POLITECNICO DI MILANO. Development of a Multi-Physics Approach to the Modelling and Analysis of Molten Salt Reactors

Transcription

1 POLITECNICO DI MILANO Doctoral Program in Radiation Science and Technology XXII Cycle Development of a Multi-Physics Approach to the Modelling and Analysis of Molten Salt Reactors VALENTINO DI MARCELLO Tutor: Supervisor: Chairman of the Doctoral Program: Prof. Lelio Luzzi Prof. Antonio Cammi Prof. Carlo E. Bottani January 2010

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3 To my parents in gratitude

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5 Acknowledgments First of all, I would like to express my gratitude to the two persons who gave me the opportunity to prepare this work under excellent conditions. Thank you Professors Lelio Luzzi and Antonio Cammi for your confidence and continuous support. Without your experience and precious advices, it would have been impossible to perform this Ph.D. work with success. Special thanks are due to Professor Leone Corradi who gave me the chance to work in the frame of thermo-mechanics. Throughout my period at Politecnico, I had the occasion to appreciate your experience and your valuable suggestions which improved my knowledge in this field. A great thank goes to my friends and colleagues of the Nuclear Power Plant group, in the past and in the present. With profound emotion and huge gratitude I thank my parents: my father Lino and my mother Luana. Thanks dad, thanks mom for your support and patience during these years, and for having encouraged me to concentrate on my study. I could always rely on your help. I would like to express special thanks to my girlfriend Sara. She helped me to concentrate on completing this dissertation and supported morally during the course of this work, especially in hard situations. Without her help, this study would not have been completed. To my grandfather Mario, affectionately. Finally, my special appreciation goes to all my friends for the joyful moments spent during these years.

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7 Contents Acronyms Introduction v vii 1 A brief overview of the Molten Salt Reactor technology History A renaissance of interest MSR description Safety Fuel cycle Materials Advantages and drawbacks of MSR technology The different concepts of MSRs Thermal-spectrum breeders Fast-spectrum breeders Fast-spectrum incinerators Thermal-spectrum incinerators Sub-critical incinerators Chloride-salt MSRs Current activities within GIF 23 2 A generalized approach to heat transfer in MSRs Heat transfer in pipe flow General solution of the heat transfer problem Method and accuracy of solution Experimental data comparison Fluids with small Prandtl number (10-2 < Pr < 1) Fluids with large Prandtl number (1 < Pr < 10 4 ) Fluids with internal heat generation 40

8 ii CONTENTS 2.3 Heat transfer in a MSR core channel Solution of the heat conduction problem Iterative procedure for the overall solution Method and accuracy of the overall solution Concluding remarks 49 3 Assessment of COMSOL Multiphysics Modelling of the MSBR core channel Thermo-hydrodynamics assessment Velocity, temperature and pressure loss inside the channel Reynolds number and turbulence modelling sensitivity Nusselt number evaluation Neutronics assessment Static fuel assessment Circulating fuel assessment Concluding remarks 71 4 The Multi-Physics Modelling approach Introduction The Multi-Physics Modelling Fluid flow and heat transfer model Neutronics model Boundary conditions Input parameters Results and discussion Steady-state conditions Transient conditions Concluding remarks 99 Conclusions 103 Nomenclature 109 References 115 Appendices A Point kinetics of Molten Salt Reactors A.1 A.1 Zero-power point kinetics model A.2 A.2 Comparison with MSRE experimental data A.4

9 CONTENTS iii A.3 Reactor stability at zero-power A.7 A.4 Complete point kinetics model A.11 A.4.1 Linear vs. non-linear approach A.14 A.5 Concluding remarks A.15 B Thermal-hydraulic analysis of the MSBR core B.1 B.1 Reference MSBR data B.2 B.2 Thermal-hydraulic parameters of the MSBR core channel B.4 C Pipe flow numerical analysis and validation C.1 C.1 Laminar flow C.2 C.2 Turbulent flow C.3 C.2.1 Turbulent heat transfer modelling C.4 C.2.2 Results of analyses C.6 C.3 Concluding remarks C.10

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11 Acronyms AHTR ALISIA AMSB ANP ATW ARE BOL CCT CFR DMSR DNP EDF FP GIF LWR MOSART MOST MPM MS ADS MSBR Advanced High-Temperature Reactor Assessment of LIquid Salts for Innovative Applications Accelerator Molten Salt Breeder Aircraft Nuclear Propulsion Accelerator Transmutation of Waste Aircraft Reactor Experiment Beginning-Of-Life Coupled Code Technique Circulating Fuel Reactor Denatured Molten Salt Reactor Delayed Neutron Precursors Electricite De France Fission Product Generation IV International Forum Light Water Reactor MOlten Salt Advanced Reactor Transmuter MOlten Salt Technology Multi-Physics Modelling Molten Salt Accelerator Driven System Molten Salt Breeder Reactor

12 vi ACRONYMS MSFR MSR MSRE NKC NPP RANS SFR SPHINX SMSR SNF SSC THSC THORIMS-NES TMSF TMSR VHTR Molten Salt Fast-neutron Reactor Molten Salt Reactor Molten Salt Reactor Experiment Neutron Kinetics Code Nuclear Power Plant Reynolds Averaged Navier-Stokes Sodium Fast Reactor SPent Hot fuel Incinerator by Neutron flux Small Molten Salt Reactor Spent Nuclear Fuel System Steering Committee Thermal-Hydraulic System Code THORIum Molten-Salt Nuclear Energy Synergetics Thorium Molten Salt Forum Thorium Molten Salt Reactor Very High Temperature Reactor

13 Introduction In the last decade, nuclear energy has gained a widespread renewal of interest as an important contributor to energy security, supply and sustainability. A number of new designs of Nuclear Power Plants (NPP) has been recently proposed in the frame of the Generation IV International Forum (GIF), in attempts to achieve advances in the following areas: sustainability; competitive economics; safety and reliability; proliferation-resistance and physical protection. In this context, innovative simulation techniques and approaches could be usefully employed for the "reactor system" description, accounting for the several aspects which involve the system behaviour either in operative or accidental conditions. Among the innovative simulation techniques, which are being the subject of a significant interest in several engineering fields, nowadays the so-called multi-physics modelling looks very promising and could be usefully employed also in the field of nuclear engineering thanks to the growing availability of computational resources and of suitable simulation environments (for instance, Abaqus Multiphysics, ADINA, ANSYS Multiphysics, CFD-FASTRAN, COMSOL Multiphysics, FlexPDE, NEi Nastran, IDC-SAC, OOFELIE Multiphysics). Actually, the multi-physics modelling approach results particularly suitable for the simulation of complex systems whose behaviour is featured by the coupling between different and simultaneous phenomena with non-negligible effects due to their time and space dependence. This is the case of the core in a nuclear reactor, whose behaviour can be properly described by taking into account the mutual interaction between neutronics and thermal-hydraulics. Up to now, this coupling is accounted for only partially by means of computational tools consisting of coupled advanced computer codes (CCT Coupled Code Technique), including a thermal-hydraulic system code (THSC) and a reactor neutron kinetics code (NKC), which are interfaced by means of more or less modifications of the software structure. The Coupled Code Technique is well assessed on conventional nuclear reactors, but may result non-fully satisfactory in the case of innovative nuclear systems in the light of the above mentioned requirements imposed by the GIF, since essential synergies

14 viii INTRODUCTION between simultaneously-occurring phenomena cannot be described in some cases with a significant lost of information. In the above outlined context, the present Ph.D. work is aimed at developing a Multi-Physics Modelling (MPM) approach, implemented in a unified simulation environment, able to describe the dynamic behaviour of a nuclear reactor of interest and, at the same time, capable of being employed for future works in order to set up the control strategy, and in the prospect of the proposal of new MSR conceptual designs. Among the several software packages available on the market, COMSOL Multiphysics has been adopted. This choice is due to the following features of COMSOL: i) flexibility, namely the possibility of implementation and resolution of non-linear and time-dependent finite element coupled partial differential equations, descriptive of the different physical phenomena in a more or less complex domain with boundary conditions of general kind; ii) the capability to offer a unified simulation environment able to be interfaced with MATLAB, which could be useful (for instance through the Simulink platform) for the development of adequate control strategies (model-based control toolbox). Among the six innovative concepts of fission nuclear reactor proposed by the GIF, the Molten Salt Reactor (MSR) is adopted for the analysis of this thesis. For such a kind of reactors, a multi-physics modelling approach is useful, if not necessary, because of the intrinsic and strong coupling existing inside the core between the hydrodynamic and temperature patterns, and the time-space distributions of Delayed Neutron Precursors (DNP) and neutrons. This coupling is due to the fact that MSRs adopt a fluoride salt mixture (in which the fissile and fertile materials as well as the fission products are dissolved) that plays simultaneously the role of both fuel and coolant, flowing through the reactor core. Moreover, according to the hydrodynamic pattern of the molten salt mixture, a certain amount of DNPs can decay outside the core (i.e., in the external primary loop) with significant consequences on the neutron balance and on the entire NPP dynamics. Therefore, an adequate description of their distribution is essential in order to study the reactor control, operation and maintenance. There is currently a strong interest in the molten salt technology from the scientific community driven by the several potentialities and advantages offered by MSRs (e.g., in terms of actinide burning, electricity and/or hydrogen production, breeding of nuclear fuel, natural resource utilization and management of radioactive wastes), and by their attractive features in terms of specific power density, intrinsic safety and ease to operate and maintain the reactor (Chapter 1). Differently from the other GIF projects, a specific reference configuration for the MSR has not been identified yet. In fact, current R&D activities on

15 INTRODUCTION ix MSRs are devoted to this subject and many reactor configurations have been proposed until now. Actually, the MSR can operate both with thermal or fast neutron spectrum as a breeder or converter or incinerator reactor by adopting either a critical or sub-critical (i.e., driven by an external neutron source) configuration. The single-fluid Molten Salt Breeder Reactor (MSBR), proposed by Oak Ridge National Laboratory (ORNL) in the 1970s was chosen as representative MSR configuration and is adopted as case study in the present thesis. The choice of the MSBR (featured by a thermal neutron spectrum and graphite moderated) relies on its detailed and quite definitive design, which is useful for feasibility studies of next generation MSRs. Actually, because of its significant progress and the exhaustive information and data delivered by ORNL, this reactor was considered as reference system for benchmark and validation analyses such as those carried out in the frame of EURATOM research programs (e.g., the MOST project MOlten Salt Technology). In addition, this reactor constituted the starting point for the development of the project THORIMS-NES (Thorium Molten-Salt Nuclear Energy Synergetics), which is based on a symbiotic system coupling fission with spallation, with several advantages in terms of proliferation resistance, safety, fuel cycle, radioactive waste management, economics and resources. Circulating fuel reactors pose a new challenge from the perspective of mathematical and numerical schemes for simulation, because of the mentioned intrinsic coupling, so that qualified numerical tools are required in order to perform reliable modelling of a complex and non-classical system such as the molten salt reactor. Therefore, the adopted multiphysics approach has required a deep and appropriate work aimed at the assessment of the capabilities of COMSOL Multiphysics for what concerns both neutronics and thermohydrodynamics. The study of heat transfer characteristics in MSRs is required since a non-classical nuclear fuel is under consideration, actually represented by a fluid with internal heat generation in turbulent flow, whose fluid dynamic behaviour has been the subject of few studies up to now. This part of the thesis is focused on the thermal-hydraulics aspects of a single-channel of the MSBR core, with particular attention to the heat transfer issues between molten salt and graphite. Due to the lack of experimental data, a generalized analytic approach was developed to evaluate the steady-state temperature distribution in a representative channel of the reactor (fuel + graphite) and for validation purposes. In particular, reference is made to the general formulation of the Graetz problem for the heat transfer in pipe flow, which has been subsequently coupled with a heat conduction model for the graphite (see Chapter 2). Such a formulation applies for fluids in hydro-dynamically developed, but thermally developing flow conditions, and includes recent advances in turbulence modelling.

16 x INTRODUCTION Moreover, boundary conditions and internal heat source can be assigned with arbitrary shape. The developed analytic approach has been assessed on the basis of a comparison with experimental data for a wide range of Reynolds and Prandtl numbers, with and without internal heat generation. In this context, a correlation for convective turbulent heat transfer in fully developed flow is proposed for simultaneous uniform wall heat flux and heat generation. The proposed model allows an analytic description of temperature distribution of both fuel and graphite in a MSR core channel, and has been exploited to assess the COMSOL potentialities exploring different Reynolds and Prandtl numbers, different turbulence models, and making also use of FLUENT as a dedicated computational fluid dynamics code in order to have a deeper insight into numerical modelling of turbulence, meshing strategy and numerical methods of solution (see Chapter 3). In addition, the use of correlating equations for the Nusselt number is analysed since it is of importance for engineering applications even with modern computational resources. By adopting classical correlations for the Nusselt number generally available in literature, the heat transfer coefficient of the MSR fuel can be overestimated by a non-negligible amount, making at least questionable their direct application to fluids with internal heat generation. In the case of thermal-neutron-spectrum MSRs, this can lead to significant consequences on the graphite temperature predictions and on the reactor dynamic behaviour. In order to confirm this idea, a brief overview of the correlations available in literature is performed, both in laminar and turbulent flows. In particular, they are compared with the numerical results obtained by means of COMSOL and FLUENT as well as with the values of Nusselt number given by the correlation proposed by the author, which in turn includes the effect of the internal heat source. As far as neutronics is concerned, COMSOL does not make use of specific modules for its description, but, thanks to the software structure flexibility, it is possible to include neutron transport equations with a diffusive approach. Therefore, during the Ph.D. activities, efforts have been spent to the extension of COMSOL applicability to neutronics, which have led to a strong interest from the software developers, since the code has been used for the first time in this research area. The developed model consists of two group neutron diffusion equations with six families of DNPs, and requires cross section values as input. These last have been calculated by means of the deterministic transport code SCALE 5.1 under the assumption of static fuel and in steady-state conditions (this is an obligation since dedicated neutronics code do not allow the analysis of circulating fuels, unless drastic modifications of software structure). The assessment of COMSOL has been performed in the case of both static and circulating fuel

17 INTRODUCTION xi (see Chapter 3). In the former, a code-to-code comparison with the Monte Carlo transport code MCNP, as well as with the numerical results provided by SCALE 5.1 is performed, in terms of neutron flux profiles, cross sections, and multiplication factor. In the case of circulating fuel, a study of effective delayed neutron fraction as a function of fuel velocity has been performed by means of COMSOL and the results have been assessed making use of a simplified neutron kinetics model, able to take into account the effect of fuel velocity in the primary loop. The results of this simplified model in terms of frequency response of the system have been compared with the experimental data available from the Molten Salt Reactor Experiment (MSRE) developed at ORNL. The model results representative of dynamic behaviour of the MSRE at zero-power, and has allowed a preliminary study of the reactor stability as a function of fuel velocity. On the basis of the assessment and set-up of the simulation environment of COMSOL and of the developed models, the Ph.D. research activity was afterwards focused on the coupling between neutronics and thermo-hydrodynamics, applying the MPM approach to a singlechannel representative of the average conditions of the MSBR core (see Chapter 4). This study is aimed at analysing the dynamic behaviour of the system, by studying the time evolution of the most important physical quantities such as the neutron flux, the DNP concentration, the graphite and fuel temperatures, the velocity field, the heat transfer characteristics and the friction pressure losses of the molten salt mixture. For the sake of clarity towards the reader, it must be mentioned that such analyses are oriented to the set-up and the assessment of the potentialities of the developed multi-physics approach, either from a theoretical or a numerical point of view. For this reason, a detailed, precise and comprehensive study of a particular MSR configuration in terms of reactor core geometry, its thermal-hydraulic characteristics and neutronics behaviour is out of the scope of the present thesis. The analysed system, even if simplified from the geometrical point of view, results complex and strongly non-linear as concerns the descriptive physical-mathematical model. In particular, as far as the molten salt thermo-hydrodynamics is concerned, Navier Stokes equations are used with the turbulence phenomena treated according to Reynolds Averaged Navier Stokes scheme, while the heat transfer is taken into account through the energy balance equations for the fuel salt and the graphite. As far as the neutronics is concerned, the two-group diffusion theory is adopted, where the group constants, computed by means of the neutron transport code SCALE 5.1, are included into the model in order to describe the neutron flux and precursor distributions, the system time constants, and the temperature feedback effects of both graphite and fuel salt. The MPM approach is applied to study the

18 xii INTRODUCTION behaviour of the system in steady-state conditions and under several transients (i.e., reactivity insertion due to control rod movements; fuel mass flow rate variations due to the changing pumping rate; presence of periodic perturbations), pointing out the main advantages offered with respect to conventional approaches generally employed in literature for the MSRs. Note to the reader The Ph.D. thesis is substantially organized into three parts. These deal with the three main contributions of the present work, detailed into three separate chapters (Chapters 2, 3 and 4), which have been written so as to be read independently of each other.

19 Chapter 1 A brief overview of the Molten Salt Reactor Technology This chapter presents a brief overview of the Molten Salt Reactor technology. After a short summary on the historical challenges, which characterized the most active development period between the late 1950s and early 1970s at Oak Ridge National Laboratories, a quick description of MSR systems is given. The attention is focused on the main MSR attractive features which lead this reactor to be reconsidered as one of six Generation IV reactor types for the future nuclear energy production. In particular, the potentialities of MSRs in terms of safety, sustainability, fuel cycle strategies and materials are discussed along with the different reactor concepts developed in past and in the last few years.

20 2 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY 1.1 History Investigation of Molten Salt Reactors (MSR) started in the late 1940s as part of the United States' program to develop a nuclear powered airplane. A liquid fuel appeared to offer several advantages, so experiments to establish the feasibility of molten salt fuels were begun in 1947 and molten fluoride salts were adopted in 1950 in the framework of the Aircraft Nuclear Propulsion (ANP) program (MacPherson, 1969) of the Oak Ridge National Laboratory (ORNL). The fluorides appeared particularly appropriate because of their high solubility for uranium, excellent chemical stability, low vapour pressure, reasonably good heat transfer properties, radiation resistance, good compatibility with air, water, and some common structural metals. A small reactor, the 2.5 MW th Aircraft Reactor Experiment (ARE), was built at ORNL to investigate the use of molten fluoride fuels for aircraft propulsion reactors and particularly to study the nuclear stability of the circulating fuel system. In 1954 the ARE operated successfully for 9 days, demonstrating high-temperature operation at 860 C and established benchmarks in performance for a circulating fluoride molten salt (NaF-ZrF 4 ) system with the uranium dissolved in the salt (Bettis et al., 1957). That MSRs might be attractive for civilian power applications was recognized from the beginning of the ANP program, and several studies were carried out considering a number of concepts over a period of several years. MacPherson and his associates concluded that graphite-moderated thermal reactors operating on a thorium fuel cycle would be the best molten-salt systems for producing economic power (Lane et al., 1958). Two types of graphite-moderated reactors were considered by MacPherson s group: i) single-fluid reactors, in which thorium and uranium are contained in the same salt; ii) and two-fluid reactors, in which a fertile salt containing thorium is kept separate from the fissile salt that contains uranium. The two-fluid reactor had the advantage that it would operate as a breeder. However, the single-fluid reactor appeared simpler and seemed to offer low power costs, even though the breeding ratio would be below 1.0 using the technology of that time. By 1960, more complete conceptual designs of MSRs had emerged. Both the two-fluid and the single-fluid concepts were studied (Alexander et al., 1961, 1965), pointing out that another reactor experiment was needed to investigate some of the technology for power reactors. The design of the Molten Salt Reactor Experiment (MSRE) was begun in 1960 (Haubenreich and Engel, 1959, 1962). The MSRE (see Fig. 1.1), a single-fluid 8-MW th reactor, demonstrated many of the features required for a power generating reactor: (1) a 7 LiF-BeF 2 salt suitable for breeding applications; (2) graphite moderator compatibility with the fluoride salt; (3) stable

21 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY 3 performance; (4) Xe/Kr removal from the fuel and trapping in the off-gas systems; and (5) use of different fuels, including 235 U, 233 U, and plutonium. Its construction began in 1962, and the reactor was first critical in 1965, whereas sustained operation at full power began in December The MSRE successfully operated for equivalent full-power hours until March of 1968, when operation stopped since the initial objectives of the project were achieved. The second phase of MSRE operation began in August 1968, when a charge of 233 U fuel was added to the same carrier salt, and on October 2 the MSRE was made critical becoming the first reactor to operate on 233 U. Fig MSRE flow diagram. The MSRE experience brought itself to several technological challenges, and basic chemistry studies of molten fluoride salts continued after the shut-down of the reactor. In particular, a significant discovery was that the lithium fluoride and beryllium fluoride in a fuel salt can be separated from rare earths by vacuum distillation at temperatures near 1000 C. As a consequence, most of the studies was focused on the two-fluid breeder reactor (Fig. 1.2) because of a good fuel utilization (e.g., higher breeding ratio). A review of the technology associated with such reactors was published in 1967 (Briggs, 1967; McCoy and Weir, 1967). In particular, new experimental information (Helm, 1967; Henson et al., 1967) pointed out that the major disadvantage of this two-fluid system was recognized as being that the graphite had to serve as a piping material in the core where it was exposed to very high

22 4 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY Fig Depiction of the two fluid molten salt breeder reactor design. neutron fluxes with detrimental consequences on dimensional stability. This made necessary to lower the core power density for the graphite, in order to have an acceptable service life, and to plan on replacement of the core at fairly frequent intervals. For this reason, study efforts were oriented also to a single-fluid breeder reactor. In particular, the advance in core design that was important in the switch to the single-fluid breeder was the recognition that a fertile "blanket" can be achieved with a salt that contains uranium as well as thorium. This together with fast chemical protactinium extraction process pursued design studies of singlefluid breeders, indicating that the fuel utilization in single-fluid, two-region molten-salt reactors can be almost as good as in two-fluid reactors, and with the present limitations on graphite life, the economics probably can be better. Consequently, in 1968 ORNL s Molten- Salt Reactor Program was directed toward the development of a single-fluid breeder reactor, introducing several new features, concerning the processing and the extraction processes with respect to the MSRE experience. The immediate effect was to bring the reactor itself to a more advanced state, and in 1971 a conceptual design of the single-fluid Molten Salt Breeder Reactor (MSBR) was proposed at ORNL (Robertson, 1971) (see Fig. 1.3). In the early 1970s, however, for reasons many would argue more political than technical, the MSBR program was terminated by the Atomic Energy Commission. Oak Ridge did continue a modest program until the early 1980s, highlighted by the designs of a Low Enriched Uranium burner (Engel et al., 1980), the "30 Year Once Through Design" and the Denatured Molten Salt Reactor (DMSR) (Engel et al., 1978), which showed that break even breeding

23 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY 5 could be accomplished using depleted 238 U and Th as the fertile constitutes. From the late 1970s until the late 1990s, little advance was made and MSRs were almost forgotten. Fig Plant layout of the single-fluid Molten Salt Breeder Reactor (MSBR). 1.2 A renaissance of interest Several voices attempted to keep the MSR concept relevant, including Charles Forsberg at ORNL and Kazuo Furukawa in Japan. In particular, the nuclear energy system named THORIMS-NES (Thorium Molten-Salt Nuclear Energy Synergetics) was proposed within the Japan activities. This concept is based on a symbiotic system coupling fission with spallation (Furukawa et al., 1990). THORIMS-NES is featured by a thorium breeding fuel cycle consisting of fuel self sustaining molten-salt reactors (fission system), and some fissile producers (accelerator) and batch chemical processing facilities (see Paragraph 1.5). Even if the concept was proposed more than 15 years ago, it has been recently reconsidered thanks to its potentialities for the realization of global-scale thorium breeding fuel cycle (Furukawa et al., 2008). Thanks to such efforts, there has been a common reconsideration of MSRs as the many advantages of the general design are recognized and the limited potential for improvement of other reactors has become evident.

24 6 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY The selection of molten salt reactors as one of the six next generation nuclear fission reactors by the Generation IV International Forum (2002, 2008) (GIF) has certainly contributed to a renaissance of interest. With changing goals for advanced reactors and new technologies, MSRs have been reconsidered under different respects. The new technologies include: Brayton power cycles (rather than steam cycles), which eliminate many of the historical challenges in building MSRs; and the conceptual development of several fast spectrum MSRs that have large negative temperature and void reactivity coefficients, a unique safety characteristic not found in solid-fuel fast reactors. Moreover, high temperature molten salts as coolants are under development for nuclear and non-nuclear applications (Forsberg et al., 2007; Williams and Clarno, 2008; Khokhlov et al., 2009), such as the employment for the intermediate heat transport loops (Ferri et al., 2008), fusion reactor technology (Moriyama et al., 1998), hydrogen production concepts (Forsberg et al., 2003), solar thermal energy (Gil et al., 2010), oil refineries and shale oil processing facilities. Since MSRs are featured by an uncommon flexibility, an incredible number of configuration have been proposed until now. Actually, the MSR can operate both with thermal or fast neutron spectrum as a breeder or converter or incinerator reactor by adopting either a critical or sub-critical (i.e., driven by an external neutron source) configuration. Moreover, different kinds of fuel can be burnt. In particular, 233 U, 235 U, transuranic elements or a mix can be used with different fuel cycle strategies. 1.3 MSR description An example of the layout of a typical graphite-moderated Molten Salt Reactor is given in Fig The molten fluoride salt (with dissolved fissile and fertile isotopes, and fission products) flows through a reactor core (moderated by graphite, if thermal spectrum reactor is under consideration) to a primary heat exchanger, where the heat is transferred to a secondary molten salt coolant. The fuel salt then flows back to the reactor core. In case of thermal spectrum reactors, the graphite to fuel ratio is adjusted to provide the optimal neutron balance 1. The heat is generated directly in the molten fuel. The liquid fuel salt typically enters the reactor vessel at 565 C and exits at 705 C, with a pressure of ~1 atmosphere (coolant boiling point: ~1400 C). The reactor and primary system are 1 Since the carrier salt itself is capable of modestly efficient moderation of neutrons, a wide variety of spectrums from soft epithermal to fairly hard can occur. Despite of this feature, graphite-moderated MSRs are indicated as thermal-spectrum reactors even if epithermal neutron spectra are actually achieved, in order to be distinguished from fast-spectrum MSRs.

25 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY 7 constructed of modified Hastelloy-N or a similar alloy for corrosion resistance to the molten salt. Volatile fission products (e.g., Kr and Xe) are continuously removed from the fuel salt. Fig Schematic representation of a graphite-moderated MSR. The secondary coolant loop transfers the heat to the power cycle or hydrogen production facility. In the former case, a Joule-Brayton cycle is common to any configuration of MSR. In particular, current proposals for MSRs use a multi-reheat helium Brayton cycle. The helium Brayton cycle has major advantages over the use of a steam Rankine cycle: simplified balance of plant with lower cost, improved efficiency, reduced potential for salt freezing in the heat exchangers, and simplified control of tritium within the reactor (trapping system is similar to that proposed for many high-temperature gas-cooled reactors). Calculations have shown that for core exit temperature of 705 C (current materials) the corresponding thermal-to-electrical efficiency is 45.5%. At temperatures approaching 1000 C, the efficiencies may exceed 60% Safety The reactor design characteristics minimize the potential for accident initiation. Unlike solidfuel reactors, MSRs operate at steady-state conditions with no change in the nuclear reactivity of the fuel as a function of time. Fuel is added as needed and, consequently, the

26 8 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY reactor has low excess nuclear reactivity. This is also a consequence of the on-line fuel feeding and reprocessing, in which the fuel is clean up from neutron poisons such as Xe. Fission products (except Xe and Kr) and nuclear materials are highly soluble in the salt and are expected to remain inside the mixture under both operating and accident conditions. The fission products that are not soluble (e.g., Xe, Kr) are continuously removed from the molten fuel salt. The chemical reaction of fluoride salts, both fuel and secondary system, with water or moisture is relatively benign. The accident source term (the quantity of radioactivity in the reactor core) is less than that in solid-fuelled reactors. The liquid heat capacity is considerably greater than other high temperature coolants, such as sodium. The primary system is low pressure with a fuel salt boiling point of ~1400 C. This eliminates a major driving force (high pressure) for transport of radionuclides in an accident from the reactor to the environment. MSRs use passive emergency core cooling systems. If the fuel salt overheats, its thermal expansion causes it to overflow by gravity and be dumped to multiple critically-safe storage tanks with passive decay heat cooling systems. As concerns temperature reactivity feedback coefficient, MSRs are characterised by a large and, above all, prompt negative reactivity coefficient of fuel salt, mainly due to the Doppler effect (Mathieu et al., 2006). On the other hand, in thermal-spectrum reactors, the total feedback coefficient could be reduced by the presence of graphite, and in some conditions can become positive with significant consequences on reactor dynamics and safety. Actually, the graphite temperature reactivity feedback coefficient can be positive according to the moderation ratio, due to the possible thermal neutron spectral shift in low energy resonance in the fission cross section of the 233 U. An exhaustive study of the impact of design parameters on constraints of MSRs can be found in (Mathieu et al., 2006), where the effect of different channel diameters (i.e., an equivalent estimation of the moderation ratio) on some significant constraints is analysed. An overview of the results from Mathieu et al. (2006) is reported in Fig. 1.5, where the main differences in terms of total feedback coefficient of reactivity, breeding ratio, neutron flux, graphite life span and fissile inventory between fast (larger channel radius) and thermal (smaller channel radius) spectrum configurations is outlined.

27 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY 9 Fig Influence of the channel radius on the total feedback coefficient of reactivity, the breeding ratio, the neutron flux, the graphite life span and the fissile inventory Fuel cycle Four major fuel-cycle options can be adopted according to different goals of reactor operation (Forsberg, 2002). The basic reactor remains unchanged except for the salt composition, salt-cleanup systems, and off-site fuel cycle operations. All 233 U fuel cycles require remote handling. The processing for all the other fuel cycles can be performed offsite with removal of the fuel salt every few years, with the exception of the breeder reactor fuel cycle. Actinide burning. This fuel cycle burns multi-recycle Pu, Am, and Cm from Light-Water Reactor (LWR) Spent Nuclear Fuel (SNF) or other sources and can produce denatured 233 U as a byproduct. The penalty for burning actinides in an epithermal neutron flux is partly offset by the greater fission neutron yield of the higher actinides. In this mode of operation, up to 10% of the electricity is produced from MSRs that burn the actinides from other reactors in the system.

28 10 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY Once-through fuel cycle. The once-through fuel cycle converts thorium to 233 U internally in the reactor and uses 20% enriched uranium as fresh fuel to the reactor. The annual fuel consumption is ~45 t/gw e, or about one-fifth that of an LWR. There would be no recovery of fissile material from the discharged salt (SNF). Denatured thorium- 233 U breeder cycle. This is a breeder reactor fuel cycle designed to maximize proliferation resistance by minimal processing of the fuel salt and by addition of 238 U to isotopically dilute fissile uranium isotopes. This lowers the breeding ratio to slightly above one and results in a very low fissile plutonium ( 239 Pu and 241 Pu) inventory of ~0.16 kg/mw e. Thorium- 233 U breeder cycle. With reference to the MSBR (2250 MW th ), the breeding ratio is ~1.06, with an equilibrium 233 U inventory of about 1500 kg. After start-up, only thorium is added as a fuel. On-line processing of the fuel salt is required for 233 Pa management and efficient removal of fission products. The choice of a fluid-fuelled reactor offer three main advantages with respect to solid-fuelled nuclear systems in the management of fuel cycle. First, isotopic blending is not necessary. Different lots of SNFs, having very different Pu, Am, and Cm isotopic, can be fed to the reactor thanks to the homogeneous liquid fuel. In contrast, in solid-fuel reactors the quantity and isotopics of the fissile materials in every location of every fuel assembly must be controlled to avoid local overpower conditions that burn out the fuel, becoming a complex and very expensive process. Secondly, no fuel fabrication is needed with advantages in terms of cost and actinide management since most of them have small critical masses and high rates of decay heat. Finally, a fuel minimal reprocessing is necessary, since only fission products are removed from the reactor during operation. This is a reverse of traditional processing, in which fissile material is extracted from SNF Materials Molten salt The idea of a liquid fuel reactor brings with itself the necessity of an adequate fuel carrier. The problem was deeply investigated during the ARE project, when fluoride molten salts were chosen, because of their high solubility for uranium, excellent chemical stability, low vapour pressure, reasonably good heat transfer properties, radiation resistance, good compatibility with air, water, and some common structural metals. For instance, the

29 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY 11 solubility of PuF 3 in Flibe (i.e., lithium and beryllium fluoride salts) ranges from 0.35% at 798 K to 1.22% at 948 K (Forsberg et al., 2004). Being the fuel carrier also employed as coolant, the heat transfer characteristics represent a significant parameter to be investigated. In Table 1.1 some properties of molten salts are compared with those of more usual coolants. Table Physical properties of molten salts and of other coolants (Forsberg et al., 2007). Coolant T melt [ C] T boil [ C] ρ [kg/m 3 ] C P [kj/kg K] K [W/mK] υ 10 6 [m 2 /s] Li 2 BeF 4 (Flibe) NaF-40.5ZrF LiF-11.5NaF-42KF LiF-31NaF-38BeF LiF-29ThF NaF-92NaBF NaNO KNO Sodium Lead Helium (7.5 MPa) Water (7.5 MPa) Salt compositions are shown in mole percent, and their properties at 700ºC and 1 atm. Nitrate properties at 500 C. Sodium properties at 550ºC. Pressurized water data are shown at 290 C for comparison. It can be seen that the high melting temperature creates additional operational constraints on reactor temperature (operation at high temperature) to avoid freezing during the normal operating conditions or during maintenance operations. On the other hand, a high melting temperature makes possible quantities of molten salt escaping from the reactor vessel to immediately freeze. Boiling points are high enough to enable a reactor to operate at atmospherical pressure. The specific weights are between sodium and lead. Conductivities are much lower than that of liquid metals and this enhances the thermal resistances in the fuel, even if there is no need to transfer heat from the fuel to the coolant, as in solid-fuel reactors. Actually, the high viscosity that limits the achievable power density for a given pumping power constraint, induces, together with the low thermal conductivity, a large film temperature drop that is enhanced by the low Reynolds number (Hejzlar et al., 2009). Specific heats are generally greater than that of sodium or lead. Another significant advantage is represented by the required pumping power, which results lower than that

30 12 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY needed for sodium. Actually, in order to transport 1000 MW th, a molten salt would require one pipe with a diameter of 1 meter, compared to the four identical pipes necessary for sodium (Forsberg, 2007). It can be concluded that molten salts represent a good compromise between other typical reactor coolants, and currently, research on molten salts is connected to many other technical challenges, such as heat transport for hydrogen production systems and fusion reactors. Primary loop structural components The absence of solid fuel elements in MSRs eliminates by design the associated material problems and constraints on fuel burn-up. Anyway, during normal operations the molten salt contains several chemical species, leading to possible compatibility problems with the structural materials, which are subjected to a significant irradiation field, being in direct contact with the fuel. The problems related to the choice of a suitable material for the primary loop of a MSR were first faced during the years preceding the ARE construction. It was found that Fe-alloy showed excessive corrosion problems. Actually, austenitic steels showed problems of mass transfer corrosion connected to the temperature-varying solubility of chromium in molten salts (Lane et al., 1958). As a consequence, nickel-based alloy were chosen for both ARE (Inconel alloy) and MSRE (Hastelloy-N 2 ) projects (Forsberg et al., 2003). Hastelloy-N showed good corrosion and temperature resistance. Two problems were encountered during the experiment: the helium hardening of the alloy and the stress corrosion cracking due to tellurium (MacPherson, 1985). In subsequent years, these problems were solved by ORNL with a modified version of Hastelloy-N able to reduce helium hardening, whereas it was found that tellurium attack could be controlled by keeping the fuel on the reducing side (MacPherson, 1985). For these reasons, if a MSR will be built in the next years, Hastelloy-N or its modified version will be certainly considered as primary loop material. In addition, carbon-carbon (C-C) materials (Peterson et al., 2003) and advanced nickel-based alloys are currently under investigation in order to support the high temperature power generation for hydrogen production. In particular, C-C materials seem to be promising for the low cost, high temperature resistance, good mechanical properties and compatibility with molten salts. Moreover, they seem to be immune to fission product noble metal deposition, problem encountered with Hastelloy-N on heat exchangers resulting in high thermal loads (due to decay heat) and limited equipment lifetimes (Forsberg et al., 2004). 2 At 973 K the main thermo-physical properties are: ρ = 8890 kg/m 3, C P = 0.43 kj/kgk, K = 9.8 W/mK, T melt = 1623 K.

31 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY 13 Inside the reactor core, the main structural component is represented by graphite that serves as moderator in thermal-spectrum MSRs and as reflector in fast-spectrum MSRs. Hence, a precise characterization of this material under thermal and radiation fields represents a good deal of the feasibility of MSRs. Graphite has a quite good moderating efficiency, good thermal and irradiation stability, adequate mechanical properties, low cost, and an excellent compatibility with molten salts (extensively tested in MSRE). An overview of graphite behaviour in reactor environment is reported in (Kasten, 1969). The main issues of this component are typical of thermal-spectrum reactor, since graphite, representing the major part of the core, is subjected to higher temperatures and neutron fluencies with respect to fast reactors. Here below the main features of graphite performance in reactor are given: 1. Dimensional changes due to temperature and irradiation field. 2. Penetration of the salts into the graphite porosities or cracks, with consequent hot spots and possible local boiling. 3. Penetration of the fission gases (like Xe) into the graphite, with significant consequences on neutron balance. 4. Deposition of noble fission products. Some chemically instable fission product (present in metallic form) tend to plate out on the graphite surface, with consequent neutron poisoning problems. 5. Thermal and mechanical properties can significantly change under irradiation. Graphite can be subjected to significant dimensional changes due to the combined action of temperature and irradiation fields. This material is characterized by an initial shrinkage as a consequence of closure of voids present after the fabrication process, followed by a rapid irradiation-induced swelling. This effect must be carefully take into account to avoid several problems such as buckling, excessive mechanical loadings, a departure from the optimum salt-to-graphite ratio, and variations of the spatial distribution of the neutron flux. Last but not least, an excessive graphite swelling can lead to a significant increase in porosity and a generation of cracks, leading to an enhancement of penetration of fuel salt and fission gasses in the material. For the previous reasons, the constraint on the graphite lifetime is generally fixed at that time required by the material to swell back to its as-fabricated volume after the initial shrinkage. According to the core power density and moderation ratio, the lifetime of graphite can significantly vary as shown in Fig. 1.5, so that the necessity of a periodical substitution of the reactor core cannot be excluded, especially in thermal-spectrum reactors.

32 14 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY 1.4 Advantages and drawbacks of MSR technology In this paragraph, a brief summary of the main advantages and drawbacks of the MSR technology is reported (see Table 1.2). A quite comprehensive treatment can be found in (Furukawa et al., 2008; Hejzlar et al., 2009; LeBlanc, 2009). Table Main advantages and drawbacks of the MSR technology. Advantages There is no scenario called "fuel melt down" (i) Formation of stable fluoride fission products (ii) High availability (iii) No excess of reactivity (iv) Strong negative temperature and void coeff. (v) Low-pressure primary cooling loop (vi) (Par ) No isotopic blending and fuel fabrication (Par ) Chemical stability and good compatibility Improved sustainability with 233 U cycle (vii) Flexibility in the fuel cycle strategy (viii) Low fissile inventory (viii) No radiation damage constraint on attainable fuel burn-up (Par ) High conversion efficiency (ix) Possibility of hydrogen production (ix) (Par ) Reactors have full passive safety Several non-proliferation advantages (x) Drawbacks (Par ) High melting temperature of molten salts (Par ) High viscosity and low thermal conductivity Graphite dimensional changes (Par ) Penetration of salt and fission products in graphite and deposition of noble fission products on graphite surface (Par ) Structural components stability under irradiation (Par ) Possible positive temperature feedback coefficient (Par ) of graphite in thermal-spectrum MSRs Long term corrosion of fuel salt on structural (Par ) materials Fast reprocessing time in thermal-breeder MSRs (xi) (i) Fluid nature of the fuel means meltdown is an irrelevant term and allows the fuel salt to be automatically drained to passively cooled, critically-safe dump tanks. (ii) Most fission products quickly form stable fluorides that would stay within the salt during any leak or accident. Volatile fission products, such as the noble gases and noble metals, come out of the salt as produced. Noble gases bubble out and are stored outside the reactor loop. Noble and semi-noble metals will plate out on metal surfaces and/or can be collected by replaceable high surface area metal sponges within the loop. So there is no danger of release of volatile radioactive FPs, even under accident conditions. (iii) High availability is mainly due to the simultaneous refuelling and reprocessing online. Moreover, the continuous removal of the noble gas Xenon means that there is no "dead time" of the reactor after shutdown or a power decrease that most solid fuelled reactors must deal with, due to the production of 135 Xe from the decay of 135 I.

33 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY 15 (iv) Fissile concentrations are easily adjusted on a continuous basis meaning no excess reactivity and no need for control rods or burnable poisons. Control rods for shutdown and start-up may be included, but are not necessary, given the ability to drain fuel out of the core into critically-safe storage tanks. (v) Most MSR designs have very strong negative temperature and void coefficients which act instantly, aiding safety and allowing automatic load following operation. (vi) No pressure vessel is needed as the salts run near atmospheric pressure. No water or sodium means no possible steam explosion or hydrogen production within the containment. In designs without graphite moderator, there is not even combustible material present. (vii) Utilization of the thorium to 233 U cycle produces several orders of magnitude less transuranic wastes than a conventional once-through cycle and significantly less than even a U-Pu fast breeder (based on 0.1% losses during fuel processing). This leads to radiotoxicity of waste being less than equivalent uranium ore levels within a few hundred years (see Fig. 1.6). Thorium is 3 times as abundant as uranium with large proven reserves even with the small current usage. As example, a single new deposit in Lemhi Pass Idaho has recently added tonnes to the world s proven reserves of 1.2 million tonnes. (viii) Without any fuel processing, MSRs can run as simple converters with excellent uranium utilization. MSRs offer large advantages for the destruction of transuranic wastes from traditional once-through reactors. TRUs may also be used as start-up fissile inventory in most designs. Fuel processing and utilization of thorium permits break even breeding with ease and ability to reach a breeding ratio of 1.06 or even up to Adding 238 U to denature the uranium content and still break even is also possible. Break even operation only requires approximately 800 kg of thorium per GW e -year added simply as ThF 4. Start-up fissile requirements can be as low as 200kg/GW e or upwards of 5.5 tonnes in harder spectrum designs, with 700 to 1500 kg more common. Thorium start-up inventory varies from 50 to 200 tonnes. Fig Radiotoxicity for various cycles. (ix) High temperature of fuel salt permits higher conversion efficiency and even holds promise for other heat based applications, such as hydrogen production. (x) The adoption of 233 U leads to several advantages in terms of proliferation resistance when compared to existing NPPs, because the fuel is always contaminated with inseparable 232 U. The radioactivity due to 232 U daughters (strong gamma rays of 2.6 MeV are emitted by 208 Tl ) makes the diversion of 233 U difficult and safeguards easy. (xi) Fast reprocessing time is required in thermal-breeder MSRs. In particular, a fast fuel cleaning from 233 Pa is necessary mainly for two reasons: 1) 233 Pa is a neutron poison having a high absorption cross section; 2) 233 Pa half life is 27 min so it must be extracted quickly from the fuel in order to have a high breeding ratio before its conversion by neutron absorption in reactor.

34 16 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY 1.5 The different concepts of MSRs This paragraph is intended to give a brief summary of the several MSR concepts proposed in the past and currently under development, in order to provide a quick idea of the wide flexibility offered by this kind of nuclear reactors, and of the large number of studied configurations aimed toward different fuel cycle strategies. Hereinafter, for the sake of simplicity, the different concepts are divided in thermal-spectrum breeders, fast-spectrumbreeders, fast-spectrum incinerators, thermal-spectrum incinerators, sub-critical incinerators and chloride-cooled reactors Thermal-spectrum breeders This has been the most studied configuration in the past by ORNL and recently in the framework of the MOST project (Renault et al., 2005). The thermal-spectrum MSR core consists of an unclad graphite elements, through which a Flibe salt (in which uranium and thorium are dissolved) flows in smooth circular channels. A typical core of a thermal breeder MSR is shown in Fig The concept was developed during the 1960s by ORNL, which proposed the single-fluid and two-fluid molten salt breeder reactors. In the single-fluid project (Robertson et al, 1971), the adopted molten salt is 72 mol% LiF, 16 mol% BeF 2, 12 mol% ThF 4, 0.3 mol% UF 4. Inlet and outlet core temperatures are 839 K and 977 K respectively, the thermal power is 2250 MW and the core a nearly right cylinder of 3.96 m in height and 5.54 m in diameter. Fig Schematic representation of vertical and horizontal sections of a thermal-spectrum MSR core.

35 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY 17 A good deal of the feasibility of a thermal-spectrum breeder reactor relies on the feasibility of fast and efficient on-line reprocessing system, in order to have a rapid extraction of 233 Pa from the fuel, which is characterised by a quite large absorption cross section and a half life of 27 days. Without 233 Pa extraction, a breeder reactor with characteristics similar to those of the original MSBR is possible only doubling the core dimension and thus reducing the neutrons leakage with a lower impact of 233 Pa absorptions (Mathieu et al., 2006). In the case of a reliable and efficient reprocessing system, the original MSBR configuration leads to a breeding ratio of 1.06 (Robertson et al, 1971). As concerns the sustainability of thermalbreeder reactors, 100 kg of natural thorium could be sufficient to produce 1 TWh of electrical energy (GIF, 2002). In addition, thanks to the 233 Pa extraction, a doubling time of 20 years is possible (compared to the 100 year doubling time of a gas cooled reactor fuelled with Th-U) (Renault et al., 2005). One of the main problems connected to a thermalspectrum MSR is the uncertainty on temperature coefficients of reactivity. In particular, the coefficient related to the variation of graphite temperature is expected to be positive (Mathieu et al., 2006), leading to possible unstable situations in accidental conditions. The addition of erbium in the graphite could overcome this problem, but this choice is characterised by a worse neutron economy (Křepel et al., 2008). The main safety aspects of thermal-spectrum MSRs are featured by the small inventory of fissile material and the high sub-criticality of the salt out of the core. Besides the well known MSBR, other concepts of thermal-spectrum MSRs have been proposed. It must be mentioned the early proposal made by Engel et al. (1980): the so-called Denatured Molten Salt Reactor (DMSR). It is a MSBR-like reactor employed as a simple once-through converter using 20 % enriched uranium as fresh fuel. In this case, the annual fuel consumption would be of 45 t/gw e, about one fifth that of an LWR (GIF, 2002). This kind of reactor can be operated without reprocessing for its 30-years graphite lifetime. Another design by Engel et al. (1978) also using 20% enriched fuel, but with processing, needed no refueling for its 30-year life. Advances in fuel cycle strategy have been recently proposed within the Japanese activities of the Thorium Molten-Salt Forum (TMSF), born with the aim of improving the concept developed at ORNL. The members of TMSF have proposed the Thorium Molten-Salt Nuclear Energy Synergetic System (THORIMS-NES), which is a new concept aiming the effective urgent usage in the global scale with a long-term perspective, and it has been designed so as to bring a new huge size nuclear industry without any big investment (Furukawa et al., 2008). This idea brings itself several advantages in terms of proliferation resistance, safety, fuel cycle, radioactive waste management, economics and natural resource

36 18 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY exploitation. A sketch illustrating the global energy environment problems and achievable solutions by THORIMS-NES is shown in Fig. 1.8, whereas a comprehensive description can be found in (Furukawa et al., 2008). The concept is based on a symbiotic system coupling fission with spallation, aiming at obtaining less than 10-years doubling time for the fission energy production growth, which cannot be achieved by ordinary fission breeding power station concept. To this purpose, the separation of fissile-producing breeders process plants and power generating fission-reactors are required. This innovative concept consists of simple power stations MSR named FUJI-series (see Fig. 1.9), fissile-producers AMSB (Accelerator Molten Salt Breeder), and batch-type process plants establishing a symbiotic Th breeding fuel-cycle system (THORIMS-NES). In particular, Small Molten Salt Reactors (SMSR) of MW e (FUJI-II, FUJI-12 and FUJI-U3) have been proposed (Mitachi et al., 2005). FUJI-12 is a simple 1-region core, and it attained a high conversion ratio of 0.92 by the batch chemical processing every 7.5 years. However, FUJI-12 has to replace the graphite moderator every 15 years, and this may cause additional maintenance work and cost. Then, an improved SMSR (FUJI-U3) has been proposed, in which a 3-region core design concept is introduced in order to eliminate graphite replacement by reducing the maximum neutron flux (no need of replacement for 30 years operation). Fig Global energy/environment problems and achievable solutions by THORIMS-NES.

37 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY 19 Fig General view of FUJI molten-salt power station Fast-spectrum breeders These reactors are based on Th-U fuel cycle, but a fast neutron spectrum is chosen. This feature allows better breeding characteristics and, consequently, a slower reprocessing time is required for 233 Pa extraction (Forsberg, 2007). Moreover, the fast spectrum determines higher negative reactivity coefficient of fuel with no problems related to graphite temperature feedback (Forsberg, 2007). This kind of reactor can be started with LWR spent nuclear fuel, beginning its life as an actinide burner, but it is expected to evolve into a Th-U cycle. An example of fast-spectrum breeder reactor is the so called Thorium Molten Salt Reactor (TMSR), which is the result of studies conducted during the MOST project. TMSR uses only LiF as fuel carrier, avoiding the highly toxic and expensive BeF 2, which avoids the production of 6 Li, thus reducing the presence of tritium. The content of heavy nuclides is about 22 mol% that determines a melting temperature of 838 K, representing the eutectic point of the mixture. Fuel salt enters the core at 903 K, and its temperature can rise until 1000 K, with a thermal efficiency of the power plant of about 40%. Radial reflectors consists of graphite channels containing 72 mol% LiF and 28 mol% ThF 4. A biannual extraction of the fertile blanket salt is needed. Gaseous fission products are extracted within 30 s, whereas a complete reprocessing of the fuel salt is required every 6 months. Breeding ratio is expected to be more than 1.1. For further information about TMSR, see (Mathieu et al., 2006; Forsberg et al., 2007).

38 20 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY Fast-spectrum incinerators The main appeal of MSRs is represented by the possibility to burn actinide without fuel element fabrication, so that problems of manipulation and problems connected to the different composition of spent nuclear fuels are avoided. The adopted molten salt will require high solubility for actinides, and must be composed by high atomic mass elements, so to harden the neutron spectrum. For this reason, sodium fluorides are usually chosen (Becker et al., 2008). Being non-breeder reactors, the fast-spectrum actinide burner reactors do not require a rapid reprocessing of the fuel, so that the development of ad hoc reprocessing systems is not strictly required. By the way, 233 U can be produced inserting thorium into a fertile blanket or directly into the fuel. An example of this kind of reactor is the russian MOlten Salt Actinide Recycler and Transmuter (MOSART). MOSART is a 2400 MW th actinide burner (Ignatiev et al., 2003). The composition of the carrier salt is the following: 15LiF, 58NaF, 37BeF 2. It is featured by high solubility for actinides and lanthanides (up to 2 mol%) and a low melting temperature (752 K). The actinide inventory will be between 7320 kg and 9346 kg, corresponding to mol%. The effective residence time of fuel in the reactor is intended to be of 300 full-power days (Forsberg et al., 2007). A schematic representation of the MOSART core is depicted in Fig Core power density is 43 kw/l and temperature will rise from 873 K to 998 K. No internals will be present in the reactor vessel, except for graphite reflectors. MOSART has shown very promising characteristics in terms of safety (Schikorr, 2005), especially for the strong negative temperature feedback coefficient, estimated between 4.1 pcm/k and 6.6 pcm/k. Once-through cycle MSR actinide burners with fast-spectrum have also been proposed by Bowman (1998) in order to burn LWR SNFs. In short, this reduces costs and proliferation problems and still allows a high transmutation efficiency of 99.8% (Forsberg et al., 2007). Finally, the SPent Hot fuel Incinerator by Neutron flux (SPHINX) concept must be mentioned for completeness. It has been under development in the Czech Republic as an actinide burner in resonance neutron spectrum and a radionuclide transmuter in a wellthermalized spectrum. Within this program, several experimental activities are being carried out for the validation of computer codes and verification of design inputs for designing of a demonstration unit of the MSR-type (Hron and Mikisek, 2008).

39 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY 21 Fig Schematic representation of the MOSART core Thermal-spectrum incinerators Even if smaller fissile inventory is encountered in thermal spectrum reactors in comparison with fast reactors, a large amount of minor actinides such as 244 Cm and 252 Cf is expected to be present. It has been calculated that several hundreds of kg of 244 Cm can be produced in a 1 GW e reactor (Renault et al., 2005). This kind of reactors requires a quite fast reprocessing (mainly to extract 233 Pa), and thorium support in order to have a good incineration efficiency. The presence of a fertile material reduces the amount of actinide burnt for unit energy produced, but allows the use of an abundant resource like thorium to produce energy. Moreover, thermal spectrum reduces problems related to structural material irradiation and safety. The following two concepts must be mentioned in this subparagraph: TIER and AMSTER projects. The former is a thermal-spectrum reactor fuelled with only transuranic elements, and is considered to be a non-viable concept (Renault et al., 2005). On the other hand, great efforts is being spent into the AMSTER incinerator by the French EDF.

40 22 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY Sub-critical incinerators In this reactor concept a continuous reprocessing and an intense neutron flux are employed in order to obtain extremely high transmutation efficiencies. The intense neutron flux is obtained exploiting spallation reactions of 1 GeV protons on a lead target. Each proton is expected to generate 20 to 50 neutrons. The target is surrounded by a sub-critical (multiplicator factor equal to ) blanket, which amplifies the neutron flux generated into the target (Woosley and Rydin, 1997). The blanket is made of a flowing molten salt solution of actinide and fission products and can contain graphite blocks or not. This configuration can generate neutron fluxes on the order of 100 to 1000 times as intense as fluxes in traditional LWRs, but its effectiveness is reduced due to the time spent by the fuel outside the multiplying region (Woosley and Rydin, 1997). The waste stream does not contain materials that can be attractive for the construction of nuclear weapons and the inventory of actinides is very small compared to that of LWRs or LMFBRs (Woosley and Rydin, 1997). It is important to note that the more complex dynamics of a liquid fuel reactor forces the designers to use a configuration with lower reactivity with respect to that of solid fuel sub-critical reactors. One of the main problems connected to the discussed concept relies on the accelerator system. Indeed, the availability of reliable, powerful and flexible accelerator is quite a controversial point, already encountered for accelerator driven systems cooled by lead-bismuth. Studies are carried out at Los Alamos National Laboratory, on the Accelerator Transmutation of Waste (ATW) reactor (Woosley and Rydin, 1997). A similar concept is presently being studied in Japan and the project is referred to as Molten Salt Accelerator Driven System (MS ADS) (Ishimoto et al., 2002) Chloride-salt MSRs An explanation of the potentialities of chloride-salt MSRs can be found in (Forsberg et al., 2007). Here, only a very quick summary is reported. The use of chloride salts instead of fluorides has several advantages. In particular, their higher atomic number will result in a hardening of neutron spectrum, which can lead to a better breeding ratio and slower reprocessing time in fast configurations. However, there are major drawbacks, such as: higher fissile inventories, higher melting points, uncertainty on materials to be used for primary systems. Moreover, natural chlorine contains 75.4% of 135 Cl, which, absorbing neutrons, becomes the long lived radioactive isotope 136 Cl. An example of chloride MSR is the French REBUS, which uses a plutonium fuel cycle with 45 mol% of uranium and transuranic elements and 55 mol% of NaCl.

41 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY Current activities within GIF The present situation of MSRs is well assessed in the 2008 annual report of the Generation IV International Forum (2008). As concerns the status of cooperation, the decision for setting up a provisional System Steering Committee (SSC) for the MSR was taken by the GIF Policy Group in May 2004, in order to enhance and harmonize international collaboration. The participating members are Euratom, France and the United States. Other countries have been represented systematically (the Russian Federation) or occasionally (Japan) as observers in the meetings of the provisional SSC. The major contribution of Euratom to MSR R&D within GIF has been the ALISIA (Assessment of LIquid Salts for Innovative Applications) project, which was part of its 6 th Framework Programme. In particular, salt systems have been critically reviewed and reference compositions have been proposed or confirmed (Fig. 1.11). After the ALISIA project, the renewal and diversification of interests in molten salts have led the MSR provisional SSC to shift the R&D orientations and objectives initially promoted in the original Generation IV Roadmap issued in 2002, and a new MSR proposal has been submitted to the 7 th Euratom Framework Programme. At the end of 2008, the following significant progress has been achieved: - Development of Molten Salt Fast-neutron Reactor (MSFR) 3 pre-conceptual designs (France). - Completion of salt selection for different applications and identification of the needs for complementary data (Euratom 7 th Framework Programme). - Significant improvement of fuel salt clean-up scheme (France). - Identification of candidate materials (Ni-W-Cr alloys) with very high corrosion resistance at temperatures above 750 C (Ignatiev et al., 2008a). - Demonstration of the Advanced High-Temperature Reactor (AHTR) 4 performance and safety (United States). - Criticality tests for the assessment of AHTR fuel and core behaviour (United States, Czech Republic). - Better understanding of the transmutation capabilities, dynamics and safety-related parameters, for fertile and fertile-free fuel concepts (Ignatiev et al., 2008b). 3 The MSFR is a long-term alternative to solid-fuelled fast neutron reactors offering very negative feedback coefficients and simplified fuel cycle. Its potential has been assessed, but specific technological challenges must be addressed and the safety approach has to be established. 4 The AHTR uses liquid salts as a coolant and the same graphite core structures with coated fuel particles as gascooled reactors like the Very High Temperature Reactor (VHTR).

42 24 A BRIEF OVERVIEW OF THE MOLTEN SALT REACTOR TECHNOLOGY The main step for the future development of a MSR can be summarized as: scoping and screening phase, which should end in 2011; the viability and performance phases, and respectively, should come. The main milestones for the demonstration phase have also been discussed in (GIF, 2008), envisioning a MSR prototype after Fig Main salts assessed in ALISIA project.

43 Chapter 2 A generalized approach to heat transfer in MSRs The study of molten salt heat transfer characteristics is a key issue in the current development of the MSRs. When employed as fuel (i.e., representing a fluid with internal heat generation), the heat transfer depends on the strength of the source whose influence on the heat exchange process is significant enough to demand consideration. At present, few studies have been performed on the subject from either an experimental or a numerical point of view. In order to provide a deeper insight into this field and to establish a useful validation framework for the assessment of computational fluid-dynamics (CFD) analyses of liquids with internal heat generation, a generalized analytic approach for both fluid velocity and temperature fields in a circular pipe surrounded by a solid region has been developed. The chapter is substantially organized into three parts. I) The general and unified solution of the heat transfer equation is applied to the turbulent Graetz problem with boundary conditions of the third kind and arbitrary heat source distribution, incorporating recent formulations for turbulent flow and convection. Fluids are flowing through smooth and straight circular tubes within which the flow is hydrodynamically developed but thermally developing (conditions of interest for MSR core channels). II) An assessment of the developed approach for a large variety of fluids is presented. Results are shown to be in a good agreement with experimental data concerning heat transfer evaluations for both fully developed and thermally developing flow conditions, over a large range of Prandtl numbers (10-2 < Pr < 10 4 ). In addition, a preliminary correlation, which includes the Prandtl number range of interest for molten salts, is proposed for the Nusselt number predictions in the case of simultaneous uniform wall heat flux and internal heat generation. III) An extension of the approach elaborated for pipe flow is proposed, in order to take into account the heat conduction in a solid domain surrounding the tube (represented by the graphite matrix in the specific case of interest). The overall solution results applicable in a more general context and provides an insight into the heat transfer characteristics of the graphite-moderated MSR core channel. Main results are published in (Di Marcello et al., 2010).

44 26 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs 2.1 Heat transfer in pipe flow The problem of heat transfer in pipe flow has been extensively investigated in the past. Many different models have been proposed and utilized to predict the velocity profile, the eddy diffusivity, the temperature distributions, the friction factor and the heat transfer coefficient (Bhatti and Shah, 1987; Yakhot et al., 1987; Schlichting and Gersten, 2000; Kays et al., 2004). However, the majority of such studies give a description of the problem for noninternally heated fluids. According to the author s knowledge, models regarding fluids with internal heat generation have been performed more than 30 years ago (Poppendiek, 1954; Sparrow and Siegel, 1958; Siegel and Sparrow, 1959; Kinney and Sparrow, 1966), giving in most cases a partial treatment of the problem in terms of boundary conditions and heat source distribution, and relying on a turbulent flow treatment that does not seem fully satisfactory in the light of recent investigations (Kays, 1994; Zagarola and Smits, 1997; Churchill, 1997, 2002). Fluids exploiting such a special characteristic are of great interest for current nuclear technology. The Molten Salt Reactor (MSR) employs indeed a non-classical fuel type constituted by a circulating molten fluoride (or chloride) salt mixture, within which the fission material is dissolved. By adopting classical correlations for the Nusselt number (e.g., Dittus-Boelter), the heat transfer coefficient of the MSR fuel can be overestimated by a nonnegligible amount (Di Marcello et al., 2008a). In the case of thermal spectrum MSRs, this has significant consequences on the graphite temperature predictions and on the reactor dynamic behaviour (Di Marcello et al., 2008b). Therefore the influence of the heat source within the fluid is important, and requires further investigation. This paragraph aims at a preliminary assessment of the heat transfer of fluids by means of a unified and general treatment, which also considers the heat generation in smooth, straight circular tubes when the flow is hydrodynamically developed, but thermally developing. This situation is consistent with the flow characteristics encountered in the MSR core channels, both in steady-state and transient operation (Mandin et al., 2005; Di Marcello et al., 2008b). A generalized analytic approach is undertaken for this purpose, consisting of boundary conditions of the third kind with arbitrary axial distribution, arbitrary inlet temperature radial distribution and arbitrary variations of internal heat source in both the radial and axial directions. Such an analytic model offers a useful validation framework for testing commercial codes and is also an excellent building-block case for testing turbulence models in view of their application to analysis of more complex and design-oriented geometries.

45 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs 27 In principle, the adopted model is applicable and valid for annular tubes and parallel plate channels but, in the interest of simplicity and practicality, the results herein are limited to circular pipes. Since the development of the model and the derivation of the solution are described in detail in Mikhailov and Őzişik (1984), only those expressions essential to understanding are reproduced herein. In order to obtain the solution of the turbulent Graetz problem (Graetz, 1883, 1885), the reinterpretation of turbulent flow and convection of Churchill (2002) is considered, so that the eddy diffusivity and thus the velocity profile is expressed in terms of the local turbulent shear stress. On the other hand, the turbulent Prandtl number (Pr T ) remains an essential (and critical) piece of information, and the effect of choosing different Pr T correlations is also investigated General solution of the heat transfer problem The problems of heat transfer by forced convection of fluids inside conduits are generally referred to as the Graetz problem, since Graetz (1883, 1885) appears to have been the first investigator who was involved in this subject. Numerous extensions of the original problem have been reported in literature. In this subparagraph, a general mathematical formulation is described (Mikhailov and Őzişik, 1984), since it proves useful in appropriately treating the heat transfer characteristics of systems containing flowing fluids with internal heat generation. The considered analytic model applies to smooth, straight and round pipes, with uniform cross-section, within which the fluid flow is hydrodynamically developed, but thermally developing, as depicted in Fig. 2.1, under the following assumptions: 1. axial-symmetric conditions are taken into account; 2. steady-state exists; 3. the fluid is incompressible with no phase change, and constant physical properties (consequently, the velocity problem is uncoupled from the temperature problem); 4. the hydrodynamic pattern is established; 5. natural convection effects are negligible, so that, because of assumption 4, there are no components of the time-averaged velocity normal to the pipe axis; 6. the average temperature at any radius does not vary with time or angular position; 7. axial conduction of heat is negligible. The last assumption has been shown by Weigand et al. (2001) to introduce a negligible error for Pe > 20 and can be considered acceptable within the scope of the present work. Nevertheless, it must be pointed out that the Peclet number can be smaller than 20 when liquid

46 28 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs Fig Geometrical configuration and coordinate system. metals with low Reynolds number are used. Furthermore, axial heat conduction effects can be significant for a short finite length of heated section also for Peclet number larger than 100 (Weigand et al., 2001). Under the previous hypotheses, the energy equation can be written as follows: T u z ( r, z) 1 υ T( r,z) Q( r,z) = r r r + ε Pr H r + ρc P (2.1) where u and ε H depend only on the radial coordinate. The boundary conditions for Eq. (2.1) at r = 0 (at the pipe centreline) must be of the second kind (see Eq. (2.2a)) because of assumption 1, while at r = r 0 (at the pipe wall) they can be taken as any combination of the boundary conditions of the first, second, and third kind, as expressed by Eqs. (2.2b), (2.2c) and (2.2d), respectively: T r T ( r, z) r= 0 = 0 ( r, z) T ( z) 0 = W, T K r ( r, z) r= r0 = j W ( z), T( r, z) 0 K h W T r ( r, z) r= r0 = T E ( z) (2.2a) (2.2b,c,d) Finally, the boundary condition at the pipe entrance (z = 0) is given by Eq. (2.3): T ( r,0) T ( r) = (2.3) IN In order to get the solution of the Eq. (2.1) with the boundary conditions (2.2) and (2.3), it is often convenient to express it in the dimensionless form by introducing the dimensionless variables, defined by the set of Eqs. (2.4):

47 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs 29 R = r r 0, (2.4a); 2z Re Pr Z =, (2.4b); θ ( R,Z) TW ( ) ( z) T0 TE θ Z =, (2.4d); ( ) ( z) T0 θ Z =, (2.4e); θ ( Z) W ( Z) T ( z) jw r =, (2.4g); K T 0 J W g υ Pr + ε υ Pr E r 0 h Wr0 Bi K H ( R) =, (2.4j); S( R, Z) T IN =, (2.4h); ( R) ( z) Q r, r0 =, (2.4k) K T 2 T = ( r,z) T = u avg T T IN ( r) 0 T T u f =, (2.4i);, (2.4c); 0, (2.4f); Substitution of the foregoing expressions into Eq. (2.1) yields the dimensionless form of the heat transfer equation represented by Eq. (2.5), while the boundary conditions are given by Eqs. (2.6): ( R, Z) ( R, Z) θ 1 θ f ( R) = R g( R) + S Z R R R ( R, Z) θ R α BC θ R= 0 ( 1, Z) = 0 β BC ( R,0) = θ ( R) IN ( R, Z) θ R R= 1 = φ ( Z) ( R, Z) in 0 < R < 1 (2.5) (2.6a) (2.6b) θ (2.6c) The generalized boundary condition expressed by Eq. (2.6b) can yield any of the boundary conditions of first, second, or third kind if the coefficients α BC, β BC, and the function φ(z) are chosen as: First kind: α BC = 1, β BC = 0, φ(z) = θ W (Z) (2.7a) Second kind: α BC = 0, β BC = 1, φ(z) = J W (Z) (2.7b) Third kind: α BC = 1, β BC = 1/Bi, φ(z) = θ E (Z) (2.7c) The solution of the boundary value problem defined by Eqs. (2.5) and (2.6) is obtained by means of the so-called "splitting-up procedure", for which details are given in Mikhailov and Őzişik (1984). Such a procedure applies if the non-homogeneous term φ(z) and the term P(R,Z) = R S(R,Z) can be expressed in terms of q-order polynomials of the axial coordinate Z, as shown in Eqs.

48 30 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs (2.8) and (2.9), and consists in splitting-up the solution of the original problem into two parts as given by Eq. (2.10): q ( ) = j φ Z φ Z (2.8) j= 0 ( R, Z) = P ( R) q j j P Z (2.9) θ j= 0 q j 2 j µ Z ( R, Z) = θ ( R) Z + C e ψ ( R) j= 0 j i= 1 i i (2.10) i θ j 1 α [ ] ( R) = φ + β P ( R) ( j + 1) R f ( R) θ ( R) + α BC R 0 1 j BC 0 1 R g 1 R g ( R ) ( R) j+ 1 [ P ( R ) ( j + 1) R f ( R ) θ ( R )] j+ 1 [ Pj( R ) ( j + 1) R f ( R ) θ j+ 1( R )] R 0 1 BC 0 R 0 j j dr dr dr dr dr (2.11) Here, j = q, q 1, q 2,,1,0 and θ q+1 (R) = 0. ψ i (R) and µ i are the eigenfunctions and the eigenvalues, respectively, of the well-known Sturm-Liouville problem represented by the differential equation (2.12) with its boundary conditions (2.13): d dr R g ( R) ( R) dψ dr dψ i = 0 dr R= 0 ( R) 2 [ µ R f ( R) ] ψ ( R) 0 i + = i i α ( 1) β g( 1) dψi dr ( R) BC ψi BC = R = 1 0 (2.12) (2.13a,b) Imposing the boundary condition (2.6c) to Eq. (2.10) and exploiting the orthogonality property of the eigenfunctions, the coefficients C i can be evaluated as follows: 1 R IN 0 i = 1 [ θ ( R) θ ( R) ] f ( R) ψ ( R) 0 R f 0 2 ( R) ψ ( R) i dr i dr C (2.14) A particular case of the examined problem occurs when all the boundary conditions are of the second kind (i.e., when α BC = 0 and β BC = 1). In this case, the first eigenvalue of the associated Sturm-Liouville problem is zero (µ 0 = 0) and the corresponding eigenfunction is

49 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs 31 constant (ψ 0 = const.). It can be demonstrated (Mikhailov and Őzişik, 1984) that this result implies the addition of a term in Eq. (2.10), which represents an average temperature over the region, according to Eqs. (2.15) and (2.16): θ θ q 2 * j * µ Z ( R, Z) = θ ( Z) + θ ( R) Z + C e ψ ( R) AV AV j= 0 j i= 1 i i (2.15) ( Z) = R f ( R) θ ( R) dr + φ( Z ) + P( R, Z ) dr dz R f ( R) i 1 Z IN dr (2.16) In the particular case considered, the expressions for θ j * (R) and C i * are given by Eqs. (2.17) and (2.19), respectively: θ * j ( R) = φ + P ( R) + 1 R 1 0 j R 1 0 j dr ( R) ( R) dr ( R ) g( R ) R 1 * [ ( ) ( ) ( ) ( )] ( ) P j R j + 1 R f R θ j+ 1 R g R 0 R ( R) * ( ) ( ) ( ) ( ) ( ) [ P + θ j R j 1 R f R j+ 1 R ] R h R g h R g 1 R h R dr dr dr dr dr (2.17) where j = q, q 1, q 2,,1,0, θ q+1(r) = 0 and h(r) is given by the following expression: ( R) R ( R ) R f dr 0 h (2.18) = R R f ( R) dr [ θ ( R) θ ( 0) θ ( R) ] f ( R) ψ ( R) IN * 0 i = 1 0 AV R f 0 C (2.19) 2 ( R) ψ ( R) i dr i dr Once the temperature distribution, θ(r,z), in the flow is determined, the Nusselt number, to which the results presented in this chapter, can be evaluated by means of Eqs. (2.20) and (2.21): Nu ( Z) θ = 2 θ ( R, Z) R R= 1 ( 1, Z) θ ( Z) Bulk (2.20)

50 32 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs ( ) 1 R f ( R) θ( R, Z) 0 θbulk Z = (2.21) 1 R f dr 0 ( R) dr Turbulent flow formulation The model previously described can be applied both to laminar and turbulent flow. In the first case, the solution can be obtained by considering the Hagen-Poiseuille parabolic velocity profile and zero eddy diffusivity, i.e.: f(r) = 2(1 R 2 ) and g(r) = 1. In the past, the differential and integral expressions for turbulent flow have generally been formulated in terms of an eddy diffusivity or a mixing length for momentum transfer, but as shown by Churchill (1997), these two concepts are both related algebraically to the local turbulent shear stress. This approach has been adopted in the present work in order to solve the model previously described, and will be hereinafter summarized. The eddy diffusivity concept, suggested by Boussinesq (1877), introduces the following definitions: u u v = ε M (2.22) y T T v = ε H (2.23) y Now, the turbulent Prandtl number can be defined as: Pr T ε ε M = (2.24) H The relationship between the eddy diffusivity for momentum and the dimensionless turbulent shear stress, ( v ) + + u, in hydrodynamically developed flow, is a "one-to-one correspondence" (Churchill, 1997), as revealed by Eq. (2.25): ε υ M = 1 ++ ( u v ) ( u v ) + + (2.25) where

51 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs ρ ( ) ( u v ) v = u τ, y τ = τ W 1, r 0 f τ 2 W = ρ u avg (2.26a,b,c) 8 From Eqs. (2.22) and (2.25), the velocity can be obtained, as follows: ++ [ ] 1 τwr0 ( R) = R 1 ( u v ) u dr (2.27) µ R In order to obtain explicitly the velocity profile and the eddy diffusivity for the heat equation, needed to solve the original problem (2.1), expressions for the dimensionless turbulent shear stress, the friction factor and the turbulent Prandtl number (this last parameter is discussed below) are required. For the first one, the correlation suggested by Heng et al. (1998), based on the recent turbulent velocity measurements of Zagarola and Smits (1997) and reported in Eq. (2.28), has been adopted for the analyses. This correlation falls within the scatter of the best experimental data (Churchill and Zajic, 2002) and its associated uncertainties are not found to influence the Nusselt number predictions significantly (Churchill et al., 2005): y y = 0.7 exp y a + a (2.28) ( u v ) This equation is valid in turbulent hydrodynamically developed flow in a circular tube for any kind of thermal boundary conditions. It becomes inapplicable for a + < 150 because of the onset of laminar flow. Eq. (2.28) confirms itself suitable also for fluids with internal heat generation, like the molten salt nuclear fuels, as will be shown in the next paragraphs. As concerns the Darcy friction factor, the recent correlation proposed by Guo and Julien (2003) has been used since it predicts the values determined experimentally by Zagarola and Smits (1997) very well, and its form is preferable in terms of explicitness and simplicity, as shown by Eq. (2.29): 7 8 f Re = (2.29) Re Turbulent Prandtl number Many empirical and semi-theoretical correlating equations have been proposed in the past for the turbulent Prandtl number (Pr T ). Reynolds (1975) reviewed more than 30 expressions

52 34 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs of this kind. More recently Kays (1994) examined, with an extensive detail, theoretical and experimental results and correlating equations for Pr T. Churchill (2002) concluded in his fundamental reinterpretation of the turbulent Prandtl number that a reliable and comprehensive expression for the prediction of Pr T does not yet exist. However, the effect of Pr T is limited to low Prandtl number fluids (Pr << 1), and it is actually significant for Pr near 0.01 (Churchill, 2002), decreasing its importance when Pr departs from this value. It can be said that the evaluation of Pr T remains a critical and unsolved issue and can be significant enough when liquid metals are under consideration as it will be shown in Subparagraph The correlation proposed by Kays (1994) and reported in Eq. (2.30) has been chosen for the analyses as representative of Pr T, since this expression was found to be in a good agreement with most experimental and computed values of the turbulent Prandtl number (Kays, 1994): ++ ( u v ) ( ) + u v Pr T = (2.30) + Pr As suggested by Kays (1994), a coefficient of 2.0 rather than 0.7 should be used for liquid metals. The adoption of Eq. (2.30) for Pr T represents just a reasonable choice among the several correlations existing in literature Method and accuracy of solution The solution of the model described in previous subparagraph has been achieved by means of MATLAB ver 6.5 (The MathWorks Inc.). The Sturm-Liouville problem is solved using the LiScEig package (Trif, 1995), which calculates a finite number of numerical eigenvalues and eigenfunctions according to the radial discretization. The package was tested by computing eigenvalues and eigenfunctions in laminar flow. Results were compared with those achieved by Hsu (1968) and the same values were obtained both in the case of uniform wall temperature and uniform wall heat flux. An analytic expression is required by this package for the term f(r), and thus for the velocity (see Eqs. (2.4i) and (2.12)). However, the integral of Eq. (2.27), by using Eq. (2.28) for the dimensionless turbulent shear stress, cannot be solved analytically. For this reason, the numerical velocity profile, obtained thanks to the adaptive Simpson quadrature, is interpolated by means of a polynomial of high order. The remaining integrals of Eqs. (2.11), (2.14), (2.16), (2.17), (2.18), (2.19) and (2.21) are solved numerically using the trapezoidal rule. An example of a solution with uniform internal heat source and boundary conditions of first kind, having a parabolic shape along axial direction, is given in Fig. 2.2.

53 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs 35 a) 1.0 µ 1 = b) µ = µ 3 = µ 4 = µ 5 = µ 6 = µ 7 = ψ / ψ R= Radial coordinate R 0.10 c) d) 200 Temperature θ Z = 1/4 L Z = L/2 Z = 3/4 L Z = L Nusselt number Radial coordinate R Axial coordinate Z Fig Solution for Re = 10 4, Pr = 1, θ W (Z) = Z Z 2 /L, θ IN (R) = 0 and S(R,Z) = const: a) first 7 eigenvalues and radial distribution of the relative eigenfunctions; b) dimensionless temperature distribution; c) dimensionless radial temperature profile at different axial positions; d) Nusselt number profile with the axial coordinate. To assess the accuracy of the present calculations, a comparison with the numerical results of Yu et al. (2001) has been performed for the cases of uniform wall temperature and uniform wall heat flux. Fully developed flow conditions have been chosen in order to eliminate possible inaccuracies introduced by considering the thermal entry region (Heng et al., 1998). Moreover, according to author s knowledge, there are no studies in literature regarding thermally developing flow with the same turbulence formulation and numerical precision of results achieved by Yu et al. (2001). As a matter of fact, the computations of Yu et al. concerning the Nusselt number can be considered essentially exact for Pr = 0 and Pr = Pr T. The uncertainty for the other Pr values arise from that related to the choice of the turbulent Prandtl number correlation (Heng et al., 1998; Churchill, 2002). The comparison, performed with the same Pr T correlation considered by Yu et al. (i.e., that one developed by Jischa and Rieke (1979)), is shown in Table 2.1, only in the case of uniform wall heat flux for brevity.

54 36 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs Table 2.1. Comparison between Nusselt number results of Yu et al. (2001) and present calculations (in bold) in fully developed flow conditions with uniform wall heat flux for various Re and Pr values. The percentage absolute errors are shown in italics. Present results are in a very good agreement with those obtained by Yu et al. (2001) with errors below 0.5%. The discrepancies reported in Table 2.1 are due to different sources of uncertainty. Namely: 1) the discretization in radial direction, and thus the numerical evaluation of integrals; 2) the limited numbers of eigenvalues considered; and 3) the interpolation of the velocity profile. 2.2 Experimental data comparison In the present paragraph, results of the analyses are compared with several experimental data of Nusselt number for fluids with small and large Prandtl numbers, and additionally for fluids with internal heat generation. Moreover, a Nusselt number correlation is proposed, which includes the internal heat source term and comprises the Prandtl number of interest for molten salts applications Fluids with small Prandtl number (10-2 < Pr < 1) The critical parameter affecting the Nusselt number predictions in low Pr fluids is the turbulent Prandtl number. An overview of its effect is given in Fig. 2.3, where results achieved by means of different Pr T correlations (Dwyer, 1966; Notter and Sleicher, 1972; Reynolds, 1975; Jischa and Rieke, 1979; Kays, 1994; Weigand et al., 1997; Cheng and Tak, 2006) are compared to experimental data from Skupinski et al. (1965), which refer to NaK (Pr = ) in uniformly heated tubes.

55 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs 37 As emerges from Fig. 2.3, the effect of Pr T on the values of Nu becomes more significant as Reynolds increases. A factor of 1.5 can be found in the Nusselt number evaluations when Re is about This discrepancy points out that Nu predictions are significantly affected by Pr T and so improved measurements are required for a better assessment of such parameter. Nusselt number Data from Skupinski et al. (1965) Dwyer (1966) Notter and Sleicher (1972) Reynolds (1975) Jischa and Rieke (1979) Kays (1994) Weigand et al. (1997) Cheng and Tak (2006) Reynolds number Fig Effect of different Pr T correlations on the evaluation of the Nusselt number. Computed values are compared with experimental data of Skupinski et al. (1965) (NaK with Pr = ) for uniformly heated tubes. For completeness, the comparison with experimental data in the case of uniform wall temperature (Abbrecht and Churchill, 1960; Awad, 1965; Sleicher et al., 1973) and in the developing thermal region with uniform wall heat flux (Awad, 1965) is shown in Figs. 2.4 and 2.5, respectively, with reference to the correlation of Kays - see Eq. (2.30). It can be concluded that good agreement exists between present calculations and the set of measured data shown in this subparagraph, both in the fully developed flow and in the developing thermal region.

56 38 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs This work NaK - Pr = Nusselt number Reynolds number Nu/Nu Na - Pr = (full symbols) Air - Pr = 0.72 (void symbols) This work - Pr = (black lines) This work - Pr = 0.72 (gray lines) Re = 6.5 x 10 4 Re = 3.02 x 10 5 Re = 6.5 x 10 4 Re = 3.02 x Fig Nusselt number with constant wall temperature. Present work results are compared with: a) Sleicher et al. (1973) experiments (NaK with Pr = 0.024) in fully developed flow; b) Awad (1965) (Na with Pr = ) and Abbrecht and Churchill (1960) (air with Pr = 0.72) measurements in developing thermal region. z/d

57 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs 39 Nu/Nu Na - Pr = (full symbols) NaK - Pr = (void symbols) This work - Pr = (black lines) This work - Pr = (gray lines) Re = 5 x 10 4 Re = 5 x 10 4 Re = 10 5 Re = 2 x 10 5 Re = 2 x 10 5 Re = 5 x 10 5 Re = 5 x 10 4 Re = 10 5 Re = 2 x 10 5 Re = 5 x 10 5 Re = 5 x 10 4 Re = 2 x z/d Fig Entry region Nusselt number for uniform wall heat flux. Experimental data of Na (Pr = ) and NaK (Pr = ) are taken from (Awad, 1965) Fluids with large Prandtl number (1 < Pr < 10 4 ) High molecular Prandtl number fluids are very insensitive to the choice of Pr T as demonstrated by Churchill (2002) in his review. Fig. 2.6 gives a wide comparison of Nusselt predictions with experimental data (Monin and Yaglom, 1971). As a result, the agreement of the present model with measured data is very satisfactory for the large range of considered Prandtl numbers (1 < Pr < 10 4 ) Nusselt number Re = 2 x 10 5 Re = 5 x 10 4 Re = 2.5 x 10 4 Re = 10 4 This work Prandtl number Fig Large Prandtl comparison of present calculations with experimental data of Monin and Yaglom (1971).

58 40 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs As concerns fluoride molten salts without internal heat generation (Ambrosek et al., 2009), present predictions follow the experimental trend of Nu with Re very well (Fig. 2.7). The predictions are also compared with the Dittus-Boelter correlation, which is usually adopted for molten salts (Mandin et al., 2005; Yamamoto et al., 2005) although a non-negligible difference can be noticed. However, the Dittus-Boelter predictions agree with measured data within ±15% over the considered range of Reynolds number. 500 FLiNaK data Dittus-Boelter This work 100 Nu/Pr Re x 10-5 Fig The experimental data of the ternary fluoride salt FLiNaK (with 4.9 Pr 17.1) reviewed by Ambrosek et al. (2009) are compared with present calculations and the Dittus- Boelter correlation Fluids with internal heat generation Few studies can be found in literature regarding the heat transfer of fluids with internal heat generation both of experimental and theoretical kind (Poppendiek, 1954; Sparrow and Siegel, 1958; Siegel and Sparrow, 1959; Kinney and Sparrow, 1966). In this subparagraph the predictions of the generalized approach of the present work are compared with the accurate measurements of Kinney and Sparrow (1966), performed with an aqueous solution of sodium chloride salt in an insulated tube. Fig. 2.8 shows that present calculations are in good agreement with measured wall-to-bulk temperature differences, whose scatter was no more than ±7% (except for a few isolated points) over the considered Reynolds number range. The computed values with Pr = 3.9 fall within the uncertainty of the measured data (Kinney and Sparrow, 1966). Such a result can be regarded as very good when turbulent heat transfer data are under consideration and the experimental difficulties to be faced during the measurement campaigns are considered.

59 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs (θ W θ Bulk )/(Qr 2 0 /K) Pr = 2.9 Pr = 3.0 Pr = 3.1 Pr = 3.6 Pr = 3.8 Pr = 3.9 Pr = 4.0 This work - Pr = 3.0 This work - Pr = Re x 10-5 Fig Experimental (Kinney and Sparrow, 1966) and computed wall-to-bulk temperature results for uniform internal heat generation in an insulated tube (NaCl aqueous solution). Correlation of the computed values The computation of Nusselt values are impractical for engineering applications even with modern computational resources. Hence, interpolative and correlative expressions are appropriate as a supplement of the computational results. Here below, fluids with simultaneous uniform internal heat generation and uniform wall heat flux are considered, since the effect of the heat source cannot be neglected when the heat transfer coefficient needs to be evaluated (e.g., in the thermal-hydraulic analysis of the MSR core). To this purpose, Nusselt number calculations in fully developed flow have been correlated by means of a simple equation so that the effect of internal heat generation can be easily taken into account. A brief description of the correlating procedure is described here. A parametric analysis has been performed by varying the heat source term, Q, in the established model. The dependence of the Nusselt number with the ratio of Q over the wall heat flux, j W, in fully developed flow conditions (the Nusselt number in the entrance region can be calculated by means of simple correlations e.g., Chen and Chiou, 1981; Kays et al., 2004) can be described as follows: Nu Nu Qr 1 + C = (2.31) jw where Nu 0 is the Nusselt number without heat generation and C is a coefficient that in general depends on both Reynolds and Prandtl numbers. Now, it is probably not possible to

60 42 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs find one simple expression for C that correlates Nusselt calculations accurately in the Prandtl number range of interest for molten salts (1 < Pr < 20). However, fitting the coefficient C for several values of Reynolds and Prandtl numbers, the computed results can be correlated, but introducing a small sacrifice in simplicity; the equation proposed is: Nu Nu 1 + = 1 + A α a Pr a ( A1 Pr + ( α1 Pr ) Re ) Q r j b B2 b1 Pr ( B1 Pr + ( β1 Pr+ β2 ) Re ) 0 2 W 0 Q r j W Pr 5 5 < Pr 100 (2.32) where the coefficients are defined in Table 2.2. Table 2.2. Coefficients of Eq. (2.32). A B A B α β α β a b a b Eq. (2.32) correlates the calculations within 3.5 % for 0.7 Pr 100, < Re < , and for fully developed flow in a smooth pipe with uniform wall heat flux and uniform internal heat source (Fig. 2.9) This work Eq. (2.32) Re 2 x Pr 100 Nu / Nu Re Pr 100 Re 5 x Pr Re 3 x Pr 15 Re 3 x Pr 5 Re 2 x Pr 5 Re 3 x Pr Qr 0 /j W Fig Comparison between computed values of Nu/Nu 0 with those given by the correlation proposed in Eq. (32) for various Reynolds and Prandtl numbers.

61 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs 43 In summary, the general analytic approach established in this paragraph, and the numerical solution obtained by means of the more recent and accurate formulation of the turbulence proposed by Churchill (1997), are appropriate and suitable as demonstrated by several comparisons with measured data in different flow conditions. It is worth noting that accurate measurements for fluids with internal heat generation, in particular with reference to the molten salt mixtures of interest for nuclear reactors and to their typical working conditions, are required for a better assessment of present calculations and of the proposed correlation. 2.3 Heat transfer in a MSR core channel This paragraph presents an extension of the generalized approach previously proposed for circular pipes, which accounts for the heat conduction in the graphite and its internal heat generation (due to gamma heating and neutron irradiation). Moreover, this model is representative of the conditions encountered in a typical graphite-moderated MSR core channel during steady-state operation, as will be shown in Chapter 3 and 4. The analysed geometry consists of a smooth circular channel with constant flow section surrounded by a solid region, within which the fluid flow is hydrodynamically developed, but thermally developing, as depicted in Fig The overall analytic solution (fluid + solid domain) of the heat transfer problem, with reference to the geometry shown in Fig. 2.10, is obtained by gathering the two separate analytic solutions for the fluid and the solid region by means of an iterative procedure. Fig Geometrical configuration and coordinate system Solution of the heat conduction problem The problem of heat conduction in a solid domain has been widely treated in the past, and different numerical methods are available for its solution. Here below, the so-called

62 44 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs separation-of-variables method (Carslaw and Jaeger, 1959), which involves a solution expressed as a series of terms, has been adopted. The considered analytic model applies for axial-symmetric geometry, constant material properties, and arbitrary axial distribution of heat generation. Under such hypotheses, the heat conduction in the solid domain is governed by the following equation, which considers also the possible anisotropy of the solid material thermal conductivity: 1 r T K rr r r ( r,z) T( r,z) + K z z z = Q s ( z) (2.33) Boundary conditions Before going into the details of the mathematical formulation for the heat transfer problem, a brief scheme of the energy balance in the considered geometry is helpful for better understanding the definition of boundary conditions. These ones are chosen in order to reproduce the typical conditions occurring in a MSR core channel, whose representation in cylindrical coordinate is outlined in Fig (symbols are defined in nomenclature). If the geometry of Fig is thought as a part of a reactor core where graphite moderator elements are adjacent each other, a further symmetry boundary condition must be introduced on the outer radius (r = R 2 ) see Eq. (2.34a). This means that no heat can be exchanged from the outer surface of the solid region. As concerns the upper boundary of the solid region (z = H), an insulation condition is adopted (see Eqs. (2.34b)). This represents an approximation, but in the analysed situation, the axial conduction of solid material is negligible and such condition results nearly exact. A uniform temperature is assigned on the inlet section (z = 0) according to Eq. (2.34c). Finally, at the interface between the two media (r = R 1 ), the continuity of heat flux and temperature should be imposed (see next subparagraph). Anyway, in order to obtain a general mathematical formulation for the heat conduction in the solid domain, a temperature of arbitrary axial distribution is considered as a first approximation (see Eq. (2.34d)). T r ( r, z) ( r,z) T z r= R 2 z= H ( r,0) TIN = 0 (2.34a) = 0 (2.34b) T = (2.34c) T ( R,z) T (z) 1 = W (2.34d)

63 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs 45 Fig Schematic representation of the boundary conditions. According to the sketch of Fig. 2.11, the total power flowing from the solid to the fluid (P Solid Fluid ) can be considered equal to the total power generated into the solid domain with a good degree of approximation (a small amount of heat can be exchanged from the inlet) because of the adopted boundary conditions. This is a reliable condition for MSR graphitemoderated core channels, since the radial temperature gradients are much greater than the axial gradients. This means that, in steady state conditions, the heat flux is directed from graphite to molten salt and, practically all the energy generated into graphite flows to the fuel salt in the radial direction (Kasten, 1969). Consequently, even if the heat source in the fuel is order of magnitude greater than that present in graphite, the temperature of the moderator results higher than that of the fuel, which practically cools down the graphite (see next chapter). A situation where the fuel is not the material subjected to the highest temperature, is non-intuitive for nuclear engineers and is only typical of graphite-moderated MSRs. Since the separation-of-variables method requires a homogeneous form of Eqs. (2.33) and (2.34c) to be applied, it is convenient to express T(r,z) with respect to the inlet temperature and to consider the resulting (normalized) temperature T * (r,z) T(r,z) T IN as the sum of two terms according to Eq. (2.35):

64 46 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs T * ( r,z) ϑ(r,z) + Φ(z) = (2.35) Substitution of the foregoing expression into Eqs. (2.33) and (2.34) yields two simpler problems for the variables Φ(z) and ϑ(r,z) according to the set of Eqs. (2.36) and (2.37), respectively: 2 d Φ(z) Qs = 2 dz K ( z) z (2.36a) Φ ( 0) = 0 (2.36b) dφ(z) dz 1 r z= H = 0 ( r,z) ϑ( r,z) ϑ K r r + K z = 0 r r z z ϑ r ( r,z) ( r,z) ϑ z r= R 2 z= H ( r,0) = 0 = 0 = 0 (2.36c) (2.37a) (2.37b) (2.37c) ϑ (2.37d) ( R,z) = T (z) T (z) ϑ (2.37e) 1 W IN Φ The solution of the boundary value problem (2.37) is given by Eq. (2.38). Φ(z) = z 0 dz z Qs K 0 z H ( z ) Qs ( z ) dz + z dz K 0 z (2.38) The problem defined by Eqs. (2.37) can be solved exploiting the separation-of-variables method, which yields the following result for details see (Carslaw and Jaeger, 1959): n= 0 ( z) F ( r) ϑ( r,z) = c ψ (2.39) n n n 1/ 2 K z I γ R 1/ 2 1 n 2 ( ) K 1/ 2 K r + γ z K z Fn r = I0 γnr K r 0 n (2.40) K 1/ 2 r K Kr K γ z 1 nr 2 Kr

65 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs 47 where ψ n (z) and γ n are the eigenfunctions and the eigenvalues, respectively, of the Sturm- Liouville problem given by the differential equation (2.41) with its boundary conditions (2.42): 1 ψ n ( z) ( z) 2 d ψ dz n 2 dψ n = 0 dz z= H ( z) 2 = γ n, ( 0) 0 n = (2.41) ψ (2.42a, b) The eigensolutions of Eqs. (2.41), (2.42a) and (2.42b) can be found analytically as follows: n ( z) sin( γ z) ψ (2.43) n π γ n = (2n + 1) n N (2.44) 2H Imposing the boundary condition (2.37e) to Eq. (2.39) and exploiting the orthogonality property of the eigenfunctions ψ n (z), the coefficients c n are evaluated as follows: H ( T (z) T Φ(z) ) W IN sin( γ nz)dz 1 0 c n = (2.45) H Fn ( R1) 2 sin ( γ z)dz 0 n Finally, the solution of the original problem (2.33) with boundary conditions (2.34) can be expressed as: T(r, z) = TIN + n= 0 c n sin( γ n z)f n ( r) + Φ( z) (2.46) Iterative procedure for the overall solution The temperature fields of the fluid region (given by the solution of Eq. (2.1)) and of the solid region (Eq. (2.46)) are linked by the condition of continuity of the heat flux and temperature at wall (r = R 1 ). Since the solution of Eq. (2.1) requires boundary conditions in a polynomial form due to the adopted "splitting-up procedure", the imposition of the boundary condition at wall introduces further unknown variables represented by the coefficients of the polynomial, which have been evaluated by means of the following numerical iterative procedure, based on a trial and error method.

66 48 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs At first, an initial guess of the wall heat flux is taken, having an axial shape identical to that of the heat source in the solid region, and considering that all the heat generated inside the solid is transferred to the fluid. The wall heat flux is interpolated in the least-square sense in order to be imposed as boundary condition of Eq. (2.1) through a polynomial form. Consequently, the temperature field inside the fluid is evaluated, and the resulting wall temperature is used as boundary condition of the heat transfer problem in the solid domain. Subsequently, the solid region temperature field is computed by means of Eq. (2.46), and thus a new wall heat flux is available to continue the iterations. The procedure is repeated until convergence is eventually reached Method and accuracy of the overall solution The overall (fluid + solid domain) solution has been achieved by means of MATLAB 6.5 (The MathWorks Inc.). As concerns the method of solution of Eq. (2.1), details are given in Subparagraph As regards the solid region, the implementation of Eq. (2.46) has been performed by means of a discretization of the domain in both the radial and axial directions. The integrals defined by Eqs. (2.38) and (2.45) have been evaluated numerically by means of the trapezoidal rule. Through the above mentioned method of solution, the sources of uncertainty are the following: (i) the method of solution adopted for the fluid region. It was shown in Subparagraph that the obtained results are in a good agreement with those given by Yu et al. (2001), with errors below 0.5%; (ii) the numerical evaluations of integrals and the limited number of terms in the series adopted to get the solution in the solid region; (iii) the polynomial form required for the wall heat flux, which is obtained by interpolation of the actual wall heat flux calculated from Eq. (2.46). A systematic sensitivity analysis has been carried out in order to get a good compromise between accuracy and computational time, by varying the number of terms of the series, the order of the wall heat flux interpolating polynomial, as well as the discretization of the domain. As a result, by adopting 120 terms in the series of Eq. (2.46) and a polynomial of 13 th order, the estimated errors introduced by the treatment of the solid region, in terms of wall temperature and heat flux, are about one order of magnitude lower than those encountered for the fluid, and can be therefore considered acceptable within the scope of the present work.

67 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs Concluding remarks The present chapter presented a general mathematical formulation for the heat transfer of fluids with internal heat generation, which have been the subject of few studies up to now both from the experimental and the numerical points of view, according to the knowledge of the author. Actually, molten salts employed as nuclear fuels are of interest for the MSR technology development. In such applications molten salts are internally heated, hence the thermal-hydraulic behaviour occurring in a typical core channel of a graphite-moderated MSR as well as the capabilities of numerical codes to analyse such kind of circulating fuel reactor (see Chapter 3) demand accurate investigation. In order to get a deeper insight into this field, a generalized approach was developed taking into account the specific feature of the system, represented by the heat generation in both the fuel salt and the graphite (fluid and solid region, respectively). The model incorporates the recent formulation of turbulent flow and convection of Churchill (1997) and, after its accuracy assessment and validation, has been applied: (i) to study the heat exchange of fluids in a wide range of Prandtl number with different boundary conditions, by comparing computations with experimental data, and (ii) to propose a Nusselt number correlation for fluids with simultaneous uniform internal heat generation and uniform wall heat flux, which are the conditions of interest for molten salts reactors. In particular, from the results achieved in the present investigation, the following main conclusions can be drawn: The assessment of the adopted approach (Subparagraph 2.1.2), performed by comparing present calculations with the essentially exact solution of Yu et al. (2001), has shown that unavoidable inaccuracies introduced by the presented mathematical formulation and method of solution are of minor importance, because they lead to a maximum error of 0.5% in the calculated Nusselt numbers. The extensive comparison with experimental data has shown a satisfactory agreement of present calculations. Nevertheless, measured data are often characterized by a large scatter of about 10-20%, due to measurement difficulties precluding a very accurate assessment of the predictions. As concerns small Prandtl number fluids, the turbulent Prandtl number remains a critical piece of information leading to non-negligible discrepancies in predicting the Nusselt number, as shown by the investigation of its effect (Fig. 2.3). However, such discrepancies are not encountered while analysing large Prandtl number fluids, and in particular molten salts (without internal heat generation), whose Nusselt number can be predicted in a

68 50 A GENERALIZED APPROACH TO HEAT TRANSFER IN MSRs suitable way by means of the Dittus-Boelter correlation. In short, the higher uncertainties are related to small Prandtl number fluids, whereas more accurate predictions are achievable when molten salts are under consideration. As concerns fluids with internal heat generation, the level of agreement between computations and experiments is entirely comparable to that generally found for non-internally heated fluids. This result suggests that the turbulent shear stress concept described by means of Eq. (2.28) is applicable and extendible to the case of fluids with internal heat generation. As these fluids are extremely important for the nuclear field in the frame of the current development of the Molten Salt Reactor of Generation IV, the correlating equation proposed is thought to be helpful since it provides a relatively simple description of the involved phenomena, even if it has to be assessed on experimental grounds. All things considered, the overall solution (fluid + solid region) set-up in this chapter is thought to be useful under the following respects: i) it represents an innovative contribution in the field of heat transfer of fluids, allowing a generalized formulation and taking into account recent advances in turbulence modelling; ii) it outlines a helpful validation framework for testing computational fluid dynamics tools and/or different turbulence models; iii) it is representative of the thermo-hydrodynamic conditions occurring in a graphite-moderated core channel (see Chapter 3); iv) it permits to evaluate in a simple and prompt way some fundamental quantities such as the distributions of temperature and velocity, and the Nusselt number; v) it will reveal as an important interpretative support of numerical solutions in steady-state conditions, when a more refined model (thermohydrodynamics + neutronics) is adopted for the study of the MSR core channel (see Chapter 4).

69 Chapter 3 Assessment of COMSOL Multiphysics The MSR is a sort of circulating fuel reactor (CFR), which adopts a molten fluoride salt mixture containing the fissile material and playing the distinctive role of both fuel and coolant. In a thermal-neutron-spectrum MSR, the reactor core is composed by a graphite matrix, through which the liquid nuclear fuel flows and leads to a strong and intrinsic coupling between thermo-hydrodynamics and neutronics. This peculiar feature requires a suitable and qualified multi-physics simulation environment for a proper description of the system (fuel/coolant+graphite) behaviour. With reference to such complex and non-linear system, the assessment of numerical results is an important goal to be fulfilled, in the present thesis prospect of a MPM approach able to accurately describe the synergy of the involved different physical phenomena. To this purpose, the present chapter is intended to give two contributions to the validation of numerical results. I) A preliminary investigation of the thermal-hydraulic behaviour occurring in a typical graphite-moderated MSR core channel. In particular, the analytic approach developed in the previous chapter has been applied to a typical MSR core channel: (i) by testing the capabilities of COMSOL Multiphysics to evaluate the heat transfer characteristics and the hydrodynamic behaviour of such system; and (ii) by investigating the applicability of correlations for the Nusselt number to fluids with internal heat generation. For a deeper insight into the numerical solutions provided by COMSOL, a code-to-code comparison has been also carried out, adopting a dedicated CFD finite volume software (i.e., FLUENT ). II) A preliminary assessment of neutronics characteristics of the MSR core channel in steadystate conditions. In particular, a two energy group diffusion model has been implemented and carefully set-up in COMSOL, using group constants calculated by means of the deterministic code SCALE 5.1, and has been validated in the case of: i) static fuel (no circulation) by mean of a code-to-code comparison with the Monte Carlo code MCNP as well as with SCALE 5.1; ii) circulating fuel, by means of a comparison with a zero-point neutron kinetic model developed separately and representative of the behaviour of the MSRE (see Appendix A). Some results are published in (Cammi et al., 2009; Memoli et al., 2009). Main results are in the following paper, submitted to Chemical Engineering Science: Luzzi, L., Cammi, A., Di Marcello, V., Fiorina, C., An Approach for the Modelling and the Analysis of the MSR Thermo-Hydrodynamic Behaviour.

70 52 ASSESSMENT OF COMSOL MULTIPHYSICS 3.1 Modelling of the MSBR core channel Among the several Molten Salt Reactor concepts developed in the past and reconsidered in the last few years, the single-fluid Molten Salt Breeder Reactor (MSBR), proposed by Oak Ridge National Laboratory (ORNL) in the 1970s (Robertson, 1971) has been chosen as reference configuration in this chapter and in the following of the present thesis. Because of its significant progress and the exhaustive information and data delivered by ORNL (Energy From Thorium, this reactor was considered (for instance, Křepel et al. 2007) as reference system for benchmark analyses and validation purposes in the framework of the MOST project (Renault et al., 2005). The MSBR was designed to produce 1000 MW e and is featured by a thermal neutron spectrum and thorium fuel cycle (see Chapter 1). The reactor core is formed of square graphite-moderated blocks, each one with a central channel, through which the fuel salt flows, as shown in Fig In particular, the core has a central zone in which 13% of the volume consists of fuel salt (zone I), an outer, under-moderated region characterized by 37% of salt (zone II), and a reflector region containing about 1% of fuel. In the MSBR core, interstitial flow passages are present between adjacent graphite elements in order to provide the required salt-to-graphite volume ratios. The salt composition in the primary circuit is the following: 7 LiF (71.7 mol%), BeF 2 (16 mol%), ThF 4 (12 mol%) and 233 UF 4 (0.3 mol%). a) b) Fig Vertical (a) and horizontal (b) sections of the MSBR core. In order to assess the numerical results provided by COMSOL, a single-channel of the MSBR core has been adopted for the analysis. The thermal-hydraulic parameters have been

71 ASSESSMENT OF COMSOL MULTIPHYSICS 53 chosen so as to reproduce the average conditions of zone I, since the fission energy is mainly released in this region that extends over the major part of the core (Robertson, 1971). The geometrical parameters and power densities of the considered core channel have been calculated on the basis of a preliminary thermal-hydraulic analysis reported in Appendix B. The main results are summarized in Table 3.1, whereas in Table 3.2 the main thermophysical properties of fuel salt and graphite are reported. Table 3.1. Main parameters of the analysed MSBR core channel. Parameters Values Average power density (kw/l) 30.6 Average fuel salt power density (kw/l) Average graphite power density (kw/l) 2.75 Channel radius, R 1 (m) Axial channel length, H (m) 3.96 Inlet fuel velocity (m s -1 ) 1.47 Inlet fuel temperature (K) 839 Outlet fuel temperature (K) 977 Volume fraction of fuel salt (%) 13.2 Graphite power density fraction to the total power (%) 9* * average value derived from ORNL calculations Table 3.2. Thermo-physical properties of fuel salt and graphite at 908 K. Properties Fuel salt Graphite ρ (kg m -3 ) C P (J kg -1 K -1 ) K (W m -1 K -1 ) η (kg m -1 s -1 ) 0.01 / Two different geometries have been considered in the validation of COMSOL for thermohydrodynamics and neutronics aspects, respectively, as shown in Fig As concerns the former, in order to benefit by the axial-symmetry of the problem, and to reduce the size of the problem to be numerically solved, the core channel has been modelled as a cylindrical shell, as shown in Fig. 3.2c. Starting from data reported in Table 3.1, the outer external radius of graphite results R 2 = m. Anyway, in order to take into account the effect of fuel flow through interstitials on heat transfer, a symmetry boundary conditions is introduced in the graphite domain. Since interstitials have hydraulic diameters approximately equal to the channel diameters in zone I, conductive heat transfer is solved on half of the graphite domain (i.e., between r = R 1 and r = R G = m), imposing a symmetry boundary condition at r = R G. As concerns power densities, a sinusoidal distribution along the axial direction is imposed on both the molten salt and the graphite, while it is maintained flat

72 54 ASSESSMENT OF COMSOL MULTIPHYSICS along r. Such an approximation is usually adopted for analysing the thermal-hydraulics of conventional nuclear reactor core channels (Duderstadt and Hamilton, 1975) and is also consistent with the conditions occurring in the MSBR (Robertson, 1971). Fig Schematic representation of the core channel modelling: (a) vertical section of the MSBR reactor core; (b) simplified view of the graphite blocks; (c) cylindrical shell approximation of the single channel; (d) ¼ of the square channel approximation. On the other hand, as concerns neutronics validation, ¼ of the representative MSBR core channel is adopted for the analyses as shown in Fig. 3.2d. This choice is due to the necessity to let COMSOL results be comparable to those obtained by means of SCALE 5.1 (DeHart, 2005a, 2005b) and MCNP (Briesmeister, 2000), which require a pin cell geometry descriptive of a square (or triangular) lattice of a reactor core. Imposing the required salt-tographite ratio, the graphite element thickness, L G, result equal to m (see Appendix B).

73 ASSESSMENT OF COMSOL MULTIPHYSICS 55 As far as the fuel composition is concerned, reference is made to the beginning of life (BOL) equilibrium composition as reported in Table 3.3 (Robertson, 1971) and therefore free of fission products. Table 3.3. BOL equilibrium composition of fuel salt and graphite. Constituent Atomic density [atom b -1 cm -1 ] 232 Th Pa U U U U Np Pu Pu Pu Pu Pu Li Li Be F Graphite Thermo-hydrodynamics assessment Here below, the analytic approach discussed in Chapter 2 is applied to the study of the thermal-hydraulic behaviour of a typical graphite-moderated molten salt reactor core channel. To this purpose, the numerical solutions (in terms of temperature and velocity profiles, heat transfer coefficient, and friction pressure loss) obtained by means of COMSOL and FLUENT are discussed with respect to the overall analytic solution and to classical correlations for the friction factor and the Nusselt number. In addition, a sensitivity study for different Reynolds numbers and different turbulence models is presented for the above mentioned thermal-hydraulic quantities. As concerns numerical analyses, calculations have been performed: (i) in steady-state conditions, with reference to a 2-D axial-symmetric (r,z) computational domain and in accordance with the hypotheses and the boundary conditions assumed in Paragraph 2.3; (ii) adopting the incompressible RANS (Reynolds Averaged Navier Stokes) equations for the fluid motion with Boussinesq s eddy viscosity hypothesis (Boussinesq, 1877); and (iii) considering as reference the standard k-ε turbulence model, which is the two-equation model generally employed among the several turbulence models available in literature. As far as the approach for modelling the near-wall region is concerned, COMSOL adopts the logarithmic wall functions, assuming that the computational domain begins at a certain

74 56 ASSESSMENT OF COMSOL MULTIPHYSICS distance, which depends on the mesh size, from the real wall (Comsol Inc.). On the other hand, the enhanced wall treatment approach (Fluent Inc.) has been chosen for FLUENT calculations. Great effort was spent in setting up the elements/cells size, particularly at the interface between salt and graphite, by means of a systematic mesh sensitivity analysis that is not reported here for brevity. Some details concerning mesh strategy and near wall modelling are reported in Appendix C Velocity, temperature and pressure loss inside the channel In this subparagraph, the operating conditions of the channel representative of the MSBR thermal-hydraulic behaviour are discussed. In particular, it must be mentioned that, corresponding to data reported in Tables 3.1 and 3.2, the Reynolds and Prandtl numbers are equal to and 11, respectively. Fig. 3.3 shows the comparison in terms of velocity profiles, while in Fig. 3.4 the axial and radial temperature distributions are depicted Velocity u [m/s] Analytic solution COMSOL FLUENT Radial coordinate r [m] Fig Velocity profile comparison (Re = ). The Nusselt number (Nu) and friction pressure loss ( p) are given in Table 3.4. As concerns p, the comparison is proposed by adopting both the Blasius (Todreas and Kazimi, 1990) and the Guo and Julien (2003) correlations. The latter, which is expressed in terms of the friction factor, has been adopted in the analytic model for the calculation of the velocity profile (see Eq. (2.29) in Chapter 2), and consequently affects the temperature distribution and the Nusselt number evaluation.

75 ASSESSMENT OF COMSOL MULTIPHYSICS 57 Normalized temperature T*(r,z) [K] Analytic solution COMSOL FLUENT r = R 2 r = R 1 r = Axial coordinate z [m] Normalized temperature T*(r,z) [K] z = H/2 Analytic solution COMSOL FLUENT Fuel salt Graphite Radial coordinate r [m] Fig Axial (a) and radial (b) temperature profile comparison (Re = ). Table 3.3. Comparison in terms of Nusselt number and friction pressure loss (Re = ). Nu* Err. (%)** p [kpa] # Err. (%) ## Analytic solution 123 / - - COMSOL FLUENT Guo and Julien correlation / Blasius correlation *calculated at channel mid plane (i.e., z = H/2) **calculated with respect to the analytic solution # calculated on the entire channel length ## calculated with respect to the Guo and Julien correlation

76 58 ASSESSMENT OF COMSOL MULTIPHYSICS As a result, there is a general good agreement of the numerical evaluations of the analysed thermal-hydraulic quantities with those obtained by means of the analytic approach, and with the correlations of Blasius and of Guo and Julien. FLUENT results are more accurate than those obtained by means of COMSOL because of the near wall treatment of the boundary layer. On one hand, the logarithmic wall functions approach of COMSOL neglects a certain fraction of the domain near the wall with consequences on the velocity profile (see Fig. 3.3) and therefore on temperature calculation. On the other hand, the use of semi-empirical formulas for the wall treatment (the so-called wall functions) does not allow to correctly account for the fluid heat generation near the wall. On the contrary, these two drawbacks are not encountered in FLUENT simulations because the enhanced wall treatment approach allows to resolve numerically the boundary layer until the laminar sublayer. All things considered, the discrepancies by which COMSOL is affected are acceptable in an engineering context, which would employ this software for multi-physics simulations (see next Chapter) and not only for the study of the thermo-hydrodynamics itself Reynolds number and turbulence modelling sensitivity Graphite-moderated MSRs can be subjected to different thermal-hydraulic conditions, according to the several designs proposed till now. Great effort is being spent to identify the optimal core configuration in terms of diameter of the channels since it affects several important design parameters, such as the total feedback coefficient of reactivity, the breeding ratio, the neutron flux, the graphite life span and the fissile inventory (see Chapter 1). Actually, considering different channel diameters and velocities, the Reynolds number ranges approximately between 10 3 and 10 5 among the several graphite-moderated MSR concepts, so that a sensitivity study has been performed. Besides, the role played by the Reynolds number is possibly connected with the use of different turbulence models, which can be more or less suitable according to the fluid velocity field. In particular, the sensitivity analysis has been carried out for: (i) Re equal to , 10 4, , , 10 5, ; and (ii) the k-ε and k-ω turbulence models. It must be mentioned that the results obtained by means of the k-ω model of COMSOL are essentially the same obtained with the k-ε model, so that only the latter ones are shown. A systematic study of this circumstance can be found in Appendix C for pipe flow, where also the adoption of different Prandtl numbers has been investigated. As concerns laminar flow, the numerical results of COMSOL and FLUENT are essentially the same given by the analytic solution, as shown in Appendix C, where a systematic analysis performed in the case of pipe flow is given.

77 ASSESSMENT OF COMSOL MULTIPHYSICS 59 The results in terms of temperature are given in Fig. 3.5, while Fig. 3.6 shows the friction pressure loss comparison, where also the McAdams correlation (Todreas and Kazimi, 1990) is reported for completeness. Numerical / analytic temperature T*(r,z) r = 0 COMSOL k-ε - Black r = R FLUENT k-ε - Grey 2 r = R FLUENT k-ω - Blue 1 +10% - 10% Reynolds number Fig Ratio of the numerical to the analytic temperature values as a function of Reynolds number at different radial position of the channel mid plane. Friction factor COMSOL k-ε FLUENT k-ε FLUENT k-ω Eq. (2.29) Blasius correlation McAdams correlation Reynolds number Fig Pressure loss comparison in terms of the friction factor. As expected, the main difficulties of the turbulence modelling occurs at low Reynolds number in the laminar-turbulent transition region, where the discrepancies in terms of temperature are within 20% 10% for COMSOL and ±10% for FLUENT. Adopting the k-ε

78 60 ASSESSMENT OF COMSOL MULTIPHYSICS model, the situation significantly improves as Reynolds number increases, in terms of both temperature and friction factor predictions. It can be noticed that a non-negligible underestimation of the graphite temperatures and of the pressure losses is achieved at very low Reynolds numbers by means of COMSOL, because of the mentioned discrepancies in the velocity profile calculation as well as the adopted logarithmic wall functions. As concerns the k-ω turbulence model, it results slightly more accurate for low Reynolds number (below 10 4 ), while the discrepancies in terms of wall temperature with respect to the analytic solution do not reduce as Reynolds increases. The k-ε confirms itself as one of the most reliable two-equations turbulence model to be adopted for practical engineering applications. The discrepancies encountered with COMSOL are related to its particular near wall modelling, but they can be considered satisfactory in view of the adoption of this code for multi-physics simulations of the MSR and more in general of other circulating fuel reactors Nusselt number evaluation Independently of thermo-hydrodynamics numerical computations, the use of correlating equations for the Nusselt number (in order to predict the heat transfer coefficient) is of importance for engineering applications even with modern computational resources. Empirical correlations are widely used in a number of papers (Yamamoto et al., 2005; Křepel et al., 2007; Kópházi et al., 2009) for the modelling of the MSR core thermal-hydraulics, in order to reduce the size of the problem to be solved numerically. However, by adopting these classical correlations for the Nusselt number, which generally refer to fluids without internal heat generation (OECD/NEA, 2007), the heat transfer coefficient of the MSR fuel can be overestimated by a non-negligible amount, making at least questionable their direct application to fluids with internal heat generation. In the case of thermal-neutron-spectrum MSRs, this can lead to significant consequences on the graphite temperature predictions and on the reactor dynamic behaviour (Di Marcello et al., 2008b). In order to confirm this idea, a brief overview of the correlations available in literature is performed, both in laminar and turbulent flows. As concerns turbulent convection heat transfer, several correlations are available for molten salts without internal heat generation, and an extensive overview in the laminar-turbulent transition region and fully developed turbulence conditions can be found in (Yu-ting et al., 2009) and in (Bin et al., 2009), respectively. Some of them, such as those developed by Bin et al. (2009), Colburn (1933), Dittus and Boelter (1930), Gnielinski (1976), Hausen (1959), Petukhov (1970), Sieder and Tate (1936), and Yu-ting et al. (2009) for fully developed flow

79 ASSESSMENT OF COMSOL MULTIPHYSICS 61 conditions with uniform wall heat flux are compared with the numerical results obtained by means of COMSOL and FLUENT, whose analyses have been performed considering a fluid with uniform heat generation. Besides, the correlation developed by the author published in (Di Marcello et al., 2010) and reported in Chapter 2 (see Eq. 2.32), which is valid for 0.7 Pr 100, < Re < and accounts for the internal heat source, is also shown in Fig 3.7. The Nusselt number without internal heat generation (Nu 0 ) is evaluated by means of the Yu et al. (2001) correlation, as follows: Nu 0 PrT = Pr 8 f Re [( 8 f ) ] PrT + 1 Pr Pr Pr T 1 3 f Re (3.1) where the turbulent Prandtl number is calculated according to the Jischa and Rieke (1979) correlation, while the friction factor (f) is given by Eq. (2.29). Actually, Eqs. (2.32) and (3.1) represent a very good approximation of Nu calculated by means of the developed analytic solution in the considered flow conditions. Nusselt number COMSOL FLUENT Dittus and Boelter (1930) Colburn (1933) Sieder and Tate (1936) Hausen (1959) Petukhov (1970) Gnielinski (1976) Yu et al. (2001) Bin et al. (2009) Yu-ting et al. (2009) Di Marcello et al. (2010) Molten salt reactors Reynolds number Fig Comparison between the correlations available for molten salts in literature, the numerical computations of FLUENT and COMSOL and the correlation proposed by Di Marcello et al. (2010) for fluids with internal heat generation. The Reynolds number range of interest for graphite-moderated MSRs is also indicated. As a result, FLUENT calculations reported in Fig. 3.7 follow quite well the trend of Eq. (2.32), while the Nusselt number is overestimated by COMSOL at low Reynolds numbers, still remaining within the uncertainty of correlations. As depicted in Fig. 3.7, these

80 62 ASSESSMENT OF COMSOL MULTIPHYSICS correlations are affected by a significant dispersion for Reynolds numbers greater than Besides these discrepancies, a slight different behaviour can be noticed between the Di Marcello et al. (2010) correlation and those ones neglecting the internal heat generation. Such difference is enhanced at low Reynolds numbers due to the increasing importance of the term Nu/Nu 0 given by Eq. (2.32). In particular, the Nusselt number calculated by means of the Gnielinski (1976), Hausen (1959), and Yu-ting et al. (2009) correlations at Re = 5000 is about 70% higher than that of Eq. (2.32). This leads to an underestimation of the graphite temperature, whose importance depends also on the channel diameter and the strength of heat source in the fuel. As far as laminar flow conditions are concerned, they are of importance for the study of the molten salt reactors mainly because of two reason: first, the effect of heat generation on the heat transfer becomes more significant as soon as the Reynolds number decreases; secondly, they represent the typical operational conditions of the MSRE (Haubenreich and Engel, 1962). Analyses are carried out considering Re = 10 3 and both fully developed and thermal developing flow conditions (the thermal length of development is indeed higher than the core height). When there is no fluid internal heat generation, the following correlation developed by Bird et al. (1960) can be used since it represents a good approximation of the analytic solution of Hsu (1965): ( Z 4) ( Z 4) 1/ for Z /3 4 3 Nu 0 = for 2 10 Z 6 10 (3.2) ( 10 Z 4) exp( 41Z 4) for Z 6 10 In order to take into account the fluid internal heat generation, the following expression derived from the analytic solution of Poppendiek (1954) for fully developed flow can be adopted: 48 jw Nu = (3.3) av 11 3 Q d + j 44 W The comparison is given in Table 3.4, where also the results of the developed analytic approach are shown together with the numerical computations.

81 ASSESSMENT OF COMSOL MULTIPHYSICS 63 Table 3.4. Nusselt number comparison in laminar flow (Re = 10 3 ). Nusselt number Thermally developing flow Fully developed flow (z = H/2) Analytic solution COMSOL FLUENT Eq. (3.3) Bird et al. correlation As expected, the importance of the heat source is enhanced in laminar flow, and the adoption of classical correlations such as the Bird et al. (1960) can lead to an erroneous predictions of Nu even by an order of magnitude, as shown in Table 3.4. In summary, the Nusselt number and hence the heat transfer coefficient can be significantly overestimated when the heat source is neglected. This has important consequences for the evaluation of the graphite temperature in the core of a thermal-neutron-spectrum MSR. Therefore, the heat source contribution should be taken into account in order to correctly evaluate the graphite thermal loads and consequently the performance of this component in terms of its residual life-time. 3.3 Neutronics assessment Since COMSOL Multiphysics does not allow to treat neutronics with a specific module, great effort has been spent to set-up a suitable model for the reactor physics at Politecnico di Milano through different preliminary works (Cammi et al., 2007; Di Marcello et al., 2008b; Memoli et al., 2009), which have encountered a growing interest by the software developers. In the case of the MSBR, a preliminary model, based on the two energy group diffusion theory with six groups of neutron precursors, in steady-state conditions, has been implemented using the Convection and Diffusion application mode of COMSOL, according to the following equations to be solved using the eigenvalue analysis: 6 2 (1 β) 1 φ1 a f1 1 2 f 2 2 i i = k eff i= 1 D ( Σ + Σ ) φ + Σ φ + ( ν Σ φ + ν Σ φ ) + λ c 0 (3.4) 2 D2 2 a = φ Σ φ Σ φ + Σ φ 0 (3.5) βi u ci = ( ν1σf1φ1 + ν 2Σf 2φ2 ) λici i = 1 6 (3.6) k β = 6 i= 1 eff βi (3.7)

82 64 ASSESSMENT OF COMSOL MULTIPHYSICS The group constants of the diffusion equations (i.e., nuclear data, fission cross sections, absorption cross sections and diffusion coefficients) have been evaluated by means of pin cell calculation performed with the code components NEWT/TRITON (DeHart, 2005a, 2005b) of SCALE 5.1 package. NEWT solves the two-dimensional multi-group neutron transport equation according to the "extended step characteristic method". The nuclear data used are based on ENDF/B version VI.7 library, available in SCALE 5.1 as multigroup energy cross section libraries (Bowman et al., 2005). In the present work, the 238 energy group library has been adopted. The reference temperature of 900 K has been assumed for the fuel salt and graphite whose composition is reported in Table 3.3. The solution of the pin cell calculation has been used to collapse the 238 group cross section library in the thermal and fast energy groups. In order to choose an adequate spatial mesh for the SCALE 5.1 model geometry, a sensitivity analysis has been carried out evaluating the effective multiplication factor (k eff ) as a function of the number of the computational cells. The results (Fig. 3.8) show that k eff is weakly dependent on the mesh size; it varies of less than 50 pcm (per cent mille) over three orders of magnitude; a good compromise between accuracy and computational time was reached by choosing a mesh structure of 590 cells k eff Number of computational cells Fig Results of the mesh sensitivity analysis for the SCALE 5.1 pin cell model. The cut-off energy for the thermal group is usually chosen sufficiently high so that upscattering out of the thermal group can be neglected. In a typical Light Water Reactor (LWR) this value ranges between 0.5 and 1 ev, and up to 3 ev for high temperature gas cooled reactors (Duderstadt and Hamilton, 1975). In order to choose a reasonable value of the cut-off energy for the MSBR, the number of upscattering events per fission neutron source as a function of neutron energy has been calculated by means of SCALE 5.1. The

83 ASSESSMENT OF COMSOL MULTIPHYSICS 65 curve is illustrated in Fig. 3.9 and shows that upscattering becomes negligible for neutron energy of about 1 ev. This value has been selected for the group constant calculation, whose results are briefly summarized in Table 3.5. Upscattering events per fission neutron [-] Neutrons energy [ev] Fig Number of upscattering events per fission source neutron vs. neutron energy calculated by SCALE 5.1 (v6-238 cross section library). Table 3.5. Group constants calculated by means of SCALE 5.1 at 900 K. Physical quantity Value Unit ν 1Σ f cm -1 ν 2Σ f cm -1 Σ a1,f cm -1 Σ a2,f cm -1 Σ a1,g cm -1 Σ a2,g cm -1 Σ 1 2,F cm -1 Σ 1 2,G cm -1 Σ 2 1,F cm -1 Σ 2 1,G cm -1 D 1F 1.29 cm D 2F 1.15 cm D 1G 0.98 cm D 2G 0.82 cm λ s -1 λ s -1 λ s -1 λ s -1 λ s -1 λ s -1 β β β β β β

84 66 ASSESSMENT OF COMSOL MULTIPHYSICS Static fuel assessment A first assessment of the agreement between COMSOL, SCALE 5.1 and MCNP is given by the multiplication factor calculation results reported in Table 3.6. This result can be considered acceptable for a preliminary evaluation of the rector core neutronics. The difference between SCALE 5.1 and MCNP, around 920 pcm, can be explained by the different nuclear data used, which are usually the main error source in neutronic calculations (Ridikas et al., 2003; Kolbe et al., 2008; Kamiyama et al., 2009; Memoli and Petrovic, 2009). It is worth to remind that the SCALE 5.1 model is based on energy multigroup approximation unlike MCNP, which uses continuous energy data. As far as COMSOL is concerned, the reason of the discrepancy can be found in the adopted diffusion theory approximation. Table 3.6. Effective neutron multiplication factors. Code k eff SCALE COMSOL MCNP ± In Table 3.7 the macroscopic cross sections for the two energy groups calculated by SCALE 5.1 and MCNP are given. As can be observed, the maximum difference (around 10%) occurs in the graphite and, more specifically, for the capture cross section. Table 3.7. Comparison of the two energy group collapsed macroscopic cross sections. νσ f [cm -1 ] Σ C [cm -1 ] Σ TOT [cm -1 ] Group Fast Thermal Fast Thermal Fast Thermal Fuel salt MCNP SCALE Diff a [%] -3.5% 4.7% 1.5% 5.6% 2.3% 1.0% Graphite MCNP SCALE Diff a [%] % 8.8% 3.0% 2.2% a with respect to the MCNP values.

85 ASSESSMENT OF COMSOL MULTIPHYSICS 67 In Fig the average axial flux profiles calculated by means of both COMSOL diffusive model and the MCNP model are shown 5, whereas in Fig the radial flux comparison is reported. In particular, fast and thermal neutron fluxes are plotted for the graphite and fuel salt materials. Neutron flux [10 14 n cm -2 s -1 ] Thermal flux - graphite Thermal flux - fuel salt Fast flux - graphite Fast flux - fuel salt COMSOL (full lines) MCNP (symbols) Axial coordinate [m] Fig Comparison between COMSOL and MCNP average axial flux profiles. Normalized neutron flux [-] Fuel salt Graphite COMSOL MCNP SCALE 5.1 Fast neutron flux Thermal neutron flux Radial coordinate [m] Fig Comparison between COMSOL, SCALE 5.1 and MCNP average radial flux profiles. 5 Being SCALE5.1, more specifically NEWT, a two-dimensional code, the axial profile cannot be calculated. However, the channel height (axial dimension equal to 396 cm) has been used for a buckling correction to calculate neutron leakage normal to the plane.

86 68 ASSESSMENT OF COMSOL MULTIPHYSICS As can be noticed, COMSOL flux profiles are in accordance with those calculated by means of MCNP and SCALE 5.1 within a maximum error of 7%. Nevertheless, the results show a slightly different behaviour of the COMSOL radial profile at the interface between fuel and graphite (see Fig. 3.11), being the gradient lower. This discrepancy is mainly due to the diffusion theory approximation, adopted in the COMSOL simulation, which cannot properly describe the neutron transport near boundaries or where material properties change significantly (Duderstadt and Hamilton, 1975). Finally, a difference within 10% has been found for average and peak neutron fluxes between the COMSOL results and those ones reported by ORNL (Robertson, 1971) Circulating fuel assessment Dedicated neutronics codes, such as MCNP and SCALE 5.1, do not generally allow to analyse neutron physics in the case of circulating fuel, unless modifications of the software structure is performed see, for instance, (Kópházi et al., 2003). For this reason, in this subparagraph, after a brief analysis of the effect of fuel velocity on neutron flux and precursor concentration carried out by means of COMSOL, the calculated loss of reactivity due to the circulating fuel is assessed in comparison with a point kinetics model (see Appendix A) used as reference for validation. The neutronic behaviour of the system has been studied by means of the COMSOL model as a function of fuel velocity, which is assumed uniform along the channel. Results are shown in Figs and 3.13, where the axial flux and the precursor concentrations are reported. The axial fluxes calculated for circulating fuel (the normal operation values are adopted i.e., u = u ref = 1.47 m/s, and τ EL = (u ref /u) τ * EL = τ * EL = 6 s) are compared with the static fuel neutron fluxes. As expected, the profiles do not show any significant variations (Lapenta, 2005; Cammi et al., 2007; Nicolino et al., 2008; Wang and Qiu, 2009), proving a posteriori that the assumption of static fuel represents an acceptable approximation.

87 ASSESSMENT OF COMSOL MULTIPHYSICS 69 Fuel salt neutron flux [10 14 n cm -2 s -1 ] Static fuel (full lines) Circulating fuel (dashed lines) Fast neutron flux Thermal neutron flux Axial coordinate [m] Fig Effect of fuel velocity on axial flux profiles. Precursor concentration, c 4 [10 12 cm -3 ] u = 0 u = 0.1 u ref u = 0.5 u ref u = u ref Axial coordinate [m] Fig Precursor concentration of group 4 (c 4 ), achieved by means of COMSOL, as a function of the axial coordinate, for different values of fuel velocity. The theoretical reactivity loss curve, based on space independent point reactor kinetics (see Appendix A), calculated according to Eq. (A.4), is compared in Fig with the results obtained by means of the two energy group diffusive model adopted in COMSOL.

88 70 ASSESSMENT OF COMSOL MULTIPHYSICS Reactivity loss [pcm] Theoretical COMSOL Fuel velocity [m/s] Fig Comparison of COMSOL results with zero-dimensional theoretical reactivity loss as a function of the fuel velocity. The COMSOL reactivity variations calculated starting from a critical configuration, k eff = 1, with the fuel velocity set to zero, are generally higher than the theoretical ones. This difference has to be found in the two different modelling approaches. In particular, the contribution to the reactivity of the delayed neutron precursors (DNP) at different axial positions of the channel is neglected in the theoretical model, which is based on a zerodimensional geometry and, for this reason, is not able to catch the effect of the axial precursor distribution that is strongly affected by the fuel transport along the channel (Wang and Qiu, 2009). On the contrary, spatial effects are embodied in the COMSOL model, where the problem is also solved in the axial dimension. In order to quantify the importance of the space dependency on the effective DNPs in the core, a calculation has been performed using a simplified model of MSR that considers an infinite external loop (i.e., τ EL ). This means that all precursors leaving the core have no chance to re-enter it. In Fig. 3.15, the results are compared with the analytic reactivity loss based on a one-dimensional model in the case of cosine precursor distribution, Eq. (3.8), and flat distribution see (Kópházi et al., 2003) for details: ρ = 6 i= 1 2 π βi 2 2 λ π i 2H + 2 u H e λ ih u (3.8)

89 ASSESSMENT OF COMSOL MULTIPHYSICS Reactivity loss [pcm] Theoretical (flat distribution) Theoretical (cosine distribution) COMSOL Fuel velocity [m/s] Fig Comparison of COMSOL results with one-dimensional theoretical reactivity loss as a function of the fuel velocity. As it can be noticed, the reactivity loss calculated by means of COMSOL is higher. This is straightforwardly explained by observing that the fuel transport causes the majority of the precursors to concentrate near the core exit (as depicted in Fig. 3.13, which shows the concentration of a precursor group for different values of velocity), where the probability for a precursor to decay outside the core becomes larger. Similar results were obtained by Kópházi et al. (2003), where a modified version of MCNP was used for modelling the transport of DNPs. In summary, the COMSOL model permitted to take into account both the finite external loop and the space-dependent effects. 3.4 Concluding remarks In this chapter, the potentialities of COMSOL Multiphysics were assessed in the case study of the MSBR core channel, with regard to the thermo-hydrodynamics and neutronics aspects. The adopted methodology is justified when innovative simulation techniques, such as the MPM approach, are foreseen in the study of the MSR of Generation IV, and more in general of other CFRs, in order to investigate their dynamic behaviour taking into account the coupling between the several involved phenomena, as will be presented in the next chapter. As far as thermo-hydrodynamics is concerned, a first contribution to the assessment of the numerical computations of the heat transfer to fluids with internal power generation was provided, leading to the following two main conclusions:

90 72 ASSESSMENT OF COMSOL MULTIPHYSICS The level of agreement between the numerical computations and the analytic solution, in the explored range of Reynolds numbers, can be considered acceptable from an engineering point of view. As a result, FLUENT simulations obtained by means of the k-ε turbulence model are more accurate than those given by COMSOL, which are affected by the particular treatment of the boundary layer. As soon as the Reynolds number increases, the difference between the two codes diminishes and the agreement with the analytic solution is improved. All things considered, COMSOL revealed itself as a reliable and robust tool for thermal-hydraulic analyses providing a good degree of accuracy, at least for the considered Prandtl number of the MSBR and in the analysed simple geometry. The Nusselt number correlations generally found in literature for molten salts appear to be unsatisfactory when fluids with internal heat generation are under consideration. In fact, the discrepancies with respect to Eqs. (2.32) and (3.1) are significant for low Reynolds numbers, and the conventional correlations result inappropriate in the case of laminar flow (that is the hydrodynamic regime occurring in the MSRE core), with a relevant consequence for the calculation of the graphite temperature. On the basis of the analysed cases, the correlation described by Eqs. (2.32) and (3.1) published in (Di Marcello et al., 2010) is thought to be helpful, because it provides a relatively simple description of the heat transfer in the case of internal heat generation. Nevertheless, accurate measurements are required for a better assessment on experimental grounds. As regards neutronics, a diffusive neutronic model was built in COMSOL, exploiting the deterministic code SCALE 5.1 for the calculation of the group constants, with the purpose to assess the COMSOL capabilities for neutronic analyses in the situation of static and circulating fuel. In the first case, a validation of the main neutronics quantities was performed in steady-state conditions by comparing the COMSOL results with those obtained by the stochastic code MCNP, which represents a reliable tool for neutronic analyses. In particular, criticality calculations were performed showing a good agreement of the flux profiles between COMSOL and MCNP models, based on diffusion and transport theory, respectively. In the case of circulating fuel, COMSOL permitted to study the fuel velocity influence on the neutron flux, precursor concentration and system reactivity. The flux profiles show no variations with the fuel velocity, proving that the assumption of static fuel for neutron distribution calculation represents an acceptable approximation. On the contrary, the DNP distribution and the system reactivity strongly depend on velocity. The calculated reactivity

91 ASSESSMENT OF COMSOL MULTIPHYSICS 73 curve was compared with the theoretical one in three cases: zero-dimensional model with recirculating fuel; one-dimensional model with flat and cosine distribution of precursors in the case of no re-circulation. The comparison showed that a significant difference occurs between the zero-dimentional approach and COMSOL results due to the spatial distribution of DNPs accounted for in the diffusive model. Since the effects related to the fuel velocity represent a delicate aspect in the neutronic design and control of such systems, spatial distribution of physical quantities should be carefully taken into account in the study of MSRs (as outlined in the next chapter).

92 74 ASSESSMENT OF COMSOL MULTIPHYSICS

93 Chapter 4 The Multi-Physics Modelling approach This chapter presents the Multi-Physics Modelling (MPM) approach developed in the present Ph.D. work for the study of the Molten Salt Reactor (MSR) behaviour in steady-state and transient conditions. After the assessment of numerical computations performed in Chapter 3, a multi-physics model is here developed. It allows the description of the coupling between heat transfer, fluid dynamics and neutronics occurring in a typical MSBR core channel, and it reveals able to take into account the spatial effects of the most relevant physical quantities (i.e., fuel velocity, fuel and graphite temperature field, neutron fluxes and delayed neutron precursor concentration). In particular, as far as the molten salt thermohydrodynamics is concerned, Navier Stokes equations are used with the turbulence phenomena treated according to RANS (Reynolds Averaged Navier Stokes) scheme, while the heat transfer is taken into account through the energy balance equations for the fuel salt and the graphite. As far as the neutronics is concerned, the well-known two-group diffusion theory is adopted, where the group constants, computed by means of the neutron transport code SCALE 5.1, are included into the model in order to describe the neutron flux and precursor distributions, the system time constants, and the temperature feedback effects of both graphite and fuel salt. The developed MPM approach is implemented in the unified simulation environment of COMSOL Multiphysics, which revealed flexible, robust and modular (e.g., able to be interfaced with MATLAB, in the prospect of control analyses by means of the Simulink platform). The MPM approach is applied to study the behaviour of the system in steady-state conditions and under several transients (i.e., reactivity insertion due to control rod movements; fuel mass flow rate variations due to the changing pumping rate; presence of periodic perturbations), pointing out the main advantages offered with respect to the conventional approaches generally employed in literature for MSRs. Main results are in the following paper, submitted to Annals of Nuclear Energy: Cammi, A., Di Marcello, V., Luzzi, L., Memoli, V., A Multi-Physics Approach to the Dynamics of Molten Salt Reactors.

94 76 THE MULTI-PHYSICS MODELLING APPROACH 4.1 Introduction In Molten Salt Reactors, the unusual characteristic of fluid fuelled core shows itself in the form of a strong non-linear coupling between the fuel motion and neutron dynamics, because DNPs created in a certain point of the core can decay in a different position of the primary circuit affecting the overall neutron balance. Besides, the fuel velocity field heavily depends on the fission source term, which determines the temperature distribution in the fuel salt, and its density variations. For this reason, a careful description of the dynamic behaviour of such a complex system should properly account for the different physical phenomena, their coupling mechanisms and the related spatial effects. In nuclear engineering, computer codes are widely used for Nuclear Power Plants (NPP) safety analysis within several purposes (licensing, safety improvements, better utilization of nuclear fuel, higher operational flexibility, etc.). Since a safety key parameter of the assessment of NPPs is represented by the ability in determining the time-space thermalhydraulic conditions especially inside the reactor core, the coupling between neutronics and thermal-hydraulics has been studied and analysed from many years and different approaches exist. In particular, in a solid-fuelled reactor, computational tools consisting of coupled advanced computer codes are employed according to the CCT approach (Salah et al., 2008), which includes a thermal-hydraulic system code (THSC) and a reactor neutron kinetics code (NKC). This coupling can be performed adopting different strategies. In particular two different methods to couple THSC and 3D NKC exist; i.e.: serial integration coupling and parallel processing coupling. The former requires modifications of the codes by implementing a neutronics subroutine into the THSC, which can be performed through internal coupling 6, external coupling 7, and parallel coupling 8. The parallel processing coupling allows the codes to run separately and exchange data during the calculation, involving minor modifications (Salah et al., 2008). The CCT approach is well assessed on conventional nuclear reactors, but may result noncompletely satisfactory in the case of MSR, because the description of physical quantities (for instance, the unusual spatial distribution of delayed neutron precursors) would require drastic modifications of the numerical and mathematical structure of the software generally adopted for solid-fuelled reactors. 6 The 3D neutron kinetics model is integrated into the core thermal-hydraulic model of the system code. 7 The core calculations include 3D neutronics and the fluid dynamic models, while the system code is used only to model the thermal-hydraulics in the primary circuit excluding the core region. 8 Both codes are run simultaneously by exchanging core power and information through the boundary conditions at the core edges.

95 THE MULTI-PHYSICS MODELLING APPROACH 77 For this reason, the current research is aimed at investigating different descriptive approaches for MSRs. Effort in this sense was carried out by different authors and for a variety of molten salt systems adopting different hypotheses and simplifications. In each case, a dedicated code to the specific reactor of interest was implemented from scratch or extended from a previous code in order to include the fuel motion effect. Lapenta et al. (2001) analysed the neutronics for fluid fuel systems by means of the point kinetic model, while Dulla et al. (2004) and Dulla and Ravetto (2007) used the quasi-static method. In both of their studies, only a prescribed velocity field oriented in one direction was considered and no temperature cross-sections feedback is taken into account. Lecarpentier and Carpentier (2003) developed the Cinsf1D code to study the AMSTER system, in which a simplified 1D thermal-hydraulic and neutron diffusion model is adopted. Yamamoto et al. (2005, 2006) and Suzuki and Shimazu (2006, 2008) performed the steady-state and transient analyses for the SMSR coupling the neutron diffusion equations with the heat transfer equations in fuel salt and graphite, in which the fluid flow model is solved in one dimensional form through the use of some empirical formulas. Křepel et al. (2005, 2007) developed the DYN1D-MSR and DYN3D-MSR codes for performing transient analysis on the MSBR and MSRE, in which the neutron diffusion equations were adopted for neutronics calculation, and the one dimension flow model was used even in the 3D code. Wang et al. (2006) focused on the fluid dynamic simulations and optimizations of the MOSART core with the code SIMMER-III by extending the thermo-hydraulic and neutronic models in steady-state conditions. The coupled thermo-fluid and neutronic dynamics of MOSART were also researched by Nicolino et al. (2008), in which the stream function-vorticity was preferred for fluid motion and diffusion theory was chosen for the neutronics. Kópházi et al. (2009) set up a 3-dimensional time-dependent calculation scheme for the MSRE (by coupling the DALTON and THERM codes in an iterative manner), in which the heat transfer in the fuel core channels is simplified by using a correlation for the Nusselt number. Zhang et al. (2009a) performed transient analysis on neutron kinetics for a single fluid fuel molten salt reactor with a simplified heat transfer and flow models. Simplified neutronics and thermal-hydraulics models were used by Zhang et al. (2009b) for safety analyses of the MOSART system. Zhang et al. (2009c) extended the models previously adopted (Zhang et al., 2009a, 2009b) by means of a complete thermo-hydrodynamics and neutronics model in steady-state conditions. Although many studies on the MSRs have been carried out, most of the previous works use rough flow models and hypotheses, such as assuming known velocities (Lapenta et al., 2001; Dulla et al., 2004; Dulla and Ravetto 2007), one-dimensional flow and simplified heat

96 78 THE MULTI-PHYSICS MODELLING APPROACH transfer models (Lecarpentier and Carpentier, 2003; Yamamoto et al., 2005, 2006; Křepel et al., 2005, 2007; Suzuki and Shimazu, 2006, 2008; Kópházi et al., 2009; Zhang et al., 2009a, 2009b), simple cross-section feedback (Nicolino et al., 2008), only steady-state conditions (Wang et al., 2006; Zhang et al., 2009c). In order to provide a deeper insight into the steady-state and transients characteristics of MSRs, this chapter presents a multi-physics approach for the description of the coupling between neutronics and thermo-hydrodynamics by means of COMSOL Multiphysics (COMSOL Inc.). This approach has been developed with the aim to study the dynamic behaviour of MSRs, taking into account the spatial effects of the most relevant physical quantities. The MPM coupling scheme consists of the computational fluid dynamics model (RANS equations + k-ε turbulence model + energy balance) coupled with two group neutron diffusion theory, completely implemented in COMSOL. The neutronics cross-sections are considered as input values for the model and are computed by means of the neutron transport code SCALE 5.1 assuming beginning of life conditions (BOL) for the fuel composition. Adopting such coupling procedure, it is difficult to label the proposed approach according to the classification employed for conventional nuclear reactors. The MPM approach can be indeed considered as a sort of "serial integration coupling approach", even if it needs the neutronics group constants (calculated at the beginning of the simulation only once). Actually, if cross-section are thought as input parameters, the model is self-sufficient and its implementation in a unified simulation environment offers several advantages and potentialities in comparison with the conventional procedures of analysis, as discussed in the following. A sketch of the MPM approach consisting of the set of partial differential equations implemented in the unified environment of COMSOL and adopted for the analyses of this chapter is outlined in Table 4.1. All the details concerning the modelling aspects of neutronics, thermo-hydrodynamic, boundary conditions and input parameters are given in the next paragraph. 4.2 The Multi-Physics Modelling The MPM approach discussed here below is aimed at an accurate description of the strong coupling between neutronics and thermo-hydrodynamics of MSRs in transient conditions, by taking into account the spatial effects related to the distribution of the most significant physical quantities (i.e., the neutron flux, precursors concentration, velocity and temperature of fuel salt and graphite). Reference is made to a simplified geometry, represented by a single-channel of the Molten Salt Breeder Reactor (MSBR) core, in order to assess the potentialities of the developed MPM approach, either from a theoretical or a numerical point

97 THE MULTI-PHYSICS MODELLING APPROACH 79 Table 4.1. Sketch of the MPM approach. of view. In particular, the channel is modelled as a cylindrical shell in order to exploit axialsymmetric condition with reference to the system description and data discussed in Paragraph 3.1. Such an approximation is generally adopted in literature for neutronics calculations (Honma et al., 2008). On the other hand, as concerns thermal-hydraulics, a precise evaluation of the temperature pattern inside graphite elements would require the analysis of the entire core of the reactor, in order to take into account geometrical effects and power density distribution, but this is out of the scope of the present Ph.D. thesis Fluid flow and heat transfer model The MSBR can be subjected to different thermal-hydraulic conditions, according to the different channel position inside the core. Great effort was spent by ORNL (Robertson, 1971) to identify the optimal core configuration in terms of diameter of the channels, since (as already remarked) it affects several important design parameters, like the total feedback coefficient of reactivity, the breeding ratio, the neutron flux, the graphite life span and the fissile inventory (Mathieu et al., 2006; Forsberg et al., 2007). Actually, considering the different channel diameters and velocities, the Reynolds number ranges approximately between and Corresponding to the calculated inlet flow velocity shown in Table 3.1, the Reynolds number is about , so that turbulent flow regime occurs. Hence, the

98 80 THE MULTI-PHYSICS MODELLING APPROACH incompressible form of the Reynolds Averaged Navier Stokes (RANS) equations with Boussinesq s eddy viscosity hypothesis (Boussinesq, 1877) is adopted considering the standard k-ε turbulence model (the empirical constants are given as C ε1 = 1.44, C ε2 = 1.92, C µ = 0.9, σ k = 1.3, σ ε = 1.3), according to Eqs. (4.1)-(4.4): [ ( )] T u ρ F + ρf( u ) u = F + pi + ( µ + µ T ) u + ( u) (4.1) t u = 0 (4.2) ρ F k µ + ρfu k = µ + t σ T k k ρ F ε + µ T 1 2 T 2 ( u + ( u) ) (4.3) ρ F ε µ T + ρfu ε = µ + t σε ε C ε2 ρ k F ε 2 + C ε1 ε µ k T 1 2 T 2 ( u + ( u) ) (4.4) As to the heat transfer, the energy balance equations for the fuel salt and the graphite are reported by Eqs. (4.5) and (4.6), respectively: C T t + [ ( KF + KT ) TF ] = QF ρf CP,F u TF F ρ F P,F (4.5) TG ρ GC P,G + ( KG TG ) = QG (4.6) t It must be mentioned that natural convection can have a non-negligible effect on the fluid fuelled reactor dynamics due to the large fuel salt expansion coefficient as demonstrated by Nicolino et al. (2008), and Křepel et al. (2007). Hence, the buoyancy effect due to gravity is accounted for through the volume force F into Eq. (4.1), as follows: F ( T ) * // gαρ F T0 =, F = 0 (4.7a,b) where the reference temperature T 0, the reference density ρ *, and the coefficient of thermal expansion α of the fuel are calculated on the basis of ORNL evaluations (Robertson, 1971), and are equal to: T 0 = 839 K, ρ * = 3374 kg m -3, α = K -1. Molten salt are characterized by high molecular Prandtl numbers (actually, Pr 11 according to data reported in Table 3.2 of Chapter 3) and thus their heat transfer characteristics are slightly affected by the choice of a particular correlation for the turbulent Prandtl number as extensively argued by Churchill (2002). In the analyses of the present thesis, the correlation

99 THE MULTI-PHYSICS MODELLING APPROACH 81 developed by Jischa and Rieke (1979) has been adopted since its form is to be preferred in terms of explicitness and simplicity, as shown by Eq. (4.8): Pr T = Pr (4.8) The heat source of the fuel salt in Eq. (4.5) depends on the neutron fission reactions and can be calculated from the neutron flux, as follows: QF = ε Σ φ + ε Σ φ (4.9) f1 f1 1 f 2 f 2 2 As concerns the graphite, the heat generation (see Eq. (4.6)) is mainly induced by gamma heating and neutron irradiation (Robertson, 1971). This contribution has been modelled assuming that the graphite heat source is a certain fraction of the fission energy released into the fuel, according to Eq. (4.10): Q = γ (4.10) G Q F Such an approximation is generally employed in literature, when a diffusive neutron approach is adopted (Křepel et al., 2005; Zhang at al. 2009c). The graphite to fuel salt power density ratio (γ) has been evaluated on the basis of ORNL neutronic calculation results (Robertson, 1971). An average value for the Zone I of the MSBR core is considered for the analyses and is given as γ = Neutronics model In order to treat the neutron behaviour, the well-known two-group diffusion theory has been adopted. The velocity field of the fuel is taken into account by introducing a convection term in the balance equations of six families of delayed neutron precursors (DNP), whereas the neutrons are not affected by the molten salt motion because of the much shorter life span with respect to the characteristic time of the fuel circulation. The neutron and precursors balance equations, in both the fuel and graphite material, can be written in terms of fluxes and precursor concentrations, according to Eqs. (4.11) and (4.12). Assuming that all fission and delayed neutrons are born in the fast group, χ 2 and f 2,i can be set to zero. The governing neutronics equations can be completed, as shown in the next subparagraph, inserting a condition that takes into account the recirculation of the precursors.

100 82 THE MULTI-PHYSICS MODELLING APPROACH Fuel region 1 φ1 = v1,f t (1 β) χ1 1 f1 1 φ2 = v2,f t (1 β) χ2 1 f1 c i = i t ( D φ ) ( ν Σ φ + ν Σ φ ) 1 ( D φ ) Σ ( ν Σ φ + ν Σ φ ) 1 Σ λ c ( u c ) + β ( ν Σ φ + ν Σ φ ) i 1,F 2 2,F 2 f f 2 f a1,f 6 6 i= 1 2 φ f 1,i i= 1 a2,f f 1 λ c φ 2,i f 2 + Σ 2 i Σ i 2 i 21,F i φ 12,F φ λ c i 1 i Σ 1 12,F + Σ φ 1 21,F φ i = 1 6 (4.11) 1 φ1 = ( D1,G φ1 ) Σa1,Gφ1 + Σ21,Gφ1 Σ12,Gφ1 v1,g t Graphite region (4.12) 1 φ2 = ( D2,G φ2 ) Σa2,Gφ2 + Σ12,Gφ1 Σ21,Gφ2 v2,g t The group constants (i.e., the cross sections and the diffusion coefficients) are usually calculated solving the neutron transport equation for infinite medium with a deterministic tool using a fine group cross section library. The calculated neutron flux spectrum is then used to collapse the fine group cross sections. In the present work, the sequence NEWT of the modular system SCALE 5.1 (DeHart, 2005a, 2005b) is chosen to perform this task using the standard cross section libraries ENDF/B-VI.7 (Bowman et al., 2005). In particular, NEWT solves the two-dimensional Boltzmann equation for neutrons using the "extended step characteristic" approach. The adopted cell geometry for the group constant calculation is obtained taking into account the effective fuel to moderator volume ratio of the MSBR zone I. The fuel composition refers to the beginning of life (BOL) as reported in (Robertson, 1971) and therefore free of fission products. In order to take into account the axial neutron leakage, a buckling correction is used in the solver. More specifically, the cross sections are collapsed using critical neutron spectra obtained by means of the critical B1 approximation (Lewis and Miller, 1984) Boundary conditions Flow and heat transfer As concerns the fuel salt the following boundary condition have been imposed. Inlet boundary. The inlet velocity is given as u z = u in = 1.47 m/s and u r = 0. The inlet values for the turbulent kinetic energy and the turbulent dissipation rate are imposed

101 THE MULTI-PHYSICS MODELLING APPROACH 83 according to the correlations suggested in (Fluent Inc.) as follows: k in = m 2 /s 2, ε in = m 2 /s 3. T F = T in = 839 K is imposed to the inflow temperature. Outlet boundary. The "local one-way method" is exploited for u, k, ε, and T F, as generally used for outflow boundary conditions (Patankar, 1980). Central boundary. Symmetry boundary conditions are imposed on the axis of the channel. Wall boundary. The boundary condition at the interface between graphite and fuel salt is treated by means of the wall function approach offered by COMSOL Multiphysics (for details see Appendix C). As regards boundary conditions of the graphite domain, T in is imposed on the inlet boundary, while the "local one way method" is used for T G at the outlet boundary. In the MSBR core, interstitial flow passages are present between adjacent graphite elements to provide the required salt-to-graphite volume ratios (Robertson, 1971). In order to take into account the effect of fuel flow through interstitials on heat transfer, a further symmetry conditions is assumed in the graphite domain. Since interstitials have hydraulic diameters approximately equal to the channel diameters in Zone I, conductive heat transfer (Eq. (4.6)) is solved on half of the graphite domain (i.e., between r = R 1 and r = R G = m), imposing a symmetry boundary condition at r = R G. Neutron fluxes and delayed neutron precursors Inlet boundary. The "vacuum" boundary condition is imposed for the fast and thermal neutron fluxes at the inlet (i.e., φ 1 = φ 2 = 0). The circulation of delayed neutron precursors is taken into account considering that a certain amount of DNPs can return into the core from the external loop according to their decay features, as follows: c i,in i,out ( λ τ ) = c exp (4.13) i EL Outlet boundary. The "vacuum" boundary condition is imposed for the neutron fluxes at the outlet, while the "local one-way method" is used for the delayed neutron precursors. Wall and central boundaries. Symmetry boundary conditions are imposed for the neutron fluxes at r = 0 and r = R 2. As concerns DNPs, the symmetry condition is chosen at r = 0, while insulation boundary condition is adopted at wall (r = R 1 ).

102 84 THE MULTI-PHYSICS MODELLING APPROACH Input parameters Neutron group constants As concerns neutronics modelling, a set of temperature-dependent two-group cross sections are produced by means of SCALE 5.1, in the assumption of BOL fuel composition with no fission products. They are included into the MPM approach as input parameters, so that no time dependence of neutron cross sections is accounted for. Such an approximation can be considered acceptable in the case of MSRs because their neutronics characteristics are relatively independent of core lifetime (Suzuki and Shimazu, 2006, 2008). More precisely, group constants are calculated over a discrete temperature range, and fitted as a function of both fuel salt and graphite temperature, according to the following equation: av F av G 0 av 1TF av 2TG Σ ( T,T ) = a + a + a (4.14) Differently from conventional homogeneous-type neutronics approaches to cross sections calculation, the proposal of Eq. (4.14) is made, since it appears more appropriate in the light of system heterogeneity between fuel salt and graphite, which are indeed characterized by very different thermal time scales. Since a stationary solution depending on the power level is needed for the analyses, a further absorbing term, Σ abs, has been introduced as control variable in order to adjust the neutron population according to the selected power level. In particular, Σ abs can be considered as a diluted thermal neutron absorber. In such a way, power transients due to reactivity insertion can be simulated by setting a certain value of the control variable. A comparison of the feedback coefficients of fuel and graphite calculated by means of SCALE 5.1 and by the diffusive model implemented in COMSOL are shown in Fig. 4.1, where COMSOL values are obtained performing stationary eigenvalue calculations at different temperatures. More specifically, if k 1 and k 2 represent the multiplication factors for temperatures T 1 and T 2, the feedback coefficient is given by α T = (1/k 2 1/k 2 )/(T 1 T 2 ). As can be observed, a general good agreement can be found between the two approaches, and, in both cases, the calculated values do not significantly differ from those given by (Křepel et al., 2007). A little discrepancy can be found in the fuel salt coefficient, since the diffusive model returns higher negative values than those calculated by SCALE 5.1. A possible reason for this difference relies on the fact that SCALE 5.1 values are obtained starting from a channel configuration that differs from the criticality of about 4000 pcm (buckling given by the nominal height of the channel). On the other hand, the two-group

103 THE MULTI-PHYSICS MODELLING APPROACH 85 cross sections used in the COMSOL model are obtained collapsing the fine group cross sections with a corrected buckling in order to have a critical neutron spectrum, which can have a non-negligible influence on the thermal feedback coefficients. A complete explanation of such circumstance would require further investigation (out of the scope of the present thesis). a) Graphite feedback coefficient [pcm/k] SCALE 5.1 COMSOL Temperature [K] b) Fuel salt feedback coefficient [pcm/k] SCALE 5.1 COMSOL Temperature [K] Fig Thermal feedback coefficients of graphite (a) and fuel salt (b) calculated by SCALE 5.1 and COMSOL.

104 86 THE MULTI-PHYSICS MODELLING APPROACH Thermo-physical properties Thermo-physical properties dependent on the temperature are necessary in the coupling analysis and calculation. Several experimental investigations were carried out at ORNL during the 1960s and 1970s in order to study the thermo-physical properties of different salt mixtures and graphite types. As concerns MSBR materials, a wide database is available and can be found in the documentation recently declassified by ORNL (Energy From Thorium, The correlations regarding density and viscosity of fuel salt and the graphite thermal conductivity are taken from (Robertson, 1971) and are reported by Eqs. (4.15)-(4.17), respectively. As regards fuel salt thermal conductivity, experimental data reported in (Rosenthal et al., 1970a) have been fitted by the author and the resulting expression is given by Eq. (4.18). 4 ( ( T 839) ) ρ = 3374 F µ K K F G 3 = exp ( T ) -0.7 ( 4090 / T ) F (4.15) (4.16) = 3763 (4.17) G F F F F = T [ C] T [ C] C T 900 C (4.18) The other thermo-physical properties of the fuel salt and graphite are considered constants and are taken from Table 3.2 of Chapter Results and discussion The behaviour of a single-channel of the MSBR is analysed in this paragraph where the results obtained by means of the proposed MPM approach are discussed both in steady-state and transient conditions Steady-state conditions The spatial distributions of the most relevant physical quantities evaluated by means of COMSOL are shown and briefly discussed here below. Actually, they give useful indications on the peculiar behaviour that is featured by the considered MSBR core channel, and more in general is typical of CFRs. As concerns velocity and temperature profiles (Figs. 4.2 and 4.3), present results are compared with the analytic solution discussed in Chapter 2 and used for validation, as argued in Chapter 3.

105 THE MULTI-PHYSICS MODELLING APPROACH Axial velocity u z [m/s] COMSOL Analytic solution Radial coordinate r [m] Fig Axial velocity profiles. The comparison has been performed in order to obtain an interpretation a posteriori of the model assumptions. Even if the analytic model does not consider the temperature dependence of the thermo-physical properties (neglecting natural convection effects), assuming a fluid in thermally developing flow conditions, the discrepancies with the numerical results are kept down. This implies that the analytic solution is well representative of the temperature distribution and consequently of the velocity pattern occurring inside the core channel in steady-state conditions, and can be therefore used for a preliminary evaluation of the most important thermal-hydraulic parameters. In addition, the typical temperature trend of MSR core channels is encountered (Fig. 4.3). The graphite is subjected to higher temperature with respect to the fuel salt, and the calculated values are in accordance with those given by ORNL (Robertson, 1971). Such behaviour is generally exhibited in steady-state conditions (Křepel et al., 2005), but during power transients the situation can be different, as will be shown in the next subparagraph. The calculated neutron fluxes are shown in Fig The fast neutron flux in the fuel results higher than that calculated in the graphite, since fission reactions can occur only in the molten salt. On the contrary, the thermal neutron flux is lower in the fuel zone with respect to the graphite moderator, analogously to solid fuel reactors. DNP spatial distribution is a relevant feature of MSRs, and should be considered properly in the analyses. For instance, a zero-dimensional approach (by means of a zero-point neutron kinetics model) would lead to an underestimation of the reactivity loss due to fuel circulation (as demonstrated in Chapter 3), with possible consequences on the system behaviour during transients. The present MPM approach allows to take into account the effects related to the

106 88 THE MULTI-PHYSICS MODELLING APPROACH spatial distribution of DNPs, both in steady-state and transient conditions. As concerns the former ones, shown in Fig. 4.5, DNP concentration is strongly perturbed by the hydrodynamic pattern inside the channel. In particular, precursors are more concentrated in the upper zone of the channel and near the interface between fuel and graphite (i.e., where fuel velocity is lower). It can be noticed that the non-uniform radial profile of precursors (Fig. 4.5(b)) is interesting to be investigated by means of analyses oriented to the reactor control strategy and safety, since, for instance, local fluctuations of neutron density can occur. a) COMSOL Analytic solution Temperature T [K] r = R G r = 0 r = R Axial corrdinate z [m] b) Temperature T [K] z = H/2 COMSOL Analytic solution Fuel salt Graphite Radial coordinate r [m] Fig Axial (a) and radial (b) temperature profiles.

107 THE MULTI-PHYSICS MODELLING APPROACH 89 a) 8x10 14 Neutron flux φ [n/cm 2 s] 6x x x10 14 Thermal neutron flux Fast neutron flux Full lines - Fuel salt (r = 0) Dashed lines - Graphite (r = R G ) Axial coordinate z [m] b) Neutron flux φ [n/cm 2 s] 9x x x x x10 14 Thermal neutron flux Fast neutron flux z = H/2 Fuel salt Graphite 4x Radial coordinate r [m] Fig Axial (a) and radial (b) neutron flux profiles.

108 90 THE MULTI-PHYSICS MODELLING APPROACH a) Precursor concentration c i [cm -3 ] 2.0x x x x10 11 c 1 c 2 c 3 c 4 c 5 c Axial coordinate z [m] b) Precursor concentration c i [cm -3 ] 2.0x x x x10 11 c 1 c 2 c 3 c 4 c 5 c 6 z = H/ Radial coordinate r [m] Fig Axial (a) and radial (b) DNP profiles.

109 THE MULTI-PHYSICS MODELLING APPROACH Transient conditions In this subparagraph, the proposed MPM approach is applied to study several transients driven by: (i) reactivity insertion due to control rod movements; (ii) fuel mass flow rate variations due to the changing pumping rate; (iii) presence of periodic perturbations. Reactivity-driven transients The first group of simulated transient conditions are driven by reactivity insertion due to control rod movements. This kind of transients are of specific interest for MSRs because the delayed neutron fraction is reduced due to the loss of delayed neutrons in the external primary loop and is smaller than that of other fissile materials. So, an addition of positive reactivity could lead to a serious transient. In the following, the reactivity prompt jumps up to 300 pcm are simulated starting from steady-state conditions at nominal power and nominal fuel flow rate. The results in terms of power variations and temperatures of graphite and fuel salt are shown in Fig Power and temperature values encountered during the transients are strongly coupled each other. In particular, the power response of the system mainly depends on the negative salt reactivity feedback coefficient, which slows down the initial power fast growth. Moreover, the temperature response relies on the adopted thermal-hydraulics parameters and thermo-hydrodynamic model. It can be noticed from Fig. 4.6(b) that the fuel experiences higher temperatures than the graphite moderator differently from steady state conditions. This behaviour is due to the strong difference between fuel salt and graphite time constants and represents a specific feature of MSR dynamics, unlike conventional solid fuel reactors. Actually, at the beginning of the transient, the heat is transferred from graphite to fuel salt (like in steady-state conditions), but a situation is reached with the radial heat flux inverted between them, due to the fast time response of the fuel. From the safety point of view, the maximum temperature reached by the fuel salt and graphite are fundamental parameters which should be discussed, even if no explicit safety criteria have yet been established for MSRs. In the case of 300 pcm reactivity jump, the greatest temperature is attained by the fuel, maintaining an adequate safety margin with respect to the boiling temperature (about 1400 C) 9. As concerns graphite, its temperature should be kept as low as possible due to undesired thermal expansions and increased thermal 9 The limiting temperature of fuel can be lower than the boiling temperature, according to a criterion for the mechanical integrity of Hastelloy-N based on the Larson-Miller parameter (Sides, 1967; Suzuki and Shimazu, 2006, 2008), which imposes the limiting outlet fuel temperature equal to 1200 C. Anyway, an accurate evaluation of the maximum outlet temperature would require the entire core analysis.

110 92 THE MULTI-PHYSICS MODELLING APPROACH stresses, which can affect the performance of this component and its residual life-time in reactor. a) pcm 100 pcm 200 pcm 300 pcm P/P Time [s] b) Temperature [K] Fuel salt Graphite 900 Full lines - average temperature Dashed lines - maximum temperature Time [s] Fig Response of the MSBR core channel to the reactivity insertion with several reactivity levels at nominal power. The power (a) is shown for all reactivity levels. The temperatures (b) are given only for 300 pcm insertion results. The achieved results are comparable to those obtained by Křepel et al. (2008), who performed the analysis of the entire MSBR core, as shown in Fig. 4.7, where also a point kinetic model (whose parameters are chosen accordingly to the average conditions of the zone I of the MSBR core) is presented. Even if the analysed geometry is simplified, the present model is well representative of the system behaviour in terms of power response. The

111 THE MULTI-PHYSICS MODELLING APPROACH 93 discrepancies with respect to Křepel results in terms of peak and final power levels are due to the different system and geometry considered, namely: a representative channel of Zone I of the MSBR core vs. the entire MSBR core. The additional piece of information of the proposed MPM approach arises from the possibility of describing the hydrodynamic pattern. This induces more pronounced oscillations in the power response, which cannot be caught by adopting a simplified thermal-hydraulic approach (one- or zero-dimentional) that leads to a more flat dynamic behaviour, as highlighted by the results of Křepel et al. (2008) and of a point-kinetics model. 7 5 COMSOL Krepel et al. (2008) Point kinetics model P/P Time [s] Fig Comparison between present results, those obtained by Křepel et al. (2008) and a point kinetics model in terms of system response to the reactivity insertion of 300 pcm. Pump-driven transients The second group of analysed transients is driven by the changing pumping rate. The system response is studied in the case of unprotected pump-driven transients (i.e., power is not maintained by control rods) considering the following two situations: a) pumping rate decrease of 80%; b) pumping rate increase of 20%. The variations of fuel flow rate are simulated assuming an exponential response of the pump with a time constant of 2 s (Křepel et al., 2007). The results in terms of power and temperature response are given in Fig At zero power, the reactor dynamics is affected by the DNP behaviour as extensively analysed in Křepel et al. (2007). Actually, because of the DNP drifting along the primary loop, pump flow rate increase causes DNP loss and induces negative reactivity, while pump flow rate decrease leads to positive reactivity. When power effects are considered, the behaviour of the MSBR core channel (Fig. 4.8) is mainly governed by the feedback reactivity coefficient of the fuel

112 94 THE MULTI-PHYSICS MODELLING APPROACH salt. Figure 4.8(a) denotes the power decrease at the transient beginning, and subsequently the stabilization at about the 35% of its initial value P 0. The decreasing fuel flow rate causes a rapid increase in the fuel temperature (Fig. 4.8(b)), which affects the power response through its feedback reactivity coefficients. The system response in the case of 20% increasing fuel flow rate (see Figs. 4.8(c) and 4.8(d)) is driven by a larger increase of graphite temperature (featured by a positive temperature feedback coefficient) with respect to that of fuel salt, leading to a positive reactivity insertion, and thus to an increase of power of about 25%. The case of a decreasing pumping rate is of interest from the safety point of view, since it can give important information in the case of pump malfunctioning or in a more severe channel blockage in the reactor core. The strong increase of fuel and graphite temperatures shown in Fig. 4.8(b) can represent a dangerous condition for the MSBR, leading to possible local core damage in case of channel blockage (Křepel et al., 2008) if not prevented by the reactor safety systems. a) 1.2 b) Γ/Γ P/P Graphite Fuel salt Time [min] Temperature [K] Full lines - average temperature Dashed lines - maximum temperature Time [min] c) 1.3 d) Γ/Γ 0 P/P Time [min] Temperature [K] Graphite Fuel salt Full lines - average temperature Dashed lines - maximum temperature Time [min] Fig Response of the MSBR core channel to fuel pump rate variations: power (a) and temperatures (b) response in case of 80% pump rate decrease; power (c) and temperatures (d) response in case of 20% pump rate increase.

113 THE MULTI-PHYSICS MODELLING APPROACH 95 Presence of periodic perturbations The circulation of the fuel, besides the main effect of delayed neutron fraction reduction, can bring to periodic perturbations caused by local changes of the fuel properties. These local variations can then propagate throughout the whole primary circuit. Examples of these perturbations are the formation of gas bubbles, which were experimentally detected in MSRE experience, and precipitation of fissile compounds, which can occur when fuel is added in excess with respect to the solubility limit in order to compensate burn-up effects. Dulla and Nicolino (2007) thoroughly analysed the problem of fissile compounds precipitation showing also the difference between a point-like approach and a threedimensional one, considering the MSRE and MOSART systems. Herein, fissile compound precipitation in the MSBR is studied. From the kinetic point of view, the main difference between the MSRE and MSBR is the presence in the latter of a positive thermal feedback of the graphite characterized by a large time constant (Mathieu et al., 2006). In order to simulate the presence of a fissile lump in the channel, it is assumed that the fuel properties, which are represented by the macroscopic cross sections, change according to a periodic square wave pulse. The spatial width of the pulse is given by the axial size of the lump and the amplitude is evaluated assuming that the fissile compound consists of UF 4 radially spread across the fuel channel. The period is given by the circulation time τ r. In this way, the fuel two-group cross sections can be written according to (Dulla and Nicolino, 2007) as: 0 F Σ (z,t) = Σ + δσ(z,t) (4.19) F [ Θ( z u ( t t )) Θ( z u ( t t ) z) ] b z 0 z 0 m τr < t m τr + τc δσ( z, t) = (4.20) 0 m τr + τc < t < ( m + 1) τr where b is a constant dependent on the cross sections of solid UF 4 and molten salt fuel, Θ is the Heaviside function, z is the axial size of the lump, and m is an integer number. The reactivity introduced into the system can be then adjusted for example by varying the UF 4 atomic fraction in the moving volume or the axial size itself. Collapsed cross sections for UF 4 are again calculated through the sequence NEWT of SCALE 5.1. Zero power transient and nominal power transients are considered. The zero power transient refers to the case of a circulating fissile lump in the channel that gives a maximum total reactivity of 50 pcm. As it can be noticed from Fig. 4.9, during the fissile lump transit, lasting about 2 s, the power is subject to a fast increase. When the fissile lump exits the channel, the power quickly decreases until the precursors (whose concentration builds up

114 96 THE MULTI-PHYSICS MODELLING APPROACH during the lump transit), start emitting delayed neutrons. As discussed by Dulla and Nicolino (2007), the zero power dynamics is characterized by an asymptotic linear evolution of the power, which is clearly visible in the Fig a) P/P Time [s] b) Precursor concentration c i [cm -3 ] 2.5x x x x x10 11 c 1 c 2 c 3 c 4 c 5 c Time [s] Fig System response to periodic perturbations (maximum total reactivity of 50 pcm) at zero power in terms of power (a) and precursor concentration (b). The nominal power case is analysed for a maximum reactivity of about 50 pcm and 500 pcm, indicated in the following as low reactivity and high reactivity cases, respectively. The latter case (high reactivity) describes a super prompt critical configuration since the maximum reactivity insertion is larger than the value of β. Fig 4.10 shows the total reactivity behaviour

115 THE MULTI-PHYSICS MODELLING APPROACH 97 during the lump transit in both cases. The maximum value is reached, as expected, when the lump is in the mid channel (this occurs after about 1.3 s according to the reference fuel velocity), where according to the perturbation theory the adjoint fluxes reach the maximum values (Meem, 1964) High reactivity Low reactivity Fitting curve Reactivity [pcm] Time [s] Fig Total reactivity behaviour during the lump transit in high and low reactivity cases. In Fig. 4.11, the system response of normalized power and precursor concentration for the case of 50 and 500 pcm, for the first three periods, is depicted together with the results achievable by a point kinetics model. Besides the presence of large oscillations, with amplitudes up to 30% and 750% for low and high reactivity cases, respectively, both the power and the DNP concentration show a slight increase of the period averaged values. However, as demonstrated by Dulla and Nicolino (2007), differently from the zero power case that is characterized by a divergent solution, the thermal feedback has the effect to stabilize the solution around a stationary value. The local perturbation of the neutron flux can be noticed in Fig. 4.12, which shows the normalized fast and thermal flux axial profiles at different times during the lump transit for the high reactivity case. The solid fissile lump acts as a localized source of fast neutrons, so that the fast flux shows a peak (Fig. 4.12). On the other hand, a corresponding absorption of thermal neutrons occurs, so that the thermal flux shows a depression peak. This local perturbation, as can be observed, is swept throughout the channel by the fuel velocity. For the same amount of reactivity, the average fuel and graphite temperature evolutions are shown in Fig As already discussed, the fuel temperature presents a faster response to power change, whereas the graphite is subject to slow variation,

116 98 THE MULTI-PHYSICS MODELLING APPROACH being its time constant around 900 s. Moreover, it is worth to note that also in the periodic perturbation analysis the description of the fluid motion by means of a fluid dynamics approach (accounting for turbulent phenomena), has relevant effect on the global parameter behaviour. This behaviour presents indeed more pronounced oscillations with respect to more simplified thermo-hydraulic schemes, such as that offered by a point-kinetics model, as can be seen in Figs. 4.11(a) and 4.11(b). A full stability analysis would be required in order to put in evidence the contribution of the different physical quantities to the transfer function that relates the power to the cross section variations. a) 1.5 b) 1.4 P/P COMSOL Point kinetics model Time [s] Normalized precursor concentration c 1 c 2 c 3 c 4 c 5 c Time [s] c) P/P COMSOL Point kinetics model Time [s] d) Normalized precursor concentration c 1 c 2 c 3 c 4 c 5 c Time [s] Fig System response to periodic perturbations at nominal power level in the low (a and b) and high (b and c) reactivity cases in terms of: (a-c) power where also the point kinetics model is shown; (b-d) precursor concentration.

117 THE MULTI-PHYSICS MODELLING APPROACH 99 Normalized neutron flux of fuel Full lines - Fast neutron flux Dashed lined - Thermal neutron flux 0.4 t = 2.6 s t = 1.3 s t = 1 s t = Axial coordinate [m] Fig Local perturbations of fast and thermal neutron fluxes at different times in the high reactivity case. Average temperature [K] Graphite Fuel salt Time [s] Fig System temperature response in the high reactivity case. 4.4 Concluding remarks In this chapter, a MPM approach for the description of the molten salt reactor behaviour both in steady-state and transients condition was developed, in order to take into account the coupling (in space and time) between the several physical phenomena whose interaction cannot be neglected when an adequate description of the system dynamics is required. In

118 100 THE MULTI-PHYSICS MODELLING APPROACH particular, the attention was focused on the interaction between neutronics and thermalhydraulics. This is a well-known problem in nuclear engineering because it occurs with a more or less evidence in conventional nuclear reactors for which several coupling approaches are available in the literature (e.g., the Coupled Code Technique between THSC and NKC), above all dedicated to safety analyses. Differently from solid-fuelled reactors, MSRs are featured by an intrinsic coupling between neutronics and thermo-hydrodynamics (typical of circulating fuel systems), so that conventional approaches like the CCT may result unpractical requiring strong modifications of the code numerical structure. In this framework, several studies were performed and different models and approaches can be found in literature. They provide a reasonable good description of the MSR behaviour, even if significant simplifications (mainly on the thermal-hydraulics model) are made. The MPM approach proposed herein was born with a different strategy, which is not intended to substitute the conventional tools of analysis, but is aimed at giving a useful support in the study of the MSR dynamic behaviour. The analyses have been oriented to the assessment of the potentialities of the developed MPM approach, either from a theoretical or a numerical point of view. To this purpose, a complex model (neutron diffusion theory + RANS + k-ε turbulence model + energy balance) implemented in the unified simulation environment of COMSOL Multiphysics was applied to a simplified geometry, represented by a single channel of the MSBR, and was exploited to investigate both steady-state and transient conditions. In particular, from the result achieved in this chapter, the following main conclusions can be drawn: 1. The steady-state analysis has allowed to study the most relevant physical quantities, by taking into account the several features of the coupled graphite-molten fuel system, namely: the spatial distribution of thermal and fast neutron flux, of precursor concentration (very important in the case of circulating fuel reactors), and of graphite and fuel temperatures. As concerns heat transfer, the analytic solution developed in Chapter 2 has offered a satisfactory interpretation a posteriori of the model assumptions, revealing as representative of the temperature distribution and the velocity pattern occurring inside the core channel, and useful for a preliminary evaluation of the most important thermal-hydraulic parameters. 2. Transient analyses have permitted to analyse the system response in the case of reactivity-driven transients, pump-driven transients and presence of periodic perturbations, giving preliminary indications on safety related issues of MSRs, and

119 THE MULTI-PHYSICS MODELLING APPROACH 101 more in general of circulating fuel systems. The comparison with the results obtained by Křepel et al. (2008) on the entire MSBR core (a one-dimensional fluid flow model is used), as well as with those achievable with a point kinetic model, has demonstrated that: on one hand, the proposed MPM approach is representative and well descriptive of the MSBR core behaviour; on the other hand, the present model is able to give significant additional information on the system behaviour represented by the presence of further oscillations in the system response, due to the adoption of a complete thermo-hydrodynamic model. This characteristic does not appear clearly in above comparison, but could make the difference when the hydrodynamic pattern results complicated, allowing a reliable description of the core performance with respect to reactor operation and safety, as in the case of nonmoderated MSRs (e.g., TMSR or MOSART systems). All things considered, the developed MPM approach is thought to be useful and innovative because: (i) neutronics and thermo-hydrodynamics are solved together in transient conditions, taking into account turbulence and buoyancy effects, as well as the heterogeneity of the system (in the calculation of group constants for molten salt and graphite); (ii) it provides additional information for the MSR dynamics, thanks to the accurate description of the fuel velocity pattern; (iii) it is in principle applicable to more complex geometries of MSRs, including the treatment of other "physics" (such as thermo-mechanics of structural components and/or permeation of fissile species and fission products through graphite), and can be used for control-oriented analyses by means of the MATLAB/Simulink interface.

120 102 THE MULTI-PHYSICS MODELLING APPROACH

121 Conclusions In the present Ph.D. thesis, a multi-physics approach to the modelling and analysis of nuclear reactor core behaviour was developed, and implemented in the unified simulation environment offered by the finite element COMSOL Multiphysics software. The MPM approach, applied to the study of the MSR dynamics, resulted able to properly catch (without excessive computational resources) the synergy between the different phenomena involved in the reactor core behaviour, whose modelling would otherwise require the adoption of dedicated simulation tools (with drastic modifications of their structure) or the development of ad hoc numerical codes for the particular analysed situation. The Molten Salt Breeder Reactor was chosen as reference configuration for the analyses. In such kind of circulating fuel reactor, the coupling between neutronics and thermohydrodynamics is a key issue. This feature cannot be neglected in order to perform an adequate description of the reactor dynamic behaviour, which shows peculiar aspects with respect to solid-fuelled conventional NPPs. The developed MPM methodology and the adopted models for neutronics and thermo-hydrodynamics have required a deep investigation for what concerns the assessment and the extension of the COMSOL simulation environment. As far as thermo-hydrodynamics is concerned, a generalized analytic approach was developed and exploited for the assessment of COMSOL simulations. This approach was built in order to carefully take into account the molten salt mixture specificities (fuel/coolant), the reactor core power conditions and the heat transfer in graphite. In particular, the overall analytic solution (fuel salt + graphite) turned out to be useful under the following respects: i) it represents an innovative contribution in the field of heat transfer of fluids, allowing a generalized formulation and taking into account recent advances in turbulence modelling; ii) it permitted to set-up a usable correlation for the Nusselt number that takes into account the fluid internal heat generation; iii) it outlines a helpful validation

122 104 CONCLUSIONS framework for testing computational fluid dynamics tools and/or different turbulence models; iv) it is representative of the thermo-hydrodynamic conditions occurring in a graphite-moderated core channel; v) it permits to evaluate in a simple and prompt way some fundamental quantities such as the distributions of temperature and velocity, and the Nusselt number; vi) it reveals as an important interpretative support of numerical solutions in steadystate conditions, when a more refined (thermo-hydrodynamics + neutronics) modelling of the MSBR core channel is adopted. On the basis of the developed analytic solution, COMSOL results were assessed exploring a wide range of flow conditions (different Reynolds and Prandtl numbers, both in laminar and turbulent flows), making also use of a dedicated finite volume CFD code (FLUENT) in order to better appreciate the differences in numerical approaches to turbulence. The numerical results (in terms of velocity profiles, temperature distributions and friction pressure losses) were found in a good agreement with the analytic solution, whereas the encountered discrepancies can be definitely considered acceptable from an engineering point of view. Throughout the analyses, a relevant aspect of the molten salt behaviour emerged, namely the importance of internal heat source on the heat exchange properties of fuel salt. This effect was accurately investigated by means of an overview of the empirical correlations available in literature for molten salts. In order to assess their applicability to fluids with internal heat generation, a comparison was performed with the numerical results of FLUENT and COMSOL, as well as with the correlation proposed in this thesis for the turbulent convective heat transfer of liquid fuels. It was shown that the effect of fluid internal power should be accurately taken into account (especially at low Reynolds numbers, and above all in laminar flow), in order to avoid the underestimation of graphite thermal loads, and consequently of the performance issues of this component during the life in reactor. As far as neutronics is concerned, a module for the "reactor physics" was built in the COMSOL environment of simulation, and allowed to extend the potentialities of this software. Numerical results were assessed by means of: i) a code-to-code comparison with dedicated neutron transport codes, in the case of static fuel; ii) a comparison with a simplified neutron kinetics model, which resulted representative of the zero-power dynamics of the MSRE, in the case of circulating fuel. In the former case, COMSOL results (in terms of multiplication factor and neutron flux profiles) were found in a good agreement with those obtained by means of SCALE 5.1 and MCNP, as well as with the analyses carried out by ORNL. The discrepancies resulted within the uncertainty associated to the cross section libraries and can be considered acceptable from an engineering point of view.

123 CONCLUSIONS 105 In the case of circulating fuel, the analyses performed as a function of fuel velocity showed that DNP concentration is strongly modified by the fuel velocity field, whereas the neutron flux is not perturbed at all. This result demonstrated a posteriori the validity of the static fuel approximation adopted for the evaluation of neutron group constants. The reactivity loss caused by DNP circulation was investigated by means of a comparison with a simplified point kinetics model. Such model was validated on the basis of experimental data available from the MSRE and was exploited to study the stability at zeropower of this reactor. In addition, it was adopted as basis for the assessment of COMSOL results in terms of reactivity loss. The comparison showed a good agreement in the case of the MSBR, and highlighted the importance of DNP spatial distribution on neutron balance and system dynamics, which can be adequately taken into account thanks to the proposed neutronics model included in COMSOL. After the assessment of the COMSOL capabilities to cope with the adopted models for the neutronics and the thermo-hydrodynamics, the MPM approach was applied to study the dynamic behaviour of a single-channel representative of the average conditions of the MSBR core. The proposed MPM modelling includes several innovative contributions with respect to the state of the art. From the thermo-hydrodynamics and neutronics point of view, the coupling of RANS, energy balance, and neutron diffusion equations was treated, for the first time, in transients conditions taking into account: i) the fluid buoyancy effect, by means of the Boussinesq approximation, and the turbulence according to the k-ε model; ii) the system heterogeneity by means of a fitting of neutron cross sections as a function of both fuel and graphite temperatures, which appears more adequate in comparison with the conventional homogeneous-type approaches (generally adopted for MSRs). From the numerical point of view, the present MPM is featured by robustness and flexibility. In all the analyses, COMSOL revealed adequate in terms of simulation accuracy and solution convergence, at reasonable computer cost. With reference to the MSBR, several different transients were analysed, such as those driven by: reactivity variations due to control rod movements; fuel mass flow rate variations due to the changing pumping rate; presence of periodic perturbations, due to local precipitation of fissile solid compounds within the molten salt mixture. The analyses gave significant information on the specific dynamic behaviour of the MSBR, which can be considered of more general interest for the current development of other MSRs. In particular, the following aspects were investigated: (i) the system stability, which is a delicate feature in MSRs, since the graphite temperature reactivity feedback coefficient can be positive according to the channel size and the neutron flux; (ii) the influence of spatial DNP distribution on the system

124 106 CONCLUSIONS response to reactivity insertion; (iii) the local precipitation of fissile compounds in the mixture and the non-uniformity of DNP distribution, which are relevant aspects from the safety point of view; (iv) the system time constants, which are a significant piece of information in order to develop the most appropriate control strategy of the reactor. The proposed MPM approach is not intended to substitute the conventional tools of analysis (e.g., the well-known CCT between THSC and NKC), but it should be considered as a useful support to investigate the MSR dynamic behaviour, according to different needs and goals (for instance, those envisaged in control-, safety-, and design-oriented studies). Several advantages and potentialities are thought to be offered by the proposed approach, in terms of both multi-physics modelling and simulation environment. The main advantage that emerged from the analyses of the present thesis appears the capability of describing (by means of a CFD modelling) the fuel velocity field, which allows a deeper insight into the system behaviour without losing significant information on spaceand/or time-dependence of involved phenomena. Actually, large non-uniformity of fuel velocity pattern occurs in several situations, e.g. the following ones: in non-moderated MSRs (like TMSR, MOSART), the molten salt is not forced to flow through core channels, hence the thermal-hydraulic pattern could be complicated with relevant effects on the reactor performance in terms of both operation and safety; the velocity field can be significantly affected in correspondence of complex geometries (e.g., inlet or outlet core flow passages) and in the occurrence of natural circulation conditions. In such cases, the MPM approach can give important piece of information for a more accurate description of the system. Among the potentialities of the proposed MPM approach in the unified simulation environment of COMSOL, the most promising seems the "modularity". In fact, it is possible to include other physical phenomena and couplings (e.g., thermo-mechanics, permeation of fissile species and fission products through graphite) thanks to the COMSOL structure, which is constituted by "modules" for the different "physics". These modules can be coupled each other, and independently developed in order to include more complex or simple models, with different degree of detail according to specific needs. Moreover, the simulation environment of COMSOL can be easily interfaced with the MATLAB software by means of the Simulink platform (e.g., in the prospect of control-oriented analyses). The MPM approach developed in this Ph.D. thesis was applied for the first time to study the behaviour of a nuclear reactor by means of COMSOL, with reference to the MSBR. Independently of this simplified case study, the approach is thought to be useful in a more general context (e.g., for the simulation of the entire MSBR core, for the analysis of other

125 CONCLUSIONS 107 MSR configurations), and the simulation environment can be in principle adopted for a more complete modelling that could exploit the "modularity" of COMSOL. In particular, the following items could be the subject of future investigation: - the application of the MPM approach to other MSR concepts (for instance, fastneutron-spectrum reactors, or the MSRE in order to exploit the availability of experimental data); - the development of more complex models (e.g., neutron diffusion theory with more energy groups in the case of fast-spectrum MSRs) and/or the adoption of different geometries for design-oriented analyses; - the adoption and the development of the COMSOL modules for thermo-mechanics and for diffusion in porous media, in order to individuate adequate constraints in the design/management of graphite elements; - the analysis of the entire NPP dynamics, in order to develop the most appropriate control strategy thanks to the possibility of interfacing COMSOL with the Simulink platform of MATLAB. The mentioned developments can be helpful in the design study of next generation MSRs, in order to optimize the NPP layout according to the requirements individuated by the Generation IV International Forum.

126 108 CONCLUSIONS

127 Nomenclature Latin symbols a + dimensionless pipe radius { = r 0 (τ W ρ) 1/2 /µ } a 0, a 1, a 2 coefficients defined by Eq. (4.14), dimensionless A heat transfer surface, m 2 A flow core passage flow are of fuel, m 2 b coefficient of Eq. (4.20), dimensionless Bi Biot number c i concentration of the i th precursor group, cm -3 c i0 steady-state concentration of the i th precursor group, cm -3 c n coefficients defined by Eq. (2.45) C coefficient of Eq. (2.31), dimensionless * C i, C i constants defined by Eqs. (2.14) and (2.19), respectively C P specific heat, J kg -1 K -1 C ε1, C ε2, C µ k-ε model empirical constants, dimensionless d pipe diameter, m D neutron diffusion coefficient, m f(r), g(r) functions defined by Eqs. (2.4i) and (2.4j), respectively f Darcy friction factor f g,i fraction of delayed neutrons of the i th family emitted in the g th group F horizontal component of volume force, N m -3 F // vertical component of volume force, N m -3 F n (r) functions defined by Eq. (2.40) g gravity acceleration, m s -2 G(s) transfer function of static fuel, dimensionless h FG -1 heat transfer coefficient between fuel salt and graphite, W m -2 K h W -1 overall heat transfer coefficient at wall, W m -2 K H axial channel length, m H(s) transfer function expressed in terms of reactivity, dimensionless

128 110 NOMENCLATURE I identity matrix, 2x2 I i modified Bessel function of first kind and i th order j W (z) wall heat flux, W m -2 J W (Z) dimensionless heat flux at wall k turbulent kinetic energy, m 2 s -2 k(t) multiplication factor, dimensionless k 0 k eff multiplication factor at steady-state, dimensionless effective neutron multiplication factor, dimensionless K thermal conductivity, W m -1 K -1 K i modified Bessel function of second kind and i th order K r -1 radial component of thermal conductivity of the solid region, W m -1 K K z -1 axial component of thermal conductivity of the solid region, W m -1 K K T turbulent thermal conductivity, W m -1 K -1 {= C P,F η T / Pr T } l axial length of pipe, m l n neutron lifetime, s L dimensionless axial length of pipe ( = 2 l / (Re Pr r 0 ) ) L(s) open loop transfer function, dimensionless L C L EL L G M(s) M F M G n core length, m external primary loop length, m graphite thickness, m transfer function expressed in terms of multiplication factor, dimensionless mass of fuel salt, kg mass of graphite, kg neutrons n(t) neutron density, m -3 n 0 neutron density at steady-state, m -3 N natural number N ch Nu Nu 0 Nu m p number of core elements in MSBR zone I Nusselt number Nusselt number without internal heat generation Nusselt number in fully developed flow integer number fluid pressure, Pa pcm per cent mille {= 10-5 } P power generated by the analysed core channel, W P(R,Z) = R S(R,Z) P 0 P j (R) nominal power generated by the analysed core channel, W j th term of the polynomial expansion of P(R,Z)

129 NOMENCLATURE 111 P TOT total power generated by zone I of the MSBR core, W Pe Peclet number ( = Re Pr ) Pr molecular Prandtl number { = C P µ / K } Pr T q turbulent Prandtl number order of the polynomial expansion of P(R,Z) and φ(z) Q heat source, W m -3 Q av average heat source, W m -3 Q s (z) heat source within the solid region, W m -3 Q s av average heat source within the solid region, W m -3 Q TOT total power density of zone I of the MSBR core, W m -3 r r 0 radial coordinate, m pipe radius, m R dimensionless radial coordinate { = r / r 0 } R 1 R 2 R G channel radius / inner radius of graphite (or of the solid region), m outer radius of graphite (or of the solid region), m symmetry radius of graphite, m Re Reynolds number { = (d u avg ρ) / µ } s variable of the Laplace transform, s -1 S(R,Z) t t 0 T dimensionless internal heat source time, s reference time, s temperature, K T + dimensionless temperature { = (Pr T /κ) ln(y + ) } T fluctuation in temperature, K T 0 T * (r,z) T av T boil T melt T E (z) T IN T W reference temperature, K normalized temperature, K average temperature, K boiling temperature, K melting temperature, K temperature of the environment, K inlet temperature, K wall temperature, K T v time-averaged value of T v u velocity vector, m s -1 u fluctuation in axial component of velocity, m s -1 u r velocity along the radial direction, m s -1 u z velocity along the axial direction, m s -1 u avg average velocity in the pipe section, m s -1

130 112 NOMENCLATURE u ref reference fuel velocity, m s -1 u v time-averaged value of u v ( v ) + + u dimensionless turbulent shear stress v neutron velocity, m s -1 v fluctuation in radial component of velocity, m s -1 V F fuel salt volume in zone I of MSBR core, m 3 V G graphite volume in zone I of MSBR core, m 3 V TOT total volume of zone I of MSBR core, m 3 y normal distance from the wall, m y + dimensionless distance from the wall { = y (τ W ρ) 1/2 /µ } z Z Greek symbols axial coordinate, m dimensionless axial coordinate α volume thermal expansion coefficient of fuel salt, K -1 α(s) α BC, β BC α F α G β β i γ gain of the closed loop transfer function, dimensionless coefficients defining the kind of boundary condition at wall, dimensionless fuel salt temperature reactivity feedback coefficient, pcm/k graphite temperature reactivity feedback coefficient, pcm/k total delayed neutron fraction, dimensionless delayed-neutron fraction of the i th precursor group, dimensionless graphite to fuel salt power density ratio, dimensionless γ n n th eigenvalue of the Sturm-Liouville problem defined by Eqs. (2.41), (2.42) γ G graphite to total power density ratio, dimensionless Γ fuel salt mass flow rate, kg s -1 Γ 0 fuel salt mass flow rate at nominal conditions, kg s -1 ~ δ k(s) Laplace transform of the multiplication factor variation from steady-state, δη i (t) dimensionless variation of the dimensionless concentration of the i th precursor group from steadystate δρ(t) reactivity variation from steady-state, dimensionless δ ~ ρ(s) Laplace transform of δρ(t), dimensionless δσ(z,t) cross section local perturbation, m -1 δψ(t) variation of the dimensionless neutron density from steady-state ~ δ Ψ(s) Laplace transform of δψ(t), dimensionless p friction pressure loss, kpa

131 NOMENCLATURE 113 T reference temperature difference, K z axial size of the fissile lump, m ρ reactivity loss, pcm ε turbulent dissipation rate, m 2 s -3 ε f heat produced per fission reaction, J ε H -1 eddy diffusivity for heat, m 2 s ε M -1 eddy diffusivity for momentum, m 2 s ζ F ζ G η i (t) fuel salt to total power ratio, dimensionless graphite to total power ratio, dimensionless dimensionless concentration of the i th precursor group ϑ(r,z) function defined by Eq. (2.39) θ(r,z) dimensionless temperature θ AV (Z) dimensionless average temperature over the region θ Bulk (Z) dimensionless bulk (or mixed-mean) temperature θ E (Z) dimensionless environment temperature θ IN (R) dimensionless inlet temperature θ j (R), θ * j (R) functions defined by Eqs. (2.11) and (2.17), respectively θ W (Z) dimensionless wall temperature Θ Heaviside function κ von Karman constant λ i decay constant of the i th precursor group, s -1 Λ neutron generation time, s µ dynamic viscosity, kg m -1 s -1 { = ρ υ } µ i i th eigenvalue of the Sturm-Liouville problem defined by Eqs. (2.12), (2.13) µ T eddy viscosity, kg m -1 s -1 {= ρ F C µ k 2 / ε } ν average number of neutrons per fission ρ density, kg m -3 ρ * fluid density at reference temperature, kg m -3 ρ(t) reactivity, dimensionless ρ 0 ρ CR (t) σ ε, σ k reactivity at steady-state, dimensionless reactivity due to control rod movement, dimensionless k-ε model empirical constants, dimensionless Σ macroscopic cross section, m -1 Σ 0 unperturbed cross section, m -1 Σ 12 downscattering cross section, m -1 Σ 21 upscattering cross section, m -1

132 114 NOMENCLATURE Σ a absorption cross section, m -1 Σ abs control absorption cross section, m -1 Σ f fission cross section, m -1 Σ C capture cross section, m -1 Σ TOT total cross section, m -1 τ total shear stress, Pa τ 0 transport time constant, s τ r fuel recirculation time, s { = τ C + τ EL } τ C τ EL τ EL τ F τ G τ W residence time in the core, s residence time out of the core, s residence time out of the core at reference velocity, s fuel salt time constant, s graphite time constant, s wall shear stress, Pa υ kinematic viscosity, m 2 s -1 φ neutron flux, n cm -2 s -1 φ(z) function defining the boundary condition at wall φ j j th term of the polynomial expansion of φ(z) Φ(z) function defined by Eq. (2.38) χ g fractions of fission neutrons born in the g th group ψ i (R) i th eigenfunction of the Sturm-Liouville problem defined by Eqs. (2.12), (2.13) ψ n (z) n th eigenfunction of the Sturm-Liouville problem defined by Eqs. (2.41), (2.42) Ψ(t) dimensionless neutron density Subscripts F fuel salt G graphite 1 fast neutron 2 thermal neutron in inlet out outlet

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141 Appendices

142

143 Appendix A Point kinetics of Molten Salt Reactors Point kinetics represents a highly simplified but useful model to analyse the kinetics of neutron multiplying systems. In this appendix, the behaviour of MSRs is studied by means of a space-independent point kinetics model, based on one neutron energy group and six families of delayed neutron precursors in the case of zero-power, and including feedback temperature effects. The governing equations for fluid fuel systems require modifications in order to take into account the circulation of fuel through the primary loop. They have been adopted to analyse the MSR dynamic behaviour by means of the transfer function of the system (core + external primary loop) exploiting the theory of linear systems. In the case of zero-power, the developed model has been validated on the basis of the experimental data from the Molten Salt Reactor Experiment (MSRE). After the validation, it has been applied to study the effects of the coolant velocity on the reactor stability. Main results are published in (Di Marcello et al., 2009).

144 A.2 POINT KINETICS OF MOLTEN SALT REACTORS A.1 Zero-power point kinetics model The adopted neutron-kinetics model is based on the neutron diffusion equation with one energy group considering six groups of delayed neutron precursors. Thus, the governing equations are the following (Ash, 1965): dn(t) ρ(t) β = n(t) + dt Λ dc i dt (t) i 6 i= 1 λ i c i (t) 1 τ 1 τ (A.1) i EL = λ c (t) + n(t) c (t) + c (t τ ) e (A.2) i i β Λ C i C In this model, which refers to zero-power and is space-independent, Eq. (A.1), involving neutron density, is the same as for the static fuel reactors, while Eq. (A.2) has two additional terms (the third and the fourth on the right hand) in order to account for the fact that delayed neutron precursors are leaving and entering the reactor core. Thus, the rate of change of precursor population in the core [dc i (t)/dt] is given by: the rate of decay in the core [ λ i c i (t)], the rate of formation in the core [β i n/λ], the rate leaving the core [ c i (t)/ τ C ], and the rate of re-entry of precursors, which left the core τ EL seconds before and did not decay in the external loop [c i (t τ EL )exp( λ i τ EL )/τ C ]. The assumptions implied in the above model are the following: 1) the fuel is thoroughly mixed in the reactor (homogeneous fuel concentration); 2) the molten salt velocity is constant and it is treated as a parameter (hence, τ C = L C /u and τ EL = L EL /u are not function of the time); 3) fissions occur in the reactor core only; 4) only the core itself is capable of criticality. As a further assumption, the system is considered at steady-state for t 0: as a consequence, the relationship between the initial neutron density, n 0, and the precursor concentration, c 0i, is described by Eq. (A.3): i L λ τ c n 0i 0 = λ i β i Λ 1 e + τ λ τ C i EL (A.3) It follows that the reactivity at steady-state, ρ 0, given by Eq. (A.4), is not equal to zero, unlike static fuel reactors, and this reactivity loss is only due to the fuel circulation:

145 POINT KINETICS OF MOLTEN SALT REACTORS A.3 ρ 0 = β 6 i= 1 λ i β λ i 1 e + τ i λiτel C (A.4) To obtain the expression for the transfer function of the system, the Laplace transform is the method to be used as suggested by the form of Eqs. (A.1) and (A.2), since they represent a system of delay differential equations. It is useful to rewrite Eqs. (A.1) and (A.2) in terms of dimensionless variables Ψ(t) = n(t)/n 0 and η i (t) = c i (t)/c 0i, and to express these ones in terms of their variation with respect to the values at steady-state, as shown by Eqs. (A.5): Ψ(t) = 1+ δψ(t) η (t) = 1+ δη (t) i i (A.5) ρ(t) = ρ 0 + δρ(t) Substituting Eqs. (A.5) into Eqs. (A.1) and (A.2), and neglecting the terms of second order, the linearization of the equation system is obtained, as follows: dδψ(t) δρ(t) ρ0 β = + δψ(t) + dt Λ Λ 6 i= 1 β λ δη (t) Λ i i i λ τ 1 e λi + τ C i EL (A.6) dδηi (t) = dt λ i 1 + τc 1 τ λiτel [ δψ(t) δη (t)] + e [ δη (t τ ) δψ(t) ] i C i EL (A.7) Applying the Laplace transform to Eqs. (A.6) and (A.7), the transfer function at zero-power of the system is obtained, as described by Eq. (A.8): ~ 6 i i ~ (s) δψ H(s) β λ = = sλ ρ0 + β ( s+λ (s) i ) (A.8) τ δρ EL i= 1 1 e s + λi + τc The transfer function can also be expressed in terms of the multiplication factor, instead of reactivity [i.e., ρ(t) = (k(t) 1) / k(t) and Λ = l n / k(t)], as given by Eq. (A.9): 1

146 A.4 POINT KINETICS OF MOLTEN SALT REACTORS M(s) ~ δψ ~ (s) = δk(s) 1 (1 β)k 1 β + i= 1 = sl n βiλi ( s+λi ) τel 1 e s + λi + τc βiλi k0 ( s+λ ) i= 1 1 e s + λi + τ C i τel (A.9) A.2 Comparison with MSRE experimental data The model presented in the previous paragraph has been validated on the basis of the experimental data available from the MSRE performed at ORNL. In particular, reference is made to a series of experiments carried out to obtain the frequency response of the reactor (i.e., the nuclear power to reactivity ratio in terms of both amplitude and phase) by means of different types of disturbance (Kerlin and Ball, 1966). A brief description of the MSRE history can be found in Chapter 1, while the reference data adopted in the analyses are briefly resumed in Table A.1. Table A.1. MSRE reference data. Physical quantity Value Unit β pcm β pcm β pcm β pcm β pcm β pcm λ s -1 λ s -1 λ s -1 λ s -1 λ s -1 λ s -1 β pcm l n s L C 2.0 m L EL 4.0 m 8.5 s # τ C # τ EL 17 s h FG A W K -1 M F C P,F J K -1 M G C P,G J K -1 C P,F J kg -1 K Γ 171 kg s -1 P MW α F 8.46 pcm K -1 α G 4.68 pcm K -1 # Value corresponding to the reference condition, taken from (Ball and Kerlin, 1965).

147 POINT KINETICS OF MOLTEN SALT REACTORS A.5 The comparison is shown in Fig. A.1 in the case of circulating fuel, and in Fig. A.2 in the case of static fuel (i.e., τ C = τ EL = 0), where the results of a more refined and complex model developed in a previous work at ORNL (Ball and Kerlin, 1965) are also shown. The ORNL model includes two energy groups for neutron diffusion, xenon poisoning reactivity feedback, heat transfer to the pipes of the primary and secondary loops and finite difference modelling to compute the thermal reactivity feedback and the behaviour of heat exchangers. It has been applied at zero-power conditions, in order to compare its predictions with the results achieved in the present work. a) b) Fig. A.1. Frequency response of k 0 M(s) in the case of circulating fuel: a) amplitude; b) phase.

148 A.6 POINT KINETICS OF MOLTEN SALT REACTORS a) b) Fig. A.2. Frequency response of k 0 M(s) in the case of static fuel: a) amplitude; b) phase. The agreement of the results of the present model with the experimental data as well as with the ORNL model is very satisfactory: the frequency response at zero-power can be successfully described by means of the simple model discussed in the previous paragraph, pointing out that the assumption of the linear response of the system is certainly adequate. Little discrepancies between experimental data and theoretical predictions are due to uncertainties of the measured values caused by equipment limitations (Kerlin and Ball, 1966).

149 POINT KINETICS OF MOLTEN SALT REACTORS A.7 A.3 Reactor stability at zero-power The stability of a CFR at zero-power has been analysed by means of the Nyquist criterion as function of the coolant velocity, which is the most relevant parameter acting in these conditions: together with the length of the primary loop (L C +L EL ), it affects the precursor circulation time in the primary loop (τ C +τ EL ), and, as a consequence, the reactivity loss at steady-state (ρ 0 ) and the system frequency response. The kinetics model of a CFR works as a closed loop system, whose transfer function, H(s), given by Eq. (A.8), can be represented in terms of an open loop transfer function, L(s), with a feedback equal to 1 (see Fig. A.3) in order to apply the Nyquist criterion: L(s) H(s) = (A.10) 1+ L(s) Fig. A.3. Flow diagram of the closed loop representation of the primary circuit. The function L(s) can be expressed as the product of the transfer function describing the static fuel, G(s), and the gain α(s), which takes into account the precursor movement along the primary circuit and depends on the coolant velocity, according to the following equations: L(s) = α(s)g(s) (A.11) 1 6 βiλ i G(s) = sλ + β (A.12) i= 1 s + λ i Substituting Eqs. (A.8) and (A.11) into Eq. (A.10), the gain α(s) can be found, and finally the open loop transfer function can be written as follows:

150 A.8 POINT KINETICS OF MOLTEN SALT REACTORS 6 i i L(s) β λ = sλ ρ0 1+ β (A.13) ( s+λi ) τel i= 1 1 e s + λ i + τc The effect of the coolant velocity has been investigated with respect to a reference system configuration characterized by the adoption of the operating coolant velocity of the MSRE (u ref = 0.24 m/s), evaluated as u ref = L C /τ C = L EL /τ EL from the circulation times given in (Ball and Kerlin, 1965) (see Table A.1), which lead to a reference reactivity (ρ ref = 246 pcm) by means of Eq. (A.4). Assuming that the system has an initial reactivity equal to the reference reactivity (ρ 0 = ρ ref ), the system stability has been analysed varying the coolant velocity (u) with respect to its reference value u ref, and consequently the circulation times (τ C = L C /u, τ EL = L EL /u) in Eq. (A.3). The Nyquist diagram of the open loop transfer function for different values of coolant velocity has been achieved by means of MATLAB (The MathWorks Inc.), adopting the Padè approximation for the exponential term, exp( sτ EL ) (Fig. A.4). 1 Fig. A.4. Nyquist diagram of L(s) for different values of coolant velocity. Since the open loop transfer function has a real positive pole, the Nyquist criterion implies that the closed loop system is stable when the velocity is greater than its reference value, and

151 POINT KINETICS OF MOLTEN SALT REACTORS A.9 unstable when it is lower, as confirmed by the reactivity step response of the system given in Fig. A.5. When the velocity is equal to u ref, the Nyquist plot intersects the point (-1,0), and thus the closed loop transfer function has a pole in the origin, due to the linearization of the equation system, as shown in Fig. A.6, where the dominant pole is plotted as a function of coolant velocity. Fig. A.5. System response to a reactivity step (1 pcm). Fig. A.6. Velocity dependence of the dominant pole value (closed loop transfer function).

152 A.10 POINT KINETICS OF MOLTEN SALT REACTORS In short, if the coolant has such a velocity that the correspondent reactivity loss is lower/higher than the initial reactivity (or reference reactivity), the system is super-critical/sub-critical. In fact, the coolant velocity has a strong impact on the reactivity loss as shown in Fig. A.7: evaluating the reactivity loss by means of Eq. (A.4) at the circulation times τ C and τ EL corresponding to different values of coolant velocity, it can be noticed that there is a strong variation in the trend up to a velocity of ~2 m/s, which induces ~400 pcm. Moreover, if the velocity tends to infinity, the reactivity loss tends to zero, that is the situation of a static fuel. Fig. A.7. Reactivity loss as a function of coolant velocity. This behaviour gives some preliminary, but useful indication for the system design as well as for the control strategy to be adopted, though in the present model the power effects relevant for an adequate evaluation of both operational and accidental transients are not considered. As a matter of fact, during the start-up of the molten salt circulation, the reactivity strongly decreases, according to Fig. A.7, and such a loss has to be counterbalanced in a suitable way by means of neutron absorbers in the reactor; vice versa, during the reactor shut-down, the significant insertion of reactivity has to be carefully removed by the same neutron absorbers. Therefore, in both these operational transients the strong variation of the reactivity for velocities lower than 2 m/s has to be opportunely managed by the control systems.

153 POINT KINETICS OF MOLTEN SALT REACTORS A.11 A.4 Complete point kinetics model In order to take into account the temperature feedback effects on MSR dynamics, the following two equations for the fuel salt and graphite average temperatures must be introduced, respectively: M M F G C C P,F P,G dt dt F dt dt G ( T T ) + h A( T T ) + ζ P = ΓC (A.14) P,F OUT IN FG ( T ) + ζ P G F = h A T (A.15) FG G A schematization of the reactor core is given in Fig. A.8. F F F Fig. A.8. Schematization of the graphite-moderated MSR core. As usual in point kinetics models, the introduction of the time constants for the fuel salt (Eq. (A.16)), the graphite (Eq. (A.17)) and the transport phenomena within the fuel (Eq. (A.18)), and the imposition of a constraint for the average fuel temperature (Eq. (A.19)), have been adopted: τ MFCP,F F = h A (A.16) FG MGCP,G τ G = (A.17) h A τ 0 M = Γ F FG (A.18)

154 A.12 POINT KINETICS OF MOLTEN SALT REACTORS T F TOUT + TIN = (A.19) 2 Substitution of the foregoing expressions into Eqs. (A.14) and (A.15) leads to the following expressions for the fuel salt and graphite average temperatures: dt dt dt dt 1 ( T T ) ( T T ) 2 ζ P F F = G F F IN + (A.20) τf τ0 h FGAτF 1 ( T T ) ζ P G F = G F + (A.21) τg h FGAτG The steady-state solution for the heat transfer problem (A.20) and (A.21) is the following: T F,0 τ0p0 = TIN + (A.22) 2τ h A F FG ζg τ0 TG,0 = TIN + P0 + (A.22) h FGA 2τFh FGA The heat transfer and the neutronics models are coupled through the power, P, which can be expressed in terms of neutron density, assuming a proportional dependence, as follows: ( t) n P = P (A.24) 0 n 0 while the system reactivity depends also on the thermal feedback contributions due to graphite and fuel salt: ( t) = ρ ( t) + α ( T T ) + α ( T T ) ρ (A.25) CR F F F,0 G G G,0 Analogously to the zero power case of Paragraph A.1, the total transfer function of the system expressed in terms of neutron density response to reactivity variation, where the inlet temperature is considered as a parameter and do not depend on time can be obtained by means of linearization of Eqs. (A.20) and (A.21), and, subsequently, by means of the application of the Laplace transform. For the sake of simplicity, a detailed formulation of the mathematical passages is omitted, also because it deals with algebraic substitutions. By adopting the above procedure, the following result is obtained:

155 POINT KINETICS OF MOLTEN SALT REACTORS A.13 ~ 6 * δψ(s) i i H (s) β λ = ~ = sλ ρ0 + β ( s ) FG F(s) GG G (s) +λi τel CR (s) α α (A.26) δρ i= 1 1 e s + λi + τc where 1 1 ζgp0 /(h FGA) ζfp0 G F (s) = + ξ(s) sτg + 1 h FGA (A.27) 1 1 ζ ζ GP0 /(h FGA) ζ FP0 GP0 GG (s) = ( sτg + 1) + + ξ(s) sτg + 1 h FGA h FGA (A.28) sτ τ G S ξ (s) = sτf (A.29) sτg + 1 τ0 Equation (A.26) can be expressed in terms of multiplication factor as follows: ~ * δψ 1 ~ (s) M (s) = = [ M(s) α ] 1 FGF(s) αggg (s) (A.29) δk(s) An example of the transfer function of the system in terms of amplitude and phase is given in Fig. (A.9), where it was obtained in the conditions of the MSRE according to data reported in Table A a) b) k 0 M * (s) Frequency (rad/s) Phase (deg) Frequency (rad/s) Fig. A.9 Frequency response of k 0 M * (s): a) amplitude; b) phase.

156 A.14 POINT KINETICS OF MOLTEN SALT REACTORS A.4.1 Linear vs. non-linear approach A brief sensitivity analysis is performed in this subparagraph in order to investigate the validity of the linear approximation previously adopted for studying the frequency response of the system as well as its stability as a function of fuel velocity. For this purpose, the nonlinear equation system represented by Eqs. (A.1), (A.2), (A.20) and (A.21) is solved numerically by means of the delayed differential equation solving tool of MATLAB (dde23) for different reactivity prompt jumps up to 300 pcm. The results are compared with those obtained by means of the linear equation system, as shown in Figs. (A.10) and (A.11) Non-linear approach Linear approach Amplitude ρ=5 pcm 0.01 ρ=1 pcm ρ=0.5 pcm Time [s] Fig. A.10 System response of the linear and non-linear approaches to reactivity jumps of 0.5, 1, and 5 pcm. 9 Non-linear approach Linear approach Amplitude 6 3 ρ=300 pcm ρ=50 pcm Time [s] Fig. A.11 System response of the linear and non-linear approaches to reactivity jumps of 50 and 300 pcm.

157 POINT KINETICS OF MOLTEN SALT REACTORS A.15 As a result, until 50 pcm of prompt jump, there are no significant differences between a nonlinear and linear approach, whereas above 50 pcm the discrepancies become larger. As expected the system response is practically linear for small variation around the steady-state solution. On the other hand, when the reactivity step becomes significant (see the 300 pcm curve in Fig. A.11) the system response cannot be described by means of a linear approximation of the system. A.5 Concluding remarks In this appendix, a preliminary approach to the simulation of the CFR dynamics has been set-up, focusing on the effects of the coolant velocity on the reactor stability. It is based on the point neutron kinetics equations modified in order to take into account the coupling with the molten salt (fuel/coolant) circulation. Results of the adopted model in terms of frequency response of the system have been compared with the experimental data available from the MSRE and with a more complex model previously developed by ORNL. As a result, the agreement is good and points out that the system dynamics at zero-power can be successfully described by means of the proposed simple model. The system stability analysis has demonstrated that the effect of the coolant velocity is important, since it influences the reactivity loss: this effect is due to the fuel/coolant circulation time in the primary loop, and has to be carefully taken into account for an appropriate control strategy in case of operational transients, like the start-up and the shut-down of the reactor. Finally, the sensitivity analysis performed to investigate the validity of linear approximation, has suggested to use the linear approach to neutron kinetics only for low reactivity variation from steady state solution (up to 50 pcm). In the case of significant reactivity prompt jumps (above 50 pcm), the system has shown a strong non-linear response. The model here proposed is highly simplified but useful to analyse the kinetics of molten salt reactors as it is revealed representative of MSRE conditions. In addition, it has offered an reference solution for the interpretation of numerical results achieved by means of COMSOL throughout this thesis as concerns both the neutronics assessment performed in Chapter 3 and the transient analyses carried out in Chapter 4.

158

159 Appendix B Thermal-hydraulic analysis of the MSBR core In this appendix, a preliminary thermal-hydraulic analysis of the Molten Salt Breeder Reactor core is carried out in order to obtain the main parameters for the single channel analysis presented in the thesis. The thermal-hydraulic data are calculated so as to reproduce the average conditions (in terms of power density, mass flow rate and channel size) achieved in the zone I of the MSBR core, starting from the early design analysis performed at ORNL in the 1970s.

160 B.2 THERMAL-HYDRAULIC ANALYSIS OF THE MSBR CORE B.1 Reference MSBR data The main reference design parameters of the reactor core (see Tables B.1 and B.2) are taken from the conceptual design study of the single-fluid MSBR performed at ORNL by Roberson (1971) and from the subsequent report on MSBR control studies (Sides, 1971). Table B.1. Main parameters of the zone I of the MSBR core. Parameters Values Total power (MW th ) 1830 Fuel salt volume (m 3 ) 7.9 Volume fraction of fuel salt (/) Number of graphite elements 1466 Inlet-outlet temperature difference (K) 138 Core height (m) 3.96 Core peak power density (kw/l) 70.4 Graphite peak power density (kw/l) 6.35 Table B.2. Thermo-physical properties of fuel salt and graphite at 908 K. Properties Fuel salt Graphite ρ (kg m -3 ) C P (J kg -1 K -1 ) K (W m -1 K -1 ) η (kg m -1 s -1 ) 0.01 / ORNL has released a consistent large amount of information for the MSBR design. Anyway, the missing data in the above tables is represented by the fraction of heat generated by graphite, which is needed for the calculation of power densities. This parameter has been calculated starting from the results of neutronics analyses performed by ORNL (Robertson, 1971), which are reported in Fig. B.1. The curves relative to graphite (curve B in the pictures) have been fitted assuming a cosine function of both the axial and radial coordinate, only in the zone I of the core, as follows: neutrons MW πr πz QG = 1.75 cos cos (B.1) 3 m gamma heating MW πr π z QG = 4.6 cos cos (B.2) 3 m gamma heating ( / 2) r ( / 2) neutrons MW π π z QG = QG + QG 6.35 cos cos (B.3) 3 m

161 THERMAL-HYDRAULIC ANALYSIS OF THE MSBR CORE B.3 a) b) Fig. B.1. Neutron and gamma heating in the core mid-plane (a) and near the core axis (b) of the MSBR.

162 B.4 THERMAL-HYDRAULIC ANALYSIS OF THE MSBR CORE As concerns the total core power density, Q TOT, a cosine trend is also assumed in both the radial and axial direction, as suggested by Robertson (1971), considering an extrapolated height and length of the core equal to H estr = cm and R estr = 506 cm, respectively, as follows: ( / 2) r ( / 2) MW π π z TOT = 70.4 cos cos (B.4) m Q 3 Now, it is possible to calculate the average value of the graphite to the total power density ratio, γ G, as the ratio of the average graphite heat generation to the average core power density, as follows: Q Q V G VTOT TOT γ G = = (B.4) Q Q V av G av TOT V TOT TOT TOT Since zone I of the MSBR core can be approximate as a cylinder with a height of 396 cm and a diameter of 440 cm, Eq. (B.4) can be evaluated, assuming the same extrapolated lengths for the total power generation and power generated into graphite, as follows: peak QG 1 π r π z γg cos cos dvtot = 0.09 peak QTOT V TOT R estr H (B.4) estr V TOT This means that in the zone I of the MSBR core about 9% of the total power density is generated into graphite. B.2 Thermal-hydraulic parameters of the MSBR core channel From data reported in Table B.1, it is possible to evaluate the total volume, V TOT, the graphite volume, V G, and the fuel to graphite volume ratio of zone I of the MSBR core, as follows: V 3 = 59.85m (B.5) F V TOT = V G = VF = 51.95m (B.6) V F = (B.7) V G

163 THERMAL-HYDRAULIC ANALYSIS OF THE MSBR CORE B.5 The average power densities of zone I of the MSBR core, of graphite and of the fuel salt are the following: P 1830MW MW = 30.6 (B.8) 3 m av TOT Q TOT = = 3 VTOT 59.85m av av MW Q G = γgqtot = 2.75 (B.9) 3 m ( V γ V ) MW = (B.10) 3 m av av TOT G G Q G QTOT = VF The zone I passage flow area, A flow, the mass flow rate, Γ, and the average fuel velocity u avg are evaluated as follows: 3 VF 7.9m A flow = = 2 = 1.995m (B.11) H 3.96m Γ = C P,F PTOT 7 T IN OUT = kg /s= kg / h (B.12) u Γ = 1.47 m / s (B.13) ρ A avg = F flow Now, the average channel radius of the zone I of the MSBR core can be calculated starting from the passage flow area and the number of the core channels, N ch, as follows: R Aflow = m (B.14) N π 1 = ch As concerns the geometrical representation of the average core channel of the MSBR core, two different geometries are adopted in this thesis, as shown in Chapter 3, i.e.: cylindrical shell approximation and square shape approximation. In the first case, the external radius of graphite can be evaluated starting from the fuel to graphite ratio of Eq. (B.7), as follows: R VG = R m (B.15) V 2 = F In the case of square shaped channel, the graphite thickness results:

164 B.6 THERMAL-HYDRAULIC ANALYSIS OF THE MSBR CORE L ( 1+ V V ) 1 2 F G = πr m (B.16) 2 V V G = F G

165 Appendix C Pipe flow numerical analysis and validation In this appendix, further results concerning the assessment of COMSOL numerical analysis of the heat transfer of fluids with internal heat generation in circular pipes are provided both in laminar and turbulent flow. With respect to Chapter 3, some additional details regarding the near wall modelling of COMSOL are also given. Results in terms of velocity and temperature profiles are presented for different values of Prandtl and Reynolds numbers, different turbulence models and exploring two kind of boundary conditions at the pipe wall, i.e.: uniform wall temperature and uniform wall heat flux.

166 C.2 PIPE FLOW NUMERICAL ANALYSIS AND VALIDATION C.1 Laminar flow The assessment of numerical solutions provided by COMSOL in laminar flow is performed by means of a comparison with the analytic solution and FLUENT results. For brevity, the presented analyses are limited to the case of fully developed flow and under simultaneously uniform wall heat flux and fluid internal heat generation. In such conditions, instead of using the generalized analytic approach discussed in Chapter 2, the analytic solution developed by Poppendiek (1954) can be adopted as alternative and is preferable for its simplicity. Under the above mentioned flow conditions, the general heat transfer problem (see Eq. (2.1)) reduces to the following one-dimensional formulation, where imposed wall heat flux is considered: d dr υ + ε Pr H dt r dr ( r) = u avg u ρc P Q 2 r 0 j W r Q r ρc P (C.1) j W K(dT dr) r= = (C.2) ( r r D ) TD r 0 T = = (C.3) where the second boundary condition, expressed by Eq. (C.3), is some reference temperature T D such as wall, centre-line, or mixed-mean fluid temperature. The solution of the boundary-value problem defined by Eqs. (C.1), (C.2) and (C.3) was achieved by Poppendiek (1954), considering that the eddy diffusivity is null in laminar regime and the fluid velocity attains a parabolic profile along the pipe radius once the hydrodynamic pattern is established. The solution is given by Eq. (C.4): ( r) T T(0) 2F 1 = Q r r r 2 0 2K r0 2 4 F r (C.4) where T(0) is the centre-line temperature and F 1 ( 2j Q ) =, namely {1 fraction of W r 0 heat generated within moving fluid that is transferred at wall} (El-Wakil, 1978). The dimensionless radial temperature profile given by Eq. (C.4) is plotted in Fig. C.1 for several values of the function F, and compared with the CFD simulations results attained by means of COMSOL and FLUENT. As expected, no problems occur in the case of laminar flow and both numerical solutions are practically superimposed to the analytic one.

167 PIPE FLOW NUMERICAL ANALYSIS AND VALIDATION C.3 (T-T(0))/(Q r 2 0 /2K) FLUENT COMSOL Eq. (C.4) F=2 F=1 F=3/4 F=1/2 F= r/r 0 Fig. C.1. Comparison between the different evaluations of the dimensionless radial temperature profile in a pipe with laminar flow. C.2 Turbulent flow The present paragraph is focused on turbulent heat transfer in smooth circular pipes with constant flow section. The adopted geometry is axial-symmetric and the use of a twodimensional domain is made possible. A typical mesh example set-up by means of COMSOL is given in Fig. C.2. Fig. C.2. Mesh example of the lower part of the pipe.