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Nuclear Reactor Core as a Multiphysic System

In this section, the concept of multiphysics of nuclear reactor cores is introduced from a physics-based point of view. In the first part, some general examples are provided for other multiphysic systems and for nuclear reactors in general. In the second one, a more detailed analysis is done for the PhD thesis context: a PWR core along irradiation. The fundamental elements that define this system are reported for every concerned discipline and the main interdependencies among physics are introduced.

Multiphysic Issues in General

The most characterizing keyword of this PhD thesis is multiphysics, which in this context refers to multiphysic modelling. The general definition of such a word is quite intuitive, as it simply corresponds to any model accounting for multiple physics. However, the actual meaning in the context of reactor physic simulations is something more specific that needs several concepts to be defined beforehand, hence, a more precise definition is given later in the introduction. The majority of engineering problems could suite the wide definition of multiphyisic problem. For instance, let us consider an electric cable: the electromagnetic field and the heat transfer are largely influencing each other. Therefore, both these physical phenomena have to be addressed in order to find an accurate solution of the problem. In this example, the transport of electrons through the wire implies a certain heat generation according to the electrical resistance of the cable. On the other hand, the electrical resistance itself depends on the heat conduction, as the resistivity of the materials is determined by the local temperature. Some examples of modelling strategies for this problem are given in [15, 16]. Another typical illustration is the simulation of the wing of an aeroplane, where fluid and structural dynamics are strongly coupled. To accurately model the fluid dynamics, the temperature and the deformations fields of the wing are needed. At the same time, the structure dynamics compute these quantities using the pressure field and the heat transfer coefficient as input. Hence, in many aerospace applications these disciplines are treated together, for instance as in [17]. One further example is the multiphysic modelling of the cardiac function, like in [18, 19]. In this case, in order to reproduce the heart’s behaviour under given conditions, the biomedical branches of fluid mechanics, solid mechanics and electromagnetism are combined together.
Nuclear reactors are intrinsically multiphysic systems. Indeed, in most of the power reactors, the neutron transport is exploited to generate heat, which for the majority is generated in the solid fuel, is transmitted to the coolant by conduction and is removed by the latter through forced convection. The heat generation naturally affects the temperature and density fields of the materials. In turn, the different neutron reactions probabilities depend on both these quantities. This is a crucial aspect of nuclear reactors, which has been exploited to obtain a system that is self-stabilizing in most of the scenarios. In the sense that a power excursion would reduce the probability for neutrons to undergo fission, hence, the power would receive a negative feedback intrinsically opposing to any divergent behaviour. Moreover, depending on the time scale, the effect of neutron transport can be also measured in terms of isotopic change mostly in the fuel, i.e. neutron transmutation. At the start-up of a reactor, effects can be seen already on a small time scale, in the order of minutes. This is linked to the intrinsic nature of fission, a heavy element (like uranium-235) is split into (in most of the cases) two fission products. These fission products can be or rapidly become neutron poison, in the sense that elements like xenon-135 and and samarium-149 are characterized by extremely high capture cross-sections1 that have a measurable impact on the reactivity (this quantity is mentioned
−1 as := ) of the core even for small variations in their concentrations. On a longer time scale, the reduction in the content of uranium-235 due to neutron absorption (i.e. capture + fission) and the production of new fissile isotopes like plutonium-239 and 241 consequently to multiple neutron captures on uranium-238 have to be considered. Furthermore, the neutron transport and the operating conditions impact also the chemistry and the mechanics of the fuel, leading to variations in the geometry and thermal-mechanical properties. These changes in turn may affect the neutronics. Therefore, depending on the scenario to be modelled, the neutronic modelling may require to address also isotopic depletion, themal-hydraulics, heat conduction, thermal-mechanics and eventually other disciplines related to the fuel modelling.

Multiphysics of PWRs along Irradiation

In this sub-section, the multiphysic modelling of PWR along irradiation is introduced. To do that, some of the most fundamental elements of PWR modelling of the core are introduced from a single-physic perspective and finally an overview of their interactions focusing on the neutronics is given.
This PhD thesis focuses on the modelling of the nuclear reactor core of the PWR type. This reactor design is largely the most widespread one for energy production. According to [20], in 2016, 289 of the 448 operational nuclear power reactors are PWRs. Nowadays, nuclear energy represents the second-largest source of low-carbon electricity after hydroelectric [21, 22]. By producing about 10 % of the total electricity generation, corresponding to 2700 TWh (data for 2018 [21]), it is one of the main actors in the fight against climate change. However, nuclear is facing a strong decline in western countries, where the number of operational reactors is decreasing. The steep raise in renewable power has been just sufficient to replace nuclear: the low-carbon share in the total energy production has not increased in the last two decades (36 % both in 1998 and 2018[21]). In France, this is even more a crucial topic, since about 79.6 % of the energy production is from nuclear reactors [23] and all of them are PWRs [24]. Only one reactor is under construction in France, the EPR of Flamanville. Designed and developed by Framatome (ex Areva-NP) and Électricité de France (EDF), it is considered as a third generation PWR.
1The cross-section quantifies the probability of a given reaction (in this case radiative capture) among neutrons and the nucleus of the considered isotope.
As the majority of the operating reactor designs, PWRs are thermal reactors using low-enriched uranium dioxide as fuel, which means that each uranium isotope is bounded to two oxygen-16 atoms (UO2). By low-enrichment, it is meant that, in the fresh fuel, uranium-235 represents from 2 to 5 % of the total uranium nuclei, while the rest is mainly uranium-238. The fundamental principles, on which thermal reactors are based, are described by the simple tough very popular Fermi’s four-factors formula, which is reported in Eq. (1.1). It was firstly used to predict the critical mass of the famous first atomic pile, while, in this paragraph, this formula is just used to introduce the main stages of neutron’s life in a thermal reactor.
The main principle behind thermal reactors is indeed the thermalization, which is the process of slowing the neutron down to thermal energy, which is about 0.025 eV and corresponds to the equilibrium speed of the medium, which is defined by the temperature. This slowing down is obtained through the use of a moderating material. In PWRs, this material is the water, the very low mass of hydrogen nuclei allows to absorb large portions of the neutron kinetic energy at every scattering collision. This property is one of the reasons that makes the water a good moderator even if its capture cross-section is not so small. Each fission produces on average 2.5 new neutrons which could be classified as fast, as their average kinetic energy is about 2 MeV.
In Eq. (1.1), the infinite multiplication factor ( ∞) represents the global neutron balance (defined as the ratio of the neutron population over two consecutive generations) without accounting for the neutron leakages. The fast fission factor ( ) is the number of neutrons slowing down below 1 MeV per each neutron produced by thermal fission. Its main role is to account for the portion of neutrons that may cause fission before slowing down to 1 MeV. The neutron slowing down needs to happen with the lowest number of collisions in order to avoid the so-called resonances. The resonances are energy intervals in which the capture probability increases sharply. For simple isotopes, the presence of these peaks of the interaction probability can be explained by the minimization of the kinetic energy of the products of the considered reaction. For this reason, they occur at slightly higher energies than the excited levels of the target nucleus. The fission and radiative capture cross-sections of uranium-235, which is the main fissile isotope in fresh fuel, are available in Fig. 1-1. Since the resonances constitute a major filter in neutron slowing down, their presence is accounted by , the probability to escape the resonances.
Once the neutrons are thermalized, the main turning point is whether they are absorbed in the fuel or in other materials, this is represented by , the thermal utilization factor. The reproduction factor ( ) is the average number of fission neutrons produced per neutron absorbed in the fuel. With the neutron’s life-cycle in an infinite thermal reactor is closed. To account for the finiteness of the reactor’s size, the escape probability should be added for fast and thermal neutrons.
However, the heat generating system of the Rankine loop is not the reactor itself. In fact, the Rankine loop constitutes a secondary circuit, while the core power of the reactor is extracted by a primary loop that transfer it to the secondary one through the steam generator. In this way, eventual radioactive leakages from the fuel rods would be kept in the primary circuit, resulting in several design simplifications for the secondary loop. The core power is removed by a mostly vertical water flow through the fuel rods, directly cooling the cladding outer surface. As anticipated by the name of the reactor type, this water flow is highly pressurized, at about 15.5 MPa and 300 oC. The water mass flow rate through the core is very large, several millions of kilograms per hour in standard operation. Such a flow rate at this high pressure requires pumps consuming few Megawatts of electricity, but it allows the extraction of few thousands of Megawatts of heat. In this way, all the power can be removed with a small temperature increase of the water across the core, about 30 K. Such a low temperature increase contributes to making the reactor more homogeneous and maximize the thermodynamical efficiency.
Under nominal conditions, the water of the primary circuit stays close to saturating conditions but always below. Just some subcooled boiling may locally happen due to the temperature gradient caused by the heat flux. This phenomenon affects just a small portion of the water, where the steam may reach few per mille in volume, while the channel average temperature is well below saturation. Nevertheless, as stated in [28], it is important to model this void formation also during normal operation as it heavily affects the heat transfer coefficient. Indeed, the subcooled boiling is a very effective heat transfer mechanism, but pushing the thermal-hydraulic conditions too far, boiling crisis may occur. When this happen, the heat transfer rapidly and strongly deteriorates and the cladding temperature may overcome the one imposed by safety limits. This problem is quite typical of nuclear power plant as the heat removal mechanism is power dominated, in the sense that the heat source is controlled and the cladding temperature depends on the cooling capability. In combustion plants instead, the maximum temperature is controlled, as it is determined by the flue gas and the heat removed depends on the efficiency of the process.
A simple model to interpret this phenomenon is given for the pool boiling by the famous ex-periment of Nukiyama [29]. Pool boiling occurs when the fluid is globally at rest, even if natural convection may induce some local movements. In PWRs, the mechanism is rather flow or forced-convective boiling, in the sense that the boiling happens with the fluid circulating through the core. However, this experiment has been central in the qualitative description of the boiling crisis in general, the main plot is reported in Fig. 1-5.
The main outcome of this study is that, when controlling the heat flux as in the figure, a sudden temperature increase of the cladding outer surface is expected above a certain power level. The quantity on the x-axis is the wall superheat temperature, which corresponds to the difference between the wall temperature and the saturation temperature of the water. The wall temperature Figure 1-5: Nukiyama curve describing the boiling crisis as a transition from different heat transfer mode. It should be noticed that PWRs heat removal mechanism is power controlled (instead of temperature controlled as in most of the conventional plants). The boiling crisis happens when moving from the to the point: for a given heat flux, the wall superheat temperature suddenly increases. Source [4].
is the temperature at the interface between the water and the solid material, which in PWRs corresponds to the temperature of the cladding outer surface. In this plot, this sudden increase happens when moving from the point to the one and the temperature increase is very large even on a logarithmic scale. Although this curve describes a boiling crisis happening while leaving the saturated boiling condition (i.e. when the average water temperature is above saturation), such a crisis may happen also during subcooled nucleate boiling under PWR’s operating thermal-hydraulic conditions. In PWRs this phenomenon is called Departure from Nucleate Boiling (DNB), as it happens when leaving the nucleate boiling.

Fuel Performance of PWRs

The integrity of the cladding of the fuel rods is very important for the safety and the economy of reactors. The age of fuel rods is generally measured with the burnup. This quantity is also called fuel utilization as it quantifies the energy that is extracted per unit mass of heavy metal, which in case of uranium dioxide, is the mass of uranium. The typical burnup range found in reactors is from 0 to 60 MWd/kg. Below 30 MWd/kg, the fuel pellets are capable of preventing the release of the majority of the radioactive fission products. The main leakage mechanism for low exposures is the “knockout”, which corresponds to the release of gaseous fission products close to the outer fuel surfaces as a consequence of collisions with other elements. The expelled elements reach the allocated free space, called plenum. Beyond 30 MWd/kg other mechanisms become predominant and cause releases up to the 5 % of the total fission products, as described in [3], pages 17 to 19. The radioactive elements accumulated in the plenum are kept within the fuel rod by the cladding, which can consequently be considered as the first barrier between the radioactivity and the biosphere. Even if several other barriers are present, like the reactor pressure vessel and the containment building, cladding failure may imply the sharp increase of the radioactivity level in the primary circuit and the reactor shutdown, which entails considerable economic losses.
Fuel rods mechanical integrity is a limiting factor in terms of plant’s power uprate, speed in reactor start-up and maximum achievable burnup. Therefore, a lot of attention is put into its modelling, which is often referred to as fuel performance or fuel behaviour. The fuel performance is a multiphysic problem in itself, because ideally it requires the full solution of neutronics, thermal-hydraulics, heat conduction, fuel chemistry, fuel mechanics and fuel transmutation. Fortunately, not every single aspect of fuel performance has an important influence on neutronics. What matters the most is the estimation of the temperature distribution, which may be strongly affected by the change of fuel thermodynamic properties. Hence, to improve the prediction of neutronic quantities it is important to model the behaviour of the fuel-cladding heat transfer coefficient along exposure. Even if the fuel and the cladding are separated just by a thin layer (a maximum of 80 m) of helium, which has a relatively high conductivity for a gas, this layer accounts for a big portion of the total thermal resistance, hence, it has a big influence on the fuel temperature. A typical radial temperature profile in a fuel rod at 200 W/cm is reported in Fig. 1-6.
Figure 1-6: Typical temperature profile assuming 280 oC of water bulk temperature, 200 W/cm of linear power and typical PWR thermodynamic properties. Courtesy of [5].
During a fuel rod lifetime, the fuel-clad heat transfer coefficient may vary from 5’000 W/m2/K to 200’000 W/m2/K, as confirmed by [10], pages 7 and 8. The peak value corresponds to the moment when the fuel and the cladding begin the mechanical contact. The main phenomena driving the gap size are described in [30]. Basically, at the very beginning of the power ramp, the fuel expands due to instantaneous thermal expansion. The same mechanism summed to the increase in the rod pressure, which derives from the raising temperature, make the cladding displace outwards. In early irradiation, the fuel volume decreases due to the densification: the higher temperature reduces the concentration of fuel structural defects. After this short phase, which lasts few MWd/kg, the cladding starts to creep inward (it reduces its radius at constant load due to stressful environment) and the fuel volume increase due to swelling. This latter consists in the accumulation of fission products which occupy a larger volume than the fissile elements and it is largely due to the noble gaseous fission products that may even cluster in bubbles. A contribution to swelling is also given by the -decay of transuranic elements, that similarly forms helium clusters (called helium bubbles). Due to these two phenomena, the gap width constantly decreases until mechanical contact is reached. For this reason the gap heat transfer coefficient consistently increases until about 30 MWd/kg. The only opposed process is the release of gaseous fission products with lower conductivity like xenon and krypton.
Once the gap is closed, the heat conduction is dominated by the fuel conductivity which degrades significantly due to the increase of structural defects. This process that begins when the fuel and the cladding enter in contact is called Pellet Cladding Interaction (PCI). It is an important safety concern as it endangers the cladding integrity and it is an important limiting factor to the speed of power ramps and to plants uprate. In France, many nuclear reactors perform the load following, which consists in slowly changing the power level to adapt to the grid needs. For this reason, in this country, the study of PCI is a key research topic.

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PWRs along Irradiation

Modelling PWR along irradiation means to reproduce the reactor’s behaviour under normal op-erating conditions over long time intervals that require to account for the isotopic transmutation. This type of simulations is also called depletion calculations as they also address the consumption of fissile material. Under the typical operating conditions, the reactor power ramp is sufficiently slow to allow to model the scenario as a sequence of steady-states characterized by evolving isotopic concentrations and power levels. Therefore, the fundamental mode of the neutron flux is researched, i.e. the flux that perfectly satisfies the steady-state equation. In respect of the thermal-hydraulics and of the heat conduction, the steady-state conditions define a state in which the energy is not accumulated in any material, hence, all the power generated in the core is removed by the coolant. In PWRs, this balance is obtained by keeping the reactor almost always under critical conditions (effective multiplication factor equal one, = 1) by adjusting the boron concentration or the control rod insertion for the fuel depletion, for the different operating conditions (e.g. power level) and for the varying thermal-mechanical properties (e.g. fuel temperature). Boron-10 is a thermal neutron poison and its presence in the water is regulated by changing the boric acid concentration. Adjusting the boron concentration allows to change the core reactivity in a rather homogeneous way, as its concentration is almost constant over the core, but it is a slow process. For quicker reactivity changes, the control rods are deployed. They are commonly assembled into blocks of rods activated together, each block has a specific purpose. The safety blocks are devoted to the core rapid shutdown. The shim blocks are used for large reactivity changes. The regulating rods are used to perform fine manoeuvres and reactivity control. The neutron absorber is generally boron or cadmium. Due to the fact that they are inserted into the core through the guide tubes, generally from above, and to their strong absorption cross-section they heterogeneously affect the neutron flux distribution in the core.
Depending on the phenomena to be modelled, the variables to be predicted and the considered scenario, different subsets of physics may be considered. In this paragraph, without descending into the modelling details, a coupling scheme for the modelling of PWRs along irradiation is introduced with the aim of showing the main interdependencies among the selected physics. The scheme is depicted in Fig. 1-7. Several schemes exchanging more variables could be conceived, but they would go beyond this introductory purpose, further analyses of additional variables to be exchanged are reported in Chapter 4.
Figure 1-7: Introductory multiphysic coupling scheme aiming at underlining the main variables that
are shared among physics. is the neutron flux, is the concentration of the -isotope, is the fuel temperature, is heat generated in the fuel, is heat generated in the water, is the water density, is the wall temperature (clad outer surface) and ′′ the wall heat flux.
The neutronics and the isotopic evolution are naturally coupled as the isotopic evolution is mainly caused by the neutron interactions with the matter and the isotopic composition is a key factor for the neutron transport. However, making abstraction, even with no neutron flux the isotopic composition of the reactor would still change due to radioactive decays, especially if Mixed Oxide Fuel (MOX) fuel is considered. By MOX, it is meant that also several plutonium isotopes are included before irradiation. Hence, they might be formalized as two different physics.
In the case of thermal-hydraulics, even if the water flow rate is very high, the heat flux deriving from the conduction through the fuel pellet and the power directly generated in the water have a significant effect on the density profiles along the channels. The water direct heating is caused by the neutrons slowing down that releases a non-negligible amount of energy. Other particles like electrons of the beta decays and photons may contribute as well as their energy may not be entirely deposited in the fuel.
In respect to neutronics, the moderator density impacts mainly two of the four factors. A density increase would imply a larger probability to escape the resonances as on average neutrons would undergo more scattering collisions before reaching again the fuel, but at the same time it would lower the fuel thermal utilization factor as more neutrons would be captured in water. Moreover, a density increase would also raise the non-leakage probability as all the collisions would become more likely. In Fig. 1-8, the two factors are reported as a function of the moderator to fuel ratio, which corresponds to the fraction of the atomic number densities (i.e. the number of atoms per unit volume) of the moderator over the fuel isotopes. These curves can be drawn asymptotically for any reactor, considering that with no moderator the probability to escape the resonances would be virtually zero, while with only moderator it would be one. The same for the fuel thermal utilization factor, which with no water it is one, while with no fuel it is zero. A maximum reactivity is found for a given moderator to fuel ratio. This point would move to higher moderator quantities if the leakage probability is included. An other perturbation is introduced if considering a varying boron concentration in the moderator, as the fuel thermal utilization factor would decrease more rapidly moving the maximum to lower ratios. Reactors with moderator to fuel ratio higher than the one that maximizes the reactivity are called over-moderated, if the contrary is true they are under-moderated. PWRs are under-moderated by design, for several reasons, mainly the following two. Under-moderation contributes to a negative power feedback, as a power increase implies a moderator density reduction that reduces the multiplication factor with a stabilizing effect opposed to power excursions. The under-moderation helps also to face Loss Of Coolant Accident (LOCA), as for the same principle the reduction in moderator density has a negative impact on reactivity. It should be noticed that, as boron concentration increases, the reactor could become over-moderated, hence, limits are imposed on the maximum levels.
Figure 1-8: Impact of density variations on the neutronics interpreted with the four-factors formula.
This plot, source [6], is also used to introduce the concept of under vs over-moderation.
The power generated in the fuel, about 97 % of the total, which is mostly coming from the kinetic energy of the fission fragments, is transferred to the cladding and finally removed by heat convection of the coolant. The remaining 3 % directly heats up the moderator as it is carried by neutral particles (a more detailed analysis is given in sub-section 4.2.1). Considering the geometry of the fuel rods, the heat is conducted to the cladding outer surface predominantly through the radial direction. As in steady state conditions all the power reaches the moderator, the cladding outer surface temperature, called wall temperature, can be determined independently from the fuel temperature profile. The direct water heating, therefore, is just affecting the heat conduction as lesser power is generated in the fuel and the wall heat flux is lower. In this context, the fuel performance has the role of modelling the evolution of the fuel geometry. In particular, it is important to forecast when there is contact between the fuel and the cladding as this has a major impact on the heat conduction.
Every cross-sections depends on the temperature. In fact, if in Fig. 1-7, the thermal-hydraulics shares only the water density with the neutronics, it is because under nominal conditions, the water pressure is constant enough to approximately associate every density to a temperature value. The reason for this simplification will become clearer when speaking about the modelling approach. Anyway, it should be considered that, for the moderator, a relative density variation has a larger impact on the neutronics than the equivalent temperature one. For the fuel, the opposite is true, because it is a solid material, hence, under normal operating conditions, density variations happen on a lower scale, but also because of the Doppler effect. This phenomenon is another negative feedback, like the moderator one, and it is of crucial importance for safety concerns. In many scenarios the fuel warms-up more rapidly than the moderator, hence, the Doppler feedback intervenes faster than the one linked to the moderator. The Doppler effect is strongly connected to the resonances, which are not so important in water cross-sections, while they are abundant in the heavy isotopes of the fuel. The global outcome of this phenomenon is that, as temperature raises, the resonant isotopes absorb more neutrons and since uranium-238 is way more abundant than uranium-235 the number of radiative captures increases much more than the one of fission events. Similarly to the famous Doppler effect of wave physics, in nuclear reactors it is driven by particles relative speed. At 0 K, the resonances appear as very sharp peaks of the absorption cross-section, few eV thick, that, as mentioned before, correspond to the small intervals of velocities that would lead the target nucleus to one of its excited levels while minimizing the kinetic energy of the products. The temperature increase flattens these peaks out, reducing the maximum values whilst preserving the integral below the curve. This flattening corresponds to an enlargement of the range of speeds that implies resonant absorptions. In theory, if the neutron flux were constant in energy within a resonance, the absorption rate would stay rigorously constant. On the contrary, the probability to be absorbed for neutrons interacting at energies close to the resonance peaks is so high that the neutron flux is strongly perturbed spatially and energetically. The sharp decrease of the neutron flux within the resonance energy intervals within the resonant media is what makes the Doppler broadening increase the absorption probability. In fact, the broadening increases the cross-section’s value where the neutron flux is higher and reduces it where it is lower. Therefore, the global outcome is a lower probability to escape the resonances, which is equivalent to a reactivity decrease. It should be noticed that the lower the kinetic energy of the incoming neutron, the more important is the nucleus speed on the relative velocity, hence the greater is the Doppler broadening. For this reason, a very important role is played by two resonances of uranium-238, which are respectively centred at 20.87 and 6.67 eV.
All this analysis is based on a sort of global approach, however, most of these quantities are in reality multidimensional fields. Therefore, the actual problem is more complex, but these elements can still help to make a simplified analysis and understand local flux variations linked to changes in other variables.

Main Issues of Multiphysic Modelling of a PWR along Irradiation

In this section, the focus is on the main issues of the multiphysic modelling of the considered type of scenario. In the first part, this is done focusing on the individual physics, while introducing the fundamental governing equations and the complexities they may hide. In the second one, the specific meaning of multiphysic modelling is given and some general issues of such an approach are introduced. Since the modelling choices are not presented yet at this stage, this part is intended to be very introductory.

Table of contents :

1 Introduction
1.1 Nuclear Reactor Coreas a Multiphysic System
1.1.1 Introduction to Multiphysic Issues
1.1.2 Multiphysics of PWRs along Irradiation
1.2 Main Issues of Multiphysic Modelling of a PWR along Irradiation
1.2.1 Single-physic Problems
1.2.2 Coupled Problem
1.3 Strategies for the Multiphysic Modelling of a PWR along Irradiation
1.3.1 Fundamental Element son Single-Physics Modelling
1.3.2 Layout of the Thesis
2 State of the Art
2.1 Modelling Choices and Data Exchange
2.1.1 Conventional Approach
2.1.2 Best-Estimate
2.1.3 High-Fidelity and Massive Parallelization
2.1.4 Main Conclusions
2.2 Numerical Methods
2.2.1 Damped Fixed-Point
2.2.2 Anderson Acceleration
2.2.3 Jacobian-freeNewton-Krylov
2.2.4 Main Conclusions
3 AvailableTools 
3.1 Neutronic Models
3.1.1 APOLLO2
3.1.2 APOLLO3
3.2 Thermal-Hydraulic Models
3.2.1 FLICA4
3.2.2 THEDI
3.3 Isotopic Evolution Model
3.3.1 MENDEL
3.4 Fuel Performance Models
3.4.2 Simplified Gap Heat Transfer Coefficient Model
3.5 Coupling Tools
3.5.1 SALOME Coupling Platform
3.5.2 CORPUS
3.6 Pre-Existing Coupling Schemes
3.7 Conclusion
4 Development of the Multiphysic Coupling Scheme for Steady-State Calculations
4.1 Problem Formalization
4.2 Modelling Choices
4.2.1 Power Generation Model
4.2.2 Thermal-Hydraulic Model
4.2.3 Depletion Model
4.3 Implementation Details
4.3.1 Neutronic Operator
4.3.2 Thermal-Hydraulic Operator
4.3.3 Depletion Operator
4.4 Chapter Conclusion
5 Analysis of the Application to a Steady-State Case Study
5.1 Definition of the Case Study
5.1.1 Definition of the Characterizing Variables
5.1.2 Analysis of the Available Case Studies
5.1.3 Description of the Chosen Case Study
5.2 Implementation of the Damped Fixed-Point Coupling Scheme
5.3 Selection of the Neutronic Model for Core Calculations
5.3.1 Decoupled Analysis of the Models
5.3.2 Selection of the Models Based on the Coupled Analysis
5.4 Application of the Complete Coupling Scheme on the Case Study
5.4.1 Implementation of More Advanced Models
5.4.2 Assessment of the importance of the Thermal-Hydraulic and Heat Conduction
5.4.3 Analysis of the impact of the Coupling Scheme on the Axial Power Profile
5.5 Chapter Conclusion
6 Numerical Optimization of the Steady-State Coupling Scheme
6.1 Analysis of the limitation of the Damped Fixed-Point Algorithm
6.2 Generalised Fixed-Point with Partial-Convergences
6.2.1 Introduction
6.2.2 Parametric Performance study
6.2.3 Analysis of the Fixed-Point Bifurcations
6.3 Assessment of the Performance of the Anderson Algorithm
6.3.1 Implementation Details
6.3.2 Comparison to the generalised Fixed-Point with Partial-Convergences
6.4 Customization of the Anderson Algorithm with Partial-Convergences
6.4.1 Strategy Based on the Single-Solver Iterations
6.4.2 Strategy Based on the Progressive Refinement of the Internal Convergence Criteria
6.5 Chapter Conclusion
7 Evolution Calculation
7.1 Integration of Burn up Dependent Thermodynamic Variables
7.1.1 Fuel Conductivity Law
7.1.2 Fuel Gap Heat Transfer Coefficient
7.1.3 Impact on the Steady-State Calculation
7.2 Research of the Target Boron Concentration
7.2.1 Definition of the Algorithm
7.2.2 Impact on the Steady-State Calculation
7.3 Depletion Calculations
7.3.1 Definition of the Multiphysic Time Evolution Scheme
7.3.2 Application to a Constant Power Irradiation Scenario
7.4 Chapter Conclusion
8 Conclusions 
8.1 Research Problem
8.2 Main Results
8.3 Discussion
8.3.1 Models Selection for Steady-State Simulations
8.3.2 Numerica Optimization of the Steady-State Scheme
8.3.3 Models Selection for Depletion Simulations
8.4 Perspectives


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