Materials Issues for Generation IV Systems
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Springer Springer Springer IOS Press IOS Press
Materials Issues for Generation IV Systems Status, Open Questions and Challenges
edited by
Véronique Ghetta CNRS, Laboratoire de Physique Subatomique et de Cosmologie (LPSC), Grenoble, France
Dominique Gorse CNRS, Laboratoire des Solides Irradiés (LSI), Palaiseau, France
Dominique Mazière CEA, DRI/DAE, Gif-sur-Yvette, France and
Vassilis Pontikis CEA, Laboratoire des Solides Irradiés (LSI), Palaiseau, France
Published in cooperation with NATO Public Diplomacy Division
Proceedings of the NATO Advanced Study Institute on Materials Issues for Generation IV Systems: Status, Open Questions and Challenges Cargèse, Corsica, France 24 September – 6 October 2007
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TABLE OF CONTENTS
Forword
ix
List of Authors
xi
The Energy Issue and the Possible Contribution of the Various Nuclear Energy Production Scenarios H. Nifenecker
1
Outlook on Generation IV Nuclear Systems and Related Materials R&D Challenges F. Carré, C. Renault, P. Anzieu, P. Brossard, and P. Yvon
25
Fundamentals of Neutronics: Reactivity Coefficients in Nuclear Reactors P. Reuss
49
Introduction to Thermodynamics G. Inden Kinetics of Phase Transformation in Multi-Component Systems G. Inden
73 113
Ab Initio Approaches to Designing Thermodynamic Properties of Materials A. Pasturel and N. Jakse
141
Correlation between Electronic Structure, Magnetism and Physical Properties of Fe-Cr alloys: Ab Initio Modeling I. Abrikosov, P. Olsson and A.V. Ponomareva
153
The Computational Modeling of Alloys: From Ab Initio Calculations and Thermodynamics to Heterogeneous Precipitation A. Caro
169
Numerical Modeling of Radiation Effects in Solids: Principal Features, Limitations and Perspectives P. Geysermans
187
v
vi
TABLE OF CONTENTS
A Multiscale Approach to Measuring and Modeling Cleavage Fracture Toughness in Structural Steels G. R. Odette, H. J. Rathbun, M. Hribernik, T. Yamamoto, M. He and P. Spätig Microstructures and Mechanical Properties of Irradiated Metals and Alloys S. J. Zinkle Multiscale Modeling of RPV Embrittlement N. Soneda Parametric Dislocation Dynamics and Boundary Element Modeling of Elastic Interaction between Dislocations and Precipitates A. Takahashi
203
227
245
263
Crystal Plasticity from Dislocation Dynamics V. V. Bulatov
275
Radiation-Induced Solute Segregation in Alloys A. J. Ardell
285
Research and Development of Oxide Dispersion Strengthened Ferritic Steels for Sodium Cooled Fast Breeder Reactor Fuels M. Inoue, T. Kaito and S. Ohtsuka
311
Some Aspects of the Structural, Mechanical and Electronic Properties of SiC P. Pirouz
327
Review: Oxidation of SiC/SiC Composites in Low Oxidising and High Temperature Environment C. Cabet
351
Fundamentals of Liquids J. -P. Hansen
367
Introduction to Interfaces and Diffusion P. Wynblatt
393
Some Aspects of Wetting at High Temperature D. Chatain and V. Ghetta
425
TABLE OF CONTENTS
Potentiometric Sensors for High Temperature Liquids J. Fouletier and V. Ghetta
vii
445
Influence of Liquid Sodium on Mechanical Properties of Steels, Refractory Alloys and Ceramics H. U. Borgstedt
461
Radiation Effects in Structural Materials for Fusion Power Plants: the Outcomes of the EU Fusion Program J. -L. Boutard, S. Dudarev and E. Diegele
481
Introduction to the Physics of Molten Salt Reactors 501 E. Merle-Lucotte, D. Heuer, M. Allibert, V. Ghetta, and C. Le Brun Physico-Chemical Properties of Molten Salts J. -C. Poignet and J. Fouletier
523
Combined Effect of Molten Fluoride Salt and Irradiation on Ni-based Alloys A. S. Bakai
537
Specific Features of Particule/Matter Interaction for Accelerator-Driven Sub-Critical Reactors S. Leray
559
Operation of High Power Liquid Metal Spallation Targets: a Challenge for the Structural Materials J. Henry, T. Auger and Y. Dai
575
Index
585
FOREWORD The NATO Advanced Study Institute, on MATerials issues for GENeration-IV systems, MATGEN-IV, was held from September 24 to October 6, 2007 at the “Institut d’Etudes Scientifiques de Cargèse” in Corsica, France. Principal objectives of the ASI were (i) the examination and critical comparison of the properties of existing structural materials, in view of their use for building the nuclear reactors of the 4th generation and (ii) the identification of the technical needs motivating the demand for new structural materials be developed. Prerequisite for reaching these objectives, is the close collaboration between academic researchers and engineers in nuclear science and engineering. Prior to and during the ASI, care was given to favor such contacts and to promote the sharing of available scientific and technical knowledge between participants, scientists or engineers, the cooperation of which would be decisive in designing and constructing safe and efficient Generation IV advanced systems. Introductory remarks by the Vice-Chairman of the French Atomic Energy Commission preceded the inaugural lecture focusing on energy issues related to mitigation of global warming via a drastic decrease of green house gas emissions during this century. The presentation of energy production scenarios, as proposed by the IIASA* and used by the WEC* and the IPCC*, has emphasized on nuclear intensive variants accounting for the sole economic and technological constraints. One particular scenario, consisting in the vast deployment of fast breeder reactors by 2050, has then been analyzed and various aspects of the fuel cycle have been considered with final conclusion that an intensive, worldwide R&D program on structural materials is crucially and urgently needed. Plenary lectures were about reactor physics, neutronics, the fundamentals of thermodynamics and few more specialized topics of “nuclear metallurgy”, such as wetting and techniques commonly used for the compositional control of coolants. Moreover, a detailed presentation of irradiation damage in materials, at different space and time scales, has focused on all aspects of ageing ranging from the micro-structural evolution to the associated macroscopic mechanical property change. Appropriately chosen
______ * IIASA: International Institute for Applied Systems Analysis ; WEC: World Energy Council; IPCC: Intergovernmental Panel on Climate Change
ix
x
FOREWORD
seminars have addressed different aspects of fracture and related experimental methods, the ductile to brittle transition (DBT), scaling approaches unifying experimental observations of fracture surfaces and environment driven changes of the mechanical properties. Additional lectures were devoted to the mechanisms of particle/matter interactions at different energy levels (from keV to GeV) and to a variety of projectile/target couples whereas others have summarized present knowledge on heavily irradiated materials in spallation-neutron sources* or in the future fusion reactors. Finally, special attention has been given to oxide-dispersion strengthened steels and SiC, both good candidates as cladding materials in Generation IV systems. Lecturers and participants, from over twenty countries, have contributed in interesting discussions and live exchanges at the occasions of scheduled events and informal meetings all along the eleven days the ASI has lasted. However, due to the lack of space, only 70% of the lectures are included in these proceedings. For the reader interested in other aspects of the ASI, program, presentation viewgraphs, pictures, …, information can be found on the web (http://www-matgen4.cea.fr). It is our hope that these proceedings provide the reader with a starting point while deeper understanding of specialized matters can be found in existing textbooks (e.g. neutronics, the fundamentals of liquids, …). The organizers thank warmly their colleagues who lectured in this ASI and/or, contributed to the proceedings. Moreover, our thanks go to the sponsoring institutions: the North-Atlantic Treaty Organization (NATO)/Scientific Affairs Division, the Commissariat à l’Energie Atomique (CEA), the Centre National de la Recherche Scientifique (CNRS), Electricité de France (EDF), The European Nuclear Education Network (ENEN), the association Ecoles en Sciences des Matériaux (ESM), the Institut National des Sciences et Techniques Nucléaires (CEA-INSTN) and the Groupement de Recherche Gestion des Déchets et Production d’Energie par des Options Nouvelles (GEDEPEON). V. Ghetta (CNRS-LPSC), D. Gorse (CEA/CNRS-LSI) D. Mazière (CEA-DRI), V. Pontikis (CEA/CNRS-LSI) Saclay, January 31st, 2008
______ *
MEGAPIE at Paul Scherrer Institute, Switzerland and SNS at Oak Ridge, USA
LIST OF AUTHORS
ABRIKOSOV I. A. Department of Physics, Chemistry and Biology, Linköping University, Sweden ALLIBERT M. LPSC, UJF Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs 38026 Grenoble Cedex, France ANZIEU P. French Atomic Energy Commission (CEA) Nuclear Energy Division Saclay Center, 91191 Gif sur Yvette Cedex, France ARDELL A. J. Department of Materials Science and Engineering, University of California at Los Angeles, Los Angeles, CA 90095-1595, USA AUGER T. CNRS, CECM-UPR 2801, 94407 Vitry-sur-Seine Cedex, France BAKAI A. S. Kharkiv Institute of Physics and Technology, 61108, Kharkiv, Ukraine BORGSTEDT H. U. FZK, Karlsruhe, Germany BOUTARD J.-L. EFDA-CSU Garching, Boltzmannstrasse 2, D-85748 Garching bei München, Germany BROSSARD P. French Atomic Energy Commission (CEA) Nuclear Energy Division Saclay Center, 91191 Gif sur Yvette Cedex, France BULATOV V. V. Lawrence Livermore National Laboratory, USA CABET C. Laboratoire d’Etude de la Corrosion Non Aqueuse, DEN/DANS/DPC/SCCME, CEA Saclay, 91191 Gif sur Yvette Cedex, France
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LIST OF AUTHORS
CARO A. Chemistry, Materials, and Life Sciences Directorate Lawrence Livermore National Laboratory, Livermore CA 94551, USA CARRÉ F. French Atomic Energy Commission (CEA) Nuclear Energy Division Saclay Center, 91191 Gif sur Yvette Cedex, France CHATAIN D. Centre de Recherche en Matière Condensée et Nanosciences CNRS, campus de Luminy, 13288 Marseille, France DAI Y. Spallation Source Division, Paul Scherrer Institut CH-5232 Villingen, Switzerland DIEGELE E. EFDA-CSU Garching, Boltzmannstrasse 2 D-85748 Garching bei München, Germany DUDAREV S. EURATOM/UKAEA Fusion Association, Culham Science Centre Oxfordshire OX14 3DB, UK FOULETIER J. Grenoble University, Laboratoire d’Electrochimie et de Physico-chimie des Matériaux et des Interfaces (LEPMI), E.N.S.E.E.G., BP 75, 38402 Saint Martin d’Hères Cedex, France GEYSERMANS P. CNRS – INSP UMR 7588, UPMC Université Paris 06, 140 rue de Lourmel, 75015 Paris, France GHETTA V. LPSC, UJF Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs 38026 Grenoble Cedex, France HANSEN J. -P. Department of Chemistry, University of Cambridge Lensfield Road, Cambridge CB2 1EW, UK HE M. Department of Mechanical Engineering, University California Santa Barbara, Santa Barbara, CA 93106, USA
LIST OF AUTHORS
HENRY J. CEA Saclay, DEN/DMN/SRMA 91191 Gif-sur-Yvette Cedex, France HEUER D. LPSC, UJF Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs 38026 Grenoble Cedex, France HRIBERNIK M. Department of Mechanical Engineering, University California Santa Barbara, Santa Barbara, CA 93106, USA INDEN G. Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany INOUE M. Advanced Nuclear System Research and Development Directorate Japan Atomic Energy Agency (JAEA), 4002 Narita-cho, Oarai-machi Higashi-ibaraki-gun, Ibaraki-ken, Zip code 311-1393, Japan JAKSE N. Sciences et Ingénierie des Matériaux et Procédés, INP Grenoble, UJF-CNRS 1130, rue de la Piscine, BP 75, 38402 Saint-Martin d’Hères Cedex, France KAITO T. Advanced Nuclear System Research and Development Directorate Japan Atomic Energy Agency (JAEA), 4002 Narita-cho, Oarai-machi Higashi-ibaraki-gun, Ibaraki-ken, Zip code 311-1393, Japan LERAY S. DAPNIA/SPhN, CEA/Saclay 91191 Gif-sur-Yvette Cedex, France LE BRUN C. LPSC, UJF Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs 38026 Grenoble Cedex, France MERLE-LUCOTTE E. LPSC, UJF Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs 38026 Grenoble Cedex, France NIFENECKER LPSC, UJF Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs 38026 Grenoble Cedex, France
xiii
xiv
LIST OF AUTHORS
ODETTE G. R. Department of Mechanical Engineering, University California Santa Barbara, Santa Barbara, CA 93106, USA OHTSUKA S. Advanced Nuclear System Research and Development Directorate Japan Atomic Energy Agency (JAEA), 4002 Narita-cho, Oarai-machi Higashi-ibaraki-gun, Ibaraki-ken, Zip code 311-1393, Japan OLSSON P. Department of Materials and Mechanics of Components, EDF R&D Les Renardières, Moret-sur-Loing, France PASTUREL A. Sciences et Ingénierie des Matériaux et Procédés, INP Grenoble, UJF-CNRS 1130, rue de la Piscine, BP 75 38402 Saint-Martin d’Hères Cedex, France PIROUZ P. Department of Materials Science and Engineering, Case Western Reserve University, Cleveland, OH 44106, USA POIGNET J.-C. Grenoble University, Laboratoire d’Electrochimie et de Physicochimie des Matériaux et des Interfaces (LEPMI), E.N.S.E.E.G., BP 75, 38402 Saint Martin d’Hères Cedex, France PONOMAREVA A. V. Moscow institute of Steel and Alloys Moscow, Russia RATHBUN H. J. Department of Mechanical Engineering, University California Santa Barbara, Santa Barbara, CA 93106, USA RENAULT C. French Atomic Energy Commission (CEA) Nuclear Energy Division Saclay Center, 91191 Gif sur Yvette Cedex, France REUSS P. Emeritus Professor Institut National des Sciences et Techniques Nucléaires, Point courrier 35, CEA/Saclay 91191 Gif sur Yvette Cedex - France
LIST OF AUTHORS
SONEDA N. Central Research Institute of Electric Power Industry, 2-11-1 Iwado-kita, Komae, Tokyo 201-8511, Japan SPÄTIG P. Fusion Technology-Materials, CRPP-EPFL Association EURATOM-Confédération Suisse, 5232 Villigen PSI, Switzerland TAKAHASHI A. Department of Mechanical Engineering, Faculty of Science and Technology Tokyo University of Science 2641, Yamazaki, Noda-shi, Chiba, 278-8510, Japan YAMAMOTO T. Department of Mechanical Engineering, University California Santa Barbara, Santa Barbara, CA 93106, USA YVON P. French Atomic Energy Commission (CEA) Nuclear Energy Division Saclay 91191 Gif sur Yvette Cedex - France WYNBLATT P. Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA ZINKLE S. J. Materials Science and Technology Division Oak Ridge National Laboratory, P.O. Box 2008 Oak Ridge, TN 37831-6138 USA
xv
THE ENERGY ISSUE AND THE POSSIBLE CONTRIBUTION OF THE VARIOUS NUCLEAR ENERGY PRODUCTION SCENARIOS H. NIFENECKER* Scientific consultant, LPSC, Université Joseph Fourier Grenoble 1, CNRS/IN2P3, INPG, 53, avenue des Martyrs, 38026 Grenoble Cedex, France Chairman of “Sauvons le Climat”
1. Introduction Since the beginning of the industrial era, less than two centuries ago, our society has relied heavily upon fossil fuels. It was, first, coal that provided ample energy for industry and transport, that allowed the generalization of electricity and even town gas obtained by reacting coal with water vapour. In the first half of the twentieth century oil took over coal as the most used fossil fuel. It was much easier to use and became intimately intertwined with the exponential development of the “automobile society”. It also started to displace coal as fuel in electric power plants. However following the 1973 oil price crisis the use of oil was restricted to transportation and petro-chemistry. Natural gas became more and more popular for electricity and heat production. In 2004 the World Total Primary Energy Supply (TPES) amounted to 11 Billion tons oil equivalent (toe) [1], of whom 34% was provided by oil, 25% by coal and 21% by gas. Thus fossil fuels provided 80% of our energy supply. It appears that the amount of oil and gas reserves discovered every year has fallen below their yearly consumption. It is predicted that the amount of extracted oil will start decreasing within the next 10 to 15 years (peak oil) and that of gas will behave similarly within 20 to 25 years. This means that the price of oil and gas will increase steadily until consumption decreases to the level of production. There might come a point where it will become cheaper to make oil and gas out of coal via chemical reactions like that of “Fischer Tropsch”. Reserves of coal are plentiful and should allow to pass this century without real energy shortage.
______ *
[email protected] V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008
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2
H. NIFENECKER
It is clearer and clearer that the most difficult challenge in the energy sector is related to the mitigation of global warming via a drastic decrease of Green House Gas (GHG) emissions. 2. The Global Warming Challenge The main Green House Gases naturally present in the atmosphere are water vapour, carbon dioxide, methane and nitrous oxide. Water vapour has a very short cycling time so that its temperature dependent equilibrium concentration in the atmosphere is reached almost immediately and has a mere amplifying influence of the effect of other determinants. Green House Gases allow the average temperature of the earth to reach a gentle 15 d°C while it would be a chilling –18 d°C if they were not present in our atmosphere. In this respect they are useful and cannot be considered as pollutants. The historical records of temperature and GHG concentrations have been reconstructed from ice cores excavated from the Antarctic and the Greenland caps. A clear and positive correlation has been observed between temperatures and GHG concentrations with a quasi-periodical behaviour. The average temperature oscillated from minima close to 8 d°C lower than present to maximums 2 d°C higher. CO 2 concentrations oscillated from lows around 180 parts per million (ppm) to highs close to 280 ppm, methane between 300 parts per billion (ppb) and 700 ppb. The driving parameter behind those oscillations is the solar irradiation, which varies periodically according to the parameters of the earth orbit around the sun (Milankovitch oscillations). However, the magnitude of the change in solar irradiation is not sufficient to account for the amplitude of the temperature changes. Amplification factors are required. It is suspected that, starting from a glacial minimum, an increase of solar irradiation of the northern hemisphere leads to decrease of the surface of the sea ice in summer, which decreases the average earth albedo, which increases further the temperature. Increase of temperature of the ocean leads to CO2 release by degassing, which increases the temperature and the extent of boreal forests, here again accompanied by a decrease of the albedo etc. We can summarize the causality chain, which leads to the phasing out of a glacial era: ΔW→ΔT→ΔA→ΔT’→ΔA’→ΔT’’→ΔGHG→ΔT’’’→ΔGHG’→ΔT’’’’ where ΔW is the increase of the solar irradiation in the northern hemisphere, ΔT, ΔT’, ΔT’’, ΔT’’’, ΔT’’’’ the temperature increases, ΔA, ΔA’ decreases of the albedo, ΔGHG and ΔGHG’ the increase of the GHG concentrations. Because of the finite limits of the temperature and concentrations excursions one concludes that the series are converging.
ENERGY ISSUE AND SCENARIOS
3
The massive injection of GHG in the atmosphere due to billions tons of fossil fuels burning displays a new driving parameter of a new increasing series for temperatures and concentrations. Is it insured that such series are also converging? Nobody is sure, because of the infinitely more rapid present pace of change as compared to that observed previously. This is one of the reason, and, perhaps, the most frightening why massive injection of GHG in the atmosphere should be halted as soon as possible. 2.1. RECENT EVOLUTION OF GREEN HOUSE GAS EMISSIONS
Globally [2], emissions of the GHGs increased by about 70% (from 8 to 13 GtCeq*) from 1970 to 2004, with carbon dioxide CO2 being the largest source, having grown by about 80%. The largest growth in CO2 emissions has come from power generation and road transport. Methane (CH4) emissions rose by about 40% since 1970, with an 85% increase from the combustion and use of fossil fuels. Agriculture, however, remains the largest source of CH4 emissions. Nitrous oxide (N2O) emissions grew by about 50%, due mainly to increased use of fertilizer and the growth of agriculture. In the following I restrict myself to the case of CO2, which is the main responsible for global warming and is of crucial concern for the energy sector. 2.2. THE EFFORT TO MAKE
Out of the 7,4 GtC-eq injected into the atmosphere by the use of fossil fuels, about 3 seem to be absorbed by the biosphere and the Ocean. This is the level of emissions we should aimed at. Note that this amount supposedly absorbed by mother Nature might change according to the temperature, especially of the ocean, and to the concentrations of GHG in the atomsphere. In this respect future might reserve bad as well as good surprises. Present lines of thought point, unhappily, towards the bad (acidification of the ocean). Table 1 shows the evolution of the world population and GHG emissions between 2000 and 2004. It also shows what could be an acceptable future in 2050. It shows that, worldwide, one should decrease our individual emissions by a factor 3.5.
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* Different gases have different warming powers and it is conventional to translate their warming power into that of CO2, and, more usually, into the mass of carbon included in the CO2. For example Methane has a warming power 20 times larger than that of CO2, thus 1 mole of methane (16 g) has a warming power equal to that of 20 moles of CO2 (880 g) and of 240 g Carbon equivalent (Ceq).
4
H. NIFENECKER TABLE 1. Observed and objective emissions from 2000 to 2050 [1]. 2000 2004 2050 Population Billions
6
6.4
9
Emissions CO2 GtCeq
6.4
7.3
3
Emissions/capita
1.06
1.15
0.33
Different countries should have different objectives, according to their present rate of emissions. A few examples are displayed on Table 2. The extreme effort would be that of US citizens who should decrease their emissions by more than 16! TABLE 2. 2004 CO2 emissions/capita in selected countries. 2004 CO2 emissions/capita USA
5.4
Germany
2.8
France
1.7
China
0.99
India
0.28
World Total
1.15
2.3. THE FACTORS TO CONTROL
Since the primary aim is the decrease of the amount of emission of CO2 per capita it is useful to isolate the main factors that influence it. This is done with the simple tautological equation:
QCO 2 GDP Energy Q = × × CO 2 N pop N pop GDP Energy
(1)
where QCO2 is the total amount of CO2 emitted by the energy sector, Npop the population, GDP the World Gross Domestic Product, Energy is the total world primary energy supply. The factor Energy/GDP is the “energy intensity” while the factor QCO2/Energy is the CO2 or Carbon intensity. It is rarely advocated that the first factor, the Gross Domestic Product per capita should decrease at the world level. The “energy intensity” has a clear tendency to decrease with increasing GDP, so that is seems reasonable to expect that, with persistent efforts, the energy consumed per capita might stay constant or increase only slightly. Therefore one expects that the total primary energy supply (TPES) might, in the best case, follow the population increase. It seems clear that the last factor will play a prominent role if one wishes to decrease strongly the amount of CO2 emitted worldwide.
ENERGY ISSUE AND SCENARIOS
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2.4. THE IMPORTANCE OF ELECTRICITY
Before trying to see if such an effort has any chance of succeeding it is necessary to have a clear knowledge of where to put it the most efficiently. Table 3 shows that the electricity production sector is the main CO2 emitter. Since competitive technologies exist to produce electricity without CO2 emissions it seems natural to put the stress on it, in the first place. TABLE 3. Shares of CO2 emissions by sectors. Emitting sector Percent of total % Electricity 33 Refineries 5 Industry 25 Transportation 24 Other (non electric home heating etc.) 13
That electricity is, indeed an important key can be shown when one compares the CO2 emissions of several countries with similar degrees of development as function of their use of hydro or nuclear energies for producing their electricity.
Figure 1. Correlation between the Carbon intensity and the share of hydro and nuclear electricity in various countries.
This comparison is seen on Figure 1 and speaks for itself. The data corresponding to the figure are given in Table 4. The difference between Denmark with no use of hydro nor nuclear electricity and a carbon intensity of 2.57 and Sweden with 100% hydro+nuclear electricity and a carbon intensity of 1.09 is striking. Since year 2000 Denmark has considerably increased its share of wind electricity production, which reached 17% by 2006, while its carbon intensity decreased to 2.54 by year 2004. It is interesting to see that the curve of Figure 1 shows a favorable increase of slope for shares of hydro and nuclear electricity exceeding 50%. That this could be related to an increase use of electricity, especially for heating, is a possible explanation.
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H. NIFENECKER
TABLE 4. Data corresponding to Figure 1. The data on carbon intensities are from “Key Energy World Statistics 2002”and relate to year 2000. Country
Hydro + Nuclear (%)
tCO2/toe
Denmark
0.0
2.57
China
16.0
2.62
UK
23.0
2.28
US
26.8
2.46
Spain
28.0
2.28
Germany
30.0
2.45
Russia
33.8
2.45
Korea
37.0
2.24
Japan
38.9
2.20
Ukrain
45.0
2.16
Canada
71.2
2.10
Brasil
87.3
1.66
France
90.3
1.45
Norway
99.5
1.31
Sweden
100.0
1.09
2.4.1. The double face of electricity Aside from its specific applications, electricity can be used for heat production and for transportation, either collective or individual. It might also be used for hydrogen production via electrolysis. As far as CO2 emission is concerned, these use of electricity can be the best or the worse, depending on the nature of electricity production. A very good example is provided by heating. If electricity is produced by fossil plants with an efficiency of, say, 33%, 1 kWh heating via an electric furnace with efficiency 70% will require 4.3 kWh and, thus produce 0.39 kgCeq. If electricity is produced with renewables or nuclear energies, there will be no emission*. Using direct gas heating with an efficiency of 60% requires 1.66 kWh of gas and produces 0.09 kgCeq. It is clear that electric heating when electricity is produced with coal is catastrophic,
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It is sometimes argued that one should count here the “life cycle” CO2 emissions which take into account, not only the CO2 emitted in the production process, but also that which is related to the mining and plant construction phases. We do not think that this method is correct since one should rather think in terms of marginal emissions, i.e. in terms of the additional emission due to the actual consumption. Construction and mining emissions are, then, allocated to the mining and building industries, not to the private consumer. Furtherrmore mining and construction emission are much larger for fossil electricity production than for the nuclear one.
ENERGY ISSUE AND SCENARIOS
7
4 times worse than direct gas use. Let us, now, assume that the electricity fossil production amounts to 10%. In most cases (especially in Europe) this production is concentrated during the winter months, say during 3 months. That means that during these winter months fossil electricity will represent almost 40% of the total. 70% of electric heating is supposed to take place during these three months. It follows that 1kWh electric heating would require 1.2 primary kWh of electricity produced by coal plants, i.e. produce 0.11 kg-Ceq. It follows that even a small fraction of fossil electricity leads to CO2 emissions due to electric heating similar to that of gas. Another example is given by electric cars. Consider a small diesel car consuming 4 liters of gasoline par 100 kms. It emits about 3 kg-Ceq for this distance. Taking efficiencies of 0.3 and 0.7 for the thermal and electric engine respectively, a consumption of 20 kWh of an electric car would allow the same service as that of the diesel car. If electricity is produced without fossils the electric car does not produce CO2. If the electricity is produced with coal the amount of CO2 produced will be 5.5 kg-Ceq, that is almost twice more than the diesel car. Thus, in order to minimize the CO2 emission, switching to electric cars is only efficient if electricity is produced without resorting to fossil fuels. 2.4.2. Learning from the past Before 1973, many countries and electricity operators thought that oil was the best choice for producing electricity since it was very cheap. After the oil price crisis of 1973, most of them decided to resort to other resources, mostly coal and (or) nuclear. As striking examples are Denmark who switched to an almost pure coal-based system and France which initiated a crash program of nuclear reactors. United States had already stopped building nuclear reactors due to the strong counteroffensive of the coal industry and to the opposition of environmentalists which gained impetus following the TMI accident in 1979. Germany who had a strong coal mining industry kept a large proportion of its electricity produced with coal but, also, started a strong program of reactor building. The Chernobyl catastrophe halted reactor construction almost in all industrialized countries with the exception of France, and Pacific States like Japan and Korea. These various choices produced a divergence between the electricity production structures of average OECD countries and countries like France, Sweden or Switzerland. This divergence is illustrated on Table 5 where the structures of electricity production of France and of the whole OECD ensemble are compared. Let us rewrite history and see what things would have looked like if all OECD countries had made the same choice as France did. Note that these countries have the technical knowledge and industrial strength to do so and
8
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TABLE 5. Comparison of the electricity mix of France and of the totality of OECD countries. Primary Energy source
share in electricity production in % for OECD countries
share in electricity production in % for France
Coal Oil Gas Nuclear and Renewables Total
43 6 21 31
5.5 1.1 3.5 89.9
100
100
TABLE 6. Comparison of actual OECD use of nuclear energy and CO2 emissions with those that would have been observed if OCDE countries would have adopted the same electricity mix as France.
Original OCDE Modified OCDE
Source of Primary Energy
Primary Energy Supply Mtoe
Total CO2 Primary Energy Emitted for Electricity Production MtCO2 Mtoe
CO2 emitted for Electricity Production MtCO2
Nuclear and Renewables
716
0
716
0
Total
5280
13311
2334
5284
Nuclear and Renewables
2097
0
2097
0
Total
5280
8922
2334
728
that the question of nuclear proliferation was not relevant in their case. Table 6 shows the result of this operation. While the total primary energy supply and the energy used for electricity production remain unchanged to 5280 Mtoe and 2334 Mtoe respectively, the CO2 emissions would have fallen by 33% from 13311 MtCO2 to 8922 Mtoe. Nuclear energy would have soared from 716 to 2097 Mtoe, i.e. almost a factor 3. The 33% reduction in CO2 emission is much larger than the 8% which have been set as a goal for industrialized countries by the Kyoto protocol. 3. Energy Scenarios for Imagining the Future In order for politicians and industrials to adjust their politics, futurologists imagine scenarios which try to foresee how the mixture of “given” evolutions and voluntary politics will shape our future. Scenarios are not “predicttions” but a way to understand how our decisions of today may influence our future.
ENERGY ISSUE AND SCENARIOS
9
3.1. THE IIASA SCENARIOS
I have chosen to give as examples the scenarios built by the Vienna International Institute of Applied Systems Analysis (IIASA) since they have been routinely used by the World Energy Council (WEC) and by the Intergovernmental Panel on Climate Change (IPCC). Three illustrative storylines, A2r, B1 and B2 are described in [3] and constitute the GGI Scenario Database, 2007. They are built at the regional level. They first make hypothesis on population evolutions. Storyline A2r assumes a continuous increase of the world population throughout this century up to more than 12 billions. Storyline B2 shows some kind of stabilization of the population to a little more than 10 billions in 2100. Finally storyline B1 goes through a maximum of 9 billions around 2050 and decreases to 7 billions in 2100. The second set of assumptions deals with the rates of increase of the GDPs. These are declined regionally. The fastest GDP growing storyline is B1 (low population) which raises to almost 350 trillion dollars in 2100 from 27 trillions in 2000 while the slowest growing is A2r raising only to 190 trillions dollars in 2100. Each storyline is then subdivided in different scenarios characterized by an assigned final concentration of CO2. The storyline A2r is subdivided into a baseline scenario and 5 additional scenarios with asymptotic CO2 concentrations between 670 and 1390 ppm. The baseline scenario exceeds 1450 ppm by 2100. These scenarios are clearly unacceptable, and would lead to catast-rophic consequences, not only climatic. The storyline B2 has 3 scenarios, the baseline which reaches more than 975 ppm by 2100, and scenarios with asymptotic CO2 concentrations of 670 and 480 ppm. The storyline B1 has a baseline scenario reaching 830 ppm of CO2 and 5 additional scenarios with asymptotic CO2 concentrations between 480 and 670 ppm. I believe that the storyline B2 is probably the most realistic as far as population projections are concerned. And I discuss it henceforth. Figure 2 shows the evolution of the CO2 equivalent GHG concentrations in the B2 scenarios. Note that, for the 480 ppm scenario, the concentration rises up to 600 ppm and, then decreases. This decrease is made possible by a very early decrease in the CO2 emissions which goes through its maximum of 12.7 GtCeq in 2020 (from 11 GtCeq in 2000) and, then decreases sharply to 9.4 GtCeq. in 2050 and 0.7 GtCeq. in 2100. The surprising decrease of the CO2 equivalent concentrations is related to a strong, probably optimistic, change of land-allocation which builds a CO2 drain well. Land use is a source of 1000 Mt CO2eq. (deforestation) in 2000 and becomes an absorbing drain well of 3500 Mt CO2eq. by 2100 (reforestation).
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Figure 2. Evolution of the GHG CO2 eq. concentrations for the three B2 scenarios.
We concentrate on the “optimum” 470 ppm B2 scenario. The total primary energy supply (TPES) increases from 9.5 Gtep in 2000 to 25 Gtep in 2100 with 19 Gtep in 2050. The regional evolutions of the TPES are shown on Table 7. By 2100 OECD countries will have a rather marginal contribution to the energy consumption. By 2050 the primary supply of non OECD Asia will exceed that of the whole OECD. TABLE 7. Regional evolutions of TPES. 2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 Asia
0.91
1.47
2.59
3.64
4.49
5.89
6.62
7.31
8.09
8.94
9.65
Form. USSR
0.63
0.94
1.56
2.12
2.90
3.76
4.67
5.50
6.32
6.91
7.43
OECD
2.87
3.18
3.79
4.19
4.39
4.38
4.32
4.21
4.07
4.02
3.99
LAM+Africa 0.61
0.75
1.03
1.35
1.76
2.08
2.17
2.32
2.41
2.50
2.53
Aside from the above mentioned changes of land use the key of the success of scenario B2-470 ppm to mitigate the increase of CO2 concentration is the increase of the electricity share and of that of fossil free electricity production methods. Figure 3 shows the spectacular increase of the shares of electricity and bio-fuels in final demand. Electricity essentially replaces coal and traditional biomass while modern biomass replaces oil. Figure 4 shows how the fossil electricity decreases from 62% to 10% with the share of nuclear electricity reaching around 60% in 2100.
ENERGY ISSUE AND SCENARIOS
11
Figure 3. Evolution of final energy demand structure The shares of bio fuels and district bio heat are added.
Figure 4. Evolution of the Electricity mix in scenario B2-470 ppm.
The decrease of hydroelectricity reflects an almost constant absolute production while the total electricity production increases by more than a factor 10 between 2000 and 2100. Wind and solar electricity reach a maximum share of 30% around 2050 but a maximum absolute production of more than 3 Gtoe by 2100. I believe that a 30% share is not realistic for intermittent energy sources as long as huge electricity storage is not available. In absolute value, nuclear energy is multiplied by more than 6 in 2050 and more than 35 by 2100. Such a surge would clearly require breeder reactors.
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3.2. THE IPCC SCENARIOS
The just presented scenarios of the IIASA belong to the family of scenarios which were built for the IPCC reports. The family counts a total of 40 different scenarios. It is to be feared that deciders will have a hard time finding their way in such a jungle. The complexity facing deciders is illustrated on Figure 5.
Figure 5. Emission profiles corresponding to the family of scenarios considered by IPCC.
3.3. SHARE OF NUCLEAR ENERGY IN “OFFICIAL” SCENARIOS
Most recent IPCC scenarios foresee a share of nuclear electricity of 18% in 2030, assuming a cost of CO2 less than 50 $/ton. The share of renewable electricity would be 35%. In 2030, the IIASA-470 ppm scenario (see Figure 4) foresaw a nuclear electricity share of 21% and a renewable electricity share of 28%, including hydroelectricity. Prediction of nuclear energy shares are highly political and reflect the state of public opinion. A few years ago it was ill behaved to even talk about it. As time passes and as the price of oil and gas increase it has become more admissible to mention nuclear energy. An example is given by the predictions of the World Energy Outlook of the International Energy Agency (IEA). WEO2004 predicted a decrease of nuclear production from 2975 TWh in 2020 to 2929 in 2030. This reflected “out of nuclear” energy policies dominance. The alternative WEO 2006 predicts a nuclear electricity production of 4106 TWh in 2030 with a share of 14% , in clear increase with the respect of the 8% in 2020 predicted by the WEO1998. It seems clear that scenario makers tend to apply auto censorship when it
ENERGY ISSUE AND SCENARIOS
comes to take fully in consideration nuclear power. I believe that important to determine how much nuclear energy could help dealing the CO2 emission question in absence of ideological opposition to it. ponsibilities of those who act this opposition will, then, be clearly luated.
13
it is with Reseva-
4. Nuclear Intensive Scenarios 4.1. REFERENCE IIASA-WEC 1998 SCENARIOS
A first approach of a nuclear intensive scenario with 3000 GWe nuclear power 30 years from time 0 and 9000 GWe 50 years from time 0 was given in 1999 [4]. A revised version is found in [5]. I will summarize the results presented there. The method was to use, as a reference, scenarios built by IIASA-WEC* in 1998 [6] and increase the share of nuclear power to the highest reasonably achievable level. The nomenclature of the 1998 IIASA scenarios are different from that used more recently in the frame of the IPCC studies. The philosophy of the 1998 scenarios was not essentially different from that of the 2007 ones. I believe that the conclusions of the studies of [4] and [5] are still essentially valid. In the IIASA reference study, storylines A correspond to a fast growth of the GDP/capita in all geographical regions of the world. It assumes a significant reduction of inequality between them. The growth is especially fast in former Soviet Union countries. Storyline C has a rather slow average GDP/capita growth but is, clearly, of the equalitarian type. For storylines A and B the energy intensities decrease as a consequence of the increase in the GDP/capita, as observed in the past. Storyline C assume a voluntary decrease of energy intensities, especially in the most developed countries. The main assumptions of the IIASA-WEC storylines are given on Table 8. The energy intensity is assumed to decrease markedly from 0.43 toe/k$ in 1990 to 0.245 for storyline A, 0.27 for storyline B and 0.19 for storyline C. The main differences between the scenarios are the energy mix producing the primary energy. In scenarios A1, A2 and A3 the relative shares of coal, oil and gas are different. In scenarios C1 and C2 the relative shares of renewable and nuclear energies are different. Note, also, the relative importance of gas in the energy mix of scenario C1. These features are displayed on Table 9. Whether the six scenarios are realistic must be evaluated with respect of the fuel reserves. This is done on Table 10.
______ *
WEC: World Energy Council
14
H. NIFENECKER TABLE 8. Assumptions of the IIASA-WEC storylines. GDP/capita k$ Population 2050 millions 362.42 494.6 148.12 394.67 141.06 838.58 924.25 1735.73 1984.17 750.55 2281.28 10055.43
North America Western Europe Pacific OECD Former Soviet Union Eastern Europe Latin America M. East N. Africa Africa Centrally planned Asia Other Pacific Asia South Asia World
1990 21.62 16.15 22.78 2.71 2.39 2.50 2.12 0.54 0.38 1.53 0.33 3.97
Scenarios B 45.84 37.06 45.80 7.48 7.83 7.07 4.03 1.03 3.36 7.86 1.33 7.24
A 54.47 45.88 58.68 14.09 16.27 8.33 5.64 1.57 6.99 12.21 2.00 10.10
C 38.79 32.95 42.80 7.14 7.97 7.39 4.09 1.19 5.40 10.20 1.75 7.46
TABLE 9. Primary energy mix for the IIASA-WEC scenarios. 1990 Coal Oil Nat. Gas Nuclear Hydro Biomass (comm) Biomass (nonc) Solar Others Total CO2(MtC)
B
2176 4136 3064 4040 1685 4499 450 2738 489 917 246 1122 849 860 0 432 17 1087 8976 19831 5932 9572
A1
A2
A3
C1
C2
3786 7901 4699 2904 993 1124 717 1858 852 24835 11619
7827 4781 5459 1092 1104 2207 747 420 1201 24840 14668
2241 4329 7913 2824 1062 2906 743 1636 1007 24661 9294
1504 2668 3919 521 1031 1481 822 1552 747 14246 5343
1472 2616 3344 1771 962 1357 824 1377 526 14250 5114
TABLE 10. Cumulative consumptions of fossil fuels for the IIASA-WEC scenarios compared to the 1990 reserves. Cumulative consumptions by scenarios Coal Gtep Oil Gtep Gas Gtep
A1 200 300 210
A2 275 260 211
A3 158 245 253
B 194 220 196
C1 125 180 181
C2 123 180 171
Reserves 1990 540 146 133
While it is true that reserves are subject to changes Table 10 shows that oil and gas will become rare, and, thus expansive. By 1999 this was still not clear and many economists foresaw a strong rise of the use of gas, especially for electricity production. It is, now, widely recognized that the Peak Oil will occur within the next 10 or 20 years and the Peak Gas 10
ENERGY ISSUE AND SCENARIOS
15
years later. It seems that coal will become, once more, the main fossil fuel: liquid and gas fuels can be synthesized from coal. This is why we have especially studied scenario A2 as an example of strong economic expansion. 4.2. NUCLEAR INTENSIVE ALTERNATIVES
The IIASA scenarios foresee a rather modest contribution of nuclear energy to the global energy mix. As said above, it seems that the main reason for this shyness is more related to “political correctness” than to economical or technological constraints. It is interesting to examine how much a deployment of nuclear energy limited only by these constraints might limit global warming and resource exhaustion. We, now, give the outline of such a treatment for the three scenarios A2, B and C2. We assume that, by 2030, (in our study, our time 0 was tyear 1999, a shift of 8 years should be made in 2007) the use of fossil fuels for electricity production will be drastically reduced. Table 11 displays the reduction factors used for several geographic aggregates. TABLE 11. Reduction factors used in the nuclear intensive scenarios in 2030 (column 2) and 2050 (columns 3 to 5). In the last column the share of Hydrogen and (or) electricity in the transportation sector is given. reduction factor for reduction factor reduction factor H2 (E) share for residual fossil use in electricity for residual coal use gas use production 2030 2050 (E) 2050 (N) 2050 (N) 2050 (H) OECD 0 0 0 0 0.8 former USSR 0.5 0 0 0.3 0.4 Eastern Europe 0.5 0 0 0 0.6 Latin America 0.5 0 0 0 0.6 Arab countries 0.3 0 0 0 0.3 South Sahara 1 0 0 0 0.3 China 0.3 0 0.3 0.3 0.3 India 0.3 0 0 0 0.2 Other Asia 0.3 0 0 0 0.3
The same reduction was applied in the three scenarios. Fossil fuels were assumed to be saved by resorting to nuclear power, although renewable energies could, equally well, be used provided they become reasonably competitive. The choice we have made of nuclear power provides a kind of existence theorem for a solution to curb Carbon Dioxide emissions, and, at the same time, tests the capabilities of nuclear power in terms of fuel availability, wastes produced and capital needs.
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H. NIFENECKER
Table 11 shows different types of reduction factors used for 2050. These reduction factors are used differently in three different scenarios with increasing use of nuclear power: •
The “electric” scenario, labelled “E”, assumes that fossil fuels are no longer used for electricity production,
•
The “nuclear” scenario, labelled “N”, assumes that coal and gas are no longer used in industry, homes or offices (in particular for heating), except, principally, in former USSR and China,
•
The “Hydrogen” scenario, labelled “H”, assumes that hydrogen and (or) electricity have largely replaced oil in the transportation sector. The share of hydrogen fuel is displayed in the last column of Table 11.
The performances for CO2 emissions of the different variant of the A2 and C2 scenarios are displayed on Figure 6. The Figure shows that politics based upon energy efficiency and use of nuclear electricity are operational to mitigate CO2 emissions. Even storylines which are rather energy proficient like the A one could be tamed with the use of nuclear power and CO2 emissions limited to less that one half of present emissions.
Figure 6. Influence of deployment of nuclear power on world CO2 emissions for different IIASA-WEC like scenarios.
The maximum extent of nuclear power would be around 14 Gtep, i.e. more than 20 times more than present. Will resources of Uranium allow such an intensive program? 4.3. DEPLOYMENT OF A WORLD BREEDER REACTOR FLEET
Light water reactors need approximately 200 tons of natural uranium each year to feed a 1 GWe reactor. A 7000 fleet would require 1.4 million tons annually. Estimates of uranium reserves depend upon the price one accepts to pay for it. For less than 130$/kg reserves are estimated around 4.7 millions tons, while for less than 260 $/kg they would reach 15 millions tons. Other non conventional reserves (phosphates, coal) might reach 25
ENERGY ISSUE AND SCENARIOS
17
additional millions tons. Finally one can note that the ocean contains 3 to 4 billions tons of uranium. Extraction of uranium from sea water has been technically demonstrated at a cost in the range of 500 to 1000 $/kg. Aside from the exploitation of oceanic uranium at a considerable economic and environmental cost, uranium reserves could not feed a fleet of 7000 standard reactors for more than a few decades. This is clearly not sustainable. There needs breeder reactors able to use natural uranium with an efficiency 100 times better than standard light water nuclear reactors. The breeding reaction consists in neutron capture by Uranium 238 to produce Plutonium 239, which is a fissile isotope as is Uranium 235. For the Uranium-Plutonium cycle, the breeding reaction reads: 238
239 239 239 β− β− U+n⎯ ⎯→ U ⎯⎯ → Np (period of 2 days) ⎯⎯ → Pu.
For the breeding reaction to be effective it is necessary that the number of neutrons emitted in the fission of the Pu239 nucleus should be large enough to ensure both a new fission and a capture by a U238 nucleus. In practice this condition is only realized for fast neutron spectra such as provided in fast neutron reactors. Another breeding reaction is based on the Thorium-Uranium cycle, which reads: 232
233 233 233 β− β− Th + n ⎯ ⎯→ Th ⎯⎯→ Pa (period of 27 days) ⎯⎯→ U
Unlike the Uranium-Plutonium case, the Thorium-Uranium cycle can work with thermal neutron spectra as well as with fast neutron spectra, although with difficulty. A rather large experience with fast neutron reactors have been collected in a number of countries like France, Japan and Russia. This is why we study more specifically how and if it is possible to deploy a fleet of 7000 fast breeders reactors. How fast such a park could be deployed will depend on the amount of Plutonium available from standard reactor fuel reprocessing, the initial inventory of the breeders, as well as on the doubling time of the breeder park. A relevant study has been made in [4] where the deployment of 3000 PWR reactors in 2030 and an additional 6000 breeders by 2050 was considered. The question addressed was whether such a deployment is compatible with Uranium reserves and doubling times of the breeders. The initial loads are assumed to be mixtures of the fertile element (U) with Plutonium taken from the spent fuels of PWR and BWR reactors. It is important to make sure that the amount of Plutonium available would be sufficient to supply all the breeding reactors by 2050. Experience with fast breeders shows [7] that a 1.2 GWe reactor requires an initial inventory of 5 tons of Pu. Such a reactor produces around 0.25 tons of Pu annually, corresponding to a doubling time of 20 years. However this value of the doubling time does not take into account the reprocessing stage. The longer the cooling time of the used fuel before reprocessing, the
18
H. NIFENECKER
longer the effective doubling time. As an example, if the residence time of the Plutonium in the reactor is 4 years, and the cooling time also 4 years, the Plutonium inventory is doubled, and so is the doubling time.
Figure 7. Size of installed nuclear power (in GWe) for the U-Pu cycle as a function of time. In a first stage a PWR park is developed which produces Plutonium used to start FR U-Pu breeder park.
The transition from a PWR(BWR) based system to a fast reactor one was studied. It was assumed that a strong PWR program starts in 2010, first breeders starting progressively in 2020*. By 2030 no new PWRs are connected to the grid, leaving the field to fast reactors. Figure 7 shows the evolution of the reactor park corresponding to a Plutonium production of 250 kg/GWe by the PWRs and 200 kg/GWe by the fast breeders. A cooling time of 1 year was assumed. The target of 9000 GWe by 2050 could then be reached. For longer cooling times it was found that the target cannot be reached. Cooling times as short as 1 year are probably not possible with standard aqueous reprocessing and would require pyro-chemical reprocessing. After 2050 the PWRs would be phased out progressively and the doubling time of the FR could be adjusted to the desirable evolution of the reactor park. In Figure 7, a 1.5% annual increase of the nuclear park was assumed. In our scenario the last PWR reactors will be phased out in 2070. At that time the total amount of natural Uranium used would reach 12 million tons, close to the presently estimated reserves. This means that the number and life time of the PWR park cannot be considered as an easily adjustable variable to achieve the strong increase of nuclear power between 2030 and 2050. This increase will be difficult to achieve and requires the early development of breeders, as well as the availability of as much as
______ *
Since this study was carried out in 1999 one should shift the time-table by 8 years in 2007
ENERGY ISSUE AND SCENARIOS
19
possible reprocessed Plutonium. The generalization of MOx incineration has to be weighed against this requirement. Similarly incinerating Plutonium in HTR reactors may be counterproductive if spent fuel reprocessing is not possible. Of course, accepting a lower value for the target in 2050 would make things easier. For example a target of 7000 GWe could be reached with a doubling time of 32.5 years. Another possibility would be to increase the share of more efficient Plutonium producing reactors such as the CANDUs. An alternative deployment scheme has, recently, been proposed by Merle et al. [8]. Aside from capitalizing the Plutonium produced in PWRs, it consists of using a limited number of Fast Reactors with Thorium blankets in order to produce the large quantities of 233U to be fed into fast neutrons molten salt converter reactors*. 450 1 GeV PWR like reactors and 300 FR suffice to feed a fleet of Molten Salt reactors reaching 3000 by 2050 and 5000 by 2100 with a consumption of 7 millions tons of natural Uranium. 4.4. FEASIBILITY AND COST
A multiplication of the number of reactors by a factor between 10 and 20 would require very important investments. Do they exceed the financial resources of the world energy sector? Such program has to be compared to the only alternative which is building a comparable fleet of coal power plants with CO2 capture and sequestration devices. For nuclear reactors, the investment cost amounts, approximately, to 2 billions euros per GeV. A program of 9000 reactors for 2050 would cost approximately 13500 billions Euros. During the 50 years when the deployment of this fleet would take place, the sales return of the electricity industry will be approximately 30000 billions Euros. Therefore the financial requirements for deploying a large fleet of nuclear reactors are within the capabilities of the power industry. The alternative to nuclear reactors are coal power plants. Their investment cost is only two thirds of that of nuclear ones, i.e. 1.3 G€/GWe. However, implementing the CO2 capture-sequestration scheme, an absolute necessity in view of the considerable CO2 emissions of coal plants, will increase the investment cost by at least 50%, which would suppress the relative advantage of coal versus nuclear plants. Furthermore the fuel and running costs of coal plants is much larger than that of nuclear reactors.
______ *
A converter reactor produces one fissile nucleus for each fissioning one. It is a breeder with infinite doubling time.
20
H. NIFENECKER
A serious problem for deployment of a large fleet of nuclear reactors will be the availability of skilled manpower and of specialized manufacturing equipment such as soldering of core containers. 4.5. THE QUESTION OF NUCLEAR WASTES
It is not possible, here, to treat the question of nuclear wastes with pertinence and completeness. Let us just stress that the use of breeder reactors reduces considerably the volume of highly radioactive wastes. For PWR or similar reactors the production of high level wastes amounts to 27.279 tons per GeV [5]. Of these, 26.047 correspond to Uranium, 266 kg to Plutonium and 20 kg to minor actinides. In a breeder fleet Uranium and Plutonium are resources and Minor Actinides may be reprocessed and incinerated. Thus the residual wastes are Fission products for a total of 946 kg. Among these only 63 are long-lived. Thus, for the same amount of energy produced wastes of breeders have a volume at least 30 times smaller than those of standard reactors. Furthermore, if Minor Actinides are reprocessed and incinerated the heat production after 300 years is almost a factor 100 less than that of UOx used fuels. 4.6. THE PROLIFERATION ISSUE
A multiplication of the number of reactors by an order of magnitude rises fears of uncontrollable proliferation of nuclear weapons. Evaluating whether such fears are justified requires examination of technical as well as political arguments. 4.6.1. Technical aspects of nuclear proliferation Any state that wants to acquire nuclear arms needs to have the technical skill necessary to make nuclear bombs and to master military grade nuclear matter production. In practice, only 235 U and 239 Pu can be used as nuclear explosive. 235 U allows simple (and low efficiency) designs of nuclear bombs. The critical mass of a simple (gun type) U235 bomb is in the range of 25 kg. Because of the spontaneous fissions of 240 Pu which tend to cause premature power excursion, 239 Pu bombs need a rather complex implosion device. But the higher neutron multiplicity of 239 Pu fission neutrons yields a smaller critical mass of around 11 kg. Military grade 235U is obtained by isotopic enrichment. The first used techniques of mass separation and gaseous diffusion are very electricity greedy. This makes detection and monitoring by IAEA inspectors rather easy, although the discovery of the importance of the Iraki program after the
ENERGY ISSUE AND SCENARIOS
21
first Gulf war, came as a surprise. The ultra centrifugation technique is much more discreet and difficult to detect, even for covert transport of equipment parts. The simplicity of conception of 235U bombs and the availability of ultracentrifuges makes this option the easiest for candidates to proliferation. 239 Pu are technically more difficult to make. Military grade 239Pu may be obtained in reactors with small fuel burn up. When reactors need to be halted for fuel extraction, monitoring by the IAEA inspector is easy since satellite observation can detect halted reactors and initiate inspection. CANDU reactors are equipped with continuous fuel discharge equipments operating while the reactor is working. It is easy to discharge weakly irradiated fuel, suited for military use. Chemical extraction of pure Plutonium may be simply carried out at the laboratory scale. Furthermore, CANDU and Graphite reactors may work with natural uranium. Most countries started their nuclear armament with such reactors. PWR or BWR reactors have never been used to obtain military plutonium because they have to be stopped for discharge. The fuel burn up is much higher than in the CANDU case (at least 35 Mega Watt days/ton versus only 5 Mega Watt days/ton), and, thus, the fraction of 240 Pu is so high that no one has ever made a bomb with plutonium extracted from used light water reactor fuel. Breeder reactors used fuels are planned to have very high burn ups (in the range of 200 Mega Watt days/ton). This will make the use of plutonium extracted from them difficult. Thus, provided the technical skill is present, any state has the possibility to produce 235 U and 239 Pu military grade material, independently of its nuclear electric power program. 4.6.2. Proliferation is a political question The non-proliferation treaty had the aim to limit the possession of the nuclear armament to those who had it already: USA, USRR, UK, France and China. Countries who did not have it pledged not to try to get it and got, in exchange, the right to have assistance in developing civil nuclear applications via the IAEA and by bilateral or multilateral agreements. Countries who had not signed the treaty were deprived of any help on nuclear matters from the IAEA as well as subscriber states. Selling strategic material to non-subscribers was forbidden to subscriber states. The main non-subscriber states were Israel, India, Pakistan, South-Africa (who signed later on). The stability of the treaty was insured by the cold war system, dominant states in the two camps keeping discipline among their allied. Recently, subscriber states tried to develop a nuclear armament: Iraq, Lybia, North Korea and, possibly, Iran. The non-proliferation treaty also
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implied that nuclear states should proceed to disarmament. Not much has been made in this direction. In general I believe that the motivation of states, who want to acquire a nuclear armament is that they want to get “sanctuarized“ by its possession. That there is some justification for that ambition can be seen if one asks what would have happened to Iraq if it had possessed the nuclear bomb. It is to be expected that states, who feel that they are threatened will, eventually, try to acquire, as discreetly and covertly as possible, nuclear capability. I believe that the present “unbalanced” situation where a few states have a monopoly over nuclear armament is basically unstable. The efforts made by “haves” to prevent “have nots” to become “haves” will, more and more, be interpreted as power policies. Already now, the multiplication of nuclear states present a risk of miscalculations and unwanted nuclear conflicts. The real danger, here, is a use of a nuclear arm without warning. I believe that a new treaty should be negotiated, treating all states on an equal footing. Since it is very doubtful that those in possession of nuclear armament will renounce it, one should accept that all states are untitled to it. In order to minimize the dangers of uncontrolled developments, all states should accept to submit their nuclear armament to international inspection. No armament apt to immediate use should be allowed. That is, airplanes in flight, submarines at sea carrying nuclear bombs should be banned. All missiles should be deprived of nuclear charge. This means that a minimum delay should be observed before use of nuclear armament could be made. Thus, clearly, nuclear arms could only be used as ultimate deterrent, and banned as surprise devices. Terrorist use of nuclear explosive devices is a really frightening prospect. Military grade 235U could be smuggled out or, even, provided to the terrorist groups by infiltrated militaries. 235U bombs are relatively easy to make, or could even be smuggled. I have no idea how to counter such threats except good secret services. It seems clear that an international and generalized inspection system of al military nuclear facilities might diminish the danger. 5. Proposal for Research and Development An important development of nuclear power requires very important R and D in three different but complementary fields: Breeder reactors, Fuel cycles including Thorium and Material and process studies. These lines of research imply specific work on the different types of reactors and their associated fuel cycle, namely.
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5.1. FOR THERMAL NEUTRON REACTORS •
for PWR reactors – Selective reprocessing: Extraction of Cs, Sr and M.A. This with the aim to reduce the cost and number of geological disposal facilities – Pyro chemical and electro metallurgical: Reprocessing techniques could be useful not only for standard reactor fuels but, also for that of advanced reactors as well as for molten salt fuels. – Th-Pu MOx fuel: In order to produce 233U. The use of 233U could be very useful for diminishing Plutonium and Minor Actinides production and limiting the technical possibility of proliferation.
•
for CANDU reactors – Use of Th-Pu and, then Th-U3 fuel: This will reduce the aptitude of CANDU reactors for proliferation and make use of their excellent neutronic properties. – Reprocessing of Th-U3 fuel: The Thorex or alternative processes should be evaluated at an industrial level. Fuel fabrication with high level gamma activity should be developed – Optimization of fuel regeneration: CANDU Th-U3 reactors are close to regeneration, giving the prospect of a very significant improvement of fuel use.
5.2. FOR FAST NEUTRONS REACTORS •
Sodium cooled: – Void coefficien: Present Sodium cooled breeders have a positive void coefficient which might be unacceptable for safety authorities. Geometries and, possibly, new fuels should be studied to prevent this unwanted property, – Core Recompaction: In case of partial or total fusion of the core present Sodium cooled breeders may lead to more reactive geometries. One should find geometries preventing such occurrence, – Th blanket: Study and experimental tests of a Thorium blanket in order to produce large quantities of 233U are recommended, – Reprocessing of Th blanket: in order to extract 233U.
•
Lead cooled reactors maybe a valuable alternative to Sodium cooled reactors since Lead do not react with air or water and since the vacuum coefficient of lead reactors is negative. The main problems that remain to be solved are:
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– Corrosion by hot lead under irradiation, – Pb-Bi alloys. Such alloys have a much lower melting temperature than lead. However a large production of the volatile polonium rises specific safety questions, as well as the availability of Bismuth. •
Molten salt cooled reactors. These are other alternatives to Sodium coolant with the nice property of being transparent. Chemical composition has to be optimized with regard to neutronic and corrosion properties as well as lowering the melting point.
•
Gas cooled reactors: – Reprocessing of refractory fuels, – Loss of coolant safety studies.
5.3. MOLTEN SALT REACTORS
Main R and D fields are based on neutron spectrum optimization, corrosionand fuel reprocessing. A demonstration reactor in the 10s thermal MW range is urgent in order to rise competences at a level comparable to that of other GEN IV reactors.
References [1] International Energy Agency (IEA): Key World Energy Statistics, http://www.iea.org. [2] Fourth Assessment Report of the Intergovernmental Panel on Climate Change (IPPC) 2007, http://www.ipcc.ch. [3] International Institute of Applied Systems Analysis (IIASA), http://www.iiasa.ac.at/webapps/ggi/GgiDb/dsd?Action=htmlpage&page=series. [4] Nifenecker, H., Heuer, D., Loiseaux, J. M., Meplan, O., Nuttin, A., David, S., and Martin, J. M., 2003, Scenarios with an intensive contribution of nuclear energy to the world energy supply, Int. J. of Global Energy Issues 19(1):63-77. [5] Nifenecker, H., Méplan, O., and David, S., 2003, Accelerator driven sub-critical reactors, IOP publishing, pp. 1-316. [6] IIASA-WEC, 1998, Global Energy Perspectives Report, http://www.iiasa.ac.at. [7] Bussac, J., and Reuss, P., 1985, Traité de Neutronique, Hermann. [8] Merle-Lucotte, E., Heuer, D., Allibert, M., Ghetta, V., Le Brun, C., Mathieu, L., Brissot, R., and Liatard, E., 2007a, Optimized transition from the reactors of second and third generations to the thorium molten salt reactor, Proceedings of the International Congress on Advances in Nuclear Power Plants (ICAPP), Nice, France (2007).
OUTLOOK ON GENERATION IV NUCLEAR SYSTEMS AND RELATED MATERIALS R&D CHALLENGES F. CARRÉ*, C. RENAULT, P. ANZIEU, P. BROSSARD AND P. YVON French Atomic Energy Commission (CEA) Nuclear Energy Division Saclay Center, 91191 Gif sur Yvette Cedex - France
Abstract:: Preoccupations of energy security and concerns about of the role that future reactors should play for minimizing long-lived radioactive waste led the French Government to formally approve a R&D strategy that gives first priority to developing a new generation of fast neutron nuclear systems and recycling technologies so as to assure a sustainable, proliferation resistant and environment friendly electricity production in the second half of the 21st century. This materializes though the goal of preparing the deployment of fast neutron reactors in the French generation fleet around 2040 that is being addressed by the CEA and industrial partners along two tracks: seeking for innovations to improve sodium cooled fast reactors current performances, and revisiting the potential merits of gas cooled fast reactors with safety features enhanced by the use of an innovative refractory fuel technology. This goal is closely linked to that of preparing the renewal of the spent fuel reprocessing plant of La Hague around 2040 through research and development on advanced recycling technologies capable of managing minor actinides so as to offer best nonproliferation features, as well as to minimize the long term burden of ultimate radioactive waste (heat load and radiotoxicity). The paper presents the R&D strategy to meet these goals, as it has been updated in 2006 to account for the presidential decision to launch design studies of a prototype of innovative sodium cooled fast reactor to be put in service by 2020. Furthermore, this strategy is devised according to the date of 2012 stipulated by the bill of June 2006 on the “management of radioactive materials and waste” as a major milestone to assess the potential of the considered fast reactor types for industrial deployment and transmutation of long-lived radioactive waste. The paper will also address the second objective of the French R&D programme on future nuclear energy systems that consists in supporting the strategy of industrial partners in the field of high temperature nuclear technologies for potential applications to the production of hydrogen and synthetic hydrogen fuels, or the delivery of process heat for other industrial applications. The prospects opened by the concrete implementation of the multilateral collaboration within the Generation International Forum and those opened by the perspectives of a strengthened cooperation on future nuclear fission systems within the 7th European R&D Framework Programme will also be covered, as well as the stakes of an efficient linkage between R&D programmes on the same nuclear system conducted at the national, European and broader international levels.
______ *
[email protected] V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008
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1. General Context of Nuclear Energy Development The rising awareness of a fast growing world primary energy demand at the beginning of the 21st century, together with considerations of energy security and concerns about greenhouse gas emissions led to acknowledge that all energy sources are to be used. Prospective studies, such as “Global Energy Perspectives” and “Energy to 2050 – Scenario for a sustainable future” respectively performed by the World Energy Council (1998) [1] and the International Energy Agency (2003) [2] show that, even with optimistic assumptions on the potential contribution of fossil and renewable energies, the use of nuclear energy will be necessary in countries where it can be developed safely and competitively (Figure 1).
Figure 1. IEA scenario of energy growth for a sustainable future from [2].
Furthermore, the safe operation of current power plants over the past 20 years, the increasing economic competitiveness of nuclear energy as fossil fuel prices escalate, as well as considerations of energy security pave the way for an active development of nuclear energy in Asia and a renaissance in the United States and Europe. This leads to anticipate an installed capacity of nuclear power of the order of 1000 to 1500 GWe by 2050, which is about four times the current installed capacity (370 GWe). Such a nuclear power capacity would require about 15 millions tons of natural uranium, if realized only with light water reactors which use less than 1% of the uranium (235U mainly) over a lifetime of 60 years. This amount, which is comparable to the estimated assured plus speculative reserves at a price below 130$/kg [3], incites to prepare the deployment by 2040 of fast neutron reactors with a closed fuel cycle that can burn more than 80% of natural uranium. Even if the situation around the middle of the century would not lead to a shortage of uranium because of additional reserves in phosphates or sea water, the rising cost of this resource that is already
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observed today (Figure 2), together with the accumulation of spent fuel, would drive the need to switch to fast neutron reactors with a proliferationresistant closed fuel cycle to achieve a more efficient use of uranium and minimize the ultimate long lived radioactive waste.
Figure 2. Recent evolution of Uranium market price (US$/lb).
Besides the goal of a sustained production electricity in the second half or the century, future prospects of cogeneration and needs of other energy products than electricity such as hydrogen, synthetic fuels and high temperature heat for industrial processes also trigger a renewed interest in high temperature reactors. Hydrogen is already needed to produce ammonia for fertilizers and to amend heavy crude oil, which will become an application of increasing importance in the next decades. Nuclear energy can also produce heat and/or steam to assist in the exploitation of tar sands or oil shells, and produce synthetic hydrocarbons as make-up fuels to gasoline from conventional oil resources. The rising perspective of the hydrogen economy and the subsequent recognition of the strategic nature of hydrogen technologies in the United States, Europe, Japan and other countries led to sign a multilateral agreement of International Partnership for Hydrogen Economy (IPHE) which paves the way for an important R&D programme on nuclear hydrogen production. In addition, continuing R&D on fuel and reactor technologies of 3rd generation light water reactors is of utmost importance to help optimizing these reactors in an evolving context over the entire century. Improving the conversion ratio of light water reactors that could permit to use up to 2% of the uranium energy content is of special interest to temporarily mitigate the consequences of rising natural uranium costs pending fast neutron reactors are deployed in the second half of the 21st century. In summary, research and development are essential to prepare the future of nuclear energy in three directions at least:
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•
Securing a sustainable electricity generation in the second half of the 21st century in a context of rising price of natural uranium, while assuring an adequate control of proliferation risks associated with the front and the back end of the fuel cycle (reprocessing and recycling);
•
Developing the production or cogeneration of other energy products than electricity (hydrogen, synthetic fuels, process heat for the industry…);
•
Innovating to adapt 3rd generation reactors to an evolving context over the 21st century.
2. The R&D Strategy in France on Future Nuclear Energy Systems Within the above context, prospective studies carried out by the CEA and industrial partners led to elaborate for France a R&D strategy on future nuclear energy systems for the medium and the longer terms (>2040), that aims at three complementary objectives: •
The development of fast neutron systems with a closed fuel cycle (sodium or gas cooled fast reactors) for a sustainable energy supply through breeding in the long term, and for managing actinides from light water reactors spent fuel in the medium term;
•
The development of key technologies for the nuclear production of hydrogen or the supply of high or very high temperature heat for the industry in close collaboration with industrial partners (high or very high temperature reactors and water splitting processes);
•
Innovations for light water reactor fuels, systems and high conversion cores to further optimize LWRs until 4th generation fast neutron systems are mature for industrial deployment around 2040.
The R&D priority is clearly put on fast neutron nuclear systems with a closed fuel cycle, the Sodium-cooled Fast Reactor (SFR) and the Gascooled Fast Reactor (GFR), owing to the general recognition of their capability to meet sustainability goals that include optimizing the use of natural uranium resources and minimizing long lived radioactive waste production (minor actinides) [4]. In addition, at a lower but significant level, R&D is conducted on high or Very High Temperature Reactors (V/HTR) in support to the AREVA project of multipurpose nuclear heat source ANTARES [5] presently viewed at 850°C for potential market needs from 2025 onwards. This R&D strategy proposed by the CEA and its industrial partners AREVA and EDF was formally approved by the French Government in March 2005. Since then, two major events came to sharpen the objectives
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and milestones of the French R&D programme on future nuclear energy systems: a presidential declaration in January 2006, and a bill promulgated on June 28, 2006 on the sustainable management of radioactive materials and waste. Both events set the objective to build and start in 2020 a prototype of 4th generation reactor intended to proceed with demonstrations of sustainable fuel cycle (“produce less waste and make a better use of uranium resources”). The decision of this prototype made it necessary to update the R&D strategy approved in 2005 through a second meeting of the French “Nuclear Atomic Energy Committee” that took place on December 20, 2006. This meeting that gathered representatives of the ministries of Research and Industry, CEA and industrial partners (EDF, AREVA) led to sharpen the R&D strategy on future nuclear energy systems along the following lines: •
Identify the sodium fast reactor (SFR) as reference fast neutron technology and select by 2012 innovative design features and technologies to be demonstrated in a 300-600 MWe around 2020;
•
Conduct in parallel a R&D programme on the gas fast reactor (GFR) with a major focus on key issues for the feasibility and performance of the concept (primarily a robust ceramic fuel and a safe management of depressurization accidents) with a view to preparing eventually a decision around 2012 for the construction of a 50-100 MWth experimental reactor abroad that could demonstrate around 2020 key principles and technologies of the GFR (the Engineering & Technology Demonstration Reactor (ETDR));
•
Proceed with the development in parallel of advanced fuel cycles that may allow recycling minor actinides and may replace current spent fuel treatment services around 2040 when fast reactors are deployed to eventually improve the ultimate waste form and the resistance to proliferation. Pre-industrial demonstrations will include pilot plants on the site of La Hague in France around 2015 to manufacture driver and experimental MOX fuels with minor actinides to be tested in the prototype of sodium cooled fast reactor.
Supporting AREVA’s industrial projects in the field of high temperature nuclear technologies for the production of hydrogen, synthetic hydrocarbon fuels, or the delivery of process heat for the industry remains an unchanged goal of the French R&D program on future nuclear energy systems. Besides, work on more prospective reactor types such as Supercritical water reactors (SCWR) and Molten salt reactor (MSR) or molten salt coolant is so far limited to assessing feasibility and performance issues and advancing special basic key technologies.
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An optimization of the global program is sought through common R&D pathways such as advanced closed fuel cycles for both fast neutron systems (SFR and GFR) and high temperature helium system technologies (materials, thermal insulation, purification…) between both gas cooled systems (VHTR and GFR). 3. Overview of R&D Programmes on Future Reactor, Fuel and Fuel Cycle Technologies The R&D programmes have been structured and funded in accordance to the strategic orientations and milestones presented above. This section develops the main challenges to be resolved by R&D and presents salient results obtained over the two past years. 3.1. SODIUM-COOLED FAST REACTOR (SFR)
The Sodium cooled Fast Reactor (SFR) is the reference technology for fast reactors. It may be considered for industrial deployment around 2040 since Europe, in cooperation with Japan, Russia and the United States, has acquired important expertise in this reactor type. However, innovations are needed for a Generation IV sodium cooled fast reactor to compete with Generation III LWRs in economics and safety. This will require systems’ simplification to reduce investment cost, enhanced safety with improved prevention and management of severe accidents, improved operability (fuel handling, maintenance and repair) to achieve high capacity factors, and advanced closed fuel cycles with multiple recycling of actinides offering appropriate resistance to proliferation and optimized waste forms. Given the maturity of the technology, the prototype reactor planned in France for 2020 will in the range of 300 to 600 MWe to demonstrate the innovations selected in 2012 to upgrade sodium cooled fast reactor performance and to open the way to a “first of a kind” commercial reactor. 3.1.1. High performance core with enhanced safety characteristics The optimized design of the SFR core combines a number of criteria potentially hard to reconcile: prevention and control of accidents (favourable reactivity coefficients), minimum recourse to breeding blankets (resistance to proliferation) and eventual incorporation of minor actinides (sustainability). The optimization of core design leads to consider dense and high thermal conductivity fuels such as carbide in place of oxide fuels, as well as high performance cladding materials as ferritic-martensitic, oxide-dispersed strengthened steels.
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Preliminary studies resulted in the definition of a 3600 MWt (~1500 MWe) reference core (Figure 3) with improved characteristics compared with those of the European Fast Reactor project (terminated in 1998). The core is self-breeding without radial fertile blankets, with improved safety parameters (reduced voiding effect). This result has been obtained while minimizing the inter-pin space which implies to make use of a cladding material with excellent swelling properties under irradiation. 10.25 9.87 9.48 9.09 8.70 8.32 7.93 7.54 7.15 6.77 6.38 5.99 5.60 5.22 4.83 4.44
Figure 3. SFR reference core design 3600 MWt (1500 MWe).
3.1.2. Innovative system and power conversion unit The motivation for investigating alternative power conversion systems to the steam turbine is twofold :simplification and/or increased compactness of the general system architecture (intermediate circuit) and elimination of water-sodium chemical interaction. The suppression of the intermediate system and direct coupling of the primary circuit with a Brayton power conversion unit using nitrogen or supercritical CO2 requires the development of specific sodium-nitrogen or sodium-CO2 heat exchangers. The use of nitrogen (or mixture of nitrogen with helium) implies increasing the temperature by 50 to 100°C in order to preserve an acceptable conversion efficiency. Issues associated to the supercritical CO2 option are (1) consequences of a possible ingress of gas in the primary system, (2) control of the chemical reaction with sodium and (3) design and qualification of specific turbines. Double-walled steam generators and the use of coolants reasonably compatible with sodium and water, such as liquid salts or metallic eutectic, are also investigated. Current investigations proceed along two tracks: a loop-type concept without intermediate circuit and gas power conversion system, and a pooltype concept with a simplified and compact design of the intermediate circuit with an alternative coolant.
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3.1.3. Overall system and component integration The main options investigated are integrated into point designs of innovative concepts to assess the consistency of all considered options and to support preliminary safety and performance assessments. Figures 4a and 4b below show typical lay-outs of innovative loop-type and pool-type designs.
Figure 4. a) Large pool-type concept (1500 MWe) with design optimizations (reduced size of the vessel), b) Modular loop-type concept (500 MWe) with gas conversion system (no intermediate circuit).
3.2. GAS-COOLED FAST REACTOR (GFR)
The helium cooled fast reactor is an innovative nuclear system with such attractive features as a chemically inert and optically transparent coolant, as well as a quasi-decoupling of the reactor physics from the state of the coolant. Other advantages of the GFR relate to its promise as a very/high temperature reactor (V/HTR) capable of producing hydrogen, synthetic fuels and process heat. On the downside, since gas is a poorer coolant than liquid metals, key aspects demonstrating the viability of the GFR include development of a refractory and dense fuel, and robust management of accidental transients, especially cooling accidents. Furthermore, the GFR has the potential to operate at high temperature (at least 850°C), which enables in principle the production of hydrogen or synthetic hydrocarbon fuels in a sustainable manner. The feasibility of the GFR is essentially linked to two demonstrations: the mastery (fabrication, thermo-mechanical behaviour) of a high fissile content refractory fuel, and the implementation of appropriate safety systems for the prevention and a robust mitigation of accidental scenarios (especially depressurization). Because there is no experience available on the GFR, a first step for demonstrating its feasibility is the operation of a
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50-100 MWth experimental reactor, the ETDR, to qualify its specific fuel, materials and operating principles. Ideally, R&D results expected by 201215 could support a decision to construct the ETDR, possibly as a European Joint Undertaking. The next step would be a prototype GFR that could come 10-15 years after. 3.2.1. A refractory fuel concept for the GFR The GFR fuel should comply with: •
an operating temperature of 1200°C in normal conditions and 1600°C in accidental conditions (to offset the gas poor efficiency as coolant);
•
a high fissile atom density and high thermal conductivity, thus triggering a renewed interest for carbide or nitride fuels;
•
a power density in the range of 100 MW/m3 as a trade-off between minimizing the plutonium content (lower boundary) and safety (slowdown of adiabatic heat-up).
Attempts to transpose attractive features of V/HTR fuel particles to fast neutron cores (fission product confinement, very high temperature resistance, thermal conductivity…) remained unsuccessful so far. Preliminary studies finally led to a reference fuel element consisting of fuel pellets arranged in cells within a ceramics clad plate (Figure 5).
Figure 5. Cellular fuel sub-assembly with composite cladding material (SiC-SiCf).
Each cell contains a fuel pellet composed of mixed uranium, plutonium and minor actinides. The cladding is made of composite silicon carbide reinforced with SiC fibres (SiC-SiCf) for an increased mechanical resistance. The sub-assembly is composed of a stack of such plates axially piled up in a triangular array and enclosed in a hexagonal wrapper. Significant progress has been made recently about the selection of constitutive materials(cladding, liner) to ensure leak-tightness to fission products and comply with requirements of thermo-mechanical integrity and adequate chemical compatibility between materials. An alternative design based on ceramics clad fuel pins is also investigated thoroughly.
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3.2.2. GFR design and safety studies The GFR reference design features a reactor of 2400 MWth unit power. Feasibility studies demonstrated the potential of high power concepts for increased margins on the fuel design features and for the optimization of both safety and economy. Indeed, larger core sizes afford to more easily accommodate lower fractions of actinide compound in the fuel assembly and do not emphasize threshold effects in the safety systems needed to safely manage accidental conditions. Initially envisioned with a direct cycle, the GFR reference design moved to the indirect-combined cycle option. The removal of residual power in case of accidental LOCA conditions is assured by gas circulating in forced convection for the short term (actuation of active systems) and in natural circulation for mid and long term. A guard containment is necessary to maintain the 5-10 bars pressure required for an efficient core cooling in natural circulation (coherent with the use of metallic guard containment). 3.2.3. Definition of the characteristics of GFR-type experimental reactor The ETDR is a small power experimental reactor (50 MWth) designed to be the pathway for direct qualification of GFR fuels (Figure 6), materials and other key specific technologies, as well as for testing GFR specific safety principles and associated systems.
Figure 6. ETDR system layout (50 MWt).
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Because of the very innovative character of the fuel type selected as a reference for the GFR, an evolutionary approach was advocated, involving two successive core configurations: a start-up core based on conventional pin sub-assembly with metallic cladding (ODS), then a demonstration core using the prototypic GFR fuel element (plate type (U,Pu)C-SiC subassembly). An intermediate stage is planned for preliminary testing of a few prototypic sub-assemblies inserted in the pin-type core. It was concluded that the power density for the demonstration core should be similar or slightly above that of the GFR. A 50 MWth reactor with a power density of 100 MW/m3 is currently retained as an acceptable trade-off between irradiation performances and economy. 3.3. R&D PROGRAMMES ON FUEL AND FUEL RECYCLING
The development and qualification of an appropriate and consistent set of technologies for spent fuel treatment and fuel re-fabrication is an essential component of a fast neutron system strategy. The head-end processes, aiming at dissolving the spent fuel refabricating fuel to be recycled are specific to the type of fuel (MOX fuel, carbide…). In return, the separation process is largely independent of the type of fuel (SFR or GFR) but it is rather determined by options selected for the recycling of plutonium and minor actinides: •
The COEX process, derived from PUREX, is able to manage uranium and plutonium together, but does not partition minor actinides which therefore are co-managed with fission products to constitute vitrified ultimate waste;
•
DIAMEX and SANEX processes which have been demonstrated in 2005 at laboratory scale to partition lanthanides and actinides in two steps can be combined with COEX to manage separately minor actinides for recycling eventually in breeding blankets or transmutation targets;
•
Based on above advances, the GANEX process has been developed to achieve a grouped extraction of actinides and enable a co-management of all actinides together (uranium, plutonium and minor actinides) so as to achieve a homogeneous recycling in core subassemblies with the intent to provide an improved resistance to proliferation.
Considering the time scale for industrial deployment of these various separation processes, it is important to keep all options open, as they could be deployed in sequence in accordance with international standards of non proliferation and with a goal of global optimization of the fuel cycle backend from the technical and economic viewpoints.
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As far as spent fuel treatment is concerned, the hydrometallurgy remains the reference technology. Pyrochemistry is an alternative option to be more deeply investigated, especially for metallic fuels. 3.3.1. Minor actinide recycling As stated above, two modes of minor actinide recycling strategies in fast neutrons reactors are being investigated (Figures 7a and 7b): homogeneous recycling in fuel subassemblies of the active core together with uranium and plutonium, and heterogeneous recycling in blankets at the core periphery.
Figure 7. a) Homogeneous recycling (Generation IV reference), b) Heterogeneous recycling minor actinides bearing blankets at the core periphery.
Important results have been obtained since 2005 on separation and fuel re-fabrication techniques. The potential of different processes for individual minor actinide partitioning (DIAMEX, SANEX) has been confirmed and a first test of oxalic co-conversion of actinides (uranium, plutonium, neptunium, americium) has been successfully realized. Exploratory experiments have also been conducted using the sol-gel technique, alternative to the oxalic technique, for the co-conversion of U-Pu (Figures 8a and 8b).
Figure 8. a) Microspheres of hydroxides U(VI)-Pu(IV) (85/15), b) Microspheres of hydroxides U(IV)-Pu(III) (85/15).
3.3.2. Fabrication of co-precipitated MOX fuel UPuO2 powders obtained by oxalic co-precipitation (COEX process) have been used for the fabrication of first samples of MOX fuel. Two tracks are being investigated:
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•
Fabrication from powder initially at the targeted plutonium content,
•
Fabrication from high plutonium content powder (+/– 45%) diluted with UO2 for adjustment to the aimed plutonium content.
Figures 9a, 9b and 9c compares the microstructure of MOX fuels obtained by co-precipitation and by more conventional fabrication processes (MIMAS, COCA).
Figure 9. Comparison of MOX fuel microstructures obtained by co-precipitation (first track) and more conventional fabrication processes (MIMA, COCA): a) UPuO2 MIMAS 11%Pu, b) (U, Pu)O2 6% Pu (COCA), c) (U, Pu)O2 27.5%Pu (co-precipitation) .
This shows the very good homogeneity of plutonium distribution in the co-precipitated MOX fuel. These results look promising in terms of thermomechanical behaviour and capability for fission production retention in the microstructure. The behaviour of MOX fuel manufactured from COEX separation will be tested under irradiation through a dedicated irradiation experiment in PHENIX beginning in 2008. 3.4. VERY HIGH TEMPERATURE REACTOR (V/HTR)
The HTR/VHTR (V/HTR) system corresponds to a thermal neutron heliumcooled reactor concept operated at high (850°C) or very high temperature (>950°C). AREVA currently develops the project ANTARES of multipurpose High Temperature Reactor (600 MWt, 850°C at core outlet) that delivers process heat through an intermediate heat exchanger (IHX). Electricity can be produced with a combined power conversion cycle that consists in a Brayton cycle (nitrogen/helium mixture 80/20) coupled to a classical Rankine cycle with steam superheat. This option takes the best benefit from a gas cycle in the high temperature range (850°C) and steam cycle below 550°C, with a potential for high efficiency: 47% or more. Essential R&D issues for V/HTR concepts are: •
The qualification of materials and components capable to sustain the service conditions of high temperature helium systems,
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•
The development of compact gas-gas heat exchangers with high thermal efficiency,
•
The mastering of the fabrication procedure of particle fuel.
Major R&D milestones to support a V/HTR prototype reactor around 2020 are a sufficient confirmation by 2012 of design features and key technologies to document a Preliminary Safety Analysis Report and a full qualification by 2015 to ground a decision to build the facility. 3.4.1. High temperature materials Typical V/HTR operating conditions require materials capable of sustaining service conditions in the temperature range of 350 to 1650°C: •
(~350-550°C) for the reactor pressure vessel and internals. Ferriticmartensitic steel 9Cr-1Mo (T91) that is selected as reference material has been qualified for thick wall welding. Further developments on T91 relate to additional qualification in areas such as welding properties, mechanical characteristics, emissivity, corrosion behaviour and irradiation effects;
•
(~650-950°C) for the materials of the primary circuit, heat exchangers and turbine. The candidate materials (Inconel 617 and Haynes 230) are being characterized and compared in terms of mechanical behaviour, ageing, weld-ability, corrosion with impure helium in order to select the reference material for the IHX in 2008. Besides, investigations are conducted on Fe-base and Ni-base ODS (Oxide Dispersed Strengthened) alloys, in order to develop high performance grades for very high temperature applications (up to 950°C);
•
(~950-1650°C) for core structural materials. The qualification of graphite grades for V/HTRs and that of composite ceramics (Cf/C, SiCf/SiC) for the VHTR and GFR core materials goes through the successive steps of screening tests and characterization, before and after irradiation.
3.4.2. Gas-gas heat exchangers A major technical issue is the development of the Intermediate Heat eXchanger (IHX) for an indirect power conversion into electricity and other energy products, and eventually also the recuperator for a direct cycle Brayton electricity generation (Figure 10). For a power conversion system that uses gas as working fluid, the challenge is to transfer several hundreds of megawatts from one gas to another (He/He or He/He+N2) while minimizing the temperature difference so as to optimize the power conversion efficiency. A heat transfer area of several 10,000s m2 is required while
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keeping outside dimensions compatible with an acceptable overall size for the reactor building. Among potential advanced technologies, plate concepts appear more promising because of their compactness. Their thermal performance is also better qualified owing to prior use in other industrial applications. Different plate concepts appear as good candidate technologies: Printed Circuit Heat Exchangers (PCHE), Plate Machined Heat Exchanger (PMHE), Plate Fin Heat Exchangers (PFHE), and Plate Stamped Heat Exchanger (PSHE). Each concept is being tested and assessed in terms of fabricability and mechanical resistance in normal and accidental conditions. These concepts are being currently investigated and compared at the scale of mock-ups in terms of thermal performances, manufacturing techniques and predicted lifetime, so as to select the most suitable technology by 2010.
Figure 10. AREVA’s ANTARES Project of multipurpose of high temperature heat source.
Molten salts are being assessed as another candidate coolant to transfer heat from the reactor to high temperature applications. 3.4.3. V/HTR TRISO fuel particle The V/HTR safety approach and subsequent design options are essentially based on the fact that fission products remain confined inside the fuel
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particle in normal operating conditions and accidental situations. In order to demonstrate adequate performances in this respect, it is necessary: •
To establish an industrial manufacturing process capable to produce fuel particles with very high quality (i.e. failure rate of particles below 10–5);
•
To qualify experimentally the behaviour of such particles in irradiation and temperature conditions relevant to HTR normal operation (1300°C) and accidental situations (1600°C) with a failure rate below 1.5 10–4.
These objectives are covered by a comprehensive program on the fabrication and characterization (before and after irradiation) of standard TRISO (UO2/SiC) particles. Uranium particle fuel is currently fabricated at CEA by the GAIA laboratory scale fabrication line (Sol-Gel and CVD processes) that was put in service in mid 2005 (Figure 11). The production has reached a sufficient quality today (diameter, sphericity) to prepare irradiation tests in the materials testing reactor OSIRIS that should begin in 2009.
Figure 11. TRISO fuel particles (natural uranium) fabricated by the GAIA facility.
A first batch of coated particles is being used for the fabrication of fuel « compacts » that will be loaded in the experimental reactor OSIRIS at the beginning of 2009. For the objective of very high temperature (>850°C), the R&D program on fuel is focused on innovative particle fuel forms that would allow an enhanced burn-up performance (20% FIMA) and a higher temperature of helium (up to 1000°C). UCO/SiC concept limiting CO production and internal gas pressure at high burn-up, or UO2/ZrC accommodating higher temperature in normal and accidental conditions, together with adequate techniques for the treatment of spent fuel of these types.
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4. Transition From Light W ater Reactors to Fast Reactors The path for Europe towards a closed fuel cycle depends on political, technical and financial contingencies. Indeed, such an evolution implies: •
Integration within the broader worldwide policy of safeguarding and proliferation resistance,
•
The renewal and addition of new plants for processing spent fuel and re-fabricating fuel to be recycled with advanced processes,
•
A funding process for the new investments required, especially in fuel cycle plants.
Figure 12 illustrates the transition from current reactors and a spent fuel treatment plant in France to Generation IV technologies around 2040. It shows how deployment of a progressive separation and recycling strategy allows the technical capabilities of fast neutron reactors and advanced recycling modes with a co-management of actinides to be implemented around 2040. This strategy is flexible enough to be adapted to the time line and type of fast neutron systems to be developed. Furthermore, it can preserve the possibility (if ever compatible with the technical and economic optimisation of the fuel cycle) of an integral recycling of actinides (Figure 13) capable of drastically reducing long term potential radiotoxicity and decay heat of the ultimate waste (Figure 14), thus eventually making the fuel cycle more resistant to proliferation.
Figure 12. Scenario of renewal of French nuclear plants and fuel cycle plants: a) 2040 – Deployment of Fast neutron systems (SFR or GFR), b) 2040 – Renewed spent fuel treatment plant at La Hague (grouped actinide extraction).
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Figure 13. Integral recycling of actinides (U-Pu + minor actinides) with Unat or Udep as make-up fuel.
Figure 14. Evolution of radiotoxicity of ultimate waste for direct disposal, recycling of U-Pu only, and integral recycling of actinides.
5. Major Stakes in International Collaboration The Generation IV International Forum that was launched by the UD-DOE in 2000 [6] and the initiative INPRO [7] under the auspices of the IAEA constitute today the most active multilateral framework of collaboration on 4th generation nuclear systems. In a context where the European contribution to the international R&D effort on future nuclear energy systems is not at the level that nuclear power currently represents for Europe, good prospects exist today to improve this unsatisfactory situation.
GEN. IV SYSTEMS AND MATERIALS CHALLENGES
43
5.1. STATUS OF 4TH GENERATION NUCLEAR SYSTEMS IN THE EUROPEAN R&D PROGRAMME
Diverging national visions of nuclear energy in Europe have limited work on future fission systems so far, although there is a visible and reasonably funded R&D programme on safety, waste management and radioprotection of current power reactors within the Euratom R&D framework programme. Strengthening the effort in Europe on future nuclear fission systems is in full agreement with Euratom formally joining the Generation IV Forum on May 11, 2006 and with the orientation of both Green Books issued by the European Commission in 2000 and 2006 for sustainable energy development in Europe [8, 9, 10 and 11]. Initiatives in this sense are all the more urgently needed in the 7th R&D Framework Programme (2007-2011). So far, the European R&D Framework Programme, together with national initiatives, has been successful in preserving competence in the nuclear field through a network of excellence, cooperative actions and by promoting the European dimension of education in nuclear engineering. Future nuclear energy systems will require all aspects of expertise involved in the design, technology development and safety demonstrations for light water reactors, nuclear fuels and fuel cycle processes including design and safety studies, fuel, materials and component technology, and fuel cycle and waste management processes and technology. Beyond this, the search for breakthroughs beyond LWR technologies, such as very high temperature and/or fast neutron systems with full actinide recycle, will require additional and non-nuclear-specific skills in materials science, very high temperature materials and components (composite C/C and ceramics), separation chemistry, and thermo-chemical and electrochemical processes for water splitting and hydrogen production. Other competence is required, either for prospective studies (uranium resources, hydrogen market) or assessments in economics, safety or proliferation resistance. Updating assessment methods may be effected nationally, through collaboration or through participation in INPRO and/or corresponding methodological groups of the Generation IV Forum (Economics modelling, Risk and safety, Proliferation resistance and Physical protection). Development of networks of experts in these fields in Europe may be considered. 5.2. RENEWAL AND EVOLUTION OF LARGE R&D FACILITIES IN EUROPE
Materials testing reactors and hot laboratories are essential R&D infrastructures to explore innovative research on fuels and fuel cycles, key technologies for 4th generation reactors:
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•
HFR, OSIRIS, LVR-15 and Jules Horowitz Reactor (JHR) in 2014…
•
ITU, ATALANTE, FZK, SCK, ACTINET core group facilities…
•
Bor-60, PHENIX.
Sizeable non-nuclear facilities are also needed to resume R&D on high temperature gas-cooled reactors, such as the particle fuel laboratory scale fabrication line, test benches for high temperature helium system technology (850, 950°C and above) and experimental loops in the 1 to 10 MW range for component mock-up tests. France is engaged in re-establishing such a basic R&D capability. Equipment costs for this effort on a European or a broader international basis are estimated to be about 100-300 M€. The cost for R&D equipment to resume the industrial development of sodiumcooled fast reactors is about the same. The decision to build a prototype of 4th generation fast neutron reactors in France by 2020 is important for Europe to remain credibly involved in R&D on fast neutron systems after PHENIX is shut-down in 2009. Other European countries, and possibly other international partners, may decide to develop experimental or prototype reactors of other Generation IV systems of specific interest, with invited external participation. Even though the costs of such facilities range from 0.5 to 2 B€, the prospect of such prototypes does not appear excessively ambitious in comparison with the number of experimental and prototype reactors operating in Europe in the 1980s: •
Sodium cooled fast reactors: – The experimental reactor KNK II (17 MWe) in Germany (1978-1991), – The prototypes PHENIX (250 MWe) in France (1973-2009), PFR (234 MWe) in the United Kingdom (1975-1994), BN600 in Russia (1980 Æ),
•
High temperature reactors (HTRs): – The experimental reactors Dragon (1964-1975) in the United Kingdom and AVR (13 MWe) in Germany (1966-1988), – The prototype THTR (300 MWe) in Germany (1983-1989).
5.3. TOWARDS A “SUSTAINABLE NUCLEAR FISSION TECHNOLOGY PLATFORM”
One of the major initiatives to integrate and strengthen R&D work on future nuclear energy systems in Europe is the proposal to organize fissionoriented R&D work in the European Union into a “Sustainable Nuclear Fission Technology Platform” (Figure 15). This is intended to address strategic R&D for European policy makers and industrial projects in the
GEN. IV SYSTEMS AND MATERIALS CHALLENGES
45
medium term such as LWR safe operation and life extension, fast neutron reactors with a closed fuel cycle, and high temperature reactors for cogeneration. Such an R&D organization on nuclear fission in Europe would help: •
directing R&D towards strategic goals,
•
identifying large experimental equipment for this research, so as to plan for investment or refurbishing consistent with the scale of Europe, and also
•
preparing decisions to realize and operate prototypes of 4th generation reactors within the framework of joint ventures.
It would also facilitate developing synergistic R&D with that for fusion systems already organized internationally. An appropriate level of R&D funding for 4th generation systems in Europe, should be comparable to the US and Japan in the same area, currently 300 MUS$/year exclusive of investments in experimental and prototype reactors. At stake are preserving the advance of the European nuclear industry in the world, preparing the deployment of sustainable fast neutron systems with advanced recycling modes in Europe by 2040, and promoting the development of key technologies for non-electricity applications of nuclear energy. Nuclear Fission Technology Platform: Operation LWR Safety & Economics
SRA and Platform
Innovative Materials and Fuels
V/HTR Process Heat, Electricity & H2
Simulations and Experiments: Reactor Design, Safety, Materials and Fuels Strategic Research Agenda; Platform Operation
TSO: Mirror Group
Training and R&D Infrastructures
Fast Systems With Closed Fuel Cycles Sustainability
EU High Level Group on Nuclear Safety & Security
Geological Disposal Technologies, design, safety assessment
Waste Management (CARD)
Figure 15. Goals and components of the “Sustainable Nuclear Fission” Technology Platform.
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6. Future Prospects Prospective studies carried out by the CEA and industrial partners led to elaborate for France a R&D strategy to support the development of nuclear reactors for the continuous and sufficient supply of energy over the 21st century. To prepare the deployment of fast neutron systems in the French reactor fleet around 2040, CEA and its industrial partners AREVA and EDF have promoted a dual approach including the search of innovations for the SFR and in parallel the evaluation of the GFR promising potentialities. Closely linked to R&D on reactor concepts, a consistent programme on fuel cycle technologies is also conducted with the objective to achieve significant progress on the nature of final waste and on proliferation resistance towards an optimized management and recycling of minor actinides. The overall R&D program is oriented towards the construction of a prototype of 4th generation reactor in 2020, subsequently to the selection of the main design options in 2012. The 4th generation nuclear energy systems also support the goal of generating other energy products than electricity such as hydrogen and synthetic hydrocarbon, or process heat for industry. In this respect, CEA supports AREVA’s ANTARES project of 600 MWth multipurpose high temperature nuclear heat source (850°C). These fast neutron and high temperature reactors both require breakthroughs beyond 2nd and 3rd generation light water reactors (hence recognition of the switch to a new generation). They pose real technological challenges for nuclear fuels, systems materials and technology, spent fuel treatment processes and non-conventional applications. Their application in recycling spent fuel for efficient use of the uranium and burning long-lived radioactive waste, leads to consideration of these 4th generation reactors as “nuclear systems” consisting of reactor, fuel and fuel cycle, optimised as a whole. Such technology challenges require cooperation among European research partners (National Laboratories, Universities and other research organizations) and industrial partners on corresponding R&D objectives. They also require development through international cooperation, to share the cost of innovation, experimental reactors and prototypes in Europe. Securing sustainable electricity generation in the second half of the century suggests the need for possible deployment of at least one type of fast neutron reactor in Europe around 2040. This in turn suggests work along two complementary lines of research: (1) innovation to develop a new generation of sodium cooled reactors, already a mature technology, and (2) diversifying risks and market opportunities by developing at least one other
GEN. IV SYSTEMS AND MATERIALS CHALLENGES
47
reactor type such as the gas cooled fast reactor with properties inverse to sodium cooled systems or lead cooled fast reactors. The French decision to build a prototype of sodium cooled fast reactor by 2020 as a successor to PHENIX, could permit a comparative evaluation of the merits of alternative fast reactor types to select a second technology and build an experimental test reactor in another European country. With a core outlet temperature of at least 850°C, the gas fast reactor concept could also represent a nuclear technology useful for high temperature applications. This stresses the need to fully integrate R&D work on future nuclear energy systems at the European level. This makes sense in view of the important share of nuclear electricity in Europe (32%) and of the fact that Euratom affiliated countries joined the Generation IV International Forum in 2006. Even though some European countries still maintain a nuclear moratorium, the preparation of the 7th European R&D Framework Program (2007-2011) offers prospects for strengthening work in this field. This could increase exchanges with the Generation IV International Forum, an essential condition to achieve balanced cooperation with major nuclear partners such as the United States and Japan, which spend about 300 MUS$/year each on future nuclear systems, and also with Russia and China. Last but not least, another goal for European stakeholders of nuclear research and industry, is to become sufficiently involved in international R&D on future nuclear energy systems to benefit from past experience in precursor reactors of Generation IV technologies (principally sodium fast and high temperature reactors), to keep current with advances in technologies such as sodium cooled fast reactors and fuel cycle processes, and ultimately to become involved in development of standards and commercial technologies strategic for Europe and international markets. Overcoming the current limitations of the European R&D Framework Program on Fission and maximizing European contributions to international R&D on advanced nuclear technologies, requires an integrated organization gathering research laboratories and industry. It is essential to identify and set strategic priorities on R&D needs and necessary competence, to define needs and elaborate plans for new large experimental facilities such as material testing reactors, hot laboratories, large experimental loops, and to take decisions for the construction of experimental or prototype reactors within the joint undertakings. Organizing R&D on nuclear fission in Europe along these lines would not only help direct R&D towards strategic goals and make Europe a major partner of international collaboration, but would also increase the European potential to profit from building and operating prototypes of 4th generation reactors. Such a strategy would offer the best prospects for European stakeholders in nuclear energy to preserve their current leadership.
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References [1] Global Energy Perspectives to 2050 and beyond, 1998, Report of the World Energy Council with scenario studies with the International Institute for Applied Systems Analysis, Cambridge University Press. [2] Energy to 2050, Scenario for a Sustainable Future, 2003, Report of the International Energy Agency. [3] Uranium 2005: Resources, Production and Demand, 2006, Joint report by the OECD/NEA and the IAEA Nuclear Energy Agency homepage http://www.nea.fr. [4] Fast Reactors R&D Strategy in France for a Sustainable Energy Supply and Reduction of Environmental Burdens, 2005, Carré, F., JAIF International Symposium, Tokyo, March 24. [5] ANTARES, 2006, Ready for the Combined Heat and Power Market, Gauthier, J. C., Lecomte, M., Carré, F., and Renault, C., HTR2006, Sandton, Republic of South Africa, October 2-4. [6] A Technology Roadmap for Generation IV Nuclear Energy Systems, Dec. 2002, GIF002-00, Issued by the U.S.DOE Nuclear Energy Research Advisory Committee and the Generation IV International Forum - http://gif.inel.gov/roadmap/. [7] IAEA International Project on Innovative Nuclear Reactor (INPRO), 2006, Final report of phase 1. [8] Towards a European Strategy of Energy Security, Nov. 2000, European Commission, COM (2000) 769. [9] Green paper on “A European Strategy for Sustainable, Competitive and Secure Energy”, March 2006, Commission of the European Community, COM (2006) 105. [10] An Energy Policy for Europe, 2007, Communication from the Commission to the European Council and the European Parliament. [11] The Role of Nuclear Power in Europe, Jan. 2007, World Energy Council.
FUNDAMENTALS OF NEUTRONICS: REACTIVITY COEFFICIENTS IN NUCLEAR REACTORS P. REUSS* Emeritus Professor Institut National des Sciences et Techniques Nucléaires, Point courrier 35, CEA/Saclay, 91191 GIF-SUR-YVETTE CEDEX, France
Abstract: The objective of this lecture is to discuss some aspects of importance for the safety performance of a nuclear fission system, related to the production of energy, the incineration of nuclear wastes and/or the fuel reprocessing. This presentation will be first devoted to the analysis of the basic principles of the fission-neutron chain reaction behaviour. The design parameters will then be introduced, and some exemplary systems described. Particularly, the aspects concerning, from the neutronics point of view, the choice of the materials (theme of this autumn school) will be emphasized: main options for the core design, reactivity effects, conditions for a working stability, neutron utilization. The coupling between the neutronics and the other branches of physics involved in the reactor design (thermal-hydraulics, irradiation of materials, safety...) will be recalled in the conclusions.
Foreword The present concerns of the public opinion which more and more emerge in our word have brought a new interest in nuclear fission energy, as this source appears to be safe, competitive and almost without any emission of carbon dioxide [3]. However, it also appears that the known reserves of uranium will be rapidly exhausted, with the present mode of utilization of uranium using merely the isotope 235 which represent only 0.72 % of the natural uranium. It is the reason why the specialists of numerous nuclear laboratories have studied, since one or two decades, new concepts of fission reactors (said “forth generation systems”), safer, more competitive, and above all more economical in natural resources [1]. The Generation 4 Forum compared the various proposals and selected the six most promising ones. The young scientists and engineers will now have to develop these projects: the objective of this autumn school is to contribute to the training they will need to carry out this important mission.
______ *
[email protected] V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008
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P. REUSS
The topics of this school concern the materials which will be needed for the forth generation reactors. The choices must be compatible with the criteria of the neutron physics (or neutronics): it is the reason why some basic knowledge of this physics, specially about the reactivity effect, is necessary. The present paper is an introduction to neutronics through analyses of safety coefficients of PWRs. 1. Fission Chain Reaction and Reactivity [5, 6, 7, 8, 9, 10, 17] The fission chain reaction in the nuclear reactors is based on the facts that fissions yields neutrons and that neutrons can induce new fissions, specially of fissile material: one natural nuclide, uranium 235, and some artificial nuclides as plutonium 239, plutonium 241, uranium 233. The neutrons are emitted by fission around 2 MeV (about 20 000 km/s). At this energy, the cross-sections are small. Then it is necessary to use a material quite rich in fissile atoms in order to reduce the number a sterile captures by other materials. This is the way of the “atomic bombs” and of the “fast (neutron) reactors”; these last ones are characterized by a good neutron balance which allows, for instance, some breeding. A second way takes advantage of the very large cross-sections of the fissile nuclides for the slow neutrons. Then, in addition to the fuel, another material is used, called “moderator”, to slow down the neutrons by successive scatterings until they come close to the thermal equilibrium at an energy of a few tenths of eV and a velocity of a few km/s. For these slow neutrons, it is possible to get a chain reaction with a much less enriched fuel, and even with natural uranium if one of the best moderators is used (heavy water, graphite, beryllium or beryllia [beryllium oxide]). Most of the world power reactors are such “thermal (neutron) reactors” and, among them, the water reactors (PWR and VVER, BWR) using water as moderator; because of a not negligible capture cross-section of hydrogen for the thermal neutrons, the fuel of these systems must then be enriched at least to a few percents in fissile nuclei. The equilibrium or imbalance of a chain reaction is characterized by the multiplication factor k defined as the mean ratio between the number a neutrons of a given generation and the number of neutrons at the preceding generation or, equivalently, the number a fissions at a given step and the number of fissions at the preceding step. Indeed, these numbers evolve like N(0) kn with the generation n: •
if k > 1, these numbers increase when time goes on: the system is overcritical;
•
if k < 1, these numbers decrease: the system is under-critical;
FUNDAMENTALS OF NEUTRONICS •
51
if k =1, these numbers remain constant: the system is critical: its power (proportional to the number of fissions per unit of time, or to the number of fissions per generation) is constant.
In its nominal state, a reactor must therefore be critical; to start it or to increase its power a transient (small) over-criticality is created; to reduce the power or to stop the system, a transient, resp. permanent, undercriticality is created. (In a nuclear weapon, a larger as possible overcriticality is looked for.) The multiplication factor k is the product π ν of the probability π that a neutron emitted by fission induces a new fission and the mean number ν of new neutrons that this fission will yield. This last number is a nuclear characteristic. To adjust k, π must therefore be adjusted: in practice, some movement of material allows to modify the respective probability of fission and (sterile) capture; generally, control rods are used: they are made with a highly capturing material (boron, cadmium, etc.): when inserted, the captures increase and k decreases, and conversely when extracted. Instead of k, the reactivity ρ is often used:
ρ = (k −1) k It allows to characterize the criticality by 0 instead of 1; as ρ is small in practice, it is generally expressed in pcm (p.c.m: pour cent-mille, i.e. 10–5). A simple modelling of the reactor, shows that its power varies like eωt or a linear combination of such functions [11, 12]. The dominant mode is characterized by a value of ω of the same sign as ρ , i.e. increasing if ρ > 0, decreasing if ρ < 0 and constant if ρ = 0. The other values of ω , always negative, correspond to transients due to the several groups of delayed neutrons (neutrons emitted a few seconds after the fission and a preliminary beta decay of a “precursor”; average delay: about 10 seconds). The key parameter for the kinetic behaviour is the proportion β of the delayed neutrons, for instance: • •
β = 680 pcm for uranium 235, β = 220 pcm for plutonium 239.
If the reactivity is less than this proportion, the kinetics remains slow and quite manageable because the delayed neutrons play the major role despite their small proportion (the life of the “prompt” neutrons is very short, because they travel rapidly through the matter: between 10−7 s and 10−4 s according to the type of the reactor). Note that a reactor using plutonium is a little more “nervous” than a reactor using uranium because β is smaller. If ρ exceeds β , the power increase very rapidly (prompt critical mode). Fortunately, in practice and for all the types of reactors, temperature feedbacks stop the explosion.
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2. Analysis of a Neutron Balance These orders of magnitude show that a precise analysis of the multiplication factor is necessary. It is the reason why it is useful to present the neutron balance obtained in practice by a computer code in a form allowing the physical analyses. The most famous and usual presentation of such a neutron balance is due to Enrich Fermi himself, who elaborated the basic neutron physics during the late 1930’s and early 1940’s. At that time, the formulae below were used to calculate the multiplication factor. Nowadays, a precise spectrum and space calculation of the neutron tux is performed by a computer code (for instance, in France, by the CEA code APOLLO2) and the factors are reconstituted a posterior for a physical analysis. First of all, Fermi distinguished the reactions occurring in the neutron multiplying medium and the neutrons escaping the reactor (leakage):
k = k∞ PNL where k∞ would be the multiplication factor without leakage, that is to say, if the reactor would be innate, and PNL the non-leakage probability (Figure 1).
Figure 1. Variation of the “effective” multiplication factor k of a reactor (here spherical) as a function of its size, showing the critical size (depending on the used materials).
For a bare homogeneous reactor, this last one can be written:
PNL =
1 1 + M 2 B2
where M 2 is the sixth of the mean square distance travelled by the neutrons and measured as the crow flies, and B2 a geometrical parameter characterizing both the dimensions and the form of the reactor; for instance: • •
B 2 = π 2 R 2 for a sphere of radius R, B 2 = π 2 H 2 + j 2 R 2 for a cylinder of height H and radius R ( j = 2.405: first zero of the Bessel function J0),
•
B 2 = π 2 a 2 + π 2 b 2 + π 2 c 2 for a parallelepiped of edges a, b and c.
FUNDAMENTALS OF NEUTRONICS
53
According to the neutron histories in a thermal neutron system*, Fermi broke down k∞ into four factors:
k∞ = ε p f η The fast fission factor ε takes into account the rare fissions of non-fissile materials as uranium 238 induced by the very fast neutrons (typically of energy greater than 1 MeV) which produce few extra neutrons beyond the main production by the fissions of fissile material induced by thermal neutrons. For instance, for a PWR, ε is about 1.07. Here, we shall not comment further this corrective term. The resonance escape probability or antitrap factor p [13-16] is the probability for a neutron to escape the numerous resonances, specially of uranium 238, which risk to capture it without fission during its slowing down. Order of magnitude: 0.75 for a PWR. The classical formula giving this factor was established by Fermi:
⎡ V f N f I eff ⎤ p = exp ⎢− ⎥ ⎢⎣ Vm N m (ξ σ s )m ⎥⎦ where f and m refer to the fuel and the moderator respectively, and where V are the volumes, N the atomic or molecular concentrations, Ieff the “effective integral” of the fuel capture cross section (see Figures 2 and 3), σ S the moderator scattering cross-section and ξ m the mean gain of lethargy† during a neutron scattering by the moderator. The effective integral depends on a parameter called “equivalent dilution cross-section” which characterize the reactions other than the resonant captures in the fuel, and which can approximately be written:
σd =
1 (1− C)b l N f 1− C + C b
where l is the diameter of the fuel element. The Bell factor b (about 1.1 for uranium 238) is a correction due to the transposition from a homogeneous geometry to a heterogeneous one. The Dancoff factor (see Figure 4) is the probability for a neutron emerging from a fuel element to cross the
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* At that time, Fermi prepared his famous experiment CP1 (Chicago pile number one) which used natural uranium and was moderated by graphite, and which reached criticality on December 2th, 1942. † In neutronics, we often use, instead of the neutron kinetic energy E, its “lethargy” u defined by u = – ln(E/Ereference); Ereference is chosen conveniently, e.g. 10 MeV.
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moderator without collision and, consequently, to re-enter a fuel element and to risk to be captured in a resonance. The smaller the lattice pitch (indicated next the curves), the greater the Dancoff effect, i.e. the self shielding of the resonances.
Figure 2. Zoom on the uranium 238 capture cross-section showing the numerous resonances in the epithermal energy range of the neutrons.
Figure 3. Effective integral of uranium 238; this integral increases if the fuel temperature increases thanks to the so called “Doppler effect” (see paragraph 3).
Figure 4. Dancoff’s factor for a square lattice of cylindrical fuel elements (clads neglected).
FUNDAMENTALS OF NEUTRONICS
55
The thermal utilization factor f is the probability for a neutron which escaped the “traps” (uranium 238 resonances) and became thermal to be absorbed into the fuel and not elsewhere in the reactor lattice (moderator, clads...). Considering, for instance these three materials (indices f, m and c), we can write:
f=
V f Σ a, f Φ f V f Σ a, f Φ f + Vm Σ a,m Φ m +Vc Σ a,cΦc
where the Φ are the values of the fluxes averaged over the thermal spectrum and over each volume respectively, the Σ the macroscopic crosssections, and the V the volumes. For a PWR, the flux tilts are small (a few percents less in the fuel than in the moderator); f is about 0.92 (without boron in the water). The reproduction factor η is the mean number of neutrons emitted by fission (induced by a thermal neutron) for one (thermal) neutron absorbed into the fuel:
η=
(ν Σ f ) f Σ a, f
(The fuel volume and the averaged flux in the fuel cancel between the numerator and the denominator.) For a PWR using enriched uranium, η is about 1.8: the product k∞ of the four factors is therefore about 1.3; such a high value is required to compensate the leakage (a few percents of the neutrons) and, above all, the large decrease of k∞ which will appear during irradiation due to the burning up of the fissile materials and to the capture by the non-fissile by-products, specially the fission products. At the end of its irradiation, the value of k∞ of a PWR assembly is less than one! Of course, the reactor always runs at criticality: that is possible because it contains a mixture of fresh and irradiated fuels*. This Fermi’s presentation of the neutron balance is quite convenient for the thermal neutron reactors; even for the PWR, is which the spectrum is rather “hard” (not well thermalized), this breakdown into four factors is relevant, and it appears that the elementary formulae we recalled can
______ *
Generally, for the PWRs, there are 3 or 4 batches of fuel elements. At each end of cycle (1 cycle: between one and two years) the reactor is stopped, the most irradiated batch is removed from the core and replaced with fresh fuel, and all the fuel elements are repositioned in the core in order to optimize the multiplication factor and the power distribution. At the beginning of the cycle, the over-criticality is compensated thanks to boron acid in solution in the primary water, and possibly burnable poisons; the end of cycle is at the moment when there is (almost) no boron in the primary circuit.
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P. REUSS
produce good orders of magnitude of the “reactivity coefficients”, that is to say the (small) variations of the multiplication factor due to (small) variations of the state of the reactor (temperatures, concentrations...) or of the cross-sections. However, such a presentation cannot apply for fast reactors. It is the reason why other breackdowns, similar to the one of Enrico Fermi but usable for any type of reactor, have been proposed, for instance by Roger Naudet and by Alain Satamarina. We will present here below the decomposition used by the author of this paper (similar to Santamarina’s one). We can go on distinguishing the reactions in the core and the leakage, consequently writing:
k = k∞ PNL with the same approximate expression of the non-leakage probability PNL. But we cannot consider that the neutrons follow two successive distinct steps, slowing down, then thermalization. Therefore, is is preferable to analyse the reaction rates not according to the neutron energy range but according to the type of material liable to absorb the neutron. Here we shall propose to distinguish four types of materials: 1. the fissile* (heavy) nuclei, 2. the fertile† (heavy) nuclei, 3. the non-fissile-non-fertile heavy nuclei‡, 4. the other (non-fissile) nuclei§. Using the following notations**: S number of neutrons emitted by fission (source), A number of neutron absorptions, P number of neutron productions (resulting from these absorptions), L number of neutron leakages, and the indices of the nuclide types (no index means “total”), we can write:
______ *
Fissile: which can be fissioned by a neutron of any energy, even (and generally easily) by slow neutrons; e.g. uranium 235, plutonium 239, etc. † Fertile: which gives a fissile nucleus by a neutron capture; e.g. thorium 232, uranium 238, plutonium 240, etc. ‡ E.g. uranium 236, plutonium 242, etc. § E.g. moderator, coolant, clads, etc. ** The numbers of reactions can refer to any unit of time, as the normalization do not appear in the definitions. Very often, we take in the codes S =1.
FUNDAMENTALS OF NEUTRONICS
k=
57
P = k∞ PNL S
with:
PNL =
A A+ L
k∞ =
A+ L P P =ζ S A A
The factor ζ =(A + L)/S takes into account the extra neutrons produced by (n,2n) and (n,3n) reactions: because of these reactions, the number A + L of neutrons disappearing is greater than the number S of neutrons emitted (in practice of a few hundreds pcm). We propose to split the remaining fraction P/A of the expression of k∞ into five factors, in order to get the six factor formula:
k∞ = ζ ε pq f η with the following definitions:
ε=
P1+2+3 P , η= 1, P1 A1
p=
A1 A 1+2
, q=
A 1+2 A 1+2+3
,
f =
A 1+2+3 A 1+2+3+4
Note: if we introduce
η 1 = P1 A1 (= η), η 2 = P2 A 2 , η 3= P3 A 3 , we can write:
ε = 1+
(1− p) q η2 + (1− q) η3 pq η1
The last five factors have a physical meaning similar to the one they had in the Fermi breakdown: ε expresses the fissions of the secondary heavy nuclides (types 2 and 3) and η expresses the reproduction by the fissile nuclides (type 1); the factors p and q take into account the loss of neutrons due to the absorptions by the heavy nuclides which generally are not fissioned, respectively the nuclides of types 2 and 3; the factor f takes into account the loss a neutrons in the captures not in the fuel (type 4 materials). The factor η is the reproduction factor of the main fissionable material, i.e. η 1 of the type 1 material. In the following tables some examples are given. Table 1 gives the Fermi four factors for PWR assemblies: a uranium oxide assembly (uranium 235 enrichment: 3.7%) and a MOX assembly (6.5% of plutonium and 93.5% of depleted uranium), both at the beginning of life without leakage (wl) and the first one at the beginning (0 GWd/t) and the end of irradiation (46 GWd/t) with critical leakage (cl).
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P. REUSS
TABLE 1. The infinite multiplication factor and the four factors of its breakdown according to the conventions of Fermi for some typical PWR cases (500 ppm of B, Xe at equilibrium).
Case
ε p f
η k∞
UOX-0 1.0698 0.7621 0.8937 1.7490 1.2743
UOX-0-cl 1.0766 0.7537 0.8936 1.7479 1.2674
UOX-46-cl 1.1045 0.6878 0.8965 1.2846 0.8749
MOX-0-wl 1.0888 0.6382 0.9498 1.7239 1.1379
Table 2 gives the six factor proposed breakdown for the same PWR cases (the columns for the MOX cases with critical leakage both at the beginning and the end of irradiation have been added). TABLE 2. The infinite multiplication factor and the six proposed factors of its breakdown for the typical PWR cases (500 ppm of boron, xenon at equilibrium). Case
ζ ε p q f
η k∞
UOX-0-wl
UOX-0-cl
UOX-46-cl
MOX-0-wl
MOX-0-cl
MOX-46-cl
1.0018 1.0679 0.6748 1.0000 0.8928 1.9770 1.2743
1.0014 1.0752 0.6669 1.0000 0.8937 1.9751 1.2674
1.0020 1.1023 0.5373 0.9673 0.7949 1.9173 0.8749
1.0020 1.0867 0.5931 0.9719 0.9546 1.8989 1.1379
1.0017 1.0905 0.5908 0.9720 0.9547 1.8998 1.1378
1.0019 1.1065 0.5320 0.9511 0.8553 1.9428 0.9321
The neutron leakage in the water reactors occurs above all for the fast neutrons: therefore the effects on ε, p and q, but no effect on the Fermi f and η factors. The irradiation induces a very big decrease of k∞ , particularly for the UOX case; at the end of life, the media are clearly under-critical. The change of nuclear fuel gives a decrease of p and q due to the great capture by the 240 and 242 isotopes of plutonium, respectively. Because of the much greater thermal absorption of the MOX, the Fermi f is better for this case; but the Fermi η ’s are similar. Table 3 gives the six factor proposed breakdown for five examples of helium cooled fast neutron media [2] (preliminary design studies of one of the Generation IV concepts: helium cooled fast reactor), calculated with critical leakage. They differ by the cladding (C: graphite; S: SiC; Z: zirconium), the percentage of plutonium in the MOX fuel (first and second digits), the volume percentage of the MOX in the compact (third and fourth digits), and the volume percentage of helium in the cell (last digits). Both the cases with carbon differ by the plutonium contents: a greater contents improves ε because plutonium 240 is more easily fissioned than uranium 238, but, above all, the improve appears on p because uranium 238 almost disappeared; there is also an improve on η because of the spectrum
FUNDAMENTALS OF NEUTRONICS
59
hardening; the first case is largely under-critical. The comparison of the third and the fourth cases show an improve of ε and η for the zirconium case because of the spectrum hardening; but the main effect is the decrease of f because of the great capture by zirconium. The last two cases differs only by the helium contents, i.e. the migration area which is twice greater for the last case: the infinite multiplication factor are almost the same, but the critical dimensions would be very different. TABLE 3. The infinite multiplication factor and the six factors of the proposed breakdown for examples of helium cooled fast neutron media.
Case
ζ ε p q f
η k∞
C-25-12-65 1.0001 1.0314 0.4420 0.9596 0.9701 1.7776 0.7545
C-99-12-65 1.0000 1.0530 0.8019 0.9651 0.9744 1.8580 1.4754
S-25-60-40 1.0006 1.1523 0.5362 0.9814 0.9852 1.9589 1.1709
Z-25-60-40 1.0012 1.1837 0.5295 0.9812 0.8789 2.0715 1.1210
Z-25-60-40 1.0012 1.1752 0.5294 0.9813 0.8701 2.0547 1.0928
3. Reactor Operation The reactor operation is characterized by two types of reactivity effects: •
actions of the operator to start the system, modify the power, or stop the chain reaction; any term of the neutron balance (fission, capture, leakage) can a priori be used for such changes of reactivity; most generally, the modification of the capture is chosen, often thanks to the movement of “control rods” (or bundles), sometimes by other means as the introduction or dilution of some boric acid in the primary circuit of the PWRs;
•
spontaneous variations of the multiplication factor due to the chain reaction and the production of heat. These spontaneous effects are themselves of two types:
•
the effects resulting from changes of the composition of the materials, above all the fuel, due to the consumption and the transmutation of the heavy nuclides, and to the accumulation of fission products; these variations of reactivity are slow: the time constants can vary between a few hours for some fission products as the xenon 135 to years for other fission products and for the heavy nuclides;
•
the effects resulting from changes of the temperatures of the materials; these last ones are almost instantaneous (for the fuel temperature which is directly linked to the power production, i.e. the fission rate) or
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P. REUSS
characterized by times constants of a few seconds (coolant and moderator temperatures whose variations need some heat transfer from the fuel element). In any case, the time constants of the power variations are of the order of a few tens of second, as governed by the delayed neutrons. Here we shall comment only these temperature effects. First of all, do note that it is compulsory to design the system in such a way that the temperature coefficient* α = ∂k/∂T (where k is the multiplication factor and T the considered temperature) is negative: indeed, then the system is stable: if there is, for instance, an increase of power, there will be an increase of the temperature, then a decrease a the multiplication factor (if α is negative), and consequently an under-critical situation, and therefore a decrease of the power. Conversely, if α was positive, any perturbation would be amplified. For example, Figure 5 shows the curves giving the convergence of the power towards its equilibrium level (k = 1) from initial states characterized by an initial power greater or smaller than this equilibrium power.
Figure 5. Evolution of the power towards the equilibrium level from some non-equilibrium initial states, less or greater than the equilibrium level, and characterized by the values at time zero (instantaneous negative temperature effect; one group of delayed neutrons).
Let us considerer the example of the PWR (the physical effects are similar for the other types of reactors). Three temperature effects can be distinguished: The Doppler effect is linked to the fuel temperature: because of the thermal agitation, the resonances, as “seen” by the neutrons in the
•
______ *
Or, at least, the coeffcient of the most rapid effect (as we shall see, it is the effect linked to the fuel temperature, i.e. the Doppler effect).
FUNDAMENTALS OF NEUTRONICS
61
laboratory (reactor) system, broaden and their peaks decrease, the integral remaining unchanged (Figure 6); but, because of the “self-shielding effect”, the associated absorption (that is to say the effective integral in the formula of the factor p) increases (see Figure 3). The main Doppler effect occurs on the uranium 238 captures and consequently decreases the factors p, and therefore k. So this effect, though not very great in absolute value ( ≅ –2 pcm/°C), is the main contribution to the reactor stability, and consequently safety in case of reactivity accident.
Figure 6. Broadening of the main uranium 238 resonance because of the Doppler effect. •
The thermal spectrum effect is linked to the temperature of the material thermalizing the neutrons, i.e. the moderator: if its temperature increases, the thermal agitation of its atoms increases, and so do the neutron spectrum which consequently becomes “harder”. Because several of the most important cross-sections do not vary exactly as the inverse of the velocity v of the neutrons* (Figure 7), some variations of the factors f and η occur, generally giving a small negative temperature coefficient for uranium 235 and a small positive temperature coefficient for plutonium 239. For the water reactors, this spectrum effect is almost negligible in comparison with the following third effect.
•
The moderator expansion effect, indeed, is preponderant for the moderator temperature coefficient of the PWRs, because water expands considerably with a temperature increase around the nominal condition (about 300°C): (1/ρ ) · ∂ ρ /∂T ≅ –250 pcm/°C, where ρ is the water density. This expansion modifies the factors p (stabilizing effect) and f (destabilizing effect). It can easily be calculated from the elementary formulae of these factors that:
______ *
The absorption (capture and fission) cross-sections for the slow neutrons generally follow the “ 1/v law ”.
62
P. REUSS
1∂p 1 1 ∂ρ = ln ⋅ , p ∂T p ρ ∂T
1∂f 1 ∂ρ = − (1− f ) ⋅ f ∂T ρ ∂T
i.e. –72 and +20 pcm/°C, if we use the values of p and f given previously. Therefore the global water temperature coefficient is negative (which is required because of the safety criteria) if (neglecting the spectrum effect):
ln(1 p) > (1 − f ) This is verified for the PWR examples.
Figure 7. Variation of some important cross-sections in the thermal range [v σ /v0 σ 0 versus v/v0; v0 = 2 200 m/s and σ 0 = σ (v0)].
But, look out! If some boron is added in solution in the primary water in order to reduce the thermal utilization factor f and consequently the multiplication factor k, this condition may no more be fulfilled: as we shall explain further, in this case solid (and burnable) poisons, modifying η and no longer f, will be used. 4. Reactor Design: Choice of the Main Parameters We will now discuss some examples of design problems: we will take these examples in the well known generation 2 and 3 pressurized water reactors (for which a very great experience has been accumulated), in order to show how various problems are interwoven. It is clear that similar interweaving will occur in the generation 4 reactors design, though it is problably still too early to completely appreciate them. We chose three examples of problems: the first one concerns the very basic choices which were to be made at the very beginning of the PWR development. The second one appeared when the utilities tried to improve the fuel utilization thanks to longer burnups; the last one cropped up when great quantities of plutonium (got thanks to the reprocessing of the standard
FUNDAMENTALS OF NEUTRONICS
63
fuels) accumulated because of the lack a fast neutron breeder reactors to use it and when it appeared more judicious to use it in the PWRs rather than to wait hypothetical breeders. The three main parameters of a PWR lattice are: the radius of the fuel pins, the pitch of the lattice and the initial isotope 235 content of uranium in the fuel. The neutron balance is not very sensitive to the radius of the pins if we compare situations differing only by a homothety ratio, i.e. without change of the volume ratios: if the dimensions increase, the effective integral in the formula of the factor p slightly decreases (because of l, though the decrease of the Dancoff factor C partly compensates) and p increases; in the formula of the factor f, the flux ratio Φ f Φ m slightly decreases and f decreases. As all these effects are small, the choice of the fuel radius is made not by neutronics but by thermal-hydraulics criteria, particularly the central temperature which must remain below the fusion temperature of the uranium dioxide (with a security margin) for the power density which is looked for. Conversely, the neutron balance is very sensitive to the so-called “moderation ratio”, that is to say the ratio of the moderator volume (or mass) to the fuel volume (or mass). Indeed this ratio directly appears in the formulae of the factors p and f. If it increases, p improves and f deteriorates: therefore, their product (and consequently the multiplication factor k) goes through a maximum between a null moderation ratio (where p goes to zero) and an infinite moderation ratio (where f goes to zero), as shown schematically on Figure 8.
Figure 8. Optimum of the moderation ratio (the numerical values are orders of magnitude for PWRs).
It can be seen that an under-moderated configuration is chosen. There are two main reasons of this choice: 1) the first one is the compacity of the core: if it is smaller, the vessel and all the associated materials are smaller and less expensive; 2) the second reason is linked to the safety: in any incidental situation (leak of water, formation of bubbles or water expansion due to a temperature increase) the mass of water decreases whereas the
64
P. REUSS
mass of fuel does not change; if the configuration is under-moderated, the multiplication factor decreases, and so does the power. In the PWRs, the potential (i.e. without compensation) multiplication factor continuously decreases with irradiation. Therefore a value greater than 1 is required at the beginning of cycle, the excess of reactivity being compensated with poisons. The end of cycle is reached when the potential excess of reactivity (and this poisoning) go to zero*. The initial multiplication factor of course increases if the initial contents in uranium 235 of the fuel increases: it is consequently chosen in order to satisfy the lenght of cycle which is looked for. 5. Reactor Design: Problem of Using a Boron Solution in the Primary Circuits of the PWRs It can be seen on Figure 8 that the volume moderation ratio of the PWRs is about 2, whereas the optimum value is about 4. This loss of profit (some 7 000 pcm!) is accepted because, as we said, it allows to reduce the size of the core and, above all, because a reactivity compensation by boron acid B(OH)3 is used. This boron poisoning reduces the value of the thermal utilization factor f... but also induces a positive component on the moderator expansion coefficient: indeed, the boron in solution expands like water and consequently its concentration (and poisoning) decreases if the moderator temperature increases. The curve of Figure 8 corresponds to a situation without boron: it can easily be shown thanks to the elementary formulae, that if the boron reduces f, therefore k, it also moves the maximum of the curve (optimum of moderation) towards the left. As a negative moderator expansion coefficient is associated to a positive slope of the curve for the chosen moderation ratio, this last one must remain smaller than the optimum one in spite of the addition of boron. The designers of the PWRs evaluated the needs of reactivity compensation, deduced the maximum boron concentration to be used, and finally chose the moderation ratio in order to keep a negative moderator temperature coefficient with this maximum concentration. At that time, the maximum fuel burn-up was about 30 000 MWd/t: that correspond to a loss of reactivity of about 30 000 pcm, or with the usual
______ *
As the poisoning is made by boron in solution, the end of cycle occurs not for a zero concentration but a minimum small concentration for which the dilution becomes ineffective. Some flexibility is possible: a cycle can be stopped before the normal instant; it can also be extended (stretch-out) thanks to the negative moderator temperature coefficient by a small decrease of the primary water temperature.
FUNDAMENTALS OF NEUTRONICS
65
core management in three batches, to 10 000 pcm of reactivity excess at the beginning of cycle, i.e. about 1 000 ppm of boron (efficienty ≅ –10 pcm/ppm). Such a order of magnitude keeps an optimum moderation ratio slightly greater than the chosen moderation ratio, that is to say, the following criterium remains satisfied:
ln(1 p) > (1 − f ) A better knowledge of the effects of irradiation (particularly the interactions between the pellet and the clad) allowed, some decades after the first operation of the PWRs, to propose longer irradiations (in spite of the greater consumption of natural uranium and separative work units per unit of mass of fuel this appears to be interesting on a economy point of view). The consequence of greater cycles is a greater initial potential reactivity and greater needs of compensation. It appeared that sometimes this needs would exceed what is possible thanks to boron in solution, that is to say the limit associated to a negative moderator temperature coefficient. Then, as this limit is compulsory, what to do? The solution of this difficulty has been the use of burnable poisons. This poisons are: •
solid, in order not to expand and not to induce a positive temperature coefficient. Boron carbide in specific pins or gadolinium oxide mixed with the uranium oxide in some pins have been used as such burnable poisons;
•
efficient, in order to be useful for the reactivity compensation;
•
burnable, in order to disappear (or, at least, almost disappear) at the end of cycle, when there is no more need of reactivity compensation.
Figure 9. Evolution of the multiplication factor of a PWR without and with burnable poisons.
66
P. REUSS
As an example, Figure 9 shows (schematically) what can be done thanks to the burnable poisons: of course, they cannot compensate exactly for the reactivity excess, but their contribution allows to reduce the use of boron and, consequently, to insure a negative moderator temperature coefficient. 6. Reactor Design: Problem of Recycling Plutonium in the PWRs The history of the plutonium use in France is interesting to recall to the young scientists preparing the next generation of reactors and particularly their breeding! [4] After irradiation, the standard fuel of the PWRs (uranium initially enriched to a little more than 3% of uranium 235) still contains about 1% of uranium 235 and about 1% of plutonium (Figure 10). Since the beginning of the developments of the pacific nuclear energy during the late 1940s, France had been a convinced supporter of the fast neutron breeder reactors: the official politics was to built a few thermal neutrons reactors (at that time natural uranium-graphite-gas (UNGG) reactors) in order to produce some plutonium, then to introduce this plutonium into breeders which will produce more and more plutonium for more and more fast neutron reactors. It is the reason why an important development was devoted to these fast reactors, and why a “close cycle” was chosen, that is to say a systematic reprocessing of the irradiated nuclear fuels and recycling of the energy-giving materials, particularly plutonium.
Figure 10. Typical evolution of the concentrations of the main heavy nuclides in a PWR fuel.
In reality, a very small number of fast reactors were built in France: after the experimental facility Rapsodie, the prototype Phénix (250 MWe, 1973, still running) and the industrial plant Superphénix (1 200 MWe, 1986, stopped in 1998 not for technical but for political reasons).
FUNDAMENTALS OF NEUTRONICS
67
Meanwhile, the UNGG program was stopped and an ambitious REP (“réacteur à eau sous pression”, i.e. PWR) program was carried out: 58 plants between 1977 and 1999. The “closed cycle” was confirmed and, after the plant which operated at Marcoule, new reprocessing plants were built at La Hague. It can be calculated that about 10 tons of plutonium per year are produced at La Hague*. Of course, the only running fast reactor Phénix cannot use such a huge amount of plutonium! It the reason why Électricité de France (EdF) decided to recycle plutonium into the water reactors: some PWRs use plutonium fuels since 1987. The recycling of plutonium in the thermal reactors are less interesting than in fast reactors, because the reproduction factor η is smaller: 2.1 (almost the same value than the uranium 235 one) instead of 2.4 in a fast spectrum, for plutonium 239; particularly, a plutonium loaded (as a uranium loaded) thermal reactor cannot be a breeder. Nevertheless, it was thought that it should be more judicious to use the available plutonium in such conditions rather than to wait hypothetical fast reactors†. In practice, the standard uranium fuels are replaced with so-called “MOX” fuels, that is to say mixed (natural or depleted) uranium and plutonium oxides, e.g. 6.5% of plutonium and 93.5% of depleted uranium instead of uranium enriched up to 3.7% in isotope 235. Remark that the plutonium content in the MOX fuel is greater than the uranium 235 content in the standard fuel, because the plutonium is fissile for about the two thirds only. When the recycling of plutonium was envisaged in the PWRs, the neutron physicists had to adapt and requalify the codes, thanks to experimental programs, and the designers had to calculate the core neutron distribution and management for such a new fuel. Indeed, the difference is great! If the standard and MOX fuels are very similar for the fast and epithermal neutrons (the amounts of uranium 238 and water remain the same) they are completely different for the slow neutrons: the MOX assembly is about 2 to 3 times more absorbing for such neutrons‡. This is the consequence: 1) of the greater concentration of plutonium than uranium 235; 2) of a greater microscopic absorption crosssection for plutonium 239 than uranium 235; 3) of the presence of resonances in (or near) the thermal range for all the main plutonium isotopes (Figures 11 and 12).
______ *
Nowadays, 1 150 tons of irradiated fuels are unloaded from the PWRs and 850 tons are reprocessed per year. † Note that, if plutonium is stored, the isotope 241 (fissile) disappears by radioactive decay with a half-life of 15 years; note also that about 70% of the initial plutonium remain in the irradiated plutonium fuels and could be recovered (presently, these fuels are not reprocessed). ‡ We compare the macroscopic mean absorption cross-sections.
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P. REUSS
Figure 11. Absorption cross-sections for slow neutrons of uranium 235 and the main isotopes of plutonium.
Figure 12. Comparison of thermal neutron spectra in the standard and MOX fuels (multigroup calculations by the code APOLLO [18]) Note the great decrease of the flux due to a large absorption in the MOX fuel and the selfshielding effect (flux depression) of the plutonium resonances.
The first consequence of a such a big absorption is a drastic reduction of the efficiency of the control rods, boron and poisons*. The core calculation which are made for the nominal and the main transient situations showed
______ *
These materials mainly absorb thermal neutrons; their efficiency is proportional to the ratio between the absorbent and the assembly macroscroscopic absorption cross-sections: therefore, the greater this last one, the less the efficiency.
FUNDAMENTALS OF NEUTRONICS
69
that a core completety loaded with MOX could not be controlled in some cases, and consequently could not be authorized by the safety authority. Conversely, a core half loaded with MOX and half with standard fuels could be managed. In order to keep some margins, EdF decided to restrict the proportion of MOX in the core (and consequently in each reload) to one third. Consequence of such a choice: the presence in the core of interfaces between very different zones with uranium 235 and plutonium fissile material respectively; for such an interface, a peak of power appears near the interface in the most absorbent zone, i.e. the MOX assembly (Figure 13).
Figure 13. Flux and power behaviour near a uranium-plutonium interface (scheme).
The flux is lower in the plutonium (more absorbent) zone, but must be a continuous function, therefore the indicated behaviour; the power is proportional to the product of the flux by the fission macroscopic crosssection, which is greater in the plutonium zone, therefore the indicated behaviour showing a power peak in the plutonium zone near the interface. Precise calculations show that such a peak cannot be accepted*. The only mean to reduce such a power peak is to reduce the plutonium content in the pins situated near the interface, and to increase the content in the internal pins in order to keep the average content. Calculations showed that at least three zones are necessary. We see that such a design leads to an even more expensive manufacturing of the MOX fuels†... The question arises: is the plutonium recycling in the PWRs still interesting on an economic point of view? The answer is a matter of controversy... In the facts, almost all the first generation plutonium produced in France from the standard PWR fuels is recycled once in most of the 900 MWe plants.
______ *
The maximum pin temperature, therefore the maximum pin power is limited for safety reasons. Consequently the smaller the form factor Pmaximum/Paverage, the greater the total power of the plant. A too high power peak therefore induces a too low efficiency of the plant. † The manufacturing of the MOX fuel is more expensive than the standard one: it has to be made in “ glove boxes ” because of the α -activity of plutonium, much greater than the uranium one.
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P. REUSS
Such a controversy, during the 1990s, and the concerns of the opinion about the radioactive wastes let sometimes think that the main interest of the plutonium recycling was to reduce the amount of such a “waste”: from a energy-giving produce the statute of plutonium became more and more a waste to incinerate as much as possible! We even saw EdF develop some design studies (notably thanks to a greater moderation ratio) to increase the consumption of plutonium, and the Commissariat à l’Énergie Atomique (CEA) lunch a programm CAPRA, which means “greater consumption of plutonium in the fast reactors”: the main idea was to reduce as far as possible the plutonium breeding by removing the blankets and reducing the uranium 238 mass in the core (some of it must be kept in order to preserve a significant negative Doppler effect). Nowadays, after the Generation IV reflexions, and the awareness of the limited ressources of uranium, such considerations sound rather strange! 7. Conclusions At the very beginning of the neutron physics, the main concern was to get the critical condition, particularly when only natural uranium was available for such a purpose. Nowadays, we are able to enrich uranium or to substitute plutonium for uranium 235, and to get criticality is no more the main difficulty. Conversely, we see that the economic optimization (particularly the flatening of the power distribution and the extension of the burn-ups) and, above all, the intrinsic safety thanks to negative reactivity feedbacks have become the major concern of the reactor designers. The example of the boron in solution in the PWRs showed the technical constraints brought by such criteria. We could also have develop the example of the recycling of second generation plutonium in the PWRs, or many examples which can be find in the fast reactor design, such as the replacement of oxide fuel with nitride or carbide fuels, the change of fissile materials, the change of coolant (lead or helium instead of sodium), etc. The young generations of physicists and engineers will have a lot of studies to perform in order to elaborate the best choices! We must underline that even a deep knowledge of neutron physics will never be sufficient to get these good choices. Indeed, they also depends on all the physical characteristics of the cores, and therefore let arise numerous domains of physics. Particularly, all the problems of heat transfers, hydraulics, effects of irradiation on the materials... are strongly linked to neutronics. Only a good knowledge of all these aspects by all the designers (and discussions between the various specialists) can lead to sound solutions. Let us hope that this autumn school will be a step in that direction!
FUNDAMENTALS OF NEUTRONICS
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In practice, the codes treating the various aspects of the core behaviour are to be linked and run in an iterative process until convergence. For instance, the neutron distribution depends on the temperature of the various materials through the Boltzmann equation and the cross-sections it contains... and these temperatures depends on the power sources through the equation of the thermal-hydraulics: therefore, for any core calculation, both the codes of neutronics and of thermal-hydraulics have to be coupled and iterated. Similarly, codes treating the whole system are necessary to take into account the counter-reactions of a part of the reactor on the other parts, for instance of the secondary circuit on the primary one. Finally, we would like to emphasize two conclusions which can be drawn from the examples relative to the PWRs that we developed and that will certainly remain pertinent for the Generation IV developments: The first conclusion is the importance to design the reactors with margins as large as possible. Indeed, new operational modes appears, which were not anticipated. If the margins are not large enough for these new modes, technological adaptations are generally possible, but they are time consuming in developments and expensive to realize. For example, we see that longer irradiations could no more be managed with boron in solution only, but necessitated the use burnable poisons – which necessitated code developments and costly experiments to qualify them. Similarly, the introduction of MOX fuel into the PWRs led to great technical and expensive difficulties, particularly the necessity to zone the MOX assemblies – which, again, necessitated many codes developments and experiments. The second conclusion is the “weight of history”. The decisions taken at a given time not only govern the next operations but often direct the technical choices for a very far future. A good example is the history of plutonium in France. The initial direction was to develop as rapidly as possible the fast neutron reactors. Immediate consequence: use plutonium, reprocess the irradiated fuel and build prototypes of fast reactors. Far and not anticipated consequence: go on with reprocessing, introduce the MOX fuel into the water reactors with all the difficulties which arose. It is the reason why it is so important to make, as far as possible at the first attempt, the good choices!
Ackownledgements The author would like to thank his colleagues who provides the APOLLO calculations [18, 19]: P. Bioux and L. Payen from EdF (Clamart) for the PWR cases, and A. Nicolas and L. Buiron from CEA (Saclay) for the GFR cases.
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References [1] Commissariat à l’énergie atomique, 2005, L’énergie Nucléaire du Futur: Quelles Recherches pour quels Objectifs?, e-den, monographie de la Direction de l’énergie nucléaire. [2] Commissariat à l’énergie atomique, 2006, Les Réacteurs Nucléaires à Caloporteur Gaz, e-den monographie de la Direction de l’énergie nucléaire. [3] Reuss, P., 2006, L’énergie Nucléaire, série: Que sais-je?, PUF. [4] Reuss, P., 2007, L’épopée de l’Energie Nucléaire, une Histoire Scientifique et Industrielle, série :Génie atomique, EDP Sciences. [5] Barjon, R., 1992, Physique des Réacteurs Nucléaires, Institut des sciences nucléaires, Grenoble. [6] Bussac, J., and Reuss, P., 1985, Traité de Neutronique, Physique et Calcul des Réacteurs Nucléaires, Enseignement des sciences, Hermann. [7] Reuss, P., 1998, La Neutronique, série: Que sais-je?, PUF. [8] Reuss, P., 2003, Précis de Neutronique, série: Génie atomique, EDP Sciences. The english translation will be published. [9] Reuss, P., 2004, Exercices de Neutronique, série: Génie atomique, EDP Sciences. The english translation will be published. [10] Tellier, H., 1898, Réactions Nucléaires Induites par les Neutrons, CEA/Institut national des sciences et techniques nucléaires, collection Enseignement. [11] Rozon, D., 1992, Introduction à la Cinétique des Réacteurs Nucléaires, éditions de l’École polytechnique de Montréal. [12] Tellier, H., 1994, Cinétique des Réacteurs Nucléaires, CEA/Institut national des sciences et techniques nucléaires, collection Enseignement. [13] Livolant, M., and Jeanpierre, F., 1974, Autoprotection des Résonances dans les Réacteurs Nucléaires, rapport CEA-R-4533. [14] Reuss, P., 1991, Théorie de l’Absorption Résonnante des Neutrons, note CEA-N-2679. [15] Coste, M. and Reuss, P., 2003, Development of computational models used in France for neutron resonance absorption in light water lattices, Prog. Nucl. Energy 12(3): 237282. [16] Coste, M., 2006, Modélisation du Phénomène d’Autoprotection dans le Code de Transport Multigroupe APOLLO-2, thèse CNAM (7 mars 2006) and rapport CEA-R6114. [17] Lewis, E. E., and Miller, W. F., 1994, Computational Methods of Neutron Transport, John Willey & Sons. [18] Hoffmann, A., Jeanpierre, F., Kavenoky, A., Livolant, M., and Lorain, H., 1972, APOLLO Code Multigroupe de Résolution de l’Equation du Transport pour les Neutrons Thermiques et Rapides, note CEA-N-1610. [19] Coste, M., Hébert, A., Sanchez, R., Stankovski, Z., Zmijarevic, I., 1999, APOLLO-2 Notice Théorique, rapport CEA/DMT/SERMA/LENR/RT/99-2719/A. [20] Fédon-Magnaud, C., 1994, Résolution de l’Equation de Transport dans le Code CRONOS, note CEA-N-2763.
INTRODUCTION TO THERMODYNAMICS G. INDEN* Max-Planck-Institut für Eisenforschung GmbH, Düsseldorf, Germany
Abstract: At first, the fundamental concepts and their mathematical formulation are recalled: integral quantities like various forms of thermodynamic potentials, exchanged energy forms, partial molar quantities like chemical potentials, activities etc.. The emphasis will be on multi-component systems of solid, liquid and gaseous phases. The thermodynamic treatment of chemical reactions is presented and applied to oxidation reactions. The basis of Ellingham diagrams and their use is illustrated. The thermodynamic description of multi-component solution phases is presented as well as the construction of phase diagrams. The restricted equilibrium in the case of immobile substitutional elements is discussed (para-equilibrium). Keywords: Thermodynamics; potentials; phase equilibria; phase diagrams; para-equilibrium
1. Reminder of Thermodynamic Principles 1.1. STATE, VARIABLE, SYSTEM
The dynamic description of physical systems and processes is based on the concept of an exchange of quantities. The physical quantities are represented by variables, indicating that only their numerical value is of primary importance, not the way an exchange is achieved. A state is defined by the values taken by the variables. A system is then an ensemble of states. It is defined by: •
the number of degrees of freedom r
•
by defining (r +1) state variables
•
by defining a mathematical relation (often called fundamental relation) between the (r +1) state variables.
At this point it may be interesting to make the link with concepts in geometry. The state corresponds to a point in geometry. The state variables
______ *
Prof. Dr. Gerhard Inden, Im Weidengrund 19, D-40878 Ratingen, Germany; e-mail:
[email protected] V. Ghetta et al. (eds.), Materials Issues for Generation IV Systems. © Springer Science + Business Media B.V. 2008
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correspond to the space variables (e.g. x, y, z), which are given by the coordinates relative to a coordinate system. A state is defined by the values of the state variables, just as the point is defined by the values of the coordinates. A system corresponds to a geometric object, like a sphere, representing an ensemble of points. A sphere is defined by: •
the dimension of space r (here r = 3)
•
by defining r +1 variables (here x,y,z and R)
•
by defining a mathematical relation between the variables (here x2 + y2 + z 2 = R2 )
The definition of a system given above is mathematical and very general and it is not certain that every system defined this way can be realized. However, it is sometimes helpful to have recourse to idealized systems in order to derive certain conclusions. The variables characterising a system may be divided into two groups: extensive variables Xi (e.g. volume V, entropy S, number of species Ni, magnetisation M,…) and intensive variables Yi (e.g. pressure P, temperature T, chemical potentials μ i , magnetic field H,… ). It is not trivial to identify the extensive or intensive character of a variable. The term extensive suggests that the variable scales with the size of a system. This is, of course, true for energy, volume, entropy, number of particles: taking e.g. two identical systems together to form a new system, these variables double their value. There are, however, many variables which cannot be classified on the basis of this heuristic scheme, such as the position in a field, the extension of a spring. Intensive variables do not depend on the size of the system. Yet, not every quantity that does not depend on the size is automatically intensive. Examples of this type are reduced quantities like molar volume, mole fractions etc.. For every system with degree of freedom r one may chose r variables, e.g. (X1,... Xi, Yi+1,…Yr), to define a coordinate system. At least one extensive variable must be within this set in order to define the size of the system. The (r +1)-th variable, Ψ , is then defined by the fundamental relation:
Ψ = Ψ ( X 1 ,... X i , Yi +1 ,...Yr ) . The function Ψ is called Massieu*- Gibbs*- Function (or simply GibbsFunction) of the system defined by the coordinate system.
______ *
François Jacques Dominique Massieu, 1832-1896, professor at Ecole des Mines de Paris; he discussed the mathematical properties of such functions.
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1.2. INTERNAL ENERGY
Axiom: Any realizable system of dimension r admits at least one set of (r+1) extensive variables such that the fundamental equation defines a Gibbs-Function, which is a first order homogeneous function of the r independent extensive variables. This first order homogeneous function is the energy E. Let us denote by Xi the r independent extensive variables. The energy may then be written as r
E ( X1,...X r ) = ∑Y j ⋅ X j
(1)
j=1
Since E is a first order homogeneous function of the independent extensive variables, it has to fulfil the following criterion:
E (λX1, λX 2 ,...λX r ) = ∑Y j ⋅ (λX j ) = λ∑Y j ⋅ X j = λ E (X1, X 2 ,...X r ) (2) Consequently, the Yi have to be of zero order in the extensive variables and thus cannot depend on the size of the system. The Yi are the energyconjugate intensive variables of the corresponding extensive ones, Xi. They are defined by
Yj =
∂E ∂X j
(3) X k ,k≠ j
1st law: Energy is a conserved quantity. Its value for a given system can change only by exchange with other systems. The change in energy caused by exchange of extensive quantities is given by r
∂E dE = ∑ j=1 ∂X j
r
⋅ dX j =∑Y j ⋅ dX j X k ,k≠ j
(4)
j=1
Systems in physical metallurgy are assumed to be at rest. In that case they do not exchange linear or angular momentum, and the energy is then called internal energy U. Common extensive variables are e.g. volume, *
Josiah Willard Gibbs, 1839-1903, professor of mathematical physics at Yale University; he introduced this fundamental concept in physics.
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entropy, number of particles of type i, Xj =V, S, Ni. The internal energy is given by
U (V,S,N i ) = ∑Y j ⋅ X j = −PV + TS + ∑ μi N i j
i
with pressure P, temperature T and chemical potential μi :
P =− def
∂U ∂V
∂U def ∂S V ,N i
T= S,N i
∂U def ∂N i V ,S ,N
μi =
j ≠i
as energy-conjugate intensive variables. For any change dXj of the extensive variables Xj one gets
dU = ∑Y j dX j = −PdV + TdS + ∑ μi dN i j
(5)
i
The various terms in Eq. (5) are called exchanged energy forms. The sign convention of the individual terms is such that if the fluxes of extensive quantity and of energy are oriented towards the same direction, the energy terms are counted positively. This is the case of entropy, particle number etc. However, in the case of mechanical work by volume change, a “flux of volume from the outside” ( dV > 0 ) means work supplied by the system to the outside. Thus the fluxes of energy and volume are oriented into opposite directions. Consequently, −P and not P is the conjugate variable of V. Equation (5) suggests to associate the term –PdV with work performed on or by the system. Indeed, considering a system with gas in a cylinder and a piston of area A, the pressure can be written as P = F/A and dV = A dx, so that PdV = F dx, i.e. equivalent to mechanical work. This, however, is not generally true, it only holds if the process of volume change is performed reversibly. The same holds for the term TdS representing the amount of heat exchange only in reversible processes. The difference between reversible and irreversible processes can be illustrated with the expansion of a gas. In Figure 1a a reversible realization is shown. The pressure from the outside is decreased by infinitesimal steps. The piston is then moved outwards by the gas, the pressure decreases until the level imposed from outside is reached. The process is reversible: it is sufficient to reset the external force from (F-ε) to F, the gas contracts until it reaches the original state. The irreversible realization is shown in Figure 1b. The volume expansion is performed by pulling out walls. The gas then has access to the increased volume without performing any work. This process cannot be reversed, since the probability for the event that a volume segment does not contain any gas, is zero. It is to be emphasized that in both cases the term -PdV appears and takes the same value for the given volume change. However, only in the first case this value represents work performed by the system. Since in the irreversible case the system does not
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exchange any energy form, the internal energy stays constant. From Eq. (5) it follows dU = −PdV + TdS = 0, TdS = PdV . In the irreversible process the quantity PdV is thus converted into T dS . Such conversion is also called entropy production.
Figure 1. a) Realization of a reversible expansion and compression of a gas. If the external force F is reduced by an amount ε, the gas expands the volume until the gas pressure equals the external pressure. The process can be reversed by resetting the external force to its original value F, b) Realization of an irreversible expansion of a gas. The volume increase is not performed by the gas. It is made available externally by removing the walls. The gas simply accesses the additional volume. This process cannot be reversed.
2nd law: Entropy cannot be annihilated: a reduction of entropy can only occur by exchange, while an increase can occur by exchange and/or production. Reversible process: process with no entropy production. Ideal gas: system which fulfils the equation PV = nRT (n=number of moles, R=8.3144 J/(K·mole)). From the PV = nRT equation it follows that
∂U(V,T,N) =0 ∂V
i.e. at constant T and N the internal energy of the ideal gas does not depend on volume. Consequently, if the reversible expansion of the gas is performed at constant temperature, the internal energy does not change. The energy loss due to the work − PdV performed by the gas is thus compensated by a heat flux TdS from the heat reservoir, which keeps temperature constant. 1.3. CONJUGATE VARIABLES, LEGENDRE TRANSFORMATION
The experimental conditions are often characterized by keeping certain variables constant. In order to be able to impose such constraints the corresponding variables must be independent. However, many practical cases require the control of intensive rather than extensive variables, e.g. temperature rather than entropy or pressure rather than volume. For such
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cases it is necessary to change from independent extensive variables to intensive ones. By definition, see Eq. (4), the intensive variables are the first order partial derivatives of the internal energy. The problem of changing from an extensive variable to its conjugate intensive variable is thus equivalent to expressing the functional relationship in terms of first order derivatives. This problem is solved by the Legendre transformation.
Figure 2. Legendre transformation of a function y(x) with only one variable.
A one dimensional illustration of the Legendre transformation is shown in Figure 2. Suppose a one dimensional functional relationship y(x) describing a curve in the (y,x)-plane. This functional relationship shall be described in terms of the first derivative ξ = dy dx which represents the slope of the tangent to the curve y(x). Every point of the curve is characterized by the slope and the intercept u(ξ ) of the tangent to the curve y(x). The variation of this intercept with the slope represents the property of the functional relationship in terms of the slope. The intercept is given by u(ξ ) = y (x ) − ξ x . In this one-dimensional example, the transformation is given by x → ξ = dy dx and y (x ) → u(ξ ) = y − ξ x . Introducing the symbolic description [y ]x for the Legendre transform of y with respect to x, we can write
[y ]x = y − ξ
x
(6)
The situation in the case of two variables is illustrated in Figure 3. Figure 3a shows the partial transformation with respect to the variable x. The construction is the same as in Figure 2, here of course performed within the section y = cst. The same procedure applies to the partial transformation with respect to y, the construction being performed within the section
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x = cst., Figure 3b and 3c shows the transformation with respect to both variables. The tangent plane is defined by the two slopes ξ1 and ξ 2 . Its intersection with the Z-axis defines the value of the Legendre transform:
[Z(x, y)]x,y = Ψ(ξ1,ξ 2 ) = Z (ξ1,ξ 2 ) − ξ1 x − ξ 2
y
(7)
The Legendre transformation is often written in a way as (6) which could suggest that the transformation consists of simply adding a product term of the two conjugate variables to the original function. Equation (7)
Figure 3. Legendre transformation of a function with two variables. a) Partial Legendre transformation with respect to x, b) Partial Legendre transformation with respect to y, c) Complete transformation with respect to both variables, d) Transformation of a cone: all tangent planes define the same intercept: the origin zero.
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specifies clearly that the transformation requires the substitution of the variables x and y by ξ1 and ξ 2 in the original function Z, which means that the relation ξ1 = ∂Z ∂x defining the intensive variable must be resolved with respect to its conjugate x. The cone is a geometrical example of a homogeneous function of first order in (x,y). In this case the tangent planes define a unique intercept, the origin zero (Figure 3d). 1.4. LEGENDRE TRANSFORMATION OF THERMODYNAMIC POTENTIALS
This last situation is indeed encountered in thermodynamics since the internal energy U is a homogeneous function of 1st order. It thus represents a hyper-cone in the space of the extensive variables. The Legendre transform with respect to all of the extensive variables thus leads to the identity zero. This is known as Gibbs-Duhem equation. It is, of course, possible to perform partial Lengendre transforms. These correspond to hyper-planes tangent to conic sections. These planes do not pass through the origin. When a partial Legendre transformation is performed, a new function is obtained which is still a homogeneous linear function of the remaining extensive variables. Let us denote the Legendre transform of a function
Φ(X1, X 2 ,... X r)
with respect to Xj symbolically by
[Φ]X
j
= Ψ(X1, X 2 ,...Y j ...X r) .
The Legendre transform is then given by
Ψ = Φ(X1, X 2 ,...Y j ,...X r )− X j Y j
(8)
In order for this transformation to be possible it is necessary that the equation
Y j = ∂Φ ∂X j
can be inverted such that one gets
X j = X j (Y j )
as a function of Yj, otherwise one cannot replace Xj in Φ as required in Eq. (8). Because of the following relations:
d Φ = ∑ Yk ⋅ dX k + Y j ⋅ dX j k≠ j
and
d (Y j ⋅ X j ) = X j ⋅ dY j + Y j ⋅ dX j
the differential of Ψ is obtained as
dΨ = d (Φ − Y j X j ) = ∑ Yk dX k − X j dY j . k≠ j
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The general case of a multiple Legendre transformation with respect to m variables, e.g. X1, X2, ... Xm, is obtained as
[Φ]X ,...X 1
m
m
= Ψ(Y1 ...Ym , X m +1 ...X r ) = Φ(Y1,...Ym , X m +1,...X r ) − ∑ X k Yk
(9)
k=1
and
dΨ =
r
∑Y j= m +1
m
j
dX j −∑ X k dYk
(10)
k=1
1.4.1. Enthalpy The Legendre transform of U with respect to V, is called the enthalpy:
H (P,S,N i ) = [U (V,S,N i )]V = U + PV = TS + ∑ μi N i
(11)
dH = VdP + TdS + ∑ μi dN i
(12)
def
i
with i
At constant pressure (!) enthalpy is again a state function since it is a homogeneous function of 1st order in the remaining extensive variables. At constant pressure and particle numbers, Eq. (12) leads to dH = TdS, i.e. the enthalpy change represents the reversible exchange of heat. 1.4.2. Free energy Free Energy is the Legendre transform of U with respect to entropy, i.e. with
F (V,T,N i ) ≡ [U (V,S,N i )]S = U − TS
dF = −PdV − SdT + ∑ μi N i i
1.4.3. Gibbs energy Gibbs energy is the Legendre transform of U with respect to both volume and entropy such that now pressure and temperature become independent variables.
G(P,T,N i ) ≡ [U ]V ,S = U + PV − TS = ∑ μi N i i
with
dG (P ,T , N i ) = VdP − SdT + ∑ μ i dN i i
(13)
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At constant temperature and pressure Gibbs energy is again a state function. There are many more Legendre transforms if one considers different starting functions, e.g. entropy. 1.4.4. Gibbs-Duhem-Relation If we consider the Legendre transform of U with respect to all extensive variables, we do not end up with a new potential function, but with a relation between the intensive variables: n
[U ]V ,S,N = U + PV − TS − ∑ μi N i ≡ 0 1
(14)
i−1
and n
d[U ]V ,S,N = VdP − SdT − ∑ N i dμi = 0 i
(15)
i=1
At constant temperature and pressure Eq. (15) takes the familiar form: n
∑ N dμ i
i
= 0.
i=1
Thus, the Legendre transform with respect to all extensive variables does not lead to a new Gibbs function, but leads to a constant which can be set to zero. Therefore, it is sometimes called “zero potential ”. It is not possible to define the size of a system with intensive variables only. In order to define a system at least one extensive quantity must be among the variables characterizing the system. 1.5. PARTIAL QUANTITIES
1.5.1. Molar quantities In most cases of physical metallurgy the size of the system is not really important. It is then common practice to refer the extensive quantities to 1 mole of atoms. Taking Gibbs energy as an example, we may write :
G(T,P,N i ) = N with
G(T,P,N i ) = N G (T,P, x i ) N n
N = ∑ N i and xi = i =1
Ni N
Setting N = 1 mole of atoms, we obtain the molar Gibbs energy:
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83
n
Gm (P, T, x i ) = ∑ x i μi i=1
where the chemical potentials have to be given in energy per mole of atoms. 1.5.2. Partial molar quantities Let us suppose a homogeneous first order extensive quantity A that has only the particle numbers as extensive variables, e.g.
A = A(T , P , N1 ,...N k )
The first order homogeneity implies
A ( P, T , λ N1 ,..., λ N k ) ⇒
= λ ⋅ A ( P, T , N1 ,..., N k )
dA ( P, T , λ N1 ,..., λ N k ) = A ( P, T , N1 ,..., N k ) dλ
Consequently,
A(P,T,N1,...,N k ) = ∑ i
∂A(P,T, λN1,..., λN k ) ∂λN i P ,T , λN
=∑ i
⋅ j
, j≠ii
∂A(P,T, λN1,..., λN k ) ∂λN i P ,T , λN
j , j≠ i
dλN i dλ
⋅ Ni
This relation must hold for any value of λ. We may thus take λ = 1 and get
A(P,T,N1,...,N k ) = ∑ N i i
∂A(P,T,N1,...,N k ) ∂N i P,T ,N
= ∑ N i ⋅ Ai i
j , j≠ii
The partial quantity Ai is thus defined as
Ai =
∂A(P ,T , N 1 ,...N i ,...N k ) ∂N i P ,T ,N
(16) j , j ≠i
As a result, any homogeneous function of first order in the particle numbers can be set up by the weighted sum of the contributions of the components. The individual contributions Ai are called partial quantities. Since we have
Ai (P,T, λN1,..., λN k ) =
∂A(P,T, λN1,..., λN k ) ∂λN i P ,T , λN
= Ai (P,T,N1,...,N k )
j
, j≠i
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the Ai are indeed intensive quantities and they are constant along lines defined by (λN1,..., λN k ) within the space spanned by axes defining the particle numbers. That means: the whole variety of Ai values can be obtained from a section in this space. If we take
λ=
1 1 = ∑ Ni N i
the coordinates of this section are given by the mole fractions x i = N i N and the partial quantity is a function of the mole fractions, Ai= Ai ( P, T , x1 ,..., xk ) . 1.5.3. Geometrical interpretation of partial molar quantities The quantity A(P,T,N1,...,N k ) defines a hyper-cone in the space defined by the coordinate system (A,N1,...,N k ). For a fixed number of N, e.g. N = 1 mole of atoms, the quantity Am (P, T, x1,..., x k ) defines the values of A for 1 mole of substance. These values are within a vertical section defined by a hyper-plane parallel to the A-axis and passing through the points for 1 mole on every axis Ni, i.e. (1,0,...0), (0,1,...0),... (0,0,...,1). This is shown in Figure 4 for the case of a binary system. The partial molar quantity Ai (P, T, N1,..., N k ) defines the slope of A(P, T, N1,..., N k ) in the direction of the Ni-axis at a point of the hypercone, defined by the coordinates (N1,..., N k ). This direction being parallel to the Ni-axis, the coordinates Nj remain unchanged, in accordance with definition (16). It will now be shown that the partial molar quantity
A i (P , T , x10 ,..., x k0 )
is obtained as intersection of the tangent plane to the surface Am (P, T, x1,..., x k ) defined in the space (x1,..., x k ) at the point
A m (P , T , x10 ,..., x k0 )
We start from equation:
Ai =
∂A ∂N i
= P ,T , N j , j≠i
∂N ⋅ Am (P, T, x1,...x k ) ∂N i P ,T , N
Since
∂Am ∂N i and
= P ,T , N j , j≠i
= Am + N j
, j≠i
∂Am ∂N i
P ,T , N j , j≠i
k ∂Am (P, T, x1,..., x k ) ∂Am ∂x i ∂A ∂x j = +∑ m ∂N i ∂x i ∂N i j≠i ∂x j ∂N i
Ni N = N − N i = 1 (1 − x ) i ∂N i N2 N
∂
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∂
85
Nj
N =−Nj =− xj ∂N i N2 N
we get
∂Am ∂N i
= P ,T ,N j , j≠i
k 1⎡ ∂A ∂A ⎤ ⎢(1− x i ) m − ∑ x j m ⎥ ∂x i j≠i ∂x j ⎥⎦ N ⎢⎣
For the partial molar quantity Ai we thus obtain
Ai = Am + (1− x i )
k ∂Am k ∂A ∂A ∂A − ∑ x j m = Am + m − ∑ x j m ∂x i j≠i ∂x j ∂x i j=1 ∂x j
(17)
In the derivation of equation (17) the particle number Ni was varied while taking the other numbers Nj constant. Yet, all the mole fractions varied since the total number varied. In fact, Ai represents a property that is not related to the vertical section of 1 mole, since it is not possible to add atoms of a specific kind, say i, if the system is forced to remain within the section defined by Σ i x i = 1.
Figure 4. Binary system with components 1 and 2. The extensive property A(T,P,N1,N2) defines a cone because of its linear homogeneity. A vertical section, defined by the condition N1 + N2=1, intersects the cone leading to the curve A(T,P,x1,x2). At point (0.1,0.9) the partial quantity A2(T,P,0.1,0.9) represents the slope of the cone into the direction of axis N2. This slope is the same as that of the triangle in the (A,N2)-plane defined by the origin, the point (0,1) and the intersection of the tangent plane with the N2-axis.
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This becomes clear in Figure 4 where it is shown that Ai represents a slope in a direction that is not within the section. Yet it is possible to derive the value of Ai within the section. For this purpose we have to take account of the constraint Σ i x i = 1 and select one of the components as dependent. Let us take x1 as dependent variable and replace it by k
1− ∑ x j j =2
in the expression Am. This implies:
∂Am = 0. ∂x1
Furthermore, the remaining xi are all independent. The partial derivatives then become in fact derivatives and we get
Ai = Am + (1− x i )
k dAm k dA dA dA − ∑ x j m = Am + m − ∑ x j m dx i j= 2 dx j dx i j= 2 dx j j≠i
k
A1 = Am − ∑ x j j= 2
i ≠1 (18)
dAm dx j
This is in fact the equation of the intercept of the (hyper)-plane tangent to the (hyper)-surface Am at the coordinates (x 2 ,..., x k ) with the Am-axis at the pure component i. 1.5.4. Examples Partial molar Gibbs energy of a binary system (Figure 5): For a binary system we take e.g. the mole fraction of component B as independent variable.
Figure 5. Binary system A-B: graphical construction of the partial Gibbs energy G i according to equations (19). The partial molar Gibbs energy is the chemical potential μ i . The chemical potential of component i is obtained as intercept of the tangent to the molar Gibbs energy curve with the axis of molar Gibbs energy of component i.
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From equation (18) we get
dGm dGm dGm − xB = Gm + x A dx B dx B dx B dG μ A = GA = G m − x B m def dx B
μ B = GB = G m + def
(19)
Partial molar Gibbs energy of a ternary System (Figure 6): In this case we take the mole fractions xB and xC as independent composition variables and obtain from Eq. (18):
dGm dGm − xC dx B dxC dG dG μC = Gm + (1− xC ) m − x B m dxC dx B
μB = Gm + (1− x B )
μ A = Gm − x B
dGm dGm − xC dx B dxC
(20)
The graphical construction is shown in Figure 6 where the molar Gibbs function Am the Gibbs energy has been taken, namely Am = Gm .
Figure 6. Ternary system A-B-C: Graphical construction of the partial Gibbs energy Gi according to Eq. (20), here demonstrated for component B. Similar constructions apply for components A and C. In a ternary system the chemical potential of a component i is obtained as intercept of the tangent plane to the molar Gibbs energy curve with the axis of molar Gibbs energy of component I.
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1.6. EQUILIBRIUM
Systems can be coupled together to exchange quantities, provided the quantities are of the same type, e.g. extensive. It will turn out that this offers a way of defining the character of a variable which cannot be identified easily by simple arguments like a scaling with the size of the system. Let us consider two systems (1) and (2) characterized by the MassieuGibbs functions
(
)
v
Φ( 1 ) X i( 1 ) = ∑ Yi( 1 ) X i( 1 )
(
)
i =1 v
Φ ( 2 ) X i( 2 ) = ∑ Yi( 2 ) X i( 2 ) i =1
The variables
X i(ν ) and Yi (ν )
represent respectively extensive variables of the two systems ν = 1, 2 and the conjugate intensive variables. The potential of the combined system (1)+(2) is given by Φ = Φ(1) + Φ(2) . Equilibria between systems are established by a free exchange of coupling variables, while the remaining variables are kept constant. A free exchange of the extensive quantity X j between the two systems means two conditions:
(
)
(1) (2) (2) 1. X (1) = 0 or ordX j j + X j = X 0 = const. and thus dX 0 = d X j + dX j
(1)
= −dX (2) j 2. Φ = const . , i.e. no change of the potential of the combined system due to the exchange, and thus dΦ = d(Φ(1) + Φ(2) )= 0. Since the remaining X i(≠n )j are kept constant, one gets:
(2) (2) d Φ = Y j(1) dX (1) = (Y j(1) − Y j(2) ) dX (1) j + Y j dX j j =0
i.e. Y j(1) = Y j(2) At equilibrium, the intensive variables conjugate to the exchanging extensive quantities take the same value. The condition dΦ = 0 defines an extremum. It can be shown that in the case of “energy type” potentials like enthalpy H, free energy F or Gibbs energy G, the extremum is a minimum. In the case of “entropy-type” potentials the extremum is a maximum. 1.6.1. Reduction of dimension due to equilibrium: Every condition for equilibrium reduces the degree of freedom of the system by one. Taking the system Φ as the combination of two subsystems
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89
(1) and (2) with dimension r1 and r2 , the number of extensive variables is r1 + r2 . The Gibbs function of the combined system is r1
r2
j =1
j =1
Φ = Φ (1) + Φ ( 2 ) = ∑ Y j( 1 ) X (j1) + ∑ Y j(2 ) X (j2 ) Let us suppose an equilibrium by a free exchange of the variables
X k( 1 ) + X m( 2 ) = const . = X 0 this implies:
dX m(2 ) = −dX k(1) and Yk(1) = Ym(2 )
Consequently:
(
(1)
( 2)
d Φ = Yk − Yk r1
) dX
(1) k
r1
r2
j ≠k
= ∑ Y j( ) dX (j ) + ∑ Y j( ) X (j 1
1
j ≠k
1 = d Φ (( r1)−1)
r2
+ ∑ Y j dX j + ∑ Y j( 2) X (j 2) (1)
2
(1)
j≠k
2)
j ≠k
2 + d Φ (( r2)−1)
Due to the equilibrium condition each of the subsystems has a dimension reduced by 1. 1.6.2. Equilibrium with a reservoir Reservoir: A system with only one degree of freedom is called a reservoir of a given energy form dE = YdX if it can exchange energy only at a fixed value of the intensive variable Y. Let us denote this fixed value by Y0. Let us take system (2) as a reservoir. We then can write
(
)
r1
r1
j≠k
j≠k
d Φ = Yk(1) − Y0 dX k(1) + ∑ Y j(1) dX (j1) = ∑ Y j(1) dX (j1) This shows that for an equilibrium with a reservoir the dimension of the system is reduced by one: the Massieu-Gibbs function is of first order in the r1 − 1 extensive variables. Remark: it is important to realize that an equilibrium is only established if a free exchange of variables is occurring. This can be confirmed in experiments. If e.g. iron is exposed at high temperatures to a high carburizing atmosphere, graphite will be deposited on the surface. If this deposition occurs with the basal plane of graphite parallel to the iron surface, then the C-atoms are so strongly bound within the basal plane that no free exchange is possible. Consequently, equilibrium between graphite and iron is not established in that case.
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2. Chemical Reactions 2.1. REACTIONS BETWEEN GASES
At a given temperature T and pressure P the Gibbs energy of an ideal gas with reference to the state at pressure P0 and n=1 mole, is given by
G(P, T, n ) − G(P0 , T, n ) =
P
P
P0
P0
∫ VdP = nRT ∫
⎛P⎞ dP = nRT ln⎜ ⎟ P ⎝ P0 ⎠
(21)
The chemical potential is obtained as
μ=
⎛ P ⎞⎤ ⎛P⎞ ∂G d ⎡ 0 = ⎢nGm + nRT ln⎜ ⎟⎥ = G 0 (P0,T,n = 1) + RT ln⎜ ⎟ ∂n dn ⎣ ⎝ P0 ⎠⎦P ,T ⎝ P0 ⎠
(22)
Let us consider a mixture of ideal gases with species A1, A2,… Ak, and mole numbers ni within a volume V. Each of the gases has to fulfil the equation Pj V=nj RT. Pj is called partial pressure of component j. It is defined as
Pj =
nj
∑n
P = xjP j
j
The system composed of the different gaseous species thus fulfils the relation
V ⋅ ∑ P j = V ⋅ P = RT ∑ n j = nRT
Let us consider a system with n1 moles of species A1, n2 moles of species A2, etc.. A reaction may be defined by the symbolic relation
n1 A1 + n 2 A2 + ... + n i Ai = n i +1 Ai +1 + n i + 2 Ai + 2 + ... + n r Ar r
or
∑ν
j
Aj = 0
(23)
j =1
with coefficients ν j = −n j for 1 ≤ j ≤ i and ν j = n j for i + 1 ≤ j ≤ r . Since the reaction proceeds with fixed values of ν j , we can define the degree of advancement ξ of the reaction in such a way that for every step dξ the amount of reactants i
∑n
j
dξ
j
dξ
j =1 r
transforms into the products
∑n j = i +1
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At constant temperature and pressure the change in Gibbs energy associated with the reaction is given by r ⎛ r ⎞ ΔG = ∑ μ j dn j = ⎜⎜ ∑ν j μ j ⎟⎟ ⋅ dξ j=1 ⎝ j=1 ⎠
(24)
The term in parenthesis is called Gibbs energy of reaction:
Δ rG =
def
r
∑ν
j
μj
j=1
According to (22) the chemical potentials are given by
⎛ Pj ⎞ ⎟ ⎝ P0 ⎠
μ j = G 0j (P0, T,1) + RT ln⎜
(25)
ΔG 0j defines the Gibbs energy of 1 mole of species j in its standard state, i.e. at pressure P j = P0 = 1 bar . The Gibbs energy of reaction thus becomes r ⎛r ν ⎞ r Δ rG = ∑ν j G 0j + RT ln⎜ Π P j j ⎟ = ∑ν j G 0j + RT ln(Q) (26) ⎝ j=1 ⎠ j=1 j=1
For Δ rG < 0 the reaction proceeds forward and for Δ rG > 0 the reaction proceeds backward. The first term on the right hand side of (26) is called standard Gibbs energy of reaction: r
Δ rG 0 = ∑ν j G 0j
(27)
j=1 r
ν
while Q = Π P j j is called reaction quotient. j=1
At equilibrium: Δ rG dξ = 0 . Here dξ represents an arbitrary positive or negative deviation from equilibrium. Consequently, Δ r G = 0 , or
Δ rG 0 = −RT ln(K )
(28)
The reaction quotient at equilibrium is called equilibrium constant K=Qequil. If the value of Δ rG 0 is given for a temperature T0 the value corresponding to a temperature T is obtained from: Δ rG 0 (P0 ,T ) = Δ rG 0 (P0 ,T0 ) − (T − T0 )Δ r S 0 (P0 ,T0 ) +
T
∫Δ c r
T0
T
P dT − T ∫ T0
Δ rc P
τ
dτ
(29)
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G. INDEN
Δ r S 0 = ∑ν j S 0j and Δ rc p = ∑ν j c (p j ) .
with
j
The specific heat capacity of species j is given by
c
=
( j) p
∂H ( j ) ∂T
P
Equation (29) shows that the Gibbs energy of reaction can be expected to be a linear function of temperature since for gases the term Δ r c P is small and virtually constant. The corresponding diagram versus temperature is called Ellingham diagram. Table 1 gives the thermodynamic data for gaseous substances in their standard states at T = 298 K. There from the standard Gibbs energies of reaction can be deduced for the following reactions:
(H 2 )g + 1 (O2 )g = (H 2 O )g 2
ΔrG0
= −228570 − ( T − 298 )(188.83 − 130.684 − 102.57 ) + ( −9.9 )( T − 298 )
(30)
= −228570 + 54.25 (T − 298 ) = −244736 + 54.25 T
J/mole H 2O
( CO ) g + ( O2 ) g = ( CO2 ) g 1
2
Δ rG 0 = −394360 + 137170 − (T − 298)(213.74 −197.67 −102.57) + (−6.7)(T − 298) = −257190 + 79.8 (T − 298) = −280970 + 79.8 T
(31)
J/mole CO2
Application: It is often required to maintain a very low partial pressure of oxygen in a gas atmosphere. This can be realized by taking an appropriate mixture of water and hydrogen. Suppose a partial pressure of 10–10 bar O2 is required at T = 2000K. Introducing the data of Eq. (30) into Eq. (28), one gets the equilibrium constant at T = 2000K:
⎛ PH O 2 − RT ⋅ ln⎜ ⎜P ⋅ P O2 ⎝ H2 PH 2O PH 2
⎞ ⎟ = −136236 ⇒ K = 3616 ⎟ ⎠ = 3.616 ⋅ 10 −2 .
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TABLE 1. Thermodynamic data of gaseous substances, The values correspond to T=298K, P=1bar and 1mole Gaseous species
(O2 ) g (H 2 ) g (H 2O) g (CO) g (CO2 ) g
ΔGi0 J/mole
S 0 J/mole·K
c P J/mole·K
0 0 –228 570 –137 170 –394 360
205.138 130.684 188.83 197.67 213.74
29.35 28.82 33.58 29.14 37.11
If the partial pressure of H2 is set to 1 bar, then
PH 2O = 0.036 bar . This partial pressure corresponds to the saturation pressure of water at 28°C. The required gas mixture can thus be obtained by bubbling hydrogen gas at 1bar through liquid water at 28°C. The partial pressure of oxygen can also be controlled using reaction (31). Suppose a pressure of 10–10 bar O2 is required at T = 1500K. The equilibrium constant is obtained from
⎛ P ⎞ CO2 ⎟ = −161270 ⇒ K = 4.132 10 5 −RT ⋅ ln⎜⎜ ⎟ ⎝ PCO ⋅ PO2 ⎠ PCO2 = 4.132 PCO
If the total pressure is 1 bar, i.e. PCO2 + PCO = 1 bar then
PCO2 = 0.805 bar
PCO = 0.195 bar
The required gas mixture can thus be obtained by taking the gases in the ratio 80 volume% CO2 and 20 volume% CO. 2.2. REACTION BETWEEN GASES AND SOLID PHASES
In the same way as shown for the reaction with gases, the standard Gibbs energies of reaction between solid and gaseous phases are obtained. However, for solid phases the pressure term in Eq. (21) is practically zero because of negligible variation of volume with pressure. Consequently Eq. (25) yields for solid phases:
μi = Gi0 (P0,T,1)
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G. INDEN
The missing pressure term for the solid phases implies that the reaction constant contains only the partial pressures of the gaseous species. This means: equilibrium according to Eq. (28) for the reaction
M
s
+ (O2 )g = MO2
s
is thus given by
⎛ 1 ⎞ ⎟⎟ = RT ln(PO2 ) Δ rG 0 = −RT ln⎜⎜ P ⎝ O2 ⎠
(32)
and named standard Gibbs energy of reaction. The Ellingham diagram represents the standard Gibbs energy of reaction as function of temperature. The essential features of the diagram shall first be discussed with schematic diagrams. In Figure 7a the case of oxidation of a solid metal to form a solid oxide is shown. The standard Gibbs energies of two reactions are represented. It is possible to include the right hand side of Eq. (32) into this diagram. This leads to lines with slope RT Ln PO2. In the range of negative energy values the negative slope induces pressures less than 1 bar. The equilibrium condition given by Eq. (32) is fulfilled at the intersection of these lines with the Δ rG 0 curves. The slope of the line gives the corresponding equilibrium partial pressure of O2. These equilibrium partial pressures can be labelled on a separate nomographic axis as shown in Figure 7a. In this figure the standard Gibbs energies of reaction are shown for two reactions:
2 M s + (O2 )g = 2 MO 2 A s + (O2 )g = 2 AO
(i)
s
(ii)
s
For a given temperature the equilibrium vapour pressure O2 can be read from the nomographic pressure axis by drawing a line from the origin 0 (T = 0K, Δ rG 0 = 0 ) to the corresponding point on the line Δ rG 0(T ) of the reaction. At the temperature Tequil of the intersection of the two lines Δ rG 0(T ) for (i) and (ii) the equilibrium for both reactions is fulfilled. The difference
Δ rG 0(iii) = Δ rG 0 (i) − Δ rG 0(ii) represents the standard Gibbs energy of reaction (iii):
(i)
− (ii) ⇔
2 M s + AO s = 2 A s + MO
s
(iii)
Since Δ rG (iii) > 0 at temperatures below T , the reaction proceeds backward, i.e. both 2 <M>S and S are stable relative to 2 S and <MO>S. Above Tequil it is the reverse. This means that if A were to be used as a reducing agent to reduce MO to form M and AO, then the reduction could only be effected at temperatures T > Tequil. 0
equil
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Figure 7. a) Schematic Ellingham diagram of oxidation reactions of solid phases to solid oxides, b) Schematic Ellingham diagram for an oxidation reactions between gases at different pressure ratios P(CO2)/P(CO).
The case of an oxidation reaction between gaseous species is illustrated in Figure 7b. The standard Gibbs energy of reaction corresponds to 1 bar partial pressure of the reacting gases. Maintaining PO2 = 1bar the ratio of partial pressures of CO2 and CO can be varied leading to an additive term 2 ⎞ ⎛ PCO RT ln⎜ 2 2 ⎟ ⎝ PCO ⎠
according to Eq. (26). This is shown in Figure 7b. The linear variation of the additive term with temperature allows drawing a nomographic scale for the pressure ratio.
Figure 8. Effect of a phase transition on the standard Gibbs energy of reaction: melting of metal A as reactant and melting of the metal oxide MO as reaction product.
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G. INDEN
In Figure 8 the effect of a phase transformation on the standard Gibbs energy of reaction is shown. Taking the melting of a species I as an example, the entropy changes abruptly by the amount of entropy of melting,
ΔS Is →liq = ΔH Is →liq Tm leading to a change in slope of the corresponding standard Gibbs energy curve. Depending on whether species I belongs to the reactants or the reaction products the slope is increased or reduced. In Table 2 the standard Gibbs energies for some reactions are given. In the case of reaction 2 C
s
( )
( )
+ O2 = 2 CO g g
the solid is reacting with 1 mole of gas to produce 2 moles of gas. This leads to a substantial value of the entropy of reaction. In the case of reaction
C
s
+ (O2 ) g = (CO2 ) g
a solid is reacting with 1 mole of gas to form 1 mole of gas. Consequently, the entropy of reaction is very small. Figure 9 shows the Ellingham diagram for these and other reactions. There are three main uses of the Ellingham diagrams: •
estimation of the ease of reducing a metal oxide into the pure metal,
•
the determination of the partial pressure of oxygen in equilibrium with a metal oxide at a given temperature,
•
the determination of the CO/CO2 ratio and of the gas composition that will allow a reduction of the oxide into the pure metal. TABLE 2. Standard Gibbs energies of reaction between gases and solid phases
Reaction 2 < C > S + (O2 ) g = 2 (CO) g < C > S + (O2 ) g = (CO2 ) g < Mn > S + (O2 ) g = 2 < MnO > S 2 < Mg > S + (O2 ) g = 2 < MgO > S < Ti > S + (O2 ) g = < TiO2 > S 2 < Ni > S + (O2 ) g = 2 < NiO > S
Δ r G 0 J/mole O2 –223425–175.3 T –394133–0.837 T –769437+145.6 T –1207921+284 T–10.73 T lnT –897468+173.2 T –489109.6+197 T
The position of the line Δ rG 0 in the Ellingham diagram determines the stability of the oxide as function of temperature. Reactions closer to the top of the diagram define oxides which are less stable and easy to reduce, e.g. Ag2O. A given metal can reduce all the oxides with Δ rG 0 lines above in the Ellingham diagram. E.g. the line of 2Mg+O2=2MgO lies below the line of Ti+O2=TiO2. Consequently, Mg can reduce TiO2 to metallic Ti.
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Figure 9. Ellingham diagram of various metal and carbon oxides. The reactions are defined for 1 mole of O2. M=melting point, A=austenite/δ-ferrite transition of Fe.
The line for 2C+O2=2CO has a negative slope. It thus intersects many of the lines of other reactions. As soon as the carbon oxidation line goes below a metal-oxide line, carbon can reduce the metal oxide to pure metal. This makes carbon so important as a reducing agent. As an example, C can reduce Cr2O3 once the temperature exceeds approximately 1225°C. The nomographic axes at the right hand side of the Ellingham diagram allow reading the partial pressures of the atmosphere for an equilibrium between a metal and its oxide. If this value is exceeded the metal will be oxidized, if it is lower the oxide will be reduced. The axis giving the ratio of the partial pressures of CO and CO2 can be used to determine the minimum partial pressure of CO needed for reduction of the oxide. The more stable the oxide, the greater must be the proportion of CO.
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G. INDEN
To illustrate the use of the Ellingham diagram the problem of reduction of MnO by solid carbon at 1200°C shall be taken as example. The oxidation reaction is
2 Mn s + (O2 )g = 2 MnO
(33)
s
Carbon in equilibrium with oxygen implies the following reactions:
C s + ( O2 ) g = ( CO2 ) g 2 ( CO ) g + ( O2 ) g = 2 ( CO2 ) g
(a) 2 C s + ( O2 ) g = 2 ( CO ) g (c)
(b)
C s + ( CO2 ) g = 2 ( CO ) g (d )
(34)
Figure 10 illustrates the procedure of determining the experimental conditions. At first the point corresponding to T=1200°C is fixed on the line of equilibrium (33). From the starting points at T=0K of the equilibrium lines defining the equilibria (34), lines have to be drawn to the point at 1200°C. These lines intersect the corresponding nomographic axes and define the equilibrium partial pressures.
Figure 10. Ellingham diagram of carbon and manganese oxides. The construction illustrates the condition for a reduction of MnO by a gaseous atmosphere of O2, CO and CO2.
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In the present instance, this construction leads to PCO = 5 10−2 , PCO2 = 2.2 10−6, PO2 = 2 10−20 (bar), PCO PCO2 = 2.310 4 Thus, at equilibrium the total pressure is practically 5·10–2 bar. If the total pressure is reduced, then spontaneous reduction of MnO will occur. 3. Thermodynamics of Alloys In the following the experimental conditions T=Const. and P=Const will be assumed. The thermodynamic description must therefore be based on Gibbs energy. As long as the size of the system does not have any effect, as in most cases of physical metallurgy, it is sufficient to consider molar quantities. Therefore, in the following only molar quantities will be described. In order to simplify the notation an index “m” will be omitted. The molar Gibbs energy of an alloy is then constructed as a sum of various contributions, starting with the contribution coming from the pure components, then the contribution due to the mixing process to form a random alloy, followed by further contributions due to the formation of special configurations like ordering or magnetic effects etc.. 3.1. REFERENCE FRAME, LATTICE STABILITY
Energies can only be defined as differences. Therefore it is necessary to define a reference frame. For a pure component i the enthalpy of the stable state, say α, at T = 273K and P = 1bar is usually taken as reference for Gibbs energy of the component:
Δ 0Giα (T ,1 bar ) = 0Giα (T ,1 bar ) − 0 H iα ( 273K ,1 bar ) The reference term is denoted 0
H iα (273K,1 bar) ≡ H iSER def
where SER stands for Stable Element Reference. Since a pressure of 1 bar is taken as the standard situation, the pressure will not be explicitly indicated in the following. If at some temperature T the state is different from the one at 273K, Gibbs energy is then described by
Δ 0Giβ (T ) = 0Giβ (T )− 0H iα (273K ) = 0Giβ (T ) − H iSER = 0Giβ (T )− 0Giα (T )+ 0Giα (T ) − H iSER = Δ 0Giα →β (T ) + Δ 0Giα (T )
The term Δ 0Giα →β represents the Gibbs energy of transformation from α to β. This term is called lattice stability. In the case of a compound θ with
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G. INDEN
fixed (stoichiometric) composition, the molar Gibbs energy of formation from the pure elements is given by
Δ f0Gθm = 0Gθm − ∑ν i 0Giα where the ν i are the stoichiometric coefficients. The value of Δ f0G mθ then corresponds to 1 mole of formula unit, i.e. to a total amount of atoms n = Σν i . If the stoichiometric coefficients are ν i = xi its value corresponds to one mole of atoms. The Gibbs energy of formation representing an energy difference, the reference state does not enter explicitly. It can, of course, be introduced in order to use tabulated data which usually are given relative to a reference state REF (which may correspond to SER):
Δ f0Gθm =0Gθm − H REF − ∑ν i (0Giα − H iREF ) 3.2. PURE COMPONENTS
In thermodynamic databases the molar Gibbs energy of a pure component is written as a power series in T and P:
Gm − H REF = a + bT + cT lnT + dT 2 + ...+ eP + fTP + gP 2 + ... If the power series is truncated at the quadratic terms the coefficients can be related to the following quantities: •
specific heat capacity c P . Since
⎛ ∂G ⎞ H (T , P ) = G + TS = G + T ⎜ − ⎟ ⎝ ∂T ⎠ P the specific heat capacity is obtained as:
cP =
∂H ∂T
= P
∂G ∂G ∂2G ∂2G − − T 2 = −T 2 ∂T P ∂T P ∂T P ∂T P
Consequently: c P = −c − 2 d T •
molar entropy: From:
S=−
∂G ∂T P
it follows Sm = −b − c − c lnT − 2 d T − f P
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101
molar volume: From:
V=
∂G ∂P T
it follows Vm = e + f T + 2 g P •
molar enthalpy: H m − H REF = a − c T − d T 2 + e P + g P 2
•
molar free energy: Fm − H REF = a + b T + c T lnT + d T 2 − g P 2
•
molar internal energy: U m − H REF = a − c T − d T 2 − f T P − g P 2
3.3. SOLUTION PHASES
The Gibbs energy of solution phases is usually built up starting from the pure components as a so-called “mechanical mixture” which is just the amount of different elements. The Gibbs energy of the mechanical mixture is thus given by
ΔG
mech.− mixt. m
n
= ∑ x i 0ΔGi i= 1
The Gibbs energy of the solution phase is then set up in the form
ΔGm = ∑ x i 0ΔGi +
M
Gm
by various additive terms which shall describe the various forms of solution phases. 3.3.1. Ideal solution The simplest model for a solution is the random mixing of the atomic species without any particular interaction between unlike atoms. In this case only the contribution of random mixing is to be added to the Gibbs energy of the mechanical mixture. This contribution is given by the entropy of mixing: n n n ideal ideal 0 Sm = − R ∑ xi ln xi ΔGm = ∑ xi ΔGi + RT ∑ xi ln xi i =1 i =1 i =1
(35)
The ideal entropy term is not only important for the hypothetical model system, but also for real solutions. At high enough temperatures every system approaches states, which are close to random mixing since due to the factor T the entropy becomes dominant against other terms due to interactions.
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G. INDEN
We may calculate the partial molar entropy according to Eq. (18): n
Siideal = Smideal − R (ln x i + 1) + R∑ x i (ln x i + 1) = − R ln x i i= 1
Similarly, we obtain for the chemical potential:
μ i = Gi = G
n
ideal m
+ Gi + RT (ln x i + 1) − ∑ x j 0
j=1
(G 0
j
)
+ RT [ln x j + 1]
(36)
= 0Gi + RT ln x i 3.3.2. Regular solution The regular solution model tries to take into account a pair-wise interaction between unlike atoms leading to an enthalpy of mixing term. in the Gibbs energy expression. Defining by (1) ε AA ,ε
(1)
(1) and ε AB
BB
the interaction energies (at 0K) between the various nearest neighbour pairs, the energy contribution upon mixing derives from a counting of such pairs. Given the coordination number z1 of atoms in the nearest neighbour shell, the total number of pairs between N atoms is z1 N 2 . The probability for a pair to be of type i-j is given by pij = x i x j . The energy is thus:
U=
(
z1 N 2 (1) (1) (1) (1) x A ε AA + xB2 ε BB + x A xB ε AB + x A xB ε BA 2
)
Taking the pure components as the reference:
U − x A 0U A − xB 0U B = U − =
(
z1 N (1) (1) x A ε AA + xB ε BB 2
(
z1 N (1) (1) (1) − ε AA − ε BB 2 ε BA 2
)
)
Defining the energy per nearest neighbour bond A-B as (1) ε (1) = ε AB −
(1) (1) ε AA + ε BB
2
the Gibbs energy of the regular solution becomes n
G
regular m
n
n
n
= ∑ x i Gi + ∑ ∑ z1εij x i x j + RT ∑ x i ln x i 0
i= 1
i≠ j j =1
(1 )
i= 1
If the nearest neighbour interaction energy is positive, this means that A-B bonds are less stable than the average between the two like bonds. At low temperatures where the energy terms are dominating the mechanical mixture is thus more stable than the solution. This is called a phase separation tendency. If the nearest neighbour interaction energy is negative, this means
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103
that at low temperatures the most stable state is with a preference of unlike bonds. This leads to ordering. The regular solution can be generalized to higher order pair interactions. Defining the coordination number z k for the k-th neighbour shell, one can write: n n n ⎛ n (k ) ⎞ Gmregular = ∑ x i 0Gi + ∑ ∑⎜ ∑ zkεij ⎟ x i x j + RT ∑ x i ln x i (37) ⎠ i= 1 i≠ j j =1 ⎝ k i= 1
Defining the interaction coefficient Lij =
∑z
k
k
n
Gmregular = ∑ x i 0Gi + i= 1
n
∑
(ε
n
(k)
ij
)
+ ε ji( k ) one can write: n
∑ Lij x i x j + RT ∑ x i ln x i
i= j +1 j =1
i= 1
3.3.3. Chemical potentials of regular solutions The chemical potential μ i of a component i is the molar partial Gibbs energy. From Eq. (18) one gets for a ternary system:
μB = 0 GB + LAB x A (1− x B ) − LAC x A xC + LBC xC ((1− x B ) + RT ln x B μC = 0 GC − LAB x A x B + LAC x A (1− xC ) + LBC x B ((1− xC ) + RT ln xC μA = 0 GA + LAB x B (1− x A ) + LAC xC (1− x A ) − LBC x B xC + RT ln x A Similarly, for a binary system:
μB = 0 GB + LAB x A (1− x B ) + RT ln x B = 0GB + LAB x A2 + RT ln x B
μA = 0 GA + LAB x B (1− x A ) + RT ln x A = 0GA + LAB x B2 + RT ln x A 3.3.4. Activity of binary regular solutions The regular solution may be described in the same format as the ideal solution if one introduces a new variable, the activity a i , instead of xi by means of the following equation:
μ i = 0 μ i + RT ln a i
(38)
The activity thus plays the same role for the regular solution as does the mole fraction for the ideal solution. The molar Gibbs energy being given by the expression
ΔGm = Gm − ∑ x i 0Giref
= x A Δ 0GA + x B Δ 0GB + x A x B L + RT (x A ln x A + x B ln x B )
we obtain for the chemical potential:
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G. INDEN
μB =Δ 0GB + L x A2 + RT ln x B According to the definition of activity (38) we get:
⎛ Δ 0GB + L x A2 ⎞ RT ln aB = RT⎜ ⎟ + RT ln x B RT ⎝ ⎠ ⎧ ⎛ Δ 0G + L x 2 ⎞⎫ B A = RT ln⎨exp⎜ ⎟⎬ + RT ln x B RT ⎠⎭ ⎩ ⎝ ⎛ Δ 0G + L x 2 ⎞ B A ⎟⎟ aB = x B exp⎜⎜ RT ⎝ ⎠
or
The reference state for the activity is the same as that for Δ0 GB , usually SER. Raoult’ian reference state: If the structural state of component B at a given temperature T is the same as that of the alloy, one may equally well adopt the state of the pure component at temperature T as a reference. In that case we have Δ0 GB = 0 .
⇒
aB
x B =1
=1
In the Raoult’ian reference system the activity of the pure component is 1. For the composition x B we get in the Raoult’ian reference system
⎛ L (1− x ) 2 ⎞ B ⎟⎟ aB = x B exp⎜⎜ RT ⎝ ⎠ In the case of an ideal solution, L = 0 , the activity is identical with the mole fraction. In activity vs composition plots this gives a line from x B = 0 to x B = 1 . For L ≠ 0 , the activity deviates from this line. For L>0 the deviation is towards higher activity values, see Figure 11b, for L0). At low temperatures a miscibility gap opens. a) Iso-activity lines of component B. The tie-lines in the miscibility gap are branches of iso-activity lines, b) Variation of activity vs composition. The activity is constant for compositions within the miscibility gap.
Figure 12. Variation of activity with composition and temperature for a system with ordering tendency between the atoms A and B (L
