ÿ0.92 ÿ1.0652 ÿ1.23 ÿ1.33 ÿ1.3595 ÿ1.61 ÿ1.82 ÿ1.98 ÿ2.65
Basic solutions 2OHÿ + Ca = Ca(OH)2 + 2e 3OHÿ + Cr = Cr(OH)3 + 3e 4OHÿ + Zn = ZnO2= + 2H2O + 2e 2OHÿ + Cnÿ = CNOÿ + H2O + 2e 2OHÿ + SO3= = SO4= + H2O + 2e H2 + 2OHÿ = 2H2O + 2e 2OHÿ + Ni = Ni(OH)2 + 2e OHÿ + Fe(OH)3 = Fe(OH)3 + e O2 + 2OHÿ = O3 + H2O + 2e
3.03 1.3 1.216 0.97 0.93 0.828 0.72 0.56 ÿ1.24
* W.M. Latimer, The Oxidation States of the Elements and Their Potentials in Aqueous Solutions (NewYork: Prentice-Hall, 2nd edn, 1952).
Thermodynamic aspects of metals processing CS ˆ (wt % S) 0001 [pO2/pS2]1/2
79 (2.100)
Similar expressions for other impurities taken up by the slag phase can be defined. The slag capacities are largely experimentally determined. Extrapolation of the data with respect to composition and temperature is enabled by suitable models. The THERMOSLAGÕ software can be used to compute the iso-Cs lines in the case of multicomponent slag systems.
2.9
Basics of electrochemical thermodynamics
The application of thermodynamic principles to electrochemical concepts has long since been known. This is a very important field in metallurgy, specifically electrometallurgy in aqueous phase as well as through the molten salt route. The latter is an important process route towards the production of reactive and refractory metals, with great significance to strategic materials. In the case of a reversible galvanic cell, chemical driving force of the cell reaction, 0001GCell reaction is equal and opposite to the electrical potential generated and is expressed by the Nernst equation as 0001GCell reaction ˆ ÿnEF
(2.101)
where `n' refers to the number of electrons participating in the reaction, `E' is the cell EMF and `F' is the Faraday constant. The validity of equation (2.101) implies the absence of electronic conduction in the cell. In the case of aqueous systems, the situation is complicated by the formation of hydrated ions and the electrolysis of H2O forming H2 and O2. A table of standard oxidation potentials could be established in the case of aqueous electrolysis by assuming that the potential for the reaction H2 ˆ 2H+ ‡ 2eÿ
(2.102)
as zero. This table is presented as Table 2.1. Stability diagrams similar to Fig. 2.12 for aqueous systems could be constructed in order to define the stabilities of various species. The diagram for Ag-O-H system is presented in Fig. 2.26.30
2.10 Conclusions As mentioned in the beginning of this chapter, thermodynamics has a wide variety of applications and is a fascinating subject. It is very ambitious to condense the subject into a small chapter, without losing the essentialities. The authors have made an attempt to present the importantant aspects of the subject for an understanding of metallurgical processes and properties, being fully aware of the limitations. For the convenience of more interested readers, a bibliography is presented at the end of this chapter, apart from the reference list. The readers
80
Fundamentals of metallurgy 20
_
Ag(OH)2
pH 10
Ag2 O Ag+2
Ag(s) Ag+ 0 0
10
0
20
1
pE E volt
30
40
2
2.26 The stability diagram for Ag-O-H system (0.1c mol Ag per litre at 25 ëC).30
are advised to consult these classical textbooks for a deeper understanding of the subject and enjoying the beauty of the logic in the same.
2.11 Further reading Chemical Thermodynamics of Materials, C.H.P. Lupis, North-Holland, Elsevier Science Publ. Co., NY, US (1983). Introduction to Metallurgical Thermodynamics, D.R. Gaskell, Hemisphere Publ. Corp., NY (1981). Physical Chemistry of Melts in Metallurgy, F.D. Richardson, Academic Press, London, UK (1974). Physical Chemistry of Metals, L.S. Darken and R.W. Gurry, McGraw-Hill Book Co. Inc., NY. US (1953). Thermodynamics of Alloys, C. Wagner, Addison-Wesley, Reading, MA (1952). Thermodynamics of Solids, R.A. Swalin, John Wiley & Sons, Toronto, Canada (1972).
2.12 References 1. A. Einstein, in Albert Einstein: Philosopher-Scientist, Tudor Publishing Co., NY (1949). 2. Introduction to Metallurgical Thermodynamics, D.R. Gaskell, Hemisphere Publ. Corp., NY (1981) 2. 3. GrundlaÈggande thermodynamik, M. Hillert, Royal Institute of Technology (1991) I:9. 4. Metallurgisk kemi, L.I. Staffansson, Royal Institute of Technology (1976). 5. Thermodynamics of Solids, R.A. Swalin, John Wiley & Sons, Toronto, Canada (1972) 214, 215.
Thermodynamic aspects of metals processing
81
6. Introduction to Metallurgical Thermodynamics, D.R. Gaskell, Hemisphere Publ. Corp., NY (1981) 287. 7. H.H. Kellogg and S.K. Basu, Trans. AIME (1960) vol 70, 218. 8. L.S. Darken, Trans. AIME (1967) vol. 239, 80. 9. Physical Chemistry of Melts in Metallurgy, F.D. Richardson, Academic Press, London, UK (1974) 182. 10. Chemical Thermodynamics of Materials, C.H.P. Lupis, North-Holland, Elsevier Science Publ. Co., NY, US (1983) 242. 11. Chemical Thermodynamics of Materials, C.H.P. Lupis, North-Holland, Elsevier Science Publ. Co., NY, US (1983) 248. 12. Physical Chemistry of Melts in Metallurgy, F.D. Richardson, Academic Press, London, UK (1974) 185. 13. G.R. St Pierre, Trans. AIME 14. M. Margules, Sitzungber, Kaiser Akadem. Wissenschaft, Wien, MathematischNaturwiss. Class. (1895) vol. 104, 1243. 15. M. Hillert, in Phase Transformations, ASM, Metals Park, Ohio (1970) 181. 16. M. Hillert and L.I. Staffansson, Acta Chem. Scand. (1970) vol. 24, 3618. 17. M. Temkin, Acta Phys. Chim URSS (1945), vol. 20, 411. 18. Thermocalc Software, version P on WinNT (2000) http://www.thermocalc.com. 19. L.D. Teng, X.G. Lu, R.E. Aune and S. Seetharaman, accepted for publication in Metall. Mater. Trans. (2005). 20. H. Flood, T. Fùrland and K. Grjotheim, Inst. Min. Metall., London (1953) 47. 21. R. Sridhar and J.H.E. Jeffes, Trans. Inst. Min. Metall. (1967), vol. 76, C44. 22. F.D. Richardson, Trans. Faraday (1956) vol. 52, 1312. 23. C.R. Masson, Proc. R. Soc. (1965) vol. A287, 201. 24. M.L. Kapoor, Mehrotra and M.G. Frohberg, Archiv. EisenhuÈttenw. (1974), vol. 45, 213 and 663. 25. M. Blander and A. Pelton, Metall. Trans. B (1986), vol. 17B, 805. 26. J. BjoÈrkvall, D. Sichen and S. Seetharaman, Ironmaking and Steelmaking, vol. 28, no. 3 (2001) 250±257. 27. P. Fredriksson and S. Seetharaman, Presented in the VII International Conference on Molten Slags, Fluxes and Salts, Cape Town, South Africa, January 2004. 28. C.J.B. Fincham and F.D. Richardson, Proc. Roy. Soc. (1954), vol. A223, 40. 29. Introduction to Metallurgical Thermodynamics, D.R. Gaskell, Hemisphere Publ. Corp., NY (1981) 551. 30. GrundlaÈggande Thermodynamik, M. Hillert, Royal Institute of Technology (1991) VI:19.
3
Phase diagrams, phase transformations, and the prediction of metal properties K M O R I T A , The University of Tokyo and N S A N O , Nippon Steel Corporation, Japan
3.1
Introduction
Phase diagrams provide a variety of thermodynamic information through equilibria among multiple phases as well as quantitative data on phase distribution in a specific system. Phase relations demonstrate that any chemical potentials of all coexisting phases are identical, imposing significant restrictions in the thermodynamic properties of the probable phases. By solving these restrictions, with simultaneous equations, the variables of compositions can be determined as a function of temperature. Nowadays, a number of phase diagrams are drawn by thermodynamic calculations by commercial PC programs using empirical parameters within some constraint. Although phase diagrams show the phase relations quantitatively by lever rules, an average composition in a multi-phase region shows no difference in chemical potentials, while sometimes there is uncertainty in the chemical potential of a single stoichiometric compound. For instance, an intermetallic compound AB has two different chemical potentials at A deficient side and B deficient side. Strictly speaking, an abrupt gap of each chemical potential should be considered at that composition. In the present chapter, phase diagrams are discussed from the aspect of thermodynamic stability. By further investigation of chemical potential diagrams, their conversion to phase diagrams is used to compare the difference in the physical meanings. On the other hand, it is important to understand the behaviour of phase transformation as well as phase relations and their phase stability in any metallurgical process. As a primary step, expression and interpretation of ternary phase diagrams are introduced, and then the fundamental features of the diagrams are discussed from thermodynamic aspects. As the easiest way in viewing a three-dimensional phase diagram, just a number of compiled isothermal sections representing various phase relations are shown. Finally, solidification behaviour is described as an example of practical application of the ternary phase diagrams.
Phase diagrams and phase transformations
3.2
83
Phase diagrams and potential diagrams
As introduced in the previous chapter, the Ellingham diagram clearly shows the most stable formulae of substances in terms of their red-ox equilibria, where the standard Gibbs energies of formation of oxides per one mole of oxygen gas are demonstrated as a function of temperature for various materials. Let us consider the oxidation of iron. There are three kinds of oxides, FexO, Fe3O4 and Fe2O3, and the following equilibria, (i)±(iv), can be considered. Here, the value of x in the non-stoichiometric compound FexO is assumed to be 0.95 when equilibrated with metallic Fe, and 0:83 < x < 0:95 when equilibrated with Fe3O4. (i)
2x Fe ‡ O2 ˆ 2FexO
0001Gë ˆ ÿ502960 ÿ 36.2 T ‡ 63.818 T log T (J/mol)1
(x ˆ 0.95) (ii) 1.5 Fe ‡ O2 ˆ 0.5 Fe3O4 0001Gë ˆ ÿ551100 ‡ 153.7 T (J/mol)2 (iii) a FexO ‡ O2 ˆ b Fe3O4 0001Gë ˆ ÿ543110 ÿ 344.1 T ‡ 169.893 T log T (J/mol)1 (0:83 < x < 0:95) (iv) 4 Fe3O4 ‡ O2 ˆ 6 Fe2O3 0001Gë ˆ ÿ405040 ÿ 277.4 T ‡ 158.081 T log T (J/mol)1 According to the above equilibria, each standard Gibbs energy change per one mole of oxygen, 0001Gë, can be drawn as a function of temperature as shown in Fig. 3.1. Strictly speaking, however, the standard Gibbs energy changes for (iii) and (iv) are not for `formation', because the reactant Fe cannot coexist with Fe3O4 at higher temperatures than 850K or Fe2O3 at all temperatures. Since the line shows the coexistence of two compounds with the relationship between temperature, T, and oxygen potential, RT ln pO2, the area in between represents the most stable substance at a given temperature and oxygen potential. When the figure is redrawn as the relationship between log pO2 and 1/T, the diagram shows the most stable phase as functions of temperature and oxygen partial pressure more clearly as shown in Fig. 3.2. The stable areas are also represented in the same manner. This typical chemical potential diagram may also be referred as a phase diagram. Compared with the Fe-O binary phase diagram (Fig. 3.31), you may notice that each line in the former, which demonstrates a two-phase boundary, corresponds to the dotted area of the latter, and vice versa. This is because oxygen partial pressures of two phases with different oxygen contents in an equilibrium state are identical, while there is a wide range of oxygen partial pressure in a stoichiometric compound. However, as is the case with FexO, nonstoichiometric oxides have their unique oxygen partial pressure as a function of composition. When another component beside oxygen is added, the chemical potential diagram becomes three dimensional. At a constant temperature, however, we can draw such a diagram in a plane, showing the most stable phase as a function
84
Fundamentals of metallurgy
3.1 Standard Gibbs energy changes for the reactions (i)±(iv) as a function of temperature.
3.2 Phase stability regions for various iron oxides as a function of temperature and oxygen partial pressure.
Phase diagrams and phase transformations
85
3.3 Phase diagram for the Fe-O system.
of two chemical potentials. As an example, let us consider the Ca-S-O system at 1000K. Probable phases in the present system are considered to be Ca, CaO, CaS and CaSO4. Stabilities of such substances are controlled by the chemical potentials of oxygen and sulfur, namely their partial pressures at a certain temperature. Hence, there must be six boundaries among four substances, e.g. (i) Ca-CaO, (ii) Ca-CaS, (iii) Ca-CaSO4, (iv) CaO-CaS, (v) CaO-CaSO4 and (vi) CaS-CaSO4. Each boundary can be determined by equilibria between the two compounds as follows. (i)
Ca-CaO Ca + 1/2 O2 ˆ CaO log PO2 ˆ 55.48 (ii) Ca-CaS Ca + 1/2 S2 ˆ CaS log PS2 ˆ 46.04 (iii) Ca-CaSO4 Ca + 2O2 + 1/2 S2 ˆ CaSO4 4 log PO2 + log PS2 ˆ 109.11
0001Gë ˆ ÿ531090 (J/mol)2 0001Gë ˆ ÿ440720 (J/mol)2 0001Gë ˆ ÿ1044550 (J/mol)2
86
Fundamentals of metallurgy
3.4 Boundaries between stable phases for the Ca-O-S system at 1000K.
(iv) CaO-CaS 0001Gë ˆ 90360 (J/mol)2 CaO + 1/2 S2 ˆ CaS + 1/2 O2 log PO2 log PS2 ˆ 9.44 (v) CaO-CaSO4 CaO + 3/2 O2 + 1/2 S2 ˆ CaSO4 0001Gë ˆ ÿ513460 (J/mol)2 3 log PO2 + log PS2 ˆ 53.63 (vi) CaS-CaSO4 CaS + 2 O2 ˆ CaSO4 0001Gë ˆ ÿ603830 (J/mol)2 log PO2 ˆ 15.77 From all the relations between log PO2 and log PS2, six boundaries can be drawn as shown in Fig. 3.4. Each boundary separates the diagram into two regions and shows more stable substance among the two. For example, boundary (i)
3.5 Chemical potential diagram for the Ca-O-S system at 1000K.
Phase diagrams and phase transformations
87
indicates that Ca is more stable than CaO in the left-hand area of the boundary, while CaO is more stable in the other side. From the six restrictions in the figure, one can finally draw the most stable phase as is shown in Fig. 3.5, the so-called chemical potential diagram. As a result, the line (iii) was not used in the determination of the diagram because Ca and CaSO4 cannot coexist at 1000K, but the line still demonstrates the difference in the order of relative stability.
3.3
Ternary phase diagrams
3.3.1 Representation of composition and Gibbs triangle While a composition can be represented by an axis, which is one dimensional, in binary phase diagrams, one more dimension must be added in order to show a composition for ternary systems. Accordingly, the composition is represented in a plane and that of x + y + z ˆ 100% can be the simplest representation as shown in Fig. 3.6. In the space of x, y, z 0015 0, the plane becomes a regular triangle and any composition (x, y, z) can be represented. This is called Gibbs triangle. Let's consider compositions for the A-B-C ternary system in Fig. 3.7. When a line is parallel to the base, line BC, concentration of A is constant, while the ratio of B/ C is constant on any lines drawn through the apex A. When two of the solutes, x and y, are dilute compared to the other solvent, z, rectangular coordinates may be useful by plotting the composition with (x, y).
3.6 The plane x + y + z = 100 (x, y, z > 0).
88
Fundamentals of metallurgy
3.7 Representation of compositions in the A-B-C ternary system.
Then another coordinate, temperature, should be added to demonstrate a three-dimensional ternary phase diagram. For simplicity, a ternary system in which all constituents are entirely miscible can be shown in Fig. 3.8. As is the case with a binary system, the triangular prism can be divided into three spaces, a liquid region (L), a solid region (S) and the mixture of both (L + S), and they are separated by liquidus and solidus planes. It is important that how you can figure out the composition locating in the mixture region. Accordingly, an isothermal cross-section would be helpful to understand the phase relations quantitatively.
3.3.2 Isothermal cross section and tie-line Here, in order to consider the phase relation of point P locating in an L + S region, an isothermal cross-section at the temperature concerned, T1, can be demonstrated in Fig. 3.9. The isothermal plane obviously intersects with liquidus and solidus planes at two intersection lines, liquidus and solidus curves. There is a set of the coexisting liquid and solid compositions on respective curves and the straight line between them must go through the point P. The lever rule also applies to the line as well as in binary systems, and this line is called a tie-line or a conjugation line. This is a typical example of two phase equilibria at the average composition of point P, and the amount ratio of liquid at L1 to solid at S1 can be identified as lL/lS as shown in Fig. 3.9. However, the position of the tieline, which will not always go through an apex, cannot be recognized just by liquidus and solidus curves and it must be represented in phase diagrams in order to show the relationship of the two phases in exact equilibrium.
Phase diagrams and phase transformations
89
Liquidus L
P
Solidus
T1
Temperature
L+S
S C
A
B
3.8 Phase diagram for the A-B-C ternary system entirely miscible in both liquid and solid phases.
When the triangle prism is cut perpendicularly through the apex A in Fig. 3.8, the cross-section which is shown in Fig. 3.10 looks weird since the lens is not closed at one end. This clearly shows the stability region does not represent phase relations in the two-phase region at all, because such perpendicular crosssections usually exclude tie-lines. As a special case, when stable congruent compounds exist in binaries, the cross-section through these extremes appears as a pseudo-binary phase diagram as is shown in Fig. 3.11, the system Mg2SiO4Fe2SiO4. In ternary systems, sometimes three phases coexist, where degree of freedom is zero at a certain temperature. Wherever two-phase regions come across, the intersection point of two boundaries and those of the other two compositions of
90
Fundamentals of metallurgy A
Tie-line IS
L+S
P IL
B
S
C
3.9 Isothermal cross-section of the A-B-C ternary system and a tie-line of average composition P between liquid and solid phases at T1.
L
Temperature
T1
P L+ S
S
A
3.10 Perpendicular cross-section of the A-B-C ternary system through A axis and point P.
the coexisting phases make a triangle surrounded by the three tie-lines. This is the so-called three-phase triangle and each phase has a certain composition throughout the region regardless of the mixing ratio. This will be illustrated in isothermal phase relations as shown in the following section. When a liquid phase exists as one of these extremes, its locus with temperature is a boundary
Phase diagrams and phase transformations
91
o
1890 C L
o
Temperature ( C)
1800
1600
1400
L+ S
S o
1205 C 1200
0 Mg2SiO4
20
40 60 Mass % Fe2SiO4
80
100 Fe2SiO4
3.11 Pseudo-binary phase diagram for the Mg2SiO4-Fe2SiO4 system.3
line dividing the primary crystals on the reflected perspective of the diagram. In most cases, the appearance of liquid phase L is related to the other two solid phases, and , by either of the following reactions, (i) and (ii). (i) L $ + (ii) + L $ When the liquid at the extreme follows reaction (i), its composition is called a eutectic point and the boundary line is named as a crystallization curve. On the other hand, the composition is called a peritectic point and the boundary line is named an alteration curve in case of reaction (ii). A crystallization curve ends at the eutectic point of the binary system, hence it must penetrate the threephase triangle composed of L, and as shown in Fig. 3.12(a). In contrast, an alteration curve ends at the peritectic point of the binary system, and it passes outside of the triangle. See Fig. 3.12(b). Accordingly, one can tell if a boundary line is a crystallization curve or an alteration one by graphical investigation.
3.3.3 Representation of ternary oxide systems For the metallic systems, which form solid solutions, tie-lines cannot be uniquely determined because the solubility changes with temperature. On the other hand, it is not necessary to show such lines for many oxide systems, since the solubility of a solid solution may be disregarded and you may project every isothermal phase relations on one figure with ease. Herewith, the phase diagram for the CaO-A2O3-SiO2 system is shown in Fig. 3.133 as an example.
92
Fundamentals of metallurgy (a)
Crystallisation curve Cooling
L
a
b
(b) L Cooling
Alteration curve a
b
3.12 Location of (a) crystallization curve and (b) alteration curve.
Liquidus lines for various temperatures are shown by the contours like those of altitude in a map. It turns out that these lines show where the coastline will be made if water is filled to the height when a certain geographical feature is seen from above, and the domain filled with water corresponds to that of liquid phase at a certain temperature. In most cases inter-compounds exist and they appear like mountains or islands. When you can discern which portion of liquidus (coast line) the compounds are in equilibrium with, ternary phase diagrams are already comprehended. In the present system at 1873K, each liquidus curve can easily be followed up and the corresponding solid oxide in equilibrium is recognized as shown in Fig. 3.14. In addition, a break point appears when the liquid is connected to two different solids with respective tielines. One should notice that these three points form a three-phase triangle surrounded by three tie-lines. Thus, since there is no width in solid composition, an arbitrary composition of 1873K is specified even in the two-phase domain. When temperature is lowered, new islands, namely congruent compounds such as CaO0001Al 2 O 3 , 2CaO0001Al 2 O 3 SiO 2 , CaO0001Al 2 O 3 00012SiO 2 , CaO0001SiO 2 ,
Phase diagrams and phase transformations
93
3.13 Phase diagram for the CaO-Al2O3-SiO2 system.3
12CaO00017Al2O3, appear above sea level and the ocean is separated into several lakes. Such a complicated situation was snapped at 1673K and shown in Fig. 3.15. At 1443K, the final lake will be dried up at the eutectic composition of the CaO0001SiO2-CaO0001Al2O300012SiO2-SiO2 ternary system. Regarding the incongruent compounds, such as 3CaO0001SiO2 and CaO00016Al2O3, etc., their summits are not visible, because these compounds do not have their own melting points but decompose into liquid and other solid phases at peritectic temperature as will be explained in the following section. The evidences of such compounds can be seen as strata appearing at slanting surfaces divided by the alteration curves as specified in Fig. 3.13. When you glance at a map, you can figure out how creeks run and summits continue. Similarly, one may imagine a diagram to be a map of bird's-eye view, and some general rules of the ternary phase diagrams can be naturally recognized.
94
Fundamentals of metallurgy A : Al2O3 C : CaO S : SiO3 SiO2
CA2 : CaO 2Al2O3 CA 2 : CaO 6Al2O3 A2S2 : 3Al2O3 2SiO2 C2S : 2CaO SiO2 C3S : 3CaO SiO2
L+S
L + A3S2
L + C3S + C2S C2S C2S
L + C2S
L + C + C2S L+C CaO
L + A + A3S2
L
L + C3S
L+A L + CA 6
2
A3S2 L + A + CA 6 L + CA 6 + CA 2
L + CA 2 CA 2
CA 6
Al2O3
3.14 Liquidus and phase relations for the CaO-Al2O3-SiO2 system at 1873K.
3.15 Liquidus and phase relations for the CaO-Al2O3-SiO2 system at 1673K.
Phase diagrams and phase transformations
3.4
95
Solidification in ternary systems and four-phase equilibria
Solidification of binary alloys can be easily understood. Congruent solidification, which is characterized by isothermal freezing point and formation of solid from liquid of the same composition, occurs mainly for pure metals and intermetallic compounds. Also, a system with maxima or minima in the solidus and liquidus shows congruent solidification at that composition. On the other hand, incongruent solidification occurs over a wide temperature range, forming solid of a different composition from liquid, which is typical for binary alloys. As shown in Fig. 3.16, liquid of composition a starts to freeze in forming a solid phase with composition b richer in component A. Then, liquid becomes enriched in component B and the liquidus temperature will be lowered. Thus the solidification can be followed as a simultaneous progress of equilibrium solid and liquid compositions along the solidus and liquidus lines. The relative amount of liquid phase, determined from the tie-line by the lever rule, decreases with temperature. At the eutectic temperature, congruent solidification, simultaneous precipitation of and phases, occurs until the liquid phase with composition c diminishes. A representative cooling curve is shown in Fig. 3.17. Solidification of ternary alloys can be understood in the same manner as that of binary alloys. The complexity is only due to the possible appearance of another solid phase during cooling, and, as mentioned in the previous section, the tie-lines do not generally lie on an arbitral vertical plane. Let's consider an example of a ternary eutectic system as shown in Fig. 3.18, where some points
Tl
L
TS
T
b
a
A
b
a
c XB
3.16 Phase diagram for the binary A-B system.
B
96
Fundamentals of metallurgy Cooling of liquid Evolution of heat by a solidification Eutectic isotherm
T
Cooling of solid
L
L+a
L+a+b
a+b
t
3.17 Cooling curve for the alloy shown in Fig. 3.16.
and lines are reflected to the bottom of the prism to show the behaviour more simply. Here, the solubility of each component in a solid phase is assumed to be negligible for simplicity. When the liquid of a certain composition, X, is cooled along path Xx, it reaches the liquidus surface at x, where the solid of composition B starts to precipitate. This precipitated phase is called a primary crystal and the boundaries which separate primary crystal regions are called boundary lines, such as E1±E0, E2±E0 and E3±E0 in Fig. 3.18. They are obviously the intersections of liquidus surfaces and correspond to the `creeks' in the map described in the Section 3.3.3. The reflected triangle in the figure can be divide into three regions of AE1E0E2, BE3E0E1 and CE2E0E3, which are named the primary phase fields of A, B and C, respectively. Once the primary crystal starts to precipitate, the liquid composition changes to one with a lowering A concentration at a constant ratio of B and C along path x±y until it reaches another liquidus surface, namely the boundary line E1±E0. While the liquid composition is cooled along path x±y, it is (singly) saturated with B and the composition x should be positioned on the tie-line A± y in the reflected plane of the figure. After the liquid composition reached another liquidus surface, it varies along the boundary line E1±E0, increasing C concentration and coprecipitating the solid phases A and B. The secondary precipitate B can be quoted as a secondary crystal, but may precipitate as a two-phase secondary microconstituent composed of A and B. Thus the liquid composition during solidification along E1±E0 is doubly saturated with A and B. Also, the composition X must be kept inside the triangle ABy surrounded by three tie-lines. This is called a three-phase triangle indicating that three phases coexist throughout the triangle keeping every chemical potential constant. Finally, it reaches the eutectic composition at a certain temperature, namely eutectic point, E0. Here, the liquid is saturated with three solid phases, and the degree of freedom becomes zero. This situation is the so-called four-phase equilibria, and the three-phase tertiary microconstituent composed of A, B and
Phase diagrams and phase transformations
3.18 Phase diagram for the ternary eutectic system A-B-C.
97
98
Fundamentals of metallurgy Cooling of liquid Evolution of heat by B solidification Evolution of heat by A + B solidification Eutectic isotherm Cooling of solid
T
L
L+B L+A+B L+A+B+C A+B+C
t
3.19 Cooling curve for the alloy shown in Fig. 3.18.
C is precipitated until the liquid phase diminishes. The cooling curve is shown in Fig. 3.19. For most of the alloys, solid phases have some solubility of the other components. During the precipitation of primary crystals, their composition in equilibrium with the liquid can be given by the tie-line at any temperature. This is demonstrated in Fig. 3.20. After the liquid becomes saturated with primary and secondary crystals, their compositions are determined by the extremities of the three-phase region as shown in Fig. 3.21. The fraction of their amounts can be given by applying the lever rule to the three-phase triangle. In the practical solidification processes of a system with more than two components, however, equilibrium solidification is exceptional and solidus curves or surfaces will be depressed due to the non-uniformity of the solid Average alloy composition
Equilibrium liquid composition
3.20 Solid composition during primary solidification.
Equilibrium solid composition
Phase diagrams and phase transformations
99
Average alloy composition
a b
Three-phase triangle (a + b + L)
3.21 Solid composition during secondary solidification.
phase. Accordingly, the average composition of the solid inside will lie below the equilibrium curves or surfaces, although the equilibrium solidus lines or curves still demonstrate the composition in equilibrium with the liquid phase (Fig. 3.22). Thus, when considering the average composition of solid phases in ternary systems, apparent solidus surfaces will be depressed and the three-phase regions will also be enlarged. The eutectic temperature will not be affected, but the apparent tie-line will be lengthened showing the average compositions of primary and secondary microconstituents in a non-equilibrium eutectic microstructure. Although phase relations in practical systems may not follow equilibria as mentioned, such alloys can be treated as in the equilibrium state, and a number of phase transformations can be described in ternary systems. Four-phase equilibrium is one of the distinctive cases. The first representative case is a decomposition of a single phase upon cooling to form three new phases, which was already pointed out in Fig. 3.18, such eutectic and eutectoid reactions. L$000b‡ ‡ $ ‡ ‡000e
100
Fundamentals of metallurgy
Liquid composition
Tl
L
a
T
Equilibrium solid composition
b
Average solid composition
A
b
B
a XB
3.22 Non-equilibrium solidification for the binary eutectic system A-B.
3.23 Phase diagram for the ternary system A-B-C based on two eutectic and one peritectic binaries.4
Phase diagrams and phase transformations
101
3.24 Phase diagram for the ideal ternary peritectic system A-B-C.4
Another type of four-phase equilibrium is a decomposition of two phases on cooling to form two new phases, which can be explained by a typical phase diagram shown in Fig. 3.23.4 The system is composed of one peritectic and two eutectic binaries. Two boundary lines (alterative and crystallization curves of L ‡ ! and L ! ‡ ), p1 and p2, descend from each binary with increasing amount of the third component and meet somewhere in the middle, where the reaction, L ‡ ! ‡ , occurs at a constant temperature until the phase diminishes. During this stage, four-phase equilibria are attained. Thereafter, three-phase equilibria among L, and will be kept on cooling. One more case is shown in Fig. 3.24.4 When two alterative curves forming phase, p3 and p4 …L ‡ ! and L ‡ ! †, ascend from binaries with increasing amounts of the third component and meet as shown in the figure, where the reaction, L ‡ ‡ ! , occurs at a constant temperature.
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3.5
Examples of solidification behaviour from a phase diagram perspective
Various metals are often refined by oxidation of impurities followed by their removal into slag phases. In steel refining treatments, several impurities, such as phosphorus, silicon, carbon, etc., are removed during oxygen blowing processes. As can be seen in the Ellingham diagram, oxidation refining of silicon is hopeless because it will be preferentially oxidized to most of the impurity elements, such as iron, titanium, etc. On the other hand, the solubilities of most impurity elements in solid silicon are extremely low. Figure 3.25 shows the Al-Si binary phase diagram5 and the solubility curve of aluminum in solid silicon is reported as shown in Fig. 3.26.6 Although there are discrepancies among the reported data, the solubility of aluminium in solid silicon can be found to be lower than 100 ppmw at 1000K. For example, when 74 mass% Al-Si molten alloy was cooled from above liquidus temperature, solid silicon with only 100 ppmw Al starts to precipitate at 1000K, which means that 99.995% of aluminium was excluded by solidification at the initial stage. This principle of solidification refining makes the ultra high purification of silicon for semiconductors easier. Although some elements, such as phosphorus and boron, have high solubility in solid silicon, most impurities can
3.25 Phase diagram for the Al-Si binary system.
Phase diagrams and phase transformations
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3.26 Solubility of aluminium in solid silicon reported by several researchers.
be removed by the one-directional solidification refining due to their small segregation coefficients, which are tabulated in Table 3.1. Herewith, an example of solidification behaviour of a ternary silicon-based alloy is considered. Since solubilities of most impurities in solid silicon are negligibly small, they tend to concentrate at the grain boundaries of primary crystals of silicon almost free from such impurities. By making use of this tendency, a new refining process10 has been proposed, combined with the acid leaching procedure. Although efficiency of impurity removal strongly depends on its segregation coefficient, pure silicon grains remain after condensed impurities at the grain boundary are washed away by acid dissolution. In order to promote the selective dissolution of the grain boundaries by leaching process, a Table 3.1 Segregation coefficients of impurities in silicon Impurity
Segregation coefficient
Impurity
Segregation coefficient
B P C Al
8.00 0002 10ÿ1 3.50 0002 10ÿ1 5.00 0002 10ÿ2 2.80 0002 10ÿ3
Fe Ti Cu
6.40 0002 10ÿ6 2.00 0002 10ÿ6 8.00 0002 10ÿ4
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leachate must corrode the boundary phase, but its amount may not be enough in the case of silicon. Hence, the addition of an acid soluble element which forms intermetallic compounds or eutectic microstructure with silicon might be effective. One of the promising elements to form such phases is calcium as shown in the following paragraph. Recently, a new mass production process of solar grade silicon has been developed in order to solve the rapidly increasing demand of solar cells. Metallurgical grade silicon (>98%) is selected as a starting material and the final purity of solar grade silicon should be higher than 6N through some metallurgical refining treatments. Among others, iron is one of the most harmful elements for the solar grade silicon since it shortens the lifetime and drastically lowers the efficiency of solar cells. As mentioned previously, however, iron is less favourable to be oxidized than silicon, and oxidation refining is not suitable as well as vacuum refining for high vapour pressure species. Hence, the only effective way is solidification refining using a segregation coefficient as low as 10ÿ6.11 However, another pretreatment at the stage of metallurgical grade silicon may be helpful for reducing the solidification refining cost. Accordingly, iron removal from metallurgical grade silicon by acid leaching was investigated.12 Figure 3.27 shows the experimental results in which Si-Ca-Fe alloys with various compositions were subjected to acid leaching procedure using aqua regia. Calcium was added to form an acid soluble grain boundary. As seen in the figure, there seems to be an optimum ratio of calcium to iron content. Optical images of Si-Ca-Fe alloys of two different compositions before and after acid leaching are shown in Figs 3.28(a), (c) and 3.29(a), (c). The sample shown in Figs 3.28(a), (b) and (c) has higher calcium to iron ratio and calcium silicide, CaSi2, seems to have precipitated as a secondary phase during cooling, after silicon precipitated as a primary phase. Iron silicide phase, FeSi2, was considered to exist as a part of the microstructure
3.27 The relationship between ratio of calcium to iron content and removal ratio of iron.
3.28 (a) Optical image of Si-Ca-Fe alloy (Si-8.64%Ca-0.756%Fe, before acid leaching). (b) Microstructure of Si-Ca-Fe alloy. (c) Optical image of Si-Ca-Fe alloy (Si-8.64%Ca-0.756%Fe, 5 min. in aqua regia).
3.29 (a) Optical image of Si-Ca-Fe alloy (Si-0.929%Ca-1.21%Fe, before acid leaching). (b) Microstructure of Si-Ca-Fe alloy. (c) Optical image of Si-Ca-Fe alloy (Si-0.929%Ca-1.21%Fe, 5 min. in aqua regia).
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107
3.30 Phase diagram for the Ca-Si-Fe system.
of the ternary eutectic of Si-CaSi2-FeSi2 which is easily soluble in aqua regia. On the other hand, when the ratio of calcium to iron is lower, FeSi2 precipitates as a secondary phase and most of the FeSi2 precipitates remained after acid leaching procedure. Figure 3.29(c) shows the optical image after acid leaching, where a large piece of undissolved FeSi2 was observed on the grain boundary. These phenomena can be confirmed by following the solidification behaviour from phase diagram viewpoint. Since reliable data for the Si-Ca-Fe system were not available, the ternary phase diagram was drawn as shown in Fig. 3.30 using `Thermo-Calc', a thermodynamic database and software. From the diagram, the secondary phase should be CaSi2 when XCa/XFe > 6.5 and FeSi2 when XCa/XFe < 6.5. This coincides with the experimental results that the optimum composition for iron removal is XCa/XFe 0019 6±9 as shown in Fig. 3.27. Accordingly, controlling the alloy composition to ensure the precipitation of CaSi2 secondary phase during cooling becomes the key for the removal of iron from silicon by acid leaching procedure. As can be seen in the present example, comprehension of phase diagram in view of solidification behaviour is helpful in developing a new technology in materials science.
3.6
Conclusions
In this chapter, fundamentals of chemical potential diagrams and phase diagrams were briefly reviewed and solidification behaviour was discussed through phase
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diagram perspective by showing simplified examples together with recent experimental work. Both diagrams are helpful in predicting a final product and in verifying the processing conditions. We often treat the systems in non-equilibrium states, such as supercooling, appearance of metastable phases, non-uniformity due to slow diffusion or slow reaction processes, etc. Such conditions are preferable or artificially created in producing various new materials, such as low TC superconductors, semiconductor compounds, new glasses and ceramics, etc. They are developed with profound consideration on relations among the phases whether they are stable or metastable. Henceforth, fundamental studies on phase diagrams should be continued for the future development of materials science and engineering.
3.7
References
1. Phase Diagram of Binary Iron Alloys, H. Okamoto (ed.) (1993), ASM International, Materials Park, OH. 2. Physical Chemistry of High Temperature Technology, E.T. Turkdogan (1980), Academic Press, New York, NY. 3. Phase Diagram for Ceramists, vol. I, E.M. Levin, C.R. Robbins and H.F. McMurdie (eds) (1964), American Ceramic Society, Westerville, OH. 4. Phase Diagrams in Metallurgy, F.N. Rhines (1956), McGraw-Hill Book Co., Columbus, OH. 5. Binary Alloy Phase Diagrams, T.B. Massalski and H. Okamoto (eds) (1990), ASM International, Materials Park, OH. 6. T. Yoshikawa and K. Morita (2003), J. Electrochem. Soc., 150, G465. 7. R.C. Miller and A. Savage (1956), J. Appl. Phys., 27, 1430. 8. D. Navon and V. Chernyshov (1957), J. Appl. Phys., 28, 823. 9. V.N. Lozvskii and A.I. Udyanskaya (1968), Izv. Akad. Nauk SSSR, Neorg. Mater., 4, 1174. 10. T.L. Chu and S.S. Chu (1983), J. Electrochem. Soc., 130, 455. 11. R.H. Hopkins and J. Rothatgi (1986), J. Cryst. Growth, 73, 67. 12. T. Sakata, T. Miki and K. Morita (2002), J. Japan Inst. Metals, 66, 459.
4
Measurement and estimation of physical properties of metals at high temperatures K C M I L L S , Imperial College, London, UK
4.1
Introduction
4.1.1 The need for thermo-physical property data Surveys of the requirements of industry show that there is an urgent need for reliable data for the thermo-physical properties of the materials involved in high-temperature processes (metals, slags and refractories). This need arises from the fact that thermo-physical property data have proved extremely useful in improving both process control and product quality. Physical property data are beneficial in two ways: 1. 2.
In the direct solution of industrial problems. As input data in the mathematical modelling of processes.
One example of the direct use of physical property data is in the case of `Variable weld penetration' in gas tungsten arc (GTA sometimes known as TIG) welding of steels. Some applications require a large number of welds (e.g. heat exchangers). In these cases the welding conditions providing good weld penetration are established in preliminary trials. However, sometimes another batch of the materials (fully matching the materials specification) had to be used and the resulting welds were very shallow and consequently, weak. Compositional differences between the two batches were very small. Subsequent work1±3 showed that: (i) good weld penetration was associated with lower surface tensions ( ) and a positive temperature dependence (d /dT) and a sulphur content of >50 ppm of the steel; and (ii) shallow weld penetration was associated with a high surface tension, a negative (d /dT) and a sulphur content of 50 ppm) and poor penetration (upper curve, S content < 30 ppm).
namely, buoyancy, Lorenz, aerodynamic drag and thermo-capillary (Marangoni) forces.1±3 However, the Marangoni forces are dominant because of the huge temperature gradients across the surface of the weld pool. In steels with S < 30 ppm (d /dT) is negative (Fig. 4.2) and thermo-capillary flow (high to low ) is radially-outward taking hot liquid to the periphery of the weld where melt-back produces a shallow weld (Fig. 4.3). For steels containing more than 50 ppm S the thermo-capillary forces are inward and the hot liquid is forced down the weld and melt-back occurs in the bottom of the pool giving a deep weld. Surface tension measurements played an important part in solving this problem. Mathematical modelling has proved a valuable tool in improving process control and product quality. There are several types of mathematical models, e.g. those based on thermodynamics, kinetics and heat and fluid flow of the process. In this review we are mainly concerned in the modelling of the heat and fluid flow in the process. Defects in a casting can result in the scrapping of a casting. The cost of scrapping has been estimated to be greater than 2 billion US$ per annum. Mathematical models of the heat and fluid flow have been developed to predict the locations of defects. It has been shown that the accurate prediction of defects requires accurate thermo-physical data for the alloy being cast.5 Similar models have been developed for the prediction of micro-structure (e.g. dendrite arm spacing) and Fig. 4.4 shows the sensitivity of local solidification time to changes in the various properties6 of aluminium alloys. It can be seen that it is particularly sensitive to the thermal conductivity value used. The properties needed for modelling fluid flow and heat and mass transfer are as follows.
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111
4.2 The effect of sulphur content on (a) surface tension ( ) at 1923K and (b) its temperature dependence (d /dT).4
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4.3 Schematic diagram illustrating the mechanism for variable weld penetration showing fluid flow in the weld pool for a steel with S < 30 ppm on the left and for a steel with S content >50 ppm on the right.1,2,3
4.4 Sensitivity of local solidification time to thermal conductivity, (k) the parameter (density 0002 heat capacity, (Cp.001a)) latent heat (denoted 0015)6 and emissivity (000f).
Measurement and estimation of physical properties of metals Heat flow
Fluid flow
Heat capacity, enthalpy Density Thermal conductivity/diffusivity Viscosity (Electrical conductivity) Surface/Interfacial Emissivity tension Optical properties (slags and glasses)
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Mass transfer Diffusion coefficient
These data are needed for alloys, slags, glasses and fluxes. The measurements are time-consuming and require considerable expertise. Consequently, it is an enormous task to provide all the thermo-physical data for all materials involved in the various industrial processes. Thus, considerable effort has been devoted to the estimation of physical properties. Usually these estimations are based on the chemical composition since this is usually available on a routine basis.
4.2
Factors affecting physical properties and their measurement
4.2.1 Structure Some physical properties are very dependent upon the structure of the sample. The effect of structure is greatest in the case of viscosity and, in fact, viscosity measurements have been used in conjunction with other data to determine the structure of melts. The effect of structure on property values is in the following hierarchy: viscosity > thermal conductivity > electrical conductivity > density (small) > enthalpy (usually has little effect). Structural effects tend to be much larger for slags and glasses than for metals. Both X-ray and neutron diffraction have proved very useful in determining the structure of solids. In crystalline solids the atoms have well-defined positions. The array of atoms interferes with the passage of X-rays which are scattered in all directions except those predicted from Braggs law: 0015 ˆ 2d sin 0012
(4.1)
where 0015 is the wavelength, d is the distance between two layers of crystals and 0012 is the angle of diffraction. X-ray diffraction patterns for a gas show a constant scattering intensity with no maxima due to the random distribution of atoms. However, X-ray diffraction patterns for liquids exhibit a few maxima and minima (Fig. 4.5) indicating that atoms are randomly distributed in an approximately close-packed array (short range order) but have little long range order due to the thermal excitation and motion.7 The principal parameters used to describe structure are derived from X-ray and neutron diffraction data, namely: · Pair distribution factor (g(r)) where r is the radius and g(r) is the probability of finding another atom at a certain position (Fig. 4.6a); in real liquids g(r) is
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4.5 X-ray diffraction pattern for a liquid.7
4.6 (a) Pair distribution function (g(r)) of a metal near its melting point and (b) radial distribution function (rdf): the number of nearest neighbours can be determined from the area of the hatched region.
Measurement and estimation of physical properties of metals
115
4.7 Schematic drawings of (a) tetrahedral arrangement of Si-O bonds and (b) silicate chain with bridging O (Oo) shown in black, non-bridging O (O-) as hatched and tetrahedrally-coordinated cations as open.
affected by attraction and repulsion forces (of other atoms). · Radial distribution factor (rdf) can be calculated from g(r); the hatched area in Fig. 4.6b is a measure of the number of nearest neighbours (i.e. coordination number). Other parameters used to describe the structure are summarised by Iida and Guthrie.7 The situation is much more complicated in alloys since for a binary alloy three distribution factors are needed, namely, (g(r))11, (g(r))22 and (g(r))12. Structure of slags and glasses Slags and glasses are polymers in the form of chains, rings etc made up of SiO44ÿ tetrahedral units. They have the following characteristics:8±11 · Each Si (with a valence of 4) is surrounded (tetrahedrally) by 4 O ions (with a valence of 2) each connecting to 2Si ions (Fig. 4.7). · In SiO2 these SiO44ÿ tetrahedra are connected in a three-dimensional polymerised structure (Fig. 4.8) and the oxygens are predominantly bridging oxygens (denoted Oo). · Cations such as Ca2+, Mg2+ etc. tend to break up the Si-O bonds and depolymerise the melt by forming non-bridging oxygens (NBO denoted Oÿ) and free oxygens (denoted O2ÿ), i.e. not bound to Si. · Other cations such as Al3+, P5+, Ti4+ can fit into the Si polymeric chain but need to maintain charge balance, e.g. if an Al3+ is incorporated into a Si4+ chain it must have a Na+ (or half* of a Ca2+) sitting near the Al3+ to maintain local charge balance. · Smaller cations such as Mg2+ tend to give a wider distribution of chain lengths than larger cations such as Ba2+. * The Ca2+ would charge balance two neighbouring Al3+ ions.
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4.8 The effect of convective flows (shown by differences between values obtained in micro-gravity (0g) and on earth (1g)) for (a) thermal conductivity19 and (b) diffusion coefficient.20
· Cations such as Fe3+ in small concentrations can act as network breakers but in higher concentrations can be incorporated into the chain in a similar way to Al3+. · The degree of polymerisation can be expressed in terms of the numbers of bridging (NOo) non-bridging (NOÿ) and free-oxygens (NO2ÿ). However, the degree of de-polymerisation is frequently represented by parameters such as the ratio of {NBO/(the number of O atoms in tetragonal coordination)} which is denoted (NBO/T)8±11 or the optical basicity11 (see Appendix A). · The structure of melts (both slags and metals) can be represented using thermodynamic quantities (e.g. excess free energy12,13) since thermodynamics provides a description of bond strengths. · B3+ has threefold coordination in borosilicates but the structure of borosilicates (used as enamels) is a complex mixture of 3- and 4coordination.14,15 · The structure of glasses are very similar to those of liquid slags.16
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· Many metallurgical slags have a high basicity (e.g. CaO/SiO2 > 2) and consequently the Si ions are predominantly in the form of monomers, i.e. completely de-polymerised.
Methods of determining structure The various methods of determining structure are summarised in Table 4.1. For further details the reader should consult other sources.7,8,9,17
4.2.2 Surface and interfacial properties It should be noted that surface and interfacial tensions are interfacial properties and not bulk properties. Consequently, they are dependent upon the composition and structure of the interface. Certain materials are surface active and have a dramatic effect on the surface tension and its temperature dependence as can be seen in Fig. 4.2. These surface-active materials have low surface tensions and have much larger concentrations in the surface layer than in the bulk. For instance, sulphur concentrations in the surface layers of steels were found to be 100x that in the bulk. Thus ppm levels of strongly surface active components can have a dramatic effect on surface and interfacial tensions. In metals the hierarchy of surface activity is Group VI > Group V > Group IV and within any group surface activity increases with increasing molecular weight, e.g. Te > Se > S = O. In slags and glasses it has been reported that B2O3, P2O5, Fe2O3, Cr2O3, Na2O and CaF2 are surface-active.18
4.2.3 Convection Measurements of both thermal conductivity and diffusivity and diffusion coefficients for liquids are greatly affected by convection.19,20 In practice it is difficult to eliminate convection. The following techniques have been used to eradicate (or minimise) convection: · Use of transient techniques where the measurements are made very rapidly such that the experiment is completed (in usually 106 Kminÿ1) and thus any changes in enthalpy arising from solid state transitions will tend to be `smeared' over a temperature range. In nickel-based superalloys the coarsening of the phase is followed closely by the dissolution of the 0 phase (Fig. 4.9). Under these circumstances it is difficult to determine to obtain an equilibrium value for the Cp of the 0 phase.23 In dynamic methods (DSC) the enthalpy of fusion is manifested as an apparent enhanced Cp value (Cpapp) (Fig. 4.9) The enthalpy of fusion can be obtained by integrating the Cp±T curve and the fraction solid at any specific temperature can be derived from the area up to that temperature divided by the
4.9 Heat capacity values for Ni-based superalloy IN 718 as a function of temperature: the dotted sections should be regarded as apparent Cp values.23
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area corresponding to the enthalpy of fusion. However, these Cpapp values are not the true values of the heat capacity and should not be used for calculating, for instance, thermal conductivity (k) values from thermal diffusivity (a) measurements by the relation, k ˆ a.Cp.001a.23 Estimated values (see Section 4.7.1) can be used for this purpose without much loss of accuracy. This would also apply to first order solid state transformations.
4.2.5 Measurements in the mushy zone Transient techniques are the most reliable way of minimising convective contributions to thermal conductivity (k) measurements on liquids. In these techniques a short pulse of energy is delivered and the change in temperature as a function of time is monitored. However, when these techniques are applied to samples containing both solid and liquid (`mushy') some of the energy is not conducted but is used to convert solid into liquid. This leads to erroneous values of the conductivity in the mushy range. It has been shown that the Wiedemann± Franz±Lorenz (WFL) rule (k ˆ 2.445 0002 10ÿ8 (T/R) Wmÿ1Kÿ1 where T is in K) linking thermal conductivity (k) and electrical resistivity (R) works well around the melting range and consequently the best way of deriving thermal conductivities is via electrical resistivity measurements in the mushy zone. Alternatively, the following formula can be used: kT ˆ fsksTsol ‡ (1 ÿ fs)kl Tliq where fs ˆ fraction solid in the sample, ksTsol is the thermal conductivity of the solid phase at the solidus temperature and kl Tliq is the thermal conductivity of the liquid at the liquidus temperature.23
4.2.6 Commercial materials Measurements on commercial materials are more difficult and are subject to greater uncertainty than those carried out on pure metals or synthetic slags for the following reasons: · Commercial materials are sometimes inhomogeneous and the compositional differences may be significant when small specimens are used. · Commercial materials frequently contain reactive elements (such as Al in alloys) which can react with containers, the atmosphere or soluble elements (e.g. oxygen) to form Al2O3 inclusions (which increase the viscosity), or surface films or oxide skins which affect the experiment.
4.3
Measurements and problems
4.3.1 Reactivity of sample The paucity of reliable values of thermo-physical properties at high temperatures is a reflection of the measurement difficulties experienced at high temperatures.
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121
These difficulties can be expressed in terms of the unofficial laws of high temperatures, namely: 1. 2. 3.
`At high temperatures everything reacts with everything else.' `They react very quickly and the situation gets worse (quicker) the higher the temperature.' `At high temperatures there are no electrical insulators.'
Consequently, in recent years there has been a rapid development of containerless methods and techniques since these minimise the contact between the sample and the container. These containerless methods encompass:29,30 · Levitation, including electro-magnetic (EML), electro-static (ESL), aerodynamic (ADL) and ultra-sonic (USL) levitation. · Measurements carried out in micro-gravity (0 g) using space flights, parabolic flights and drop towers. · Pendent drop where the tip of a sample rod is melted with an electron beam or other source of energy (consequently the liquid is only in contact with the remainder of the sample). · Exploding wire techniques where measurements are carried out between room temperature and up to 5000K in a period of ms and values are obtained for both the solid and liquid phases before the sample explodes. Other techniques used for high temperature measurements should be (i) robust; (ii) involve minimum contact between sample and container; and (iii) have a minimum of moving parts. Typical examples are: · Laser pulse method for measuring the thermal diffusivity of solid and liquid samples. · Surface laser light scattering (SLLS) where measurements of viscosity and surface tension have been derived from the reflected beam hitting the surface of a liquid. · The draining crucible method where the viscosity, surface tension and density have been derived from the drainage rates through an orifice in the base of a crucible. When using a technique which involves some contact between sample and container, the selection of container materials is an important issue in high temperature measurements. It is customary to fabricate containers and probes from oxides when making measurements on metals. Similarly, metals are used as containers when making measurements on liquid oxides (slags, glasses and fluxes). · The usual oxides used for containing molten metals are those with high thermodynamic stability, namely Al2O3, MgO, ZrO2 and CaO. · The usual metals used for containing molten slags and glasses are Pt (for
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oxidising or neutral atmospheres) Mo, W, Ta and Fe or Ni (with reducing or neutral atmospheres). · Graphite (carbon) and carbide (e.g. SiC) containers are sometimes used where metals have a low solubility for carbon (e.g. copper and aluminium) but should not be used for metals with a high solubility for carbon (e.g. Fe, Co and Ni). Carbon has also been used for containing slags and glasses but may react with some components (e.g. PbO) of the sample. · Boron nitride has also been used for holding metals but tends to oxidise to form an oxy-nitride at high temperatures and should only be considered for use where low partial pressures of oxygen in the apparatus can be guaranteed. Other nitrides have also been used where an oxide has been thought unsuitable.
4.3.2 Different methods for metals and slags and glasses Property values for slags are sometimes very different from those for alloys. Consequently, different methods are needed for measurements on metals and alloys to those used for slags. For instance, the viscosities of slags lies in the range 100±104 mPas and the rotating cylinder method is frequently used for measurements. In contrast, the viscosities of metals and alloys lie in the range 1±5 mPas where capillary and oscillating viscometers are the most suitable techniques.
4.4
Fluid flow properties
The principal properties involved in fluid flow are density (001a), viscosity (0011) and surface tension ( ). It should be noted that their temperature coefficients are equally important since (d001a/dT) determines the buoyancy-driven convection and (d /dT) determines the strength of thermo-capillary flow.
4.4.1 Density (001a) and thermal expansion coefficient (000b, ) The density (001a) and molar volume (V) are linked through the relation: (001aV) ˆ M
(4.2)
where M is the molecular mass. The linear (000b) and volume ( ) thermal expansion coefficients are defined through equations (4.3) and (4.4), respectively, where L and V are the length and volume. (Kÿ1) ˆ (1/L)(dL/dT) ÿ1
(K ) ˆ (1/V)(dV/dT)
(4.3) (4.4)
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Methods There are several well-established methods for measuring density; these are listed below and further details are given elsewhere.7,30±34 The experimental uncertainty is usually 00062% or less. · Dilatometry is the established method for determining the thermal expansion coefficient from length changes in the sample; the `piston method' has recently been used to determine the densities of liquid metals and is particularly successful where the metal readily forms an oxide which tends to seal the apparatus.35 · Sessile and large drop methods36 have provided more accurate values in recent years as a result of the introduction of software to fit the profile of the drop or the free surface. Some errors may arise from asymmetry of the drop if it is viewed from one direction only. · Pycnometry involves filling and weighing a vessel with known volume containing a liquid sample after cooling to room temperature.7,34,36 · The Archimedean method provides reliable results but corrections have to be made for the effect of the surface tension of the sample on the suspension wire; there are several ways of compensating for this effect.7 · The hydrostatic probe works on similar principles and records the apparent weight of a probe at different immersion depths in the melt.37 · The maximum bubble pressure method is principally used for measuring surface tension but densities can be obtained by measuring maximum bubble pressures at different immersion depths.7 Several new techniques have been introduced recently. · The exploding wire method24,25 has been used to determine densities and thermal expansion coefficients for both solid and liquid states. Its principal drawbacks are (i) when studying materials exhibiting solid state transitions (see Section 4.2.4) and (ii) in the measurement of temperature. However, it is capable of obtaining measurements up to 5000K though experimental uncertainties for the thermal expansion values of 000610% have been reported.25 · The levitated drop method views a drop horizontally and vertically and spherical images of the drop are selected and the volume of a drop (of known mass) determined; since the drops are only a few mm in diameter errors in edge detection can lead to experimental uncertainty but this is usually overcome by processing a large number of images.38 · In the draining crucible method the rate of draining through an orifice is determined and the density is determined from a hydrodynamic analysis of the data.39 In general, density measurements have sufficient accuracy for their subsequent use in mathematical models. The worst problems seem to be those caused by the
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formation of tenacious oxide films or `skins' such as those encountered in measurements on molten alloys containing significant amounts of Al and Mg. Density data for metals, alloys slags, glasses Sources of density data are given in Table 4.2 and density data for metallic elements are given in Table 4.3.
4.4.2 Viscosity (0011) There is a considerable difference between the viscosities of slags and glasses (0011 ˆ 100 to 10000 mPas) and those for metals and alloys (0011 ˆ 0.5 to 10 mPas). Consequently, different techniques are required to measure the viscosities of metals7 to those used for slags and glasses. Slags Methods Several established methods are available: · The concentric cylinder method exists in two forms in which the torque is measured when (i) the outer cylinder (crucible) is rotated and (ii) the inner cylinder (bob) is rotated. Although the rotating crucible method7,31±33 is capable of measuring to lower viscosities, the rotating bob method7,33 is usually preferred because it is easier to align and operate. · In the falling ball method the time for a sphere to either fall or be dragged through the liquid sample is determined.33 It is difficult to apply at high temperatures due to the limited length of the uniform hot zone under these conditions. Table 4.2 Sources of property data for metals, alloys, slags and glasses Property
Metals and alloys
Slags, glasses and fluxes
Density, 001a Viscosity, 0011 Surface tension, Cp, (HT ÿ H298) 0001fus H Thermal conductivity, k diffusivity, a Electrical conductivity, R Emissivity and optical constants Diffusion coefficients, D
7: 111, 112, 113, 114, 32, 34 7: 111, 116, 112 7: 111, 58: 114 117: 114, 118
33: 111, 18: 115 33: 111, 18: 115 33: 111, 18: 115
111, 80, 119, 120
33: 111, 18: 115
7, 111, 119, 120
33: 111, 18: 115
111, 119, 120
33: 111, 18
7, 111: 121, 122
33: 111
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125
· In the oscillating plate method the amplitudes of an oscillating plate are determined in the melt and in air.7 The apparatus constants are determined by calibration with liquids of known viscosity. Data for the product (gq) are obtained, so it is necessary to know the density of the melt for the given temperature.
Recently, two methods have been reported which may be suitable for measurements on slags and glasses: · In the surface laser light scattering (SLLS) method `ripplons' are monitored.40±42 The surface of a melt may appear smooth but it is being continually deformed by thermal fluctuations of the molecules. Capillary waves (ripplons) have small amplitude (circa 1 nm) and a wavelength of circa 100 0016m which is dependent upon the frequency. Ripplon action depends upon surface tension for restoration and the kinematic viscosity (0017 ˆ 0011=001a) for oscillation damping. The spectrum of the ripplons is derived using a Fourier spectrum analyser allowing the surface tension and the viscosity to be determined. The method has been successfully used for measurements on liquid silicon42 and LiNbO3 up to 1750K.41 It has been estimated that it can be used for liquids with viscosities in the range 0.5±1000 mPas.40 · In the draining crucible method39 the rate of drainage through an orifice is determined and the viscosity is derived from a hydrodynamic analysis of the data. Although it has been applied to H2O and liquid Al it would appear that the method is ideal for carrying out measurements on liquid slags and glasses. Problems in measurements on slags and glasses The experimental uncertainties associated with viscosity measurements on slags is of the order of 000625% but these uncertainties can be reduced to 0006 90ë, Fig. 4.12b). At high temperatures `reactive wetting' is common. Consequently, in many applications wetting is promoted by reactions between the liquid and the substrate. Some typical applications making use of reactive wetting are highlighted below: · Good bonding (i.e. wetting) is needed between the fibres and the metal is needed in metal-matrix composites, this is usually achieved through introducing a reactive component (e.g. Mg) into the metal which reacts with the fibre. · To ensure good wetting in metal±ceramic (e.g. Si3N4) joining, Ti is added to the braze to react with the ceramic (to form a silicide).
It is very difficult to measure interfacial tensions involving solid phases. However, Young's equation, which can be derived from Fig. 4.12, allows us to determine the difference between the surface and interfacial tensions of the solid phase where s, l and g denote the solid, liquid and gas phases, respectively:
lg cos 0012 ˆ ( sg ÿ sl)
(4.5)
The following terms are useful in understanding interfacial phenomena affecting
4.12 Schematic diagrams showing (a) a liquid wetting a solid inclusion and (b) non-wetting liquid on a solid inclusion (I = solid in this diagram).
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the removal of inclusions (I) from liquid metal (M) by gas bubbling, the spreading of one liquid over another, etc. · The work of adhesion (WA) is a measure of the energy change (per unit contact area) in separating two media at the solid±liquid interface. WA ˆ lg ‡ sg ÿ ls ˆ lg(1 ‡ cos 0012)
(4.6)
· The spreading coefficient (S*) is a measure of the tendency of a non-reacting liquid to spread across a solid surface. S* ˆ sg ÿ ls ÿ lg ˆ lg(cos 0012 ÿ 1)
(4.7)
· The flotation coefficient (0001) provides a measure of the ease of removing solid particles (e.g. inclusions) by gas bubbling; the best conditions are obtained when 0001 is both positive and has a high value.70 0001 ˆ IM ‡ MG ÿ IG ˆ MG(1 ÿ cos 0012)
(4.8)
Interfacial tension ( MS) Many metal-producing processes use a molten slag to protect the metal from oxidation and to promote the removal of undesirable impurities from the metal (e.g. S, P and non-metallic inclusions). The metal±slag interfacial is important since a low interfacial tension will promote the formation of emulsions and foams which provide very fast refining reactions (because of the huge surface area/mass ratio) and high productivity. However, low interfacial tension also encourages slag entrapment in the metal which is a major problem in the continuous casting of steel.71 The interfacial tension is linked to the surface tensions of the metal (M) and slag phases (S) through the Good±Girafalco equation, where ' is an interaction parameter.
MS= M + S - '( M. S)0.5
(4.9)
The surface tension of the metal is usually the biggest term in this equation (e.g. for steel M ˆ 1700±1400 mNmÿ1 depending on S content compared with S ˆ 450 0006 50 mNmÿ1 for the slag). Thus the S content of the steel has a marked effect on the interfacial tension. Methods The following methods have been used to determine the interfacial tensions of metal±slag systems ( MS): · The most widely-used method is the X-ray sessile drop method18,72,73 in which a metal drop is placed in a crucible of molten slag. The crucible must be transparent to X-rays and Al2O3 and MgO are frequently used. The interfacial tension is determined using software which derives the parameters
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giving the best fit with the measured profile of the drop. The X-ray pendent drop could also be used for measurements at high temperatures. · Another form of the sessile drop method (which does not require X-rays) has also been reported.18,74,73 A crucible is filled with molten metal and a drop of slag is placed on the surface of the metal and forms a sessile drop. The drop is photographed and the contact angle derived. · The maximum drop pressure method (MDP)73 is similar to the MBP method and consists of a crucible containing one liquid into a second vessel containing the other liquid which is joined to the first liquid via a capillary. The second vessel is evacuated in a controlled manner and the liquid from the crucible rises up the capillary and forms a drop which, subsequently,
4.13 The effects of time (a) on interfacial tension and (b) on the mass transfer of Al from steel (to the slag).75,76
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detaches. The maximum pressure is determined and used to calculate the interfacial tension. This method has not been widely used for measurements at high temperatures. Results Slag±metal reactions are ionic in nature (e.g. equation 4.10): Smetal ‡ O2ÿslag ˆ Ometal ‡ S2ÿslag
(4.10)
Rapid mass transfer between the metal and slag has been found to be associated with low interfacial tension (Fig. 4.13) and when the rate of mass transfer slowed down the interfacial tension was found to increase to a high value once more.75,76 It has been suggested that that the decrease in interfacial tension occurs when the rate of oxygen transfer from slag to metal exceeds a critical value. Similar phenomena have been observed in organic systems77 and it was found that the decrease in interfacial tension did not occur when the volumes of the two liquids were similar. Consequently, there is a possibility that the observed phenomenon is due to the fact that the volume of the metal is much lower than that of the slag in the X-ray sessile drop method.
4.5
Properties related to heat transfer
4.5.1 Heat capacity (Cp), enthalpy (HT ÿ H298) The temperature dependence of the heat capacity is usually expressed in the form: Cp ˆ a ‡ bT ÿ c/T2
(4.11)
The enthalpy at a specific temperature, T, relative to 298K, can be derived by integrating Cp between 298 and T. Z T …HT ÿ H298 † ˆ CpdT ˆ a…T ÿ 298† 298
‡ …b=2†…T2 ÿ 2982 † ‡ …c=T ÿ c=298†
…4:12†
which can be expressed as a power series: …HT ÿ H298 † ˆ aT ‡ …b=2†T2 ÿ …c=T† ÿ d
…4:13†
where d contains all 298K terms. Methods The following methods have been used to measure heat capacities and enthalpies: · Calorimetry is a well-established technique but is difficult and timeconsuming at higher temperatures. Detailed descriptions of the methods
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adopted have been reported elsewhere.78 Enthalpies have been studied extensively by drop calorimetry. In this technique a crucible containing the sample (of known mass and temperature) is dropped into a copper or silver block and the temperature rise of the block (typically 1±5K) is measured very accurately. In levitated drop calorimetry the sample is levitated by EML, ESL, ADL or USL and in the case of high melting metals (e.g. W) a laser heating can be used to augment the EM heating. The experimental uncertainty in enthalpy measurements is usually around 00061%. · In recent years the development of commercial, differential scanning calorimeters (DSC) has resulted in the wide use of these instruments to measure to Cp and enthalpies. There are two types of DSC: (i) in which the temperature difference between the sample and reference pans are monitored (differential temperature scanning calorimeter (DTSC)) and (ii) in which the power required to keep the two pans at the same temperature (differential power scanning calorimeter (DPSC)). DPSC can be used to temperatures to 3. For optically thick conditions, kR can be calculated using equation 4.18 where n is the refractive index and 001b is the Stefan Boltzmann constant with a value of 5.67 0002 10ÿ8 Wmÿ2Kÿ4. kR ˆ 16001b.n2.T3/3000b*
(4.18)
However, it is difficult to calculate kR if it lies outside the optically thin region. The effective conductivity keff is given by equation 4.19. keff ˆ kc ‡ kR
(4.19)
where kc is the phonon (or lattice) conductivity. Radiation conductivity is diminished by the presence of crystallites in the sample which scatter the radiation (i.e. photons). Radiation conductivity is also decreased by the presence of transition metallic oxides such as FeO, NiO or Cr2O3 which increase the absorption coefficient significantly. For solids the
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absorption coefficient in equation 4.18 should be replaced by the extinction coefficient E ˆ * ‡ s* where s* is the scattering coefficient. Although, steady state methods can be used to calculate the effective conductivity of semi-transparent media such as glasses in the solid state it is essential that the sample be optically thick. Furthermore, it is necessary to measure the absorption coefficient and refractive index of the sample if the contributions from kR and kc need to be determined. However, transient techniques are preferred for measuring keff since contributions from kR tend to be much smaller. The thermal conductivities of slags and glasses are dependent upon the polymeric structure of the melt. The thermal resistance (1/km, i.e. reciprocal thermal conductivity at the melting point) has been shown to be a linear function of structural parameters (e.g. (NBO/T) or the corrected optical basicity).11,83 The thermal resistance was found to decrease as the polymerisation increased (or the conductivity increases as the chain length of the slag increases (i.e. SiO2) increases). Susa et al.83 showed that the thermal resistance terms were smaller for the Si-O bonds in the chain than those for ionic (NBO) bonds at the end of the chain (denoted as O-O bonds). There is also some evidence that a change in slope in the k-temperature plot for some solid slags may be due to the change of a glass into a supercooled liquid which occurs at the glass transition temperature. Thermal effusivity (e) is a measure of the ability of a body to exchange thermal energy with its surroundings (thermal impedance) and is given by the following equation and has units of Js0.5mÿ2Kÿ1.84±86 e ˆ (k.Cp.001a)0.5
(4.20)
It has been reported that thermal effusivity measurements carried out on a drop of liquid at ambient temperatures were free from convection contributions84 and this may provide a way in the future of obtaining thermal conductivity/ diffusivity values which are free from convective contributions. Methods for thermal conductivity and diffusivity measurement 1. Steady state techniques The conventional steady state techniques such as the concentric cylinder, parallel plate and axial and radial flow methods78 are all suitable for measuring thermal conductivities of metals in the solid state. However, they are not particularly suitable for measurements (i) in the liquid state because of the difficulties of minimising convection at high temperatures, and (ii) on semitransparent solid unless the samples are optically thick (see above). 2. Non-steady and transient techniques The laser pulse apparatus is manufactured commercially by several companies and is a very popular technique.78 The sample is a disc-shaped specimen 10±12 mm in
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diameter and usually 2±3 mm thick with parallel faces. For measurements on solids the sample is placed horizontally and a pulse of energy is directed onto the front face of the specimen. The temperature of the back face is monitored continuously (usually with an infra-red detector). The temperature±time curve goes through a maximum (0001Tmax) because of radiation losses and it is customary to determine the time (t0.5) needed for 0.5 (0001Tmax). The thermal diffusivity is derived from the relation a ˆ 0.1388L2/t0.5 where L is the thickness of the specimen. For measurements on liquids and semi-transparent media a metallic (e.g. Pt or C) disc is usually placed on the upper surface of the sample and the temperature transient can be measured by monitoring the temperature of the disc or by the temperature of the back face.87 Liquid samples are usually held in sapphire or silica crucibles88,89 and it is desirable to carry out the measurements very rapidly (< 1 sec) to avoid the initiation of convection. In the transient hot wire (THW or line source) method a current is applied to a fine wire (circa 0.1 mm diameter) of known length which acts as both a heating element and a resistance thermocouple.90,22,33,80 The wire is immersed in the melt and a current is applied then the temperature rise of the wire (0001T) (or strip) is measured continuously as a function of time. The thermal conductivity is derived from the reciprocal of the slope of the linear portion of the plot of 0001T versus ln (time). Convective contributions to kc can be detected as departures from the linear relation. The current should be applied for < 1 sec to avoid the establishment of buoyancy driven convection. Experiments have been carried out in 0 g using a drop tower to minimise convection.22 Thermo-capillary convection can be minimised by floating a lid on top of the test liquid.22 There is some evidence indicating that radiation conduction contributions in semitransparent liquids (and solids) are smaller in the THW method than in the laser pulse method. This is probably due to the much smaller surface area of the wire compared with that of the metal disc used in the laser pulse. When the method is applied to metals it is necessary to insulate the metallic probe from the melt and coatings of Al2O3 or other oxides are applied to the wire or strip. Recent work has shown that thermal conductivity values for liquid metals must be corrected for the thickness of the insulating coating on the probe.91 In the radial temperature wave (RTW) method,92,80 a modulated heat flux is applied along the centre of a cylindrical sample. The variations in temperature are monitored on the outside of the specimen. There is a phase lag between the input and output and this is related to the thermal diffusivity. Thermal diffusivities can be calculated from (i) the amplitude of temperature oscillations and (ii) from phase differences. Values calculated by the two methods are usually in good agreement.92 In the plane temperature wave (PTW) method93,80 the plane temperature waves are generated by bombarding the specimen with a harmonicallymodulated electron beam. PTWs are directed onto one face of a disc-shaped sample and the temperature transient is recorded on the other face. The method
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can be used either at constant temperature or in a dynamic mode (and heating rates of up to 1000K per second have been used) to derive thermal diffusivity values. For measurements on liquids only the central portion of the disc was allowed to melt and high heating rates ensure that measurements are carried out very quickly thereby avoiding the problem of buoyancy-driven convection.93 With temperature gradients across the free surface some contributions to the thermal diffusivity from thermo-capillary convection might be expected. Problems Solids · In metallic alloys phonons can be scattered by grain boundaries, precipitated particles, etc. and consequently, the thermal conductivity or diffusivity value depends on microstructure of the sample and this is dependent upon both the heat and mechanical treatment. Differences in values of k or a of 10±20% can occur when repeating the measurements on the same sample. · Corrections should be applied to account for the effect of the insulating oxide layer on the wire (or strip) used in the THW method91 when making measurements on metals. · The determination of the magnitude of the contribution from radiation conduction (kR)94 is the principal problem with measurements on slags and glasses in the solid state (and there may be considerable difference in the contributions to kR for the experimental and the industrial conditions). Some indication of the magnitude of kR in the experiments could possibly be obtained by: (i) doping the slag or glass with FeO which increases the absorption coefficient and decreases kR for optically thick specimens (but for optically thin specimens it could also have the effect of making the specimen more optically thick), and (ii) by using materials with different emissivities for the wire in the THW method on the disc or the top of melt in the laser pulse method (e.g. Pt, C, W). Liquids The principal problems with thermal conductivity and diffusivity measurements lie in eliminating convection and minimising radiation conduction (in the case of molten slags and glasses). These problems and ways of overcoming them have been described above.
4.5.3 Electrical resistivity (R) Electrical resistivity increases with increasing temperature in both the solid and liquid phases. The resistivity of the liquid metal at the melting point is m approximately twice that of the solid for many metals but (Rm l /Rs ) ratios for Fe, Co and Ni are between 1 and 1.4.
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Methods Electrical resistivity (R) measurements on melts can be classified as either direct or indirect measurements. Direct measurements The electrical resistivity is determined from measurements of current and potential and the use of Ohm's Law. Capillary methods have been used for measurements on molten metals. The 4-probe method is usually adopted in which the potential drop across the liquid column is determined with two electrodes and the other two electrodes are used to measure the current.7,78 The cell constant is usually determined using a liquid of known resistivity (e.g. Hg). The two principal problems lie in the selection of (i) the capillary material (e.g. glass, quartz) and (ii) the electrode since C, W or Pt may dissolve in (or react with) the metal (one solution is to use a solid electrode of the same material as the sample by cooling the ends of the cell). The exploding wire method has been used for measurements on the solid state and also for measurements on the liquid phase.24,25 Values obtained for the liquid are frequently in good agreement with values obtained with 4-probe studies.25 Capillary methods have been little used for measurements on slags. This is due to the fact that resistance values are much smaller and are also more prone to polarisation and other effects. Two-electrode probes have been used with different geometries: · central electrode where one electrode is immersed in a crucible and the crucible serves as the other electrode;18,33 · two electrodes are immersed in a cell;18,33 · ring electrode cells where the electrodes are in the form of two concentric cells.18,33 However, 4-electrode cells are preferred since they largely eliminate the effects of polarisation on the current-carrying electrodes (by measuring the potential drop and the current at different electrodes). Resistances have been measured on molten slags using (i) AC bridges (ii) potentiometric methods and (iii) phase sensitive techniques.18,33 The effect of frequency on the measured resistance/conductivity has been discussed by several workers.33 Indirect method The rotating magnetic field method is based on the principle that a torque is developed when a liquid metal is subjected to a rotating magnetic field; the torque is inversely proportional to the electrical resistivity.7,96 Values of the liquid density are needed to calculate the resistivity from the torque.
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4.5.4 Emissivity (000f), absorption coefficient (000b*) Emissivity (000f) is defined as the ratio of radiated energy emitted by a surface to that emitted by a black body. However, the radiated energy is dependent upon the nature of the surface, temperature, wavelength (0015) and the direction. Spectral emissivity (000f0015) denotes the emissivity of a body at a specific wavelength. Total emissivity (000fT) is the ratio of (intensity of radiation emitted at all wavelengths by a body/intensity of radiation emitted at all wavelengths by a black body). With respect to direction emissivities these can be measured either (i) normal to the plane of the surface (denoted by subscript, N) or (ii) for the hemisphere containing all the radiated energy (denoted by subscript H). Thus (000f0015N) and (000fTH) refer to the normal spectral emissivity and the total hemispherical emissivity, respectively. The nature of the surface also affects the measured value, and measured values depend upon whether a surface is rough or smooth or whether the surface has been oxidised (or nitrided) by reaction with the atmosphere. Most cited values of emissivity refer to polished surfaces. Methods Emissivities are usually measured (using a spectroscope) by determining the ratio of the energies emitted by the surface divided by that emitted by a neighbouring black body. Values have been obtained using both electromagnetic97 and cold crucible levitation.98 Emissivities for the solid and liquid phases have also been obtained using the exploding wire method using a submicrosecond-resolution laser polarimeter.99 However, they can also be determined by measurements of other optical constants (refractive indices, reflectivities, etc.) using rotating analyser ellipsometry to determine the polarisation state of monochromatic light reflected from the surface at various angles.100 Values of (Cp/000fTN) can be derived from the cooling curves of levitated spheres in vacuum.54,101 Values of emissivity given in Table 4.2 were obtained from various sources. Absorption coefficient (000b*) and extinction coefficients (E) Absorption and extinction coefficients of semi-transparent media are needed for calculations of the magnitude of the radiation conductivity (kR) in glasses and slags. Methods There are two methods of measuring absorption coefficients.
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Transmission methods The absorption coefficient can be determined by using a spectrophotometer to measure the intensities of the incident (Io) and emerging beams after transmission through a sample of known thickness (d).33,102 I ˆ Io exp (ÿ000b*d)
(4.21)
In solids the extinction coefficient (E) is measured since radiation can be scattered by crystallites, grain boundaries, particles, etc. E ˆ * ‡ s*
(4.22)
where s* ˆ scattering coefficient. Reflectance methods Absorption coefficients have been determined from measurements of transmittance and reflectance.33,103
4.6
Properties related to mass transfer
4.6.1 Diffusion coefficient (D) There are several types of diffusion coefficient. Self diffusion involves the movement of various species present in the melt by random motions. There is no net flux of any species and no chemical potential gradients within the melt. It is also customary to quote self-diffusion values for impurity diffusion but which is strictly chemical diffusion since a concentration gradient is produced. Tracer diffusion is essentially the same process as self-diffusion but some of the species are radioactive. Consequently, there is a net flux and chemical potential; but this gradient refers solely to the radioactive species. Chemical diffusion is the movement of a species in the melt in response to a gradient of chemical potential arising as a result of either concentration or temperature gradients in the melt. Diffusion occurs in a direction that results in a reduction of the concentration gradient. Chemical diffusion in response to a temperature gradient is referred to as Soret diffusion. When diffusion involves the movement of two or more species it is referred to interdiffusivity. For example, if the cation is more mobile than the anion an electrical field is established which retards the cation and enhances the anion mobility in order to prevent a space charge being established in the melt. Fick's first and second laws apply for single component diffusion J ˆ ÿD (dC/dy) 2
(4.23) 2
dC/dT ˆ D (d C/dy )
(4.24)
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where J is the flux across a plane, (dC/dy) is the concentration gradient, D is the diffusion coefficient, and t is the time. The average value of the square of linear displacement (d) of a species after time t is given by d2 ˆ 2Dt
(4.25)
The temperature dependence of the diffusion coefficient is usually expressed as an Arrhenius relation: D ˆ Do exp (ÿED/R*T)
(4.26)
4.15 Relation between log10 D0 and (a) activation energy for diffusion of various impurities in solid obsidian104 and (b) reciprocal temperature for Co, Ni and Zn in CaO.MgO.2SiO2.105
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where Do is a constant and ED is the activation energy for diffusion. These parameters Do and ED are inversely proportional to one another (Fig. 4.15a) and this is known as the compensation effect. It is the diffusion equivalent to Urbain's assumption regarding the relation between A and E0011 for viscosity. One effect of the compensation rule is that diffusion coefficients for different species tend to come to a common value (Fig. 4.15b) at a specific temperature (around 1650K in Fig. 4.15b). Methods7,106±108,33 In the capillary reservoir method the sample containing the solute to be studied is contained in a capillary tube (of uniform diameter and known length) with one end sealed. The capillary is then immersed in a large container holding the solvent sample. The capillary tube is removed after a set time and the concentration in the capillary tube determined. Convection is minimised because of the small diameter of the capillary. In the diffusion couple method a long capillary of known cross-section is half-filled with the solvent and the solute samples. The capillary is then rapidly heated to the required temperature and maintained at this temperature for a specific time and is then quenched. Diffusion coefficients can then be determined by determining solute concentrations at various positions along the specimen. One difficulty is that some diffusion may occur during the heating and cooling periods. The shear method avoids the problems by only aligning the samples at the start of the run and then misaligning the samples at the end of the measurement period. It has not been used for high-temperature measurements. In the instantaneous plane method the solute is in the form of a thin disc and while the solvent consists of a long thin specimen. At the start of the experiment the thin disc disperses into the solvent sample. The electrochemical method has the advantage that time-consuming chemical analysis is replaced by measurements of current and voltage. However, it cannot be applied to some systems. Rayleigh scattering has been used to measure diffusion coefficients in liquids at room temperatures; this technique has the advantages of being a noncontact method and is very rapid.109 Electro-magnetic fields have recently been used to minimise contributions from convection to the diffusion coefficient.110
4.7
Estimating metal properties
It is apparent from the above text that the accurate measurement of thermophysical properties at high temperatures is both time-consuming and requires considerable expertise. Furthermore, process control often requires a rapid input of data for the relevant properties of the materials involved, for instance how
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much CaO must be added to a coal slag to obtain a viscosity where the slag can be poured from the reaction vessel. Thus it is not surprising that models or routines have been developed to estimate the properties from the chemical composition, for this is frequently available on a routine basis. Many models have been reported in the literature. This review does not claim to cover all these published models but covers only the models and routines with which the author has used or tested or which have come to his notice. The partial molar method is a simple method that has proved very useful for estimating some physical properties for alloys and slags. Using the case of a property (P) of an alloy, the value can be calculated from the sum of the product of the (mole fraction 0002 the property value) for each constituent of the alloy. The 0016 partial molar property for each constituent is denoted P. X 0016 1 † ‡ …X 2 P 00162 † ‡ …X3 P 00163 † ‡ …X4 P 0016 4 †Š ‰…X1 P …4:27† Pˆ where X ˆ mole fraction and 1, 2, 3, 4 denote the different components. However, some properties (such as viscosity) are very dependent upon the structure of the alloy or slag and in these cases the structure has been taken into account. Consequently, the partial molar method is particularly effective in estimating those properties which are least affected by structure (e.g. Cp and density).
4.7.1 Estimation of Cp and (HT ÿ H298) The heat capacity±temperature relation for solid materials (alloys and slags) can be represented in the form: Cp ˆ a ‡ bT ÿ (c/T2)
(4.28)
Integration of Cp dT between 298K and the temperature of interest, T gives Z T Cp dT ˆ …HT ÿ H298 † ˆ aT ‡ …b=2†T2 ‡ …c=T† ‡ d …4:29† 298
where d contains all 298K terms. The values of the constant a for the alloy or slag can be calculated by: X aˆ ‰…X10016a1 † ‡ …X20016a2 † ‡ …X30016a3 † ‡ …X40016a4 † ‡ :::Š …4:30† Thus the constants b and c can be calculated in a similar way so the Cp and enthalpy for the solid can be calculated for any required temperature. The heat capacity of the liquid is usually assumed to be constant, i.e. independent of temperature and consequently values for Cp of the alloy can be 00162 , etc. are the Cp values of liquid for the 00161 , P calculated from equation 4.27 where P various components. The only parameter now required is the enthalpy of fusion 0001fus H. This can be calculated in a similar way for the entropy of fusion, 0001fus S which represents the disorder which occurs when a solid transforms to a liquid.
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Fundamentals of metallurgy 0001fus H ˆ Tliq :0001fus S
…4:31†
Thus it is necessary to have a value for the liquidus temperature (Tliq) or have an estimated value. The enthalpy of fusion can then be calculated by the relation: X0002ÿ 0001 0001 00161 ‡ …X2 0001fus S 00162 ‡ …X3 0001fus S 00163 † ‡ . . .Š …4:32† 0001fus S ˆ X1 0001fus S The enthalpy for a temperature, T, where the liquid is the equilibrium phase can be calculated from Z Tliq Z T fus Cp…s† 0001 dT ‡ Tliq 0001 S ‡ Cp…l† …4:33† …HT ÿ H298 † ˆ 298
Tliq
This method works well for alloys and slags and the estimated values for the most part lie within 2 to 5% of the measured values. Problems · The above approach does not take into account any solid state transitions which might occur (such as those occurring in steels, Ni-based superalloys and Ti-alloys). Although the (HT ÿ H298) values will be slightly low in the transition ranges the values of (HT ÿ H298) for the liquid will not be affected since the entropy change can be considered as a small step in the disordering associated with the change from a fcc or cph solid to a liquid. · With glasses there is a transition from a glass to a supercooled liquid at the glass transition temperature (Tg) and this is accompanied by a step increase (0001Cp) in Cp of about 30% (and a threefold increase in ). The Cp±T relation can be modelled18 by assuming that the Cp for the supercooled liquid and liquid region remains constant and that there is no enthalpy of fusion involved. · Another weakness of the approach outlined above is that it assumes that the fusion process occurs at a discrete temperature (Tliq) whereas in practice it occurs over a melting range. · Another difficulty with the partial molar method lies in the fact that some alloys, (e.g. Ni-based superalloys) contain components such as Al (mp 933K) and W (mp 3695K) with melting points far away from the Tliq of the alloy (circa 1720K). This raises a question of whether the property values used in the 0016m) or (ii) the calculation should be (i) the property value at the melting point (P 0016m) can property value extrapolated to 1720K. The value at the melting point (P be adopted because it is considered that this was the point when all the atoms (including the Al and W) become disordered. Recent evidence on the densities of superalloys indicates that no serious errors are caused by this assumption.36 The values calculated with the partial molar approach are usually within 2±5% of measured values (which are themselves subject to experimental uncertainties). Commercial thermodynamic packages such as MTDATA, Thermocalc and FACT are available. These packages are capable of predicting
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solid state transitions and the melting range and their associated enthalpies.125 Furthermore, the fraction liquid can be calculated through the Scheil equation.125 Consequently, where a high degree of accuracy is required, Cp and enthalpies should be calculated with these packages
4.7.2 Density (001a) molar volume (V) The structure of the melt does not have a large effect on the density of the alloy or slag. The density can be calculated from molar volume: V ˆ M/001a
(4.34)
where M ˆ molecular mass of the sample (ˆ 0006xi Mi) and the molar volume can be calculated from the partial molar volumes of the constituents: X 0016 1 † ‡ …X2 V 0016 2 † ‡ …X3 V 0016 3 † ‡ …X4 V 0016 4 † ‡ . . .Š ‰…X1 V …4:35† Vˆ The substitution of density for molar volume in equation 4.35 results in only a small error. The temperature dependence of the volume is calculated from the partial molar (volume) thermal expansion coefficients (usually 0019 3 where is the linear thermal expansion coefficient). 0001 …dV=dT† X0002ÿ 0016 0001 ˆ X1 1 ‡ …X2 00162 ‡ …X3 00163 † ‡ . . .Š …4:36† ˆ V The linear thermal expansion coefficient (000b) of a glass or slag can be calculated with the model due to Priven126 or using the relation due to Yan et al.155 106000b(Kÿ1) (293±573K) ˆ ÿ18.2 ‡ 48.90003cor) where 0003cor is the optical basicity corrected for charge balancing of Al3+. Density±temperature relations for the solid state can then be calculated from the molar volumes (or densities) of the solid at 298K and the thermal expansion coefficients. Similarly, values of density of liquids as a function of temperature 0016 m and 0016 for the constituents in the liquid states. can be calculated from V However, the density is affected slightly by the structure of the melt. As we have seen in the section `Methods of determining structure' on p. 118, one way of accounting for the effect of structure is through the use of thermodynamics. Take, for example, the densities of superalloys, values calculated from partial molar volumes are consistently 2±5% lower than measured values and the shortfall increases with increasing Al content. The chemical activities of the constituents in the Ni-Al system show marked negative departures from Raoult's law (i.e. the atoms like each other). This results in tighter bonding and decrease in molar volume (0001Vxs) or an increase in density (0001001axs) of the melt. Since the effect of structure on the molar volume is relatively small it can be accounted for by adding (0001Vxs) to rquation 4.35. In the case of Ni-superalloys, 0001Vxs can be expressed in terms of the Al content (K(% Al), where K is the correction term) without much loss in accuracy.
152
Fundamentals of metallurgy
In a similar manner the activities of the CaO-SiO2 system show negative departures from Raoult's law. Correction terms in this case were determined by 0016 1 ) as a function of SiO2 content from back-calculating the parameter (XiV measured molar volumes of melts containing CaO and SiO2 and could be 0016 SiO ˆ (19.55 ‡ 7.97XSiO ) 0002 106 m3molÿ1. accounted for by the relation, V 2 2 0016 Al O3 ˆ (28.3 Similar relations are used for determining the effects of Al2O3 (V 2 2 6 3 ÿ1 ‡ 32XAl2O3 ÿ 31.45X Al2O3) 0002 10 m mol ) and P2O5 (65.7 0002 106 m3molÿ1). In most cases the values calculated using these corrections are within 00062% (and nearly always within 00065%) of the measured values. If more accuracy is required the use of thermodynamic packages such as MTDATA,126 Thermocalc and FACT are recommended. Details of published models for calculating densities of alloys and slags are given in Table 4.4.
4.7.3 Viscosity (0011) The approaches taken to calculate the viscosities of liquids fall into two classes:129 1. 2.
by treating the liquid as a dense gas;130 by assuming the liquid structure is similar to that of the solid except that it contains holes (e.g. Frenkel,131,132 Weymann,133 Eyring134 and Furth135).
In the Weymann±Frenkel approach the atoms are in thermal oscillation but for them to move from their present position into another equilibrium position it is necessary that (i) their energy should be greater than the activation energy required to move from site 1 to 2 and (ii) the next position should be empty (i.e. the site of a hole). Thus viscosity can be calculated from the probabilities that molecule can (i) jump from one position to another and (ii) find a `hole' in the liquid. Weymann133 derived the following equation: 0011 ˆ (kT/000f*)0.5 {(2kmT)0.5/(v0.667 Pv) exp (000f*/kT)
(4.37)
where k ˆ Stefan Boltzmann constant, 000f* is the height of potential barrier (associated with activation energy for viscous flow) m and v are the mass and volume of the structural unit and Pv is the probability of finding the next equilibrium site empty. The biggest drawback in developing reliable models for the viscosities of molten alloys and slags (and glasses) lies in the experimental uncertainties in the experimental data (see the section `Slags' on p. 154). For instance, Iida and Guthrie7 plotted the reported values for molten Fe and Al and showed that these values varied by 000650% and 0006100%, respectively, around the mean (although the lower values are more likely to be correct unless the crucible is non-wetting to the melt). Thus with uncertainties of this magnitude it is difficult to determine whether factors such as atom radius or structural effects are influencing the
Table 4.4 Details of models to calculate densities of alloys and slags Reference
System
Details of Method
Mills et al.127
Alloys
1. Solids: 001a298 = 0006 (X1001a298)1 + (X2001a298)2 + (X3001a298)3 + . . . (d001a/dT) = X1(d001a/dT)1 + X2(d001a/dT)2 + X3(d001a/dT)3 + . . . 001aT = 001a298 + (T ÿ 298K) (d001a/dT) Correction needed when Al > 1% e.g. Ni-alloys
2%
2. Liquids: Vm = 0006 X1V1m + X2V2m + X3V3m + . . . 2. V ÿ 0001Vxs = 0006 X1 1 + X2 2 + X3 3 + . . . where = (1/V)(dV/dT) VT= Vm (1+ (T ÿ Tm)) and 0001Vxs = V + K (% Al) for Ni alloys with Al Numerical analysis of measured data: (%) = b0 + b1(T ÿ TR) + b2(T ÿ TR)2 + b3(T ÿ TR)3 where bo, b1, b2, b3 and TR are constants: values given for various elements 2. V ÿ 0001Vxs = 0006 X1V1 + X2V2 + X3V3 + . . . 3. 0001Vxs = ÿ1.50 + 4.5XAl + 5.2(XCr + XTi) + 0.43(XW + XRe + XTa + XMO) 1. Use of thermodynamic software to predict, V and data stored for different phases of alloy: e.g. steels 001a = 001aFe + 0006 k1C1 where k is effect of different solutes and C = % 2. Values calculated for different phases V = 0006 X1V1 + X2V2 + X3V3 Values of V given Corrections to V values for compositions of Al2O3, Na2O, K2O, CaO, MgO, FeO: equations for also corrected for chemical compositions of these oxides Method 1: 001a (kgmÿ3) = 2460 + 18 (% FeO + % Fe2O3% + % MnO + % FeO) Method 2: V = 0006 X1V1 + X2V2 + X3V3 + . . . at 1773K V (10ÿ6m3molÿ1) values for: CaO = 20.7; FeO = 15.8; Fe2O3 = 38.4: MnO = 15.6; MgO = 16.1; Na2O = 33; K2O = 51.8; TiO2 = 24; P2O5 = 65.7; SiO2 = (19.55 + 7.97XSiO2) Al2O3 = (28.3 + 32XAl2O3 ÿ 31.45X2Al2O3) (dV/dT) = 0.01%Kÿ1 Use of thermodynamic software: Molar volumes stored for oxides and correction based on molar Gibbs energy of system V/0006 X1V1 = (K0001mix H/R*T) where is the relative integral enthalpy of mixing (determined with thermodynamic software) and K = constant Solids: V = 0006 X1V1 + X2V2 + X3V3 V Values and range of applicability given. Special procedures for boro-silicates
2%
Sung et al.113
Ni-based alloys
Robinson et al.125 Alloys Bottinga and Weill128
Slags
Mills and Keene18
Slags
Robinson et al.125 Slags Hayashi and Slags Seetharaman147 Priven126 Glasses
Uncertainty
2%
0006 10%) should see the development of reliable models for the prediction of the viscosities of alloys. Details of extant models are summarised in Table 4.7.
4.7.4 Electrical resistivity (R) Metals and alloys As mentioned in Section 4.7.3 the electrical resistivity (R) of solid alloys is affected by the microstructure of the sample; the model is based on the microstructure resulting in the minimum resistance (or highest conductivity). The electrical resistivity values at 298K of commercial alloys can be calculated from the relations given in Table 4.8. Approximate values for the liquid alloy can be obtained by assuming the electrical resistivity and its temperature dependence are calculated from partial molar quantities. 0016m 0016m 0016m Rm el ˆ …X1 Rel †1 ‡ …X2 Rel †2 ‡ …X3 Rel †3 ‡ . . . 0016 el =dT† ‡ X2 …dR 0016 el =dT† ‡ X3 …dR 0016 el =dT† dRel =dT ˆ X1 …dR 1 2 3
…4:42† …4:43†
Table 4.8 Coefficients for the calculation of klat and Rel for the calculation of thermal conductivities of solid alloys at 298K by Rel ˆ 0006(%. Rel)1 and klat ˆ 0006(%.klat)1 and note that terms Rel and (1/klat) for Al are not multiplied by 102 due to high conductivity of Al; no values given for klat of Ti-alloys; Bal = balance. No values reported for klat for Ti alloys Alloy
Al
C
Cr
Cu
Fe
Mn
Mo
Nb
Ni
Si
V
W
Others
Steel 102Rel
7.9
8.28
1.92
7.48
0.11
3.53
0.97
0.1
2.1
9.9
3.34
0.6
102klatt
1.0
246
0.35
347
8.1
0.11
0.39
0.81
ÿ0.28
ÿ96.8
1.14
4.63
As = 0.1; S = 2; P = 4.45 As = 1; S = 1; P=1
Ni-alloy 6.2 102Rel 102klatt ÿ16
0.1 0.1
1.6 ÿ36.4
ÿ4.0 ÿ1.56
0.6 8.31
3.8 187.5
2.7 5.5
0 0
0.1 15.7
10.2 181
9.4 9.4
ÿ38 ÿ818
ÿ26.1
200
9.1
ÿ173
ÿ11.4
9.2
10
51
8.3
ÿ8.38
5.4
ÿ7.75
52.7
0.076
0.39
2.90
Ti-alloy 102Rel
12.2
Al-alloy 3.08 Re
(1/klat) 0.046
7.4
7.1
0.076
Co = 0.08; Ti = 3 Co = ÿ0.2 Ti = 7.5 Sn = 65; Ti = 0.75; Bal = 10 Mg = 57.3 Zn = ÿ30.9; Li = 3.4; Bal = 30 Mg = 0.12; Zn = 0.38; Li = 3.36; Bal = 1.1
162
Fundamentals of metallurgy
The thermal conductivity of the liquid at temperature, T, can be calculated from the electrical resistivity value at that temperature using the WFL rule.
4.7.5 Thermal conductivity (k), diffusivity (a) Metals and alloys Liquid alloys At high temperatures heat transferred in metals is predominantly through the movement of electrons. Consequently, thermal conductivity and diffusivity values are usually calculated via the Wiedemann±Franz±Lorenz (WFL) relation shown in equation 4.44 k ˆ 2.45 0001 10ÿ8 (T/R)
(4.44)
where R is the electrical resistivity, which must be either measured or calculated. This equation works well for liquids around the melting point80 but it has been suggested81 that the following relation should be applied with measurements over extended ranges of temperature. k ˆ (A. Lo T/Rel) ‡ B
(4.45)
where A, B and Lo are constants and the subscript `el' refers to electronic conduction. However, it has been reported that the thermal conductivities of Ge, Si, Sn and Pb calculated from equation 4.44 show the opposite temperature dependence to the measured values. The absence of reliable thermal conductivity data for liquid alloys has hindered the development of models. One possibility for the calculation of thermal conductivities of commercial alloys is: (i) to derive a value of the thermal conductivity (km s ) of the solid alloy* at the liquidus temperature (or alternatively, 0.5(Tsol ‡ Tliq)) and then (ii) m m multiply km s by (Rl /Rs ) for the parent metal (for steels (Fe) by 1.05 and Nialloys by 1.40). Alternatively, approximate values for the liquid alloy can be obtained by assuming the electrical resistivity and its temperature dependence (dRel/dT) can be calculated from partial molar quantities. 0016m 0016m 0016m Rm el ˆ …X1 Rel †1 ‡ …X2 Rel †2 ‡ …X3 Rel †3 ‡ . . .
…4:46†
0016 el =dT† ‡ X2 …dR 0016 el =dT† ‡ X3 …dR 0016 el =dT† . . . …4:47† …dRel =dT† ˆ X1 …dR 1 2 3 The thermal conductivity of the liquid at temperature, T, can be calculated from the electrical resistivity value at that temperature using the WFL rule.
* Most families of alloys tend to come to a constant value of Ks at higher temperatures, e.g. for steels ks at 1673K has a value of 33 Wmÿ1Kÿ1 (Table 4.9).
Measurement and estimation of physical properties of metals
163
Solid alloys The resistance to thermal transfer at ambient temperatures can be considered to be made up of two contributions, Rel and Rlat, the resistances associated with electronic and phonon (or lattice) heat transfer. The overall thermal conductivity (keff) is dependent upon the microstructure; in modelling the thermal conductivity it is customary to derive values for the microstructure corresponding to the minimum resistance for the alloy (or maximum conductivity). There are few models for alloys and these have mostly been directed to the estimation of thermal conductivities of specific families of commercial alloys, e.g. steels, Ni-based superalloys, etc. It was noted that for individual families of alloys (e.g. steels) the thermal conductivity at higher temperatures (e.g. >1100K in steels) seems to be independent of composition of the alloy (e.g. steels), i.e. the conductivity of all alloys at a certain temperature has an identical value. Details of reported models are given in Table 4.9. The uncertainty in the estimated values is around 000610%. Mushy phase alloys It has been found that thermal conductivity measurements in the mushy phase are subject to considerable error since some of the heat is not conducted but is used to produce further melting of the alloy. Thus values are best estimated using the relation, keff ˆ fs ks el ‡ (1 ÿ fs) kl el or R ˆ fs Rs el ‡ (1 ÿ fs) Rl el. Slags Since it is difficult to estimate the magnitude of the radiation conductivity (kR) it follows that it is difficult to calculate the effective conductivity (keff) of solid and liquid slags. The radiation conductivity contribution increases markedly at higher temperatures. However, a correlation of the thermal conductivity data for liquid slags at the melting point (km) obtained using transient methods (where kR might be expected to be small) led to equation 4.48 which was applicable over the range (NBO/T) ˆ 0.5 to 3.5.11 The (NBO/T) can be calculated from the chemical composition (see Appendix A) (1/km) ˆ 0.7 ‡ 0.66 (NBO/T) mK Wÿ1
(4.48)
Thus the thermal conductivity increases with increasing polymerisation.
4.7.6 Surface tension The principal difficulties encountered in modelling surface tensions are: 1.
Surface active elements (even when present at ppm levels) can have a dramatic effect on the surface tension ( ) and (d /dT) (this is an especial problem with O, S, Se, Te in metals and alloys).
Table 4.9 Details of models for calculating the thermal conductivities of solid commercial alloys Alloy
Reference
Details and comments
Steel
Mills et al.127
1. Rel298 = (%.Rel298)1 +(%.Rel298)2 + (%.Rel298)3 + where 1, 2 etc. = different elements (Table 4.8) 2. kel298 = 2.45 0002 10ÿ8 (T/Rel298) 3. klat298 = (%.klat298)1 +(%.klat298)2 + (%.klat298)3 + (Table 4.8) 4. keff298 = kel298 + klat298 5. For 298 < T < 1073 K: Join keff298 to keff1073: kTeff = keff298 + {(T ÿ 298)/775}(25 ÿ keff298) Wmÿ1Kÿ1 6. For 1073 < T < 1573 K: keffT = 25 + 0.013 (T ÿ 1073K) Wmÿ1Kÿ1 Usually within 000610%
Ni-alloys
Powell and Tye161
1. keff = 6 + 2.2 0002 10ÿ6 (T/R) where R is resistivity at specific temperature Requires resistivity measurements: works within 00065%
Mills et al.127
Steps 1 to 4 see steel above 5. For 298 < T < 1073 K: Join keff298 to keff1073: kTeff = keff298 + {(T ÿ 298)/775}(25 ÿ keff298) Wmÿ1Kÿ1 6. For 1073 < T < 1573 K: keffT = 23 + 0.018 (T ÿ 1073K) Wmÿ1Kÿ1 Usually within 000610%
Ti-alloys
Mills et al.127
Steps 1 to 4 similar to steels and Ni-alloys but ! transformation occurs 973ÿ1273K 5. 298 < T < 973 K: kT = k298 + (23- k298) {(T ÿ 298)/675} Wmÿ1Kÿ1 6. 298 < T < 973 K: kT = k298 + 15.2 + 0.0273 (T ÿ 973) Wmÿ1Kÿ1 7. 1273 < T < 1923 K: kT = k298 + 15.2 + 0. 0273 (T ÿ 1273) Wmÿ1Kÿ1
Al alloys
Mills et al.127
As for steps 1 to 4 for steels: kel298: 5. Calculate a298 = k298/Cp298. 001a298: 6. Calculate aT: 298 < T < 573: aT = a298 {1+ 0.02 [(T ÿ 298)/275]} 7. 573 < T < Tsol :aT= a298 {1 ÿ 0.02[(T ÿ 573)/300]}
Measurement and estimation of physical properties of metals 2. 3.
165
The surface tensions of some `pure' metals are not well-established because of problems with O contamination (e.g. Ti and Zr). For alloys exhibiting marked departures from Raoult's law there is a marked effect on (e.g. Ni-Al shows negative departures which means there is less Al at surface than that calculated assuming ideal solution and hence (calc. ideal) < (actual).13
Several models have been developed and details are given in Table 4.10. Slags Tanaka et al.166,13 have applied their model (outlined in Table 4.10) to the calculation of the surface tension of slags and molten salts using commercial thermodynamic software to calculate GXS,B where the superscript B and S refer to the bulk and surface, respectively. It was found that it was necessary to allow for the fact that the ionic distances in the surface differ from those in the bulk in order to maintain electrical neutrality. This was taken into account using the parameter, 0018, which was (GXS,S/GXS,B) ˆ (ZS/ZB)/00184 ˆ 0.94/00184 with 0018 having a value of 0.97. Other models have been reported for the calculation of the surface tension of slags.7,18,169,170 The model due to Zhang et al.170 makes use of the excess surface tension ( xs = meas ÿ 0006X1 1) and derives constants to express ( xs) as a function of composition; good agreement was found for the calculated results with measured values. Interfacial tension ( MS) Interfacial tensions can be calculated using the following relation
ms ˆ m ‡ s ‡ 2001e ( M 0001 S)
(4.49)
where 001e is an interaction coefficient. The parameter 001e was found to have a value of 0.5 for slags free of FeO but increased with FeO additions.171 Tanaka13 proposed the following equation 001e ˆ 0.5 ‡ 0.3XFeO
(4.50)
Qiao et al.172 have outlined a model for estimating the interfacial tension using the excess interfacial tension ( xsMS) and considering the components of a binary alloy separately with regard to the slag. Values were derived for coefficients by fitting experimental data.
4.7.7 Optical properties Absorption coefficients (000b*) The change in absorption coefficient (0001000b*) of slags containing transition metal oxides (MO) can be estimated from the following: 0001 (mÿ1) ˆ K (%MO)
Table 4.10 Details of models developed to calculate the surface tensions of alloys References
Systems
Details of model and comments
Hajra, Lee, Frohberg162±164
Binaries, ternaries
1. Calculate molar volume of alloy V = 0006X1V1 + X2V2 + X3V3 2. Calculate surface area, si = (0.921 Vi2/3) 0002 104 for each component, i 3. Solve: 1 = 0006{Xi exp ( T ÿ 0 T) si/RT} where T and 0 T are surface tensions of alloy and element, respectively
Small165
Fe-O-S
1. T ÿ = R*TÿO ln {(1 + KOaO + KSaS)/(1 + KsaS)} + R*TÿS ln {(1 + KOaO + KSaS)/(1 + KOaO)} where a is the chemical activity and ÿ = surface excess concentration 2. Fitted experimental data for surface tension
Tanaka et al.166
Binaries, ternaries, molten salts
1. Based on Butler Eqn: = 0006 I + (R*T/si) ln (X.Si/XBI) + (1/si)(GES ÿ GEB) where GE is the excess free energy which is function of (T, X)B) S and B are the surface and bulk. 2. GESi = alloy(GEB) where = (ZS/ZB) = ratio of coordination in surface and bulk 3. = (ZS/ZB) = 0.83 for alloys = (ZS/ZB) = 0.94 for ionic mixtures Model successfully calculates effect of non-ideality on surface tension (e.g. negative departures, Fe-Si, positive departures, Cu-Pb). Good agreement with measured values
McNallan, Debroy167
Fe-Ni-Cr-S
Based on calculation of of Fe-Ni and then effect of Cr and then the effect of S is determined 1. Fe-Ni = Fe + {(R*T/sFe) ln (XSFe/XBFe)} + ln (fSFe/fBFe) where s = surface area, f = activity coefficients and superscripts S and B refer to the surface and bulk 2. ln fSFe = (ZS/ZB) ln fBFe 3. Fe-Ni = Fe ÿ 0.2XNi and effect of Cr on Fe-Ni determied and fS calculated from interaction coefficients. 4. = Fe ÿ A(T ÿ To) ÿ R*ÿS ln (1+ [k.aS exp {ÿ0001Habs/R*T}]) where 0001Habs = heat of absorption and k is equivalent entropy factor, aS is activity of S; A = constant, To = reference temperature
Su et al.168
Commercial alloys, steels Ni-based superalloys
1. Calculate `pure alloy' ( o) as in Hajra et al. above 2. Calculate effect of Al (with negative departures from ideality) on o using correction based on Tanaka's data ! ocorr 3. Calculate soluble O and S contents 4. Calculate effect of S and O on . L* = R*Tÿo(1 + Ksas + Koao)/(1 + Ksas) M* = R*Tÿs(1 + Ksas + Koao)/(1 + Koao) and 0001 = L* + M* c: (alloy) = ocorr ÿ 0001 Calculated values for steels, Ni-based alloys within 00065% of measured values
Moisou, Burlyev169
Steels, Nisuperalloys
Empirical relations based on measurements of binaries ± does not include effect of O or S
168
Fundamentals of metallurgy
where K had the following values 910 for FeO, 5 for MnO and 410 for NiO and 0001000b* (mÿ1) ˆ 390 (%Cr2O3) ‡ 370 (%Cr2O3)2.173 Refractive indices (n) Susa et al.174 reported that the refractive index (n) of a glass or slag (M2O-SiO2) can be estimated from the following relation n ˆ 2(2±3XM2O)nBO ‡ 2XM2OnNBO ˆ (2nNBO±6nBO)XM2O ‡ 4nBO ˆ a.nBO ‡ 0006b.nNBO (4.51) where values of nBO and nNBO were derived from measured data and a and b are the mole numbers of Oo and Oÿ, respectively. Values calculated were found to be within 0.4% of measured values. Priven126 reported that refractive indices were dependent upon molar volumes and thus could be calculated using the model for molar volume given in Table 4.4.
4.7.8 Diffusion constants (D) Most relations for calculating diffusion constants refer to self-diffusion or impurity diffusion at low concentrations. Several theoretical equations have been developed for calculating self-diffusion coefficients and these have been reviewed by Iida and Guthrie.7 However, calculations of self- and impuritydiffusivities are usually made relating D to other physical properties. Relation to molar volume (V) The following relation has been found to work well for self-diffusivities:7 Dm (m2sÿ1) ˆ 3.5 0002 10ÿ10 (Tm/M)0.5 (Vm)2/3
(4.52)
where M and V are the molecular mass and the molar volume, respectively, of the solvent. The activation energy for diffusion can be calculated from the relation: ED ˆ b(Tm)1.15
(4.53)
where b has a value of 2.50 for metals and 2.0 for semi-metals.7 Relation to viscosity (0011) The Stokes±Einstein equation gives a relationship with the viscosity: D ˆ kT/b.0019.0011.r
(4.54)
where k ˆ Boltzmann constant and r is the radius of the diffusing species and b has a value of 4 when r (diffusing species) 0019 r (solvent) and has a value of 6
Measurement and estimation of physical properties of metals
169
when r (diffusing species) > r (solvent). In the modified Stokes±Einstein equation7 the diffusion coefficient is given by: (kT/12.6 0011r) 0014 D 0014 (kT/10.5 0011r)
(4.55)
The Stokes±Einstein relation is not considered to work well in slags and glasses since the diffusing species (cations are small compared with the anions). However, this may not be the case for basic slags where the silicate ions are monomers. The Eyring relation also relates the D to the viscosity D ˆ kT/00110015
(4.56)
where 0015 ˆ interatomic distance and a value slightly greater than 2r has to be adopted to obtain a good agreement with measured values. Electrical conductivity (0014) The Nernst±Einstein equation relates D to the electrical conductivity (0014) and concentration (C): 0014 ˆ z2. F2.DC/R*T
(4.57)
where z ˆ valence of diffusing species, R* ˆ gas constant and F ˆ Faraday constant.
4.8
Acknowledgements
I would like to thank Dr Alistair Fox, Yuchu Su and Dr Zushu Li (Imperial College) and Dr Peter Quested of the National Physical Laboratory for his help and comments.
4.9
References
1. CA Heiple and JR Roper: Welding J. 61 (1982) 975s. 2. KC Mills and BJ Keene: Intl. Mater. Review 35 (1990) 185. 3. KC Mills, BJ Keene, RF Brooks and A Shirali: Phil. Trans. Roy. Soc. A (London) 356 (1998) 911/925. 4. RF Brooks and KC Mills: Mater. Sci. Eng. A178 (1994) 77/81. 5. S Oxley, PN Quested and KC Mills: Proc. Amer. Vacuum Soc. Conf., Santa Fe, NM February (1999) Amer. Vac. Soc. (1999). 6. RW Hamilton, D See, S Butler and PD Lee: Proc. Inst. British Foundrymen Conf. (IBF Castcon) held Stratford, June (2000). 7. T Iida and RIL Guthrie: The physical properties of liquid metals, Oxford Science (1988). 8. BO Mysen: Structure and properties of silicate melts, Elsevier, Amsterdam (1988). 9. BO Mysen: Earth Science Reviews 27 (1990) 281/365.
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66. I Egry , G Lohofer and G Jacobs: Proc. 4th Asian Thermophysical Properties Conf, held Tokyo, September (1995). 67. D Cummings and D Blackburn: J Fluid Mech. 224 (1991) 395/416. 68. S. Sauerland, RF Brooks, I Egry and KC Mills: Proc. TMS Annual conf. on Containerless Processing, WH Hofmeister and R Schiffman (eds), TMS (1993) 65/ 69. 69. H Fujii, T Matsumoto and K Nogi: Acta Mater. 48 (2000) 2933/39. 70. R Minto and W G Davenport: Trans. Inst. Min. Metall. 81C (1972) C36. 71. S Feldbauer and AW Cramb: Proc. 79th Steelmaking Conf. (1996) 655. 72. SI Popel, OA Esin and PV Geld: Dokl. Akad. Nauk SSSR 74(6) (1950) 1097 see also ibid. 83(2) (1952) 253. 73. BJ Keene, Slag Atlas, VDEH, Dusseldorf (1995) 407/409. 74. JL Brettonet, LD Lucas and M Olette: CR Hebd. Seances Acad. Sci. Ser. C 285 (1977) 45. 75. P V Riboud and L D Lucas: Can. Met. Quart. 20 (1981) 199/208. 76. H Gaye, L D Lucas, M Olette and P V Riboud: Can. Met. Quart. 23 (1984) 179/ 191. 77. J Liggieri, F Ravera, M Ferrari, A Passerone and R Miller: J Coll. Interface Sci. 186 (1997) 46/52 and 40/46. 78. KD Maglic, A Cezairliyan and VE Peletsky, Compendium of Thermophysical Property Measurement Methods, Vol. 1 Survey of measurement methods: Vol. 2 Recommended measurement techniques and practices, Plenum Press, New York (1984). 79. HB Dong and JD Hunt: Proc. 12th Intl Conf. Thermal Analysis, held Copenhagen, August (2004). 80. KC Mills, BJ Monaghan, BJ Keene: Intl Materials Reviews 41 (1996) 209/242. 81. RA Overfelt et al.: High Temp. ± High Pressure 34 (2002) 401/409. 82. R Gardon: Proc. 2nd Intl Conf Thermal Conductivity, held Ottawa (1962) 167. 83. M Susa, S Kubota, M Hayashi and KC Mills: Ironmaking and Steelmaking 28 (2001) 390/395. 84. M Gustavsson, H Nagai and T Okutani: Proc. 15th Symp. on Thermophysical Properties, held Boulder, CO June (2003). 85. K Hatori and H Ohta: Materia Japan 43 (2004) 129/135. 86. JA Balderas-Lopez: Revista Mexicana de Fisic 49 (2000) 353/357. 87. H Ohta, K Nakajima, M Masuda and Y Waseda: Proc. 4th Intl Conf Molten Slags and Fluxes, held Sendai (1992) ISIJ, Tokyo (1992) 421/426. 88. BJ Monaghan and PN Quested: ISIJ Intl 41 (2001) 1524. 89. H Szelagowski and R Taylor: High Temp. ± High Pressures 30 (1998) 343/350. 90. K Nagata, and KS Goto: 2nd Intl Conf. on Metallurgical slags and fluxes, held Lake Tahoe, NV (1984) Metall. Soc. AIME, Warrendale, PA 863/873, see also Trans. JIM 29 (1988) 133/142. 91. K Takashi and H Fukuyama, CAMP-ISIJ 17 (2004) 156. 92. LP Filippov: Intl J Heat Mass Transf. 16 (1973) 865/885. 93. VY Zinovyev, VE Polev, SG Taluts, GP Zinovyeva and SA Ilinykh: Phys. Met. Metallog. 61(6) (1986) 85/92. 94. E Yamasue, M Susa, H Fukuyama and K Nagata: J Cryst. Growth, 234 (2002) 121/ 131. 95. H Ohta, H Shibata, T Emi and Y Waseda: J Jap. Inst. Metals 61 (1997) 350/357.
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96. SG Teodorescu, RA Overfelt and SI Bakhtiyarov: Intl J Thermophys. 22 (2001) 1521/1535. 97. S Krishnan, GP Hansen, RH Hauge and JL Margrave: High Temp. Sci. 26 (1990) 143. 98. H Watanabe, M Susa, H Fukuyama and K Nagata: Intl J Thermophys. 24 (2003) 223/237. 99. C Cagran, C Brunner, A Seifter and G Pottlacher: High Temp. ± High Pressures 34 (2002) 669/679. 100. M Susa and K Nagata: Metall. Trans. 23B (1992) 331/337. 101. RK Wunderlich and HJ Fecht: Proc. Symp. on Research and Microgravity Conditions, held Nordeney, Germany (1998). 102. FJ Grove and PE Jellyman: J Soc. Glass Technol. 39 (1955) 3/15T. 103. JR Aronson, IH Bellotti, SW Eckroad, AG Emslie, RK McConnell and PC van Thuna: J Geophys. Res. 75 (1910) 3443/3456. 104. A Jambon: J Geophys. Res. 87 (1982) 10797/10810. 105. KW Semkow, R Rizzo and LA Haskin and DJ Lindstrom: Geochim. Cosmochim. Acta 46 (1982) 1879/1889. 106. HA Walls (Chapter 9C) and TS Lundy (Chapter 9A) in Physicochemical measurements in metals research, RA Rapp (ed.), Interscience (1970). 107. AE Le Claire: Diffusion and mass transport measurements in the characterisation of high temperature materials. Physical and elastic characteristics, MA McLean (ed.), Inst. of Metals, London (1989) Chapter 5 139/176. 108. JB Edwards, EE Hucke and JJ Martin: Metall. Rev. 120 (1968) 13. 109. T Nagata, K Takeo and Y Nagasaka: Proc. (abstracts) of 16th Europ. Conf. on Thermophysical properties, held London, September (2002) 104/106. 110. N Ma and JS Walker: Phys. Fluids 9 (1997) 1182 and 2789. 111. Y Kawai and Y Shiraishi: Handbook of Physico-chemical properties at high temperatures, Special Issue 41, ISIJ, Tokyo (1988). 112. KC Mills: Recommended values of thermophysical properties for commercial alloys, Woodhead, Abington (2002). 113. PK Sung, DR Poirier and E McBride: Mat. Sci and Eng. A231 (1997) 189/1197. 114. B Vinet, L Magnusson, H Fredriksson and PJ Desre: J Colloid and Interface Sci. 255 (2002) 363/374. 115. KC Mills and BJ Keene: Intl Metals Review 1 (1981) 21/67. 116. L Battezzati and AL Greer: Acta Met. 37 (1981) 1791/1802. 117. A Dinsdale: SGTE data for pure elements, CALPHAD 15 (1991) 317/425. 118. JD Margrave: Mater. Sci. Eng. A178 (1994) 83/88. 119. YS Touloukian: Thermophysical properties of high temperature solid materials, Vol. 1 Elements, Macmillan, New York (1967). 120. YS Touloukian et al.: Thermophysical properties of matter, Volumes 1±12 , IFI/ Plenum (1970 onwards). 121. P Kubicek and T. Peprica: Intl Metals Reviews 28 (1983) 131/157. 122. Landolt Bornstein: Physikalische Chemische Tabellen, Springer (1991). 123. GW Kaye and TH Laby: Tables of physical and chemical constants, 15th edn, Longmans, London (1995). 124. PF Paradis, T Ishikawa and S Yoda: J Mater. Sci. 36 (2001) 5125/5130. See also PF Paradis, T Ishikawa and S Yoda: Intl J Thermodyn. 24 (2003) 1121/1136; PF Paradis and WK Rhim: J Mater. Res 14 (1999) 3713/9.
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125. J Robinson, AT Dinsdale, LA Chapman, PN Quested, BJ Monaghan, JA Gisby and KC Mills: Proc. 2nd Intl Conf. of Science and Technology of Steelmaking, held Swansea, April 2001, IOM, London, 149/160. 126. AI Priven: Sov. J Glass Phys. Chem. 14 (1988) 321. 127. KC Mills, AP Day and PN Quested: Proc. of Nottingham Univ. ± Osaka Univ. Joint Symp., held Nottingham, October (1995). 128. Y Bottinga and DF Weill: Amer. J Sci. 272 (1972) 438. See also Geochim. Cosmochim. Acta 46 (1982) 909. 129. A. Kondratiev: Development of viscosity models for multiphase slag system. PhD Thesis, Univ. of Queensland, Brisbane (2004). 130. SG Brush: Chem. Reviews 62 (1962) 513/548. 131. J Frenkel: Z. Phys. 35 (1926) 652/669. 132. J Frenkel: Nature 136 (1935)167/168 and Trans. Farad. Soc. 33 (1937) 58/65. 133. HD Weymann: Kolloid Z Polymer. 138 (1954) 41/56 and 181 (1962) 131/137. 134. H Eyring: J Chem. Phys. 4 (1936) 283/291 and 5 (1937) 726/736. 135. R. Furth: Sci. Prog. 37 (1949) 202/218. 136. AK Doolittle: J Appl. Phys. 22 (1951) 1471/1475. 137. PV Riboud, Y Ropux, LD Lucas and H Gaye: Fachber. Huttenprax. Metallweiterverarb. 19 (1981) 859/869. 138. G Urbain: Steel Res 58 (1987) 11/116, see also Trans. J Brit. Ceram. Soc. 80 (1981) 139/141. 139. T Iida, H Sakai, Y Kita and K Murakami: High Temp. Mater. Processes 19 (2000) 153/164. 140. CL Senior and S Srinivasachar: Energy and Fuels 9 (1995) 277/283. 141. KC Mills and S Sridhar: Ironmaking and Steelmaking 26 (1999) 262/268. 142. VK Gupta, SP Sinha and B Raj: Steel India 21 (1998) 22/29, see also Steel India 17 (1994) 74/78. 143. K Koyama: Nippon Steel Tech. Report 34 (1987) 41. 144. JW Kim, J Choi, OD Kwon, IR Lee, YK Shin and JS Park: Proc. 4th Intl Conf on Molten Slags and Fluxes, held Sendai (1992) ISIJ, Tokyo 468/473. 145. TA Utigard and A Warczok: Proc. Copper 95 Intl Conf. (1995) vol. 4, Metall. Soc. of CIM 423/437. 146. RG Reddy, JY Yen and Z Zhang: Proc. 5th Intl Conf. Molten Slags, Fluxes and Salts, held Sydney (1997) ISS, Warrendale, PA 203/213, see also High Temp. Sci. 20 (1990) 195/202. 147. M Hayashi and S Seetharaman: CAMP- ISIJ 16 (2003) 860/863. 148. Ling Zhang and S Jahanshai: Metall. Trans. B 29B (1998) 177/186 and 187/195. 149. T Tanaka, M Nakamoto, J Lee and T Usui: Sci. Tech. Innovative Ironmaking, Aiming Energy Half Consumption, held Tokyo, November (2003) 161/164. See also CAMP-ISIJ 16 (2003) 864/867. 150. A Kondratiev and E Jak: Metall. Trans. B 32B (2001) 1015/1025. 151. KC Mills, L Chapman, AB Fox and S Sridhar: Scand. J Metall. 30 (2002) 396/403. 152. T. Lakatos, LG Johansson, B Simmingskold: Glass Technol. 13 (1972) 88 and Glasstekn. Tidstr. 30 (1976) 7. 153. X Feng and A Barkatt: Mat. Res. Soc. Symp. Proc. 112 (1988) 543. See also X Feng, E Saad and IL Pegg: Nuclear Waste Management III 457. 154. Y Sasaki and K Ishii: CAMP-ISIJ 16 (2003) 867. 155. FY Yan, FJ Wood and KC Mills: Proc. XVI Intl Glass Conf., held Madrid, October
Measurement and estimation of physical properties of metals 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174.
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(1992) vol. 2 177/182. M Hirai: ISIJ Intl. 33 (1993) 281/298. Du Sichen, J Bygden and S. Seetharaman: Metall. Mater. Trans B 25B (1990) 519. EA Moelwyn-Hughes: Physical Chemistry, Pergamon, Oxford (1961). RP Chhabra and DK Sheth: Z. Metallkunde 81 (1990)264/271. M. Kucharski: Z. Metallkunde 77 (1986) 393/396 and 79 (1988) 264/266. RW Powell and RP Tye: The Engineer (1960) 729/732. JP Hajra, HK Lee and MG Frohberg: Z. Metallkunde 82 (1991) 603/608. HK Lee, JP Hajra and MG Frohberg: Z. Metallkunde 83 (1992) 638/643. JP Hajra, MG Frohberg and HK Lee: Z. Metallkunde 82 (1991) 718/720. WM Small, P Sahoo and K Li: Scripta Met. et Mater. 24 (1990) 1155/1158 and 645/649. T Tanaka and T Iida: Steel Research 1 (1994) 21/28. MS McNallan and T Debroy: Metall. Trans. B 22B (1991) 557/559. Yuchu Su, KC Mills and AT Dinsdale: paper presented at the High Temp. Capillarity Conf, held San Remo, Italy (2004). LP Moisou and BP Burlyev: Svar. Proizvodsto (1997) (June) 18/20. J Zhang, S Shu and S Wei: Proc. of the 6th Intl Conf. Molten Slags, Fluxes and Salts, held Stockholm, June (2000). AW Cramb and Jimbo: Steel Research 60 (1989) 157/165. ZY Qiao, LT Kang, ZM Cao and BY Yuan: CAMP- ISIJ 16 (2003) 868/871. M Susa, K Nagata and KC Mills: Ironmaking and Steelmaking 20 (1993) 372/378. M Susa, Y Kamijo, T Kimura and T Yagi: CAMP- ISIJ 16 (2003) 872.
4.10 Appendix A: calculation of structural parameters NBO/T and optical basicity 1. (NBO/T) 1. 2. 3.
yNB ˆ 00062[XCaO ‡ XMgO ‡ XFeO ‡ XMnO ‡ XCaO ‡ XNa2O ‡ XK2O ‡ 6(1ÿf)XFe2O3 ‡ ÿ2XAl2O3 ÿ 2XfFe2O3] XT ˆ 0006XSiO2 ‡ 2XAl2O3 ‡ 2fXFe2O3 ‡ XTiO2 ‡ 2XP2O5 NBO/T ˆ yNB/XT where f ˆ Fe3+(IV) / (Fe3+(IV) ‡ Fe3+(VI)
where Fe3+(IV) and Fe3+(VI) represent the fraction of Fe3+ in four- and sixfold coordination, respectively. 2. Optical basicity (0003) 1. 2.
0003 ˆ 0006(X1n100031 ‡ X2n200032 ‡ X3n300033 ‡ . . .)/0006(X1n1 ‡ X2n2 ‡ X3n3 ‡ . . .) where n ˆ number of oxygens in oxide e.g. n ˆ 2 for SiO2 or 3 for Al2O3. 0003corr is calculated by deducting the mole fraction of cation required to charge balance the Al3+ in the chain, this is usually deducted from the largest cation present, e.g. Ba2+ thus XcorrBaO = (XBaO ÿ XAl2O3).
The following values of optical basicity are recommended: Al2O3 ˆ 0.60: B2O3 ˆ 0.42: BaO ˆ 1.15: CaO ˆ 1.0: FeO ˆ 1.0: Fe2O3 ˆ 0.75: K2O ˆ 1.4; Li2O ˆ
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1.0: MgO ˆ 0.78: MnO ˆ 1.0: Na2O ˆ 1.15: P2O5 ˆ 0.40: SiO2 ˆ 0.48: SrO ˆ 1.10: TiO2 ˆ 0.61: Various 0003 values have been cited for CaF2 0.43, 0.67 but a value of 1.2 should be used for viscosity calculations.
4.11 Appendix B: notation a = Thermal diffusivity (m2sÿ1) Cp = Heat capacity (JKÿ1kgÿ1) D = Diffusion coefficient (m2sÿ1) E = Extinction coefficient (mÿ1) e = Thermal effusivity (Js0.5mÿ2Kÿ1) = Fraction solid fs (HT ÿ H298) = Enthalpy relative to that at 298K (J Kÿ1kgÿ1) g = Gravitational constant (m sÿ2) k = Thermal conductivity (Wmÿ1Kÿ1) M = Molecular mass (g molÿ1) NA = Avogadro number R = Electrical resistivity ( m) R* = Gas constant (JKÿ1molÿ1) T = Temperature (K) X = Mole fraction = Linear thermal expansion coefficient (Kÿ1) * = Absorption coefficient (mÿ1) = Volume thermal expansion coefficient (Kÿ1) 0001fusH = Enthalpy of fusion (J kgÿ1) 000f = Emissivity
= Surface tension (mNmÿ1) 0011 = Viscosity (Pas) 0014 = Electrical conductivity ( ÿ1mÿ1) 0003 = Optical basicity 0015 = Wavelength (m) 001a = Density (kg mÿ3) ADL = Aerodynamic levitation DSC = Differential scanning calorimetry DTA = Differential thermal analysis EML = Electro-magnetic levitation ESL = Electro-static levitation mp = Melting point (K) NBO/T = Number of non-bridging atoms/no. of tetragonal atoms SLLS = Surface laser light scattering THW = Transient hot wire USL = Ultrasonic levitation WFL = Wiedemann±Franz±Lorenz rule
Measurement and estimation of physical properties of metals 0g 1g
= Micro-gravity measurements = Terrestrial measurements
Subscripts c = eff = el = g = l = lat = H = M = N = R = S = s = T = R = 0015 =
Phonon conductivity effective electronic gas liquid lattice hemispherical metal normal Radiation conductivity slag solid total Radiation conductivity spectral
Superscript liq = liquidus m = property value at melting point sol = solidus xs = excess
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5
Transport phenomena and metals properties A K L A H I R I , Indian Institute of Science, India
5.1
Introduction
Extraction and refining of metals involve the transfer of mass and heat within each phase and between the phases, one or more of which are in motion. For example, in blast furnace iron making, Fig. 5.1, iron oxide in the form of lumpy ore, sinter or pellet along with coke and limestone, are charged on the top of the furnace continuously. The charge slowly moves down and hot reducing gases produced by burning of coke in front of tuyeres move up through the bed of solids and finally escape through the gas outlet. The heat is transferred from hot gases to solid and at the same time the reducing gases reduce the oxides to metallic iron. At high temperature reduced iron melts and forms liquid metal and gangue in ore, coke ash and lime form the slag. The final product is molten iron containing dissolved carbon and other impurities like silicon, manganese, phosphorous and sulfur and liquid slag, which are tapped from the furnace. Other metallurgical operations like melting, alloying, casting, joining, heat treatment and metal deformation processes like forging, rolling, etc. also involve heat, mass and momentum transfer. So transport phenomena play a key role in all metallurgical operations. The present chapter deals with the fundamentals of transport phenomena within a phase and between phases.
5.2
Mass transfer
Transfer of a species from one location to another in a mixture of non-uniform composition is known as mass transfer. When a homogeneous solution flows, the species present in the solution moves from one location to another but there is no mass transfer since concentration of species is the same everywhere. So a concentration difference is an essential requirement for mass transfer. Mass transfer takes place because of molecular motion and bulk flow. Atoms in fluids are in random motion and this causes transfer of a species from one place to another. However this could be noticed only when mixture is not
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5.1 Schematic representation of blast furnace.
homogeneous or few radio active atoms are present in the system. The movement of atoms in fluid is easy because the positions of atoms are not rigid. But in solids, atoms move from one lattice site to another by exchanging their position with a neighbouring vacancy, Fig. 5.2. So the movement of atoms in solids is accompanied by movement of the vacancy in the opposite direction. Other mechanisms of movement of atoms are also possible but the vacancy mechanism is most predominant for substitutional solid solutions.
5.2 Diffusion in solid. Atom marked A jumps into vacancy and vacancy moves from left to right.
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5.2.1 Mass flux Figure 5.3(a) shows two non-reacting species A and B which are in contact with each other. The left side has species A and right side has B. Since the concentration of species A and B are not uniform throughout the system, atoms of A and B move in opposite directions. This leads to the decrease of concentration of A in the left side and corresponding increase of the concentration of A in the right side as shown in Fig. 5.3(b). The reverse is the case for concentration profile of B. The process continues till it attains the steady state when concentrations do not change with time. At steady state, the concentration of A becomes the same everywhere, Fig. 5.3(c). Let vA and vB be the velocities of A and B respectively. The flux of A, defined as the amount of A crossing the unit area per unit time in a direction normal to the area is NAx ˆ vA.CA
(5.1)
where NAx is the molar flux of A in x direction and CA is the molar concentration of A. Equation 5.1 can be written as NAx ˆ CA(vA ÿ vav) ‡ CA.vav av
(5.2) av
where v is the average velocity and (vA ÿ v ) is the velocity of A with respect to average velocity. So the first term on the right-hand side measures flux due to molecular motion with respect to average velocity and the second term is the flux due to bulk flow. The flux due to molecular motion with respect to average velocity, JAx, is given by Fick's law of diffusion as dCA …5:3† dx where DAB is the diffusivity of A in A±B. The negative sign in Fick's law shows that the flux is in the direction of decreasing concentration. Fick's law of diffusion gives the flux with respect to average velocity of all the species or with respect to a moving coordinate that moves with velocity vav. Combining equations 5.2 and 5.3, JAx ˆ CA …vA ÿ vav † ˆ ÿDAB
t=0
short time CB
CA x=0 (a)
A
B
steady state CB
CA
CB
CA
x=L x=0
x=L x=0
(b)
(c)
x=L
5.3 Schematic representation of diffusion of two species A and B. (a) Before diffusion. (b) Concentration profiles of A and B after a short time. (c) Steady state concentration profile.
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5.4 Diffusion of dye in water flowing through a tube.
dCA ‡ CA vav …5:4† dx To understand the physical meaning of the diffusion and bulk flow term in equation 5.4, let us perform a simple experiment. We take a long horizontal glass tube and allow water to flow through it at a very low flow rate and inject some dye along the axis of the tube. If we follow the tip of the dye as it moves through the tube, we will find that a faintly coloured zone just ahead of the deep coloured streak, Fig. 5.4. The velocity of the tip of the streak is vav and the spread of the colour beyond the tip is due to molecular diffusion. By definition, the average velocity vav is NAx ˆ ÿDAB
vav ˆ (CA. vA ‡ CB.vB)/(CA ‡ CB)
(5.5)
Using equation 5.1, equation 5.5 becomes vav ˆ (NAx ‡ NBx)/(CA ‡ CB)
(5.6)
Combining equations 5.4 and 5.6 dCA ‡ xA …NAx ‡ NBx † …5:7† dx where xA is the mole fraction of A. Definition of average velocity vav, indicates that bulk flow term can be due to molecular motion or external force or both. In solids, and stagnant fluid, there is no bulk flow due to external forces, but the diffusion of atoms and molecules itself lead to bulk flow. In these cases equation 5.7 is the appropriate form of the flux equation. When the bulk flow is primarily due to external forces as in case of Fig. 5.4, the molecular contribution in the bulk flow can be neglected and equation 5.4 is the appropriate form of the equation. The average velocity vav in this case is determined by external forces and is obtained by solving fluid flow problem. Equations 5.1±5.7 have been written in terms of molar fluxes, molar concentration, and mole fraction, but these are valid when the word mole is replaced by mass, i.e. mass flux, mass concentration and mass fraction. In the three-dimensional case, the flux of A is given by NAx ˆ ÿDAB
NA ˆ ÿDABrCA ‡ xA(NA ‡ NB)
(5.8)
Bold letters indicate vectors. The first term on the right-hand side of equation 5.8 is the pure diffusion flux JA given by Fick's law and second term is the flux due to bulk flow. The components of the molar fluxes due to molecular motion, JA, in different coordinates are given in Table 5.1.
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Table 5.1 Components of flux due to binary diffusion Rectangular
Cylindrical
Spherical
Jx ˆ ÿDAB
@CA @x
(A)
JAr ˆ ÿDAB
Jy ˆ ÿDAB
@CA @y
(B)
JA0012 ˆ ÿ
Jz ˆ ÿDAB
@CA @z
(C)
Jz ˆ ÿDAB
@CA @r
@CA @r
(D)
JAr ˆ ÿDAB
DAB @CA r @0012
(E)
JA0012 ˆ ÿ
DAB @CA r @0012
@CA @z
(F)
JA001e ˆ ÿ
DAB @CA (I) r sin 0012 @001e
(G) (H)
According to Fick's law driving force for diffusion is concentration gradient and diffusion takes place only from higher concentration to lower concentration. Darken (1949) found that when two steel bars containing the same carbon but one having silicon are welded together, carbon diffused from the bar containing silicon to the other side. Finally the bar that is free from silicon became richer in carbon than the other one. This is known as uphill diffusion. Figure 5.5 schematically shows the uphill diffusion. According to Fick's law, no diffusion of carbon should take place since carbon concentration in both the bars was initially the same. Darken's experiment clearly demonstrated that the activity gradient is the driving force for diffusion and not the concentration gradient. Silicon increases the activity of carbon and is a driving force for diffusion of carbon from the bar containing silicon to the other bar and the diffusion continues until the activity of the carbon becomes the same in both bars. However, for all usual analysis, Fick's law is used and the activity concept is invoked only to explain the anomaly.
3% Si 3% Si
0.4% C
(a)
0.4% C
0.4% C
(b)
5.5 Schematic diagram of uphill diffusion of carbon. (a) Before diffusion. (b) After diffusion.
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Relation between fluxes Equation 5.7 for NAx, contains two unknowns NAx and NBx, but in fluids a similar equation for NBx is not an independent equation (see Example 5.2, page 185). Therefore to obtain NAx, the relationship between the NAx and NBx must be known from other physical considerations or laws. (1) Graham's Law The ratio of diffusion fluxes in gases are inversely related to the square root of molecular weight or NAx/NBx ˆ (MB/MA)1/2
(5.9)
This equation applies to effusion of gases in a vacuum and equal pressure counter diffusion. (2) Chemical reaction Let us consider that oxygen is reacting with carbon according to the reaction C ‡ O2 ˆ CO2
(5.10)
Figure 5.6 shows schematically the fluxes on the surface of the carbon block. Oxygen diffuses at the carbon±gas interface for reaction and carbon dioxide
CO 2
Carbon
C + O = CO 2 O2
5.6 C-O reaction on the surface of carbon.
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formed diffuses away from the reaction interface. Since one molecule of oxygen forms one molecule of CO2 NO2,x ˆ ÿNCO2,x
(5.11)
The above condition indicates that per unit time the number of molecules of oxygen going in one direction is equal to number of molecules of CO2 going in the opposite direction. This type of diffusion is termed as equimolar counter diffusion. In the previous case one mole of reactant gas formed one mole of product gas. Let us consider, carbon reacts with oxygen to form CO C ‡ O2 ˆ 2CO
(5.12)
If one mole of O2 comes to the surface of carbon and reacts with it, two moles of CO is produced and go out in the opposite direction. Or the flux of CO is two times that of O2 and the relationship between the fluxes is NO2,x ˆ ÿ 12NCO,x
(5.13)
For a reaction of the following type nA (g) ˆ mB (g)
(5.14)
the fluxes are related by NAx ˆ ÿ(n/m) NBx
(5.15)
(3) Flux of inert species Let us consider that C-O2 reaction, equations 5.10 or 5.12, takes place in air. Obviously nitrogen present in air will not take part in the reaction so the flux of nitrogen will be zero. In other words, flux of an inert species is always zero. (4) Dilute solution If the concentration of the diffusing species A is very small (xA is very small), the bulk flow term can be neglected since its contribution is very small. In dilute liquid and solid solutions the bulk flow term is neglected. Example 5.1 Show that for a binary mixture of gases diffusivity of A is same as that of B. Solution Let DAB and DBA be the diffusivities of A and B respectively in the mixture. Let us consider that A and B shown in Fig. 5.3 are two gases and the system is isothermal and isobaric. So (CA ‡ CB) is constant, or
Transport phenomena and metals properties @CA @CB ‡ ˆ0 @x @x From equation 5.7 @CA ‡ xA …NA ‡ NB † @x @CB ‡ xB …NA ‡ NB † NBX ˆ ÿDBA @x Since xA ‡ xB ˆ 1, adding equations (b) and (c) NAx ˆ ÿDAB
@CA @CB ‡ DBA ˆ0 @x @x From equations (a) and (d) DAB ˆ DBA DAB
185 …a†
…b† …c†
…d† nnn
In the case of diffusion in binary substitutional solid solution, there are three fluxes, that of A, B and vacancies so the above derivation is not applicable. Although concentration of vacancy is very small, the flux is not small. It can be shown that Jvx ‡ JAx ‡ JBx ˆ 0. Example 5.2 Show that the following two equations are not independent for diffusion in fluids.
Solution Let
NAx ˆ ÿDAB
dCA ‡ xA …NAx ‡ NBx † dx
…5:7†
NAx ˆ ÿDAB
dCB ‡ xA …NAx ‡ NBx † dx
…5:7a†
CA ‡ CB ˆ C
(a)
where C is the total concentration, a constant. Substituting the value of CA from equation (a) in equation 5.7 d…C ÿ CB † C ÿ CB ‡ …NAx ‡ NBx † CA ‡ CB dx dCB CB ÿ ˆ DAB …NAx ‡ NBx † ‡ NAx ‡ NBx dx CA ‡ CB
NAx ˆ ÿDAB
…b† …c†
Rearranging dCB ‡ xB …NAx ‡ NBx † dx Hence equations 5.7 and 5.7a are not independent equations. NBx ˆ ÿDAB
…5:7a† nnn
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Example 5.3 Carbon monoxide and nitrogen at 2 atm pressure and 300K are separated by a wall of 2 mm thickness, Fig. 5.7. The separating wall has a small hole of 0.5 mm. Calculate the maximum leakage rate of carbon monoxide and nitrogen. The profile of oxygen and nitrogen can be assumed to be linear in the hole. At 300K and 2 atm pressure DCO-N2 ˆ 1.03 0002 10ÿ5 m2/s. 2 mm
CO 0.5 mm
N2
5.7 Leakage of CO through a hole.
Solution The diffusion of N2 and CO are in opposite directions so it is counter diffusion. Furthermore, the pressure and concentration of gases are the same and the flow of gas inside the hole is only due to molecular motion. So the ratio of fluxes can be calculated by Graham's law, equation 5.9. Since the molecular weights of both CO and N2 are same, the ratio of molar fluxes is unity. Or NN2 ˆ ÿNCO. So equation 5.7 reduces to dCN2 …a† dx The concentration of N2 in nitrogen side ˆ P/RT ˆ 2/(0.082300) ˆ 0.081 kmol.mÿ3. Leakage rate is maximum when concentration of nitrogen on CO side is zero. Since concentration profile is linear, dCN2/dx ˆ (CN2 x ÿ CN2 0)/x. Taking the face on the nitrogen side as x = 0, NN2 ˆ ÿDCOÿN2
NN2 ˆ ÿ1.03 0002 10ÿ5 [(0 ÿ 0.081)/(2 0002 10ÿ3)] ˆ 4.17 0002 10ÿ4 kmol.mÿ2sÿ1 So the leakage rate of nitrogen = NN2 0002 area of the hole = 4.17 0002 10ÿ4 0002 0019 0002 (0.5 0002 10ÿ3/2)2 = 8.19 0002 10ÿ11 kmole.sÿ1 The leakage rate of carbon monoxide will be the same.
nnn
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5.2.2 Diffusivity of gas, liquid and solid Evaluation of fluxes requires the knowledge of diffusivities. Over the years, considerable efforts have been made to calculate these values based on first principles. But the success has been rather limited. So the primary source of diffusivity is the measured values. However, in the absence of experimentally measured values, empirical and semi-empirical co-relations are used to calculate these. Diffusivity of gases According to the kinetic theory of gases, diffusivity of gases are proportional, T1.5/p where T is temperature and p is pressure. This pressure dependence is valid for less than 10 atmospheres but the exponents of temperature for real gases are higher. Chapman±Enskog equation gives a good estimation of binary diffusivity of gases. For details see Reid et al. (1977). Fuller, Schettler and Giddings (1966) proposed the following empirical relationship: DAB ˆ
10ÿ7 T1:75 ‰1=MA ‡ 1=MB Š1=2
…5:16†
p‰…VA †1=3 ‡ …VB †1=3 Š2
where DAB is in m2/s, temperature T is in K and p is pressure in atmospheres. VA and VB are diffusion volumes of A and B respectively. Diffusion volumes of simple molecules are given in Table 5.2. The above equation predicts diffusivity within 10% in most of the cases. Diffusivity of gas phases is usually in the range of 10ÿ5 to 10ÿ3 m2/s. Diffusivity through porous solid Porous solid consists of a number of interconnecting pores (Fig. 5.8). If we take unit area on the surface of the solid, the diffusion takes place only through the area occupied by the pores. Again, gas cannot move in a straight line in the porous solid. It takes a tortuous path. To take into account these two special features of porous solid, diffusivity of gas in porous solid is defined as effective diffusivity Table 5.2 Diffusion volumes for simple molecules (Fuller et al. 1966) Gas
VA
Gas
VA
Gas
VA
H2 He N2 O2 Air
7.07 2.88 17.9 16.6 20.1
Ne Ar CO CO2 N2O
5.59 16.1 18.9 26.9 35.9
NH3 H2O Cl2 Br2 SO2
14.9 12.7 37.7 67.2 41.1
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5.8 Schematic diagram of a porous solid.
Deff ˆ 000fDAB/001c
(5.17)
where 000f is porosity of solid and 001c is the tortuosity factor. 001c measures the ratio of actual path travelled by gas to geometrical diffusion distance. When gases diffuse through pores it diffuses either by ordinary gas diffusion or Knudsen diffusion or a combination of both. When the pore diameter is much larger than the mean free path, diffusion is ordinary gas diffusion. But if the pore size is smaller than the mean free path, then gas molecules have a higher probability of collision with the pore wall than with each other. Gas flow under this condition is known as Knudsen diffusion. Knudsen diffusion coefficient for diffusion through a cylindrical pore is DK ˆ 97 r (T/M)1/2 m2/s
(5.18)
where r is the radius of pore in m, T is the temperature in K and M is the molecular weight. In general, pores are not cylindrical so a correction of tortuosity is required for Knudsen diffusion. Since in porous solid, nature of diffusion depends on the size of pores, effective diffusivity of porous solid is often taken as 1/D0 eff ˆ 1/DAB ‡ 1/DK
(5.19)
Deff ˆ D0 eff. 000f/001c
(5.20)
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189
Example 5.4 Find out the pore diameter where molecular and Knundsen diffusivity are comparable for diffusion of H2-H2O mixture through a porous solid at 1000K and 1 atm pressure. Solution We use equation 5.16, to calculate molecular diffusivity of H2-H2O. From Table 5.2, VH2 ˆ 7.07 and VH2O ˆ 12.7 and molecular weights of H2 and H2O are 2 and 18 respectively. Substituting in equation 5.16 DH2 ÿH2 O ˆ
10ÿ7 …1000†1:75 ‰1=2 ‡ 1=18Š1=2 ‰7:071=3 ‡ 12:71=3 Š2
ˆ 7:3 0002 10ÿ4 m2 =s
…a†
Equation 5.18 shows that Knudsen diffusivity of H2 is greater than that of H2O. So we should compare the molecular diffusivity of H2-H2O mixture with Knudsen diffusivity of H2O. Knudsen diffusivity of H2O is DH2O ˆ 97 r (1000/18)1/2 ˆ 723r m2/s
(b)
Both the diffusivities will be same if the pore radius r is r ‡ 7.3 10ÿ4/723 ˆ 1.01 0002 10ÿ6 m So if pore radius is about 1 0016m, the contribution of both Knudsen and molecular diffusion will be almost same. If the pore radius is 10 0016m, equation 5.19 indicates that the contribution of Knudsen diffusion will be only about 10% of total diffusion. On the other hand, if pore radius is 0.1 0016m, the contribution of molecular diffusion will be only 10%. nnn Diffusion in liquids The experimentally measured values of diffusivity are often expressed in the form of the Arrhenius equation D ˆ D0exp(ÿQ/RT)
(5.21)
D0 is the frequency factor, Q is the activation energy for diffusion and R is gas constant. The value of Q for metallic system is mostly less than 16 kJ molÿ1. Table 5.3 shows the diffusivity of different solutes in liquid iron at 1873K. Table 5.3 Diffusivity of solutes in liquid iron at 1873K (Morita 1996) Solute Diffusivity 0002 109 m2/s
C 4±20
Si Mn 2.5±12 3.5±20
Solute Diffusivity 0002 109 m2/s
N 6±20
Ni 4.5±5.6
Cr 3±5
S O H 4.5±20 2.5±20 80±200 V 4±5
Mo 3.8±4.1
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Fundamentals of metallurgy
Because of difficulties in measurement at high temperature, uncertainty in measured values is often large. Note that diffusivities of different solutes except hydrogen in liquid iron are of the same order of magnitude irrespective of size of atoms. These values compare quite well with diffusivity of CO2 in water, 1.9 0002 10ÿ9 m2/s at 298K. Diffusion in solids In solid solutions we define both intrinsic and inter or mutual diffusivities. The intrinsic diffusivity of an element is the diffusivity of that element in the solution. In a mixture of gases, intrinsic diffusivities of A and B in A-B is denoted by DAB and DBA respectively, but for solids these are denoted by DA and DB respectively. In a gas mixture, DAB ˆ DBA (see Example 5.1 on pages 184±5) but in a solid (in general), DA 6ˆ DB. Inter or mutual diffusivity in solid is the diffusivity of A in B or vice versa and is denoted by DAB. For example, in Cu-Zn alloy, intrinsic diffusivities of Cu and Zn are denoted by DCu and DZn respectively and inter or mutual diffusivity by DCuZn. Inter and intrinsic diffusivities are related by DAB ˆ xBDA ‡ xADB
(5.22)
xA and xB are mole fractions of A and B in the solution. Diffusivity in solids follow the Arrhenius equation 5.21 and activation energy of diffusion is quite large. Table 5.4 gives diffusivity of different solutes in iron. It shows that both activation energy for diffusion and diffusivity are strongly related to size of atoms.
5.2.3 Conservation of mass Let us consider an elemental volume 0001x0001y0001z as shown in Fig. 5.9. Since the mass is conserved, the balance equation is Rate of accumulation of species A in 0001x0001y0001z = Rate in ÿ Rate out ‡ Rate of generation of A in 0001x0001y0001z volume by reaction Table 5.4 Diffusivity of solutes in iron (Kucera and Stransky 1982) Diffusing elements Hydrogen Boron Carbon Nitrogen Chromium Nickel
D0 0002 107 m2.sÿ1 8.1 2.0 738 480 4080 1090
Q kJ molÿ1
D at 1200K m2.sÿ1
43.2 87.92 158.98 159.1 286.8 296.8
1.07 0002 10ÿ8 2.98 0002 10ÿ11 8.86 0002 10ÿ12 5.70 0002 10ÿ12 1.34 0002 10ÿ16 1.31 0002 10ÿ17
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191
N Az NAy (x + Dx,y + Dy,z + Dz) z
Dz NAx
N Ax
y
Dy x
NAy
(x,y,z,)
Dx N Az
5.9 Mass balance in a control volume.
In this case there are six faces. Mass flux enters through three faces and goes out through the opposite three faces. NAx enters at x though the face of area 0001y0001z and goes out through the face at x ‡ 0001x. Similarly, NAz enters at z through the face of area 0001x0001y and goes out at z ‡ 0001z and NAy enters at y through the face of area 0001x0001z and goes out at y ‡ 0001y. Hence mass balance equation can be written as @…0001x0001y0001zCA † ˆ …0001y0001zNAx j x ÿ 0001y0001zNAx j x‡0001x † @t ‡ …0001x0001zNAy j y ÿ 0001x0001zNAy j y‡0001y † ‡ …0001x0001yNAz j z ÿ 0001x0001yNAz j z‡0001z † ‡ 0001x0001y0001zRA
…5:23†
where RA is the rate of generation of A per unit volume by reaction. Dividing throughout by 0001x0001y0001z and taking the limit 0001x!0, 0001y!0, 0001z!0, @CA @NAx @NAy @NAz ˆÿ ÿ ÿ ‡ RA @t @x @y @z
…5:24†
Equation 5.24 is the general mass balance equation of a species in rectangular coordinate. Table 5.5 gives the equation in different coordinate systems. These differential equations along with appropriate boundary conditions are solved to find out the concentration profiles and fluxes in a system. If we are interested in finding out the concentration profile in a cylinder or a system having cylindrical symmetry we use equation B of Table 5.5. Equation C of Table 5.5 is used for
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Table 5.5 Equation of continuity of A in different coordinate systems Rectangular coordinates @CA @NAx @NAy @Az ‡ ‡ ‡ ˆ RA @t @x @y @z
‰AŠ
Cylindrical coordinates @CA 1 @ 1 @NAz @NAz ‡ ‡ ˆ RA …rNAr † ‡ @t @z r @r r @0012
‰BŠ
Spherical coordinates @CA 1 @ 2 1 @ 1 @A001e ‡ 2 …r NAr † ‡ ˆ RA …NA0012 sin 0012† ‡ @t r @r r sin 0012 @0012 r sin 0012 @001e
‰CŠ
concentration profiles in a sphere. Although any problem can be solved in a rectangular coordinate system, appropriate choice of coordinate system makes the final equation simpler.
Diffusion in solid Let us consider carburizing of steel as an application of diffusion in solid. A steel plate containing uniform carbon concentration, C0, is exposed in a carburizing gas atmosphere. The carburizing atmosphere maintains carbon concentration on the top surface of plate at CS, Fig. 5.10. We want to find out the carbon concentration profile in the plate. Carbon diffuses from the top to the interior of the plate in y direction. So concentration gradient of carbon in x and z directions is zero. Furthermore, carbon does not take part in any reaction in the steel plate, so RA ˆ 0. Thereby, the problem involves diffusion of carbon only in y direction and equation 5.24 simplifies to
Cs
z y
x
5.10 Carburizing of steel.
Transport phenomena and metals properties @NCy @CC ˆÿ @t @y
193 …5:25†
where CC is the concentration of carbon and NCy is the carbon flux in steel in y direction. Since concentration of carbon in steel is small, we neglect the bulk flow term in the definition of flux given by equation 5.7 and simplify it as NCy ˆ ÿDC
@C @y
…5:26†
where DC is intrinsic diffusivity of carbon in steel. Substituting equation 5.26 in equation 5.25 and assuming that diffusivity of carbon is constant @CC @ 2 CC ˆ DC @t @y2
…5:27†
Equation 5.27 is known as Fick's second law. The initial and boundary conditions are At t ˆ 0, At t 0015 0,
y > 0, CC ˆ C0 y ˆ 0, CC ˆ CS
(5.28a) (5.28b)
Normally, the thickness of carburized layer is a fraction of a millimetre only and carbon concentration far away from the top surface remains at C0. Thereby, although the plate is of finite thickness, it can be considered as infinity and another boundary condition can be written as At t > 0,
y ˆ /, CC ˆ C0
(5.28c)
To solve equation 5.27 along with equations 5.28a, b and c, we define two new variables 0011 ˆ y/2(DCt)1/2 pˆ
(5.29)
dCC d0011
(5.30)
We know, @CC @CC @0011 y @CC 0011 @CC ˆ ˆÿ ˆÿ 1=2 @0011 @t @0011 @t @00112 2t 4t…DC t† @CC 1 @CC ˆ 1=2 @y @0011 2…DC t†
and
@ 2 CC 1 @ 2 CC ˆ @y2 4DC t @00112
Using the above relationships and equation 5.30, equation 5.27 can be written as ÿ20011p ˆ
dp d0011
…5:31†
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Fundamentals of metallurgy
The variable 0011 given by equation 5.29 has converted the partial differential equation into an ordinary differential equation. This transformation, which reduces the number of independent variables, is known as the similarity transform. The boundary conditions given by equations 5.28 become At 0011 ˆ 0, At 0011 ˆ /,
CC ˆ CS CC ˆ C0
(5.32a) (5.32b)
Integrating equation 5.31, p ˆ A exp (ÿ00112) where A is a constant. Substituting p from equation 5.30 dCC ˆ A exp …ÿ00112 † d0011 Integrating in the limit 0011 ˆ 0 to 0011 and using the boundary condition equation 5.32a, Z 0011 exp…ÿ00112 †d0011 …5:33† CC ÿ C S ˆ A 0
Using the boundary condition equation 5.32b p Z 1 0019 2 C0 ÿ CS ˆ A exp…ÿ0011 †d0011 ˆ A 2 0 Or,
p A ˆ 2(C0 ÿ CS)/ 0019
Hence equation 5.33 becomes (CC ÿ CS) ˆ (C0 ÿ CS)erf(0011) where
2 erf(0011) ˆ p 0019
Z
0011 0
(5.34)
exp (ÿ00112 ) d0011
Figure 5.11 shows the dimensionless concentration (CC ÿ CS)/(C0 ÿ CS) as a function of y/(4Dct)1/2. This is essentially a plot of erf(0011) with 0011. The figure shows that erf(0011) 0019 1 when 0011 ˆ 2. This indicates that if y/(4Dct)1/2 > 2, CC ˆ C0 or that carbon concentration is the same as the initial concentration. So if maximum time of interest tmax and length in the direction of diffusion, L, are such that L/(4Dctmax)1/2 > 2, the system can be considered as semi-infinite. Example 5.5 A steel plate with 0.2% carbon is exposed to a carburizing atmosphere at 1223K. Carburizing atmosphere maintains 0.5% carbon on the surface of steel. Calculate
Transport phenomena and metals properties
195
1.0 0.9 0.8
Co - Cs
Cc - Cs
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
y/(4DCt)1/2
5.11 Variation of (CC ÿ CS)/(C0 ÿ CS) with y/(4DCt)1/2.
(a) the concentration of carbon at 1 mm away from the surface after 1 hour, (b) the layer thickness where carbon is greater than 0.3% after 1 hour and (c) what should be the carburizing time if the layer thickness of 0.3% carbon is to be doubled? Solution Initial carbon concentration C0 ˆ 0.2, and surface concentration CS ˆ 0.5%. From Table 5.4, at 1223K, diffusivity of carbon DC ˆ 1.2 0002 10ÿ11 m2/s. (a) After 1 hour or 3600s, at y ˆ 1 mm 0011 ˆ y/2(DCt)1/2 ˆ 10ÿ3/{2(1.210ÿ11 0002 3600)1/2} ˆ 2.4 erf(2.4) ˆ 1 From equation 5.34, CC ˆ 0.5 ‡ (0.2 ÿ 0.5) 0002 1 ˆ 0.2% or no change in % carbon. (b) From equation 5.34, (0.3 ÿ 0.5) ˆ (0.2 ÿ 0.5)erf(0011) or erf(0011) ˆ 0.666. Using Fig. 5.11, 0011 ˆ 0.67, from the definition of 0011, y ˆ 2 0002 0.67 0002 (1.2 0002 10ÿ11 0002 3600)1/2 ˆ 0.28 mm. (c) Thickness of the layer having %C 0015 0.3% is twice that of (b) or y ˆ 0.56 mm. Obviously the value of y/(4t)1/2 must be same both for (b) and (c). 0.28/(3600)1/2 ˆ 0.56/t1/2 or t ˆ (0.56/.28)2 0002 3600, or 4 hours. nnn
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Evaporation of liquid Figure 5.12 shows a volatile liquid A in a cylindrical container of cross-sectional area S. A gas stream of A and B is flowing over container. The gas B is insoluble in liquid A. The system is isothermal and the rate of evaporation is so slow that the liquid level in the container does not change appreciably. We want to find out the rate of evaporation and concentration profile of A in the container at steady state. The problem involves diffusion in a cylindrical container, so equation B of Table 5.5 is the appropriate equation. But in this case diffusion of A is in z direction only, so the final equation will be the same both in rectangular and cylindrical coordinate systems. If we consider rectangular coordinates, fluxes NAx and NAy are zero and for cylindrical coordinate, fluxes NAr and NA0012 are zero. Table 5.5 shows that equations A and B become identical for this condition. There is no reaction, so rate of generation of A, RA ˆ 0. We are interested in a concentration profile at steady state so @CA/@t ˆ 0. Hence the conservation equation of species A simplifies to ÿ
dNAz ˆ0 dz
(5.35)
Integrating NAz ˆ C1
(5.36)
From the definition of flux, equation 5.7 dCA ‡ xA …NAz ‡ NBz † dz Since the gas B is insoluble in liquid A, the flux of B is zero. NAz ˆ ÿDAB
(5.37)
NBz ˆ 0
(5.38)
5.12 Evaporation of a liquid A from a container and concentration profiles of A and B.
Transport phenomena and metals properties
197
Using the relation CA ˆ C.xA where C is the total concentration of A and B and equation 5.38, equation 5.37 can be written as NAz ˆ ÿ
DAB C dxA 1 ÿ xA dz
(5.39)
Substituting the above relationship in equation 5.36 ÿDAB C 0001
dxA ˆ C1 dz 1 ÿ xA
Integration of the above equation gives DABC ln(1 ÿ xA) ˆ C1z ‡ C2
(5.40)
Taking the top surface of the liquid as z ˆ 0, the boundary conditions are At z ˆ 0, At z ˆ L,
xA ˆ xA0 xA ˆ xAL
(5.41a) (5.41b)
The values of xA0 and xAL are determined by the vapour pressure of liquid A and the concentration of A in the gas stream that is flowing above the container respectively. From equations 5.40 and 5.41 C2 ˆ DABC ln(1 ÿ xA0)
(5.42)
DAB 1 ÿ xAL ln L 1 ÿ xAo
…5:43†
C1 ˆ C
Substituting equations 5.42 and 5.43 in equation 5.40 and rearranging 0012 0013 1 ÿ xA 1 ÿ xAL S=L ˆ 1 ÿ xAo 1 ÿ xA0
(5.44)
In terms of mole fraction of B, the above equation can be written as xB/xB0 ˆ (xBL/xB0)z/L
(5.45)
Although the flux of B is zero, the above equation shows that the concentration gradient of B is not zero. Diffusive flux due to concentration gradient is exactly balanced by the flux due to bulk flow. Figure 5.12 shows the concentration profiles of A and B. The rate of evaporation of A ˆ S.NAz zˆ0
(5.46)
Using equations 5.36 and 5.43 Rate of evaporation of A = S C
DAB 1 ÿ xAL ln L 1 ÿ xAo
(5.47)
When concentration of A in the gas phase is very small, ln(1 ÿ xAL) 0019 ÿxAL and ln(1 ÿ xA0) 0019 ÿxA0 and the above equation simplifies to
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Fundamentals of metallurgy Rate of evaporation ˆ SDABC
xA0 ÿ xAL L
(5.48)
The above equation can also be obtained from equations 5.36 and 5.37 by neglecting the bulk flow term from the later equation.
5.2.4 Interface mass transfer Quite often we are interested in finding out the rate of transport of a species from one phase to another one. For example, evaporation of a liquid, dissolution of a solid in a liquid, absorption of a gas by liquid, etc. involves transport of a species from one phase to another. These are determined by the flux at the interface between two phases NAx xˆ0 or NA0. Figure 5.13 shows the flux at interface. This interphase transport between phases I and II, is calculated using the concept of mass transfer coefficient. 0012 0013 dCA NA0 ˆ ÿDAB ‡xA0 …NA0 ‡ NB0 † dx xˆ0 ˆ kMA …CA0 ÿ CAb † ‡ xA0 …NA0 ‡ NB0 †
(5.49)
where NA0 is the flux of A from phase I to II, kMA is the mass transfer coefficient of A, CA0 and CAb are concentrations of A in phase II at the interface and bulk respectively and xA0 is the mole fraction of A in phase II at the interface. The interface concentration of A in phase II, CA0, is related to the concentration of A
NA0
Phase I
Phase II
CA0 CAb
NA0
d
C - CAb DA ( dC A ) = DA A0 dx x = 0 d
Stagnant film
NA0
5.13 Transfer of a species from phase I to II.
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199
in phase I. For example: for evaporation of water at 300K, CA0 is determined by vapour pressure of water (phase I) at 300K. For removal of nitrogen from liquid steel by argon purging, CA0 is determined by the partial pressure of nitrogen in equilibrium with nitrogen dissolved in steel (phase I). In a number of practical systems xA0 is very small and equation 5.49 simplifies to NA0 ˆ kMA(CAw ÿ CAb)
(5.50)
A physical picture of mass transfer coefficient can be obtained from film theory. According to this model, at the interface there is a stagnant film of thickness 000e. The concentration of A varies linearly in the film as shown by dotted line in Fig. 5.13 and beyond the film; concentration of A is the same as the bulk concentration. So from equation 5.49 we get kMA ˆ DAB/000e. Using this concept quite often we say that at the interface there is a stagnant film. Mass transfer coefficient is a system property; in the sense it depends on the shape and size of the system from which mass transfer takes place. Besides, it is determined by density, viscosity, diffusivity and velocity of fluid. So kMA = f(001a, 0016, DAB, v, L)
(5.51)
where 001a, 0016 and v are respectively density, viscosity and velocity of fluid and L is the characteristic length of the system. Dimensionless analysis shows that equation 5.51 can be expressed as Sh ˆ f(Re, Sc)
(5.52)
where Sh, Re and Sc are respectively the Sherwood number, Reynolds number and Schmidt number and are defined as Sherwood number
Sh ˆ kMAL/DAB
(5.53)
Reynolds number
Re ˆ 001avL/0016
(5.54)
Schmidt number
Sc ˆ 0016/(001aDAB)
(5.55)
The Sherwood number is the ratio of interface or convective mass transfer to diffusive mass transfer rates. The Reynolds number is the ratio inertial force to viscous force and the Schmidt number is the ratio of momentum diffusivity (0016/001a) to mass diffusivity Correlations for mass transfer coefficient The mass transfer from a sphere is mostly calculated by Ranz and Marshall correlation Sh ˆ 2 ‡ 0.6(Re)1/2(Sc)1/3
(5.56)
The characteristic length for both Sh and Re is the particle diameter. So Re appearing in equation 5.56 is called the particle Reynolds number. When there is
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no fluid flow, the above relation predicts Sh ˆ 2 which is obtained by theoretical calculation. This correlation is often used for calculation of mass transfer in a packed bed. Example 5.6 A spherical drop of water of 0.5 mm diameter is falling through dry still air at a velocity of 0.75 msÿ1. Assuming that the temperature of water droplet and air is at 300K, calculate the instantaneous evaporation rate. At 300K vapour pressure of water is 0.035 bar. Density and viscosity of air at 300K are 1.177 kg mÿ3 and 18.53 0002 10ÿ6 kg mÿ1 sÿ1 respectively. 1 bar ˆ 105 Pa and R ˆ 8.314 Pa.m3.molÿ1 Kÿ1, Atmospheric pressure ˆ 1.013 0002 105 Pa. Solution Rate of evaporation ˆ surface area of droplet 0002 NA0 ˆ 0019(0.5 0002 10ÿ3)2 NA0 From equation 5.49, NA0 ˆ [kM(CA0 ÿ CAb) ‡ xA0(NA0 ‡ NB0)] In the present case A is H2O and B is air. Since air is not soluble in water, NB0 ˆ 0, NA0 ˆ kM(CA0 ÿ CAb)/(1 ÿ xA0) Rate of evaporation ˆ 250019 0002 10ÿ8 [kM(CA0 ÿ CAb)]/(1 ÿ xA0) CA0 ˆ pA0/RT ˆ 0.035 0002 105/(8.314 0002 300) ˆ 1.4 mole/m3 CAb ˆ 0 xA0 ˆ pA0/atmospheric pressure ˆ 0.035 0002 105/1.013 0002 105 ˆ 0.0345 DH2O-air at 300K ˆ 0.255 0002 10ÿ4 m2/s from equation 5.16 Re ˆ 001avL/0016 = 1.177 0002 0.75 0002 0.5 0002 10ÿ3/18.53 0002 10ÿ6 ˆ 23.8 Sc ˆ 0016/(001aDAB) ˆ 18.53 0002 10ÿ6/(1.177 0002 0.255 0002 10ÿ4) ˆ 0.617 From equation 5.56, Sh ˆ 2 ‡ 0.6 0002 (23.8)1/2 0002 (0.617)1/3 ˆ 4.49 KM = Sh.DAB/L ˆ 4.49 0002 0.255 0002 10ÿ4/0.5 0002 10ÿ3 ˆ 0.229 m2/s So rate of evaporation ˆ 250019 0002 10ÿ8 0002 0.229 0002 1.4/(1 ÿ 0.0345) ˆ 26.1 0002 10ÿ8 mole/s
5.3
nnn
Heat transfer
When a hot object is kept in atmosphere, heat is transferred from the surface of the object to the surroundings by convection and radiation. As the surface is cooled, heat is transferred from the interior of the object to the surface by conduction. These processes continue until the temperature of the object becomes the same as that of the surroundings. In all the three modes of heat transfer namely, conduction, convection and radiation; the driving force is temperature difference. The conduction of heat in solid and liquid takes place due to molecular vibration and that in gas by molecular collision. Convective heat transfer has
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201
two limiting cases: forced and natural. In forced convection the flow is due to external forces whereas in natural convection, it is due to density difference. When a hot object is kept in stagnant air, the heat transfer is by natural convection. The air in contact with the hot object becomes hot and thereby lighter and moves up and cold air from the surroundings occupies the space. This leads to fluid motion that is termed as natural convection. On the other hand, if the hot object is cooled by blowing air over it, the mechanism of heat transfer is forced convection. Both conduction and convection of heat takes place through a material medium, but energy is transported through empty space by radiation.
5.3.1 Conduction Let us consider a plate that is at a uniform temperature T0. At time, t ˆ 0, the left face is suddenly raised to a temperature T0, Fig. 5.14(a). Due to difference in temperature, heat flows from left to right by conduction and temperature along the length increases with time as shown in Fig. 5.14(b). After sufficiently long time, temperature profile attains the steady state Fig. 5.14(c). Both during steady and unsteady state, heat flows from left to right. This flow of heat is measured in terms of heat flux defined as the amount of heat flowing through unit area per unit time in a direction normal to the area. The heat flux by conduction is given by qx ˆ ÿk
dT dx
…5:57†
t=0
Short time
T0
T0
T0
x=0 (a)
x=L
Steady state
T0
T(x,t)
T
x=0 (b)
x=L
T(x)
0
x=0
x=L
(c)
5.14 Temperature profile in a slab. (a) The left face of the slab is suddenly raised to T0 at t ˆ 0. (b) Temperature profile after some time. (c) Temperature profile at steady state.
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where qx is the heat flux in x direction W/m2, k is the thermal conductivity of material Wmÿ1Kÿ1 and dT/dx is the temperature gradient K/m. The negative sign in equation 5.57 indicates that heat flux and temperature gradient are in opposite direction. In Fig. 5.14 temperature decreases with increase of x, dT/ dx is negative, and heat flux is positive or it is in direction of increasing x. It should be noted that the flux is a vector and its sign gives the direction of flow. Equation 5.57 is the one-dimensional form of Fourier's law of heat conduction. In the example above, temperature depends only on its distance from the face or T is a function of x only. But if temperature varies from point to point, i.e., if T is a function of x, y and z, heat will flow in all directions. It is given by q ˆ ÿkrT
(5.58)
The components of heat flux in different coordinate systems are similar to that for molar flux JA given in Table 5.1. The above equation is the threedimensional form of Fourier's law of heat conduction and is valid for a medium whose conductivity is the same in all direction. These types of materials are known as isotropic material. But a number of materials, for example laminated composites, unidirectional fibrous composite material like bamboo etc., are not isotropic, i.e., conductivity is different in different directions. For non-isotropic materials equation 5.58 takes the form 0014 0015 @T @T @T …5:59† ‡ j ky ‡ k kz qm ˆ ÿ i kx @x @y @z where kx, ky and kz are thermal conductivity of the material and i, j, k are unit vectors in x, y, and z directions respectively. Example 5.7 Two faces of a stainless steel plate of 0.1 m2 area and 4 mm thickness are kept at 723K and 323K respectively. The temperature profile in the plate is linear. Calculate the heat flux and total heat transferred in one minute through the plate. Thermal conductivity of stainless steel is 19 Wmÿ1Kÿ1. Solution The problem involves heat transfer only in one direction and heat flux is given by equation 5.57. Let us assume that the face at temperature 723K as x ˆ 0 and that at 323K as x ˆ 0001x. Since the temperature profile is linear, dT/dx ˆ (T 0001x ÿ T 0)/0001x ˆ (323 ÿ 723)/(4 0002 10ÿ3) = ÿ105 K/m. So heat flux ˆ 19 0002 105 Wmÿ2. The direction of heat flux is +ve direction of x. Total heat transferred in one minute = area of the plate 0002 heat flux 0002 time in seconds ˆ 0.1 0002 19 0002 105 0002 60 ˆ 1.14 0002 107 J
nnn
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5.3.2 Thermal conductivity According to kinetic theory of gases, thermal conductivity of monatomic gases is independent of pressure and is proportional to square root of temperature. The predicted pressure dependence is valid up to 10 atmospheres but temperature dependent is too weak. The temperature dependence of the thermal conductivity of gas can be expressed as k ˆ k0(T/T0)n
(5.60)
where k0 is thermal conductivity at T0 K. Eucken's equation k ˆ (Cp ‡ 1.25R/M)0016
(5.61)
is widely used for estimation of thermal conductivity of gases. Cp, R, M and 0016 are respectively specific heat, gas constant, molecular weight and viscosity. Table 5.6 gives the thermal conductivity of some common gases. It shows that thermal conductivity of hydrogen is much higher than other gases. Thermal conductivity of liquid Thermal conductivity of liquid depends on the nature of the liquid. Liquid metals have a much higher thermal conductivity compared with water or slag. Table 5.7 shows the thermal conductivity of different liquids. Thermal conductivity of solids Energy is transferred due to elastic vibrations of the lattice in solids. In the case of metal, besides the above mechanism, free electrons moving through the lattice carry energy. Heat transferred by the latter mechanism is greater than that Table 5.6 Thermal conductivity of some common gases (Wmÿ1 Kÿ1) Gases
H2
H2O
CO
CO2
Air
k 0002 103 at 400K k 0002 103 at 800K
226 378
26.1 59.2
31.8 55.5
24.3 55.1
33.8 57.3
Table 5.7 Thermal conductivity of liquids (Wmÿ1 Kÿ1) Material
Temp. K
k
Water Glycerol Slag
293 293 1873
0.59 0.29 4.0
Material Aluminum Copper Iron
Temp. K
k
933 1600 1809
91 174 40.3
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Table 5.8 Thermal conductivity of solids at room temperature Material k Wmÿ1Kÿ1
Al
Cu
Brass
Fe
237
398
127
79
Steel Concrete 52
0.9
Brick 0.6
by the former. So thermal conductivity of metal is much higher than that of nonmetals. Thermal conductivity of pure metal decreases with temperature. Table 5.8 gives thermal conductivity of some solids. Thermal conductivity of porous solid is given by keff ˆ k(1 ÿ 000f)
(5.62)
where keff and k are the thermal conductivity of porous solid and solid respectively and 000f is the void fraction in solid.
5.3.3 Conservation equation Let us consider an elemental volume 0001x0001y0001z as shown in Fig. 5.15. Since energy is conserved, the balance equation is Rate of accumulation of energy in 0001x0001y0001z ˆ rate in ÿ rate out ‡ rate of generation of energy in 0001x0001y0001z volume
(5.63)
In Fig. 5.15, heat enters through three faces and goes out through the corresponding opposite three faces. In the case of stationary solid, heat flux is only by conduction. So following the procedure given in Section 5.2.3, we get qz qy (x + Dx,y + Dy,z + Dz) z
Dz qx
qx
y
Dy x
qy
(x,y,z,)
Dx qz
5.15 Heat balance in a control volume.
Transport phenomena and metals properties @ @qx @qy @qz ÿ ÿ ‡ HG …001aCp T† ˆ ÿ @x @y @z @t
205 (5.64)
where 001aCpT is the heat content per unit volume and HG is the rate of heat generation per unit volume. Assuming 001a, Cp and k are constant and using the definition of flux equation 5.58, the above equation can be written as 0014 2 0015 @T @ T @2T @2T ‡ HG ˆk ‡ ‡ (5.65) 001aCp @t @x2 @y2 @z2 In the case of fluid, the terms `rate in' and `rate out' in equation 5.63 include heat transfer due to bulk flow as well. Similarly if the solid is in motion, in the direction of motion, the terms `rate in' and `rate out' include heat flux due to bulk motion. Table 5.9 gives the general heat balance equation in different coordinate systems. These equations are valid for solids and incompressible fluids. Heat generation within a solid can be due to phase transformation, chemical reaction or electrical heating. In the case of fluid besides the above mechanisms, heat generation can be due to viscous dissipation as well. But normally heat generation due to viscous dissipation is very small and is neglected. Steady state heat conduction Let us consider heat transfer in the plate shown in Fig. 5.14. The appropriate energy balance equation is equation A in Table 5.9. In this case heat transfer takes place only along the x axis that is normal to the plate face. So, heat flux in y and z directions is zero. The plate is stationary so vx, vy and vz are also zero. At the steady state, @T/@t ˆ 0 and there is no heat generation in the plate, HG ˆ 0. Hence equation (A) of Table 5.9 becomes Table 5.9 Energy equation for incompressible media in different coordinate systems Rectangular coordinates 0014 0015 0014 0015 @T @T @T @T @qx @qy @qz 001aCp ‡ ‡ ‡ HG ‡ vx ‡ vy ‡ vz ˆÿ @x @y @z @t @x @y @z Cylindrical coordinates 0014 0015 0014 0015 @T @T v0012 @T @T 1 @ 1 @q0012 @qz 001aCp ‡ ‡ HG ‡ vr ‡ ‡ vz ˆÿ …rqr † ‡ r @0012 @z @t @r @z r @r r @0012 Spherical coordinates 0014 0015 @T @T v0012 @T v001e @T 001aCp ‡ vr ‡ ‡ r @0012 r sin 0012 @001e @t @r 0014 0015 1 @ 2 1 @ 1 @q001e ‡ HG ˆ ÿ 2 …r qr † ‡ …q0012 sin 0012† ‡ r @r r sin 0012 @0012 r sin 0012 @001e
‰AŠ
‰BŠ
‰CŠ
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dq ˆ0 dx Integrating, qx ˆ C1 where C1 is the integration constant. Using the definition of heat flux, equation 5.57, the above equation becomes ÿk
dT ˆ C1 dx
(5.66)
Integrating, ÿkT ˆ C1x ‡ C2
(5.67)
Integration constants C1 and C2 are evaluated from the boundary conditions. At x ˆ 0, T ˆ T0
and
x ˆ L, T ˆ TL
Using these conditions in equation 5.67 C1 ˆ k(T0 ÿ TL)/L
(5.68)
C2 ˆ ÿkT0
(5.69)
and T ˆ T0 ÿ (T0 ÿ TL)
x L
(5.70)
Quite often we are interested in the case where one face is maintained at temperature T0 but the other face is exposed to the atmosphere. In this case the boundary conditions are dT ˆ h(T ÿ Ta) At x ˆ 0, T ˆ T0 and x ˆ L, ÿk (5.71) dx where h is the heat transfer coefficient and Ta is the ambient temperature. C2 is same as that given by equation 5.69. Using equations 5.67 and 5.71 0014 0015 ÿC1 L ‡ kT0 ÿ Ta C1 ˆ h k Rearranging C1 ˆ
T 0 ÿ Ta 1 L ‡ h k
…5:72†
So T ˆ T0 ÿ
x T 0 ÿ Ta k 1 L ‡ h k
…5:73†
Transport phenomena and metals properties
5.16 Electrical analogy for heat loss through a plate. T outer temperature of outer face exposed to atmosphere.
face
207
is the
Heat is lost through the face at x ˆ L, so the rate of heat lost per unit area is 0012 0013 dT HLoss ˆ ÿk dx xˆL At steady state dT/dx is constant so heat loss per unit area is same as heat flux and is equal to C1, see equation 5.67, hence T 0 ÿ Ta …5:74† 1 L ‡ h k Using an electrical analogue, one can say that heat loss is similar to current, the numerator of above equation is the driving force and the denominator is the resistance for heat loss. Two resistances, one due to plate, L/k, and other one due to the interface, 1/h, are in series. Figure 5.16 shows the electrical analogue. HLoss ˆ
Heat loss through composite wall Let us consider heat flux at the steady state through a composite wall made up of two layers shown in Fig. 5.17(a). The thickness of layers 1 and 2 and their thermal conductivities are L1, L2 and k1, k2, respectively. Obviously, at the steady state temperature profiles of the layers are given by equation 5.67 and can be written as:
5.17 (a) Temperature distribution in composite wall and (b) electrical analogue of heat flux.
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Fundamentals of metallurgy For layer 1 For layer 2
TI ˆ C1x ‡ C2
(5.75a)
II
T ˆ C3x ‡ C4
(5.75b)
The integration constants, C1, C2, C3, C4, are determined by the following boundary conditions At x ˆ 0,
TI ˆ T0
(5.76a)
The outer face is exposed to atmosphere, so At x ˆ L1 ‡ L2,
ÿk2(dTII/dx) ˆ h(TII ÿ Ta)
(5.76b)
The walls are in perfect contact, so temperature and heat flux at the interface must be the same. Or At x ˆ L1
TI ˆ TII k1(dTI/dx) ˆ k2 (dTII/dx)
(5.76c) (5.76d)
From equations 5.75 and 5.76, C1 ˆ (k2/k1)C3 C2 ˆ T0 T0 ÿ Ta C3 ˆ ÿ k2 k2 ‡ L2 ‡ L1 h k1 C4 ˆ L1C3(k2/k1 ÿ 1) ‡ T0 Heat loss ˆ (ÿk2dTII/dx)x=L1+L2 ˆ ÿk2C3 ˆ ÿ
T0 ÿ Ta 1 L1 L2 ‡ ‡ h k1 k2
(5.77)
Figure 5.17(b) shows the electrical analogue for heat loss through the composite wall. Using the electrical analogue, we could directly obtain equation 5.77. Temperature distribution in a hollow cylinder Let us consider a hollow cylinder with inner radius r1 and outer radius r2, Fig. 5.18. Inner and outer faces are at T1 and T2 respectively. Obviously, the appropriate equation for energy conservation is equation B in Table 5.9. In the hollow cylinder, heat conduction is in the radial direction only; thereby qz and q0012 are zero. Furthermore, there is no heat generation and all velocity components in equation B are zero, so at steady state the equation simplifies to d …rqr † ˆ 0 dr Integrating rqr ˆ C1
…5:78†
Transport phenomena and metals properties
5.18 Schematic diagram of a hollow cylinder.
Since qr ˆ ÿk ÿk
dT ; dr
dT ˆ C1 =r dr
Integrating ÿkT ˆ C1 ln r ‡ C2 The boundary conditions are At r ˆ r1, At r ˆ r2,
T ˆ T1 T ˆ T2
Using the above boundary conditions 0012 0013 r ln r1 ˆ T1 ÿ …T1 ÿ T2 † 0012 0013 r2 ln r1
209
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Fundamentals of metallurgy
5.19 Temperature distribution in a hollow cylinder and sphere. Ratio of inner to outer radius (r1/r2) is 0.3.
The above equation can be written in a dimensionless form as 0012 0013 r ln T ÿ T1 r1 ˆ 0012 0013 r2 T2 ÿ T 1 ln r1
…5:79†
Figure 5.19 shows the dimensionless temperature profile in a hollow cylinder. It shows that temperature profile deviates considerably from linearity. Steady state temperature distribution in a spherical shell Let us consider a hollow spherical shell of inner radius r1 at temperature T1 and outer radius r2 at T2. Heat conduction takes place only in the radial direction. The appropriate equation is equation C in Table 5.9 which simplifies for the steady state conduction problem without heat generation to d 2 …r qr † ˆ 0 dr and boundary conditions are At r ˆ r1, At r ˆ r2,
T ˆ T1 T ˆ T2
The solution of the differential equation along with the above boundary
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211
condition is 1 1 ÿ T ÿ T1 r r ˆ 1 T 2 ÿ T1 1 1 ÿ r1 r2
…5:80†
Figure 5.19 shows the dimensionless temperature profile in a hollow sphere. Comparison of temperature profiles in a hollow cylinder and sphere shows that the curvature of the latter is greater.
5.3.4 Heat transfer coefficient At solid±fluid interface, heat flux from solid to fluid or fluid to solid is defined as q ˆ h(Ts ÿ Tb)
(5.81)
where Ts is the temperature of solid surface in contact with fluid, Tb is the bulk fluid temperature and h is the heat transfer coefficient, Wmÿ2Kÿ1. Equation 5.81 is also used for heat transfer between two fluids. h is often termed as convective heat transfer coefficient to differentiate it from radiative heat transfer coefficient. The value of h depends on characteristic length of body, fluid properties like thermal conductivity, density and specific heat and velocity of fluid. By dimensional analysis it can be shown that Nu ˆ f(Re, Pr)
(5.82)
where Nusselt number Prandtl number
Nu ˆ hL/k Pr ˆ Cp0016/k ˆ 0017/000b
where L is the characteristic length of the system and ˆ k/001aCp is thermal diffusivity and 0017 ˆ 0016/001a is momentum diffusivity or kinematic viscosity. The Nusselt number measures the ratio of interface or convective heat flux to conductive heat flux and the Prandtl number is the ratio of thermal diffusivity to momentum diffusivity. The heat transfer from a sphere is often calculated by Ranz and Marshall correlation Nu ˆ 2 ‡ 0.6(Ref)1/2(Prf)1/3
(5.83)
The characteristic length for both Nu and Re is the particle diameter. The suffix f in Re and Pr indicates that these should be evaluated at the film temperature which is taken as average of surface and bulk temperature. This relationship is also often used for calculation of heat transfer in the packed bed. Comparison of equations 5.56 and 5.83 shows that the relationship between Sh, Re and Sc
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Fundamentals of metallurgy
number for mass transfer is the same as that between Nu, Re and Pr for heat transfer. This is true not only for a sphere but for other systems as well. Example 5.8 Show that when a hot sphere is placed in stagnant air, in the absence of natural convection, Nu ˆ 2. Solution Let T0 be the temperature of the sphere or temperature of air in contact with the sphere and T/ is the temperature of air far away from it. Stagnant air around the sphere is an infinitely large hollow sphere whose inner radius is the radius of the sphere r0 and the outer radius r2 ˆ /. So equation 5.80 simplifies to 0012 0013 1 1 T ˆ T0 ÿ …T0 ÿ T/ † ÿ r0 r0 r r0 r Heat flux at the outer face of the sphere is 0012 0013 dT T0 ÿ T / ÿk ˆk dr rˆr0 r0 T ˆ T/ ‡ …T0 ÿ T/ †
Equating the above equation with equation 5.81, h ˆ k/r0 ˆ 2k/D since in the case of the sphere the characteristic length is the diameter of the sphere, D, Nu ˆ hD/k ˆ 2
nnn
5.3.5 Radiation A hot body emits radiation in a continuous band of wavelength in the range of 0.1 to 100 microns. This is known as thermal radiation and it includes ultra violet, visible range and infra red of the electromagnetic radiation spectrum. The proportion of different frequencies of radiation emitted by a hot body depends on its temperature and thereby its colour changes with temperature. Like all electromagnetic waves, heat radiation travels through space at the velocity of light. Emissivity and absorptivity When radiation falls on an object, a part of it is reflected, a part is transmitted through the object and the rest is absorbed, Fig. 5.20. If G is the flux of radiant
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213
5.20 Schematic illustration of absorption, reflection and transmission of radiation falling on a body.
energy Wmÿ2, then or
G ˆ G ‡ 001aG ‡ 001cG ‡001a‡001c ˆ1
(5.84)
where is the fraction of incident radiation absorbed or absorptivity, 001a is the fraction reflected or reflectivity and 001c is the fraction transmitted or transmissivity. Most of the solids and liquids are opaque to thermal radiation, so the above equation simplifies to ‡001aˆ1
(5.85)
For any real body absorptivity is less than unity and varies with the frequency of radiation. A body for which is a constant over the entire range of frequency is known as a grey body. This is a hypothetical body but we idealize all real bodies as grey bodies for thermal radiation calculations. A limiting case of grey body is ˆ 1 for all frequencies and temperature. This is known as a black body. A cavity absorbs all radiation falling on it and hence is a black body. Let q and qb be, respectively, total radiant energy emitted per unit area per unit time (Wmÿ2) by a real surface and a black body when they are at the same temperature. The ratio q/qb is known as emissivity 000f of the real surface. Or 000f ˆ q/qb
(5.86)
Let us consider that a grey body of surface area A is enclosed in a cavity and both grey body and cavity are in thermal equilibrium. Obviously, for thermal equilibrium, the energy absorbed by the grey body must be equal to that emitted
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by it. Since the cavity is a black body, radiation emitted by it is qb Wmÿ2 and energy absorbed by the grey body is A000bqb. Hence Aq ˆ A000bqb
(5.87)
Comparing equations 5.86 and 5.87 ˆ000f
(5.88)
So, at a given temperature emissivity and absorptivity are equal for any solid surface. This is Kirchhoff's law. Highly polished metal surfaces reflect most of the radiations falling on them so their emissivity is very low. It is in the range of 0.015±0.06. Emissivity of refractory bricks lies between 0.85 and 0.95. The total energy emitted per unit area per unit time by a black surface is given by the Stefan±Boltzmann law. qb ˆ 001bT4
(5.89)
The Stefan±Boltzmann constant 001b ˆ 5.67 0002 10ÿ8 Wmÿ2Kÿ4. For non-black bodies, the radiant energy emitted per unit area per unit time is q ˆ 001b000fT4
(5.90)
View factor Let us consider exchange of radiation between two black surfaces 1 and 2 shown in Fig. 5.21. Obviously only a fraction of radiation emitted by surface 1 will be intercepted by surface 2 and vice versa. So radiation energy transferred from 1 to 2 and 2 to 1 can be written as Q1!2 ˆ qb1A1F12 Q2!1 ˆ qb2A2F21 where F12 is the fraction of radiation energy emitted by the unit area of surface 1 which is intercepted or viewed by surface 2 and F21 is the fraction of radiation energy emitted by the unit area of surface 2 which is intercepted by surface 1. These are known as the view factor. The view factor is a geometric factor and can be calculated analytically. The plots of these values for different geometries
5.21 Radiation exchange between two black bodies.
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are available in textbooks on heat transfer. It can easily be proved that the fraction of total energy emitted by surface 1 and intercepted by surface 2 is equal to fraction of total energy emitted by surface 2 and intercepted by surface 1. Or A1F12 ˆ A2F21. This is known as the reciprocity relation. Hence net exchange of radiation energy between 1 and 2 is Q1ÿ2 ˆ Q1!2 ÿ Q2!1 ˆ A1F12(qb1 ÿ qb2) ˆ A2F21(qb1 ÿ qb2) (5.91) The definition of the view factor indicates that the view factor for two large parallal plates of equal dimension separated by a small distance is unity since radiation emitted by one plate is fully intercepted by the other. Also, if an object 1 with surface area A1 is surrounded by object 2 having surface area A2, F12 ˆ 1 since all radiations emitted by surface 1 are intercepted by 2, but F21 < 1. Using the reciprocity relation, F21 ˆ A1/A2. Since the view factor is the fraction of radiation emitted by a surface and intercepted by another surface, then if a surface 1 is enclosed by surfaces 2, 3, . . . n F11 ‡ F12 ‡ F13 ‡ . . . F1n ˆ 1
(5.92)
If the surface 1 is convex, it will not intercept any radiation emitted by itself and F11 ˆ 0. Heat exchange between grey bodies Real bodies are considered as grey bodies where < 1. So when radiation falls on a grey body, a part is reflected back and treatment becomes complex because of multiple reflection of radiation between surfaces. To overcome this problem, we define G as total incident radiation flux on a surface (Wmÿ2) and J as the total radiation flux leaving the surface (Wmÿ2). Obviously J has two components, namely radiation emitted by the surface, 000fqb, and radiation reflected by the surface, 001aG. J is known as radiosity. So J ˆ 000fqb ‡ 001aG Using equation 5.85 and 5.88, 001aˆ1ÿ000f Hence, J ˆ 000fqb ‡ (1 ÿ 000f)G Net heat flux leaving the surface is qnet ˆ J ÿ G Substituting the value of G from equation 5.93 000f …qb ÿ J† qnet ˆ 1ÿ000f
(5.93)
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5.22 Electrical analogue of radiation heat exchange between grey bodies. (a) Net heat flow from a surface of area A. (b) Heat exchange between two grey bodies.
Therefore net heat flow from a surface of area A is 000f A…qb ÿ J† (5.94) Qnet ˆ 1ÿ000f Using an electrical analogue, net heat flow can be considered as current. The driving force for net heat flow from a surface is (qb ÿ J) and the resistance is (1 ÿ 000f)/(A000f). Figure 5.22(a) shows the electrical analogue for a grey surface. Now let us consider radiative energy exchange between two grey bodies 1 and 2. Q1ÿ2 ˆ Q1!2 Q2!1 ˆ A1F12J1 ÿ A2F21J2 Using the reciprocity theorem, the above equation can be written as Q1ÿ2 ˆ
J1 ÿ J2 1 A1 F12
…5:95†
Using the electrical analogue, the numerator of the above equation can be identified as the driving force and the denomenator as the resistance for heat exchange. To evaluate J1 and J2 and thereby Q1±2, we draw the electrical analogue of heat exchange between two grey bodies, shown in Fig. 5.22(b). It shows that the driving force heat flow from surface 1 is (qb1 ÿ J1) and resistance is (1 ÿ 000f1)/(A1000f1) (equation 5.94). The driving force for heat exchange between surfaces 1 and 2 is (J1 ÿ J2) and resistance is (1/A1F12) (equation 5.95) and finally the driving force for heat flow to surface 2 is (J2 ÿ qb2) and the resistance is (1 ÿ 000f2)/(A2000f2). So qB1 ÿ qb2 …5:96† Q1ÿ2 ˆ 1 ÿ 000f1 1 1 ÿ 000f2 ‡ ‡ A 1 000f1 A 2 000f2 A1 F12 It can be noted that the above equation reduces to equation 5.91 for black bodies. Let us consider two large parallel plates of equal size separated by a short distance. Here A1 ˆ A2 ˆ A and F12 ˆ 1, hence equation 5.96 simplifies to Q1ÿ2 ˆ
A…qb1 ÿ qb2 † 1 1 ÿ1‡ 000f1 000f2
…5:97†
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Quite often, we encounter the problem where a small convex grey body is enclosed in an isothermal enclosure. The latter can be treated as a cavity or a black body. Considering the grey body as surface 1 and the enclosure as surface 2, F12 ˆ 1 and 000f2 ˆ 1, so equation 5.96 simplifies to Q1±2 ˆ A1000f1(qb1 ÿ qb2)
(5.98)
In a real system, gases are mostly present. So heat exchange by radiation between surfaces takes place through them. Gases with symmetric molecules like He, H2, O2, N2, etc. are transparent to thermal radiation so their presence does not affect heat exchange by radiation. But CO, CO2, H2O, SO2, NH3, HCl, etc. are not transparent to thermal radiation. They absorb and emit thermal radiation in some narrow bands of wavelengths. For example, CO2 absorbs radiation in wavelength ranges of 2.4±3, 4±4.8 and 12.5±16.5 0016m. So if these gases are present in significant amount as in the case of fuel-fired furnaces, their effect on radiation heat exchange cannot be neglected.
5.4
Fluid flow
If we allow water to flow through a long horizontal glass tube and inject a dye into it, we will notice that the dye moves in a straight line at low flow rates of water. But the dye gets dispersed throughout the cross-section of the tube at a short distance from the entry point at high flow rates. Figure 5.23 schematically shows behaviour of dye. This experiment was first performed by Osborne Reynolds in 1883 to show that there are two types of flow behaviour of a fluid. At low flow rates, fluid elements maintain their own path. But at high flow rate, swirling or circular motion is generated within small packets of fluids, known as eddies. These eddies move in random fashion within the fluid and are continuously formed and destroyed by interaction with the surrounding fluid.
5.23 Behaviour of dye streak injected with water in a horizontal tube.
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5.24 Schematic diagram of laminar and turbulent velocity at a point at steady state.
This leads to rapid mixing of dye. The size of eddies can vary from fractions of a millimetre to fairly large values. The first type of flow is known as laminar flow and the second type is known as turbulent flow. Figure 5.24 shows schematically the velocity at a point in laminar and turbulent flow at steady state. Turbulent velocity at a point fluctuates about a mean value because of movements of eddies. This fluctuation is often 20±30% of the mean value. The transition from laminar to turbulent does not take place suddenly. As velocity is increased a transition region occurs, when the flow becomes unstable and forms local turbulent spots or eddies. With further increases in velocity, local turbulent spots spread and finally the flow becomes fully developed turbulent flow. In the transition region, the dye streak in the pipe flow becomes wavy. Although most of the flows of metallurgical systems are turbulent, in the present section we will primarily discuss laminar flow.
5.4.1 Newton's law of viscosity Let us consider a liquid between two parallel plates as shown in Fig. 5.25(a). Initially the liquid is at rest and at time t ˆ 0, the top plate is given a constant velocity vx. The fluid, which is in contact with the top plate, moves with the velocity vx. The top layer of liquid drags the layer just below it. Similarly this layer in turn drags the layer below it and this continues. Figure 5.25(b) shows the velocity profile after a short time where velocity profile is changing with time. Figure 5.25(c) shows the steady state velocity profile. The velocity of any layer is less than that of the upper one and the velocity of the bottom layer,
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5.25 Velocity profile of fluid between two parallel flat plates. (a) At t ˆ 0, top plate is set to motion with velocity vx. (b) Velocity profile after a short time, unsteady state. (c) Velocity profile at steady state.
which is in contact with the stationary plate, is zero. The motion of the top plate has resulted in x momentum (or x component of momentum) which travelled in y direction due to viscous drag. This momentum flux (amount of momentum transferred per unit area per unit time in a direction normal to the area) is given by: 001cyx ˆ ÿ0016
dvx dy
…5:99†
where 0016 is the viscosity of fluid. The negative sign in the equation shows that the direction of momentum transfer is from high velocity to low velocity. Equation 5.99 is Newton's law of viscosity. The above equation has another meaning. To move the top plate with velocity vx, a shear force has to be applied. This shear force per unit area or shear stress is 001c yx. The subscript yx indicates that the shear force is in x direction and is acting on a plane of constant y. The unit of shear stress is Pa or kgmÿ1sÿ2, the same as that of momentum flux, and the unit of viscosity is kgmÿ1sÿ1 or Pa.s. The fluids that obey this law, i.e. the fluids for which 0016 is independent of shear rate, (dvx/dy), is known as Newtonian fluid. All gases and most of the simple liquids follow Newton's law. When 001c yx is considered as momentum flux, y gives the direction of momentum flux and x the component of momentum. On the other hand when 001c yx is considered as shear stress, it is in the x direction and is acting on a fluid surface of constant y. So the direction of momentum flux and shear stress are not the same. Shear stress is always in the direction of velocity or momentum component. Although equation 5.99 has the same form as that of mass flux given by equation 5.3 or heat flux (equation 5.57), there is a basic difference between this equation and other two. Velocity is a vector but temperature and concentration are scalar. So the heat and mass fluxes are vectors and have three components but the double suffix in 001c yx indicates that in three-dimensional problem 001c has nine components and it is the stress tensor.
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Example 5.9 Two flat plates of area 0.06 m2 are 0.05 mm apart (Fig. 5.25). The space between the plates is filled by a lubricating oil of viscosity 0.2 kgmÿ1sÿ1. The lower plate is stationary and the upper plate moves with a velocity of 20 mm/s. Calculate the momentum flux from the upper to lower plate at the steady state. What is the force required to keep the upper plate moving? Solution At the steady state, the velocity profile is linear as shown in Fig. 5.24. Taking the top plate as y ˆ 0, dvx ˆ …0 ÿ 20 0002 10ÿ3 †=…0:05 0002 10ÿ3 † ˆ ÿ400 sÿ1 dy Momentum flux 001c yx ˆ ÿ0.2 0002 (ÿ400) ˆ 80 kgmÿ1sÿ2 001c yx is the shear force on the top plate. So the force required to keep the plate moving is F ˆ 001c yx 0002 area of plate ˆ 80 0002 0.06 ˆ 4.8 N
nnn
5.4.2 Viscosity of gases and liquid The kinetic theory of gases shows that viscosity of gases is directly proportional to the square root of temperature and independent of pressure. The latter is found to be true when pressure is less than 10 atmospheres but exponents of temperature for real gases are higher. The temperature dependence of viscosity is 0016 ˆ 00160 (T/T0)n
(5.100)
where 0016 and 00160 are viscosity at temperature T and T0 respectively. The exponent n lies in the range 0.6±1.0. Table 5.10 gives the viscosity of some common gases
Table 5.10 Viscosity of gases (kgmÿ1 sÿ1 or Pa.s) Gas 00160002 105 at 400K 00160002 105 at 800K
H2
H2O
CO
CO2
Air
1.09 1.77
1.32 2.95
2.21 3.54
1.94 3.39
2.29 3.64
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Viscosity of liquids Viscosities of liquids vary widely depending on the nature of liquid. Viscosities of water and liquid metal are very low but that of slag and some of the organic liquids are very high. Viscosity of liquid decreases with temperature. Table 5.11 shows the viscosity of several liquids. Table 5.11 Viscosity of liquids (kgmÿ1 sÿ1 or Pa.s) Material
Viscosity
Water at 293K Glycerin at 293K Lead at 700K Iron at 2000K Slags
0.86 0002 10ÿ3 1.49 2.15 0002 10ÿ3 5.6 0002 10ÿ3 0.1ÿ10
5.4.3 Conservation of momentum Quite often, we need to find out the velocity distribution in a system. This is obtained by solving the equation of continuity or conservation of mass and conservation of momentum or equation of motion. These equations are Rate of mass accumulation ˆ rate of mass in ÿ rate of mass out Rate of momentum accumulation ˆ rate in ÿ rate out ‡ sum of forces acting on system In Sections 5.2.3 and 5.3.3 we have derived the conservation of mass and heat in a differential volume. Following the same procedure, the equation of continuity and conservation of momentum can be derived. The final form of these equations for incompressible Newtonian fluids in different coordinate systems is given in Tables 5.12 and 5.13. The equation of motion given in Table 5.13 is known as Navier±Stokes equation. Since all liquids are incompressible and even gases can be considered as incompressible if the velocity is much less than that of sound, these equations have a very wide applicability. The equation of motion contains four unknowns, three components of velocity and pressure. The three equations of motion given in Table 5.13 along with the equation of continuity given in Table 5.12 are the required four equations for the four unknowns. Flow through a pipe Let us consider steady state liquid flow through a horizontal pipe as an example of the application of the equation of motion. The flow is laminar and we are
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Table 5.12 Equation of continuity for incompressible fluid Rectangular coordinates @vx @vy @vz ‡ ‡ ˆ0 @x @y @z
‰AŠ
Cylindrical coordinates 1 @ 1 @v0012 @vz ‡ ˆ0 …rvr † ‡ @z r @r r @0012
‰BŠ
Spherical coordinates 1 @ 2 1 @ 1 @v001e ˆ0 …r vr † ‡ …v0012 sin 0012† ‡ r2 @r r sin 0012 @0012 r sin 0012 @001e
‰CŠ
interested in the velocity profile away from the entrance zone where the entrance effect is absent and the flow is fully developed. Since the system is cylindrical, we use a cylindrical coordinate system. Fluid flows in the z direction, so velocity components in the radial and 0012 direction, vr and v0012 are zero. Using the above considerations, the equation of continuity in Table 5.12, equation B, simplifies to @vz ˆ0 …5:101† @z The above equation indicates that vz is independent of z. The relevant equation of motion is equation F in Table 5.13. This equation gets simplified for the following reasons: (a) Cylindrical symmetry suggests that vz is independent of 0012 and the equation of continuity, equation 5.101, shows that vz is independent of z. So all derivatives of vz with respect to z and 0012 are zero. Table 5.13 Equation for motion for Newtonian fluid with constant 001a and 0016 Rectangular coordinates x component 0014 2 0015 0014 0015 @vx @vx @vx @vx @ vx @ 2 vx @ 2 vx @p ÿ 001a ‡ ‡ ‡ vx ‡ vy ‡ vz ˆ0016 ‡ 001agx @t @x @y @z @x2 @y2 @z2 @x y component 0014 2 0015 0014 0015 @vy @vy @vy @vy @ vy @ 2 vy @ 2 vy @p 001a ‡ ‡ ÿ ‡ vx ‡ vy ‡ vz ˆ0016 ‡ 001agy @t @x @y @z @x2 @y2 @z2 @y z component 0014 2 0015 0014 0015 @vz @vz @vz @vz @ vz @ 2 vz @ 2 vz @p 001a ‡ ‡ ÿ ‡ vx ‡ vy ‡ vz ˆ0016 ‡ 001agz @t @x @y @z @x2 @y2 @z2 @z
‰AŠ
‰BŠ
‰CŠ
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Table 5.13 Continued Cylindrical coordinates r component 0014 0015 @vr @vr v0012 @vr v20012 @vr 001a ‡ vr ‡ ÿ ‡ vz @t @r r @0012 r @z 0014 001a 0015 001b 2 @ 1 @ 1 @ vr 2 @v0012 @ 2 vr @p ˆ0016 ÿ ‡ ÿ …rvr † ‡ 2 ‡ 001agr @z2 @r r @r r @00122 r2 @0012 @r 0012 component 0014 0015 @v0012 @v0012 v0012 @v0012 vr v0012 @v0012 001a ‡ vr ‡ ‡ ‡ vz @t @r r @0012 r @z 001b 0014 001a 0015 @ 1 @ 1 @ 2 v0012 2 @vr @ 2 v0012 1 @p ‡ ˆ0016 ‡ ÿ …rv0012 † ‡ 2 ‡ 001ag0012 @z2 @r r @r r @00122 r2 @0012 r @0012 z component 0014 0015 @vz @vz v0012 @vz @vz 001a ‡ vr ‡ ‡ vz @t @r r @0012 @z 001a 001b 0014 0015 1 @ @vz 1 @ 2 vz @ 2 vz @p ‡ 2 ˆ0016 ‡ 2 ÿ r ‡ 001agz @r @z r @r r @00122 @z Spherical coordinates r component ' # @vr @vr v0012 @vr v001e @vr …v20012 ‡ v2001e † 001a ‡ vr ‡ ‡ ÿ r @t @r r @r r sin 0012 @001e 0014 0015 2 2 @v0012 2 2 @v001e @p ÿ 2 v0012 cot 0012 ÿ 2 ÿ ˆ 0016 r2 vr ÿ 2 vr ÿ 2 ‡ 001agr r r @0012 r r sin 0012 @001e @r 0012 component ' # @v0012 @v0012 v0012 @v0012 v001e @v0012 vr v0012 v2001e cot 0012 001a ‡ vr ‡ ‡ ‡ ÿ r @t @r r @0012 r sin 0012 @001e r 0014 0015 2 @vr v0012 2 cos 0012 @v001e 1 @p ˆ 0016 r2 v 0012 ‡ 2 ÿ ÿ ÿ ‡ 001ag0012 r @0012 r2 sin2 0012 r2 sin2 0012 @001e r @0012 001e component 0014 0015 @v001e @v001e v0012 @v001e v001e @v001e vr v001e v0012 v001e cot 0012 001a ‡ vr ‡ ‡ ‡ ‡ @t @r r @0012 r sin 0012 @001e r r 0014 0015 2 @vr v001e 2 cos 0012 @v0012 1 @p ÿ 2 ‡ ÿ ˆ 0016 r2 v001e ‡ 2 ‡ 001ag001e r sin 0012 @001e r sin2 0012 r2 sin2 0012 @001e r sin 0012 @001e r2 ˆ
0012 0013 0012 0013 1 @ [email protected] 1 @ @ 1 @2 r ‡ sin 0012 ‡ 2 2 2 2 r @r @r r sin 0012 @0012 @0012 r sin 0012 @001e2
‰DŠ
‰EŠ
‰FŠ
‰GŠ
‰HŠ
‰IŠ
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(b) Since we are considering steady state, the time derivative is zero. (c) For a horizontal pipe, the component of acceleration due to gravity in z direction gz is zero. (d) Velocity components in radial and 0012 direction, vr and v0012 are zero. So the z component of the equation of motion in the cylindrical coordinate simplifies to 0012 0013 1 d dvz dp 0016 ˆ r …5:102† dr r dr dz Since vz is a function of r only, the partial derivative is replaced by the total. Integrating, dvz r2 dp ˆ ‡ C1 dr 2 dz Integrating again, 0016r
r2 dp ‡ C1 ln r ‡ C2 4 dz The boundary conditions are 0016vz ˆ
At r ˆ 0, vz is finite At r ˆ r0, vz ˆ 0 Using the above conditions, C1 ˆ 0
and
C2 ˆ ÿ
r20 dp 4 dz
Hence vz ˆ
r2 ÿ r20 dp 40016 dz
…5:103†
If the length of pipe is L, and inlet and outlet pressures are P0 and PL respectively, dp/dz ˆ (PL ÿ P0)/L. So the above equation can be written as vz ˆ
…r20 ÿ r2 †…P0 ÿ PL † 40016L
…5:104†
Figure 5.26 shows the parabolic velocity profile. The maximum velocity is at r ˆ 0 and has the value vz;max ˆ
r20 …P0 ÿ PL † 40016L
Volumetric flow rate is given by Z r0 0019…P0 ÿ PL †r40 20019rvz dr ˆ Qˆ 80016L 0
…5:105†
…5:106†
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5.26 Velocity profile of laminar flow in a pipe.
Equation 5.106 is known as the Hagen±Poiseuille law. The shear force of fluid on the pipe wall is 0012 0013 dvz ˆ 0019r20 …P0 ÿ PL † Fz ˆ 20019r0 L ÿ0016 dr rˆr0
…5:107†
This shows that viscous force counterbalances the force acting on the fluid. When fluid flows through a vertical pipe, viscous forces counterbalance force due to gravity and pressure. The above equations are valid for laminar flow, which is obtained for Reynolds number Re ˆ 001avz,avD/0016, where D ˆ 2r0 and average velocity vz,av ˆ Q/(0019r20 ), is less than 2100. Besides, there is no entrance effect. A distance of Le ˆ 0.0567DRe is required for the development of the parabolic profile given by equation 5.104. Although the critical Reynolds number for transition from laminar flow is normally taken as 2100, pipe flow can be maintained laminarly even at much higher Reynolds numbers in controlled experiments where all external disturbances are avoided. When the Reynolds number is more than 10,000 the flow is fully turbulent. However, even when the flow is turbulent, it is not turbulent near the wall. Figure 5.27 schematically shows the velocity profile near the wall. Close to the wall, the viscous forces dominate and the flow is laminar. This layer is known as the viscous sub-layer or laminar sub-layer. Some distance away from the wall, the velocity profile is relatively flat. This region is the turbulent core or fully developed turbulent flow. The intermediate region is known as the buffer or intermediate zone. Eddies are generated here. The
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5.27 Velocity profile near wall in turbulent flow.
thickness of buffer layer is about six times more than that of viscous sub-layer and total thickness of these two layers is less than a millimetre. When the flow is fully turbulent velocity distribution follows the relation 0012 0013 vz r 1=7 ˆ 1ÿ …5:108† vz;max r0 In the case of laminar flow vz,max/vz,av ˆ 2 and for turbulent flow it is 5/4. Example 5.9 Calculate the maximum flow rate of water through a pipe of diameter 20 mm when the parabolic velocity profile given by equation 5.104 is applicable. What is the corresponding pressure drop per metre length of pipe. Assume viscosity and density of water as 0.86 0002 10ÿ3 kgmÿ1sÿ1 and 1000 kgmÿ3 respectively. Solution Equation 5.104 is valid if Re is less than 2100 or 001avz,avD/0016 ˆ 2100, vz,av ˆ (2100 0002 0.86 0002 10ÿ3)/(1000 0002 20 0002 10ÿ3) ˆ 0.09 m/s ˆ 90 mm/s Flow rate Q ˆ 0019r02vz,av ˆ 28.3 0002 10ÿ6 m3/s From equation 5.106
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(P0 ÿ PL)/L ˆ (80016Q)/(0019r04) ˆ (8 0002 0.86 0002 10ÿ3 0002 28.3 0002 10ÿ6)/(0019 0002 0.014) ˆ 6.2 Pa Since kinematic viscosity of water 0016=001a is very low about 10ÿ6 m2/s, flow is turbulent at a very low flow rate. Liquid metals also have similar low values of kinematic viscosity, so the flow of liquid metal is mostly turbulent. nnn Flow over a flat plate When a fluid flows over an immersed body, such as a flat plate, the velocity of fluid at the solid±fluid interface is same as that of the immersed body. Viscous forces affect the flow near the solid surface but further away the flow is almost inviscid and velocity is same as that of the free stream. This layer where the velocity of fluid changes from that of the solid to the free stream is known as the boundary layer and this type of flow is known as the boundary layer flow. Figure 5.28 schematically shows the velocity profile over a flat plate. The boundary layer thickness is zero initially and it grows as the fluid moves along the plate. In the case of flow through a duct, the viscous boundary layers grow from the walls and fill the entire duct. So, except at the entrance region, there is no boundary layer separating viscous and inviscid flow, as shown in Fig. 5.29. Flow beyond the entrance region is fully developed flow. Let us assume that the fluid stream is approaching the stationary flat plate at a uniform velocity v/. The flow is in the x direction. The z direction is normal to the paper. On the left side of the plate the fluid velocity is uniform, vx ˆ v/. But over the entire surface of the plate (y ˆ 0), vx is zero because there is no slip condition and in the entire boundary layer vx < v/. The fluid whose velocity has decreased moves up and this gives rise to the small vy component in the
5.28 Flow over a flat plate.
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5.29 Velocity profile at the entrance region in a duct.
boundary layer. Thereby within the boundary layer there are two components of velocity vx and vy and outside the boundary layer there is only vx. There is no velocity gradient in the z direction, so @vx/@z ˆ @vy/@z ˆ 0. Since it is a steady state problem, the time derivatives are zero. Besides, in the x direction gravity force is zero and we neglect gravity in the y direction. With these simplifications, the equation of continuity, equation A in Table 5.12, and the equations of motion for the x and y components, equations A and B in Table 5.13, in the boundary layer become @vx @vy ‡ ˆ0 @x @y 0014 2 0015 0014 0015 @vx @vx @ vx @ 2 vx @p 001a vx ÿ ‡ vy ˆ0016 ‡ @x @y @x2 @y2 @x 0014 2 0015 0014 0015 @vy @vy @ vy @ 2 vy @p ÿ ‡ vy ˆ0016 001a vx ‡ 2 2 @x @y @x @y @y
…5:109a† …5:109b† …5:109c†
These equations should be solved for the unknowns vx, vy and p subject to no slip boundary condition on the surface of the plate and inlet and exit boundary conditions. These equations are too complex for analytical solution. However, the equations can be simplified from the following considerations: 1. 2.
3.
vy 001c vx, so the y component of the momentum balance equation, 5.109c can be neglected. @vx @ 2 vx and 0016 2 in equation 5.109b are the net rate of x The terms 001avx @x @x component of momentum transfer in the x direction due to bulk flow and @ 2 vx viscous flow respectively. The term 0016 2 can be neglected since viscous @x diffusion is much smaller than the bulk flow. @p/@x is negligible.
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So the above equations can be simplified to @vx @vy ‡ ˆ0 @x @y 0014 0015 @vx @vx @ 2 vx ˆ0016 2 ‡ vy 001a vx @x @y @y
…5:110a† …5:110b†
The boundary conditions for all x are At y ˆ 0,
vx ˆ vy ˆ 0
(5.111a)
At y ˆ /,
vx ˆ v /
(5.111b)
At x ˆ 0 for all y,
vx ˆ v /
(5.111c)
To solve the above equations let us define `stream function' vx ˆ
@ @y
and
vy ˆ ÿ
as
@ @x
The substitution of vx and vy, defined above, in equation 5.110a shows that the stream function satisfies the equation of continuity so we need to solve only equation 5.110b, which becomes @ @2 @ @2 @3 ÿ ˆ 0017 @y @[email protected] @x @y2 @y3
…5:112†
where 0017 ˆ (0016=001a) is the kinematic viscosity. To solve the above equation, let us define two dimensionless variables f ˆ /(x0017v/)1/2
(5.113)
1/2
(5.114)
0011 ˆ y(v//x0017)
In terms of these new variables, equation 5.112 becomes an ordinary differential equation. where
ff00 ‡ 2f000 ˆ 0 f0 ˆ
(5.115)
df d0011
The boundary conditions given by equations 5.111(a, b, c) become At 0011 ˆ 0, At 0011 ˆ /,
f ˆ f0 ˆ 0 f0 ˆ 1
Equation 5.115 indicates that f and thereby velocity is a function of 0011 only. The equation was first solved by Blasius in 1908 by power series expansion. Figure 5.30 shows the results of numerical solution of the equation. The dimensionless velocity vx/v/ rapidly increases with 0011 and becomes almost unity at 0011 ˆ 5. The
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5.30 Dimensionless velocity vs 0011.
boundary layer thickness 000e is taken as the point where velocity is 99% of the free stream velocity or vx/v/ ˆ 0.99. This corresponds to 0011 0019 5, so 000e(x) ˆ 5(0017x/v/)1/2
(5.116)
The boundary layer thickness grows along the length of flat plate following the square root law and the thickness is inversely proportional to the square root of the free stream velocity. In terms of the Reynolds number, equation 5.116 can be written as 000e(x)/x ˆ 5/Rex1/2
(5.117)
where Rex ˆ 001av/x/0016. The above equation is valid when the Reynolds number is smaller than 5 0002 105 and the flat surface is very smooth. As the fluid moves along the flat surface the Reynolds number increases and the flow becomes unstable which finally leads to a turbulent boundary layer when Re > 5 0002 106. Figure 5.31 shows the turbulent boundary layer. The turbulent boundary layer grows much faster along the length of the plate. Because of the fluctuating nature of turbulent flow the edge of boundary layer is not smooth. Example 5.10 Calculate the boundary layer thickness at the trailing edge of a 0.5 m long plate, for (a) air, (b) water, and (c) mercury at 20 ëC when the free stream velocity is 0.2 m/s. 0017 for air, water and mercury are 1.5 0002 10ÿ5, 0.9 0002 10ÿ6 and 1.1 0002 10ÿ7 m2/s, respectively.
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5.31 Turbulent boundary layer.
Solution (a) For air, Rex ˆ v/x/0017 ˆ 0.2 0002 0.5/1.5 0002 10ÿ5 ˆ 0.67 0002 104 Since Rex < 5 0002 105, equation 5.117 is applicable and 000e ˆ 5 0002 0.5/(0.67 0002 104)1/2 m ˆ 30.5 mm (b) For water, Rex ˆ 0.2 0002 0.5/0.9 0002 10ÿ6 ˆ 1.11 0002 105 Since this is less than 10ÿ6, 000e ˆ 5 0002 0.5/(1.11 0002 105)1/2 m ˆ 7.5 mm (c) For mercury, Rex ˆ 0.2 0002 0.5/1.1 0002 10ÿ7 ˆ 0.91 0002 106 This value lies in the transition range, so the calculated value may not be correct if the plate is not very smooth and the flow is not quiet. However, assuming the equation is valid 000e ˆ 5 0002 0.5/(0.91 0002 106)1/2 m ˆ 2.6 mm The boundary layer thickness is highest for air and lowest for mercury. 000e for air is about four times more than that of water and about 11.5 times more than that of mercury. This is because 0017 air/0017 water is 16.6 and 0017 air/0017 mercury is 136. n n n Example 5.11 A 5 mm thick film of water is flowing over a flat plate of 1 m long. At what distance from the leading edge will the boundary layer thickness be the same as the film thickness. The free stream velocity at the leading edge is 0.5 m/s. 0017 for water is 0.9 0002 10ÿ6 m2/s Solution We want to know the value of x when 000e ˆ 5 mm ˆ 5 0002 10ÿ3 m and v/ ˆ 0.5 m/s. From equation 5.116 000e(x) ˆ 5(0017x/v/)1/2
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So 5 0002 10ÿ3 ˆ 5(0.9 0002 10ÿ6 x/0.5)1/2 or x ˆ 0.5/0.9 ˆ 0.55 m So 0.55 m is the length of entrance zone. After this the flow is fully developed. nnn
5.4.4 Friction factor and drag coefficient When a fluid flows through a pipe, shear force acts on the pipe wall. This is called friction force and is defined as Fk ˆ 12 001av2av fA
(5.118)
where vav is the average velocity of the fluid, f is the friction factor and A is the surface area on which shear force is acting. For flow through a pipe, A ˆ 0019DL. Since Fz defined by equation 5.107 and Fk defined by equation 5.118 are the same, fˆ
D …P0 ÿ PL † 4L 12 001av2av
…5:119†
By dimensional analysis it can be shown that for fully developed flow, friction factor f depends only on the Reynolds number. For flow in a long tube Re < 2.1 0002 103
f ˆ 16/Re f ˆ 0.079/(Re)
1/4
3
2.1 0002 10 < Re < 10
(5.120a) 5
(5.120b)
Equation 5.120(a) is obtained from the Hagen±Poiseuille equation. When the cross-section of pipe is not circular, the effective diameter is calculated using the concept of hydraulic radius. Hydraulic radius is defined as Rh ˆ (cross-sectional area)/(wetted perimeter)
(5.121)
For circular tubing hydraulic radius is D/4. For non-circular tube equation 5.119 becomes fˆ
Rh …P0 ÿ PL † L 12 001av2av
…5:122†
The concept of hydraulic radius is not applicable for laminar flow. Flow around a sphere When fluid flows over an immersed object, shear force acting on the fluid is called drag force and is defined as
Transport phenomena and metals properties Fk ˆ 12 001av2/ CD Ap
233 (5.123)
where v/ is the velocity far away from the immersed object, CD is the drag coefficient and Ap is the projected area of the solid on the fluid. When fluid flows over a sphere of radius r at a very low flow rate such that the Reynolds number defined as 001av/dp/0016 < 0.1 Fk ˆ 600190016rv/
(5.124)
This is known as Stokes law. From equations 5.123 and 5.124 and using Ap ˆ 0019d2p /4 where dp is the diameter of the sphere, we get CD ˆ 24/Re
Re < 0.1
(5.125)
For higher flow rates, CD ˆ 18.5/(Re)3/5 CD ˆ 0.44
2 < Re < 5 0002 102 2
5 0002 10 < Re < 2 0002 10
(5.126a) 5
(5.126b)
When the Reynolds number lies in the range 0.1 to 2 equation 5.125 is often used for estimation of drag force. Equation 5.126b is termed as Newton's Law. Terminal velocity Stokes law is often used to calculate the terminal velocity of particles or bubbles in a fluid medium. Let us assume that the density of the particles is greater than that of the fluid medium, so the particle falls down. The downward force acting on the particle is due to gravity and upward forces are buoyant force and drag force. Initially the velocity of the particle will go on increasing because of net downward force. But as velocity increases, the drag force also increases since it is directly proportional to velocity (see equation 5.124), and finally the upward and downward forces are balanced. This velocity, which is attained when the total force acting on the particle is zero, is the terminal velocity. The force balance equation is 3 4 3 0019r (001a
ÿ 001af)g ˆ 600190016rvt
001a and 001af are, respectively, density of particle and fluid and vt is the terminal velocity. The left-hand side of the above equation is the difference between gravity and buoyant force and the right-hand side is the drag force. So the terminal velocity is vt ˆ
2 r2 …001a ÿ 001af †g 9 0016
…5:127†
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Example 5.12 Calculate the terminal velocity when a sand particle of 10 0016m diameter is falling through a water column. Density of sand and water are respectively 2500, 1000 kgm3 respectively and viscosity of water is 9.6 0002 10ÿ4 kgmÿ1 sÿ1. Solution Using equation 5.127 and r ˆ 0.5 0002 10ÿ5 m vt ˆ [2 0002 (0.5 0002 10ÿ5)2 0002 (2500 0002 1000) 0002 9.8]/(9 0002 9.6 0002 10ÿ4) m/s ˆ 0.085 mm/s Re ˆ 2500 0002 0.85 0002 10ÿ4 0002 10ÿ4/9.6 0002 10ÿ4 ˆ 0.022 Since Re is less than 0.1, calculated value of terminal velocity is correct. nnn Correlations for packed bed The packed bed is made up of solids of different shapes and sizes. The fluid flows through the voids in the bed. In general, the distributions of voids and the particle sizes are not uniform throughout the bed. But for our discussion we assume that distributions of particles, voids and gas are uniform. If the bed is made of spheres of diameter dp, the contact area between the fluid and solid per unit bed volume is Sp ˆ n 40019(dp/2)2
(5.128)
where Sp is the specific surface area (m2/m3) and n is the number of particles per unit volume. Let 000f be the void fraction in the bed. Then out of unit bed volume (1 ÿ 000f) is occupied by the particles and the rest is void. So 1 ÿ 000f ˆ n 4/30019(dp/2)3
(5.129)
From equations 5.128 and 5.129 Sp ˆ 6(1 ÿ 000f)/dp
(5.130)
If the bed is made up of mixed particle size, dp is the mean particle size. In the case of the packed bed, we define superficial velocity as the velocity of gas in the empty tube. Superficial velocity v0 is related to true average velocity vb of the gas through the void as vb ˆ v0/000f
(5.131)
If we consider that gas flow through the packed bed is similar to gas flow through a pipe of variable cross-section, the pressure drop and friction factor are related by equation 5.122. Since the cross-section of tube is not constant, hydraulic radius can be defined as the ratio void volume per unit bed volume to
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235
surface area of particles per unit bed volume, Rh ˆ
dp 000f 000f ˆ Sp 6…1 ÿ 000f†
…5:132†
Substituting the value of Rh and vb in equation 5.122 fˆ
…P0 ÿ PL † dp 000f3 3…1 ÿ 000f†L 001av20
The bed friction is defined as fb ˆ
f …P0 ÿ PL † dp 000f3 ˆ 3 …1 ÿ 000f†L 001av20
…5:133†
In the laminar and turbulent flow region, fb is related to the Reynolds number by 150 Reb fb ˆ 1:75 fb ˆ
where Reb ˆ
Reb < 20
…5:134a†
103 < Reb < 104
…5:134b†
db 001av0 is the Reynolds number for the packed bed. 0016…1 ÿ 000f†
Equations 5.134a and 5.134b are respectively Kozeny±Carman and Burke± Plummer equations. The former is applicable in laminar flow. Ergun combined the above two equations: fb ˆ
150 ‡ 1:75 Reb
1 < Re < 104
…5:135†
This equation is known as Ergun's equation. Combining equation 5.133 and Ergun's equation 5.135 and substituting for Reb, 0014 0015 2 P0 ÿ PL 1500016…1 ÿ 000f† 001av0 1 ÿ 000f ˆ ‡ 1:75 …5:136† L dp 000f 3 db 001av0 The above equation shows that the pressure drop in a packed bed is very sensitive to bed porosity and particle size. A decrease in bed porosity or particle size increases pressure drop very significantly.
5.5
Further reading
R.I.L. Guthrie (1989) Engineering in Process Metallurgy, Oxford. F.P. Incropera and D.P. DeWitt (1990) Fundamentals of Heat and Mass Transfer, John Wiley, New York. D.R. Poirier and G.H. Geiger (1994) Transport Phenomena in Materials Processing, Publication of TMS.
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5.6
References
Darken, L.S. (1949) Trans. A.I.M.E., vol. 180, 430. Fuller, E.N., Schettler, P.D. and Giddings, J.C. (1966) Ind. Eng. Chem. Vol. 58 no. 5, 18. Kucera, J. and Stransky, K. (1982) Mater. Sc. Eng., Vol. 52, 1. Morita, Zen-ichiro (1996) ISIJ Int., Vol. 36, Supplement, S6. Reid, R.C., Prausnitz, J.M. and Sherwood, T.K. (1977) Properties of Gases and Liquids, 3rd edn. McGraw-Hill, New York.
6
Interfacial phenomena, metals processing and properties K M U K A I , Kyushu Institute of Technology, Japan
6.1
Introduction
We have obtained great technological and academic benefits from describing the nature of metals processing and also by controlling such processes with the aid of thermodynamics and kinetics of bulk phases such as gas, slag and metal. In order to achieve further progress in this area, it is necessary to understand the surface and interface, and to clarify the participation of interfacial phenomena in metallurgical processes. Surface and interfacial tensions of high temperature melts such as slag and steel in iron and steelmaking processes are about 5 to 20 times as large as that of water. In addition, the high temperature melts have remarkably strong surface active agents such as oxygen, sulfur, etc., which are inevitably included in the melts, as shown in Section 6.3. The subject of interfacial phenomena in relation to iron and steelmaking processes has recently been one of the most attractive fields in metallurgy. There are a number of reasons for this. Further understanding of the kinetics in heterogeneous reactions in metallurgical systems requires more detailed information on interfacial phenomena that occur during the progress of these reactions. These phenomena can reveal more of the microscopic stages of kinetics than did the conventional treatments used in previous studies. Smelting and refining processes developed or improved recently often include various dispersion actions such as the injection of powder agents into liquid steel, degassing of liquid steel with RH, bubble injection through porous plug refractories and slag foaming in bath smelting or during pre-treatment of pig iron in torpedo cars. Control of non-metallic inclusion during the steel refining process is important for the production of much higher quality steels. Detailed information on interfacial phenomena is essential to advance practical operations in these processes. It is very important to understand the fundamental treatments of the surface and the interface by using interfacial physical chemistry in order to clarify the interfacial phenomena and to apply the results to the improvement and development of metallurgical processing. Interfacial physical chemistry used here
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means an academic field which treats interfacial phenomena by using surface chemistry, thermodynamics and kinetics. This is the subject matter for Section 6.2. In Section 6.3 typical interfacial properties obtained by previous investigators are concisely introduced for the metallurgical systems containing liquid metal and slag. Section 6.4 focuses briefly on several interfacial phenomena which have been recently shown to participate or possibly participate in the processes of metal smelting and refining. Many technological problems have been linked with the interfacial phenomena.
6.2
Fundamentals of the interface
6.2.1 Thermodynamics of interface Gibbs' dividing surface Gibbs supposed a dividing surface S (see Fig. 6.1), in order to describe macroscopic surface properties. The dividing surface is always taken parallel to the surface of tension defined later in this section and the thickness of the surface is taken as zero. Figure 6.1 shows concentration distribution of component i, ci (mol/m3) in the real surface region and bulk phases of and . ni is the total mole number of the component i, in the volume of V ‡ V . If we suppose that c0017i (concentration in 0017 (ˆ or phase)) keeps constant from bulk
6.1 Gibbs' dividing surface.
Interfacial phenomena, metals processing and properties
239
phase 0017 to the dividing surface S, mole number of the component i in this area is equal to c0017i V 0017 . Then, nsi given with equation 6.1 means excess mole number of component i at surface and corresponds to the shaded portion of Fig. 6.1: nsi ˆ ni ÿ …c000bi V ‡ c i V †
…6:1†
Excess quantity per unit surface area, that is, nsi =A is called a surface excess quantity of component i, ÿi (mol/m2), and expressed as equation 6.2: nsi ˆ Aÿi
…6:2†
where A is the surface area. Similarly the other surface excess quantities are defined in equations 6.3 to 6.6: U s ˆ U ÿ …u V ‡ u V †
…6:3†
0017
where U is internal energy and u (0017 ˆ ; ) is internal energy per unit volume of the phase 0017. S s ˆ S ÿ …s V ‡ s V †
…6:4†
where S is entropy and s0017 is entropy per unit volume of phase 0017. F s ˆ F ÿ …f V ‡ f V †
…6:5†
where F is Helmholtz energy and f 0017 is Helmholtz energy per unit volume of phase 0017. The surface excess quantities mentioned above can be understood to be the excess quantities which are altogether brought to the dividing surface with infinitesimal thickness. Therefore those excess quantities, for example, nsi , vary with the position of the dividing surface within the real surface region as shown in Fig. 6.1. The surface excess quantitiy is not the absolute value, besides that depends on the position of the dividing surface. In order to solve the abovementioned inconveniences, Guggenheim introduced two dividing surfaces. However, his treatment is complicated and nowadays is not so popular. Surface tension Thermodynamic interpretation When the surface area A of pure liquid increases by dA, dU s, the change in internal energy at the surface, is given by equation 6.6: dU s ˆ dQs ‡ dA s
…6:6†
where dQ is the heat absorbed by the surface, dA means the work which acts on the surface at constant temperature by the change in the surface area dA, and
is the surface tension which is described later.
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Fundamentals of metallurgy
For a reversible process, dQs ˆ TdS s
…6:7†
therefore dU s ˆ TdS s ‡ dA
…6:8† s
s
Surface excess Helmholtz energy F , and its total differential, dF , are expressed by equations 6.9 and 6.10, respectively: F s ˆ U s ÿ TS s
…6:9†
dF s ˆ dU s ÿ TdS s ÿ S s dT
…6:10†
Equation 6.11 is derived from equations 6.8 and 6.10: dF s ˆ dA ÿ S s dT
…6:11†
Therefore,
0012 s0013 @F f 0011 ˆ @A T s
…6:12†
Equation 6.12 means that surface tension is the surface excess Helmholtz energy per unit surface area. Therefore,
ˆ f s ˆ us ÿ Tss
…6:13†
Equation 6.13 indicates that surface tension contains the entropy term, Tss, as well as the internal energy term, us …ˆ U s =A†, that is known as surface energy. The relation between f s and for an r -component liquid±gas system is given by equation 6.14: fs ˆ ‡
r X
ÿi 0016i
…6:14†
iˆ1
0012 where 0016i ˆ
@G @ni
0013 T; P; nj …jˆ1...r; j6ˆi†
is the chemical potential of the component i.
If we choose the dividing surface so as to make r X
ÿi 0016 i ˆ 0
iˆ1
f s is equal to from equation 6.14, even for an r-component liquid±gas system. Position of dividing surface It can be shown on thermodynamic grounds that the surface tension of a plane surface at thermodynamic equilibrium is not dependent upon the position of the
Interfacial phenomena, metals processing and properties
241
6.2 Relation between surface tension and radius of the dividing surface.
dividing surface. The surface tension of a curved surface, however, varies with the position of the dividing surface as shown in equation 6.15 and Fig. 6.2. 0012 2 0013 r 2r …6:15†
ˆ s s2 ‡ 3r 3rs where r is the radius of the dividing surface. As shown in Fig. 6.2, the surface tension has a minimum s at r ˆ rs . The dividing surface at r ˆ rs is called the surface of tension. The surface of tension is within the surface layer 000e (Fig. 6.2). Therefore when rs is much larger than 000e, should be regarded as almost constant within the surface layer from equation 6.16:
ˆ s …1 ‡ 000f2 ‡ O…000f3 ††
…6:16†
where 000f ˆ 000e=rs . Radius of curvature The surface tension s at the surface of tension varies with the change in the radius of curvature rs for the incompressible liquid as expressed with Tolman's equation 6.17:
s ˆ 1=…200150 =rs ‡ 1† …6:17†
1
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Fundamentals of metallurgy
where 1 is the surface tension at rs ! 1 and 00150Pis the distance between surface of tension and the dividing surface at which riˆ1 ÿi 0016i ˆ 0. If we take the value 00150 ˆ 1:6 0002 10ÿ10 m for water, s = 1 is calculated as follows from equation 6.17: s = 1 ˆ 0:997 at rs ˆ 100 nm, s = 1 ˆ 0:97 at rs ˆ 10 nm. These results indicate that we should take account of the dependency of surface tension on the radius of curvature for materials with the radii less than 100 nm (10ÿ5 cm), such as a droplet, nucleus at nucleation, etc. We should also notice that the thermodynamic model cannot be expected to hold for a system containing so few molecules, such as water droplets of radii smaller than 1 nm, for example. Temperature For pure liquid, we can obtain equation 6.18 from equation 6.13: d df s ˆ ÿss ˆ …6:18† dT dT For pure liquid, d /dT has generally negative values, which indicate ss > 0. ss > 0 means that entropy at the surface is larger than for the bulk liquid. Several empirical relations between surface tension and temperature are proposed, for example, as expressed by equations 6.19 and 6.20:
ˆ …k=v1
2=3
†…Tc ÿ T†
…6:19†
1
where v is molar volume, Tc is critical temperature, k is constant and is about 2.1 0002 10ÿ7 J/deg for most liquids. Equations 6.19 was proposed by EoÈtvoÈs in 1886. Equation 6.19 does not hold accurately in the vicinity of Tc . Katayama and Guggenheim proposed equation 6.20 in order to improve equation 6.19:
ˆ o …1 ÿ T=Tc †11=9
…6:20†
where is constant and depends on the kind of liquid. Surface stress In the case of a solid, we have to distinguish between surface tension and surface stress. Surface tension is defined in terms of work, dA (see equation 6.6) retaining the same surface condition, such as keeping the number of atoms at the unit surface area constant. Another way to expand the surface by dA is to stretch the distance between atoms at the surface. In this case the number of atoms at the surface is kept constant and the work is expressed by gdA. g is called surface stress, and is related to surface tension by equation 6.21 (Shuttleworth's equation): g ˆ ‡ …@ [email protected]!†N
…6:21†
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243
where ! is the elastic surface strain and N is the total number of atoms at the surface. Since a liquid surface deforms in a completely plastic manner, g is equal to for the liquid surface.
6.2.2 Mechanical aspects of surface tension Mechanical definition As pointed out by Young, a system composed of two fluids such as liquid and gas which contact each other, behaves, from a mechanical standpoint, as if it consisted of two homogeneous fluids separated by a unifomly stretched membrane of infinitesimal thickness. Surface tension has been defined from a macroscopic standpoint as the tractional force, , acting across any unit length of line on this fictitious membrane. Surface tension has the dimensions of force per unit length and is usually expressed in N0001mÿ1. The real surface in the above mentioned system has a finite thickness, that is, the surface layer (or region) as shown in Fig. 6.3. For this system, the surface tension is defined from a mechanical standpoint by equation 6.22 (the Bakker equation): Z 1 …pN ÿ pT …z††dz …6:22†
ˆ ÿ1
where pN is normal pressure to the surface and equal to p, the hydrostatic pressure in the system, pT …z† is tangential pressure. pT …z† exerts normally across a plane parallel to the z axis and varies with the value of z as shown in Fig. 6.3,
6.3 Variation of tangential pressure PT (z) with z.
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Fundamentals of metallurgy
while p is required, from the condition of hydrostatic equilibrium, to be the same value even in the surface layer. If we can describe pT …z† from the molecular distribution functions of the fluid by using `molecular dynamics', for example, the surface tension can be rigorously expressed with the aid of equation 6.22. Surface tension, , as a force of traction, can be available even for the system which is not in a thermodynamic equilibrium state. Laplace's equation When a spherical surface of radius with curvature r maintains mechanical equilibrium between two fluids and phases at different pressures p and p and the interface is assumed to be of zero thickness, the condition for mechanical equilibrium provides a simple relation between p and p : p ÿ p ˆ 2 =r
…6:23†
Equation 6.23 is known as the Kelvin relation. If, instead of a spherical surface, we consider any surface, the condition of mechanical equilibrium at each point in the surface is given by Laplace's equation (6.24): 0012 0013 1 1 p ÿp ˆ …6:24† ‡ r1 r2 where r1 and r2 are the two principal radii of curvature of the surface as shown in Fig. 6.4.
pb
pa
r1
r2
6.4 Pressure difference between phase and in mechanical equilibrium.
Interfacial phenomena, metals processing and properties
245
Laplace's equation (6.24) provides a fundamental base for measuring surface tension in static state such as the capillary rise method, the sessile drop method, the pendant drop method, the maximum bubble pressure method, etc. Marangoni effect Surface tension as a tractional force is available for describing the behaviour of the system which is not in a thermodynamical equilibrium state. The surface (or interfacial) tension difference, or gradient, on the surface (or interface) of liquid, for example, in the direction x, can change the motion of liquid due to the surface shear stress 001cs , written as 001cs ˆ
d @ dT @ dc @ d' ˆ 0001 ‡ 0001 ‡ 0001 dx @T dx @c dx @' dx
…6:25†
Equation 6.25 indicates that the surface (or interfacial) tension gradient is caused by the gradients of temperature T, concentration c of the surface active component in the liquid and electric potential ' at the interface between two liquids. In hydrodynamics, the surface (or interfacial) tension difference or gradient participating in the above dynamics is called the Marangoni effect.1 The motion of liquid induced by the Marangoni effect is called Marangoni flow or Marangoni convection. The non-dimensional number defined by equations 6.26 and 6.27 is called the Marangoni number, Ma , which characterizes the Marangoni convection: Ma ˆ …@ [email protected]† 0001 0001T 0001 L=a0011
…6:26†
Ma ˆ …@ [email protected]† 0001 0001c 0001 L=D0011
…6:27†
where 0001T is temperature difference, L is the characteristic length of the system, a is thermal diffusivity, 0011 is viscosity, 0001c is the concentration difference of the surface active component and D is the diffusion coefficient of the surface active component. Since liquid metals and slags generally have high surface or interfacial tension and also have strong surface active components such as oxygen and sulfur in liquid iron (see Section 6.3.1), both factors are favorable to the occurrence of Marangoni convection in systems where these are present. Even in the field of gravity on the Earth, occurrences of the Marangoni effect in metallurgical systems have been observed in the following: (a) Marangoni convection of molten silicon2 and salts3 due to temperature gradient, (b) spreading and shrinking of slag droplets on the metal due to changes in applied potential,4 and (c) Marangoni flow of slag film5 and metal surface6,7 due to concentration gradient, which is described in further detail in Sections 6.4.1 and 6.4.2. Since the Marangoni convection is most intensive at and around liquid±gas and liquid±liquid interfaces, it effectively promotes mass transfer in the region
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Fundamentals of metallurgy
of the concentration boundary layer, resulting in the acceleration of the heterogeneous reaction rate when the reaction rate is limited by the mass transfer process (see Scetion 6.4.2), The Marangoni effect also participates in the surface dilational viscosity which is closely related to many interfacial phenomena in metallurgical processes (see Section 6.2.3). The interfacial tension gradient, which is induced along the interface between a liquid and foreign particle by the concentration gradient or temperature gradient, would propel the foreign particle in the direction of decreasing interfacial tension.8 This phenomenon is also taken as a kind of Marangoni effect (in a broad sense). The movement of fine particles driven by the interfacial tension gradient should have a close relationship to certain phenomena in iron and steelmaking processes, such as the occurrence of bubble and inclusion-related defects and nozzle clogging (see Section 6.4.5).
6.2.3 Physical chemistry of interfacial phenomena Here interfacial phenomena will be treated from a physicochemical standpoint. Adsorption The composition of the surface layer is usually different from that of the two bulk phases. This phenomena is called `adsorption'. The real concentrations in the surface layer are not uniform but vary continuously through its thickness. A macroscopic definition of adsorption can be arrived at by employing Gibbs' dividing surface described in Section 6.2.1. Gibbs' adsorption equation Gibbs' adsorption equation in general form is given by equation 6.28: d ‡ ss dT ‡
r X
ÿi d0016i ˆ 0
…6:28†
iˆ1
Equation 6.28, which is due to Gibbs, is derived by employing thermodynamics and the concept of the dividing surface. At constant temperature, equation 6.28 reduces to d ˆ ÿ
r X
ÿi d0016i
…6:29†
iˆ1
Since ÿi depends on the position of the dividing surface, it is possible and convenient to make a special choice for the position of this surface in such a manner as to make ÿ1 zero. We shall denote ÿi defined in this way as ÿi…1† . Then we have equation 6.30 instead of equation 6.29:
Interfacial phenomena, metals processing and properties
247
6.5 Adsorption of component 2 at the dividing surface which makes ÿ1 zero.
d ˆ ÿ
r X iˆ2
ÿi…1† d0016i
…6:30†
Equation 6.30 is known as Gibbs' adsorption equation. For the system composed of components 1 and 2, equation 6.30 reduces to d ˆ ÿÿ2…1† d00162
…6:31†
ÿ2…1† means the adsorption of component 2 at the surface which makes the adsorption of component 1 zero, as shown in Fig. 6.5. The value ÿ2…1† can be calculated from the experimentally obtained slope of vs ln a2 (or c2 ) based on the equations 6.32 and 6.33: 0012 0013 1 @ ÿ2…1† ˆ ÿ …6:32† RT @ ln a2 T 0012 0013 1 @ (for ideal solution) …6:33† ÿ2…1† ˆ ÿ RT @ ln c2 T where a2 is the activity of the component 2. It should be noticed that ÿ2…1† is not absolute concentration but the relative quantity, the excess surface quantity (see Fig. 6.5). Adsorption of surface active elements such as oxygen and sulfur at the liquid iron surface (see Section 6.3.1) reduces the reaction rate between nitrogen gas and liquid iron (see Section 6.4.2). Adsorptions at slag surface and slag±metal interface also closely relates to surface dilational viscosity (the area viscosity), 0010s , given by equation 6.34:9
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Fundamentals of metallurgy
1 @A …6:34† A @t where 0001 is the corresponding change in the surface tension to that in the surface area A and t is time. Equation 6.34 is analogous to Marangoni viscosity, 0010M , given by the following equation: 0001 ˆ 0010s
1 @A …6:35† A @t Therefore, 0010s and 0010M include the rate of attainment of equilibrium between the bulk phase and adsorbed surface (which contains, Gibbs elasticity10,11) as well as its own properties of adsorbed surface layer. In other words, the 0010s and 0010M may be called `Gibbs±Marangoni' viscosity. Surface dilational viscosity is closely related to the following interfacial phenomena:12 foaming, coalescence of bubbles, droplets and solid particles in liquid and also dispersion of bubbles, droplets as well as solid particles into liquid. The involvement of mold powder or slag by the molten steel should be also influenced by 0010M . All of the above mentioned phenomena include the elemental processes of (a) surface (interface) expansion or shrink, (b) drainage and (c) break of the melt film. 0001 ˆ 0010M
Wetting Wetting of solid (s) by liquid (l) without any kinds of reaction is characterized by contact angle 0012 as shown in Fig. 6.6, or with the quantity of Helmholtz energy changes for wettings shown in Fig. 6.7 and equations 6.36, 6.37 and 6.38: WS ˆ sg ÿ lg ÿ sl sg
sl
lg
sg
WI ˆ ÿ
WA ˆ ‡ ÿ
…6:36† …6:37†
sl
…6:38†
6.6 Contact angle 0012 and interfacial tension ij for the system of gas±liquid± solid.
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6.7 Three types of wetting for gas±liquid±solid system.
where g means gas phase and ij is the interfacial tension between phases i and j. Modes of wetting, (a), (b) and (c) in Fig. 6.7, correspond to different forms of wetting, namely spreading wetting, immersional wetting and adhesional wetting, respectively. Two terms which are widely used in this context are the spreading coefficient, WS (defined in equation 6.36) and the work of adhesion, WA (defined in equation 6.38). Equations 6.40, 6.41 and 6.42 are derived from equations 6.36, 6.37 and 6.38 when Young's equation (equation 6.39) can be applied to the systems in Fig. 6.6.
sg ˆ sl ‡ lg cos 0012
…6:39†
WS ˆ lg …cos 0012 ÿ 1†
…6:40†
WI ˆ lg cos 0012
…6:41†
WA ˆ lg …1 ‡ cos 0012†
…6:42† lg
Equations 6.40, 6.41 and 6.42 are valuable for practical use because and 0012 can be determined experimentally. For 0012 ˆ 0ë, hence WS ˆ 0, spreading wetting occurs spontaneously. For 0012 0014 90ë, hence WI 0015 0, immersional wetting occurs and for 0012 0014 180ë, hence WA 0015 0, adhesional wetting occurs spontaneously. Young's equation indicates that contact angle 0012 depends on the interfacial tension sg , sl and lg which are closely related to the interfacial Helmholtz energy f s as shown by equation 6.12. Since the Helmholtz energy is a function of chemical composition of the system and temperature, the contact angle also depends on the chemical composition of the system and temperature. The chemical composition of the drop varies with the dissolution of the substrate materials into the drop, resulting in the change in the contact angle. When a new phase or compound are generated at the drop±substrate interface by the reaction
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between the drop and substrate, the contact angle corresponds to the system of the drop and generated phase or compound in equilibrium. For the case where the surface of the substrate is not smooth, the Wenzel equation (6.43)13 is used instead of Young's equation (6.39): R0 … sg ÿ ls † ˆ lg cos 00120
…6:43†
where R0 is roughness factor and given with A=Ao . A and Ao are actual surface area and geometrical surface area of the substrate, respectively, and 00120 is the apparent contact angle for the rough surface. When the wettability between the drop and substrate is poor, the rough interface between the drop and the substrate tends to form a composite interface which consists of the liquid±solid interface (wetting area) and the interface including small gas phase between the liquid and solid (non-wetting area). The Wenzel equation (6.43) is not available for the above case and we have to find another expression14 on 00120 . We often observe the advancing contact angle 0012a and the receding contact angle 0012r , that is, a hysteresis of wetting. The following causes for the hysteresis have been proposed: (1) friction between drop and solid surface, (2) adsorped layer (film) at solid surface, (3) surface roughness. We may add one more cause, that is, surface stress of the solid surface (see Section 6.2.1) as a mechanism for the hysteresis, and these four causes may participate together in the hysteresis. For the wetting in non-equilibrium state, we should add the kinetic factor for describing the contact angle 0012. In the case of very rapid spreading of the drop on the solid surface, the spreading velocity may be limited by the hydrodynamic flow of the drop at the advancing front of the drop, and hence the contact angle may be dominated by the flow characteristics. When adsorption, dissolution and chemical reaction between the drop and substrate material do not reach the equilibrium state and the reaction rate is determined by the mass transfer process, the contact angle may be influenced by the mass transfer rate and time. Influence of radius of curvature We will consider one component system composed of two phases which are separated by the interface with radius of curvature r in a thermodynamic equilibrium state. We also suppose that no chemical reactions occur other than the transfer of matter from one phase to the other. For mechanical equilibrium, the pressure of phase (Fig. 6.4), p is larger than that of phase as indicated by equation 6.24. Since Gibbs energy G and hence chemical potential 0016i increases with increasing pressure, G and 0016i of phase are larger than those of phase. The above difference in Gibbs energy and chemical potential between and phases influences the physicochemical properties of phase such as vapor pressure, melting temperature, solubility, etc.
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On vapor pressure Vapor pressure of pure liquid droplet with radius r increases with decreasing r. The quantitative description of the phenomena is known as the Kelvin equation (6.44): 2 lg 0017 1 …6:44† r RT where pr and po are the vapor pressure at radius r and 1, respectively, lg is the surface tension of the droplet and 0017 1 is the molar volume of the droplet. 0017 1 is supposed to be constant at any radius. In the case of a bubble with radius r in liquid, equation 6.45 can be derived: ln pr =po ˆ
ln
pr 2 lg 0017 1 ˆÿ pi;o r RT
…6:45†
Equation 6.45 indicates that the vapor pressure in the bubble decreases with decreasing r, which is the reverse mode to that of droplet. For the droplet of mixture, the following analogous equation 6.46 is given: ln
Pi 2 lg 0017i1 ˆ pi;o r RT
…6:46†
where pi and pi;o are the partial pressures of component i at the radius r and 1, respectively, 0017i1 is the partial molar volume of component i. On heat of evaporation The heat of evaporation of pure liquid droplets decreases with decreasing r as expressed by equation 6.47: 2 lg 0017 1 …6:47† r where 0001e hr and 0001e ho are the molar heat of evaporation at radius r and 1, respectively. 0001e hr ÿ 0001e ho ˆ ÿ
On boiling point Equation 6.48 expresses the relation between boiling point of pure substance and radius of curvature of liquid droplets under constant pressure of the gas phase: ln Tr =To ˆ ÿ
2 lg 0017 1 0001 r 0001e h
…6:48†
where Tr is the boiling temperature of the droplet at radius r, To is the boiling temperature of bulk liquid (r ! 1) at given constant pressure. Equation 6.48 indicates Tr < To . This is the phenomenon of supercooling of a saturated vapor.
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On melting point Equation 6.48 may be extensively applied to the system of small solid particle with the radius r and bulk liquid. The melting point T of the small solid particle will be given by equation 6.49: ln Tr =To ˆ ÿ
2 sl 0017 s 0001 r 0001f h
…6:49†
where Tr is the normal melting point at the some external pressure, 0017 s is the molar volume of solid and 0001f h is the molar heat of fusion. On solubility The solubility of pure solid particle (component 1) depends on radius r of the particle as expressed by equation 6.50: 2 sl RT ˆ s;o ln f… 1 x1 †=… 1;o x1;o †g r 00171
…6:50†
where 00171s;o is the molar volume of pure solid 1, x1 and x1;o are the mole fractions of component 1 in the solution which, at the same T and P of the solution, is in equilibrium with the solid particle with radius r and a large particle (r ! 1), respectively. 1 is the activity coefficient of the component 1. For the ideal solution, equation 6.50 reduces to equation 6.51: 2 sl RT ˆ s;o ln …x1 =x1;o † r 00171
…6:51†
Equation 6.51 is known as the Freundlich±Ostwald equation. Equation 6.51 predicts that the solution in equilibrium with a solid particle of component 1 increases in the concentration, x1 , as the solid decreases in its radius. In other words, small solids are more soluble than large ones. In a solution in contact with solids of different radii of curvature, the larger solid will grow at the expense of the smaller one as shown in Fig. 6.8. The above mentioned phenomena is known as Ostwald ripening. In relation to phase rule For very small particles, intensive properties such as vapor pressure, boiling point, melting point, solubility, etc. depend on its radius of curvature as described above. The results indicate that the phase rule for the system where the contribution of surface quantity should be taken account of must be different from the one for the system composed of bulk phases, neglecting the contribution of the interface. We suppose the following system: number of independent components, r, number of bulk phases, 0017, number of types of surface, 0018, number of independent
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6.8 Ostwald ripening.
chemical reactions, q. For simplicity, we also suppose that each type of surface has only one surface phase. For this system, the number of degrees of freedom is given by equation 6.52 instead of equation 6.53 for the system composed of bulk phase, neglecting the interface: f ˆr‡1ÿq
…6:52†
f ˆr‡2ÿ0017ÿq
…6:53†
The equations from (6.44) to (6.51) are subject to the phase rule (6.52). Thermodynamics of nucleation Homogeneous nucleation We shall consider the change in Helmholtz energy when a nucleus of phase generates in phase (g or l). We suppose that the system after nucleation has the same total number of molecules in the same total volume and temperature to those of the system before the nucleation. The change in Helmholtz energy, …0001F†T;V , is given by equation 6.54: …0001F†T;V ˆ F ÿ Fo
…6:54†
where F is the total Helmholtz energy of the system after nucleation and Fo is that before the nucleation. For simplicity, we assume that the bulk phase is sufficiently large that the removal of the small amount of material needed to form the nuclei does not alter appreciably either the pressure or composition of
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6.9 Free energy of formation of an embryo (000b) as a function of embryo size r.
the bulk phase. Under the above assumptions, we have, finally, the following equation (6.55): …0001F†T;V ˆ
r X iˆ1
n000bi 00010016i ‡ A
…6:55†
where 00010016i ˆ 0016000bi …T; P ; x000b2 ; . . . x000br † ÿ 0016 i …T; P ; x 2 ; . . . x r † However, we have to notice that the above assumption does not hold in general. For the case of Al2O3 nucleation in liquid iron, for example, we have to take account of the composition change of the liquid iron (bulk phase) which alters the Helmholtz energy of the bulk phase.15 We shall apply equation 6.55 to the case where a droplet of component 1 ( phase) nucleates in the phase. …0001F†T;V is given by equation 6.56 and Fig. 6.9: …0001F†T;V ˆ
4 0019r3 00010016o1 ‡ 40019r2 3 00171000b;o
…6:56†
For equilibrium state, that is, at the critical nuclei (we denote *), (dF)T,V ˆ 0. When the nucleus grows, we will have the following differential: 0012 0013 @F 40019r00032 ˆ ;o 00010016o1 ‡ 80019r0003 ˆ 0 …6:57† 00171 @r T;V Therefore,
Interfacial phenomena, metals processing and properties r0003 ˆ ÿ
200171000b;o 00010016o1
255 …6:58†
…0001F 0003 †T;V ˆ 43 0019r00032
…6:59†
For the nucleation of the droplet of pure component 1 from gas phase at the above equilibrium state, 00010016o1 ˆ RT ln po1;o =po1;r
…6:60†
where po1;o and po1;r are the equilibrium vapor pressures of the pure liquid at r ! 1 and r, respectively. po1;r is then regarded as the vapor pressure at supersaturated state for the nucleation. Substituting equation 6.60 into equation 6.58, we obtain Kelvin's equation (6.44). Heterogeneous nucleation We will consider the heterogeneous nucleation of 000e phase at the interface between phase (gas or liquid) and solid(s) as shown in Fig. 6.10. The nucleus forms a lens on the plane interface between phase and solid. We suppose the same assumptions to homogeneous nucleation that pressure and composition both of and solid phases do not change before and after the nucleation. At the equilibrium, that is, critical nucleus, 2
…0001F 0003 †T;V ˆ 23 000e 0019r0003000b000e …1 ÿ cos 0012 ÿ 12 cos 0012 sin2 0012†
…6:61†
where r000b000e is the radius of the lens and 0012 is the contact angle (Fig. 6.10). For 0012 < 180ë, 1 ‡ cos 0012 ‡ 12 cos 0012 sin2 0012 > 0 then we can derive the following relation 2
…0001F 0003 †T;V ;het < …0001F 0003 †T;V ;hom ˆ 43 0019r0003 000e
…6:62†
The relation (6.62) explains that heterogeneous nucleation is thermodynamically more advantageous than homogeneous nucleation for 0012 < 180ë.
6.10 Lenticular nucleus (000e) on a plane interface between phase and solid.
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Kinetics of nucleation We shall define the number, I, of critical nuclei of phase (component 1) which are formed in phase, is unit time in unit volume of bulk phase in a state of supersaturation at a temperature T. The nucleation proceeds in a quasi-stationary state in which critical nuclei are formed at the expense of embryos, followed by their removal from the embryo population by their growth into the nucleated phase, finally the bulk phase. Equations 6.63 and 6.6416,17 are usually used for condensed systems as the quantitative expression of I. I ˆ A exp …ÿ0001F 0003 =kT†
…6:63† 0
A ˆ n0003 … =kT†1=2 …200171000b;o =90019†1=3 n…kT=h†
…6:64†
where n0003 is the number of molecules on the surface of critical nuclei given by 0 equation 6.65, 00171000b;o is the volume of one molecule of the nucleated phase given by equation 6.66 and n is the number of molecules of component 1 per mole of parent phase . 2=3 n0003 ˆ 40019r00032 =fM1 =…001a000b;o 1 No †g
0017 ;o ˆ …M1 =001a000b;o 1 †=No 0
…6:65† …6:66†
001a000b;o 1
where M1 and is the molecular weight and density of phase of component 1, respectively. No is the Avogadro's constant. Dispersion and coalescence18 Since the dispersion system composed of fine particles (dispersion phase 000e) and dispersion medium (phase ) is thermodynamically unstable, the dispersed particles tend to coalesce with each other or to be absorbed into the same phase (to the particle) which contacts with the dispersion medium. When the phase contacting with the dispersion medium is different from the phase of the particle (000e), from the thermodynamic point of view, (a) 0001Fas < 0 s and 0001Fb;i > 0 should be satisfied for the particle in the dispersion medium to s < 0 are adhere to the ± interface or (b) the relations of 0001Fas < 0 and 0001Fb;ii needed for the particle to go through the interface and transfer into phase . 0001Fas and 0001Fbs are given by equations 6.67 and 6.68, respectively. The above behaviors of the particle are also explained by Fig. 6.11. 0001Fas ˆ Ai 000e ‡ Ai 000e ÿ Ai ÿ A 000e000b
…6:67†
0001Fbs ˆ A 000e ÿ …Ai 000e ‡ Ai 000e ÿ Ai †
…6:68†
where 000e and 000e is the interfacial tension between the particle (phase 000e) and phase and that between the particle and phase , respectively. is the interfacial
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6.11 Thermodynamic illustration for the system of fine particle in phase and or at the ± interface.
tension between the phases and . A and A is the interface area between the particle and phases and , respectively and Ai is the interface area between phases and . When the particle straddles the interface between two phases, the original interface will disappear and be replaced by two separate interfaces. In practical operations, we often encounter kinetic problems, that is, (a) how to drive the particle toward the interface and or into phase via the ± interface, or (b) how to keep the dispersed system stable as long as possible. Figure 6.12 shows schematically thermodynamic and kinetic processes of the dispersed particle separation. The driving energy in Fig. 6.12 is caused by the viscous drag force when the particle moves toward the phase , electric double layer, Suffman force, van der Waals force, etc. The energy for overcoming the driving energy can be supplied by gravity (for example, density difference between the particle and phase ), mechanical agitation, applied potential, the force caused by interfacial tension gradient between the particle and the phase (see the section on the Marangoni effect, page 245), etc.
6.3
Interfacial properties of a metallurgical melts system
6.3.1 Surface tension The group of liquid metal generally has the highest surface tension among all kind of liquids.19 The surface tension of liquid slag is lower than that of liquid
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6.12 Schematic illustration of kinetic and thermodynamic processes of dispersed particle separation.
metal although it is larger than those of water and organic materials. Since the surface tension has the internal energy term as expressed by equation 6.13, it reflects bond energy and then depends on the bond types in liquids, that is, metallic bond (liquid metal), covalent bond (liquid slag), ionic bond (liquid slag), and van der Waals bond (molecular liquid). Pure liquid usually has a negative temperature coefficient of surface tension, which indicates that the surface excess entropy ss (see equation 6.18) is positive. ss > 0 means that the degree of order of atom or molecule configuration at the surface is lower than that in bulk phase (see equation 6.4). We can evaluate the value of the entropy term in equation 6.13 for liquid iron as about ÿ700 mN/m from its temperature coefficient, d =dT ˆ ÿ0:40 mN/(m0001K), which is the average value of those summarized by Keene20 from recently obtained experimental data. The absolute value of the entropy term is equivalent to 40% of the surface tension value 1800 mN/m. The result may indicate that we should take account of the contribution of entropy term for the prediction of surface tension of liquid metal. Liquid metal usually has strong surface active elements. Table 6.120 shows surface activities of several solutes in liquid iron. Especially oxygen, sulfur and nitrogen, which are inevitably included in liquid steel during the iron and steelmaking process as shown in Table 6.1,20 are remarkably surface active to
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Table 6.1 Surface activities of solute19 Solute (i) in Fe
C Mn N O Si S
Mean value of approximate surface activities (mNmÿ1[mass%i ]ÿ1) relative to iron
Approximate range (in mass%) over which surface activity was derived
ÿ19 ÿ51 ÿ5580 ÿ26190 ÿ26 ÿ10990
0ÿ2.2 0ÿ4.9 0ÿ0.025 0ÿ0.0086 0ÿ2.5 0ÿ0.029
liquid iron in comparison with surface activeness for the system of water± surfactant (C8H17SO3Na:i) solution, ÿ88 mNmÿ1[mass%i]ÿ1,(0~0.1mass% i).21 The temperature coefficient of surface tension of liquid iron alloy increases with increasing oxygen and sulfur concentration and becomes a positive value after reaching zero at around 70 ppm (on a mass basis) for oxygen in liquid iron22 and 60 ppm (on a mass basis) for sulfur in 304 and 306 stainless steels.23 The above behavior in the temperature coefficient of surface tension may cause a drastic change in the metal pool shape for TIG welding due to the change in the direction of Marangoni convection in the metal pool.22±24 The temperature coefficient of surface tension of liquid silicate slag also has a positive value in the range of high SiO2 concentration.
6.3.2 Interfacial tension between slag and metal Interfacial tension between slag and liquid iron alloy measured experimentally has generally as large a value as surface tension of liquid metal, and also has a strong surface active element such as oxygen and sulfur.25 We have to notice that many interfacial tension values reported previously are measured in non-equilibrium state. When the slag±metal system, for example, SiO2-CaO-FeO slag±iron alloy, is in equilibrium state, the number of freedom for the system is 2. Therefore, if temperature and pressure are fixed, the number of freedom is equal to zero, which means that all of the composition of slag and metal phases are fixed and we cannot change the composition of the metal phase independently of slag composition. When we apply the Gibbs' adsorption equation (6.30) to the system containing the slag±metal interface, in principle, we should pay attention to confirming whether the system is in equilibrium state or not. In addition, for non-equilibrium state, during sulfur transfer from liquid iron drop to slag phase via the interface, for example, the iron drop in the slag phase is observed to be deformed into a depressed flat shape.26 If we apply the Laplace equation to the
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non-equilibrium system in order to obtain the interfacial tension between the drop and slag phase, we will obtain an unusual low interfacial tension value27 from the analysis of the depressed drop shape. Here we also pay attention to confirming whether the depressed drop is in equilibrium of statics or not. If the deformation of the drop is dominantly caused by the Marangoni convection at the interface induced by the transfer of the surface active element sulfur, we cannot apply, in principle, the Laplace equation to the system and the unusual low interfacial tension value, obtained with the aid of the Laplace equation, is regarded as just an apparent one.
6.3.3 Wetting of ceramics by liquid metal and slag at high temperature In general, the liquids which react well with ceramics show good wettability between the liquid and the solid, that is, 0012 is smaller than 90ë. On the other hand, when the ceramic is substantially inert against the liquid, 0012 tends to be larger than 90ë, which means poor wettability. Almost all the liquid iron alloy±solid oxide systems show poor wettability, except for a few, for example, the Al(l)-SiO2(s) system.28 Oxygen in the liquid iron reduces the contact angle 0012 between the iron and alumina. However, when oxygen concentration is smaller than about 100 ppm (on a mass basis), the wetting behavior tends to become complicated.29 SiC, Si3N4 graphite and diamond are not wetted by those metals which have a very low solubility in those materials.28 On the other hand, SiC is wetted well by liquid iron, cobalt and nickel which have a pretty high solubility for carbon on SiC.28 Liquid slag usually dissolves solid oxides well and also wets them well. On the other hand, in general, graphite is not wetted well by liquid slag which hardly dissolves carbon.
6.4
Interfacial phenomena in relation to metallurgical processing
In this section, we will consider several interfacial phenomena which do, or may, participate in the metal smelting and refining processes, studied mainly by our research group. Many papers have already been published on this topic by various investigators besides ourselves.
6.4.1 Local corrosion of refractories at slag±gas and slag±metal interfaces5,30 It has been well known that refractories composed of oxides, oxides±graphite and oxides±graphite±carbide are corroded locally at slag±gas (slag surface) or slag±metal interface in the fields of glass technology and iron and steelmaking.
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261
The local corrosion of refractories is a serious problem for these industries because it limits the life of the refractories. Several kinds of ideas had been proposed on the mechanism of this local corrosion, such as interfacial turbulence, vaporization, oxygen potential, electrochemical reaction. Recent investigations have led to the following conclusions by combining optical or X-ray radiographic techniques, aiming by direct observation of the phenomena which occur in the local corrosion zone, with the conventional immersion test. (1) Local corrosion of oxide refractories and trough materials composed of oxide and SiC at the slag±gas and slag±metal interfaces is essentially caused by the active motion of the slag film formed by the wettability between the refractory and the slag. The slag film motion accelerates the dissolution rate of the refractory by breaking down the diffusion layer of the dissolved component from the refractory. The active film motion is dominantly induced by the Marangoni effect and/or change in the form of slag film (slag meniscus) due to the variation of interfacial tension and its density. The flow pattern of the slag film (or slag meniscus) is different between the following two systems: (i) For the system where the dissolved component from refractory into the slag increases interfacial tension, the typical flow patterns are as those shown in Fig. 6.13 which was observed for the systems of SiO2(s)± (PbO±SiO2)slag. The following systems are regarded as having a similar flow pattern to that of SiO2(s)±(PbO±SiO2) slag system: SiO2(s)±(PbO-SiO2)slag± Pb(l), magnesia±chrome refractory ± (CaO±Al2O3±SiO2±FeO)slag, and blast furnace trough material ± (CaO±Al2O3±SiO2)slag±(Fe±C) alloy. (ii) For the
6.13 A typical flow pattern of slag film for the rod silica specimen dipped in PbO±SiO2 slag.
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6.14 A rotational slag meniscus motion for the rod silica specimen dipped in FeO±SiO2 slag.
system where the dissolved component from refractory into the slag decreases interfacial tension, the typical motions of the slag meniscus are shown in Fig. 6.14 which was observed for the system of SiO2(s)±(FeO±SiO2)slag system. The local corrosion zone of this system forms a steeper groove and narrower vertical zone than those of the case (i) above. The horizontal cross-section of the prism specimen of this system remains square during the entire corrosion process, while in the system described in (i), the cross-section of the prism specimen changes its shape from square to round. The flow pattern of the following practical refractory systems belongs to system (ii): blast furnace trough material±(CaO±Al2O3±SiO2)slag and magnesia-chrome refractory ±(CaO± Al2O3±SiO2±FeO)slag ±Fe(l). SiO2 scarcely decreases the surface tension of Na2O-SiO2 slag. When the SiO2 specimen is partially immersed in this slag, a slag film is also formed above the slag level. However, neither film motion nor local corrosion is detected experimentally in this system. (2) Local corrosion of refractories composed of oxide and graphite at the slag± metal interface is caused by the cyclic dissolution of oxide and graphite into the slag and metal phase, respectively. As shown in Fig. 6.15, when the wall of the Al2O3-C or MgO-C refractory is initially covered with a slag film (Fig. 6.15a), the film not only wets the oxides, but dissolves them in preference to graphite. This changes the interface to a graphite-rich layer. Since the metal phase wets graphite better than the slag, the metal phase creeps up the surface of the specimen, as indicated in Fig. 6.15b, and dissolves graphite in preference to the oxides. Once the graphite-rich layer disappears due to dissolution into metal, the slag can again penetrate the boundary between the metal and the specimen, and the process is repeated. This cycle produces a local corrosion zone at the metal±slag interface. The up-and-down motion of the slag±metal interface shown in Fig. 6.15 was clearly observed using X-ray radiographic techniques. The Marangoni flow of the slag film is also considered to play an important role in the local corrosion of oxide±graphite refractory at the metal±slag interface during the stage shown in Fig. 6.15a and also the evolution of gas bubbles influence the dissolution process.
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6.15 Schematic representation of the manner in which local corrosion of oxide±graphite refractory proceeds.
The local corrosion of oxide±graphite refractory also occurs at the interface of slag±oxidative gas. The manner in which the local corrosion occurs in this system is essentially the same as that for the oxide refractory±slag±liquid steel system when the oxidative gas phase, which removes the graphite in the refractory by oxidizing it, is replaced with a liquid steel phase and the present system is turned upside down.
6.4.2 The rate of heterogeneous reaction between gas and metal or slag and metal The local corrosion of refractory described in Section 6.4.1 is a typical case of the heterogeneous reaction of the systems, refractory (solid)±slag±gas and refractory±slag±metal where the Maragoni effect accelerates the reaction rate. The rate of heterogeneous reaction of gas±metal and slag±metal is also seriously influenced by the addition of surface active agents such as oxygen, sulfur, etc. In particular, the reaction rate between nitrogen gas and liquid iron has been studied by many investigators. As a result, there is general agreement that surface active elements such as oxygen, sulfur, etc., inhibit the absorption rate of nitrogen into molten iron and also the desorption rate of nitrogen from liquid iron. Surface active elements at the surface of the liquid iron can be explained to affect reaction rates through a surface blocking mechanism.31,32 Almost all the previous investigations were conducted under actively stirred conditions of metal phase, such as induction heating of metal, levitation of metal drop, free falling of metal drop, etc. When the metal is heated by an electric resistance furnace, that is, non-inductive heating, it is reasonably explained that the rates both of nitrogen absorption and desorption are dominantly influenced by the Marangoni convection.33,34 Nitrogen is a pretty strong surface active agent
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although it is not as strong as oxygen or sulfur. Local differences in nitrogen concentration on the surface of liquid iron, which is generated by the blowing of nitrogen onto the surface or blowing argon gas onto the surface of liquid iron containing nitrogen, induces Marangoni convection (interfacial turbulence), resulting in the acceleration of the gas±metal reaction rate. Oxygen in the metal reduces the extent of surface activeness of nitrogen,35 which suppresses the Marangoni convection caused by the local difference in nitrogen concentration on the surface and, as a result, decreases the rate of nitrogen±metal reaction. We may deduce that oxygen acts on the rate of nitrogen±gas reaction in the following two ways: (1) surface blocking by the adsorbed oxygen on the liquid iron surface, and (2) reduction of the Marangoni convection induced by local differences in nitrogen concentration on this surface due to weakening of the surface activeness of nitrogen by the oxygen.
6.4.3 Bubble injection and slag foaming Bubble injection Bubble volume VB (cm3) detached from the orifice plate immersed in liquid can be described by equation 6.69,36 which was obtained experimentally, when orifice diameter do (cm) in equation 6.69 is replaced by d1m , the maximum diameter of the periphery of a bubble adhering to the plate during its growth stage: VB ˆ 0:0814 …VG do0:5 †0:867
…6:69†
3
where VG is gas flow rate (cm /s). d1m increases with increasing 0012, resulting in an increase in the bubble volume VB as predicted by equation 6.69. 0012 is the contact angle between liquid and orifice. When bubbles are formed, for example, on the porous plug refractory, an increase in d1m due to poor wettability between liquid steel and the oxide porous plug leads to the coalescence of adjacent bubbles on the surface of the plug. This coalescence may result in the formation of large bubbles in liquid steel. X-ray in situ observation revealed that the bubble formed on the oxide porous plug immersed in liquid Fe-C alloy is much larger than that on the same plug immersed in water.37 Argon gas injected through porous refractory of the inner wall of the immersion nozzle may form a gas curtain (or film) between the boundary of liquid steel and the inner wall due to the poor wettability between liquid steel and the porous oxide refractory.37 The gas curtain may effectively prevent the clogging of immersion nozzle in continuous casting of liquid steel. Slag foaming The main cause of slag foaming in iron and steelmaking processes is considered to be the high speed evolution of fine CO bubbles from the interface between
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265
slag and metal due to metal±slag reaction.38,39 The fine CO bubble can be formed due to the good wettability between slag and liquid Fe-C alloy. Interfacial turbulence caused by the Marangoni convection may also participates in facilitating the fine CO bubble generation. Foam stability can be expressed using the foam index 0006(s) given by equation 6.70,40 which was obtained semi-empirically: 0006 ˆ 1150011s1:2 = s0:2 001as dB0:9
…6:70†
where 0011s , s and 001as are viscosity (N0001s/m2), surface tension (N/m) and density (kg/m3) of slag, respectively. Equation 6.7141 predicts that the foam height hf is facilitated by the high speed evolution of the CO bubble, that is, large us , the superficial gas velocity and generation of CO bubbles with small diameter dB : h f ˆ us 0001 0006
…6:71†
6.4.4 Penetration of slag or metal into refractory Penetration of slag into refractory usually accelerates the rate of dissolution or abrasion of refractory into liquid slag and also the penetration layer of the refractory induces structural spalling, resulting in the shortening of the refractory life. Liquid slag penetrates very rapidly into MgO refractory because of the good wettability between the slag and MgO. Penetrated height h is linked with time, t by equation 6.72, which was obtained for the initial stage of the slag penetration by in situ observation with the aid of high temperature X-ray radiographic techniques:42 h ˆ kt1=2
…6:72†
where k is constant and should depend on pore radius, surface tension and viscosity of slag, contact angle between slag and refractory, pore structure of refractory (for example, labyrinth factor), etc. Refractories composed of oxide and graphite, such as magnesia±carbon and alumina±carbon are very effective for suppressing the slag penetration because the carbon acts as an inhibiting material for slag penetration due to the poor wettability between carbon and slag. Usually liquid metal does not penetrate spontaneously into oxide refractory because of the poor wettabiliety between the metal and oxide. In general, porous purging plugs made of oxide are used to blow an inert gas into the molten steel. The life of a porous purging plug is determined by how deep the molten steel penetrates into the plug. The observed quantitative results obtained with the aid of high temperature X-ray radiography43 indicates that the liquid steel penetration into porous refractory under external hydrostatic pressure is influenced by the structure and size distribution of pores in the
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Fundamentals of metallurgy
refractory, and also the chemical reaction between the steel and refractory, which mainly reduces the size of pore and contact angle between the liquid steel and refractory.
6.4.5 Interaction of foreign particles with solid±liquid interface In the steelmaking process, the interaction between foreign particles in liquid steel and a solid±liquid interface should participate in the occurrence of bubble and inclusion-related defects in steel products and nozzle clogging in continuous casting of steel. Many studies have been done theoretically and experimentally on the engulfment and pushing of foreign particles by the solidifying interface.44,45 The following factors were thought to have great influences on the engulfment or pushing of foreign particles: the size of foreign particles, solidifying velocity and the difference between the thermal conductivity of foreign particles and liquid. Various analytical models have been proposed for describing a critical solidifying velocity Vc in relation to the above mentioned factors by taking account of the forces of buoyant, viscous drag, Saffman and van der Waals forces acting on the particle in the vicinity of the advancing interface. The particle will be pushed ahead of the interface or engulfed by the interface depending on whether its velocity normal to the local interface is smaller or greater than some critical velocity, Vc . However, those models could not describe precisely the observed results.45 During the solidification of molten steel, a boundary layer, in which the concentration and temperature gradients exist, is formed in front of the solidifying interface. Once inclusions and bubbles enter the boundary layer, concentration gradients of O, S, Ti, etc., can produce an interfacial tension gradient around the particle, and in turn, could result in a driving force acting on the particle. This driving force could make the particle move towards the solidifying interface and promote the engulfment of the foreign particles.21 The model taking account of the force caused by the surface tension gradient provides a reasonable explanation of the fact that bubble defects in practical continuous casting operations increase with the increase of surface active element concentrations in molten steel.46 The clogging of the immersion nozzle in the continuous casting process results mainly from alumina build-up on the inner wall of immersion nozzle. Recently various models for explaining the mechanism of nozzle clogging were proposed. However, none of them could fully succeed in clarifying the mechanism. The force caused by the interfacial tension gradient between inclusion and liquid steel acting on the inclusion may hold a key to solve this technological problem of nozzle clogging. That may be the reason why very little inclusion is built up on the inner wall when the inner wall of the alumina± graphite nozzle is covered with high Al2O3 material, for the high Al2O3 material
Interfacial phenomena, metals processing and properties
267
intercepts the SiO and CO gas (generated in the nozzle) transfer via the inner wall±liquid steel interface into the steel, resulting in the reduction of the concentration gradient of Si and C in the vicinity of the interface.46
6.5
Further reading
The thermodynamic part of Section 6.2 has been compiled with reference to publications 1 and 2 and quoting partially from references 3 and 4.
1. R. Defay and I. Prigogine (translated by D.H. Everett), Surface Tension and Adsorption (1966) New York, John Wiley and Sons, Inc. 2. S. Ono and S. Kondo, `Molecular Theory of Surface Tension in Liquids', in S.FluÈgge, Handbuch der Physik, Band X (1960) Berlin, Springer-Verlag, 134±280. 3. J.G. Kirkwood and I. Oppenheim, Chemical Thermodynamics (1962) New York, McGraw-Hill. 4. C. Herring, `The Use of Classical, Macroscopic Concepts in Surface-Energy Problems', in R. Gomer and C.S. Smith, Structure and Properties of Solid Surfaces (1953) Chapter 1, Chicago, Univ. Chicago Press, 5±72.
6.6
References
È ber die Ausbreitung der Tropfen einer FluÈssigkeit auf der 1. C. Marangoni, `U OberflaÈche einer anderen', Ann. Phys. Chem. (1871) 143, 337±354. 2. T. Hibiya, S. Nakamura, K. Mukai, Z.G. Niu, N. Imaishi, S. Nishizawa, S. Yoda and M. Koyama, `Interfacial phenomena of molten silicon: Marangoni flow and surface tension', Phil. Trans. R. Soc. Lond. A (1998) 356, 899±909. 3. T. Nakamura, K. Yokoyama, F. Noguchi and K. Mukai, `Direct Observations of Marangoni Convection in Molten Salts', Materials Science Forum (1991) 73±75, 153±158. 4. K. Mukai, J.M. Toguri, I. Kodama and J. Yoshitomi, `Effect of Applied Potential on Interfacial Tension between Liquid Lead and PbO-SiO2 Slags', Canadian Metallurgical Quarterly (1986) 25(3), 225±231. 5. K. Mukai, `Marangoni flows and corrosion of refractory walls', Phil. Trans. R. Soc. Lond. A (1998) 356, 1015±1026. 6. J.K. Brimacombe and F. Weinberg, `Observations of Surface Movements of Liquid Copper and Tin', Metallurgical Transactions (1972) 3 (Aug.), 2298±2299. 7. S. Kimura, Y. Nabeshima, K. Nakajima and S. Mizoguchi, `Behavior of Nonmetallic Inclusions in Front of the Solid±Liquid Interface in Low-Carbon Steels', Metallurgical and Materials Transactions B (2000) 31B (Oct.), 1013±1021. 8. K. Mukai and W. Lin, `Motion of Small Particles in Solution with an Interfacial Tension Gradient and Engulfment of Particles by Solidifying Interface', Tetsu-toHagane (1994) 80(7), 527±532. 9. F.V. Vader, T.F. Erkens and V. Tempel, `Measurement of Dilational Surface Properties' Trans. Faraday Soc. (1964) 60, 1170±1177. 10. J.A. Kitchener, `Confirmation of the Gibbs Theory of Elasticity of Soap Film', Nature (1962) 194, 676±677. 11. J.A. Kitchener, `Elasticity of Soap Films: an Amendment', Nature (1962) 195, 1094±1095.
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12. S. Seetharaman, K. Mukai and D. Sichen, `Viscosities of slags ± an overview, VII Int. Conf. Molten Slags Fluxes and Salts, Cape Town, The South African Institute of Mining and Metallurgy (2004), 31±41. 13. R.N. Wenzel, `Resistance of Solid Surfaces to Wetting by Water' Ind. Eng. Chem. (1936) 28, 988±994. 14. A.B.D. Cassie and S. Baxter, `Wettability of Porous Surfaces' Trans. Faraday Soc. (1944) 40, 546±551. 15. K. Wasai and K. Mukai, `Thermodynamics of Nucleation and Supersaturation for the Aluminum-Deoxidation Reaction in Liquid Iron', Metallurgical and Materials Transactions B (1999) 30B(Dec.), 1065±1074. 16. D. Turnbull and J.C. Fisher, `Rate of Nucleation in Condensed Systems', J. Chem. Phys. (1949) 17(1), 71±73. 17. J.H. Hollomon and D. Turnbull, `Nucleation', Chapter 7 in B. Chalmers, Progress in Metal Physics, London, Pergamon Press (1953) 4, 342±343. 18. K. Mukai, T. Matsushita and S. Seetharaman, `Motion of Fine Particles in Liquid Caused by Interfacial Tension Gradient in Relation to Metals Separation Technologies', Metal Separation Technologies III, Copper Mountain, Colorado, Helsinki Univ. of Technology (2004), 269±273. 19. R.E. Boni and G. Derge, `Surface Tensions of Silicates', Trans. Met. Soc. AIME (1956) 206, 53±59. 20. B.J. Keene, `Review of data for the surface tension of iron and its binary alloys', International Materials Reviews (1988) 33(1), 1±37. 21. Z. Wang, K. Mukai and I.J. Lee, `Behavior of Fine Bubbles in Front of the Solidifying Interface', ISIJ International (1999) 39(6), 553±562. 22. K. Mukai and N. Shinozaki, `Melting and Flow Behavior of Fe-O Melts Heated by Plasma Arc', Materials Transactions, JIM (1992) 33(1), 45±50. 23. K.C. Mills and B.J. Keene, `Factors affecting variable weld penetration', International Materials Reviews (1990) 35(4), 185±216. 24. K. Ishizaki, N. Araki and H. Murai, `Interfacial Tension Theory on the Phenomena of Arc Welding (Chapter 9)', J. Jpn Weld. Soc. (1965) 34(2), 146±153. 25. B.J. Keene, `Interfacial Tension between Ferrous Melts and Molten Slags', NPL Report DMM(D)115 (Nov.) (1991) 1±206. 26. P. Kozakevitch, G. Urbain and M. Sage, `Sur la tension interfaciale fonte/laitier et le meÂchanisme de deÂsulfuration', Revue de Metallurgie (1955) LII(2), 161±172. 27. A.A. Deryabin, S.I. Popel and L.N. Saburov, `Non-equilibrium Interfacial Tension and Adhesion in the System of Liquid Metal-Oxide', Izv. A. N. SSSR Metally (1968) 5, 51±59. 28. K. Nogi (2001), `Wettability between Liquid Metals and Ceramics', Chapter 3, in T. Toshio et al., Wettability Technology Handbook (2001) Techno System Co. Ltd, 127±145. 29. N. Shinozaki, N. Echida, K. Mukai, Y. Takahashi and Y. Tanaka, `Wettability of Al2O3-MgO, ZrO2-CaO, Al2O3-CaO Subatrates with Molten Iron', Tetsu-toHaganeÂ' (1994) 80(10), 743±752. 30. K. Mukai and Z. Tao, `Local corrosion of refractories in iron and steelmaking processes', Recent Res. Devel. in Metallurg. and Materials Sci. (1998) 2, 17±24. 31. M. Byrne and G.R. Belton, `Studies of the Interfacial Kinetics of the Reaction of Nitrogen with Liquid Iron by the 15N-14N Isotope Exchange Reaction', Metall. Trans. B (1983) 14B, 441±449.
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32. J. Lee and K. Morita, `Interfacial Kinetics of Nitrogen with Molten Iron Containing Sulfur', ISIJ International (2003) 43(1), 14±19. 33. Z. Jun and K. Mukai, `The Influence of Oxygen on the Rate of Nitrogen Absorption into Molten Iron and Marangoni Convection', ISIJ International (1998) 38(3), 220± 228. 34. Z. Jun and K. Mukai, `The Rate of Nitrogen Desorption from Liquid Iron by Blowing Argon Gas under the Condition of Non-inductive Stirring', ISIJ International (1999) 39(3), 219±228. 35. Z. Jun and K. Mukai, `The Surface Tension of Liquid Iron Containing Nitrogen and Oxygen', ISIJ International (1998) 38(10), 1039±1044. 36. L. Davidson and E.H. Amick Jr., `Formation of Gas Bubbles at Horizontal Orifices', A. I. Ch. E. J. (1956) 2(3), 337±342. 37. Z. Wang, K. Mukai and D. Izu, `Influence of Wettability on the Behavior of Argon Bubbles and Fluid Flow inside the Nozzle and Mold', ISIJ International (1999) 39(2), 154±163. 38. Y. Ogawa and N. Tokumitsu, `Observation of Slag Foaming by X-ray Fluoroscopy', Sixth Int. Iron and Steel Congress, Nagoya, ISIJ (1990), 147±152. 39. K. Mukai, `Some Views on the Slag Foaming in Iron and Steelmaking Processes', Tetsu-to-Hagane (1991) 77(6), 856±858. 40. Y. Zhang and R.J. Fruehan, `Effect of the Bubble Size and Chemical Reactions on Slag Foaming', Metallurgical and Materials Transactions B (1995) 26B (Aug.), 803±812. 41. K. Ito and R.J. Fruehan, `Study on the Foaming of CaO-SiO2-FeO Slags', Metallurgical Transactions B (1989) 20B (Aug.), 509±514. 42. K. Mukai, Z. Tao, K. Goto, Z. Li and T. Takashima, `In-situ observation of slag penetration into MgO refractory', Scandinavian Journal of Metallurgy (2002) 31, 68±78. 43. T. Matsushita, T. Ouchi, K. Mukai, I. Sasaka and J. Yoshitomi, `Direct Observation of Molten Steel Penetration into Porous Refractory', J. Technical Association of Refractories, Japan (2003) 23(1), 15±19. 44. K. Mukai, `Engulfment and Pushing of Foreign Particles Such as Inclusions and Bubbles at Solidifying Interface', Tetsu-to-Hagane (1996) 82(1), 8±14. 45. R.W. Smith, X. Zhu, M.C. Tunnicliffe, C.L. Russell and W.M.T. Gallerneault, `The Capture of Suspended Particles as a Crystallising Phase Advances into Its Melt', 1st Int. Sympo. Microgravity Research & Applications in Physical Sciences & Biotechnology, Sorrento, ESA SP-454 (2001), 613±620. 46. K. Mukai and M. Zeze, `Motion of Fine Particles under Interfacial Tension Gradient in Relation to Continuous Casting Process', Steel Research (2003) 74(3), 131±138.
7
The kinetics of metallurgical reactions
S S R I D H A R , Carnegie Mellon University, USA and H Y S O H N , University of Utah
7.1
Introduction
Metallurgical processes such as extraction, refining, casting or annealing, almost always involve multiple phases and the kinetics are often coupled with the rate of movement of boundaries (surfaces and interfaces) between phases. Therefore, the emphasis of this chapter is on the fundamentals of heterogeneous reactions.
7.2
Fundamentals of heterogeneous kinetics
The recovery of metal values from their ores requires heterogeneous reactions between phases. Pyrometallurgical reactions such as the reduction of metal oxides and sulfides, the various types of roasting of metal sulfides, and hydrometallurgical operations such as leaching and solvent extraction are some examples. Heterogeneous reactions take place at a phase boundary between the reacting phases, unlike homogeneous reactions that take place over the entire volume of a given phase. Therefore, heterogeneous reactions always accompany the transfer of mass between the reaction interface and the bulk phase, in addition to the rate of the chemical reaction. Additionally, many chemical reactions are accompanied by the absorption or liberation of heat, and hence the transport of heat must also be considered. For this reason, most heterogeneous reactions involve a rather complex set of steps. Most reactions in metallurgical processes are between solids and fluids in which the solid participates as a reactant that undergoes chemical changes. Although reactions between immiscible liquid phases and between a gas and a liquid are also heterogeneous in nature, the discussion in this section will largely involve the analysis of fluid±solid reactions. Most gas±liquid and liquid±liquid reactions are rate-controlled by mass transfer, and thus their analysis reduces to a problem of mass transfer combined with chemical equilibrium at the interface. The individual component steps of the overall heterogeneous reaction process are described in this section.
The kinetics of metallurgical reactions
271
7.2.1 Reactions involving adsorption and desorption Most heterogeneous reactions occurring in metallurgical systems are described with rate expressions that are of first order with respect to the fluid reactant and product. Although often justified, they are not always correct, and even when the use of the first-order dependence is reasonable, it applies only in the range of concentrations in which the data have been collected. Extrapolation beyond this range of concentrations is risky. The reason for this is that heterogeneous reactions involve the adsorption and desorption of the reactants and products present in the fluid phase. These processes make the general rate expression nonlinear (Szekely et al., 1976). Furthermore, the rates of the intrinsic chemical reactions cannot be estimated to any reasonable degree of reliability and are highly specific to the chemical as well as the physical nature of the substances involved. Thus, these rates can only be obtained by experiments. This is in contrast to transport properties, which can be estimated to reasonable accuracy without the need of experiments. This applies in large measure to turbulent flows and mass transfer through the pores of a porous solid. Let us consider the following fluid±solid reaction to represent heterogeneous reactions: A…f † ‡ b 0001 B…s† ˆ c 0001 C…f † ‡ d…s†
…7:1†
Rather comprehensive review and derivation of the rare expressions for reactors involving adsorption, desorption, and surface reaction have been presented by Szekeley et al. (1976). The resulting rate expressions have somewhat different forms depending on which among the above steps controls the overall rate. However, they all are special cases of the following expression, in terms of the molar rate of consumption of the fluid reactant per unit area of the reaction interface: …ÿm_ A † ˆ or ˆ
k…CAm ÿ CCn =KC †
1 ‡ K1 CAi ‡ K2 CCj n kp …pm A ÿ pC =Kp †
…7:2†
1 ‡ K10 piA ‡ K20 pjC
The terms in the denominators arise from the adsorption and desorption of the fluid species. For liquid±solid reactions, the molar concentration is used in the rate equation. For gas±solid reactions, either the molar concentration or the partial pressure has been used in the literature. Partial pressure is preferred for gas±solid reactions because it more accurately represents the activity of a gaseous species and Kp in the rate equation is the thermodynamic equilibrium constant. In this chapter, however, the molar concentration is used to simultaneously treat both the gas±solid and liquid±solid reaction systems. For a gas± solid reaction at equilibrium, m_ A ˆ 0 and
272
Fundamentals of metallurgy KC ˆ
CCn …pC =RT†n pnC ˆ …RT†mÿn ˆ Kp …RT†mÿn m ˆ CA …pA =RT†m pm A
…7:3†
For a liquid±solid reaction, KC is related to the thermodynamic equilibrium constant through the activity coefficients. Many reactions can be described by a simple kinetic expression with m ˆ n ˆ 1 and K1 ˆ K2 ˆ 0 in equation 7.2, which represents a first-order reversible reaction. This further simplifies to the often used first-order rate expression when KC or Kp is large, i.e. the reaction is irreversible.
7.2.2 Heat and mass transfer Diffusion in a binary mixture The transport of species in the fluid phase occurs as a result of concentration differences. The movement of a species in a binary mixture of species A and C in the presence of a concentration gradient and in the absence of turbulence is described by Fick's first law of diffusion: Ni ˆ ÿCDAC rxi ‡ xi …NA ‡ NC †
…7:4†
Here the molar flux is defined with respect to fixed coordinates and is expressed as Ni ˆ xi Cui
…7:5†
It is noted that, for equimolar counter diffusion, NA ˆ ÿNC
…7:6†
In equation 7.4, the second term on the right-hand side refers to the flux of species A resulting from the bulk motion of the fluid, and the first term is due to the diffusion superimposed on the bulk flow. The binary molecular diffusivity DAC is a function of temperature and pressure. For gases at low density the molecular diffusivity is relatively independent of mole fraction, whereas for liquids the dependence on concentration can be significant. Extensive literature is available on the estimation of diffusivities. The diffusivities of gases can, in general, be estimated more accurately (Bird et al., 2002; Reid et al., 1973; Satterfield, 1970) than those in a liquid phase (Bird et al., 2002; Reid et al., 1973). Multicomponent diffusion Although theories of binary diffusion are sufficient to describe most metallurgical systems, there are situations where multicomponent diffusion must be considered. The rigorous equation to describe the mass flux may be found elsewhere (Bird et al., 2002). However, a simplified equation for the molar flux results if the species are assumed to behave ideally (Bird et al., 2002):
The kinetics of metallurgical reactions rxi ˆ
n X 1 …xi Nj ÿ xj Ni † CD ij jˆ1
273 …7:7†
where Dij is the binary diffusivity. The calculation of the concentration gradient in multicomponent diffusion through the solution of the rigorous Stefan± Maxwell equations is rather complex. In some instances, it is convenient to define an effective binary diffusivity Dim so that the flux of i in a multicomponent system can be expressed by a relation analogous to equation 7.4 (Bird et al., 2002): X Nj …7:8† Ni ˆ ÿCDim rxi ‡ xi Certain approximations can be made to evaluate Dim for special cases, and one that has proved useful when the variation of Dim is considerable is to assume a linear dependence on concentration (Reid et al., 1973). Satterfield (1970) provides the following equation: !ÿ1 X xj …7:9† Dim ˆ …1 ÿ xi † Dij j6ˆ1 Conduction The conduction of heat is governed by Fourier's law: q ˆ krT
…7:10†
Here, q is the heat flux vector and k is thermal conductivity. Methods for estimating thermal conductivities of gases, liquids, and solids are available in the literature (Bird et al., 2002; Reid et al., 1973). Transfer in porous media Many solid reactants have some porosity, allowing the fluid phase to diffuse in while reacting with it. Additionally, in recent times much attention has been directed towards feed preparation so that solids have sufficient porosity and yet have sufficient strength to withstand the physical forces in the reactor. Extensive investigations have been made on heat and mass transfer in porous solids (Szekely et al., 1976; Satterfield, 1970). The conduction of heat in a porous solid is also described by Fourier's law of conduction, as stated in equation 7.10, but the thermal conductivity of a porous solid is not well understood. The usual approach is to define an effective thermal conductivity that relates the heat flux to the temperature gradient. Experimentally measured values of effective thermal conductivity are most reliable. Some methods of estimating this parameter, however, are available. Two extreme types of behavior are considered, with real systems occupying an intermediate position. The first type is a solid made up of a continuous solid
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phase containing closed and isolated pores filled with a fluid that has a thermal conductivity much lower than that of the solid. Most porous pellets are described by this case. The effective thermal conductivity for such a solid is expressed as (Francl and Kingery, 1954) ke ˆ k…1 ÿ 000f†
…7:11†
In the second case, the fluid is considered to be a continuum, and the solid is made up of small particles that are in point contact with the neighboring particles (Kunii and Smith, 1960). This picture best describes loose compacts of fine powders. The effective thermal conductivity depends largely on that of the fluid. The transport of matter in a porous solid matrix is an important step in heterogeneous reactions since fluid molecules must diffuse through the pores to gain access to the solid surface. Factors that complicate the description of pore diffusion are: (a) The volume occupied by the solid is not available for mass transfer. (b) The diffusion path is quite tortuous. (c) Molecular diffusion or Knudson diffusion or both play an important role, depending on the pore size. (d) Bulk flows, sometimes with large pressure gradients, may develop within the solid (Evans and Song, 1973; Evans, 1972). This may occur when there is a net generation of product gases with the chemical reaction. When the pores are small, a large pressure gradient may develop. The effective diffusivity of the fluid in the pores of the solid is in general much smaller than the molecular diffusivity and includes the various factors discussed above. Comprehensive discussions on the earlier work can be found in (Szekely et al., 1976; Satterfield, 1970; Smith, 1970; Mason and Marrero, 1970). A widely used treatment of pore diffusion is attributed to Mason and coworkers (1970; 1967). For a binary mixture in an isothermal system, and
NA ˆ ÿDAeff rCA ‡ xA 000eA N ÿ xA A …CT B0 =0016†rP
…7:12†
0012 00130012 0013 DBK pB0 1‡ N ÿ A NA ˆ ÿ rP RT DK
…7:13†
N ˆ NA ‡ NC ;
…7:14†
where i.e., total flux
1=DAeff ˆ 1=DAK ‡ 1=DACeff
…7:15†
000eA ˆ DAeff =DACeff
…7:16†
A ˆ DAeff =DAK
…7:17†
A ˆ 1 ÿ DCK =DAK
…7:18†
The kinetics of metallurgical reactions
275
and 1=DK ˆ xA =DAK ‡ xC =DCK
…7:19†
Here, B0 is a parameter characteristic of the solid, and 0016 is the viscosity of the fluid. In equation 7.12, the first term on the right-hand side refers to the diffusive flux resulting from the concentration gradient, the second term is the bulk flow term, and the third is the viscous flow term. Under certain circumstances the bulk flow and viscous flow terms may be neglected. When equimolar counterdiffusion is encountered or when the mole fraction of the transferred species is small, the bulk flow term may be neglected. The effective diffusivities are estimated as follows: DACeff ˆ DAC …000f=001c†
…7:20†
where 000f is the porosity and 001c is the tortuosity factor characteristic of the solid; and 0012 00130012 0013 4 8RT 1=2 DAK ˆ K0 …7:21† 3 0019MA where K0 is a constant characteristic of the solid that has a unit of length. The dusty gas model of Mason et al. (1967), which assumes the solid to be made up of spherical particles of radius rg , gives 0012 0013 128 0010nd 001c 0011 2 0010 00190011 K0ÿ1 ˆ rg 1 ‡ …7:22† 9 000f 8 Additional developments and reviews on diffusion in porous media are given in the literature (Abbasi et al., 1983; Ohmi et al., 1982; Gavalas and Kim, 1981; McCune et al., 1979; Chen and Rinker, 1979; Alzaydi et al., 1978; Pismen, 1974; Youngqvist, 1970; Wakao and Smith, 1962). Convective heat and mass transfer In a heterogeneous reaction system, the transfer of heat and mass between the interface and the bulk fluid is an important step in the overall process. The rate of mass transfer between the bulk fluid and the surface of a solid is expressed as NA ˆ km …CAs ÿ CAb †
…7:23†
where CAs and CAb are respectively the concentration of the species A at the surface of the solid and that in the bulk fluid, and km is the external mass-transfer coefficient. The above equation assumes that the flux is proportional to the concentration difference. Bird et al. (2002) have suggested that a more rigorous equation that includes a net flow normal to the interface is NA ˆ km …CAs ÿ CAb † ‡ xA …NA ‡ NC †
…7:24†
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where the second term is the net flow normal to the interface. In most fluid±solid systems, xA 001c 1 or equimolar counterdiffusion takes place so that NA ˆ ÿNB . In either case, equation 7.24 reduces to equation 7.23. Considering that xA may vary significantly between the interface and the bulk fluid, Han and Sohn [2004] developed the following expression for the average mole fraction to be used in place of xA in equation 7.24: 1 ‡ …0017 ÿ 1†~xA ˆ
‰1 ‡ …0017 ÿ 1†xAs Š ÿ ‰1 ‡ …0017 ÿ 1†xAb Š 1 ‡ …0017 ÿ 1†xAs ln 1 ‡ …0017 ÿ 1†xAb
…7:25†
where 0017ˆÿ
NC NA
…7:26†
Frequently, the mass-transfer coefficient is expressed as the Chilton±Coburn j factor, defined as Sh …7:27† ReSc1=3 Various correlations of this form are available in the literature (Evans, 1979; Malling and Thodos, 1967; Wilson and Geankoplis, 1966; Rowe et al., 1965). A correlation for mass transfer involving single spheres that has found wide application is that by Ranz and Marshall (1952): jD ˆ
Sh ˆ 2:0 ‡ 0:6 Re1=2 Sc1=3
…7:28†
It must be pointed out that at a low Reynolds number the effect of natural convection can become significant, and correlations including this effect must be used (Steinberger and Treybal, 1960). The description of external heat transfer is similar to that of external mass transfer. The heat flux to the solid surface is expressed as q ˆ h…Ts ÿ Tb †
…7:29†
where h is the heat-transfer coefficient and Ts and Tb are the temperatures of the solid surface and the bulk fluid, respectively. The form of the empirical correlations for heat transfer can be obtained in a manner similar to that of the mass-transfer correlation. By analogy with mass transfer, the Ranz±Marshall equation for heat transfer may be written as Nu ˆ 2:0 ‡ 0:6 Re1=2 Pr1=3
…7:30†
and additionally it may be assumed that the Chilton±Coburn j factors are identical for mass and heat transfer, i.e., jD ˆ jH where
…7:31†
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277
Nu …7:32† RePr1=3 At high temperatures, the solid may receive heat from its surrounding by radiation in addition to convection described above. The rate of radiative transfer is described by the Stefan±Boltzman equation. As an example, the expression for simple radiative heat transfer between two surfaces of equal emissivity may be written for the surface with a unit view factor as follows: jH ˆ
qr ˆ 001b000f…Te4 ÿ Ts4 †
…7:33a†
where qr is the radiative flux, 000f is the emissivity and 001b is the Stefan±Boltzman constant. The description of radiative transfer in general is quite complex and is strongly dependent on the system geometry. While the above equation is a fairly good approximation, it is generally suggested that the relative importance of radiation and convection be determined prior to including a description of radiative transfer. For this purpose, it is convenient to define a radiative heattransfer coefficient, hr, hr ˆ 001b000f…Te2 ‡ Ts2 †…Te ‡ Ts †
…7:33b†
qr ˆ hr …Te ÿ Ts †
…7:33c†
and Then, hr may be combined with h to give the total heat-transfer coefficient. When two surfaces that are not oriented parallel to one another exchange radiative heat, the radiation incident on each surface is only part of the total emissive power and depends on the surface geometries and orientations. This is accounted for by a so-called view factors (Fij ) which can be computed but are tabulated in textbooks for simple cases (Poirier and Geiger, 1994). If two surfaces (1 and 2) have arbitrary shapes and orientations, the net radiative exchange will be: q1!2 ˆ F12 001b000f1 T14 ÿ F21 001b000f2 T24
…7:33d†
The view factors have two important properties; (i) the reciprocity relation: Ai 0001 Fij ˆ Aj 0001 Fji and (ii) that the sum of all view factors for a given surface equals unity, i.e. n X
Fij ˆ 1
jˆ1
Radiative heat transfer in an enclosure consisting of surfaces of different temperatures and radiative properties requires more complex treatment (Modest, 1993). The procedure becomes even more difficult when involving suspended particulate and temperature variations within the volume of the enclosure (Hahn and Sohn, 1990b; Perez-Tello et al., 2001b).
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7.3
Solid-state reactions
7.3.1 Reaction in a single phase Point defect formation and elimination In metals, point defects are not charged and thus the condition of electro-neutrality does not arise as it does for ionic crystals. Thus in metals single defects form rather than pairs of oppositely charged defects. In general, single defects such as vacancies or interstitials form at sites such as dislocation, grain boundaries (low and high angle) and surfaces according to the following equilibria: Mesite ‡ Vi ˆ Mei
…7:34†
MeMe ˆ Mesite ‡ VMe
…7:35†
and Here, the subscript `site' refers to a defect generation site. Homogenization of compositional gradients These reactions are diffusional processes where inequalities in concentrations and thus chemical potential gradients are equilibrated through transport of mass. In the cases where the non-equilibrium is caused by a single interstitially dissolved element at dilute amounts, the process can be readily described by an appropriate mathematical solution to Fick's first and second laws for diffusion: J0016 ˆ ÿD 0001 rC
…7:36†
@C …7:37† ˆ Dr2 C @t Here J is the flux of atoms per unit area, D is the diffusion coefficient and C is the concentration of the diffusing species. For isotropic solids, the diffusion coefficient can be taken as a constant whereas, for non-isotropic solids, D will be described by a 3 0002 3 matrix. The appropriate solution to Fick's laws can be found by solving the equations above, using the geometry of the system and boundary conditions. For more complex solutions, monographs on the mathematics of diffusion (Crank, 1956) or thermal conduction (Fourier's laws are mathematically identical to Fick's laws for these cases) can be used to find solutions (Carslaw and Jaeger, 1959). An example of simple one-dimensional diffusion processes and their appropriate mathematical descriptions is discussed below. In order to make an electronic component, boron has to be implanted into silicon. To do this, a thin silicon film containing boron is deposited on one surface of a 2 mm thick silicon wafer. The thickness of the film is 000e 0016m and boron concentration is C0 atoms/m3. Diffusion coefficient of B in Si is D m2/s. If the silicon wafer can be assumed to be semi-infinite compared to the deposited
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film, the concentration of B in the Si wafer will be described by the following solution to Fick's second law: 0012 0013 000eC0 x2 …7:38† C ˆ p exp ÿ 4Dt 0019Dt In non-dilute binary substitution systems such as diffusion couples several complications arise with respect to the mathematical treatment of the problems. First, if fluxes of the two elements are substantially different there will be a net movement of vacancies, which can cause the so-called Kirkendall shifts (Smigelskas and Kirkendall, 1947). This is a process by which mass is transported due to stress gradients caused by the vacancies without involving diffusional processes. Second, large concentration gradients invalidate the equality of concentration gradients to chemical potential gradients, i.e. activity coefficients need to be considered. Third, the intrinsic diffusion coefficients themselves are likely to vary. In the case of a binary semi-infinite diffusion couple problem like the one shown in Fig. 7.1, the two intrinsic diffusion coefficients that vary with position can be replaced with a so-called chemical diffusion coefficient (Darken and Gurry, 1953): 0012 0013 ÿ 0001 d ln 1 0003 0003 ~ …7:39† D ˆ N2 D1 ‡ N1 D2 1 ‡ d ln N1 Here, N1 and N2 are the mole fractions, 1 is the activity coefficient of component 1 and D00031 and D00032 are the tracer diffusion coefficients of 1 and 2. The tracer diffusion coefficients differ from the intrinsic ones in that they are not affected by activity coefficients but they may nevertheless vary with composition. The chemical diffusion coefficient can now be used in Fick's second law but since it is a function of composition and space, analytical solutions cannot be readily obtained. Evolution of grains In polycrystalline metal alloys, a random grain structure is inherently unstable since grains will more often than not have a net curvature and thus all the grain boundary tension forces are not balanced. Therefore, when annealed grain boundaries will tend to migrate towards their centers of curvature. The force on
7.1 Binary diffusion couple.
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7.2 Two-dimensional grain model.
a boundary of curvature r, will be of the magnitude =r. The effect of curvature can be exemplified by the two-dimensional model shown in Fig. 7.2. At the junction between three grains of equal boundary energy, a force balance necessitates that a 120ë angle is formed. Therefore, a grain is only stable when it has six boundaries. If the number of boundaries around a grain is fewer than six, the boundaries will all curve in a convex manner towards the grain (see Fig. 7.2a) and the grain will tend to shrink. The opposite will occur when the number of boundaries is greater than six (see Fig. 7.2b). As a result, during annealing there will be an increase in the mean grain size and a decrease in the number of grains. This process is known as grain growth. Atomistically, atoms detach themselves from the high pressure concave side of the boundary (the shrinking grain) and attach themselves on the low-pressure convex side (the growing grain). The pressure difference across a boundary can be related to a potential ± or free energy difference: 2 Vm …7:40† r where Vm is the molar volume, and 0001G is the free energy difference, per atom, between two grains adjacent to a curved boundary. It should be mentioned that while 0001G is the driving force behind all grain boundary motion, it is governed by dislocation strain-energies rather then grain curvatures in the case of recrystallization. The effect of 0001G on the grain migration kinetics can be described in the same way for recrystallization and grain growth. The flux of atoms from grain 1 to grain 2 can be expressed as an activated process (Porter and Easterling, 1992): 0012 00130012 0012 00130013 0001G0003 0001G 1 ÿ exp ÿ …7:41† J1!2 ˆ A2 n1 00171 exp ÿ RT RT 0001G ˆ
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Here, J is the net flux, A2 is the probability that a migrating atom from grain 1 will attach to grain 2, n1 is the number of atoms in a favorable position to jump in grain 1, 00171 is the atom vibration frequency and 0001G0003 is the activation energy needed for an atom to escape from grain 1. If the boundary is to move at a velocity, the flux J should be equal to v=…Vm =Na †, where Na is the Avogadros number and we get, assuming a sufficiently high temperature such that RT 001d 0001G: 0012 0013 A2 n1 001d1 Vm2 0001G0003 0001G _ ÿ 0001 …7:42† 0017ˆ exp Na RT RT Vm which means that the velocity should be, at a constant temperature, proportional to the driving force, v/0001
0001G Vm
…7:43†
The proportionality constant is defined as M ˆ mobility of the grain boundary.
7.3.2 Multiphase reactions Phase transformations can be grouped into those that require long-range diffusion and those that do not. The majority belong to the former category. Diffusional transformations Two examples of phase transformations are indicated in the phase diagrams in Fig. 7.3. In the first case, the precipitation transformation is: 1 ! 2 ‡
7.3 Examples of phase transformation.
…7:44†
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where of a certain composition transforms to of a different composition and of yet a different composition. The second case is a eutectoid transformation where,
!000b‡
…7:45†
where of a certain composition transforms to and of different compositions. In both the cases, solute redistribution is needed for the transformations and this is achieved through diffusion. These types of transformations proceed through, first, nucleation and then growth, and when applicable this could be followed by coarsening (Ostwald ripening). This text is only intended to provide a brief introduction to the subject of phase transformations and thus the discussion that follows on nucleation and growth will focus on the first type, i.e. precipitation of a second phase from a matrix. The eutectoid case is naturally more complicated since two new phases are formed simultaneously. Nucleation When a small amount of new phase forms from a matrix of the free energy will change according to: 0001G ˆ V …0001GS ‡ 0001GV † ‡ A =
…7:46†
In this equation, V is the volume of the -phase embryo and A is its interfacial area with the matrix. The terms 0001GV is the free energy change (ˆ 0001H ÿ T0001S) for the reaction, ! . Assuming that we are at a temperature and composition where is thermodynamically favorable, this term will be negative and thus contribute to a lowering of the energy. The term 0001GS is the strain energy due to lattice misfit between and and contributes with a positive term. is the interfacial energy between and and this also contributes with a positive term. It should be mentioned that the equation above is written by assuming that the interfacial energy is isotropic. If this was not the case, the term should be replaced by a summation of contributions from all areas. If the -embryo assumes a spherical shape, with radius r, the equation above can be written as: 4 …7:47† 0001G ˆ ÿ 0019 0001 r3 …0001GS ÿ 0001GV † ‡ 40019r2 0001 = 3 This equation is plotted as a function of radius in Fig. 7.4. It can be seen that the free energy goes through a maximum (0001G0003 ), at a critical r ˆ r0003 , and then decreases with size. Thus, -embryos that are larger then this critical size will spontaneously grow. Depending on the nature of the interface between the phases, 0001GS and = will contribute in different ways. If the interface is coherent, then the interfacial energy will be low whereas the strain energy will be high. On the other hand the opposite will be the case for incoherent interfaces. In general, the interfacial energy plays a greater role and thus
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7.4 Energy change during nucleation.
homogeneous nucleation is favored primarily as precipitates that are coherent with the matrix. While this is not possible due to the difference in most systems between matrix and precipitate crystal structures, a coherent meta-stable precipitate often nucleates first. The formation of GP zones is an example of this. By differentiating the equation and finding the maximum, 0001G0003 and r0003 can be evaluated as: r0003 ˆ 0001G0003 ˆ
2 = …0001GV ÿ 0001GS † 3 160019 =
3…0001GV ÿ 0001GS †2
…7:48† …7:49†
The number of -like clusters per unit volume of size r0003 , in a system of totally C0 atoms per unit volume, is: 0012 0013 ÿ0001G0003 0003 …7:50† C ˆ C0 exp kT A fraction, f, of these critical clusters is assumed to grow per unit time, by receiving an additional atom from the matrix. 0012 0013 0001Gm …7:51† f ˆ ! exp ÿ kT where, 0001Gm is the free energy of migration and ! is a factor that depends on the area of the critically sized cluster and atom vibration frequency. The nucleation rate per unit volume is then: 0012 0013 0012 0013 1 dN ÿ0001G0003 0001Gm exp ÿ …7:52† ˆ !C0 exp kT kT V dt
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When nucleation of a precipitate occurs at a location where the precipitate replaces a high energy area, the nucleation can be facilitated. This is due to the fact that a high energy interface will be covered by the precipitate and this will in part compensate for the interfacial free energy term in equation 7.47. This will result in a lowering of the critical free energy for nucleation. The degree of lowering is dependent on the specific heterogeneous site, e.g. grain boundary, dislocation, vacancy cluster, non-metallic inclusion, etc. as well as precipitate shape. This is called heterogeneous nucleation and is the primary mode of nucleation for non-coherent precipitates. Growth A precipitate's final size and shape is dependent on the rates of growth of the various interfaces it forms with the matrix. The growth rate is often a diffusion problem and most attention has been on transformations that occur upon cooling, i.e. growth of an undercooled stable phase in a thermodynamically unstable matrix phase. The following treatment developed by Zener (1949) is for a planar interface (incoherent interfaces are most likely planar) that grows in one dimension. A planar precipitate grows according to Fig. 7.5. The region adjacent to the precipitate (which is rich in solute) has been depleted of the solute. The interface is assumed to be in equilibrium and the concentration vs distance in the depleted region is approximated to be linear. It is, furthermore, assumed that the precipitate is sufficiently far from other precipitates that soft impingement (overlap of diffusion fields) does not occur. If the interface advances at a rate, v (ˆ dx=dt), a mass balance at the interface gives, 0017 0001 …C ÿ Ce † ˆ Jsolute ˆ ÿD
…C0 ÿ Ce † L…t†
7.5 Schematic precipitate growth.
…7:53†
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At any time, the diffusion length L…t† can be evaluated from a solute balance between precipitate and depleted zone: …C ÿ C0 †x ˆ 12 L…t†…C0 ÿ Ce †
…7:54†
Combining the two equations above results in: 0017ˆ
dx D…C0 ÿ Ce †2 ˆ dt 2x…C ÿ Ce †…C ÿ C0 †
…7:55†
Integration results in: xˆ
…C0 ÿ Ce † 0:5
…C ÿ Ce † …C ÿ C0 †
0:5
p Dt
…7:56†
If the molar volume remains relatively constant, the concentrations in the equations above can be replaced by mole-fractions (N). In the case of a one-dimensional growth of a non-planar front such as a needle, the effect of curvature on the equilibrium interface concentration (the Gibbs±Thompson effect) needs to be considered. The resulting growth rate can be shown to be (Jones and Trivedi, 1971): 0012 0013 …X0 ÿ Xe † 1 r0003 0017ˆD …7:57† 1ÿ r k…X ÿ Xr † r Here, Xr is interface concentration in the matrix, which is different from the planar equilibrium value due to the curvature. r0003 is the critical nuclei radius and r is the radius of the advancing tip. The diffusion length is proportional to the growing tip radius r. k in the above equation is the proportionality constant and from the diffusion solution is close to 1 (Porter and Easterling, 1992). The thickening of a plate where edges are facetted often occurs through a ledge mechanism. This means that atoms can only attach themselves at the edges. The growth rate can be shown to be (Laird and Aaronson, 1969): 0017ˆ
D…X0 ÿ Xe † k…X ÿ Xe †0015
…7:58†
Here, 0015 is the spacing between ledges. While the models discussed so far assume that diffusion is the rate-limiting step, it should be mentioned that there is also the possibility of interface reaction controlling the transformation rate. This is the case in single component (pure) metals where the transformation is governed by an activation energy barrier analogous to a first order chemical reaction. Combining nucleation and growth kinetics In general, the fraction transformed during a reaction ! will depend on the temperature history since both nucleation and growth are strong functions of temperature. The volume at time t of a grain that nucleated at time 001c will be
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(for three-dimensional spherical growth): 4 …7:59† V ˆ 00190017 3 …t ÿ 001c†3 3 At low transformation fractions, the volume contributed from all grains per unit total volume is N (nucleation rate): Z t X 4 0019 3 f ˆ V ˆ 0019N 0017 …t ÿ 001c†3 d001c ˆ N 0017 3 t4 …7:60† 3 3 0 The nucleation rate (N) has been assumed to be constant. If one considers impingement, which will slow down the rate, 0010 0019 0011 …7:61† f ˆ 1 ÿ exp ÿ N 0017 3 t4 3 This is known as the Johnson±Mehl±Avrami equation (Porter and Easterling, 1992) and will result in the classic C-shaped curve when the fraction transformed (f) is plotted in a diagram with temperature (T) as the y-axis and time (t) as the x-axis. A schematic of such a diagram, called a TTT-diagram (for time±temperature± transformation) is shown in Fig. 7.6. The C-shape is caused by a slow nucleation rate at high T and a slow growth (caused by slow diffusion) at low T. Phase transformation upon heating So far the discussion has focused upon phase transformations that occur upon cooling. The combined nucleation and growth behaviors, resulted in the TTT type behavior, i.e. the transformation is limited by nucleation at low undercooling (high temperature) and by growth at low temperatures. Also, the growth rates themselves have terms for both supersaturation (which increases with undercooling) and diffusion (which decreases with undercooling). The resulting rate of transformation vs temperature would then exhibit a maxima. In
7.6 Time±temperature±transformation (TTT) diagram.
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the case of heating above a transformation temperature, both the driving force and diffusion rate increase with temperature. Therefore the effect of increasing temperature is continuously to accelerate the transformation rate. Coarsening In order to lower the total interfacial energy, a multiphase alloy will strive to shift the size distribution of precipitates towards as small number of large precipitates as possible. The rate of this process is strongly dependent on temperature and, thus, is a concern in high temperature applications and processes. Depending on the process history, a certain size distribution of the second phase precipitates will be present in the matrix as a result of nucleation and growth. The chemical potential of solutes in the matrix adjacent to a precipitate will vary depending on the precipitate±matrix interface curvature due to the Gibbs±Thomson effect. Therefore there will be a chemical potential difference between two spherical precipitates of different sizes. This difference will be: 0012 0013 0012 0013 N1 1 1 ˆ 2 Vm …7:62† ÿ 0001G ˆ 00010016 ˆ RT ln N2 r1 r2 As can be seen, the solute concentration will be higher in the matrix near the smaller particle. As a result, the solute diffuses from the regions adjacent to smaller particles to the regions near larger ones and the larger particles grow at the cost of the smaller ones that shrink and eventually disappear altogether. As a result, the average particle radius ra among the particle population increases with time whereas the total number of particles decreases with time. If diffusion is the slowest step the rate of coarsening has been found to be (Wagner, 1961): ra3 ÿ ro3 ˆ kt
…7:63†
where ro is the mean starting radius. k is a kinetic constant that equals D 0003 0003 Ne , where Ne is the mole fraction of solutes at equilibrium with very large precipitates. Spinodal decomposition In the discussion of nucleation and growth of a second phase, it was assumed that a sharp boundary existed between the matrix and precipitating phase. In the case of spinodal decomposition, the transformation proceeds while maintaining a coherent and non-distinguishable boundary. In the case of spinodal decomposition, the free energy vs composition has a behavior as that is shown in Fig. 7.7, which is characteristic of a miscibility gap. Note that, inside the so-called spinodal region, the second derivative of Gibbs energy is negative. This means that any infinitesimal fluctuation in composition would lead to an energy decrease for the alloy and would be expected to grow spontaneously. Let us consider a one-dimensional binary decomposition case in
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7.7 Free energy vs composition for spinodal decomposition.
the spinodal region. Sinusoidal fluctuations may or may not result in decomposition into two phases as shown in Fig. 7.7. Based on Fick's second law solution, it can be shown (Cahn, 1968) that fluctuations will be stable under certain condition. A small free energy change can be written as a sum of the chemical (0001Gc ), strain (0001Gs ) and gradient (0001G ) parts. The chemical energy is: 1 d2 G …0001N †2 2 dN 2 The strain energy: 0001Gc ˆ
0001Gs ˆ 00112
E Vm …0001N †3 …1 ÿ 001d†
…7:64†
…7:65†
1 da Here E is Young's modulus, 001d is Poisson's ratio and 0011 ˆ , where a is the a dN lattice parameter. The gradient energy arises due to non-similar nearest neighbors and is thus related to unlike atom pairs: …0001N †2 …7:66† 0015 Here K is the so-called gradient energy, which depends on the differences in bond energies between like and unlike pairs. 0015 is the wavelength of the sinusoidal fluctuation. The total energy change can then be written: 0001G ˆ K
0012 0001G ˆ
The kinetics of metallurgical reactions 0013 d 2 G 2K E …0001N †2 2 ‡ ‡ 20011 V m 2 dN 2 00152 …1 ÿ 001d†
289 …7:67†
Fluctuations will according to this equation be stable when the term within the brackets is negative. By using a flux equation with a chemical potential gradient and solving Fick's second law, the kinetics of the decomposition are obtained (Cahn, 1968) as (after neglecting the non-linear terms): 0012 00132 0012 00134 ! 20019 20019 20019 ‡B 0001N ˆ N …t† ÿ N …t ˆ 0† ˆ cos x 0001 exp ÿA t …7:68† 0015 0015 0015 Here
00120012 AˆM
0013 0013 @2G 2 2E ‡ 0011 @N 2 1ÿ001d
and B ˆ 2KM=Vm M is the mobility Mˆ
ND RT
Ordering In some binary systems such as Cu-Zn or Cu-Au there is a strong negative deviation of the activities. This corresponds to a negative enthalpy of mixing. In general this is indicative of the fact that the components tend to strongly attract one another. This leads, at low temperatures, where the dis-ordering effect of entropy and thermal motion is weak, to the formation of the so-called super structures. In these structures the atoms organize themselves to maximize the dissimilar bonds and minimize the similar ones. In general, the ordering tendency increases with decreasing temperature below a critical temperature Tc . The internal energy and enthalpy are continuous across this temperature and thus this is a second order transformation. The common mechanism (Porter and Easterling, 1992) is through nucleation and growth. An ordered region is referred to as a domain. Both interfacial energy and strain energy are expected to be low, and therefore nucleation is relatively easy and occurs homogeneously. Martensitic transformations Martensitic transformations are brought about by a movement of the interface between parent and product phases. As the interface advances, atoms in the parent lattice re-align into the more energetically favorable martensite structure. The displacement of atoms is relatively small (less than one inter-atomic spacing) in magnitude and no compositional changes occur. Instead, the atomic re-alignment results in a shape deformations and a change in symmetry from the
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original lattice. The martensitic transformation that FCC lattice undergoes to form a BCT is described by a process called the Bain distortion (Reed-Hill and Abbaschian, 1992). Since no diffusion is involved, these transformations are called diffusionless. The growth rates of martensite are extremely fast and approach the speed of sound and therefore nucleation is what primarily controls the grain size. Nucleation is thought to occur heterogeneously (Reed-Hill and Abbaschian, 1992). The interfacial energy is relatively low but the shear associated with the Bain transformation contributes to an elastic strain energy that hinders homogeneous nucleation. The free energy of an ellipsoidal martensite embryo platelet with radius a and thickness 2c is: 0001G ˆ 20019a2 ‡
2…2 ÿ 001d† 4 2…s=2†2 0016ac2 ÿ 0019a2 c 0001 0001Gv 8…1 ÿ 001d† 3
…7:69†
The second term is the elastic strain energy term where s is the strain, 001d is Poisson's ratio and 0016 is the shear modulus. The most likely heterogeneous nucleation sites are thought to be dislocations whose strain energy field assists nucleation. The nucleation energy barrier can be reduced by the dislocation interaction energy (Porter and Easterling, 1992): 0001Gd ˆ 20016s0019 0001 ac 0001 0016 b
…7:70†
Here b is the burgers vector of the dislocation.
7.4
Gas±solid reactions
Gas±solid reactions, being heterogeneous reactions, occur at phase boundaries, and thus always accompany the transfer of mass and heat between the reaction interface and the bulk phase. The overall gas±solid reaction involves a combination of the following individual steps: 1. 2. 3. 4. 5. 6.
Transfer of the gaseous reactants and the gaseous products between the bulk gas and the external surface of the solid particle. Diffusion of the gaseous reactants and the gaseous products within the pores of the solid, if the solid contains open porosity. Chemical reaction between the gaseous reactant and the solid at the gas± solid interface. Transfer of the reaction heat within the solid. Transfer of heat between the external surface of the solid and the surroundings by convection and radiation. Changes in the structure of the solid due to chemical reaction and heat.
The rate-controlling step can change depending on reaction conditions, and thus the rate information obtained under a given set of conditions may not be applicable under another set of conditions. Furthermore, there may not be a single rate-controlling step because several steps may have comparable effects on determining the overall rate. The relative importance of these steps could also
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change in the course of the reaction. Therefore, understanding how the individual reaction steps interact with each other is important in determining not only the rate-controlling step under given reaction conditions but also whether more than a single step must be considered in expressing the rate over the entire duration of the reaction. The treatments of the above individual component steps were discussed in Section 7.2 Fundamentals of heterogeneous kinetics on page 270. Because the transfer of mass and, to a lesser extent, heat constitutes an important aspect of a gas±solid reaction, the quantitative analysis of the overall rate must necessarily take into account the geometry and structure of the solid before, during, and after the reaction. Although there are other types of gas±solid reactions, the most important group of reactions in metallurgical and materials processing operations consists of those in which a solid reacts with a gas to produce a coherent layer of porous products. Therefore, this section will mainly be concerned with this type of reaction, which is given by equation 7.1. For the analysis of other types of gas±solid reactions, the reader is referred to other references (Szekely et al., 1976).
7.4.1 Reaction of an initially non-porous solid producing a porous product layer In this type of gas±solid reaction, the reaction progresses in a topochemical manner from the outer surface of the solid towards its interior, as depicted in Fig. 7.8. The reaction forms a coherent porous product (or ash) layer around the unreacted portion of the solid, with the chemical reaction taking place at the sharp interface between the two zones. This shrinking-core picture has been applied to a wide range of reactions. Szekely et al. (1976) generalized the results of many previous investigators on reactions that are isothermal and of first order with respect to the gaseous reactant. They formulated the following equation incorporating chemical kinetics, diffusion through porous product layer, and external mass transfer, as well as the three basic geometries of the solid: gFp …X † ‡ 001b2s ‰pFp …X † ‡ 4X =Sh0003 Š ˆ t0003
…7:71†
7.8 Reaction of an initially non-porous solid that forms a coherent layer of porous product (the shrinking-core reaction system).
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where Fp is the shape factor (ˆ 1, 2, or 3 for a slab, a long cylinder, or a sphere), and 0013 0012 00130012 00130012 Ap bk CCb t …7:72† t0003 0011 CAb ÿ Fp Vp K B 001aB 0012 00130012 00130012 0013 Vp k 1 2 1‡ …7:73† 001bs 0011 Ap 2DeA K 0012 00130012 0013 0012 0013 …DA :000e†dp kmA dp DA DA ˆ Sh …7:74† ˆ Sh0003 0011 DeA DeA DA DeA gFp …X † 0011 1 ÿ …1 ÿ X †1=Fp
…7:75†
and pFp …X † 0011 X 2 0011 X ‡ …1 ÿ X † ln …1 ÿ X † 2=3
0011 1 ÿ 3…1 ÿ X †
‡ 2…1 ÿ X †
for Fp ˆ 1
…7:76a†
for Fp ˆ 2
…7:76b†
for Fp ˆ 3
…7:76c†
It has been shown (Szekely et al., 1976) that chemical reaction controls the overall rate when 001b2s < 0:1 or 0.01 depending on the range of tolerable error ( T2 ). In this diagram decreasing temperature stabilizes solid C. If one begins with a solution of A and B at temperature T1 and then one adds B at some concentration above that indicated by the dashed line, solid C will precipitate and grow until the concentration of A and B in solution is that given by the diagram, taking into account the stoichiometry of the reaction. The supersaturation necessary to initiate precipitation of solid C is related to the position of the equilibrium line and can be manipulated by decreasing temperature. Thus increasing composition and decreasing temperature can both lead to a critical supersaturation and the formation of solid C.
10.2.2 Issues of thermodynamics In order to understand solidification, we must first understand phase diagrams of the type schematically shown in Figs 10.1 and 10.2. These diagrams are
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10.2 Equilibrium diagram for the precipitation of C from a solution containing A and B.
constructed using bulk thermodynamic parameters and denote regions of stability where bulk parameters are important; however, during solidification, the first solid to form occurs by atom clustering either within the bulk (homogeneously) or on surfaces (heterogeneously). In addition, during solidification, there is formation of a solid that has both volume and a surface. Thus, if we look at the energy change during the solidification process, we can immediately see that for homogeneous nucleation of solid within a liquid mass, then the total change in free energy (0001G) must be related to the volume free energy change plus the energy necessary to create the surface:11 0001G ˆ 0001Gv 0001 Vs ‡ s=l 0001 As
…10:1†
where 0001Gv is the free energy change per unit volume created and s=l is the interfacial energy per unit area between the solid and the liquid. As the volume free energy can be calculated in two different ways, depending upon whether temperature or chemical change is driving precipitation, we can write 0001Gv as follows: 0001Gv ˆ 0001Hv;l!s ÿ T0001Sv;l!s 0019 0019ÿ
0001Hv;l!s 0001T TM
0001Hv;s!l 0001T 0019 0001Sv;l!s 0001T TM
…10:2†
Solidification and steel casting 0001Gv ˆ 0001Go ‡ RT ln Q ˆ RT ln
Q ˆ ÿRT ln S KE
403 …10:3†
where 0001Hv and 0001Sv are enthalpy and entropy change per unit volume respectively of the phase transformation; TM is the equilibrium temperature for the phase transformation; 0001T ˆ TM ÿ T and is the undercooling; 0001Go is the standard Gibbs free energy for the reaction; KE is the equilibrium constant for the reaction; Q is the reaction product calculated using the initial activities of the components and S is the supersaturation ratio (Ke =Q). Thus one can relate 0001Gv to the undercooling (0001T), if we assume that for a small range of undercoolings that the thermal contribution due to the phase transformation (0001S ˆ 0001H=TM ) accounts for more than 99% of total change of entropy, or to the natual logarithm of the supersaturation ratio (S). Equation 10.2 is Volmer's equation from 1939 and equation 10.3 is that of Guggenheim from 1957. If we assume that the solid preciptates as a sphere, then equation 10.1 must go through a maximum at small radii, as the volume term must become smaller than the area term, as r3 < r2 , when r < 1. A schematic of this situation is given in Fig. 10.3 where the maximum free energy is 0001G0003 and the radius corresponding to the maximum is r0003 . From equation 10.1, assuming solidification is driven by undercooling, the critical radius, r0003 , and the critical free energy, 0001G0003 can be written as follows: r0003 ˆ 0001G0003 ˆ
2 s=l 2 s=l TM ˆ 0001Gv;s!l 0001Hs!l 0001T
…10:4†
16
3 16
3 TM2 0019 ˆ 0019 2 2 3 0001Gv;s!l 3 0001Hv;s!l 0001T 2
…10:5†
Thus both the critical radius and the critical free energy decrease with increasing undercooling. For a reaction driven process, r0003 and 0001G0003 can be written as follows:
10.3 Schematic of the total free energy as a function of radius for a sphere.
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Fundamentals of metallurgy r0003 ˆ 0001G0003 ˆ
2 s=l 2 s=l ˆ 0001Gv;s!l RT ln S
…10:6†
16
3 16
3 0019 ˆ 0019 2 3 0001Gv;s!l 3 …RT ln S†2
…10:7†
where r0003 and 0001G0003 decrease with decreasing temperature and increasing supersaturation. The nucleation of a phase was viewed by Gibbs11 and Volmer and Weber12 as a heterophase fluctuation in an undercooled or supersaturated phase where numbers of atoms must cluster together to form the potential starting points for the phase transformation. In Fig. 10.3, at small radii 0001G is positive and reaches a maximum as a function of r. Thus if we use Gibbs view of fluctuations, this maximum represents an unstable condition as, if a fluctuation occurred in the system that leads to the formation of a sphere of radius r0003 , then any change to the size of the droplet due to evaporation or condensation, would lead to the droplet either diminishing or growing, as both processes would lead to a decrease in the free energy of the cluster. Thus r0003 is the smallest radius that has the potential to increase in size. Equations 10.4 to 10.7 are related to one's assumption of the geometry associated with the solidification process and would change if one assumed a different geometric shape and a number of such relationships can be developed, if one changes geometry. The most obvious change in geometry is to assume that the solidification process occurs on a pre-existing surface (heterogeneous nucleation). If one assumes a spherical cap which forms an equilibrium shape on a pre-existing surface, then Young's relation can be used to relate the various surface energies via an equilibrium contact angle (0012) where:
cÿl ˆ cÿs ‡ sÿl cos 0012
…10:8†
and cÿg is the surface energy between the pre-existing solid and the liquid, cÿs is the interfacial energy between the pre-existing solid and the newly solidified material and sÿl is the interfacial energy between the newly formed solid and the liquid. This leads to the following relationship for 0001G0003 : 0001G0003het ˆ 14 ‰2 ‡ cos 0012Š‰1 ÿ cos 0012Š2 0001G0003hom ˆ S…0012†0001G0003hom
…10:9†
The function S…0012† varies from 0 to 1 as 0012 varies from 0 to 180. Thus wetting conditions lead to very small values of 0001G0003 . Of course, as r0003 is related to the shape of the curve (where d0001G=dr ˆ 0), not its maximum value, heterogeneous nucleation does not change the critical radius; however, it strongly affects the volume of the critical nucleus and thus the number of atoms that must cluster together in order to form a critical nucleus: 0003 Vhet 0001G0003het ˆ S…0012† ˆ 0003 Vhom 0001G0003hom
…10:10†
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10.4 Nucleation at a liquid±liquid interface.
In the above discussion, it was assumed that the pre-existing solid was flat; however, if one assumes that surfaces are rough, the concave areas are more effective in decreasing the necessary volume for nucleation than either flat or convex areas, thus heterogeneous nucleation is favored in valleys rather than hills. Another interesting issue is to consider solid nucleation at a liquid±liquid interface11 as shown schematically in Fig. 10.4. In this figure the various interfacial energies are related as shown. In this case: 0001Ghet ˆ f …000b† 0001Ghom '
0012
1 ÿ cos2 1 ˆ 2 ÿ 3cos 1 ‡ cos 1 ‡ 1 ÿ cos2 2 3
00133=2
# 3
…2 ÿ 3cos 2 ‡ cos 2 † …10:11†
10.2.3 Effects of size Another way of looking at this thermodynamic problem is to understand the effect of surface energy on the equilibrium position of the reaction. The first attempts to understand this issue was in the formation of liquid droplets from the vapor where surface energies and contact angles are apparently easy to measure. It is well known that the effect of surface energy on a droplet is to increase the
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pressure inside the droplet. This change in pressure is described by the Young± Laplace equation which can be written for a sphere as: 0012 0013 1 1 2 0001P ˆ Pin ÿ Pout ˆ 0014 ˆ ˆ …10:12† ‡ r1 r2 r d0001P ˆ d0014
…10:13†
where 0014 is curvature and assuming that is a constant. In general: dG ˆ VdP ÿ SdT
…10:14†
thus, assuming that the total pressure is not affected by the formation of a droplet, it is easy to show that the change in the equilibrium temperature due to radius for a sphere is: 0001Tr ˆ
VTM 2ÿ 0014 ˆ ÿ0014 ˆ 0001Hlÿg r
…10:15†
2 V …10:16† r and ÿ is the Gibbs±Thompson coefficient and equals V =0001Slÿg . Equation 10.16 is often called the Thompson±Gibbs relation. Thus, the equilibrium condensation temperature for a droplet decreases with decreasing radius. In fact, this is a general finding. When one nucleates small droplets or particles, the equilibrium phase diagram will change and the equilibrium position for the reaction will always occur at a lower temperature than that given in a phase diagram that was determined from bulk samples. Of course, this means that undercooling is necessary for the formation of droplets and particles from a vapor or a liquid. A similar equation can be developed for the effect of radius on equilibrium partial pressure. This equation is generally referred to as the Kelvin equation: 0014 0015 P 2 2 V ‡ ‰p ÿ p0 Š 0019 …10:17† RT ln 0 ˆ V P r r 0001S 0001T ˆ
and relates the equilibrium partial pressure of a droplet (P0 ) to its size. Equation 10.17 indicates that as radius decreases, the equilibrium partial pressure must increase. In equation 10.17, P0 is the equilibrium partial pressure for a flat interface. The similarities between equations 10.16 and 10.17 indicate different manners by which the chemical potential can be derived. If the liquid is wetting to the solid, Kelvin also showed that in capillaries: P00 2 V ˆÿ …10:18† P0 r the equilibrium partial pressure for condensation will be less in a capillary than on a flat surface, again indicating why concavities can be better sites for the onset of nucleation. RT ln
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Equation 10.16, Kelvin's equation, also indicates an unstable equilibrium position, if one considers the result of small fluctuations in radius for a droplet in equilibrium with its surroundings. First, assume that one has a population of mono-sized droplets in equilibrium with the vapor phase. If the droplet spontaneously becomes larger by condensation, the new equilibrium partial pressure for the new larger droplet size will be lower than the pressure that previously was in equilibrium with the smaller droplet. The new larger droplet is now in a situation where the actual pressure of the vapor from which the droplet condensed is now above its new equilibrium partial pressure. This condition will cause the droplet to further grow to reduce its local pressure to the equilibrium pressure corresponding to its size. This process will continue and the droplet will grow. Similarly, if a droplet decreases due to evaporation, the new equilibrium partial pressure necessary for the droplet will now be higher than the actual pressure and the droplet will continue to evaporate. Obviously, there must be mass balance as the total mass of droplets will not change, thus some droplets will grow and others will decrease. Of course, these changes in local equilibrium will lead to gradients in chemical potential between droplets and the driving force for mass transfer between the droplets. In materials science, this phenomenon is known as Ostwald ripening, and the coarsening of structures is also very common in solidification as the system reduces its total surface area and moves to its lowest energy state by reduction of surface area. Thompson derived the effect of temperature on precipitation of droplets as a function of radius, where the effect of radius was to reduce the equilibrium temperature for the transformation in the following manner: ln
T 2 V n ˆÿ 0001Hvap r To
…10:19†
showing the necessity for undercooling. In the derivation of Thompson's equation, it is assumed that 0001S ˆ 0001H=T; if, however, 0001S ˆ 0001H=TM , is assumed, equation 10.15 results which is a clearer derivation and is preferred. Gibbs' approach to nucleation and the Kelvin equation are yielding results that are similar in that an unstable position occurs during nucleation. These two views can be easily reconciled by considering the precipitation of droplets using the natural variables of volume, pressure, area and number of moles. Thus rewriting our thermodynamic equations for nucleation in terms of the Helmholtz free energy (F) where: dF ˆ ÿPdV ÿ SdT ‡ dA ‡ 0016i dni
…10:20†
it can be shown that at low values of supersaturation, that: 0001F ˆ 40019r2 ÿ
40019r3 P0 RT ln 0 n P 3V
…10:21†
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If the maximum of this function is found (dF=dr ˆ 0), Kelvin's equation results. Of course, the maximum must occur at r0003 and thus Kelvin's equation is satisfied for droplets of a critical radius. Using Kelvin's equation, it is possible to calculate the size of a critical radius for droplet formation. For example, at 0ëC and at a supersatuation of 4.2, water droplets are observed in water vapor. This leads to a calculation of r0003 of 0.8 nanometers which is a cluster of approximately 70 water molecules. For solidification of a solid phase one must change the various constants to those of the solid. For example, equation 10.16 would be written with the solid±liquid interfacial tension and the entropy change for solidification. This discussion of droplets leads to some interesting issues in solidification: 1.
2.
The formation of very small particles necessitates that surface energy must be included in our description of equilibrium. Thus, current equilibrium phase diagrams cannot predict the onset of solidification as they are based upon bulk observations or calculations using bulk properties. Thus equilibrium phase diagrams as a function of particle size are necessary and a diagram of this type was recently calculated by Tanaka et al.13 for the system copper±lead. Similar affects of particle size have been measured by Sambles14 who noted that the melting temperature of pure gold decreased markedly (by more than 100K) when the particle size decreased below 20 nanometers. Solidification structures will naturally coarsen due to the influence of curvature on local equilibrium.
The discussion of the thermodynamics of solidification so far has lead us to understand that the equilibrium phase diagram does not necessarily help us in our understanding of the initiation of solidification and that, when solidification occurs by fluctuations of small groups of atoms or molecules, the phase diagram must be altered due to the affect of radius on the equilibrium position.
10.2.4 Nucleation rate and the formation of non-equilibrium solids In addition to the effect of radius, one must also include the effect of cooling rate when discussing solidification, as it is possible to undercool liquids to the point that the liquid begins to exhibit solid-like behavior. For example, the undercooled liquid can fail in a brittle mode at low temperatures or deform plastically at higher temperatures. Thus, not only can the phase diagram be changed by particle size, it should also take into account the effect of cooling rate on the potential for metastable phase formation and glass formation. Clearly this is beyond the realm of thermodynamics alone. Following the view that small fluctuations are responsible for the initiation of solidification,11,12 it was postulated that the homogeneous nucleation rate (J)
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should be proportional to the number of critical size nuclei (n0003c ), that is a nuclei with a radius r0003 , where: J / n0003c
…10:22†
and that the number of these nuclei would follow the Frenkel size distribution equation (10.23), where one assumes that the entropy change of the solid±liquid system must be included by calculating the free energy of mixing of embryos (potential nuclei) with the liquid atoms (nl ) by assuming an ideal mixing model: 0012 0013 ÿ0001G0003c 0003 nc ˆ nl exp …10:23† kb T If the clusters evolve by a bimolecular reaction with forward and reverse rates of monomer addition where critical size nuclei grow by addition of atoms at the interface, an expression for the steady state nucleation rate can be derived assuming an activated process: J ˆ Ns 0017 n0003c
0012 0013 0012 0013 kT ÿ0001Gd ÿ0001G0003c nl exp J ˆ Ns exp kb T kb T h
…10:24† …10:25†
where Ns is the number of atoms adjacent to the interface and in a position where a potentially successful jump is possible and 0017 is the jump frequency of an atom from the liquid to the solid, 0001Gd is the activation energy at the solid±liquid interface for a successful jump of an atom in the liquid to the surface of the nuclei and h is Planck's constant. 0001Gd is the kinetic barrier to nucleation and 0001G0003c is the thermodynamic barrier to nucleation. This can also be written (from equations 10.5 and 10.7) as: ! 0012 0013 kT ÿ0001Gd 16
3 TM2 0019 nl exp ÿ …10:26† J ˆ Ns exp 2 h 3kT 0001Hv;s!l kb T 0001T 2 ! 0012 0013 kT ÿ0001Gd 16
3 nl exp ÿ 0019 J ˆ Ns exp kb T h 3kT …RT ln S†2
…10:27†
Equation 10.26 is plotted schematically in Fig. 10.5 where the nucleation rate goes through a maximum as temperature decreases. 0001Tn in Fig. 10.5 is the undercooling necessary for an observable nucleation rate of 1 nuclei per cm3 per second. In metals it is very difficult to actually measure the maximum in Fig. 10.5 due to difficulties in accessing extreme undercoolings; however, such results are often seen in liquid oxides and in polymers. Equation 10.27 is plotted schematically in Fig. 10.6, where in this case, the nucleation rate continues to increase with increasing supersaturation. The time to nucleate is proportional to the reciprocal of the nucleation rate. Thus time±temperature±transformation (TTT) curves can be derived from the
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10.5 Nucleation rate (J) as a function of temperature.
nucleation rate and such a curve is shown schematically in Fig. 10.7. In this figure the dashed line indicates the position of the first observable solid in an experiment. Other curves could also be drawn to denote the fraction of solidification and to indicate the progression of solidification with time. In Fig. 10.7 the equilibrium position of the phase transformation is noted, as is the glass transition temperature. Thus this figure schematically shows the position of the possible phases and indicates the stability regimes (with temperature and time) of the liquid, the glass and the area where the solid crystallizes from the liquid. Theoretically all liquids should be able to be formed at high cooling rates; however, the realm of achievable cooling rate is well defined and only those liquids that can be cooled quickly enough to avoid solid precipitation can be made as glasses.
10.6 The effect of supersaturation on the nucleation rate.
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10.7 Schematic TTT curve.
Alternatively,1,8 if the jump frequency is calculated from: Dl 40019…r0003 †2 and N ˆ s a2 a2 then, for typical metals, where J is measured per cm3 per second: ! 0012 0013 0012 0013 0012 0013 Dl 40019…r0003 †2 0001G0003 0001G0003 34 0019 10 exp nl exp Jˆ a2 a2 kT kT 0017ˆ
…10:28†
…10:29†
The above discussion of nucleation rate concerns homogeneous nucleation, if one takes into account the fact that a heterogeneous nucleation site exists, then equation 10.29 can be rewritten as: 00120014 0015 0013 2 D 20019rcr …1 ÿ cos 0012† 0001G0003hom f …0012† Jhet ˆ 2 na exp ÿ kb T a a2 00120014 0015 0013 0001G0003hom f …0012† …10:30† 0019 1024 exp ÿ kb T where na is the number of atoms in contact with the surface of potential nucleating sites (approximately taken as 1024 in this calculation5). Of course, this means that J / Na where Na is the area of the nucleant and related to the number and size of the nucleating particles. From equation 10.30, heterogeneous nucleation is promoted by: · A stable solid material that exists in the liquid above the solidification temperature. This solid can be physically added and mixed or precipitated from solution during cooling. · A low contact angle between the precipitated solid and the heterogeneous nucleating surface.
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· A large surface area of the potential nucleant. The contact angle between solids in a liquid is not a readily measurable contact angle in non-transparent liquids. In addition liquid±solid interfacial energies are not easily determined, thus the use of Young's relation (equation 10.8) to determine nucleating species is not particularly useful. Thus other criteria are necessary to allow the identification of potential nucleating species. Knowledge of surface structure is useful in the determination of potential nucleants. If one assumes that epitaxy is important then nucleation should occur on planes that have a low misfit or disregistry with the plane to be precipitated. Bramfit calculated the disregistry (000e) as follows: ' # 3 X j…d‰uvwŠin cos † ÿ d‰uvwŠis j …hkl†n 000e…hkl† ˆ 0002 100% …10:31† 000eˆ s d‰uvwŠin iˆ1
where (hkl)s and (hkl)n are low-index planes in the solid and nucleant, [uvw]s and [uvw]n are low-index directions in the (hkl)s and (hkl)n planes respectively, d[uvw] is the spacing along [uvw], is the angle between [uvw]s and [uvw]n and i is one of the three directions of the crystal with minimum index. Bramfit15 used this parameter to explain the difference in efficiency of heterogeneous nucleants of 000e ferrite by measuring the degree of undercooling during solidification as shown in Fig. 10.8. Of course, measurement of the undercooling for initiation of solidification is the most accurate assessment of the effectiveness of a nucleant.
10.8 Effect of lattice disregistry on undercooling.16
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10.3 The growth of solids In the discussion of growth rates there are a number of issues. The first is, what is controlling the growth rate? There are three potential general answers: interface kinetics, mass transfer or heat transfer; however, mixed results are also possible. In addition there are two major types of interface: flat and diffuse. If the interface is diffuse as in most metals, growth will be continuous, while if the interface is flat as in oxides, for example, the interface will grow laterally and the interface will be faceted. The entropy of fusion is often used to determine which type of interface will predominate and if ˆ 0001S=R < 2; the interface will be diffuse.
10.3.1 Interface dominated growth rates Once nucleated, particles must grow and although the nucleation process is outside of our abilities to observe in liquids, one can certainly observe the growth stage after nucleation. In the above discussion of nucleation, it was assumed that growth was in an undercooled liquid and atom attachment to an interface controlled the growth rate and the jump frequency at the interface was very important. If one follows this viewpoint and assumes that growth is an activated process, then the Wilson±Frenkel relation results: 0014 0012 00130015 0014 0012 00130015 0001Gcrystallization 0001Hcrystallization 0001T D D …10:32† ˆ 1 ÿ exp 1 ÿ exp Rˆ kb TTm kb T a a and, if diffusivity is related to viscosity via the Stokes±Einstein relation, it follows that: 0014 0012 00130015 0001Hcrystallization 0001T kb T …10:33† 1 ÿ exp Rˆ kb TTm 30019a2 0011 This relation can be used to predict the growth rate as a function of temperature for crystobollite precipitation from fused silica, as shown in Fig. 10.9. If one assumes that not all sites on a surface are appropriate and that a successful jump occurs only in specific positions, then a correction factor f can be attached to equation 10.33 and for a growing monolayer, if f ˆ 0001T=20019TM : 0014 0012 00130015 0001Hcrystallization 0001T 0001T kb T 1 ÿ exp Rˆ 2 …10:34† kb TTm 60019 TM a2 0011 If equations 10.33 and 10.34 are approximated by the following approximation, if 1 ÿ exp…ÿx† ˆ x, then equation 10.33 indicates that: R / 0001T
…10:35†
and equation 10.35 indicates that: R / 0001T 2
…10:36†
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10.9 The growth or melting rate of crystobollite as a function of temperature.6
If one assumes that growth is by a screw dislocation emerging at a surface, a similar relation to equation 10.36 is found. In metals the undercoolings associated with surface growth mechanisms are very small and only at high growth rates do interface kinetics become an issue. As already noted the growth phase of solidification is easily observed and if one assumes that there is a steady state nucleation rate, that the growth rate is constant and the liquid is consumed by the growth of the particles, the following general Kolgmogorov, Johnson, Mehl, Avrami (KJMA) equation can be developed:5 ' 0015d # Z t 0014Z t Xt ˆ 1 ÿ exp ÿcg Jt Gt dt dt …10:37† 0
0
where Xt is the fraction transformed, cg is a shape factor, Jt is the nucleation rate, Gt is the growth rate and d represents the dimensionality of growth. For example, if we had growth of a sphere at constant growth rate and a constant nucleation rate (Jv ), then as cg is 40019=3, d ˆ 3, and Gt is a constant, the Johnson±Mehl equation results: 0012 0013 0019Jv R3 4 t …10:38† 1 ÿ Xt ˆ exp ÿ 3 From this equation, if Jv and R are known as a function of temperature and we assume small values of Xt , it follows that: Xt 
0019Jv ‰RŠ3 4 t 3
…10:39†
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If the limit of observation for Xt is assumed to be 10ÿ6 the time when the first observation of solidification at a given temperature T could be calculated from a knowledge of Jv and R. For example, using equations 10.29 and 10.32, for homogeneous nucleation followed by growth by the monolayer model, the TTT curve could be calculated. Although the above growth models seem reasonable in pure liquid oxides where undercooling is easy and heat transfer is not an issue; however, in metals with a high latent heat of solidification or during highly exothermic reactions, heat transfer tends to control the growth rate after the initial formation of solid. In multicomponent solutions mass transfer in the liquid can become rate controlling rather than atom attachment.
10.3.2 Heat transfer dominated growth rates In heat transfer dominated solidification, the rate of solidification is determined solely by the ability to transfer heat away from the interface. In undercooled liquids heat can be transferred either to the liquid or into the growing shell. The amount of solidification is thus related to the heat flux removed from the interface. If the amount of heat released during solidification is measured per unit mass and the thermal gradient in the solid (dT=dx) is Gs and the thermal gradient in the liquid is Gl , then the interface heat balance for a planar interface is as follows: ks Gs ÿ kl Gl ˆ 0001H 001aR
…10:40†
where 001a is the density and ks and kl are the thermal conductivities of the solid and liquid respectively. Thus the maximum growth rate occurs when Gl is negative (undercooled) and Gs is positive (normal solidification). The relation also shows that an interface can be stabilized (R ˆ 0) by ensuring that ks Gs ˆ kl Gl . If convection heat transfer controls thermal transport in the liquid then equation 10.40 would be rewritten as follows: ks Gs ÿ hl …Tb ÿ Tt † ˆ 0001H 001aR
…10:41†
To avoid the necessity of solving for hl it is not uncommon in calculations to substitute a fictitious effective conductivity in equation 10.40 to account for a higher rate of heat transport than would be calculated from conduction heat transfer. The first issue in any heat transfer problem in solidification is to define the interface temperature, as it is that temperature that will define the temperature that the heat must flow from. The starting point is, of course, the equilibrium phase diagram and to calculate the interface temperature one must first take into account the effect of chemistry (0001Tc ), and, if the interface has curvature, then this undercooling must also be included (0001Tr ), such that: Tint ˆ TMpt ÿ 0001Tc ÿ 0001Tr
…10:43†
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10.10 Schematic of thermal profile in liquid and solid for undercooled growth of an alloy.
It is also assumed that when solidification is initiated, the interface temperature immediately attains Tint as calculated in equation 10.37. Of course, there must also be a kinetic undercooling (0001Tk ) to allow heat transfer and there can also be a pressure undercooling (0001Tp ); however, both of these corrections are usually small and are normally ignored.9 A schematic of a thermal profile for undercooled solidification is given in Fig. 10.10 where the various undercoolings are marked. An example of undercooled growth can be seen in Fig. 10.11 where liquid iron is undercooled until solidification occurs on an alumina substrate. In this
10.11 Example of undercooled solidification.15 Photograph by H. Shibata.
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case the undercooling for initiation of solidification is 211ëC and the droplet fully recalesces in the time of 1 frame of the video camera. Transient moving boundary problems of this type are Stefan problems and, if one looks at the situation where a semi-infinite solution of the transient heat transfer equation is appropriate and can be used to determine thermal gradients: @T @2T ˆ 2 …10:44† @t @x it follows that the growth rate for a plane front growing into an undercooled liquid is: r …10:45† Rˆ0015 t where p Cp 0001Tt ˆ St …10:46† 00190015 exp…00152 †erfc 0015 ˆ 0001H where erfc is the error function complement, 0001Tt is the thermal undercooling of the liquid and St is the Stefan number. It should be noted that when the Stefan number equals or is greater than 1, this function has no solution. If the Stefan number equals 1, Cp 0001T ˆ 0001H and this means that the heat given out during solidification can recalesce the liquid to its equilibrium point. At Stefan numbers greater than 1, the solidification process cannot recalesce the liquid to its equilibrium point and this is termed hypercooling. Under this condition the interface temperature would not be easily determined and this solution is not appropriate. Upon nucleation of a solid the first solid formed is a sphere and the interface conditions must include the fact that the interfacial area is increasing. Equation 10.40, the interface heat balance must be rewritten for spherical growth as follows: 1 d…A† ˆ …0001H 001a ÿ 0014 †R …10:47† A dt Where the growth rate is increased due to the formation of surface, i.e. less energy needs to be conducted as it is consumed in surface creation. Mullins showed that spheres in undercooled liquids become unstable very quickly and rather than grow as spheres begin to grow as cells or dendrites. If we assume that the sphere is growing in an undercooled liquid of temperature T1 , then the growth rate of the sphere will be: 0012 0013 0012 0013 0012 0013 dr kl dT kl 1 1 …Ti ÿ T1 † Rˆ …10:48† ˆÿ ˆ ‡ p 0001H 001a dr rˆrp 0001H 001a dt rˆrp rp 0019000bt ks Gs ÿ kl Gl ˆ 0001H 001aR ÿ
where Ti is the interface temperature. At longer times equation 10.48 will tend to the steady state solution where:
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Fundamentals of metallurgy Rˆ
kl …Ti ÿ T1 † 0001H 001a rp
The progression of radius with time will then be: s Kl …Ti ÿ T1 † t rp ˆ rp;o ‡ 0001H 001a
…10:49†
…10:50†
where rp;o is the starting radius. To determine the length of time it takes to reach a steady state solution, the characteristic time (001c) can be determined where 001c0019
rp2 l
…10:51†
in metals where l is approx 10ÿ5m2/sec, for a 001c of less than 0.1 sec, rp must be less than 0.1 mm. Thus for very small spheres the steady state solution would be appropriate; otherwise the full transient solution is necessary. As we have already noted, Ti is a function of r for very small radii and: 2ÿ r Therefore equation 10.49 can be rewritten as: 0012 0013 2ÿ 0001T ÿ rp kl Rˆ rp 0001H 001a Ti ˆ Tmpt ÿ
…10:52†
…10:53†
where 0001T ˆ TMpt ÿ T1 . Thus the growth rate will become zero when 0001T ˆ 2ÿ=rp as Ti will tend to T1 and the driving force for heat transfer will be eliminated. As previously noted, Mullins has shown that spherical growth is unstable, perturbations form and grow where these perturbations develop with a tip radius that is constant. A general problem in solidification is to determine the relationship between the radius and growth rate. If one considers this problem as a cylinder growing with a hemispherical cap into an undercooled liquid, one develops a relationship very similar to equation 10.49 for steady state growth: rR Cp …Ti ÿ T1 † …10:54† ˆ 0001H 2 which, if taking into account the effect of radius on the interface temperature, is written as: 0012 0013 0001Tr PeT ˆ 1 ÿ …10:55† 0001T where PeT is the thermal peclet number and is the inverse of the Stefan number. Thus in equation 10.54 or 10.55, if the radius is known the growth rate can be calculated and vice versa. A more complete solution for this diffusion
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10.12 Schematic of the temperature profile for normal solidification.
problem was first given by Ivantsov for a geometry that is closer to that of a real dendrite where, if we correct for capillarity: 0012 0013 0001Tr …10:56† I…P† ˆ 1 ÿ 0001T where I…P† ˆ PeT exp …PeT †E1 …PeT † and E1 …PeT † ˆ ÿ0:577 ÿ ln…PeT † ‡
4PeT PeT ‡ 4
The above discussion concerns undercooled growth; however, solidification is often initiated by cooling against a mold. A schematic of the thermal profile for normal solidification is given in Fig. 10.12. Normal solidification is the most common mode of growth and, if one assumes perfect contact at the mold±shell interface and thermal gradients in both the shell and the mold, Schwartz's semi-infinite solution of the Stefan problem results, where: r shell …10:57† Rˆ0015 t and 0015 is calculated as follows: Tm ÿ To ŠCps p ˆ 0015 exp 00152 0001H 0019
's # …k001aCp †shell ‡ erf 0015 …k001aCp †mold
…10:58†
And the mold shell interface temperature is calculated from the following relation:
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Fundamentals of metallurgy 2 s …k001aCp †shell 6 6 …k001aC p †mold 6 Ts ˆ To ‡ ‰Tm ÿ To Š6s 6 …k001aC † p shell 4 ‡ erf …k001aCp †mold
3 7 7 7 7 7 00155
…10:59†
Of course, reality is always more complicated than these simplifications and, if one considers the situation of undercooling a liquid against a mold, there would be heat transfer in both directions at the interface. In this case, equation 10.60 gives the expression for the growth rate where 0015 can be determined as follows:7 Sts Stinitial p ‡ p ˆ1 2 0015 0019 exp…0015 †erf …0015† 001d0015 0019 exp…001d00152 †erf …001d0015†
…10:60†
and cp…s† …Tm ÿ Ts † 0001H r s 001dˆ l
Sts ˆ
and
Stinitial ˆ
cp…l† …Tm ÿ Tinitial † 0001H
The situation in Fig. 10.11 is typical of an undercooled solidification problem where the system cools to a temperature before solidification is initiated and then when solidification occurs the growing front immediately attains the equilibrium interface temperature and, heat is conducted into the liquid and into the shell. In this case there is heat transfer through the shell and the alumina pedestal that the iron droplet rests upon, thus this solution must be a combination of a Schwartz solution and the undercooled solution as shown in Fig. 10.13. Experimentally measured results versus this simplified model are shown in Fig. 10.14 where fair agreement between experimentally measured results is observed at higher undercoolings but not at lower undercoolings due to complete recalescence of the droplet before completion of solidification. Another more practical condition for undercooled growth would be against a cooled metal mold where, if we assume that solidification occurs by nucleation, then initially the liquid against the mold would be undercooled until the temperature reaches the level necessary for activation of a heterogeneous nuclei. Once the front is nucleated there will be a transfer of heat into the liquid and through the steel. Once the undercooled liquid has been reheated, all thermal gradients will be through the shell and there will be a transition from undercooled growth to normal solidification. There is no analytical method to represent this situation therefore a numerical solution is necessary. Results from such a calculation are given in Fig. 10.15 where as the critical undercooling for nucleation is increased there is a longer time before solidification is initiated.
Solidification and steel casting
10.13 Schematic of the temperature profile for undercooled growth.15
10.14 Growth of a liquid iron shell as a function of undercooling.15
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Shell thickness (m)
0.0010 50oC critical undercooling 100oC critical undercooling 150oC critical undercooling 200oC critical undercooling 250oC critical undercooling 300oC critical undercooling
0.0008
0.0006
0.0004
0.0002
0
0
0.1
0.2
0.3 Time (s)
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10.15 Shell profile calculations as a function of the critical undercooling for nucleation against a mold wall.
This initiation time is followed by a rapid rise in growth rate due to the combined modes of heat transfer. As the undercooled liquid is recalesced, the solution becomes that for normal solidification. It can be seen that as the ability to undercool increases, the time to solidification is decreasing. This occurs because the efficiency of heat transfer is greatest when the liquid contacts the mold. Once the shell is formed the total heat flux decreases. Clearly this effect is most obvious when casting small sections. The assumption of perfect contact at the shell±mold interface is not normally accurate and in reality there is a complex situation that is found where the shell is in intermittent contact with the mold leading to a complex conduction± convection and radiation problem. This complex situation is overcome by assuming that Newton's law can be invoked and that a heat transfer coefficient for the mold can be determined. In actual molds it is common that all of the resistances to heat transfer are important and it is common to write the heat flux equation as a sum of resistances using Newton's law and assuming an analog to electrical resistance, the heat balance can be written as: …Tp ÿ To † ˆ 001as 0001HR …10:61† < where Tp is the temperature of the liquid and To is the temperature of the medium removing heat from the mold. Resistance for heat transfer coefficients will be 1=hA and for conduction will be L=kA where l is the distance over which heat is conducted. Q ˆ hA…Tp ÿ To † ˆ
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Thus, for a water-cooled mold where there is a mold thickness and a growing shell with intermittent contact, the total resistance to heat transfer < would be calculated as follows: 1 kg (industry) 1 g (laboratory) 1000 (industry) 3300 (laboratory) 1±10
1±10 10±100 g
0.1±10 100 kg (industry) 100 g (laboratory) 60 (industry) 300 (laboratory) 6±10 70% 1:10 powder to ball weight ratio 5±100
0.1±10 100 kg (industry) 100 g (laboratory) 1800
> 10 > 1 ton
1±10
80±400 6±30 50±70% 1:10 powder to ball weight ratio 5±100
6±10 30% 1:10 powder to ball weight ratio 5±100
10±50 6±25 50% 100% of ball interstices 10±50
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On a larger scale, the economic point of view prevails and the efficient use of the equipment and raw materials, as well as energy consumption, become important. Coldstream process In this process, a high-pressure air-stream (e.g. 7 MPa) containing the particulate material, enters a vacuum chamber through a venturi nozzle where it impacts a tungsten carbide target.9 The pressure drop at the exit of the nozzle induces a temperature drop thus chilling and embrittling the material inducing its fracture when it impacts the tungsten carbide target. The material is then transported to a first classifier, which allows oversized products to drop into a storage vessel for subsequent impact against the target. This process applies to many body-centred cubic metals that go through a ductile to brittle transition at low temperatures. It is used for hard, abrasive, relatively expensive materials such as tungsten carbide, tungsten alloys, molybdenum, tool steels, beryllium, and other alloys (Inconel, nickel and cobalt high temperature alloys). This process allows a rapid production (1 t/day) of irregular micron-sized particles.
12.2.2 Atomisation route Atomisation is often used to produce metal powders and especially pre-alloyed powders (brasses, stainless steels, superalloys, NiAl . . .). Moreover its high cooling rates (102 to 107 K/s) allows producing alloys that cannot be obtained by casting. The flexibility of this method, coupled to its applicability to many alloys, easy process control and high production rate makes it a very interesting route.10,12 The world capacity of production is estimated to be higher than one million tons per year.11 Atomisation occurs when a jet of liquid is converted to very small droplets. The different atomisation processes are (see Fig. 12.2): · Two-fluid atomisation, where a molten metal is broken up into droplets by impingement of high-pressure jets of water, oil or gas (a). · Centrifugal atomisation, where a liquid-stream is dispersed into droplets by the centrifugal force of a rotating electrode (c), disc or cup (b). · Vacuum or soluble-gas atomisation, where a molten metal is supersaturated with a gas that causes atomisation of the metal in a vacuum chamber (d). · Ultrasonic atomisation, where a film of liquid metal (e) or the atomising fluid (f) is agitated by ultrasonic vibration. Atomised metal powders are generally found to follow a log normal distribution. The particles size distributions are generally from few 0016m up to 500 0016m and the mean particle size from 10 to few 100 0016m. The geometric standard deviation 001bg measures the spread of particle size around the median mass diameter dm and
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12.2 Atomisation processes. (a) gas, water or oil atomisation, (b and c) centrifugal atomisation, (d) vacuum atomisation, (e and f) ultrasonic atomisation.
typically varies from about 1.7 to 2.3 for water- and gas-atomised metal powders. Compared to other types of powders, the atomised powders are relatively compact (apparent density up to 65% of the theoretical density), with a high packing density and a low specific surface area. This induces good flow characteristics, good compressibility but low sintering activity. Particles shape varies from irregular to
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spherical for respectively water and gas atomisation (Fig. 12.3c, d). The required particle shape depends on the following process used to obtain the final product. Spherical powders are used for thermal spraying loose sintering or hot consolidation (e.g. extrusion, isostatic pressing). But irregular powders are needed to guarantee a high enough green strength after cold pressing. The cleanliness of the powder is also a very important parameter. The possible contaminants are bulk dissolved impurities, surface impurities and inclusions (mainly from melt practice). Surface oxidation is often considered as the main purity index. The process parameters controlling the powder characteristics (size, distribution, shape and oxygen content) are related to the atomiser design (nozzle defining
12.3 Morphology of powders obtained by (a) milling, (b) mechanical alloying, (c) water atomisation, (d) gas atomisation, (e) electrolysis (courtesy of Ye Xingpu) and (f) precipitation (courtesy of Rudy de Vos).
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the geometry of the atomisation), the atomiser operating conditions (nature of the fluid, pressure, temperature, atmosphere) and the material (melting temperature, viscosity, surface tension, superheat). Some of these parameters have a great influence on the powder characteristics. For example, the production of fine powders is favoured by low metal viscosity, low metal surface tension, high superheat of the metal (difference between the temperature of the molten metal and its melting temperature), small nozzle diameter, high atomising pressure, high flow rate and high velocity of the atomising fluid, short metal stream and short jet length.13 Spherical atomised powders are obtained by increasing the superheating of the liquid metal (reduction of viscosity), increasing its surface tension by addition of deoxidisers, such as B, P and Si and increasing the solidification time of the particle (atomising with gas instead of water). More details about the atomising parameters can be found in references 1 and 3. In the following, the different atomisation methods will be described in more detail. Water atomisation ± oil atomisation Water atomisation, with its capacity of at least 700 000 tons per year and production rates up to 500 kg/min (8 kg/s), is the process mostly used for the commercial production of irregular (Fig. 12.3c) metal powders (iron, stainless steels, tool steels, soft magnetic alloys, nickel alloys and copper).11 The production costs are lower than the other atomisation methods. However, limitations exist in relation to powder purity (relatively high oxygen content (1 wt%)), particularly with reactive metals and alloys. The median particle size decreases with increasing water pressure. Most industrial plants use a water pressure in the range of 5 to 20 MPa, that results in mass median particle sizes of 30 to 150 0016m. Finer powders (median particle size from 5 to 20 0016m) can be produced by using much higher water pressures (50 to 150 MPa). Oil atomisation is similar to water atomisation. Oil is used as atomising medium in order to decrease the irregular character of the particle shape and to reduce oxidation, especially for molten metal containing elements such as Cr, Mn or Si, which are readily oxidised.12,14 Gas atomisation The production of gas atomised powders is lower than water-atomised ones because of its higher cost. It is about 300 000 tons per year for air atomisation and around 50 000 tons per year for inert gas (nitrogen, argon and helium) atomisation, the later ensuring a low oxygen content (100 ppm) of the spherical powders (ferrous and non-ferrous alloys) (Fig. 12.3d).11,12 In gas atomisation the powder size distribution strongly depends on the nozzle design (`free-fall' configuration or `confined'). Confined nozzle designs
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increase the amount of fine powder particles ( 10 kPa), the temperatures of the gas and the electrons are comparable and are of many thousands of degrees. Starting materials are typically atomised at these temperatures and the powder synthesis occurs during condensation outside the plasma when the gasses cool.30 In contrast, in a low-pressure (p < 200 Pa) lowtemperature plasma (cold plasma) the electron temperature is much higher than the gas or ion temperature, which is close to room temperature because of the poor collisional coupling between electrons and heavy particles at reduced pressures.31 As the growth of particles may be completely determined by chemical kinetic factors, thermal and low-temperature plasmas may produce materials with different structures and properties. Plasma methods are very popular as they enable the preparation of twocomponent compounds as well as multicomponent powders with a high purity. Besides ceramic powders (SiC, SiN, BN), metallic nanoparticles are produced. Nano-powders are used for their light emission properties (Si) or magnetic recording tapes. Nanostructured thin films, consisting of nanoparticles embedded in a metal matrix have also been produced.
12.3 Forming processes towards near-net shape Near-net-shape components can be obtained by a conventional two-step process consisting in first giving the shape of the component (compaction) followed by its consolidation (sintering). Other processes allow the shaping and the consolidation of the component in one step (hot isostatic pressing, hot extrusion, hot forging, etc.). Because of its economical importance infiltration is also shortly described in this part, as well as the fast developing rapid prototyping processes.
12.3.1 Conventional route Compaction Compaction is a widely used method to form semi-finished or near-net-shape components from a powder. The quality of compaction (density gradients, micro-cracks) has a great influence on distortion of the product shape after sintering step as well as on the properties (defect-free components) of the final product. Powder compacts can be obtained by uniaxial pressing, isostatic pressing, metal injection moulding (MIM) or by less used processes such as powder rolling, extrusion and dynamic or explosive compaction.
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Uniaxial pressing Uniaxial pressing (dry or die pressing) is the most common method of compaction to form PM components. This low-cost process is adapted to highvolume (up to few hundreds of parts per minute) production of `relatively simple' geometry powder compacts.32 It consists in compacting a dry powder (i.e. < 2 wt% water) in a die at a pressure ranging from 20 to 700 MPa by operating one or more rigid punches (higher numbers of punches are needed with increasing complexity of the component). A sufficiently high pressure is required to guarantee a sufficient strength of the green compact for subsequent handling and processing. Compaction consists in the following three steps: filling the die with the powder, compacting the powder and ejecting the powder compact from the die. Die filling and compaction control the uniformity of the green density in the powder compact, which is a crucial parameter. Green density gradients have to be minimised in order to reduce differential shrinkage during sintering and shape distortion, as well as to avoid induced defects that limit the properties and reliability of the part. Ejection of the compact is also a critical step because macroscopic defects can be created. Uniform and high packing density of a powder during die filling favours high green density with low green density gradients after compaction. This implies the use of good-flowing powders, which are also necessary to guarantee reproducible die filling. High-flowability powders have a relatively large particle size distribution lying between 40 and 400 0016m and are usually spherical or equiaxial. Powders with a low flowability (fine (1±10 0016m) irregular powders) have to be granulated before dry pressing (e.g. by spray drying) with 1 to 5 wt% organic additives such as binders, plasticisers and lubricants, which also improve the handling of the compact and/or the compactibility of the powder. Organic binders (wax, polyethylene glycol) allow the increase the green strength of the compact. Plasticisers (water, ethylene glycol) are used in combination with binders to improve the deformability of the powder during compaction. Lubricants (wax, magnesium stearate, stearic acid) reduce the interfacial frictional forces between individual particles favourable to powder compaction and/or between particles and die surfaces. They also reduce the required ejection pressure of the compact, thus avoiding macro-defect formation. Those additives will be eliminated by thermal decomposition before sintering or in a special zone of a continuous sintering furnace. During powder compaction, the applied pressure induces particle rearrangement, deformation, in some cases fracturing, and finally consolidation of the particulate assembly. Green density and compact strength increase with increasing compaction pressure.12 Two scales of green density gradients can occur during powder compaction: (i) macroscopic density gradient induced by non-uniform die filling and/or pressure gradients during compaction and (ii)
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microscopic density gradient due to packing defects, hollow particles (granules), and/or insufficient particle (granule) deformation during compaction. Nonuniform pressure/stress gradient is due to the non-uniform pressure/stress transmission at particle±particle and particle±die (die wall friction) contacts. Green density gradients are minimised by using lubricants, smooth surface die, low compaction ratio, double-action or floating die pressing instead of singleaction pressing and by adapting the numbers of punches to the number of different thickness areas in the part. During ejection, the compact integrity is favoured by a sufficiently high green strength to withstand the applied forces on the compact required to eject it from the die, a low ejection pressure, favoured by the use of lubricant to reduce die wall friction and a low elastic springback (expansion of the powder compact after ejection from the forming die). The negative effect of a differential springback (the axial springback being higher than the radial one) can be reduced by maintaining a small axial pressure on the compact during ejection (punch hold-down ejection or withdrawal of the die).14 Uniaxial pressing can be performed with a single-action, double-action or floating die press (Fig. 12.4). A single-action press uses only one moving punch. As it induces high green density gradients in the axial directions, it may be used for parts with a very low height to diameter aspect ratio and simple shape. A double action press has at least two independently moving punches that induce a more uniform compaction
12.4 Uniaxial pressing: (a) single action, (b) double action, (c) floating die and double action multiple punches.
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especially for simple large and thick parts. In pressing of multi-level compacts, the several punches have to be operated independently so that the different areas in the die are equally compacted. In a floating die press, a similar effect can be obtained, the movement (floating) of the die corresponding to an upwards movement of the lower punch.14 In more sophisticated floating die press, the punches and the die can move independently in order to reduce the green density gradients. This allows the production of complex parts with tapers, holes and multiple steps. If higher working speeds and precise control of the movements are required, the die is directly moved by the press. Isostatic pressing Isostatic pressing (cold isostatic pressing) consists in compacting a powder in an elastomeric container submersed in a fluid at a pressure of 20 to 400 MPa. Cold isostatic pressing allows the production of simple-shaped small or large powder compacts (up to 2000 kg) with a uniform green density even for large height/ diameter ratio part (impossible by uniaxial pressing), but with the sacrifice in pressing speed and dimensional control, requiring subsequent machining in the green compact. Cold isostatic pressing is used for powders that are difficult to press such as hard metals. Metal injection moulding Metal injection moulding (MIM) is adapted to the production of small complex near-net-shape compounds (wall thickness down to 0.3 mm) with very good tolerances (0.3 to 0.5%), especially for medium (thousands of parts/year) to high volume (millions of parts/year) production.33 Because of the high raw-material costs, MIM is usually limited to the production of parts lighter than 100 g. This process consists in filling a die cavity with a viscous mixture of powder and binder at around 130 to 200ëC under a pressure up to 150 MPa.12 Metal injection moulding parts are produced in four different steps: feedstock formulation, moulding, debinding and sintering. The feedstock is a homogeneous pelletised (granulation to a special shape) mixture of fine spherical metal powder with a mean size ranging from 5 to 15 0016m and 30 to 45 vol.% organic binder.33 A binder system usually has three components, a backbone that provides green strength and that will stay after debinding, a filler phase that is easily extracted during initial stages of debinding and surfactant to control feedstock rheology. During moulding, the feedstock is introduced to the heated injection unit by, for example, a rotating screw, the mixture is then injected in the die by axial movement of the screw and finally the part is ejected from the die after quick cooling. The main parameters are injection temperature, pressure and speed, as well as thermal conductivity and viscosity of the feedstock and the mould temperature.
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The debinding can be done by thermal decomposition, solvent extraction or a combination (also called catalytic decomposition). A typical binder removal rate is 2 to 3 mm wall thickness per hour. Finally, the part undergoes a densification up to 95% to 99% of the theoretical density as well as a shrinkage of 14% to 20% during the sintering step (see below). The MIM process is applied to produce parts in stainless steels (304L, 316L), tool steels (M2), soft magnetic alloys (Fe-50 %Ni, Fe-3 %Si, 430L), alloys for glass-to-metal sealing applications (kovar), cobalt-, nickel- and titanium-based alloys used for wear medical, automotive and aerospace applications. Powder rolling ± extrusion ± dynamic and explosive compaction Powder rolling (roll compacting) consists in compacting a powder continuously passing between two turning rolls.14 A binder is usually added to the powder to favour densification. For a given thickness of the strip (defined by the distance between the two rolls), the density of the compacted strip can be increased by the increasing the diameter of the roll and reducing the rolling speed (maximum speed lying about 0.5 m/s). The main advantage of powder rolling compared to conventional casting and rolling process is a low amount of rolling passes necessary to produce a thin strip. The main disadvantages are a high powder price and a low production rate. Extrusion consists in forcing a viscous mixture of powder and binder through a die. The obtained, shaped product is sintered with a slow heating rate in order to remove the binder. During dynamic and explosive compaction (high-energy rate compaction), that is only applied at lab-scale, powders are compacted at very high velocities (200 m/s) by the propagation of a high-pressure wave. One set-up uses the conventional die compaction with an upper punch moving at high velocity through the action of an explosive charge or compressed gas. Another one consists in encapsulating a powder in a mild steel tube and subjecting this tube to the action of sheet explosives taped to it. The advantages are a high green density, a higher green and sintered strength and lower density gradient.13,14 Green density of 99% of theoretical density has been achieved for aluminium, stainless steel, amorphous powders,12 even for tungsten compacts a relative density of 97.6% has been obtained. Sintering Sintering originally used to produce clay pots35 is nowadays involved in the fabrication of net-shape components in ceramics, cermets, metals and composites. The application fields are automobile and aeronautic industries (valves, bearings, aircraft wings weight), electrical and electronic industries (tungsten wires, ultrasonic transducers), medical industries (dental or hip implants).
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During sintering porosity and the microstructure change irreversibly from contacting particles to almost dense material. This induces improvement of many properties such as strength, ductility, conductivity, magnetic permeability, and corrosion resistance. Sintering can be defined as a thermal treatment for bonding particles into a coherent, predominantly solid structure via mass transport events that often occur on the atomic scale.34 The sintering induces the consolidation (increase of strength) of a loose or compacted powder and is usually accompanied by densification (shrinkage). During sintering the reduction of total surface energy (usual driving force for mass flow during sintering) is due to the decrease of surface area by formation of inter-particle bonds and the reduction of surface curvature. The path of the atomic motion occurring during the mass flow in response to the driving force is called the sintering mechanism.12 For metal powders, the mechanisms are usually diffusion processes with surface, grain boundary or lattice paths. Sintering progresses in different stages; for each stage (i.e. for each driving force type corresponding to a particular particle-pore geometry), different ways of mass flow (i.e. sintering mechanism) are possible. The knowledge of the relations between the different sintering parameters and each sintering mechanism during the different stages allows modelling and thus optimising the sintering parameters. The main sintering parameters are: · particle size (reduction of particle size increases the surface energy per unit volume of the powder and so the driving force associated with sintering, thus the sintering rate); · particle size distribution (large difference in curvature of the grains, due to grain size difference, will promote coalescence, i.e. the growth of the large grain at the expense of the smaller one); · temperature (exponential influence on sintering because it is involved in the activation energy of the sintering mechanisms (e.g. diffusion processes)); · time (influences diffusion); · green density (density gradients occurring during compaction will induce differential shrinkage during sintering and maybe distortion of the part because higher green density induces lower shrinkage); · applied external pressure (to obtain fully dense material, HP or HIP processes); · amount of liquid, if any; · sintering aids (favouring diffusion in the solid or liquid state); · atmosphere. A particular sintering atmosphere can be used to protect the metal powder from oxidation (argon, vacuum), to remove the oxidation layer on the powder (reducing atmosphere such as hydrogen, carbon monoxide, dissociated ammonia or natural gas), to control the carbon content of the powder, to remove the
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lubricants and binders introduced during compaction or even to react with the powder (formation of nitride). Sintering can be classified as solid state, liquid phase, reaction sintering and microwave sintering. Solid state sintering In solid state sintering the microstructure changes are divided into three different stages. The first stage corresponds to the growth of the bond, called neck, between two particles, independently of the growth of the neighbouring necks. The pores are interconnected with an irregular shape. The intermediate stage occurs as the merging necks shrink the pores to form interconnected pores with a more smooth usually cylindrical shape. Most of the densification and change in properties occurs in this intermediate stage. The final stage corresponds to pore closure, where the pores become spherical and isolated. By definition the driving force changes for each sintering stage. During the initial stage the driving force is the curvature gradient between the particle and the neck, during the intermediate stage it is the curvature around the cylindrical pore and during the final stage the curvature around the spherical pore. For every sintering stage, the mass transport process (sintering mechanism) can be described by a characteristic equation. There are sintering mechanisms that induce shrinkage (i.e. densification) such as volume diffusion, grain boundary diffusion, plastic flow and viscous flow (for the amorphous solids) and some that do not such as surface diffusion, evaporation±condensation and volume diffusion from a surface source to a surface sink. Different sintering mechanisms are involved at different moments during the sintering process. For example, a finer particle size usually favours sintering by surface or grain boundary diffusion compared to volume diffusion. During the initial stage, the different sintering mechanisms can be diffusion (surface-, volume- or grain boundary-diffusion), evaporation or dislocation motion. The sintering mechanism is usually described by the size of the growing neck between the particles but shrinkage or relative change in surface area can also be used. The intermediate stage is characterised by densification usually coupled with grain growth during the latter phase of the intermediate stage. The smaller grains, having a higher curvature, are progressively incorporated to the neighbour grain by grain boundary motion. This grain boundary motion induces drag forces on the pores, which can move by volume, surface diffusion or evaporation±condensation mechanism across the pore. When the moving rate of the grain boundary is too high (e.g. favoured by high temperature), the pores cannot impinge the grain boundary any more and become isolated inside the grain (beginning of the third stage). In this case, the rate of densification is much smaller because volume diffusion is less fast than grain boundary diffusion.35 Consequently, it is important to minimise pore±grain boundary separation by careful temperature control, the incorporation of second-phase inclusion such as
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oxide particles into the microstructure to impinge the grain boundary, or the use of narrow initial particle size distribution.37,38 The usual parameter to follow the sintering is the rate of densification. The two sintering mechanisms involved in densification are grain boundary and volume diffusion. Surface diffusion or evaporation±condensation mechanisms, inducing no shrinkage, are also expected to be active in smoothing the pore structure and in pore migration with grain boundaries during grain growth. Long sintering times (compared to the first stage) are required to achieve significant property or density changes. Temperature has a complex effect on the sintering because diffusion, grain growth and pore motion are all thermally activated. The third stage is characterised by the presence of isolated spherical pores. If the closed pores are mobile enough to stay coupled to the grain boundary, then continued shrinkage is expected. This is favoured by a homogeneous grain size, which lowers the curvature of the grain boundary and so decreases their motion rate. If not, after separation from the grain boundary, the pore must emit vacancies that move by volume diffusion, which is a slow process, towards the distant grain boundary. This leads to a drop of the densification rate. With prolonged sintering, the larger pores grow at the expense of the smaller ones (that emit more vacancies in the grain because of higher curvature). This is called pore coarsening or Ostwald ripening. In addition to pore coarsening, the pore size can increase by coalescence, due to grain growth by grain boundary motion dragging pores towards each other. If the pore has trapped gas, an internal pressure is induced inside the pore, which limits densification. If this gas is soluble in the matrix, the densification rate is controlled by the internal gas pressure and not by the limit of solubility. The usual parameter to follow the sintering during the final stage is the rate of densification, but the rate of shrinkage, surface area change, or neck growth could also be used. The rate of densification depends on the pore amount, pore radius, volume diffusion, grain size distribution and stress effects (compressed trapped gas working against pore shrinkage).39 Liquid phase sintering (HSS) Liquid phase sintering is involved when powders of different composition are mixed. Usually the constituent, that remains solid during sintering, should have a relatively high solubility in the formed liquid and inversely the solubility of the liquid in the solid should be low to ensure that this liquid phase is not transient. Common systems are: WC-Co, Fe-Cu, Cu-Sn, etc.36 The main advantage of liquid phase sintering is the lower sintering time required compared to solid state sintering. During heating, the mixture of powders first undergoes solid state sintering, which can induce significant densification, before the formation of the liquid at the sintering temperature. After the liquid is formed, the sintering depends on the amount of liquid and is
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usually divided into three stages: rearrangement, solution±reprecipitation, and the third stage. If the amount of liquid is sufficient to fill all the interparticle spaces, the theoretical density can be obtained during the rearrangement stage. For lower liquid contents, the solid skeleton slows down the densification and the contribution of the last two sintering stages becomes significant. In fact, less than 15 vol.% of liquid is usually used to avoid distortion of the part during sintering. During the first stage (rearrangement) and in case of a wetting liquid, the liquid spreads as soon as the liquid is formed between the solid particles under the influence of capillary forces. The rate of densification controlled by viscous flow is very high at the beginning and then continuously slows down. As the densification rate governed by rearrangement decreases, another mechanism called dissolution±diffusion±reprecipitation, characteristic for the second stage prevails. The solubility of a grain in its surrounding liquid increases with the curvature of the grain, i.e. with a decrease of the grain size. The difference of solubility as a function of grain size induces a concentration gradient of solute species in the liquid, that diffuse from the small grains to the large ones, on which the solute species precipitate (reprecipitation) when the solubility limit in the liquid is reached. So during this stage, grain coarsening occurs (Ostwald ripening). Simultaneously the elimination of the high-energy vapour interface is obtained by grain shape accommodation (flattening) during solution± reprecipitation events inducing densification by higher packing of the grains. The third stage corresponds to the densification of a rigid solid skeleton with a rate similar to the one obtained in solid state sintering. Reaction sintering Solid state reaction sintering of metal powder mixtures is a way to produce alloys such as carbon steel, Fe-Ni, Fe-Mn, Fe-Si, Fe-Cr, Fe-Mo and Cu-Ni. The principal phenomenon occurring during sintering is the solid state interdiffusion between the different compounds. The driving force is the chemical potential gradient due to concentration differences. This phenomenon superimposes the metal powder self-diffusion caused by surface and interfacial tension forces, occurring in solid state sintering of pure or pre-alloyed powders. Reaction sintering is favoured by fine particle size (smaller the diffusion distance) and high temperature (higher coefficient of diffusion). Reaction sintering can also occur during liquid phase sintering (Mo + 2Si ! MoSi2) or by reaction of the sintering atmosphere with the powder (3Si + 2N2 ! Si3N4). Microwave sintering Sintering of metal powders is a surprising recent development in microwave applications because bulk metals reflect microwaves. However, compacted
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metal powders at room temperature will absorb microwaves and will be heated very effectively and rapidly (above 100ëC/min), inducing sintering. Metals such as iron, steel, copper, aluminium, tin, nickel, cobalt, tungsten have been sintered to high density by microwaves. Cylinders, rods, gears, and other automotive components with until now a maximum size of 10 cm have been produced in 30 to 90 min.40 Contrary to the conventional heating where the transfer of thermal energy is done by conduction to the inside of the part, microwave heating is a volumetric heating consisting in an instantaneous conversion of electromagnetic energy into thermal energy. The mechanisms involved in microwave sintering are not completely understood at the moment. However, the sample size and shape, the distribution of microwave energy and the magnetic and electromagnetic field radiation are important parameters.
12.3.2 Other routes This part is mainly dedicated to full density sintering processes such as hot pressing, hot isostatic pressing, hot extrusion, hot forging and field assisted sintering. However, other techniques such as infiltration and rapid prototyping techniques will also be presented. Full density is required to improve product properties such as rupture and fatigue strength, toughness, thermal or electrical conductivity. This can only be achieved when stress and temperature are simultaneously applied during densification to close the pores.13 Because the powder usually has to be protected from reaction with air, this makes the hot compaction processes complex and costly. So these techniques are reserved for expensive materials with special properties such as beryllium and magnesium alloys (finer grain size), superalloys (elimination of segregation), high speed steels and dispersion strengthened alloys (homogeneous distribution of the second phase). The main parameters are temperature, applied stress, strain rate and grain size. `Near-net-shape' components, implying material saving, combined with higher properties can make these processes competitive compared to the conventional casting, forging, machining route. Hot pressing Hot pressing consists in applying pressure with a hot punch on the metal powder placed in a heated die usually under a protective atmosphere.14 The main problem with hot pressing is to find a suitable die material, which has to withstand the applied pressure without reaction with the metal powder. Although the total amount of deformation of the compact is relatively limited compared to hot extrusion or hot forging, complete densification is generally achieved.
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Hot isostatic pressing In hot isostatic pressing, a hydrostatic pressure is applied to the powder by the action of a gas (Argon, nitrogen) with a pressure up to 300 MPa simultaneously with a temperature up to 2000ëC.14 Before HIPing, the powder is placed into a metallic or a glass container (`can') which is out-gassed and sealed. Spherical powders are the best suited for hot isostatic pressing, which is a long time process (around two cycles per day). The cycle duration can be reduced to a few hours for temperatures lower than 1250ëC, which allow the opening of the isostatic press without requiring the cooling down of the furnace and so reloading with a preheated can. In spite of the high capital cost, hot isostactic pressing is used to produce large semi-finished components in hardmetals,41 high speed steels, superalloys (most common technique),42 titanium alloys and beryllium. Moreover a can with a complex form allows the production of near-net-shape components. This ability has been used for superalloys and titanium alloys. Hot extrusion There are three basic methods for the hot extrusion of powders. The first method consists in filling a heated extrusion container with a loose powder (magnesium alloys), which is heated during the 15 to 30 seconds before starting its extrusion. In the second method the powder has been compacted and sintered before extrusion (aluminium alloys)43 and molybdenum, superalloys and high-speed steels. The third method, which is the mostly used (beryllium, stainless steel, aluminium, copper and nickel dispersion strengthened alloys), consists in filling a metallic capsule (`can') with the powder, to protect the powder from contamination by the atmosphere. The can is out-gassed and sealed before heating and extrusion. The material of the can (e.g. copper, low carbon steel) should have a stiffness similar to that of the powder at the extrusion temperature, should not react with the powder and should be removable by etching or mechanical stripping. Circular, elliptical or rectangular cross-section parts can be obtained by hot extrusion. For a more complex cross-section shape the `filled billet' technique was developed.44 The mild steel filler contains a cavity with the shape of the desired cross-section, which is filled by the powder, the whole is placed in a carbon steel can and then extruded. Seamless tubing of stainless steel has been produced with this technique.45 Hot forging Metal powders that have been previously compacted and sometimes also sintered (preform) can be hot forged into parts of high density (elimination of
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the residual porosity) and closely controlled dimensions.13 A height strain of 50% is at least needed for pore elimination and good interparticle bonding.12 During hot forging the preform, which has a simpler shape than the final product, undergoes a lateral flow especially at the beginning of the deformation. This process allows subsequent quenching that makes it attractive for hardened steel. A production rate of 20 to 40 strokes per minute can be reached. The main applications of this process concern the automotive industry (gears, connecting rods). Electric field assisted sintering ± `spark' sintering Electrical field assisted sintering is an emerging technology to densify powders with nanosize or nanostructure while avoiding the coarsening which accompanies standard densification routes thanks to its very short sintering time. However, it can also be used for coarser powders. This process is applicable to metals (Al, Fe, W, Mo, Be, Ti), metallic alloys (Al-Si-X, Fe-Co), intermetallics (Al-Ti, Al-Fe) and ceramics. It could be used to produce cutting tools, metal forming dies, gears, pump components, electric motors, household equipment. Electrical field assisted sintering (FAST) consists in applying simultaneously a pressure with a punch and an electrical field to the powder that fills a conductive die. The electrical discharge induced by the electrical field at the contact of the particles and in the gaps between them is responsible for the physical activation of the powder particle surface (melting and vaporisation, inducing the cleaning of the particle surface) that favours sintering. The main parameters are related to the powder characteristics, the compaction (e.g., uniformity of the packing of the die) and the characteristics linked to the electrical field (intensity, voltage, pulse pattern that induces the electrical discharge). The reached density is typically 98% to 99% of the theoretical density in very short time (10 min).46 Infiltration Infiltration consists in filling the pores of a sintered material with a liquid metal or alloy, having a (much) lower melting point.13 The driving force of infiltration is the reduction of surface free energy as in sintering. Infiltration is promoted by a low wetting angle of the liquid with the solid and by establishing a pressure gradient, e.g. putting the solid (porous skeleton) in a vacuum and applying pressure to the infiltrating liquid. The infiltration parameters are the infiltration temperature (reduction of the viscosity of the infiltrating liquid) and the dwell time above the liquidus of the infiltrating liquid. Short times are preferred to limit eventually extensive reaction of the liquid with the skeleton (Fe-Cu). Infiltration can be advantageous
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compared to liquid phase sintering when the liquid is insoluble in the skeleton material (W-Cu). Rapid prototyping Rapid prototyping (RP) concerns techniques that produce a complex part or a prototype to net or near-net shape by building the object layer by layer (additive processing) from a computer aided design (CAD) file without the need of specific expensive tooling (direct processes).47 In many cases, a post-processing step such as, for example, co-firing, sintering or infiltration, etc., is needed to ensure sufficient strength of the part. Rapid prototyping techniques can be classified in direct, also referred as solid free-form fabrication (SFF), and indirect processes, the latter being the most widely used. Indirect processes use a pattern (model) or a mould made by a rapid prototyping technique to produce the final object, which can be metallic. The following concerns only the direct processes. RP techniques mainly used for plastic products have been adapted for metals, ceramics or composites. The main advantages are the ability to produce complex parts and the absence of specific tooling that reduces the production time. However, the low dimensional accuracy and the high roughness of the part are the main drawbacks. The layer formation procedure is specific to each RP technique. The most common RP techniques are stereolithography (SL or SLA), fused deposition modelling (FDM or FDC and FDMet, respectively for ceramic and metal), ink jet printing (IJP) or three-dimensional printing (3DP) and selective laser sintering (SLS) or direct metal laser sintering (DMLS) or selective laser melting (SLM). Laser technology is involved in more than a half of all the RP techniques: laser photo-polymerisation (stereo-lithography, . . .), laser fusion or sintering (SLS), laser cladding (laser generating, controlled metal build-up (CMB), laseraided powder solidification with powder jet (LAPS-J), LENS, . . .), laser cutting (LOM, . . .), laser-induced CVD (SALD, LCVD). The applications of the metallic products are injection moulds, tools, dies, implants, etc. Stereolithography (SL or SLA) consists in photo-polymerisation of a liquid monomer by an UV laser. The laser is scanned in selected areas of a liquid monomer layer, defined by a CAD file, to cure them.47 After one layer has been processed another layer of resin is coated on the top of it (called re-coating) until the part is finished. This process has been adapted to produce metals.50 Fused deposition modelling (FDM) consists in depositing a continuous hot extruded (low viscosity) thermoplastic filament with an elliptical cross section on a fixed substrate, following a path defined by a CAD-file.47 In this case the not heated part of the filament acts as a punch. When one layer is completely cooled down the build platform indexes down and the deposition of the next
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layer is performed. Adaptation of this process by using a particle-loaded thermoplastic filament followed by debinding and sintering allows the fabrication of ceramic (FDC) and metallic (stainless steel) components (FDMet). Multiphase jet solidification (MJS) is similar to the fused deposition process of metals and ceramics.47 The low melting metal or the powder-binder mixture is introduced in a patented extruder, melted (viscosity < 200 Pa s) and subsequently deposited on a substrate. Part of stainless steels, high speed steels, FeNi, Ti and SiC have been produced. Three-dimensional printing (3DP or IJP) consists in printing with an ink jet printer a low viscosity binder, which forms a droplet, on a layer of deposited ceramic or metallic powder.47 After drying of the binder (e.g. latex, wax emulsions, homogeneous solution phase binders), the next powder layer is deposited and so on until the part is finished. The part is then ultrasonically washed and its unprinted regions are removed through redispersion in the ultrasonic bath. The 3DP processes can be divided into dry powder processes and wet slurry processes, where the slurry of the powder is sprayed onto a substrate, dried, followed by the printing of the binder. The dry powder process has been used to produce injection moulds in stainless steel and tool steel parts (latex binder), which are subsequently sintered and infiltrated with Cu or Cualloys. The dimensional accuracy and the removal of the loose powder in the narrow regions are the main problems. The advantages of 3DP are its ability to produce very complex shapes with fine features (100 0016m), its flexibility (many materials) and its well-controlled process. Selective laser sintering (SLS) consists in scanning a thin layer (100± 200 0016m) of powder with a laser, that induces local sintering of powder particles corresponding to a path defined by a CAD-file.47 The non-sintered particles act as support and so allow complex forms such as hollow sections, overhangs or undercuts, as for 3DP. After completion of the sintered layer, a new thin layer of powder is spread on top of it and then laser-scanned. The non-sintered particles are removed when the part is finished. SLS processes are mainly used for thermoplastics materials and less for metals and ceramics, because they require a higher energy input. Metals and ceramics can be produced by indirect or direct sintering approach. In the indirect sintering approach, a binder is used with the powder to be laser-sintered and a post-processing step is needed to obtain full density. For example, a polymer coated steel powder was processed by SLS and subsequently infiltrated with bronze or copper for tool making applications.49 In the direct sintering approach, a high power laser (e.g. 1 kW, CO2 laser) is used to sinter a preheated powder feed (reduction of thermal stresses) to reach the high temperature required for sintering. Liquid phase laser sintering was successfully performed for powder mixture such as Fe-Cu, stainless steel-Cu, WC-Co, TiB2-Ni and Fe3C-Fe. The SLS technique has a relatively fast building rate compared to the SLA or 3DP techniques for ceramics.
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Direct metal laser sintering (DMLS) and selective laser melting (SLM) are similar to selective laser sintering (SLS). But in the case of DMLS most of the powder layer is melted whereas in SLM it is completely melted. As a consequence, there is a good metallurgical bonding between the layers and densities up to 100% can be reached. Laser cladding (LC) includes processes such as laser engineered net shaping (LENS) and a variant of shape deposition manufacturing processes (SDM).48 Laser engineered net shaping (LENS) and direct metal deposition (DMD) processes belong to laser powder deposition processes. They comprise focusing a laser beam (Nd:YAG) on a metallic substrate to create a weld pool while simultaneously injecting the metal powder directly into the laser beam, where it melts. The moving substrate along a defined path produces the component lineby-line and layer-by-layer. The high cooling rate allows the deposition of nonequilibrium phases (e.g. extended solubility alloys) and/or fine-grained microstructure, as full density and high mechanical properties can be achieved. The fabrication of multi-material graded and layered structures is possible.
12.4 Conclusions Powder metallurgy is a mature industrial activity used to produce metallic and alloy components. There is a wide variety of powder production techniques, ranging from high output atomisation processes to high-value nano-powder production facilities. A broad spectrum of consolidation routes leads to simple or complex components. The high number of powder production techniques and consolidation methods allows enhancement of the relation between processing, microstructure and properties. Thanks to their high flexibility, PM processes find applications in the transport, energy, medical, machinery, and many other sectors.
12.5 References 1. Koch CC, `Milling of brittle and ductile materials', ASM Handbook: Powder metallurgy technologies and applications, 1998, 7, 53±66. 2. Hogg R and Cho H, `Grinding', Encyclopedia of Materials: Science and Technology, 2001, 3652±58. 3. LuÈ L and Lai MO, Mechanical alloying, Boston/Dordrecht/London, Kluwer Academic Publishers, 1998. 4. Soni PR, Mechanical alloying ± Fundamentals and applications, Cambridge International Science Publishing, 2000. 5. Suryanarayana C, `Mechanical alloying', ASM Handbook: Powder metallurgy technologies and applications, 1998, 7, 80±90. 6. Benjamin JS, `Mechanical alloying. Aprespective', Met. Powder Rep., 1990, 45(2), 122±27. 7. Suryanarayana C, Bibliography on mechanical alloying and milling, Cambridge
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International Science Publishing, 1995. 8. Bakker H, Zhou GF and Yang H, `Mechanically driven disorder and phase transformations in alloys', Progress in Mat. Sci., 1995, 39(3), 159±41. 9. Bradstedt SB, `New powder making process may expand P/M's capabilities', Precision Metal, 1969, 27, 52±55. 10. Dunkley JJ, `Atomization', ASM Handbook: Powder metallurgy technologies and applications, 1998, 7, 35±52. 11. Lawley A, `Atomisation', Encyclopedia of Materials: Science and Technology, 2001, 387±393. 12. German RM, Powder Metallurgy Science, Princeton, NJ, Metal Powder Industries Federation, 1984. 13. Lenel FV, Powder Metallurgy Principles and Applications, Princeton, NJ, Metal Powder Industries Federation, 1980. 14. Schatt W and Wieters KP, Powder Metallurgy Processing and Materials, EPMA, Shrewsbury, Liveseys Ltd, 1997. 15. Anderson IE, `Boost in atomiser pressure shaves powder ± particle size', Advanced materials and processes, 1991, 140(1), 30±32. 16. Gerking L, `Powder from metal and ceramic melts by laminar gas streams at supersonic speeds', Powder Metallurgy International, 1993, 25(2), 59. 17. Tornberg C, `Gas efficiency in different atomization systems', Advances in Powder Metallurgy and Particulate Materials, Volume 1: Powder production and spray forming, Metal Powder Industries Federation, 1992, 127±35. 18. Kerker M, `Laboratory generation of aerosols', Adv. Colloid Interface Sci., 1975, 5, 105. 19. Raabe OG, `The generation of aerosols of fine particles', in Fine Particle: aerosol generation, measurement, sampling and analysis, symposium in Minneapolis, Minnesota, May 28±30 1975, Liu BYH. New York, Academic Press, 1976, 57±110. 20. Plyum TC, Lyons SW, Powell QH, Gurav A, Kodas T and Wang LM, `Palladium metal and palladium oxide particle production by spray pyrolysis', Mater. Res. Bull., 1993, 28(4), 369±76. 21. Eroglu S, Zhang SC and Messing GL, `Synthesis of nanocrystalline Ni-Fe alloy powders by spray pyrolysis', J. of Materials Research, 1996, 11, 2131±34. 22. Sasaki Y, Shiozawa K, Kita E, Tasaki A, Tanimoto H and Iwamoto Y, `Fabrication of metal nanocrystalline films by gas-deposition method', Mater. Sci. Eng. A, 1996, 217, 344±47. 23. Daub O, Langel W, Reiner C and Kienle L, `QSM-controlled production of nanocrystalline metals by inert gas condensation in q flow system', Ber. BunsenGes. Phys. Chem., 1997, 101, 1753±56. 24. Choy KL, `Vapor processing of nanostructured materials', Handbook of Nanostructured Materials and Nanotechnology, Volume 1: Synthesis and Processing, Nalwa HS, Academic Press, 2000, 533±77. 25. Klar E, reviewed by Taubenblat PW, `Chemical and electrolytic methods of powder production', ASM Handbook: Powder metallurgy technologies and applications, 1998, 7, 67±71. 26. Riman RE, `Solution synthesis of powders', Encyclopedia of Materials: Science and Technology, 7800±10. 27. Berger S, Schachter S and Tamir S, `Photoluminescence as a surface-effect in nanostructures', Nanostruct. Mater., 1997, 8(2), 231±42.
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28. Ayers J D and Anderson I E, `Very fine metal powders', Journal of Metals, 1985, 37, 16±21. 29. Boulos MI, Fauchais P and Pfender E, Thermal Plasmas ± Fundamentals and Applications ± vol. 1, New York, Plenum Press, 1994. 30. Fauchais P and Vardelle A, `Thermal Plasmas', IEEE Transactions on Plasma Science, 1997, 25(6), 1258±80. 31. Costa J, `Nanoparticles from low-pressure, low-temperature plasmas', Handbook of Nanostructured Materials and Nanotechnology, Volume 1: Synthesis and Processing, Nalwa HS, 2000, 57±158. 32. Ewsuk KG, `Grinding', Encyclopedia of Materials: Science and Technology, 2001, 3652±58. 33. Tandon R, `Metal injection moulding', Encyclopedia of Materials: Science and Technology, 2001, 5439±42. 34. German RM, `Sintering', Encyclopedia of Materials: Science and Technology, 2001, 8640±43. 35. German RM, `Sintering: modeling', Encyclopedia of Materials: Science and Technology, 2001, 8643±47. 36. German RM, `Liquid phase sintering: metals', Encyclopedia of Materials: Science and Technology, 2001, 4601±03. 37. Brook RJ, `Pores and grain growth kinetics', J. Amer. Ceramic Soc., vol. 52, 1969, 339-40. 38. Hsueh CH, Evans AG and Coble RL, `Microstructure development during final/ intermediate stage sintering-1. Pore/grain boundary separation', Acta Met., 1982, 30, 1269±79. 39. Markworth AJ, `On the volume-diffusion-controlled final-stage densification of a porous solid', Scripta Met., 1972, 6, 957±60. 40. Agrawal DD, `Metal parts from microwaves', Materials World, 1999, 7(11), 672± 73. 41. Hodge ES, `Elevated-temperature compaction of metals and ceramics by gas pressure', Powder Metallurgy, 1964, 7(14), 168±201. 42. Fischmeister H, `Isostatic hot compaction ± a review', Powder Metallurgy Int., 1978, 10, 119±23. 43. Lyle JP Jr. and Cebulak WC, `Fabrication of high strength aluminum products from powder', in Powder Metallurgy for High-performance Applications, Syracuse, Burke JJ and Weiss V, 1972, 231±54. 44. Bufferd AS, `Complex superalloy shapes', Powder metallurgy for high-performance applications, Syracuse, Burke JJ and Weiss V, 1972, 303±16. 45. Aslund C, `A new method for producing stainless steel seamless tubes from powder', 5th European symposium on powder metallurgy, 1978, 1, 278±83. 46. Groza JR `Field activated sintering', ASM Handbook: Powder metallurgy technologies and applications, 1998, 7, 583±89. 47. Safari A, Danforth SC, Allahverdi M, Venkataraman N, `Rapid prototyping', Encyclopedia of Materials: Science and Technology, 2001, 7991±03. 48. White D, `Rapid prototyping processes', Encyclopedia of Materials: Science and Technology, 2001, 8003±09. 49. Kruth JP, Leu MC, Nakagawa T, `Progress in additive manufacturing and rapid prototyping', Annals of the CIRP, 1999, 47(2), 525±40. 50. www.dsmsomos.com
13
Improving steelmaking and steel properties T E M I , Royal Institute of Technology, Sweden
13.1 Introduction Steel is the most popular metallic material and nearly 1,000 Mt of it was produced in 2003. This amount accounts for over 90% of annual total metals production in the world. Human life is greatly supported by steel due to its inherent advantages in production and properties compared with other materials. The production advantages include: (1) plentiful availability of raw materials (iron ore and steel scrap), reducing agents (coke and pulverized coal) and refining fluxes (mostly lime and dolomite), (2) high output rate per plant (10 Mt/year), and (3) relatively low total material demand (00185 t/t-steel) and ore reduction energy (22 GJ/t-steel). The advantages in properties encompass (1) high Young's modulus, (2) outstanding combination of high strength and ductility, and hence good formability, (3) good corrosion resistance on alloying, and (4) excellent magnetic properties. These properties have been significantly improved at competitive cost by alloying and controlling structure and texture, yet leaving more room for further improvement. The production process and properties of steels have developed quite interactively as triggered by market demand. The key driver of the development in recent years has been, in many cases, market. In fact, assembly industries have required advanced steels with superior properties at shorter delivery times to produce better products at less cost for end users. The requirement has driven the steel industry to develop steels with improved or new properties. Such steels have prompted the emergence of new production processes in the steel industry. Conversely, the new steel production processes have sometimes unveiled avenues to produce more advanced steels with better properties at lower cost and/or better productivity. The advanced steels have prompted the assembly industry to refine their design and assembling process. Examples are plentiful and some representative ones will be discussed in this chapter.
13.1 Steel manufacturing processes from raw materials to semis (top) and semis to products (bottom) (courtesy of AISE Steel Foundation).1
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13.2 Developing processes and properties with reference to market, energy, and environment An illustration of steel manufacturing processes is shown in Fig. 13.1,1 the upper for upstream processing from raw materials to semis, and the lower for downstream processing from semis to products. Major iron sources for steelmaking are hot metal and steel scrap. Hot metal is made in blast furnaces (BF) by reducing at high temperatures sintered or pelletized iron ore with CO gas, i.e., Fe2O3 + 3CO ! 2Fe + 3CO2. CO gas is formed via the reaction of charged coke and hot blast blown into the blast furnace, i.e., 2C + O2 ! 2CO. Hot metal is saturated with C and contains some Si and impurity elements P arising from gangue in iron ore and S from coke. Hot metal is charged with steel scrap (0014 25%) into the basic oxygen furnace (BOF), desiliconized and decarburized by impinging pure oxygen gas jet from top lance and converted into steel. This is named the BF±BOF route. On the other hand, the majority of steel scrap, sometimes with a small fraction of hot metal and/or direct reduced iron (DRI), is charged into electric arc furnaces (EAF), melted and decarburized with injected oxygen gas and converted into steel. This is called the scrap±EAF route. Decarburized and oxygen-bearing steel melt is tapped into a ladle with alloying elements and deoxidizing agents, Si-Mn, Fe-Si and/or Al, and then processed for final removal of H, S and deoxidation products, i.e. oxide inclusions like Al2O3. The final removal and fine tuning of temperature and alloying element compositions for quality steels are done in various secondary refining furnaces (ladle furnace (LF) denoted Steel refining facility in Fig. 13.1). Refined melt is cast via tundish into the mold of a continuous casting machine (CCM), and withdrawn as semis. Semis are then reheated, hot rolled, pickled, cold rolled, heat treated, annealed and surface finished into products. Major applications of steels are for construction, engineering works, automobile, ship, machinery, containers, etc. Automobiles consume a sizable fraction of total steel production. As the design, structure, manufacturing processes and fuel economy of automobiles advance, demands on steel materials have become more stringent and multifold, chasing extremes of properties at an affordable cost. Recent moves to suppress global warming have emphasized the weight reduction of automobiles and hence thickness of steel for automobile parts. The thickness is to be determined by the strength and corrosion loss of steel sheet and the shape rigidity of steel parts. Strengthening steel helps reduce sheet thickness for auto body (panels, frame, reinforcements, members, pillars, side sills, seats, etc.) and traction system. However, strengthening must be made without impairing various formabilities (e.g. deep drawability, hole expansibility, stretch formability and bendability) that are specific to each part, and often inversely proportional to the strength. Thus, optimization of the balance between the
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strength and formabilities has become a crucial issue for steels. More so for being challenged by competing materials, aluminum alloys and engineering plastics.
13.2.1 Properties driven by the market, environment and energy Steels used in various parts of automobiles call for different properties depending on their applications, and hence best fit micro/nano-structure and texture have been developed for each part. Development has been in the following directions: · Ultimate formability for exterior panels that are subject to transfer press forming (e.g. bake-hardenable, interstitial free steel (BH-IF)). · Ultimate strength for seat frames and door impact beams (e.g. tempered martensitic steel). · Intermediate but optimized strength and formability combination for various members and pillars (bainitic ferrite steel, dual-phase (DP) steel and transformation-induced plasticity (TRIP) steel). These characteristics are shown on elongation-strength coordinates in Fig. 13.2.2 To meet the demands, innovative streamlining of the conventional sheet production process was mandatory for both upstream sectors of steelmaking/ casting and downstream sectors of rolling/heat treatment. Exterior panels for door sides, engine hood, ceiling etc. are typical examples for which excellent formability and fine surface finish are of primary importance, since the panels determine the quality of autobody.
13.2 Elongation and strength of steels used for automobile (Komiya).2
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Among them, development of BH-IF steel is of particular interest in view of process vs product interaction. Process development for BH-IF steel includes preliminary removal of P and S in hot metal pretreatment and minimization of C, P, S and O in basic oxygen furnace blowing and secondary refining. It also includes heavy reduction in hot- and cold-rolling, high temperature continuous annealing and rapid cooling of cold rolled sheet to form fine grained texture ( axis of ferrite iron crystals aligned normal to rolling plane) for superior formability. Properties required for BH-IF steel sheet BH-IF steel for automobile body exterior panels is subject to press forming where deep drawability and stretch formability count most for the quality of the body. Deep drawability is expressed in terms of limiting drawing ratio (LDR), which is a ratio of the diameter of the blank before drawing to that of tube after drawing. LDR shows good correlation with plastic strain ratio (r) which is defined by a ratio of (the logarithmic ratio of sheet width before (wi ) to that after the deformation (w)) to (the logarithmic ratio of sheet thickness before (ti ) to that after the deformation (t)), i.e., r ˆ ln …w=wi †=ln …t=ti †. Stretch formability is expressed in terms of limiting dome height (LDH) which is determined by uniform elongation (El) or work hardening coefficient. Good drawability securing sufficient metal flow without causing wrinkles is achieved at increased values of r and El. Past development of steels for deep drawing application is shown in Fig. 13.3.3 For superior press formability, r 0015 2.5 and El 0015 45% are required that can be achieved when fine grained //ND texture (called -fiber) is developed in the matrix of steel sheet. For that, dissolved interstitial elements, C and N, are to be decreased and stabilized as precipitates of carbides and nitrides. An example of the chemical composition of such steel is C 20, Si 200, Mn 1500, P 100, S 30, Al 400, N 15, Ti 400 and Nb 150 all in mass ppm, for which r 0018 2.5, El 0018 50%, yield strength (YS) 0018 140 MPa and tensile strength (TS) 0018 290 MPa have been materialized by controlling the texture and microstructure mentioned above. These criteria were not met while decarburization of liquid steel below 0.04% (400 ppm) was impractical with BOF. Accordingly, box annealing (BA) of the low carbon Al-killed (LCAK) steel strip after hot rolling was practiced (low CBA in Fig. 13.3) to reduce dissolved C, N and produce deep drawing quality (DDQ) steel. The BA makes N atoms precipitate as fine particles of aluminum nitride (AlN) to enhance the formation of //ND texture. The BA also makes C atoms precipitate in coarse carbides and hence softens the steel matrix. Cooling after BA is slow due to the large thermal inertia of the process. Accordingly, only trace amounts of C remains as dissolved in the matrix, preventing age-hardening. However, BA is a very time- and energy-consuming,
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13.3 History of the development of cold rolled deep drawing steel sheet (Obara and Sakata).3
costly process of low productivity. The heating/cooling rate has little allowance for control, limiting precipitation control. The values of r and El thus achieved were about 1.6 and 50%, and is called super extra deep drawing quality (SEDDQ) steel. During the development of SEDDQ steel, it was recognized that impurity elements like P, S, and O need to be minimized. Metallurgical factors that have contributed to the advance of deep drawability are summarized in Table 13.1. Recently, even better SEDDQ steel with r ˆ 3:0 and El > 50% has been developed with Ti and Nb added ULC-IF steel by incorporating lubricated rolling of the steel in the ferrite temperature range. The advantage of the ferrite Table 13.1 Unit processes in an integrated system to produce BH-IF SEDDQ sheet Unit process
Key operations and equipment
Steelmaking
· Removal of P and S by hot metal pretreatment · Decarburization by combined blowing BOF · Final decarb. C, N 2:0 Blur in etching Punch crack
Sour gas pipe LNG plate Lamellar tear
HIC Embrittlement Z-crack
Shape control Shape control
S 60% Concrete filler 10% Road 17% Balance, own use, civil engineering
HMPT + BOF
110
· Recycle to sinter + HMPT* · HMPT + zero slag BOF blow* · Hydration/carbohydration*
Asphalt filler Temporary road Civil engineering/ land reclaim
EAF
70
None
Same as above
LF
40
None
In house
*see text
(b) increasing productivity of ironmaking and steelmaking processes, (c) improving quality of steel products, and (d) progress of process modification. As an example, the occurrence of slags in steel industry in Japan is listed in Table 13.11. Measures taken so far to minimize the occurrence are detailed elsewhere.29 Most of them, up to 99%, are utilized in the civil engineering sector. Better utilization is under development. BF slag is near neutral, composition being quaternary 40±45%CaO± 5%MgO±35±40%SiO2±20%Al2O3. Utilization of BF slag for cement reduces energy, that is otherwise consumed for calcining limestone, resulting in a considerable decrease in CO2 emission. Instability exists in this application, however, as challenged by less expensive replacements that are coal ash occurring from coal fired power plant for cement clinker and recycled concrete from demolished buildings and asphalt filler. The slag has also been industrially converted to rock wool and fertilizer, but on a small scale. The balance has found only low-end applications in civil engineering, road and port construction. Attempts to utilize the sensible heat of molten slag were made extensively, but without much industrial success. Gangue materials in the ore and ash in the coke and coal, together with additives to sinter and pellet determine the chemistry of BF slag. Recent notable improvement under increasing PCI up to 230 kg/ton-hot metal is that the BF slag has been successfully decreased from 280±330 to 265±300 kg/ton-hot metal without impairing BF operation by reducing SiO2 content in sinter from circa 5.1% to 4.6% (see refs 3, 4, and 5 in [30]).
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Table 13.12 Reduction of waste slag in steelmaking sector by incorporating HMPT in BF±BOF route Type of HMPT
BOF slag recycle process
Zero BOF slag process
Slag before implementation* Slag after implementation* Slag reduced** Occurrence of waste slag***
138 (BOF slag 92) 121 (BOF slag 42) 59 + (desiliconized slag) 79-000b
130 60 (BOF slag 10) 70 60001870
*For details, see text. **138 ÿ [121 ÿ42(recycle) ÿ ] ˆ 59+000b, ***121 ÿ (42 ‡ ) ˆ 79 ÿ
Figures are in kg/t-steel
For HMPT and BOF slag, two typical methods have been in operation to optimize the total production system for reducing P, S and waste BOF slag as briefly mentioned before.30 One method is to recycle the BOF slag to sintering to recover iron and CaO in the slag in the BF. This naturally causes the increase of P in hot metal from the BF, beyond the dephosphorizing capability of the BOF, and hence implementation of HMTP for dephosphorization is mandatory. However, if 85% or more of hot metal was dephosphorized at HMPT to 0.04%, P in hot metal could be kept below 0.14% even at 100% recycling of BOF slag. In this case, all hot metals are desiliconized at the runner and tilting tundish in the BF shop, dephosphorized and desulfurized in sequence by injecting post-mixed fluxes into the desiliconized hot metal in torpedo cars at the pretreatment center, and blown in the BOF. As shown in Table 13.12, occurrence of waste slag in this case decreased to about 80 kg/ton-steel, about 60% of 138 kg/ton-steel for ordinary BOF operation (see ref. 6 in [30]). The other method to minimize the BOF slag without such recycling is to reduce Si in hot metal to a minimum. Si in hot metal at tap from BFs is reduced to a low 0.2%. The hot metal is desiliconized (to 0014 0.1%Si), desulfurized and dephosphorized in the transfer ladle. The pretreated hot metal is blown in the BOF without the addition of slag materials. Occurrence of slag (HMPT slag inclusive) in the steelmaking sector is much reduced from 130 kg/ton-steel for an ordinary BOF shop to 60 kg/ton-steel, less than 50%, by this method. BOF slag itself is reduced to only 10 kg/ton-steel (see ref. 7 in [30]). In both cases, the slags after dephosphorization (CaO/SiO2 0014 2, Fe  10%, P2O5 3±4%) and desulfurization (CaO/SiO2 > 3, Fe 0014 1%, some S and F, and/or Na) are not much recycled, considerable portions being disposed of for landfill at some expense. Only small fractions of BOF slag has been used as a supplemental raw material for cement. Otherwise, use is limited to landfill and gravel for temporary paths. Recently, however, HMPT and BOF slag are finding possible applications to:
Improving steelmaking and steel properties
549
(a) Improve the sea bed by capping, forming fishery ranches. (b) Protect the seashore from breaker waves or strengthen land foundations for heavy structures by forming the slags into large blocks with hydration or carbohydration. (c) Promote the growth of biomass of biofouling organisms (phytoplankton) on the block (as fertilizer) to fix CO2 and feed fish. (d) Preform the mixture of HMPT waste slag with BF slag, fly ash and water into large blocks for a similar application to (b). Although (a) to (c) are in the trial stage, (d) has been commercialized in large quantities. EAFs have been operating largely for melting and oxidizing with single slag practice. Thus, properties and the use of EAF slag are similar to BOF slag. Secondary refining calls for a variety of slag compositions to meet product requirements as mentioned before. However, there are not many applications available for LF slag except for some internal use and civil engineering. Sludge consisting of iron oxides, metallic iron, water and oil is usually separated from water, heated to 1200ëC to remove oil, and returned to sinter plant. Dust is mostly sent to Wealtz kilns where iron is recovered; evaporated Zn and Pb are concentrated and sent to non-ferrous refiners for recovery. As a new development, the Fastmet process, with rotating hearth furnaces, has emerged to recover iron and Zn from dust pelletized with C. It also utilizes recycled waste oil as fuel.
13.5.3 Lifecycle analysis of steels The process development referred to in previous sections contributes to the reduction of CO2 emissions. Properties development does the same via product stewardship which has become popular through two-way interactions with the customers. High strength steels to reduce the weight of cars, trains and ships, electric steels to reduce transmission loss of electricity, pre-coated sheet steel for electric appliances that make oiling, degreasing and painting unnecessary, Table 13.13 Reduction of CO2 emission by use of high performance steel products Application
Production (kt)
Automobile Ship Electric train Building Transformer Power plant boiler Total Note: Peta Joule (PJ) = 1015J
631 1531 8 1529 55 1.4 3757
Energy saving (PJ)
CO2 reduction (Mt)
89.5 30.7 3.5 14.0 74.4 26.2 238.3
6.4 2.4 0.1 1.1 2.9 2.4 15.4
550
Fundamentals of metallurgy
eliminating polluting organic solvents, all fall into this category. Here, the energy required to produce these steels must be kept reasonably lower than the energy saved by lifetime use of the steels. Detailed LCA indicates that the use of high performance steel products resulted in a total reduction of CO2 emissions of 15.4 Mt for the year 2000 in Japan as shown in Table 13.13.28
13.6 Future trends 13.6.1 General trend of future development of processes and properties Needs for steels for various applications will be upgraded, driven largely by energy and environmental concern and international competition. Strength, fatigue strength, ductility, weld ductility, various formabilities, corrosion resistance, resistance to SSC, HIC, SOHIC, machinability, etc. will further be improved. Regarding mechanical properties, upgrading of the balance between strength and ductility/formability at a higher level without impairing weldability will remain a moving target to chase with continuous effort. Obviously, not all of these properties are required simultaneously. In-depth cooperation with user industries from the planning stage to identify what properties are required under what conditions becomes essential to clarify real needs and enables the best specification to be developed, suited to and optimized for processing both in the user and steel industries. User needs have driven the progresses of steel materials and processes in the steel industry. Conversely, the progress in the steel industry has provided user industries with opportunities for new and advanced design and manufacturing processes and products of better performance. Such cooperation and interactive developments will continue more closely in future for the benefit of both parties. The needs for steel materials will be met only by semis lower in impurities, inclusions, segregation, and internal and external defects with precise control of alloy components. The semis are to be processed for more advanced control of micro/nano-structure and texture with better size and profile accuracy to fulfill requirements. All these will be achieved in the presence of degrading raw materials and increasing demand to reduce energy consumption, GHGs and waste emission. To comply with these circumstances, steel manufacturing processes have developed by: (a) Splitting the function of smelting reduction of iron ore and refining of hot metal into `pelletizing/sintering + coke making + BF ironmaking + HMPT + BOF steelmaking + secondary refining', although it has been made in an integrated way to keep up smooth material flow through the unit processes. (b) Converting batch ingot casting to CC or thin slab CC + inline rolling. (c) Converting hot- and cold-rolling mills into tandem mills.
Improving steelmaking and steel properties
551
(d) Integrating the function of hot rolling, microstructure control and texture control into TMCP + AcC + (DQ). (e) Converting batch annealing into CAL. It is interesting to note that in the upstream processing, functions have been split, while in downstream processing they have been integrated, and in casting and annealing, they have been made continuous. Overall, optimization of the unit processes will continue in the future toward the two opposite directions but in integrated ways with emphasis always placed on continuous processing. The following is a perspective on possible future progress.
13.6.2 Future development of processes to reduce energy consumption/CO2 emission and deal with downgrading raw materials In raw materials and ironmaking, utilization of downgraded raw materials will proceed. To deal with the unfavorable raw materials conditions, we foresee the progress in: (a) Advanced agglomeration for sintering of iron ores containing more water and gangues. (b) Increased rate of charging soft coal and waste plastics into coke ovens. (c) Injection of soft coal up to 300 kg/t-hot metal and waste plastics into BFs to reduce coke (d) Combined operational efforts to decrease RAR in BFs. (e) Industrialization of new ironmaking processes to supplement the mainstay BF±BOF/scrap±EAF route at reduced occurrence of CO2 even by using downgraded materials. (f) Cooperation with customer industries to dismantle parts that contain tramp element metals and enhance the separation of tramp element metals from shredded scrap for recycling.
13.6.3 Future development of upstream processes to reduce impurities, inclusions and defects on semis for better material properties (a) Metallurgical reactors in the upstream of the steel manufacturing system will be developed toward better mixing of liquid metal with fluxes, aiming at shorter cycle time, lower heat loss and improved efficiency of flux utilization for decreased impurities and inclusions. The utilization ratio in units of the phosphate and sulfide capacities of the fluxes in current practice would be at best somewhere around 50% for a top slag operation, and much lower for an injection operation. In addition to process thermodynamic considerations, improvement of process kinetics is necessary to improve the
552
(b)
(c) (d) (e) (f)
(g)
Fundamentals of metallurgy utilization ratio. Shorter cycle time and reduced flux amount by the enhanced utilization and improved kinetics will reduce heat loss and increase productivity of both HMPT and secondary refining. That will make implementation of HMPT economical for better integration of the refining system in countries other than Japan and Korea (where HMPT for dephosphorization and desulfurization has been practiced). After the above has been achieved, the BOF will be shifted toward a high speed decarburizer with minimum occurrence of slag. Preliminary and finishing refining function is transferred to HMPT and secondary refining. It appears that better HMPT systems will emerge to allow increased use of scrap in BOFs. A variety of combinations of iron sources (including hot metal) with high performance EAFs will continue to be industrialized, supplemented with efficient scrap heating systems. Utilization of HMPT slag and BOF slag will be developed for better application. Improved melt delivery system from ladle to tundish to mold will be developed to avoid exogenous macro inclusions caused by reoxidation and slag entrainment. CC machines to cast high performance steels will be converted to those with vertical bending (progressive bending) profile. Optimization of electromagnetic flow control in mold will proceed to reduce subsurface inclusions, Ar bubbles, slag emulsions and enhance inclusion flotation. Macroscopic center segregation of solutes in semis has been practically reduced by the combination of EMS and soft reduction of the pool end, but more effective ways to reduce semi macroscopic center segregation of solutes will be developed.
13.6.4 Future development of downstream processes for better material properties (a) TMCP-AcC-(DQ) with microalloying elements will proceed further in a continuous way toward microstructure control for higher strength and ductility for plate steels for buildings, ships, and bridges. (b) Fine control of oxides or nitride/carbonitride precipitation will develop more through precise composition control in steelmaking followed by TMCP-AcC-(DQ) for microstructure control of HAZ for better weldability. (c) Fine graining of IF-, DP- and TRIP-steel strip to lower single 0016m sizes will be pursued by intercritical rolling and precipitates control with microalloying elements. (d) Above all, better clarification is necessary of microstructure control for achieving extremes of materials properties, in particular, changes in microstructure and corresponding properties during working.
Improving steelmaking and steel properties
553
Some examples related to the above issues for specific steel grades that represent a part of super performing steels are: (a) Process to prevent delamination of 5 GPa tire cord wire. (b) More advanced reliable minimization and control of non-metallic inclusions for longer rolling fatigue life of ball bearing steel, thermal fatigue life of turbine rotor steel for large scale power plants and better premium yield of PET laminated stretcher draw can sheet steel. (c) More precise alignment of Goss texture toward rolling direction of grain oriented 3% Si steel. Steel materials are often misunderstood to have matured in their properties and production processes. However, the processes and properties have been substantially improved by continuous efforts, yet leaving more room to advance in the future.
13.7 Further reading Readers are recommended to refer to the following articles for their basic understanding of the subjects referred to in this chapter. 1.
2.
3.
For steel production processes, (1) Steel Manual, (1992), Verlag Stahleisen, Dusseldorf; (2) Making, Shaping and Treating of Steel, Steelmaking and Refining Volume, 11th edn R. Fruehan (2001) AISE Steel Foundation, Washington, DC; (3) Making, Shaping and Treating of Steel, Casting Volume, 11th edn, A. Cramb (ed.), CD-ROM publication (2003) AISE Steel Foundation, Washington, DC; (4) B. Deo and R. Boom, Fundamentals of Steelmaking Metallurgy, 2nd edn (2005) Prentice Hall International, UK; (5) Advanced Physical Chemistry for Process Metallurgy, N. Sano, W.-K. Lu, P.V. Riboud (eds), Academic Press (1997). For steel product properties, (1) T. Gladman, The Physical Metallurgy of Microalloyed Steels, The Institute of Metals, London, UK, 1996; (2) W.C. Leslie, The Physical Metallurgy of Steels, Hemisphere Publishing Co., Washington, USA, 1981. For more details of each section, (1) Proceedings Internat. Symp. for LC and ULC Sheet Steels (1998); (2) Proceedings Internat. Conf. TRIP aided High Strength Ferrous Alloys (2002); (3) Thermec 88, Ed. I. Tamura, (1988, ISIJ, Tokyo); (4) Accelerated Cooling of Steel, Ed. P.D. Southwich, (1986, TMS-AIME, Warrendale).
13.8 References 1. AISI (Washington, DC), http://www.steel.org. 2. Y. Komiya, R & D Kobe Steel Engineering Report, 52(3), (Dec. 2002), 2.
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3. T. Obara and K. Sakata, Proceedings of 39th Mechanical Working and Steel Processing Congress, Indianapolis, ISS, Warrendale, (1997), 307. 4. H. Senuma and K. Kawasaki, ISIJ Internat., 34 (1994), 51. 5. K. Sakata, S. Matsuoka, T. Obara, K. Tsunoyama and S. Shiroishi, Materia, (JIM, Sendai), 36 (1997), 376. 6. F. Kitano, T. Urabe, T. Fujita, K. Nakajima and Y. Hosoya, ISIJ Internat., 41 (2001), 1402. 7. Z. Morita and T. Emi (eds), Introduction to Iron and Steel Processing, Kawatetsu 21st Century Foundation, Text, Tape and Transparency Package, Tokyo (1998). 8. C. Ouchi, ISIJ Internat., 41 (2001), 542. (Also, I. Kozasu, Controlled Rolling and Controlled Cooling, Chijin Shokan Co., Tokyo (1997).) 9. K. Oosawa, Tetsu-to-Hagane, ISIJ, Tokyo, 81 (1995), 449. 10. H. Ogawa, N. Kikuchi, T. Yamauchi and H. Nishikawa, ISSTech 2003 Conf. Proceedings, Pittsburgh, (2003), 335. 11. Y. Okada, K. Yada, T. Nagahata, K. Maya, H. Ikemiya, S. Fukagawa and K. Shinme, Tetsu-to-Hagane, ISIJ, Tokyo, 80 (1994), T9. 12. T. Uesugi, Tetsu-to-Hagane, ISIJ, Tokyo, 74 (1988), 1889. 13. K. Kume, K. Yonezawa, M. Yoshimi, H. Motowatari and M. Kumakura, CAMPISIJ, 16 (2003), 116. 14. T. Emi, T. and S. Seetharaman, Scand. J. Metall., 29 (2000), 185. 15. K. Ahlborg, R.J. Fruehan, M.S. Potter, S.R. Badger and G.S. Casuccio, ISSTech 2003 Conference Proceedings, Pittsburgh, (2003), 177. 16. J. Lehman, P. Rocabois and H. Gaye, J. Non-Cryst. Solids, 282 (2001), 61. 17. M. Okimori, Nippon Steel Technical Report, No. 361 (1996), 67. 18. J. Schade, R.J. O'Malley, F.L. Kemeny, Y. Sahai and D.J. Zacharias, Chapter 13 `Tundish Operations', in Making, Shaping and Treating of Steel, Casting Volume, 11th edn, A. Cramb (ed.), CD-ROM publication (2003). 19. K. Isobe, H. Maede, K. Syukuri, S. Sato, T. Horie, M. Nikaidou and I. Suzuki, Tetsuto-Hagane, 80 (1994), 42. 20. T. Emi, J. Korean Ceram. Soc., 40 (2003), 1141. 21. I. Ogata and M. Sanui, Ferrum, ISIJ, Tokyo, 8 (2003), 818. 22. M. Naito and S. Matsuzaki, CAMP-ISIJ, 17 (2004), 2. 23. T. Emi and O. Wijk, Steelmaking Conf. Proceedings, ISS, Warrendale, 76 (1996), 551. 24. P. Scaife, J. Nunn, A. Cottrell and L. Wibberley, ISIJ Internat., 42 (2002) S5. (Also, Report of ACARP Project C8049 Revision 2.00, February 29, (2000).) 25. N-S. Hur, Proceedings Internat. Symp. Global Environment and Steel Industry, CSM, Beijing, (2003), 50. 26. G. Denier, A. Kremer and J. L. Roth, 25th Symp. New Melting Technologies, St Petersbourg, FL, ISS, Warrendale, May 11±14, (1997). 27. J.K. Shoop, J.C. Simmons and J.M. McClelland, ISSTech 2003 Conf. Proceedings, Pittsburgh, (2003), 379. 28. O. Iimura, Proceedings Internat. Symp. Global Environment and Steel Industry, CSM, Beijing, (2003), 38 (originally from Japan Iron and Steel Federation). 29. T. Emi and D.-J. Min, Proceedings of 2nd Internat. Green Processing Conf., Freemantle, Australia, AIMM , May 11±14 (2004). 30. T. Emi, High Temp. Materials and Processes, 20(3-4), (2001), 167.
Index
absorption coefficients 145±6 estimation 165±8 absorptivity 212±14 accelerated cooling (AcC) 516±18 acid leaching 103±7 activation area 458 activation energy, falsification of 304±5 activity 59, 60 activity coefficients 61, 64, 65±6 additive reaction times, law of 298±304 additives 473 uniaxial pressing 487 adhesion, work of 134, 248, 249 adhesional wetting 249 adiabatic processes 44 adsorption 246±8, 271±2, 391 aerosol-based processing techniques 482±3 alloying elements, processes to optimise 524±9 alloys oxidation of 13 reactive/high requirement 25±7 thermal conductivity liquid alloys 162 mushy phase alloys 163 solid alloys 163 alteration curve 91, 92, 100, 101 alumina activity in slag models 76, 77, 379±80 CaO-Al2O3-SiO2 system 91±4 inclusions 429±30, 431, 529±31 spheroidised particles 430±1 aluminium 13 Al-Si binary system 102, 103 deoxidation using 374, 382±4, 429±30, 512±14 electrolytic production 317±18 Ni-Al system 151
annealing sector 536 anode 315 anode furnace (fire-refining furnace) 18±20 API5LX pipeline 518±20 Archimedean method 123 area viscosity (surface dilational viscosity) 247±8 argon injection 395 purging and dehydrogenation 386±7 Arrhenius equation 147±8, 154, 189 atomisation 28, 30±1, 477±82 attrition 472 attrition mills 474, 475, 476 austenitic stainless steel 459, 460 axial and radial flow methods 141 backmixing, degree of 337±40 Bain distortion 290 bake-hardenable interstitial-free (BH-IF) steel 507, 508±16 process development to produce BH-IF SEDDQ steel sheet 511±16 properties required for BH-IF steel sheet 508±11 Bakker equation 243±4 ball bearings 521±3 ball mills 474, 475, 476 basic oxygen furnace (BOF) 371, 511±12, 513 blast furnace-basic oxygen furnace (BF-BOF) route 504, 506, 538±9 processes to optimise impurities and alloying elements 524±5, 526±7 slag 547, 548±9 batch casting processes 28 Bauschinger effect 468 bed porosity 234±5
556
Index
belt furnaces 385±6 BergstroÈm model 455 binary systems 51, 52, 53 binary diffusion couple 279 diffusion in binary mixtures 272 solidification 95 binders 487 black bodies 213 blast furnace (BF) 4±5, 6, 178, 179, 370, 371 blast furnace-basic oxygen furnace (BF-BOF) route 504, 506, 538±9 combination of BF and EAF 541±4 slag 76, 77, 378±80, 547 blister copper (crude copper) 18 bloom caster 534±5 boiling point 251 bond energies 62 bottom-gas-injected solvent extraction process 331±40 bottom tapping 522, 523 boundary layer diffusion and vacuum degassing 25±6 flow over a flat plate 227±32 solidification of molten steel 266 boundary lines 90±1, 92, 96, 97, 100, 101 box annealing (BA) 508±9 Bragg's law 113 bridging oxygens 115, 116 bubble flotation 391±3 bubble injection 264 bubbles foreign particles and bubble defects 266 gas bubbles rising in metals 319±20 buffer zone 225±6 Burke-Plummer equation 235 calcium 13 Ca-S-O system 85±7 solidification refining of silicon 104±7 calcium oxide CaO-Al2O3±SiO2 system 91±4 CaO-SiO2 system 152 calorimetric analysis 362±3 calorimetry 136±7 CALPHAD approach 71±2 capillary methods 144 capillary reservoir method 148 capillary viscometers 128 capillary wave atomisation 478, 482 carbide inclusions 534±5 carbon Cr-C binary system 72
injection 390 reaction with oxygen on the surface of carbon 183±4 reduction of metal oxides by 4±10, 306±8 uphill diffusion 182 carbon dioxide emissions 549±50, 551 carbon monoxide, reduction of metal oxides by 4±10 carbonyl process 485 carburisation mass transfer 192±5 reactions 7 Carnot cycle 45 casting 27±31, 370, 373, 428±49 heat conduction in the mould controls heat transfer 29±30 heat transport across the mould-solid metal controls the heat transfer 30±1 integrated optimisation 536 interactions between mould and steel shell 447±9 precipitation of oxides 428±32 solidification of mould slags 432±9 surface of steel cast in an oscillating mould 439±42 undercooling and initial solidification 443±7 cathode 315 cells 424, 426±7 growth 427 central atom model 68±9 centrifugal atomisation 477, 478, 481±2 ceramic moulds 29±30 charcoal-water-gas system 392±3 chemical diffusion 146 chemical diffusion coefficient 279 chemical driving force of the cell reaction 79 chemical equilibrium 40, 44±8 effect on gas-solid reaction kinetics 304±5 chemical potentials 40, 59±60, 82 potential diagrams 83±7 chemical powder metallurgical processes 483±5 chemical reaction fluxes 183±4 chemical vapour decomposition (CVD) 485 chemical vapour synthesis 15±17 chill zone 439 Chilton-Coburn j factors 276±7 chlorides 15±16
Index chlorination of rutile 326±30 chromium 13 Cr-C binary system 72 Clapeyron equation 49±51 Clausius-Clapeyron equation 50±1 climb 455, 459 closed systems 39 `cloud' solidification structure 433, 434, 435 CMn-steel 460, 461 coal-based DRI processes 7±9 coalescence 256±7 coarsening 287, 407, 408, 429, 430, 493, 494 coking 13±14 cold isostatic pressing 489 cold plasmas 486 cold rolling 464±7 Coldstream process 477 columnar structures 433, 434 combined blowing converters 372 commercial materials 120 compaction 486±90 compensation effect 148 complete thermodynamic equilibrium 40 complex gas-solid reactions 305±11 components of a system 39 composite wall 207±8 compression 472 computerised fluid dynamic (CFD) models 386, 393 concentration gradient 182 concentration profiles 191±8 concentric cylinder method 124, 141 condensation 485 conduction 200±2, 273 conservation equation 204±11 steady state conduction 205±7 heat conduction in the mould 29±30 see also thermal conductivity configurational entropy 47 congruent solidification 95 conjugation line (tie±line) 88±91, 92 conservation of heat 204±11 heat loss through composite wall 207±8 steady state conduction 205±7 temperature distribution in a hollow cylinder 208±10 temperature distribution in a spherical shell 210±11 conservation of mass 190±8 diffusion in solids 192±5 evaporation of liquids 196±8 conservation of momentum 221±32
557
Consteel process 542, 543 contact angle 133±4, 248±50 container materials 121±2 containerless methods 121 contamination of powders 472, 475, 479 continuous annealing (CA) 509±10 continuous annealing line (CAL) 514, 515 continuous casting 28, 370, 373, 428, 437±9, 522, 523 surface of steel cast in an oscillating mould 428, 439±42 continuous stirred tank reactor (CSTR) 332±40 controlling reactions 306±8 convection 116, 117±19, 139, 200±1 heat and mass transfer 275±8 converter 370, 371±2 converting copper 17±20 sulphide 20, 321 cooling rate 445±7 copper 10 smelting and converting 17±20 Corex process 541 corrosion local corrosion of refractories at interfaces 260±3, 314 of zinc in a de-aerated environment 315±17 counter diffusion 186 critical free energy 403±4 critical nuclei, number of 256 critical radius for droplet formation 408 solidification 283, 403±4 critical velocity 266, 388 cross-slip 455, 459 crude steel production 370, 371±2 crystallisation curve 91, 92, 100, 101 crystals, specific heat of 351±2, 353 crystobollite 413, 414 cuspidine 435±7 cylindrical coordinate systems 182, 191±2, 205, 222, 223 evaporation of liquid 196±8 flow through a pipe 221±7 temperature distribution in a hollow cylinder 208±10 Dalton's law of partial pressures 54 Debye characteristic temperature 351±2 decarburisation 511±12 decomposition reactions 14±15
558
Index
deep drawability 508±16 defects in casting 110 point defect formation and elimination 278 reducing on semis 551±2 see also bubbles; impurities; inclusions deformation processing 32±4 degassing, vacuum 25±6 dehydrogenation 386±7 dendritic fragmentation 434, 435 dendrite morphology models 313 dendrites 417, 419, 424, 426±7 alumina 429, 430 growth 427 denitrogenation 391 densification 491, 492, 493 density 122±4, 126±7 estimation 151±2 models for calculating 152, 153 deoxidation 374 steel casting 428±32 using aluminium 374, 382±4, 429±30, 512±14 dependent variables 40±1 dephosphorisation 548 de-polymerisation, degree of 116, 175±6 desiliconisation 548 desorption 271±2, 391 desulphurisation 373, 520±1 ladle treatment 381±2 micro-modelling 393±6 detachment method 132 dew point detectors 364 dicalcium silicate 435±7 die casting 30±1 die pressing 487±9 differential power scanning calorimeter (DPSC) 137 differential scanning calorimetry (DSC) 137, 355, 360±1 differential temperature scanning calorimetry (DTSC) 137 differential thermal analysis (DTA) 355, 356, 358±60 calorimetric analysis 362±3 DSC and 360±1 diffraction 118 diffuse interfaces 413 diffusion in a binary mixture 272 grain-pellet system with intra-grain diffusion effect 302±3
homogenisation of compositional gradients 278±9 multicomponent 272±3 pore diffusion 274±5 in solids 179, 192±5 see also mass transfer diffusion coefficients 146±8 estimating 168±9 diffusion couple method 148 diffusional transformations 281±2 diffusivities 187±90, 272 gases 187 gases in porous solids 187±9 liquids 189±90 in solids 190 dilatometry 123 dilute solutions mass flux in 184 multicomponent 66±70 direct measurements for electrical resistivity 144 direct metal deposition (DMD) 500 direct metal laser sintering (DMLS) 500 direct reduction (DR) technology 6±9 direct strike precipitation 485 direct thermal methods 353±4 dislocations 454±7 disorder 47 dispersion 256±7, 258 disregistry 412 dividing surface, Gibbs' 238±9 position of 240±1 dolomite 14±15 double-action die pressing 488±9 double ultrasonic atomisation 482 downstream processes 505, 506 future development of 551, 552±3 see also under individual processes drag force 232±5 drainage 24±5 draining crucible method 121, 123, 125, 128 driving energy 257, 258 driving force 266 drop calorimetry 137, 362 drop weight method 132 droplets experiments and superheating 443±7 formation of 405±8 dry pressing 487±9 dry powder 3DP processes 499 DTG curves 357 dual-phase (DP) steel 507 DP 600 468
Index dynamic dynamic dynamic dynamic
compaction 490 interfacial tension 312±13 strain ageing 459, 460 techniques 119±20
economic optimisation 537±46 market and management 545±6 production and investment 544±5 raw materials and energy 538±44 eddies 217±18 effective diffusivity 187±9 Einstein's mass-energy relationship 42 ejection pressure 447 electromagnetic stirring (EMS) 534±5 electric arc furnaces (EAFs) 6, 76, 77, 370, 371, 372 combination of BF and EAF 541±4 scrap-EAF route 504, 506, 539±40 slag foaming 389±90 slags 378±80, 384, 547, 549 electric field assisted sintering (FAST) 497 electric ironmaking furnace (EIF) 542±4 electrical analogy for heat transfer 207, 216 electrical conductivity 169 electrical resistivity 126±7, 143±4 estimation 160±2 and thermal conductivity 138±9, 162 electro slag remelting 26 electrochemical method 148 electrochemical reactions 314±18 electrochemical thermodynamics 78, 79, 80 electrodes 315 electrolysis 479, 484 electrolysis cells 315, 316 electrolyte 315 Ellingham diagrams 54±6, 83 elongation 508 emanation thermal analysis (ETA) 355, 365 emissivity 126±7, 145, 212±14 empirical slag models 75±6, 376 emulsification 313 of steel in slag 388±9 energy developing steelmaking processes and properties 506±23 processes driven by properties, environment and energy 516±21 properties driven by market, environment and energy 507±16 first law of thermodynamics 41±2
559
raw materials and 538±44 reducing energy consumption 551 energy balance equation 204±11 enthalpy 43, 47, 58, 59, 126±7 DTA and 362 estimation 149±51 relative integral molar enthalpy 63 enthalpy of fusion 119±20, 126±7, 149±50 enthalpy interaction parameter 69 entropy 45, 48, 58, 59 and disorder 47 entropy interaction parameter 69 environment developing steelmaking processes and properties 506±23 processes driven by properties, environment and energy 516±21 properties driven by market, environment and energy 507±16 environmental optimisation 546±9 abatement of hazardous wastes 546 minimisation, recycling and reuse of wastes 546±9 equation of continuity 221±32 equation of motion 221±32 equiaxed crystals 433, 434 equilibrium constant 59±60 equilibrium partial pressure 406±7 equimolar counter diffusion 183±4 equivalent ionic fractions 73 Ergun's equation 235 Eucken's equation 203 eutectic diagrams 51, 53 eutectic point 91, 95, 96 eutectoid transformations 281±2 evaporation 447±8 conservation of mass 196±8 heat of evaporation 251 melt 319 evolved gas analysis (EGA) 355, 356, 363±5 evolved gas detection (EGD) 355, 363±5 excess properties 61±2 excess stabilities 65±6 exogenous oxide inclusions 531±3 expendable moulds 28, 29±30 exploding wire method 119, 121, 123, 137, 144 explosive compaction 490 extensive properties 40 extinction coefficients 145±6 extra deep drawing quality (EDDQ) steel 509±10 SEDDQ steel 510±16
560
Index
extrusion 34, 490 Eyring relation 169 faceted crystals 433, 434±5 falling ball method 124 Fasteel process 542±4 Fastmelt process 542 Fastmet process 8, 542, 549 ferrite 458±9 ferritic perlitic steels 456 ferritic steels 462 Fick's laws of diffusion first law 146±7, 180, 182, 272, 278 second law 146±7, 193, 278, 279 filled billet technique 496 film theory 198, 199 fine powders 480 fine superalloy powder 481 Finex process 541 FINMET-EAF 451 fire-refining furnace 18±20 first law of thermodynamics 41±4 flame ionisation detectors 364 flash smelting process 322±6 flat grains 297 flat interfaces 413 flat plate, flow over 227±32 floating die presses 488±9 Flood et al.'s ion activity concept 73 flotation coefficient 134, 392±3 flow stress 456±7, 460, 461 fluid flow 217±35 conservation of momentum 221±32 flow over a flat plate 227±32 flow through a pipe 221±7 friction factor and drag coefficient 232±5 Newton's law of viscosity 218±20 properties related to 113, 122±36 viscosity of gases 220 viscosity of liquids 221 fluidised-bed processes 8, 9 fluidised-bed reactors for gas-solid reactions 326±30 foaming 264±5, 320±1, 389±90 forced convection 201 foreign particles 266±7 forging 32, 33 four-phase equilibria 95±101 Fourier Transformed Infra Red (FTIR) method 365 Fourier's law of heat conduction 201±2, 273 fraction solidified 437, 438
free energy change 352 and nucleation 402±4 free oxygens 115 Freundlich-Ostwald equation 252 friction factor 232±5 fugacity 54 fused deposition modelling (FDM) 498±9 galena 302 galvanic cells 315, 316 gas analysers 364 gas atomisation 477, 478, 479, 480±1 gas-based DRI processes 7±9 gas chromatography 365 gas-liquid reactions 318±21 gas pipelines 516±20, 521 gas purging 319±20 gas-solid reactions high-temperature processes 4±17 kinetics 290±311 complex 305±11 initially non-porous solid producing a porous product layer 291±4 porous solids 294±311 modelling fluidised-bed reactors 326±30 simultaneous 310 successive 309±10 gas tungsten arc (GTA) welding 109 gases diffusion of binary mixtures 184±5 diffusivity 187 diffusivity in porous solids 187±9 gas mixtures 54 heat exchange by radiation and 217 oxide reduction to produce powders 483±4 precipitation of powders from 485 rate of heterogeneous reaction between gas and metal 263±4 reaction of a porous solid with a gas accompanied by a volume change in the gas phase 303±4 thermal conductivity 203 viscosity of 220 geometrical slip distance 455, 456±7 Gibbs' adsorption equation 246±8 Gibbs' dividing surface 238±9, 240±1 Gibbs-Duhem equation 58 Gibbs energy 45 ideal mixing of silicates 74 integral and partial molar 57±8 relative integral and partial 58±9 unary systems 49, 50
Index variation with pressure and temperature 46 Gibbs-Helmholtz equation 46 Gibbs-Marangoni viscosity 248 Gibbs phase rule 51±4 Gibbs-Thompson effect 285, 406 Gibbs triangle 87±8, 89 Giddings equation for diffusivity of gases 187 glass transition temperature 150, 410, 411 glasses 122 structure 115±17 viscosity 125, 154, 157 Good-Girafalco equation 134 Graham's law 183 grain-pellet system with intra±grain diffusion effect 302±3 grain refined and precipitation hardened SEDDQ sheet 511 grains evolution of 280±1 grain model of porous solid 296±8 grey bodies 213 heat exchange between 215±17 grinding 472±3 growth 284±6, 413±28 combining nucleation and growth kinetics 286, 287 gas-solid reactions 300±2 heat transfer dominated growth rates 415±25 interface dominated growth rates 413±15 mass transfer dominated growth rates 425±8 solidification of mould slags 435, 436 Guggenheim's equation 403 Gulliver Scheil equation 426 haematite 4 Hagen-Poiseuille law 224±5 hazardous wastes, abatement of 546 heat balance equation 204±11 heat capacity 126±7, 136±8 estimation 149±51 heat of evaporation 251 heat flux conduction 201±2 heat loss through composite wall 207±8 steady state heat conduction 205±7 heat flux mode DSC 361 heat generation 205 heat transfer 40, 200±17, 290±1 casting moulds 28±31
561
conduction see conduction conservation equation 204±11 convection see convection growth rates dominated by 415±25 heterogeneous kinetics 272±8 properties related to 113, 136±46 radiation 200±1, 212±17, 277±8 resistance to 423±4, 437±8 steelmaking process design 385±7 thermal conductivity see thermal conductivity heat transfer coefficients 31, 211±12, 423 heat treatment sector 536 heating, phase transformation upon 286 Helmholtz energy 45 Henry's law 63±4, 66 Hess's law of constant heat summation 42±3 heterogeneous kinetics 270±8 adsorption and desorption 271±2 heat and mass transfer 272±8 rate of reaction between gas and metal or slag and metal 263±4 see also kinetics of metallurgical reactions heterogeneous nucleation 255, 284, 402, 404±5 nucleation rate 411±12 heterogeneous systems 39 high-energy rate compaction 490 high manganese steels 448 high performance steel products 549±50 high requirement/reactive alloys 25±7 high strength low alloy (HSLA) steel 462, 516 high-temperature processes 3±37 casting processes 27±31 heat conduction in the mould 29±30 heat transport across the mould 30±1 reactions involving gases and solids 4±17 chemical vapour synthesis of metallic and intermetallic powders 15±17 coking 13±14 decomposition reactions 14±15 oxidation 10±13 reduction of metal oxides 4±10 reactions involving liquid phases 17±27 processes for reactive/highrequirement alloys 25±7 slag refining 21±5 smelting and converting 17±20 thermomechanical processes 31±4
562
Index
deformation processing 32±4 Hill yield criterion 465 HIsmelt 541 HMPT 520, 524±7, 552 incorporation into BF±BOF route 548 slag 547, 548±9 hollow cylinder 208±10 hollow spherical shell 210±11 homogeneous nucleation 253±5, 282±4, 402±4 nucleation rate 408±11 homogeneous systems 39 homogenisation of compositional gradients 278±9 horizontal casting 447 horizontal mills 474, 475, 476 attritor 474, 476 ball mill 474, 476 hot extrusion 496 hot forging 496±7 hot gas atomisation 481 hot isostatic pressure (HIP) 496 hot metal charging 541±2 hot pressing 495 hot rolling 463±4, 465, 510±11 hydrated barium hydroxide 364 hydraulic radius 232 hydride-dehydride process 485 hydrogen reduction of metal oxides by 4±10 vapour-phase reduction of metal chlorides 15±16 hydrogen induced cracking (HIC) 519±20 hydrogen sulphide 519 hydrophilic non-solvent addition 484 hydrostatic probe 123 HYL process 8, 9 hypercooling 417 hysteresis of wetting 250 ice calorimetry 362 ideal mixing 74, 376±7 ideal-reactor-network model 332±40 ideal solutions 60±1, 62 immersional wetting 249 impaction 472 impurities 21±2 future development of processes to reduce 551±2 processes to optimise 524±9, 530 removal for reactive/high requirement alloys 25±7 slag refining 21±5 impurity diffusion 146
inclusions 22, 266±7 critical inclusion sizes 525 effect of interfacial energy on separation of inclusions by bubble flotation 391±3 future development of processes to reduce 551±2 oxide inclusions in ball bearings 522±3 precipitation of oxides in steel casting 428±32 processes for controlling 529±35 removal 23±5 incongruent solidification 95 independent variables 40±1 indigenous oxide inclusions 529±31 indirect methods for electrical resistivity 144±6 indirect precipitation 484±5 indirect rapid prototyping 498 inert gas condensation (IGC) 483, 485 inert gas purging 320, 386±7 inert species, flux of 184 infiltration 497±8 ingot casting 370, 373, 428, 447 ink jet printing (IJP) 499 inner state variable 41, 42 instantaneous plane method 148 integral molar properties 57±8, 60 relative 58±9, 61, 63 integrated iron and steel plants 528, 529, 539, 544 integrated optimisation processes 535±7 intensive properties 40 interaction parameters 68±9 estimations 69 and solubility of oxides in metallic melts 70 interdiffusivity 146, 190 interface instability 426±7 interface temperature 415±18 interfacial phenomena 117, 237±69 fundamentals of the interface 238±57 mechanical aspects of surface tension 243±6 physical chemistry 246±57 thermodynamics 238±43 interface dominated growth rates 413±15 mass transfer 198±200 metallurgical melts system 257±60 interfacial tension between slag and metal 259±60, 312±13 surface tension 257±9
Index wetting of ceramics by liquid metal and slag at high temperature 260 metallurgical processing and 260±7 bubble injection 264 interaction of foreign particles with solid-liquid interface 266±7 local corrosion of refractories 260±3 penetration of slag or metal into refractory 265±6 rate of heterogeneous reaction 263±4 optimisation of interfacial reactions in steelmaking 387±93 slag foaming 264±5, 320±1, 389±90 slag-metal mixing 388±9, 394±6 interfacial tension 117, 129, 133±6, 248±50 estimation 163 slag-metal 259±60, 312±13 intermediate zone 225±6 intermetallic powders 15±17 interrupted accelerated cooling (IAC) 516±18 intra-grain diffusion effect 302±3 intrinsic diffusivity 190 inverse rate cooling 354 inverse thermal methods 353±4 investment 544±5 investment casting 29±30 ion activities 73 ionic melts 72±9 KTH model 76, 77, 377±8, 384 Lumsden's description of silicates 75 Richardson's theory of ideal mixing of silicates 74 slag capacities 23, 76±9, 381±2 slag models 75±6, 77, 376±80 Temkin's and Flood et al.'s description 73 iron direct reduced iron (DRI) 6±9 removal from silicon by acid leakage 104±7 iron oxides phase diagrams 83, 84, 85 reduction 4±9 iron silicide phase 104±7 irregular powders 472, 479, 480 irreversible processes 44±5 IRSID model 76, 376 isolated systems 39 isostatic pressing 489 isothermal cross-section 88±91, 92 isothermal processes 44 isothermal thermogravimetry 358
563
isothermal transformations 352, 353 isotropic materials 202 ITmk3 process 8, 541 Jacob-Alcock empirical equation 69 Japanese steel industry 547, 549, 550 Johnson-Mehl-Avrami equation 286, 414 Kelvin equation 244, 251, 406±7, 408 kinetics of metallurgical reactions 270±349 comprehensive process modelling 321±40 bottom-gas-injected solvent extraction process 331±40 flash smelting process 322±6 fluidised-bed reactors for gas±solid reactions 326±30 gas-liquid reactions 318±21 gas-solid reactions 290±311 initially non-porous solid producing a porous product layer 291±4 reaction of a porous solid 294±311 heterogeneous kinetics 270±8 adsorption and desorption reactions 271±2 heat and mass transfer 272±8 kinetic block in micro±modelling 394±5 liquid-liquid reactions 311±13 solid-liquid reactions 313±18 solid-state reactions 278±90 multiphase reactions 281±90 single phase reactions 278±81 steelmaking process design 385±7 Kirchhoff's law 214 Kirkendall shifts 279 Knudsen diffusion 188±9 Kolmogorov-Johnson-Mehl-Avrami (KJMA) equation 414 Kozeny-Carman equation 235 KTH model 76, 77, 377±8, 384 ladle degassing 522 ladle furnaces 76, 77, 527±9 slags 378±80, 547, 549 ladle treatment 373±5 dehydrogenation 386±7 desulphurisation 381±2 micro-modelling of sulphur refining 393±6 thermodynamics and mass balance 382±4 laminar flow 217±18
564
Index
in a pipe 221±7 laminar sub-layer (viscous sub-layer) 225±6 Langmuir-Hinshelwood rate equation 271, 294 Laplace's equation 244±5 large drop method 123, 131 laser cladding (LC) 500 laser engineered net shaping (LENS) 500 laser powder deposition processes 498, 500 laser pulse method 121, 141±2 latent heat 352±3 lattice disregistry 412 Laval nozzle 481 law of additive reaction times 298±304 lead smelting 20, 375 leakage rates 186 ledge mechanism 285 levelling 467±8 levitated drop calorimetry 137 levitated drop method 123 levitation 121 lifecycle analysis 549±50 lime 308±10 limestone, decomposition of 14 limiting drawing ratio (LDR) 508 liquid alloys, thermal conductivity of 162 liquid-gas reactions 318±21 liquid-liquid reactions 311±13 liquid metal solution calorimetry 362±3 liquid metals interfacial properties of a metallurgical melts system 257±60 thermodynamic description for steelmaking 376, 380 liquid phase sintering 493±4 liquid-solid reactions see solid-liquid reactions liquids diffusivity in 189±90 evaporation 196±8 high-temperature processes involving liquid reactions 17±27 nucleation at liquid-liquid interface 405 thermal conductivity 203 viscosity of 220 local solidification time 110, 112 logarithmic rate of oxidation 12 lubricants 487 Lumsden's description of silicates 75 macro inclusions 531±3 magnesite 14
magnesium 10 reduction of metal chlorides 16±17 magnesium carbonate 14±15 magnesium oxide erosion of refractories 314 reduction of 10 magnesium sulphide 519 management, market and 545±6 management system for integrated processes 537 manganese oxide 448 Marangoni effect 245±6 Marangoni flows 245±6, 261±3, 314 Marangoni viscosity 248 marginal stability criterion 426±8 market developing steelmaking processes and properties 506±23 properties driven by market, environment and energy 507±16 and management 545±6 martensite formation 456, 457 martensitic steel, tempered 507 martensitic transformations 290 mass balance 190±4 constraints and steelmaking 381±2 ladle treatment 382±4 sub-processes and 374 mass flux 180±6 components of 182 relation between fluxes 183±6 mass spectroscopy (MS) 365 mass transfer 40, 178±200, 211±12 conservation of mass 190±8 diffusivity of gases, solids and liquids 187±90 gas purging 319±20 gas-solid reactions 290±1, 304 growth rate dominated by 425±8 heterogeneous kinetics 272±8 high temperature oxidation 11±12 interface mass transfer 198±200 mass flux 180±6 properties related to 113, 146±8 steelmaking process design 385±7 mass transfer coefficient 198±9 correlations for 199±200 matte smelting (mattemaking) 17±18 maximum bubble pressure (MBP) method 123, 132 maximum drop pressure (MDP) method 135±6 Maxwell's relations 46 mechanical alloying (MA) 472, 473±4, 479
Index mechanical equilibrium 40 mechanical milling 473 mechanical powder production processes 472±7 medium carbon steels 439, 440 melt evaporation 319 melt reoxidation 318±19 melting point 252 meniscus solidification 439±42 metal injection moulding (MIM) 489±90 metal working processes 453±70 development trends 468±9 examples of material behaviour during processing 463±8 cold rolling 464±7 hot rolling 463±4 levelling 467±8 skin-pass rolling 467 interaction with phase transformations 462±3 rate effects 457±61 work hardening 32, 454±7 metallic systems, thermodynamic modelling of 70±2 metallurgical process modelling 321±40 bottom-gas-injected solvent extraction processes 331±40 flash smelting process 322±6 fluidised-bed reactors for gas-solid reactions 326±30 metals 122 electrical resistivity 160±2 surface tension 129±33 thermal conductivity 162±3 viscosity 125±8, 159, 160 metastable phases 408±12, 474 method of mixtures 362 micro-gravity 121 micro-modelling 393±6 microporosity models 313 microwave sintering 494±5 MIDREX 8, 9 MIDREX-EAF 541 milling 479 mechanical 473 technologies 474±7 mixer-settler systems 331, 340 mixing 57 Mizukami's technique 443, 444 molar concentration 271±2 molar heat capacity 43±4, 350 molar rate of consumption of fluid reactant per unit area of reaction interface 271
565
molar volume 151±2 relation of diffusion constants to 168 molecular dynamics (MD) 118 molybdenum 10 momentum, conservation of 221±32 momentum flux 219±20 Mond process 9 mould slags, solidification of 432±9 moulds 27±31 cooling against a mould 419±25 expendable ceramic 29±30 heat transport across the mould-solid metal 30±1 interactions between mould and steel shell 447±9 oscillating 28, 428, 439±42 rotating 26, 428 see also casting multicomponent diffusion 272±3 multicomponent systems 51±7 multicomponent dilute solutions 66±70 central atom description 68±9 estimation of interaction parameters 69 interaction parameters and solubility of oxides in metallic melts 70 Wagner's equation 66±7 multiparticle systems 310±11 multiphase jet solidification (MJS) 499 multiphase materials, work hardening of 455±7 multiphase reactions 281±90 coarsening 287 combining nucleation and growth kinetics 286, 287 diffusional transformations 281±2 growth 284±6 martensitic transformations 290 nucleation 282±4 ordering 289±90 phase transformation upon heating 286±7 precipitation of multiple phases 435±7 spinodal decomposition 288±9 mushy zone 120, 139, 163, 424 mutual (inter) diffusivity 146, 190 nanoparticles 486 natural convection 201, 276 Navier-Stokes equation 221, 222±3 near-net shape forming processes 486±500 compaction 486±90 electric field assisted sintering 497
566
Index
hot extrusion 496 hot forging 496±7 hot isostatic processing 496 hot pressing 495 infiltration 497±8 rapid prototyping 498±500 sintering 490±5 negative strip time 439±42 Nernst-Einstein equation 169 Nernst equation 79, 315 new iron and steelmaking processes 540±1 Newmann Kopp rule 48 Newton's law 233 Newton's law of viscosity 218±20 nickel 9, 13 Ni-Al system 151 smelting 20 nickel-based superalloys 119 nickel carbonyl 15 nickel sulphide 56, 57 nitride inclusions 534±5 nitrogen 263±4 non-bridging oxygens (NBO) 115, 116 NBO/T 116, 175 non-equilibrium solidification 98±9, 100 nucleation rate and 408±12 non-isothermal transformations 352, 353, 354 non-isotropic materials 202 non-metallic inclusions see inclusions normal solidification 419±20 normalising process 516 nozzle clogging 266±7 nozzle design 480±1 nucleation 290, 402±5 effects of size 405±8 gas-solid reactions 300±2 kinetics 256, 282±4 combining with growth kinetics 286, 287 rate 284 alumina formation 429 and formation of non-equilibrium solids 408±12 subgrains and recrystallisation 460 thermodynamics of 253±5 Nusselt number 211±12 oil atomisation 477, 478, 480 open systems 39 optical basicity 116, 175±6 optical properties 145±6 estimation 165±8
optical thickness 140±1 ordering 289±90 oscillating drop method 128, 132 oscillating plate method 125 oscillating viscometers 128 oscillation marks 439±42 oscillation mould casters 28, 428 surface of steel 428, 439±42 Ostwald ripening 252, 253, 407, 493, 494 overflow oscillation marks 439±42 overpotentials 316 oxidation 315 high temperature 10±13 of metal sulphides with lime in the presence of water vapour 308±9 oxide-dispersion strengthened (ODS) materials 474 oxide-graphite refractories 262±3 oxide inclusions 529±33 ball bearings 522±3 exogenous 531±3 indigenous 529±31 oxide metallurgy 431±2 oxide scale 11±13 oxides Ellingham diagrams 54±6, 83 precipitation in steel casting 428±32 reduction of 4±10, 306±8 and powder production 483±4 representation of ternary oxide systems 91±4 solubility in metallic melts 70 oxygen 264 bridging oxygens 115, 116 Ca-S-O system 85±7 free oxygens 115 injection 390 and measurement of surface tension 129±31 non-bridging oxygens see non-bridging oxygens reaction with carbon on the surface of carbon 183±4 packed bed 234±5 pair distribution factor 113±15 parabolic time dependence 11±12 paraboloid 427, 428 parallel plate method 141 partial molar method 149, 150 partial molar properties 57±8, 60 relative 58±9 partial pressure 271 Dalton's law of partial pressures 54
Index equilibrium partial pressure 406±7 particle pushing models 313 particle shape 471±2, 478±9 particle size 234±5 Peierls-Nabarro barrier 458 pelletising 370, 371 pendent drop method 121, 131±2 peritectic diagrams 51, 53 peritectic point 91 peritectic steels 439, 440 phase diagrams 51, 52, 53, 82±108 CALPHAD approach 71±2 and potential diagrams 83±7 solidification 95±107, 400±2, 408 examples of solidification behaviour from a phase diagram perspective 102±7 ternary systems and four-phase equilibria 95±101 ternary phase diagrams 87±94 phase rule 252±3 phase stability diagrams 56, 57 phase transformations 281±90 coarsening 287 combining nucleation and growth kinetics 286, 287 diffusional 281±2 growth 284±6 upon heating 286 hot deformation and 462±3 martensitic 290 nucleation 282±4 ordering 289±90 specific heat of crystals and 351±2, 353 spinodal decomposition 288±9 in ternary systems 99±101 see also phase diagrams phases 39 phenomenon block of micro-modelling 394, 396 phosphate capacity of a slag 23 physical properties at high temperatures see thermophysical properties physical property block in micromodelling 394 physical vapour deposition (PVD) 483 Pilling-Bedworth ratio (PB ratio) 12±13 pipe, flow through a 221±7 pipelines 516±20, 521 plane temperature wave (PTW) method 142±3 planetary ball mills 474, 475, 476
567
plasma techniques 485±6 plastic strain ratio 508 plasticisers 487 point defects 278 polymerisation, degree of 116 polymers 365 pores 22 solid state sintering 492±3 porosity, bed 234±5 porous pellets 273±4, 310 grain-pellet system with intra-grain diffusion effect 302±3 porous product layer 291±4 porous purging plugs 265±6 porous solids 187±9 heat and mass transfer 273±5 kinetics of gas-solid reactions 294±311 complex gas-solid reactions 305±11 effect of chemical equilibrium on and falsification of activation energy 304±5 grain model 296±8 law of additive reaction times 298±304 nucleation-and-growth kinetics 300±2 potentials 40, 59±60, 82 potential diagrams 83±7 powder metallurgy (PM) 471±502 chemical vapour synthesis of metallic and intermetallic powders 15±17 near-net shape forming processes 486±500 compaction 486±90 electric field assisted sintering 497 hot extrusion 496 hot forging 496±7 hot isostatic processing 496 hot pressing 495 infiltration 497±8 rapid prototyping 498±500 sintering 490±5 production processes for powders 471±86 aerosol routes 482±3 atomisation 28, 30±1, 477±82 chemical routes 483±5 mechanical routes 472±7 physical routes 483 plasma techniques 485±6 powder rolling 490 power compensation DSC 360±1 Prandtl number 211±12 precipitation 400, 401, 402, 462, 479
568
Index
of oxides in steel casting 428±32 of powders from a gas 485 of powders from a solution 478, 484±5 precipitation from homogeneous solution (PFHS) 484 precipitation hardened steel 507 precipitation transformations 281±2 predominance area diagrams (phase stability diagrams) 56, 57 primary crystals 96, 98 primary phase fields 96 process control agents (PCAs) 473, 474 process control systems 468±9 process design 369±98 kinetics 385±7 mass balance constraints 381±2 micro-modelling 393±6 optimisation of interfacial reactions 387±93 denitrogenation 391 foaming 389±90 separation of inclusions by bubble flotation 391±3 slag-metal mixing 388±9 overview of 369±75 sub-process 373±5 whole process 369±73 thermodynamic description 375±80 thermodynamics and mass balance in ladle treatment 382±4 process integration integrated iron and steel plants 528, 529, 539, 544 integrated optimisation of processes 535±7 processing routes 369, 370 production, and investment 544±5 production line 369±73 protective oxide scale 12±13 pseudo-binary phase diagram 89, 91 pseudo-steady-state approximation 294 pycnometry 123 pyrometer 353 quality 537 radial distribution factor (rdf) 114, 115 radial temperature wave (RTW) method 142 radiation 200±1, 212±17, 277±8 emissivity and absorptivity 126±7, 145±6, 212±14 heat exchange between grey bodies 215±17
view factors 214±15, 277 radiation conductivity 139±41, 143 radiosity 215 radius of droplets 405±8 and growth rate 418 radius of curvature 241±2 influence of 250±3 Ranz-Marshall correlation heat transfer 211, 276 mass transfer 199±200, 276 Raoult's law 60±1, 63 rapid prototyping (RP) 498±500 rate effects 457±61 low to ambient temperatures 458±9 warm to hot working temperatures 459±61 raw materials 538±44 downgraded 551 Rayleigh scattering 148 reaction rates 375 reaction sintering 494 reactive/high-requirement alloys 25±7 reactive wetting 133 reactivity of sample 120±2 rebound force 24 reciprocity relation 215, 216, 277 recrystallisation 459±61 rectangular coordinate systems 182, 191±2, 205, 222 steady state heat conduction 205±7 recycling of wastes 546±9 reducing agent rate (RAR) 538±9 reduction 314±15 of metal oxides 4±10, 306±8 reference states for thermodynamic properties 47±8 refining 535±6 slag 21±5 solidification refining 102±7 zone refining 26±7 reflectance methods 146 reforming reactions 7 refractive indices 168 refractories local corrosion at slag-gas and slagmetal interfaces 260±3 metal-melt refractory reactions 314 penetration of slag or metal into 265±6 regular solution model 63, 377 relative integral molar properties 58±9, 61, 63 relative partial molar properties 58±9 reoxidation, melt 318±19
Index research and development (R&D) 546 residence-time distribution (RTD) 332±40 resistances to heat transfer 423±4, 437±8 reuse of wastes 546±9 reversible processes 44±5 Reynolds number 199, 225, 230, 232 Richard's rule 47 Richardson's theory of ideal mixing of silicates 74 roll compacting 490 rolling 32, 33, 536 rotating bob method 124 rotating crucible method 124 rotating electrode process (REP) 481±2 rotating magnetic field method 144 rotating mould casters 28, 428 roughness of cast surface 444, 445 Ruhr Stahl-Hausen (RH) process 509±10, 512, 514, 527, 529 desulphurisation of steel in RH vessel 520±1 and oxide inclusions 522±3 rupture 24±5 rutile, chlorination of 326±30 sand casting 29±30 Schmidt number 199 scrap hot metal vs in economic optimisation 538±40 recycling 370, 372 scrap-EAF route 504, 506, 539±40 scrap-EAF-thin slab CC-Hot Rolling Mill 545 second law of thermodynamics 44±5 secondary dendrite arm spacing 445±6 secondary refining furnaces 504, 506 secondary refining processes 370, 372±3, 523, 524 optimisation of impurities and 527±9 secondary solidification 96, 98, 99 segregation 426 control of inclusions 534±5 selective laser melting (SLM) 500 selective laser sintering (SLS) 499 self diffusion 146, 168 semi-empirical slag models 75±6, 376 sessile drop method 123, 131, 135 shaft furnaces 8, 9 shape casting 29±30, 31 see also near-net shape forming processes shape changes: deformation processing 31, 32±4
569
shear 472 shear method 148 shear stress 219 shell, steel 28±9, 30±1 heat-transfer dominated growth and mould-shell interface 419±24 interactions between mould and 447±9 Sherritt Gordon process 484 Sherwood number 199±200 shrinking-core reaction system 291±4 Shuttleworth's equation 242±3 Sievert's law 63±4 silica CaO-Al2O3-SiO2 system 91±4 CaO-SiO2 system 152 silicates 73, 377 indigenous oxide inclusions 529±31 Lumsden's description of 75 Richardson's theory of ideal mixing of 74 see also ionic melts silicon 102±3 Al-Si system 102, 103 reduction to minimise BOF slag 548 segregation coefficients of impurities in 103 solidification behaviour of a ternary silicon-based alloy 103±7 silicon oxide/metal oxide ratio 22 similarity transform 194 simultaneous gas-solid reactions 310 single-action die pressing 488 single pan calorimeter 137 single phase materials, work hardening of 454±5 single phase reactions 278±81 sintering 370, 371, 490±5 in a belt furnace 385±6 mechanisms 491, 492, 493 parameters 491 size effects, and solidification 405±8 skin-pass rolling 467 slag 122 compositions in different processes 378±80 electro slag remelting 26 interfacial tension between slag and metal 259±60, 312±13 local corrosion of refractories at slaggas and slag-metal interfaces 260±3 minimisation, recycling and reuse of wastes 546±9 penetration into refractory 265±6
570
Index
rate of heterogeneous reaction between slag and metal 263±4 reactions between molten metals and slags 312±13 refining 21±5 solidification of mould slags 432±9 structure 115±17 surface tension 129±33, 165, 166±7 thermal conductivity 163 viscosity 124±5, 154, 155±7, 160 wetting of ceramics at high temperature 260 slag capacities 23, 76±9, 381±2 slag foaming 264±5, 320±1, 389±90 slag line attack 260±3, 314 slag meniscus (slag film) 261±2 slag-metal mixing 388±9 micro-modelling 394±6 slag models 75±6, 77, 376±80 smelting 17±20 of lead 20, 375 modelling the flash smelting process 322±6 sulphide smelting and converting reactions 321 soda-iron smelting system 375 SOHNEX process 331±40 solid alloys, thermal conductivity of 163 solid-gas reactions see gas-solid reactions solid-liquid reactions application of law of additive reaction times 302 kinetics 313±18 electrochemical reactions 314±18 metal-melt refractory reactions 314 solidification 313 solid-solid reactions proceeding through gaseous intermediaries with a net production of gases 306±8 proceeding through gaseous intermediaries with no net production of gases 308±9 solid-solution hardened steel 507 solid-state reactions 278±90 multiphase 281±90 single phase 278±81 solid state sintering 492±3 solidification 27±8, 313, 399±452 casting of steels 428±49 interactions between mould and steel shell 447±9 precipitation of oxides 428±32 solidification of mould slags 432±9
surface of steel cast in an oscillating mould 428, 439±42 undercooling and initial solidification 443±7 experimental observations 399±400 fundamentals 400±12 conditions necessary 400±1, 402 effects of size 405±8 nucleation rate and formation of non-equilibrium solids 408±12 thermodynamics 401±5 growth of solids 413±28 heat transfer dominated growth rates 415±25 interface dominated growth rates 413±15 mass transfer dominated growth rates 425±8 heat transfer resistances 28±9 phase diagrams see phase diagrams solidification refining 102±7 solids diffusion in 179, 192±5 diffusivities in 190 thermal conductivity 203±4 solubility 252, 253 change during solidification for impurities 21 soluble-gas atomisation (vacuum atomisation) 477, 478, 481 solutions ideal 60±1, 62 mass flux in dilute solutions 184 precipitation of powders from 478, 484±5 regular 63, 377 solution models for liquid metals 376, 380 see also slag models thermodynamics of see thermodynamics solvent extraction 331±40 solvent removal (sol-gel) precipitation 484 Soret diffusion 146 `spark' sintering 497 specific heat 350, 354 of crystals and phase transition 351±2, 353 DTA and 362±3 spectral emissivity 145 spectroscopy 118 sphalerite 9 sphere, flow around a 232±3
Index spherical atomised powders 471±2, 479, 480 spherical coordinate systems 182, 191±2, 205, 222, 223 temperature distribution in a spherical shell 210±11 spinel inclusions 523 spinodal decomposition 288±9 sponge iron 6±9 sponge powders (oxide-reduced powders) 483±4 spray column contactors 331 spreading coefficient 134, 248, 249 spreading wetting 249 stabilities 65±6 stability diagrams 56, 57, 79, 80 stagnant film 198, 199 standard electrode potentials 78, 79 standard states 64 state 39 state properties 40±1 steady state techniques 141 steel casting see casting steelmaking processes and properties 503±54 developing with reference to market, energy and environment 506±23 processes driven by properties, environment and energy 516±21 properties driven by market, environment and energy 507±16 properties driven by processes 521±3 economic optimisation 537±46 market and management 545±6 production and investment 544±5 raw materials and energy 538±44 environmental optimisation 546±50 abatement of hazardous wastes 546 lifetime analysis of steels 549±50 minimisation, recycling and reuse of wastes 546±9 future trends 550±3 downstream processes for better material properties 552±3 energy consumption, carbon dioxide emissions and downgrading raw materials 551 general trend for processes and properties 550±1 upstream processes to reduce impurities, inclusions and defects 551±2 optimisation of processes to meet properties and productivity 523±37
571
controlling inclusions 529±35 integrated optimisation of refining, casting, rolling and heat treatment 535±7 optimising impurities and alloying elements 524±9, 530 overview of process 504±5 process design see process design Stefan-Boltzmann equation 214, 277 Stefan number 417 Stefan's law 423 stereolithography (SL or SLA) 498 Stokes-Einstein equation 168±9 Stokes law 233 strain rates 453 rate effects 457±61 stretch formability 508 strip casting 30±1 strontium carbonate 15 structural slag models 75±6, 376 structure methods of determining 117, 118 and physical properties 113±17, 118 subgrains 460 sub-processes 373±5 successive gas-solid reactions 309±10 sulphate decomposition 15 sulphide capacity 23, 76±9, 381±2 sulphides decomposition 15 hydrogen reduction of in the presence of lime 309±10 inclusions 534±5 modelling the flash smelting process 322±6 oxidation with lime in the presence of water vapour 308±9 smelting and converting reactions 321 sulphur Ca-S-O system 85±7 effect on steel's surface tension 109±10, 111 refining in ladle treatment 393±6 in slag 22±3 see also desulphurisation sulphur partition ratio 381±2 sulphurisation 389 super extra deep drawing quality (SEDDQ) steel 510±11 process development to produce BH-IF SEDDQ steel sheet 511±16 super structures 289±90 superalloys 150 superheat 443±7
572
Index
supersaturation 400, 401, 402 nucleation rate 409, 410 surface roughness 444, 445 of steel cast in an oscillating mould 428, 439±42 surface activity 117 solutes in liquid iron 258±9 surface coating sector 536 surface dilational viscosity (area viscosity) 247±8 surface energy: solidification and 404, 405±8 surface excess quantities 239 surface laser light scattering method (SLLS) 121, 125, 128, 132 surface of tension 241 surface stress 242±3 surface tension 126±7, 129±34, 239±42 effect of sulphur on steel's 109±10, 111 estimation 163±5, 166±7 liquid metals 257±9 mechanical aspects 243±6 mechanical definition 243±4 position of dividing surface 240±1 radius of curvature 241±2 slag 129±33, 165, 166±7 and surface stress 242±3 temperature 242 thermodynamic interpretation 239±40 systems, in thermodynamics 39 Szegvari attrition mill 475 tandem atomisation 481 tangential pressure 243±4 tarnishing (high temperature oxidation) 10±13 Tecnored 541 Temkin's ion activity concept 73 temperature distribution in hollow cylinder 208±10 distribution in spherical shell 210±11 rate effects 457±61 low to ambient temperatures 458±9 warm to hot temperatures 459±61 and surface tension 242 typical values for metal working processes 453 tempered martensitic steel 507 terminal velocity 233±4 ternary phase diagrams 87±94 isothermal cross-section and tie-line 88±91, 92
representation of composition and Gibbs triangle 87±8, 89 representation of ternary oxide systems 91±4 ternary systems solidification in and four-phase equilibria 95±101 ternary oxide systems 91±4 testing methods 453 thermal conductivity 126±7, 138±43, 203±4 estimation 162±3, 164 gases 203 liquids 203 porous solids 273±4 solids 203±4 thermal diffusivity 138±43 estimation 162±3 thermal decomposition 485 thermal effusivity 138±43 thermal equilibrium 40 thermal expansion coefficient 122±4, 126±7 thermal plasmas 486 thermoanalytical methods 350±66 calorimetric analysis 362±3 DSC 137, 355, 360±1 DTA 355, 356, 358±60 EGA and EGD 355, 356, 363±5 estimation of thermal effects 352±5 principal thermoanalytical methods 355±6 specific heat 350±2, 353, 354 thermogravimetry 355, 356±8, 364, 365 Thermo-Calc system 70, 71, 72 thermocouples 353, 432±3 thermodilatometry (DA) 355 thermodynamic modelling 118 thermodynamically stable intermediate phases 310 thermodynamics 38±81, 151 basic concepts 39±44 first law 41±4 state and state functions 39±41 chemical equilibrium 40, 44±8 constraints and sub-processes 374 description of steelmaking 375±80 ladle treatment 382±4 slag models 376±80 solution models for liquid metal 380 electrochemical thermodynamics 78, 79, 80 of interfaces 238±43 interpretation of surface tension 239±40
Index of ionic melts 72±9 modelling of metallic systems 70±2 of multicomponent dilute solutions 66±70 of nucleation 253±5 reference states for thermodynamic properties 47±8 second law 44±5 and solidification 401±5 of solutions 57±66 thermodynamic block in micromodelling 394 thermodynamic compilations 48 third law 46 unary and multicomponent equilibria 49±56 thermogravimetry (TG) 355, 356±8, 364, 365 and DTG curves 357 isothermal TG 358 sources of error 357±8 thermomagnetometry (TM) 355, 358 thermomechanical control rolling process (TMCP) 516±18 thermomechanical processes 31±4 deformation processing 32±4 thermo-physical properties 109±77 estimating metal properties 148±69 density and molar volume 151±2, 153 diffusion constants 168±9 electrical resistivity 160±2 heat capacity and enthalpy 149±51 optical properties 165±8 surface tension 163±5, 166±7 thermal conductivity and diffusivity 162±3, 164 viscosity 152±60 factors affecting physical properties and their measurement 113±20 commercial materials 120 convection 117±19 dynamic methods 119±20 measurements in the mushy zone 120 structure 113±17, 118 surface and interfacial properties 117 fluid flow properties 113, 122±36 density and thermal expansion coefficient 122±4, 126±7 interfacial tension 129, 133±6 surface tension 126±7, 129±34 viscosity 124±9 heat transfer properties 113, 136±46
573
electrical resistivity 126±7, 143±4 heat capacity and enthalpy 126±7, 136±8 optical properties 126±7, 145±6 thermal conductivity, thermal diffusivity and thermal effusivity 126±7, 138±43 mass transfer properties 113, 146±8 measurements and problems 120±2 different methods for metals, slags and glasses 122 reactivity of sample 120±2 need for thermo-physical property data 109±13 THERMOSLAG software 76, 79 thin films semiconductor technology 365 thin slab casting 30±1 third law of thermodynamics 46 Thompson-Gibbs relation 406 Thompson's equation 407 three-dimensional printing (3DP) 499 three-phase triangle 89±91, 92, 93, 96 tie-line 88±91, 92 time-temperature-transformation (TTT) curves 286, 287, 409±10, 411 mould slags 432±4, 435±7 titanium inclusions 523 titanium nitride 431, 432 titanium treated steels 430±1 Tolman's equation 241±2 torpedo furnace 373 total emissivity 145 tracer diffusion 146 tramp elements 539±40 transformation-induced plasticity (TRIP) steel 456±7, 507 transient hot wire (THW) method 142 transient techniques 120, 141±3 transition region 218 transmission methods 146 transport phenomena 178±236 fluid flow see fluid flow heat transfer see heat transfer mass transfer see mass transfer transport block in micro-modelling 394 Trouton's rule 47 TTT diagrams see time-temperaturetransformation (TTT) curves tumbler ball mills 474, 475, 476 tungsten 10 turbulent boundary layer 230, 231 turbulent flow 217±18, 225±6 twin roll casting 447, 449 two-film theory 393±4
574
Index
two-fluid atomisation 477, 478, 480±1 two-sublattice model 70±1, 376 ultra low carbon (ULC) steels 439±42, 462±3, 510 ultrafine multicomponent powders 482±3 ultrafine powder (UFP) 15±17 ultrasonic atomisation 477, 478, 482 ultrasonic gas atomisation (USGA) 478, 482 unary systems 49±51 undercooling 400±1 effect of lattice disregistry on 412 heat transfer dominated growth rates 415±19 and initial solidification in steel casting 443±7 nucleation rate 409, 410 uniaxial pressing 487±9 Unified Interaction Parameter Model (UIPM) 380 UO pipe 516±20, 521 uphill diffusion 182 upstream processes 504, 506 future development of 551±2 see also under individual processes vacancy mechanism 179 vacuum atomisation 477, 478, 481 vacuum degassing 25±6 VAI-Q 468±9 vapour-phase reduction 15±17 vapour phase species 483 vapour pressure 251 variable weld penetration 109±10, 112 vertical attritor 476 vibrating mills 474, 475, 476 vibrational heat capacity 351 vibrational specific heat 351 view factors 214±15, 277 viscosity 124±9 estimations 152±60 of gases 220 of liquids 221 models 154±60 Newton's law of 218±20 problems in measurement 128±9
relationship of diffusion constants to 168±9 viscous sub±layer (laminar sub-layer) 225±6 Volmer's equation 402±3 Wagner's equation 66±7, 380, 384 wastes abatement of hazardous wastes 546 minimisation, recycling and reuse 546±9 see also scrap water atomisation 477, 478, 479, 480 weld penetration, variable 109±10, 112 Wenzel equation 250 wet reduction 484 wet slurry 3DP processes 499 wettability 449 wetting 248±50 of ceramics by liquid metal and slag at high temperature 260 Weymann-Frenkel approach 152 Weymann relation 152, 154 Wiedemann-Franz-Lorenz (WFL) rule 120, 138±9, 162 Wilson-Frenkel relation 413 wire casting 30±1 wire drawing 33 work 41±2 work of adhesion 134, 248, 249 work hardening 32, 454±7 multiphase materials 455±7 single phase materials 454±5 wustite 5 X-ray diffraction 113, 114 X-ray pendent drop 135 X-ray sessile drop method 134±5 yield stress 32 Young-Laplace equation 406 Young's equation 249±50, 404 zinc 9±10, 12 corrosion of zinc in a de-aerated environment 315±17 zirconium 12 zone refining 26±7