Bren Jacopo

Guanyu, Matevz, Victor, Ran, Kailun, Jerry, Haeseong, Limiao, Chi, Jee, Madhumitha Anupam Ravi

Simone Lukas

Alix Richard

Alex Nuno

Bassel, Gregoire, Hans, Isabella Ana Janet

Julia Lewis

Nick Patricia Lukas

Martijn, Isabelle, Bente, Lore, Luana, Rémi, Giovanni, Giulia, Diego, Kishen, Despi, Gégé, Hakan Ana Valentin

Niels Roger

Elena Florian

Steven

Andrea Daniel

Marianne Thank

for having chosen me as a board member and taking me on board for these unforgettable moments.

Last but not least, I would like to thank my friends and my family. A doctoral project is such a demanding investment that the out-of-the-work support is a signicant plus to endure and appreciate the adventure. Among all, I have kind thoughts to some of my friends from the kindergarden from their continuous support and friendship. Thanks you my friends from ENSTA Paris and also the Board of European Students on Technology, the next cantus is on me. Finally, I would like to thank my family and particularly my parents for their continuous love, patience and trust. This manuscript is in a way the result of their 27-year long support. vi Heat Flux for conditions applicable to experimental reactors. "What are the mechanisms involved in the boiling crisis in such conditions?" "What is the relevant parameter characterizing the ow for this problem? "What means 'transient' for the ow Boiling Crisis?"

We will seek on answering these questions, and many others, along this manuscript.

The present work is organized in four chapters. Chapter 1 is a general introduction to this work problematic. It presents the nuclear safety issue of the BORAX-type accident

One thesis. Three supervisors that I had the opportunity to work with. Thank you Marie-Christine for being my "Directrice de thèse". Your patience, dedication and priceless experience were the strong levers that helped me going through this adventure until its achievement which is this work. Thank you Nicolas, for having accepted me in this journey. Thank you for your constant enthusiasm, accurate suggestions and availability... especially even after being nominated to become the head of the laboratory. Accepting to continue the supervision of a Ph.D. student while having such a new responsibility was a tough challenge, and you fullled it perfectly. Thank you Matteo for having enabled me to integrate such an unique working environment. Your expertise and your willingness to debate and to always dig deeper were invaluable driving forces that made this work what it is. To the three of you, I would like to express my innite gratitude for your support throughout these three years. My doctorate experience, scientically and humanly speaking, would have been signicantly dierent without you. You are the big contributors that had made this hard journey not only endurable, but pleasant. I can hardly imagine better supervisors.

One thesis. Three labs and many more people to thank. I had the opportunity to live unique events and to meet many people, priceless in term of scientic discussion or friendly sharing. By now, some of you have became my colleagues and some others my friends, but all will remind unforgettable in my memory. In CEA, there are the Ph.D. students Florian, Qing-Qing, Emmanuel, Aymeric, Werner, Aurore... and my oce-mates Antoine, with whom I share weekly debates and excitement on the new One Piece chapter releases, and Nicolas with whom I initiated the "Wall of memes", an unique piece of art catharsizing our Ph.D. adventures. I also would like to have a word for the interns, who stayed only a small while but remain in my memory. v There are the rst round with Jean (my rst mate and the only one I met eating more than me), Étienne and Stéphane... The team "poulailler": Axelle, Loria, Clement, Rémi, Annamisia, Salvo, Estanislau... and nally, the last comers: Andrew, Antonin, Clément, Gabriel, Julien, Mathis and Popo. Of course I want to thank the colleagues of LMSF that I had the opportunity to know and to appreciate: Anne, Pierre-Emmanuel, Ulrich, Alain, Michel, Erell, Nikos, Pascal, Adela, Violette, Jean-Christophe, Philippe, Guillaume, Maria-Giovanna, Alan, Claire, André, Jérome, Julie, Vinod and Harshit. Beyond, I cannot forgot the LGLS who shared the building 451 and thus many many delightful coee breaks, Pierre, Daniel and all the others that I had the opportunity to meet. A particular word for Elie, behind whom in a way, I made my path: Ph.D. between CEA and LIMSI and then continuing at CEA. I expected to take your title of the best Ph.D. defense cocktail, but the pandemic saved you! Talking of LIMSI, I had the opportunity to meet so many attaching people. The Ph.D. students Raphaël, Andres, Hugues (and his taste expertise), the members of the "cage à lapins": Mebarek (and the theories), Ismail (and his arrivals), Antoine (and his jokes), Guy (my twin-mate), Jianfei, Nan and Sabrina. What about the post-docs and interns like Ziad, Etienne, Savioche, Thomas, Erwan (thanks for your Matlab bro) and all the others which whom I shared coees and meetings! I would like to thank you all! The particularly friendly and warm atmosphere that exists at LIMSI, as well as its marvelous "pots" and General Assemblies, will make me have an unhealable feeling of nostalgia. I also brought back unforgettable memories from the other side of the Atlantic. Thanks [START_REF] Weisman | Theoretical prediction of maximum heat ux in power transients[END_REF]) . . . 1.24 Mechanism scheme of Liquid Sublayer Models (Celata et al. [START_REF] Celata | Rationalization of existing mechanistic models for the prediction of water subcooled ow boiling critical heat ux[END_REF]) . . . . . 1.25 Schematic descritiption of the thinning mechanisms involved in the model of Pasamehmetoglu. The blue line represents the case when the hydrodynamic mechanism dominates and the red line the case when the thermal mechanism dominates (Adapted from Pasamemhetoglu al. [START_REF] Pasamehmetoglu | Critical Heat Flux modeling in pool boiling for steady-state and power transients[END_REF]) . . . . . . 1.26 Application of the model to experimental data of Kataoka et al. [START_REF] Kataoka | Transient boiling heat transfer under forced convection[END_REF] and its sensibility to the exponent n (Pasamemhetoglu al. [START_REF] Pasamehmetoglu | Critical Heat Flux modeling in pool boiling for steady-state and power transients[END_REF]) . . . . . . . . . 1.27 Transient CHF classication proposed by Chang et al. (Ahmed et al. [START_REF] Ahmed | Towards understanding the critical heat ux for industrial applications[END_REF] adapted from Chang et al. [START_REF] Chang | Transient-eects modeling of critical heat ux[END_REF]) . . . [-] as a function of the normalized period τ /τ x [-] for the eight operating conditions (∆T sub , Re), represented by the points with error bars. The tting function (dashed line) corresponds to δ V = 180 (see legend). . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 E ¦ (x ¦ , t ¦ chf ) [¡] for 50 K and 75 K of subcooling, for Reynolds number of 35000 and a time-period ranging within 5 ms to 500 ms. The analytic solution corresponds to the steady-state case, given by Eq. (3.17). . . . . . 3.12 Dimensionless groups N x and N y [¡] versus the excursion period τ [s] . . . Introduction "Have you ever seen boiling water?" If you are reading these lines, the answer is without a doubt "YES". Boiling water is one of these common phenomena which are part of our daily life, in our kitchen or at the tea break. It is familiar to observe that by dint of heating the water, bubbles start to form at the bottom of the kettle. Small and shy at the beginning, bubbles grow and fuss, and nally become numerous and rise to the surface. This is usually the moment when the boiling water fullls its mission: cooking pasta or infusing tea leaves. With such a familiar phenomenon, we could believe that boiling has no secret for science, or at least not anymore. Absolutely not! Boiling is a complex phenomenon which has been intensely investigated for a century without revealing all its secrets. It involves many physical mechanisms taking place at dierent scales: uid mechanics and heat transfer at the macroscopic scale, interface physics and phase change at the microscopic scale... these are only a few examples. Boiling may also manifest under dierent regimes. The regime we are familiar with is the so-called "nucleate boiling". Vapor is generated at the heating wall in the form of discernible bubbles which usually depart to join the bulk water. At higher heat power and higher temperature, another form of boiling occurs: "lm boiling". Instead of having bubbles which depart from the heating surface allowing its rewetting, a lm of vapor is generated and separates the heating surface from the liquid phase. This lm of vapor works as a thermal insulator and drastically degrades the heat transfer. The transition from nucleate boiling to lm boiling is called Boiling Crisis and is still particularly delicate to describe and to predict. The maximum heat ux before this transition is called the Critical Heat Flux.

Conclusion and Perspectives

∆T sat = T w ¡ T sat Wall superheat [K] ∆T sub = T sat ¡ T b Subcooling [K] δ Mantle thickness [m] δ ν Viscous sublayer unit [m] µ Dynamic viscosity [kg.m ¡1 .s ¡1 ] ν Kinematic viscosity [m.s ¡2 ] ρ Density [kg.m ¡3 ] τ Period of the exponential power excursion [s] τ x = L/u m,V Characteristic time [s] Θ w Normalized wall temperature [¡] ε Eusivity [J.m ¡2 .K ¡1 .s ¡1/2 ] k = γ λ
"So what? It's not a big deal, is it? We know how to boil water to make pasta, isn't it enough?" Well, no! Boiling is used in numerous applications in industry as it is a highly ecient mechanism to transfer heat. It is involved in plants in heat exchangers, but also in high-power electronic systems for cooling purposes. Boiling is even used in rocket engines to cool the combustion chamber with liquid cryogenic ergols. Consequently, the ne understanding of the boiling heat transfer is a key challenge for engineering applications. The perfect description of the mechanisms involved in the nucleate boiling heat transfer would enable to optimize the processes and to improve the designs. Seeking for higher performances, and thus higher heat transfer, the heat exchanger designs tend to always increase the heat ux involved. However, when the latter reach the Critical Heat ux, the transition from nucleate boiling to lm boiling causes an important temperature rise whereas the heat ux does not particularly evolve. The temperature of the heating surface can reach such important values that it can even threaten the heat exchanger integrity: structural fragility, meltdown... For this reason, the Critical Heat Flux is usually taken as the acceptable upper limit for the heat ux value when it comes to the design of engineering systems involving nucleate boiling. Consequently, the Boiling Crisis has been widely investigated: hundreds of experiments and correlations have been generated in the past half-century in order to estimate the critical heat ux in a wide range of conditions. Even with this massive amount of data, the modeling of this phenomenon is still unclear and Introduction discussed, explaining the low level of condence we have on the estimated values of the Critical Heat Flux. To overcome this, large margins are usually imposed in the designs. These blurry margins potentially imply big losses of eciency and maximum capability of the heat exchanger, and somehow impact the economical revenues. Moreover, this low level of condence on the Critical Heat Flux estimation may be a real safety threat when it involves sensitive systems. A nuclear reactor being a big heat exchanger, the question of the Critical Heat Flux arises for nuclear safety.

In the safety design of experimental reactors, which are for example used for material testing under irradiation or generating medical radioisotopes, one of the main dimensioning accidental scenario is the so-called BORAX-type accident. It assumes a fast and unexpected increase of the core power, until possibly reaching the Critical Heat Flux. If it happens, it could result in the deterioration of the nuclear core, and in worse case scenarii to a steam explosion and a fuel meltdown. For this reason, the French Alternative Energies and Atomic Energy Commission (CEA) investigates the Boiling Crisis and the prediction of the Critical Heat Flux under conditions applicable to the BORAX-type accident. In experimental reactor cores, water ows at a high velocity (a few to a few tens of meters per second) at moderate pressure (a few atmospheres) and with a high subcooling temperature (the water temperature is about tens to two hundreds of Kelvins under its saturation temperature). Beyond these thermal hydraulic conditions, the most particular characteristic of a BORAX-type accident is that the power is not constant in time, but undergoes a fast escalation. This power escalation has an exponential form characterized by a short period, as short as a few milliseconds, depending on the nuclear core characteristics. If the Boiling Crisis has been widely investigated in steady-state conditions, this has not been the case for the transient situations, especially under these thermal hydraulic conditions. Therefore, in order to better understand boiling processes under transient power escalations and particularly to best-estimate this key quantity for the safety design, CEA decided to develop a partnership with the Massachusetts Institute of Technology (MIT) to experimentally investigate the phenomenon using high resolution diagnostics. The present work takes place in the context of this collaboration, concluding a wider partnership between the two institutes aiming for a better understanding of all the boiling heat transfer phenomena involved in a BORAX-type accident. As the boiling heat transfer involves various phenomena at dierent scale and with two phases, it is currently impossible to realize a direct numerical simulation of the phenomena applied to industrial conditions and scales in order to have robust "numerical experiment" data, like in other elds of thermal hydraulics. Moreover, concerning the Boiling Crisis, we lack mechanistic understanding and triggering criteria to theoretically model the phenomenon. The present work aims at gathering high resolution data and observations of the transient ow Boiling Crisis at high subcooling. The experiments will permit to determine the impact of the four main working parameters: pressure, water temperature, ow conditions, exponential power escalation period. From the analysis of the experimental data and the observation of the records, we will attempt to understand the underlying mechanisms in order to propose a model able to describe the Boiling Crisis and to estimate the Critical Introduction and an overview on its neutronic background. The state of the art of the boiling heat transfer and particularly of the boiling crisis and transient boiling crisis are presented. This chapter ends with the presentation of the detailed objectives of this thesis. Chapter 2 provides an overview of the experimental investigations. The experimental apparatus and its compounds are described as well as the adaptations and implementations that have been done for this particular project and the methods of post-processing. It also presents a general overview of the wider collaboration between CEA and MIT and how this work is included. Chapter 3 discusses the possible mechanisms involved in the transient ow boiling crisis at high subcooling. From ne observations accessible thanks to the high resolution visible and infrared diagnostics, we propose a model describing the phenomena involved in the transient Boiling Crisis. Chapter 4 extends the investigation to high mass uxes and moderate pressures. It presents the parametric results of transient CHF and extends the application of the model developed in the latter chapter at these extended range of conditions. Finally the conclusion recalls the new ndings of this work and proposes working directions to develop in the near future. This chapter is a general introduction to this present work. First of all, we introduce the industrial background including the nuclear safety challenge of the BORAX-type accident.

It is followed by a short introduction to notions of nuclear physics in the core, which is important to understand the importance of the boiling heat transfer in such accidental conditions. A state of the art of the boiling heat transfer is then presented. More particularly, we will emphasize on the state of the art of the modeling of the subcooled ow boiling crisis and the transient subcooled ow boiling crisis. Finally, the objectives of this work are then presented followed by the detailed plan. In order to fulll its purposes, a RTR needs to generate a high neutron ux. Consequently, the nuclear core is compact and the fuel is enriched in ssile isotopes, more than nuclear power plant reactors such as Pressurized Water Reactors (PWRs). Also, RTRs do not need to be designed to generate high power. For instance, a PWR generates about 3000 MW of thermal power, when a RTR generates less than 100 MW. To be able to perform experiments eciently, the core is designed to be easily accessible. Accordingly, the most common designs are pool-type reactors: the core is composed of several fuel assemblies and experimental boxes disposed in an about 10-meters deep pool, as illustrated in gure 1.2.

As RTRs do not need to reach a high thermodynamic eciency, their working thermal hydraulic conditions are signicantly lower than for a PWR: the water core temperature is about 30 o C and the core pressure is about a few bar, whereas a PWR is operated 1.1. The BORAX-Type Accident Figure 1.2: Cooling Scheme of OSIRIS reactor [START_REF]INB 40 réacteur OSIRIS et ISIS: Evaluation complementaire de la surete au regard de l'accident survenu a la centrale nucleaire de Fukushima Daiichi[END_REF] at 300 o C and 150 bar. Besides, as the core does not generate a lot of thermal power, a high coolant mass ow is not necessary: under a ton per second, whereas the mass ow in a PWR reaches several hundreds of tons per second. These particularities have an eect on neutronics as will be presented in section 1.1.2 and section 1.2. In table Among the dierent types of RTR, the present work will focus on pool-type reactors.

Phenomenology of the BORAX type accident

Among the various scenarii of nuclear reactor accidents studied by the safety authorities, one is called Reactivity Insertion Accident, or RIA: it occurs when the nuclear reaction rate in the core unexpectedly rises, for example because of the ejection of a control rod. The BORAX-type accident is the name of the RIA for a pool-type research reactor. The reactivity insertion leads to a high and fast escalation of the nuclear power followed by a fuel temperature escalation. If the temperature reaches a too high level, a fast thermal interaction can occur between the warm fuel and water and possibly induces an explosion. More precisely, the reactor behaves as follow:

Power excursion

Because of the insertion of a high quantity of reactivity (more than 1$ , see section 1.2.2.2), Chapter 1. Nuclear Safety Challenge and State of the Art the core reaches a prompt-critical state. The neutron population rises exponentially, inducing a similar power excursion. For pool-type reactors, the period of the exponential excursion has generally an order of magnitude of 10 ¡2 s. More details about the neutronic phenomenology will be given in section 1.2.2.

Boiling Inception

The power excursion induces a temperature rise in the fuel. As explained in section 1.1.1, experimental research reactors generally use a more enriched fuel. Thus, the ratio 238 U/ 235 U can be too small to enable a sucient Doppler-feedback eect (see section 1.2.3) and the fuel temperature continues to increase exponentially while the coolant stays liquid. Moreover, as the subcooling is high, the liquid water temperature does not increase enough for the moderator eect to stop the power excursion before boiling inception.

After boiling inception, heat transfer at the wall increases signicantly during nucleate boiling and void appears in the channels. Then, two possibilities are encountered :

Power oscillations

The power excursion is terminated by the stronger moderator feedback due to void production while the system is still in nucleate boiling regime. The power excursion being mitigated, the heat ux drops down and so the void produced. Consequently, the neutronic moderation also decreases allowing a new power excursion. This oscillatory mechanism generates secondary peaks which can be observed on gure 1.3 until the reactor stops. Transition to lm boiling If lm boiling locally occurs in the reactor, two consequences can appear. First, the lm boiling regime generates generally less void than the nucleate boiling regime, leading to a smaller moderator feedback. In a second hand, the vapor lm acts like a thermal insulator blanket which drastically decreases the heat transfer at the wall. As there is a drop of the heat release while the core power keeps increasing exponentially, these phenomena induce 1.1. The BORAX-Type Accident an enhancement of the fuel temperature, possibly leading to a fuel meltdown along with an explosive reaction. This is the main concern in the case of a BORAX-type accident. Moreover, at high temperature, the melted Aluminium can be oxidized by water: this reaction is exothermic and generates gaseous hydrogen which can also initiate an explosion.

Consequences

According to IRSN [START_REF] Couturier | Prise en compte de accidents de type[END_REF], the energy released by the induced explosion can have many consequences:

The primary circuit can be deteriorated ;

The lower part of the pool can be deteriorated and thus the connement ; Ejection of water which can attack the containment building ; Enhancement of the temperature and the pressure in the containment building ; High radioactive dose in the reactor building ; Fallout in the surrounding environment.

Consequently, studying the BORAX-type accident is of key importance for RTR safety. For example, the CEA systematically includes safety investigations of such scenarii in the design of its experimental research reactors.

History of the BORAX type accident

In this section, we give a short historical overview of the eld and introduce the main experiments which generated a signicant part of the database regarding BORAX-type accidents.

The BORAX-1 Reactor

The BORAX-1 reactor, illustrated in gure 1.4, was the rst reactor of the BORAX program. This program, BORAX standing for BOiling wAter Reactor eXperiment, conducted by Argonne National Laboratory USA in Idaho, performed various safety experiments on ve boiling reactors during the 1950s and 1960s. The experiments in 1954 on BORAX-1 aimed to investigate self-reactivity-feedback during a fast insertion of reactivity in subcooled conditions [START_REF] Dietrich | Experimental investigation of the self-limitation of power during reactivity transients in a subcooled, water-moderated reactor[END_REF]. The reactor was made of 18 fuel plates enriched at 90 % of 235 U. An electromagnet mechanism was built in order to eject the control rod in less than 0.25 second in order to inject promptly a great quantity of reactivity. Many experiments with exponential excursion periods greater than 13 ms were performed. They showed that for this reactor, a period greater than 13 ms induced a power oscillation regime, as described in section 1.1.2.

On July 22, 1954, a destructive test was performed to generate an excursion able to induce the meltdown of a few percents of the fuel and the reactor destruction. The experiment exponential power excursion period was 2.6 ms. The power increased up to a maximum of 14 GW and generated 135 MJ of energy. In the fuel, the temperature reached 1650 o C and the explosion was equivalent to 32 kg of TNT. On gure 1.5, we present successive images illustrating the explosion generated by this destructive test. For a maintenance of the control rods, in 1960, these control rods were disconnected from their command device and put inside the core to stop it. On January 3, 1961, an accident happened during the reconnection of the control mechanism : in the evening, re and radiation alarms started their warning. Two personals were found dead and a third one was found wounded and died soon later in the hospital. A picture of the aftermath of the accident is presented in gure 1.6b. After analysis, it was found that the core power had probably reached 20 GW and the pressure in the vessel exceeded 30 bars. The core was highly deformed, but did resist, and one of the control rods was ejected up to the reactor roof. However, the reactor building perfectly resisted and 99,99 % of the radioactivity stayed conned in the building. One hypothesis is that a control rod was blocked and a worker tried to unblock it manually, inserting too much reactivity, leading to the explosion. This is the only example of an unvoluntary BORAX-type accident in the history of research reactors. conducted in order to explore further parameters values, but also to investigate the consequences of the core destruction. From a small deterioration of the fuel plate, to the meltdown of a fraction of the core, the "destructivity" of the test was increased... until the last destructive test where 3.5 $ of reactivity was injected in 3.3 ms, which generated a peak power of 2.3 GW . The richness of this destructive test is the data acquisition during the whole sequence of events preceding the explosion. On gure 1.7, we present an overview of the data gathered during destructive test (gure 1.7a) and a picture illustrating the destruction of the reactor (gure 1.7b). Indeed, on the BORAX test, acquisition ended only a few moment after the power peak, whereas on this experiment, a second peak of pressure was observed after the power peak. This second peak is most likely the one which induced the explosion.

The SPERT-IV experiment aimed to study the response of the core to a reactivity insertion. The core is mostly similar to SPERT-I. But the particularity of this facility is that it allows a parametric investigation of the response in regard of a variation of the power excursion period, the water ow rate and the hydrostatic pressure above the core. Therefore, the SPERT-IV database provides really rich experimental data for thermalhydraulics and thermal-hydraulics & neutronic coupling. It is widely used to validate many theoretical and simulation models.

As seen in this section, the BORAX-type accident is of key importance in the RTR safety design. In order to have a better understanding of its behavior, it is crucial to understand its main dimensioning phenomena -the neutronic and the boiling behavioras well as their coupling. For this reason, necessary neutronics concepts will be presented in section 1.2 and the boiling regimes will be presented in section 1.3.

Reactor Physics

To understand correctly the Borax-type accident, some elements of reactor physics are necessary. In this section, a short overview of the subject is presented. Further details can be found in the books of Coste-Delclaux et al. [START_REF] Coste-Delclaux | Une monographie de la Direction de l'énergie nucléaire[END_REF] and Bussac and Reuss [START_REF] Bussac | Traité de neutronique: physique et calcul des réacteurs nucléaires avec application aux réacteurs à eau pressurisée et aux réacteurs à neutrons rapides[END_REF].

Elements on neutrons matter interaction 1.2.1.1 Fission

Fission is a nuclear reaction in which a heavy nucleus breaks into several fragments, most usually two. This phenomenon can be spontaneous or induced by an impact with a projectile such as a neutron. For nuclear reactors, the last case is the most interesting because it can be used to control the core power as explained in section 1.2.1.2. Illustrated in gure 1.13, we present the dierent sequences of the ssion of 235 U due to a neutron capture:

Neutron capture A 235 U nucleus captures a neutron and becomes a 236 U nucleus. Destabilization The 236 U nucleus is unstable and its spherical shape deforms. Fission The 236 U nucleus shape becomes so deformed that it breaks and generates fragment nuclei and neutrons. The average number of neutrons generated is 2.47 for this case.

The nature of the ssion products is not deterministic but follows a probability law. For the case of the ssion of a 235 U nucleus induced by a neutron capture, this law is presented The ssion reaction conserves the nucleons number. At 2 or 3 neutrons deviation, the ssion fragments are on the A/Z constant line as presented in gure 1.10, A being the number of nucleons in the nucleus and Z being the number of protons in the nucleus. Therefore, these product nuclei are above the zone where nuclei are stable, called the stability valley, so they will undergo β ¡ decays to reach stability. This phenomenon is of key importance for reactor core control, as developed in section 1.2.2.1. As ssion can be seen as a step by step phenomenon, a concept of generations of neutrons can be instinctively introduced as presented in gure 1.11. If n i is the number of neutrons of the generation i, a neutron multiplication factor k can be introduced as :

Chain Reaction

k = n i n i¡1 (1.1)
It characterizes the evolution of the neutron population in a specic medium. The following behavior can be encountered according to the value taken by k :

Subcriticality : k 1. The chain reaction stops after a few generations. Criticality : k = 1. The number of neutrons in the medium is stable.

Supercriticality : k ¡ 1. The medium gains neutrons at each generation. Therefore, the chain reaction increases its intensity and the power increases too.

The reactivity ρ is then introduced as ρ = k¡1 k or k = 1 1¡ρ . In case of a homogeneous neutron ux, k can be computed considering the dierent reactions cross sections. Concerning the whole nuclear reactor core, it is then characterized by its eective multiplicative coecient k ef f which can be qualitatively presented as :

k ef f = Generation Capture Escape (1.2)
The term capture includes the ssion-induction capture and the sterile capture.

Neutron-Matter Interactions

For nuclear reactions, the cross section is generally used to characterize the probability of a reaction neutron-nucleus to happen. It is counted in cm ¡2 or barns (b) and depends on the energy of the neutrons.

Figure 1.12 presents the cross sections of various interactions as a function of the incident neutron energy. Even though 238 U is not ssile, it is generally predominant in the fuel and so its interactions with neutrons in the core are of key importance. Indeed during nuclear reactions, neutrons are emitted in the fast domain, but in order to reach criticity, they must generally be slowed down to reach the thermal domain where the cross section ratio between the two isotopes is more favorable. This is made by the means of a moderator permitting to slow the neutrons by successive collisions. In the present work, the moderator is water, and approximately 30 collisions are needed to thermalize the neutrons. The overall eciency of the thermalization depends on the density and temperature of the coolant uid, as will be developed in section 1.2.3.3.

.2.1 Neutron Populations

As explained before, some neutrons are emitted in the core by ssion of 235 U. The ssion process lasts around 10 ¡14 seconds. For this reasons this population of neutrons is called prompt neutrons. The time between two generations of prompt neutrons is called l and is a characteristic of the core. However, emission by ssion is not the unique source of neutrons. Some ssion products, called precursors, also contribute to it. For example, some Bromine and Iodine isotopes undergo a β ¡ decay, producing nuclei in an excited state. If their energy is sucient, instead of γ ray radiation, these can emit a neutron.

A Z X β ¡ ¡¡¡Ñ A Z 1 Y ¦ e ¡ (1.3) A Z 1 Y ¦ ¡¡¡Ñ A¡1 Z 1 Y n (1.4)
This process is not as fast as ssion but takes in average a few seconds. This population of neutrons is called delayed neutrons. Even though the delayed neutrons gather a small fraction of the total number of neutrons, this population is of key importance for the nuclear core stability as explained in section 1.2.2.3.

The total population of delayed neutrons is characterized by the delayed neutron fraction β :

β = Delayed Neutrons Delayed Neutrons + Prompt Neutrons (1.5)
The characteristics of delayed neutrons depend on the precursors. A classic approach, called the 6-groups model, is to use 6 groups of delayed neutrons. For N groups of precursor we have :

1.2. Reactor Physics

β = N i=1 β i and β λ = N i=1 β i λ i (1.6)
Where:

β i The proportion of the group i λ i The characteristic time of decay of the group i (s ¡1 )

β The total proportion of delayed neutrons λ The characteristic time of decay of the delayed neutrons (s ¡1 )

The average emission time of a delayed neutron is given by 1/λ. Finally, for the whole population of neutrons, the average emission time, and thus the life-time of a generation is :

Λ = (1 ¡ β)l β(l 1 λ ) (1.7) 1.2.2.

Criticity Types

The neutron multiplication factor k can be split into 2 contributions :

k = k(1 ¡ β) kβ = k prompt k delayed (1.8)
The k prompt reects how prompt neutrons evolve. If k prompt ¡ 1, the prompt neutron population increases "by itself" fastly and the chain reaction becomes uncontrollable. In terms of reactivity insertion :

1 = k prompt = k(1 ¡ β) = 1 ¡ β 1 ¡ ρ (1.9)
If ρ β There is not enough reactivity inserted to make the prompt neutron population grows "by itself". The chain reaction is controllable.

If ρ ¡ β The chain reaction becomes uncontrollable. It is said that the core state is prompt critical.

.2.3 Neutronic Kinetic Equations

To be able to predict quantitatively the behavior of the nuclear core, the equations governing the population of neutrons and precursors must be established and solved. The rst approach is to establish 0D equations: the populations are counted independently of space. For that, let us introduce: n: the total number of neutrons c i : the number of precursors of the group i During a time dt, the variation dn of the number of neutrons comes from : the birth of prompt neutrons: k l (1 ¡ β)n(t)dt the birth of delayed neutrons: °N i=1 λ i c i (t)dt the loss of neutrons: n(t) l dt in some cases a source term: S. This source can be positive like during the starting of the reactor, or negative like when neutronic absorbers are put in the core to slow down the reaction.

During a time dt, the variation dc i of the number of precursors of the group i is due to: birth of precursors : k l β i n(t)dt interaction with neutrons:

λ i c i (t)dt
By combining these particule balances, one obtains the N-group punctual kinetic model equations:

5 dn(t) dt = k(1¡β)¡1 l n(t) °N i=1 λ i c i (t) S dc i (t) dt = kβ i l n(t) ¡ λ i c i (t) (1.10)
In some cases, some terms of the system can be neglected. For instance, when a lot of reactivity is injected in a short time in the core, such as during a BORAX-type accident, the precursors contribution can be neglected and the system is simplied to :

dn(t) dt = k(1 ¡ β) ¡ 1 l n(t) (1.11)
As ρ = k¡1 k , the equation can also be written:

dn(t) dt = k(ρ ¡ β) l n(t) (1.12)
and by injecting the denition of Λ from equation 1.7, one can obtain the following equation describing the prompt-criticity:

dn dt = ρ ¡ β Λ n (1.13)
It shows that for such reactivity conditions, the neutron population temporally follows an exponential evolution proportional to e t/τ with τ = Λ ρ¡β . This explains the exponential input presented in section 1.1.2 and the power excursion period can be calculated through the knowledge of Λ, ρ and β. In the case of a BORAX-type accident, a fast insertion of reactivity gives τ 10 ¡2 s.

Reactor Physics

Reactivity Eects

In order to control the reactor, one needs to understand the various phenomena impacting the kinetics of the core. As Λ and β correspond to the core conguration, it is important to understand which phenomena can impact the reactivity ρ. We present in the following the reactivity eects applicable for a short term control of the core. Indeed other eects, such as fuel consumption or Xenon poisoning, also have to be taken into account for reactor control but their eect appears signicantly after several hours or days. For this reason these eects do not have any impact on a Borax-type accident.

Neutron absorber eects

A nuclear reactor is designed to be super-critical (ρ ¡ 0) when no neutron absorber is used. To reach criticity, one must inject negative reactivity using neutron absorbers. This is generally done using control rods and borine injected in the water. Control rods are composed of highly neutron absorbent materials which are inserted among the fuel rods. According to their insertion level in the core, one can control the negative reactivity inserted. For most designs of nuclear reactors, they are the main means for emergency shutdown: they can be fully inserted in less than a second, terminating the chain reaction. Conversely, their ejection, for example during a prompt over-pressurization in the core, can lead to a fast insertion of reactivity possibly leading to a RIA. For smoother control of the reactor, borine is preferably used. Borine is a neutron absorber and can be dissolved in the water. Its use disseminates negative reactivity homogeneously in the core, whereas the control rods aect the reactivity only locally and then can lead to heterogeneities of the fuel consumption. However the use of borine is done over a longer timescale than the use of control rods and cannot be used in emergency situations.

Doppler Eect

The cross section spectra (g. 1.12) presented in section 1.2.1.3 are dened as functions of the incident neutron energy relative to the concerned nucleus. When the temperature rises, nuclei agitation increases around their equilibrium position following a Maxwell distribution [START_REF] Bussac | Traité de neutronique: physique et calcul des réacteurs nucléaires avec application aux réacteurs à eau pressurisée et aux réacteurs à neutrons rapides[END_REF]. This implies an enlargement of the resonance spikes. This phenomenon, called Doppler broadening or Doppler eect, leads to an enhancement with temperature of the probability for a neutron to interact with a nucleus in the resonance domain during its thermalization. Because of the cross section ratios in this domain, it particularly increases the probability of absorption by a 238 U nucleus, implying a drop of reactivity. In low enriched fuel (3% or 4%) like in PWRs, the Doppler eect is really important because of the proportion of 238 U in the fuel. The eciency of this negative feedback in PWRs is sucient to terminate any power excursion before the core failure. However in RTRs where the enrichment is signicant higher (20% to 95%), the Doppler eect is less ecient.

Moderator Eect

As explained in section 1.2.1.3, to reach a favorable enough ratio σ 235 f /σ 238 , neutrons need to be thermalized by successive chocs with the moderator. Consequently, the moderator density has an impact on the reactivity: when its density decreases, the probability for the moderator and a neutron to impact decreases, and so the reactivity. This happens for instance when the temperature rises and this eect is drastically enhanced in case of phase change during boiling. In such case, the quantity of negative reactivity injected in the core is directly related to the quantity of void (i.e. vapor) produced. For BORAX-type Chapter 1. Nuclear Safety Challenge and State of the Art accidents in pool-type reactors, because of low pressure conditions, the rise of temperature generates a lot of void during nucleate boiling, mitigating the power excursion. In such scenarii, the moderator eect via the void production is the dominant neutronic feedback contribution and the only one signicant enough to terminate the exponential power excursion.

As seen in this section, in case of a fast reactivity insertion signicant enough to reach prompt criticity, the core power excursion follows an exponential evolution. In commercial reactors like PWRs, the Doppler eect is strong enough to mitigate the excursion. However, in pool-type reactors, only the production of void through nucleate boiling is signicant enough to stop the excursion. If the nucleate boiling transits to lm boiling, the void produced becomes drastically lower and consequently degrades the feedback eect. The excursion could then continue until it possibly ends with an explosive escalation. For this neutronic reason, it is important to understand what are the physical phenomena leading to boiling crisis.

Boiling Regimes

As seen previously, the transition from nucleate boiling to lm boiling during a BORAXtype accident can lead to dramatic consequences due to thermal and neutronic coupling. In term of boiling heat transfer, the latter case of study is the transient ow boiling crisis. In this section, an overview on the boiling regimes will be presented as well as the state of the art concerning the boiling crisis.

Boiling regimes and boiling curve

Boiling heat transfer has been widely investigated as it is involved in many industrial processes from heat exchangers to quenching in metallurgy. However, its complexity due to the dierent involved mechanisms (heat transfer, phase change, multiphase ow,...) induces many challenges to understand the underlying physics. Two main types of boiling congurations are usually distinguished: Pool boiling and ow boiling. Pool boiling corresponds to the situation where boiling occurs in absence of forced convection and ow boiling corresponds to the cases with forced convection. Due to the high eciency of heat transfer in ow boiling, the latter is the most used in industry and has been the most investigated.

Boiling heat transfer for a given conguration is characterize by the so-called boiling curve proposed by Nukiyama [START_REF] Nukiyama | The maximum and minimum values of the heat q transmitted from metal to boiling water under atmospheric pressure[END_REF] in 1934. A boiling curve gives the relation between the heat ux transferred from the heating wall to the uid (q P w in W.m ¡2 ) as a function of the wall superheat (∆T sat = T w ¡ T sat in K) dening the dierence between the wall temperature T w and the saturation temperature of the liquid T sat at the considered pressure. In steady-state heating, the boiling curve plotted in log-log axes has the following trend (g.

1.14):

The usual approach to experimentally explore boiling heat transfer is by imposing the heat ux. A wall is heated, for instance electrically by Joule eect, transferring heat ux to the liquid and the corresponding wall superheat is then measured. The couple (∆T sub , q P w ) gives the boiling curve. When the heat ux rises, the system undergoes the following behavior:

Single Phase

At low heat ux, the conditions are not met to generate bubbles and heat transfer is realized by conduction and convection. The heat transfer coecient (HTC) dened as h = q P w /(T w ¡ T b ) is low, represented by the slant slope on the boiling curve (T b is the bulk temperature, i.e. far from the wall).

Onset on Nucleate Boiling (ONB)

When a minimum wall superheat is reached, bubbles start to nucleate at the surface of the heated wall. Nucleation occurs preferentially at some specic locations of the surface called nucleation sites. These are microscopic imperfections on the wall, usually cavities, where the thermodynamic processes for the nucleation are eased. The activation of the nucleation sites is a complex phenomenon depending on the wall superheat: large cavities may nucleate at lower temperature. For these reasons, ONB is strongly inuenced by the surface conditions: the roughness and the wettability [START_REF] Koizumi | Chapter 1 -Outline of Boiling Phenomena and Heat Transfer Characteristics[END_REF]. After ONB, the importance of boiling phenomena rises and becomes signicant on the evolution of wall temperature and heat ux: this is the Onset of Boiling Driven (OBD) regime.

Nucleate Boiling

During the boiling heat transfer regime, HTC is drastically enhanced as it can be observed in Fig. 1.14 with the steeper slope of the boiling curve. The rst reason of this enhancement is the generation of vapor which necessitates a signicant amount of latent heat. Furthermore, the bubbles formation and motion generate advection at the vicinity of the wall, bringing cold water and enhancing even more the conductive heat transfer. The understanding and the quantication of the dierent contributions of nucleate boiling heat transfer, usually named heat ux partitioning, is an important challenge for industry. The well-known model of Kurul and Podowski [START_REF] Kurul | Multidimensional eects in forced convection subcooled boiling[END_REF] proposes three contributions: evaporation, wall to liquid conduction and transient conduction due to the inow of cold water after a bubble departs (so-called quenching). Recently, other authors Basu et al. [START_REF] Basu | Wall Heat Flux Partitioning During Subcooled Flow Boiling: Part 1Model Development[END_REF], Yeoh et al. [START_REF] Yeoh | Modeling subcooled ow boiling in vertical channels at low pressures Part 2: Evaluation of mechanistic approach[END_REF] or Gilman et al. [START_REF] Gilman | A self-consistent, physics-based boiling heat transfer modeling framework for use in computational uid dynamics[END_REF] have proposed a fourth contribution taking into account the bubble sliding. Under nucleate boiling, the increase of the heat ux intensies the boiling process as more nucleation sites are activated and the nucleation frequency is enhanced.

Chapter 1. Nuclear Safety Challenge and State of the Art

Boiling Crisis

Nucleate boiling intensity increases with the heat ux until reaching a maximum value, named the Critical Heat Flux (CHF). Qualitatively, the boiling process becomes so intense that the system cannot stay in nucleate boiling. Bubbles start to coalesce and produce large pockets of vapor working as heat insulators. The phenomena leading to the boiling crisis has been widely studied since Nukiyama [START_REF] Nukiyama | The maximum and minimum values of the heat q transmitted from metal to boiling water under atmospheric pressure[END_REF] as its represents a heat transfer limit for most industrial applications, but the underlying mechanisms are still to be understood. A detailed review will be presented in section 1.3.4. Once CHF is reached, the system undergoes a transition to lm boiling: the heat ux does not increase whereas the wall temperature increases drastically (rightward blue dashed arrows in gure 1.14). The temperature rise can be sucient to damage the heating system: at the critical heat ux for water at atmospheric pressure, the transition makes the wall superheat passes from a few tens of Kelvins to more than a thousand.

Film Boiling

Once the transition ends, the system reaches the lm boiling regime: a thin lm of vapor separates the whole heating wall from the liquid, working as a thermal insulator. Consequently, HTC is low. In this regime, the wall temperature is very high : convection and conduction heat transfer processes are still present and radiation starts to have a signicant contribution.

Once Film Boiling has been reached, one can decrease the imposed heat ux. The system follows the boiling curve until it reaches the Leidenfrost point (L point in g. 1.14).

Then the system undergoes a transition to nucleate boiling at constant heat ux (leftward blue dashed arrow in g. 1.14). The Leidenfrost point represents the minimum heat ux of the lm boiling branch and is always under CHF.

This description shows that boiling under an imposed heat ux undergoes a hysteresis behavior. According to the increase or the decrease of the heat ux, the path taken by the system on the boiling curve diers, with CHF being the maximum of the nucleate boiling branch with an increasing heat ux and the Leidenfrost point being the minimum of the lm boiling branch with a decreasing heat ux. It is possible to explore the temperature domain between CHF and the Leidenfrost point by realizing a temperature-imposed experiment: the heating wall temperature is controlled for instance by a temperaturecontrolled uid. In this case, when the temperature rises, the system follows the same boiling curve until CHF, but instead of transiting to lm boiling at constant heat ux, the heat ux drops until reaching the Leidenfrost point. This is the Transition Boiling regime, represented by the red dashed line in gure 1.14.

After a general overview of the existing boiling regimes in steady-state, the following sections will present the inuence on the various regimes of the major parameters involved in the present study. As the context of the present work is power excursion and boiling crisis in ow conditions, an insight will be presented only for the left branch for ow boiling. The development on the ow boiling crisis will be presented in section 1.3.4. Further information concerning the lm boiling branch can be found for instance in [START_REF] Koizumi | Chapter 4 -Minimum Heat FluxFilm Boiling[END_REF] and [START_REF] Kolev | Multiphase Flow Dynamics Part 3 -Thermal Interaction[END_REF]. N u = h L k l with h being the heat transfer coecient, L the characteristic length of the problem and k l the thermal conductivity of the liquid. The Nusselt number is then quantied through correlations involving the Reynolds number and the Prandtl number. For instance, two correlations commonly used in industry are the Dittus-Boelter correlation [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF] N u DB = 0.023 Re 0.8 P r 1/4 and the modied Sieder-Tate correlation [START_REF] Sieder | Heat transfer and pressure drop of liquids in tubes[END_REF] N u ST m = 0.0296 Re 0.8 P r 1/3 ¢ µ bulk µ wall 0.14 which also takes into account the variation of the near wall liquid viscosity due to heating.

In single phase, an enhancement of the ow intensity will increase the heat transfer via an enhancement of the Reynolds number. An enhancement of the subcooling will increase the heat transfer via an enhancement of the Prandtl number and the term (T w ¡ T b ) = ∆T sat ∆T sub . The other parameters have a slight or a negligible impact on the heat transfer.

Onset of Nucleate Boiling

A liquid is superheated when its temperature is above its saturation temperature at the considered pressure. In this state, the liquid is metastable, meaning that it tends to stability by becoming vapor after reaching a certain activation energy. In case of a liquid heated by a wall, a thin layer at the vicinity of the wall is superheated. Hsu [START_REF] Hsu | On the size range of active nucleation cavities on a heating surface[END_REF] proposes a criterion for the activation of the nucleation sites. He stipulates that for a bubble to grow out its nucleation site of radius r, the surrounding liquid should be at the same temperature as the embryo inner vapor. For a hemispheric embryo of radius r, the considered temperature is: The dierence of temperature with the bulk saturation temperature is due to the bubble curvature implying an internal pressure rise. This expression is obtained by combining the Laplace law and the Clapeyron law as for example presented in the books of Carey [START_REF] Carey | Liquid-vapor phase-change phenomena[END_REF] or Todreas and Kazimi [START_REF] Todreas | Nuclear Systems I: Thermal Hydraulic Fundamentals[END_REF]. The prediction of boiling inception is based on two knowledges: the size of the nucleation sites and the temperature eld at the close vicinity of the heating plate.

T (r) = T sat 2σ r T sat h lv ρ v (1.
The temperature eld close to the heater is both aected by the subcooling and the forced ow conditions. A rise of the subcooling reduces the thermal boundary layer thickness as the liquid has to be more warmed up. Under a stronger forced convection, the establishing thermal boundary layer is advected, also reducing its thickness. For these reasons, an increase in the subcooling or the forced convection rises the ONB temperature. In equation 1.14, a pressure increase decreases the superheat term as it is dominated by the increase of the vapor density ρ v . Consequently, the rise of the pressure shifts the ONB temperature leftwards the boiling curve, and the rest of the nucleate boiling branch.

Finally, the surface condition has a major importance on the onset of nucleate boiling. A smooth surface, where the cavities are small, has its ONB at higher wall superheat, whereas for a rough surface the ONB is at lower wall superheat. Other surface eects like porosity or wettability are also impactful.

Nucleate Boiling

Once ONB has been reached, the nucleation sites will consecutively be activated, increasing the nucleation site density. This leads to the Fully Developed Nucleate Boiling regime (FDNB), the heat ux is highly enhanced due to the contribution of new mechanisms of heat transfer, detailed in the heat ux partitioning [START_REF] Richenderfer | Investigation of subcooled ow boiling and CHF using high-resolution diagnostics[END_REF]: q P = q P e q P c q P q q P sc , with 6 9 9 9 9 9 9 9 9 9 9 8 9 9 9 9 9 9 9 9 9 9 7

q

P e = f N P π 6 D 3 b ρ v h lv q P c = ¡ 1 ¡ N P K π 4 D 2 b © h c (∆T sub ∆T sat ) q P q = 2ε l f N P K π 4 D 2 b (∆T sub ∆T sat ) t w π q P sc = 2ε l f N P R f a sl (∆T sub ∆T sat ) t w π (1.15)
where:

q P e , the heat ux contribution of evaporation q P c , the heat ux contribution of single phase conduction q P q , the heat ux contribution of quenching q P sc , the heat ux contribution of bubble sliding conduction f , the bubble departure frequency N P , the nucleation site density D b , the bubble departure diameter h c , HTC in single phase convection K, the area inuence factor 1.3. Boiling Regimes ε l , the eusivity of the liquid t w , the waiting time (time needed after a bubble departure for a new nucleation) R f , the nucleation site reduction factor a sl , the area inuenced by the bubble sliding.

The heat ux partitioning has gathered in the past decades many interests as it enables the physical understanding of heat transfer in nucleate boiling and gives access to important insight such as the void production (through the term q P e ). Nevertheless, heat ux partitioning involves the ne knowledge of nucleation and surface conditions (N P , D b , a sl ,...) which makes this approach delicate to apply to industrial situations.

Besides heat ux partitioning, many correlations giving a relation between the wall superheat and the heat ux exist. An overview can be found in the survey of Guglielmini et al. [START_REF] Guglielmini | Survey of Heat Transfer Correlations in Forced Convection Boiling[END_REF]. For instance, Su et al. [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part I: Pool boiling[END_REF] found a satisfactorily estimation of the heat ux at atmospheric pressure with a power law

q P F DN B = C∆T 4 sat (1.16)
with the factor C increasing with the subcooling as presented in table 1.2.

∆T sub [K] C [W.m ¡2 .K ¡4 ] 10 1.33 25 2.7 75 7
Table 1.2: Value of coecient C of the correlation 1.16 at dierent subcooling degrees [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part I: Pool boiling[END_REF] This correlation shows that the heat transfer is enhanced by the subcooling. Quantitatively, this can be explained by the fact that being in contact with a colder liquid enhances two terms of the heat ux partitioning: conduction/convection and quenching. Indeed, as the bulk temperature is colder, the temperature gradient is important and so the heat conduction. The quenching is also enhanced as the the water inowned by the bubble departure is also colder.

Concerning the inuence of the ow, in the survey published by Guglielmini et al. [START_REF] Guglielmini | Survey of Heat Transfer Correlations in Forced Convection Boiling[END_REF], the correlation for FDNB does not take into account any convection eect. In their experiment, Su et al. [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part I: Pool boiling[END_REF] also did not nd any signicant inuence of the forced convection for the range of Reynolds number between 25,000 and 60,000.

Finally, the heat ux in the FDNB regime increases with pressure, with an exponential dependency. For instance, in Jens and Lottes' correlation [START_REF] Jens | Analysis of Heat Transfer, Burnout, Pressure Drop and Density Data for High-Pressure Water[END_REF] q P F DN B W e 0.064p with p in bar and p r6, 172s bar in Thom's correlation [START_REF] Thom | Boiling in subcooled water during ow up heated tubes or annuli[END_REF], we have q P F DN B W e 0.23p with p in bar and p r5, 138s bar.

Impact of the transient nature of the heat input on the boiling curve

A transient heat input drastically changes the behavior of the whole boiling curve as presented in the following (gure 1.15). In the present work, the transients of interest are the exponential power excursions of periods τ . In single phase, the temperature eld near the wall is squeezed for a given heat ux due to the heat conduction in the liquid. Qualitatively, it implies a stronger temperature gradient in the thermal boundary layer generating an enhanced heat transfer. Moreover, Su et al. [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part I: Pool boiling[END_REF] suggested that in ow conditions, heat transfer is made of two contributions: transient conduction and turbulent forced convection, the latter being leaded by turbulent mixing. By dening the normalized HTC as h = h w /h f c,exp with h w the considered HTC to water and h f c,exp the measured forced ow HTC in steady-state condition and by dening τ = τ /τ vor with τ vor the time scale of the turbulent vortices, the heat transfer in such conditions can be characterized by:

Boiling Regimes

A thinner thermal boundary layer implies that ONB appears at a higher wall superheat due to the Hsu criterion. It has been experimentally observed in [START_REF] Visentini | Experimental investigation of heat transfer in transient boiling[END_REF]- [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part II: Flow boiling[END_REF]- [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part I: Pool boiling[END_REF]- [START_REF] Schei | Transient ow boiling in a semi-annular duct: From the onset of nucleate boiling to the fully developed nucleate boiling[END_REF]. After nucleation inception, the wall is cooled by the boiling process and the transient boiling curve reaches the steady-state boiling curve. This variation of temperature leads to a temperature overshoot (OV). This phenomenon is also widely reported in the literature ( [START_REF] Rosenthal | An experimental study of transient boiling[END_REF]- [START_REF] Sakurai | Transient pool boiling heat transfer Part 1: incipient boiling superheat[END_REF]- [START_REF] Sakurai | Transient pool boiling heat transfer. Part 2: boiling heat transfer and burnout[END_REF]- [START_REF] Visentini | Experimental investigation of heat transfer in transient boiling[END_REF]- [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part II: Flow boiling[END_REF]- [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part I: Pool boiling[END_REF]- [START_REF] Schei | Transient ow boiling in a semi-annular duct: From the onset of nucleate boiling to the fully developed nucleate boiling[END_REF]- [START_REF] Kossolapov | The Boiling Crisis of water under exponentially escalating heat inuts in subcooled ow boiling at atmospheric pressure[END_REF]). The temperature OV increases with fast transients. After OV, no inuence on FDNB is observed.

Flow Boiling Crisis

Because of the high eciency of boiling heat transfer, industrials tend to work under this regime at high heat ux. However, the boiling crisis, usually characterized by its critical heat ux, represents the upper limit for the safe operation. Its prediction is an important industrial concern as over the past 60 years, the number of correlations exceeds a thousand [START_REF] Groeneveld | The 2006 chf look-up table[END_REF]. The multitude of correlations and the scarce number of mechanistic models show a general misunderstanding of the physical phenomena governing the boiling crisis. In the present section, we present a description of the boiling crisis and the current state of the art.

Types of boiling crisis

In case of ow boiling, various heat transfer regimes can occur depending on the operating conditions. They can be summarized in a equilibrium quality -heat ux diagram as presented in gure 1.17. The equilibrium quality is dened as X = h l ¡hsat h lv , with h l the bulk liquid enthalpy and h sat the liquid enthalpy at saturation. For a long heat pipe, i.e.

for a ratio L/d 4 1, L being the heating length and d a characteristic diameter of the pipe, the equilibrium quality will progressively increase with the streamwise coordinate. Consequently, one can qualitatively be assimilated to the other in gure 1.17. The boiling crisis, represented by the colored line in gure 1.17, delimitates the nucleate boiling region from the lm boiling one and the required heat ux decreases with the equilibrium quality. The boiling crisis can occur following two mechanisms as illustrated in gure 1.18. At moderate heat ux and high equilibrium quality (red line in gure 1.17), the liquid is Chapter 1. Nuclear Safety Challenge and State of the Art progressively heated up along the wall until saturation and phase change. The void fraction progressively rises and the vapor is entrained to the middle of the channel forming an annular ow. The liquid lm surrounding the wall progressively narrows because of vaporization and liquid entrainment, until its depletion. This boiling crisis mechanism is called "dryout" and is presented in 1.18.b. At higher heat ux and lower quality (blue line in gure 1.17), the vapor generation rate can be so high that the vapor is not suciently eliminated from the wall, because of a low vapor entrainment further from the heating wall and/or a low condensation rate. This leads to the development of a vapor lm at the wall whereas the bulk ow is still liquid. This boiling crisis mechanism is called Departure from Nucleate Boiling (DNB). In the present work, we investigate the occurrence of boiling crisis in the case of pool-type reactors working under very high subcooling and the BORAX-type exponential power excursion which involves really high heat uxes. In these conditions, the DNB mechanism will be more likely to occur.

Boiling crisis correlations and look-up tables

The need to predict CHF in industry produced a massive amount of experimental data on CHF for a wide range of operating conditions. Because of the extreme conditions usually required to reach CHF in industry (pressure, mass ux,...), experiments have a high cost and mostly low-resolution diagnostics with only time and space averaged measurements. Consequently, the physical mechanisms of DNB are not accessible, but these experiments enable the generation of correlations and look-up tables.

Numerous CHF correlations were proposed during the last decades. They allow the determination of the CHF by the means of dimensionless groups having physical meanings linked to the boiling crisis. A typical CHF correlation assesses the Boiling number B o = q P Gh lv at the boiling crisis and has the following form [START_REF] Mudawar | Recent Advances in High-Flux, Two-Phase Thermal Management[END_REF]:

q P CHF Gh lv = f ¢ c p,l ∆T sub h lv , L d , G 2 L σρ l , ... = f pΠ 1 , Π 2 , Π 3 , ...q , with Π i rΠ i,min , Π i,max s
When single phase heat transfer correlations have relatively simple forms (see section 1.3.2.1), CHF correlations are more elaborated, showing the complexity of the involved 1.3. Boiling Regimes mechanisms. Moreover, for a given set of experimental data, only a narrow range around the operating conditions of interest are generally investigated due to the experimental constraints. This implies that the correlations issued are only valid in a narrow range of Π i and for specic geometries. Therefore, the use of a given correlation beyond its validity ranges leads to highly inaccurate predictions [START_REF] Mudawar | Recent Advances in High-Flux, Two-Phase Thermal Management[END_REF].

An example of a CHF correlation for subcooled ow boiling is the Tong-75 [START_REF] Tong | A phenomenological study of critical hear ux[END_REF] correlation:

q P CHF,T ong¡75 G h lv = 0.23 f £ 1 0.00216 Ja Re 0.5 ¢ p out p crit 1.8
with:

f = 8 Re ¡0.6 ¡ D h D 0 © 0.32
, with D 0 = 12.7 mm

Ja = ρ l ρv cp ∆T sub h lv
, the Jakob number for subcooling conditions characterizing the transfer eectiveness between the liquid and the vapor phase. Re = G D h µ l (1¡α) , the Reynolds number characterizing the turbulent mixing. α is the void fraction calculated with the Thom's correlation [START_REF] Thom | Boiling in subcooled water during ow up heated tubes or annuli[END_REF]. p out /p crit the reduced pressure characterizing the bubble size. p out is the outlet pressure and p crit , the thermodynamic critical pressure.

The valid operational range is:

p r70, 140s bar D h r3, 10s mm L/D r5, 100s G r700, 6000s kg.m ¡2 .s ¡1 X in r¡1, 0s
A detailed comparison of the performance of the existing correlations can for example be found in Celata et al. [START_REF] Celata | Assessment of correlations and models for the prediction of CHF in water subcooled ow boiling[END_REF] or in Inasaka and Nariai [START_REF] Inasaka | Evaluation of subcooled critical heat ux correlations for tubes with and without internal twisted tapes[END_REF]. A correlation establishment and analysis can be found in the book of Tong and Tang [START_REF] Tong | Boiling heat transfer and two-phase ow[END_REF].

Many CHF correlations were commissioned by industrials and concern the conditions of commercial nuclear reactors, such as PWRs (see Table 1.1.1). They are not publicly accessible. It implies that the range of operating conditions for RTRs is poorly covered. Finally, correlations for transient heat ux conditions are also poorly developed in the literature, and almost all of them consider a transient correction factor enhancing the CHF value beyond the steady-state value. More details are presented in section 1.3.4.5.

Look-up tables (LUT) are data banks which gather numerous experimental data. They propose a CHF value associated to given values of the corresponding operating parameters. The widely-used Groenveld LUT [START_REF] Groeneveld | The 2006 chf look-up table[END_REF] considers as operating parameters the pressure p (from 1 to 210 bars),the mass ux G (from 0 to 8500 kg.m ¡2 .s ¡1 ) and the thermodynamic quality X (from -0.5 to 1). A sample extracted from this LUT is presented in gure 1.19. It is generated by interpolation from about 25,000 experimental data points from more than eighty 80 data sets, and a statistic study has been realized in order to quantify the data deviations. LUTs have the major advantage to be really simple to use and gather in a unique table massive experimental data covering a wide range of operational conditions. However, gathering such an amount of diversied data in a reduced information feature mixes up other important characteristics such as geometry, length, transient eects... which may change the value of CHF. Besides, the statistical deviations on the data make its use lowly reliable for safety investigations. Finally, no LUT exists for transient conditions.

Experimental investigation on ow boiling crisis

Boiling involves phenomena happening during a short time scale (a few milliseconds) from the local space scale (a few microns) to the integral scale (a few meters) for industrial applications. To investigate the global behavior of a system, integral tests have been designed. For example, the reactors SPERT-I [START_REF] Miller | Report of the SPERT I Destructive Test Program On An Alumnium, Plate-Type, Water-Moderated ractor[END_REF], SPERT-IV [START_REF] Crocker | Reactor Power Excursion Tests In The SPERT IV Facility[END_REF], CABRI [START_REF] Papin | French studies on high-burnup fuel transient behavior under RIA transient[END_REF] or NSRR [START_REF] Ishikawa | The Nuclear Safety Research Reactor[END_REF] provide realistic overviews of the system behavior during a RIA. Nonetheless, beyond the fact that they are extremely expensive to design and build, local information on temperature or heat ux elds is inherently unavailable and no photographic access is given to observe bubble dynamics. The need to understand the mechanisms triggering the boiling crisis leaded to the development of small-scale experiments with ne control of the conditions and diagnostics of the phenomena.

The simplest geometry used to investigate boiling crisis is the metal wire heated by Joule eect. It allows to have optical accesses from all the directions and is relatively easy to heat up. Many well-known boiling crisis experiments have been made in this conguration, for example Nukiyama [START_REF] Nukiyama | The maximum and minimum values of the heat q transmitted from metal to boiling water under atmospheric pressure[END_REF] in steady-state pool boiling or Sakurai and Shiotsu in transient pool boiling [START_REF] Sakurai | Transient pool boiling heat transfer Part 1: incipient boiling superheat[END_REF]- [START_REF] Sakurai | Transient pool boiling heat transfer. Part 2: boiling heat transfer and burnout[END_REF]. Thin ribbons were also used in order to be closer to a planar geometry (Rosenthal and Miller [63] or Johnson [START_REF] Johnson | Transient boiling heat transfer to water[END_REF]). Yet, these congurations imply external ows which are not particularly representative of the potential industrial applications. Besides, the wall phenomena are not accessible as they are obstructed by bubbles. Closed ow geometries have also been developed and are as diverse as their applications: tube (e.g. Celata et al. [START_REF] Celata | A photographic study of subcooled ow boiling burnout at high heat ux and velocity[END_REF]), planar plate in a rectangular duct (e.g. Kossolapov et al. [START_REF] Kossolapov | The Boiling Crisis of water under exponentially escalating heat inuts in subcooled ow boiling at atmospheric pressure[END_REF]) or semi-circular plate in a semi-annular duct (e.g. Visentini et al. [START_REF] Visentini | Experimental investigation of heat transfer in transient boiling[END_REF], Baudin et al. [START_REF] Baudin | Turbulent ow and transient convection in a semi-annular duct[END_REF] or Schei et al. [START_REF] Schei | Transient ow boiling in a semi-annular duct: From the onset of nucleate boiling to the fully developed nucleate boiling[END_REF]). These geometries do not provide optical access except if a part of the duct wall is transparent, allowing front and side visualization. If the heating wall is transparent, it allows a full access to the boiling processes. Note that the latter is quite scarce because of the complexity of building a transparent heater able to support the mechanical and thermal constraints. Besides, the relative fragility of common transparent materials implies that high pressures (beyond 30 bars) were until recently inaccessible for 1.3. Boiling Regimes optical diagnostics. Finally, even though a signicant part of the industrial applications of these experiments concern water, many experiments have been realized with other uids. Indeed, thermal properties of water (specic heat, latent heat, boiling temperature) lead to the need of a signicantly high power input to reach CHF. Plus, the metal heater can undergo chemical reactions in the presence of water (e.g. oxidation) below the temperature required for CHF. For this reason, it is usual in the literature to see experiments with low boiling temperature uids such as HFE7000 [START_REF] Visentini | Experimental investigation of heat transfer in transient boiling[END_REF]- [START_REF] Baudin | Turbulent ow and transient convection in a semi-annular duct[END_REF]- [START_REF] Schei | Transient ow boiling in a semi-annular duct: From the onset of nucleate boiling to the fully developed nucleate boiling[END_REF] (T sat = 35 o C at atmospheric pressure) or Novec 649 [START_REF] Bloch | A phenomenological study on eects leading to the departure from nucleate boiling in subcooled ow boiling[END_REF] (T sat = 49 o C at atmospheric pressure). For the same reason, there are also few experiments with high subcooling and/or high forced convection which are a factor of CHF increase.

Diagnostics of boiling phenomena can be distinct in two areas: bulk diagnostics recording the phenomena in the volume of liquid, and the wall thermal diagnostics recording the temperature and the heat transfer at the heating wall. Regarding the bulk diagnostics, the easiest approach is the photographic study: through a camera, preferentially a high speed video (HSV) camera. With suciently high space and time resolutions, one may observe the bubble dynamics and can access the bubble-size distribution, growth and collapse dynamics,... This diagnostic has been widely exploited throughout the years: Fiori & Bergles [START_REF] Fiori | Model of Critical Heat Flux in Subcooled Flow Boiling[END_REF], Galloway-Mudawar [START_REF] Galloway | CHF mechanism in ow boiling from a short heated wall-I. Examination of near-wall conditions with the aid of photomicrography and high-speed video imaging[END_REF], Sturgis-Mudawar [START_REF] Sturgis | Critical heat ux in a long, rectangular channel subjected to one-sided heating-I. ow visualization[END_REF], Zhang-Mudawar [START_REF] Zhang | Photographic study of high-ux subcooled ow boiling and critical heat ux[END_REF], Celata [START_REF] Celata | A photographic study of subcooled ow boiling burnout at high heat ux and velocity[END_REF]. However, this approach allows to access to 2D images but provides no information on the ow or on the temperature eld. Moreover, the boiling phenomena is cloaked by bubbles when the bubble density becomes high which can happen close to boiling crisis. To reach 3D information, X-ray tomography can for example be realized (e.g. by Johnson [START_REF] Johnson | Transient boiling heat transfer to water[END_REF]), but as of today, this method is still unreliable. This technique gives access to the void production, a quantity essential in industry for example to compute the neutronic feedback (see section 1.2.3.3). A ne diagnostic of the temperature eld in the liquid is currently almost impossible. However, having a rough qualitative overview is possible by shadowgraphy or shlieren visualization: these techniques make possible the visualization of the refraction index heterogeneity due to temperature dierences and thus temperature patterns. Attempts of ne temperature eld assessment have been realized for the nucleation of a single bubble via rainbow Schlieren visualization (Narayan et al. [START_REF] Narayan | International Journal of Heat and Mass Transfer Rainbow schlieren-based investigation of heat transfer mechanisms during isolated nucleate pool boiling phenomenon : Eect of superheat levels[END_REF]). These techniques would be really complicated to apply on numerous bubbles and/or ow boiling as the temperature eld would be totally mixed up. Finally, the ow can also be visualized through many possible techniques: Particle Image Velocimetry (PIV), interferometry... a review can be found in the thesis of Bloch [START_REF] Bloch | An Experimental Study on Vertical Subcooled Flow Boiling Under the Inuence of Turbulence and Secondary Flows[END_REF]. Concerning the wall thermal information, the most widely used technique is the resistance based determination used for metal heaters (wire, ribbon and tubes) electrically heated. For instance in case of a wire, one usually knows the electric power and the wire temperature, throughout the resistivity which depends on the temperature. The heat ux to water is then obtained as the dierence between the joule power and the rate of rise of the sensible heat in the wire. This approach is widely used because of its simplicity, but the latter can only provide space-averaged information for the temperature and heat ux. In order to get spatial thermal information on the wall, it is necessary to use a high speed infrared camera (IRC), for example chosen by [START_REF] Visentini | Experimental investigation of heat transfer in transient boiling[END_REF]- [START_REF] Baudin | Turbulent ow and transient convection in a semi-annular duct[END_REF]- [START_REF] Schei | Transient ow boiling in a semi-annular duct: From the onset of nucleate boiling to the fully developed nucleate boiling[END_REF]- [START_REF] Kossolapov | The Boiling Crisis of water under exponentially escalating heat inuts in subcooled ow boiling at atmospheric pressure[END_REF], allowing the record of the time-dependent temperature and heat ux elds. This diagnostic enables the access to ne physical phenomena and quantitative information such as heat transfer during bubble nucleation, HFP... However, the diculty to set such an experiment up, the potentially heavy post-processing techniques needed and the price of the associated IRCs make this option scarcer in the literature.

Chapter 1. Nuclear Safety Challenge and State of the Art Steady-state boiling crisis has been the most investigated situation, as it is the most needed for industrial applications. However, the eld of transient boiling crisis has also been investigated for safety design purpose, particularly in the eld of nuclear applications. The eld of transient boiling crisis may refer to three kinds of transients: Flow, pressure and power transients. In ow or pressure transients, the ow rate or the pressure is decreased or instantaneously stopped corresponding to a Loss-Of-Coolant Accident (LOCA), and in a power transient, the heat ux is increased corresponding to a RIA. In the present work, only power transients are considered and will be referred to as transient boiling crisis experiments. An overview of the ow and pressure transient boiling crisis experiments can be found in the review of Ahmed et al. [START_REF] Ahmed | Towards understanding the critical heat ux for industrial applications[END_REF]. The experimental diculties of transient boiling crisis consist in the limitation of the instruments. The electrical supplier needs to provide a high power input (a few kiloWatts) at a high power generation rate (time scale of a few milliseconds) and the recording instrumentation should be fast enough to produce exploitable data. In the present work, we will focus on the transient ow boiling crisis. Experimental investigations will be conducted at moderate pressure under exponential power escalations with water. An overview of the existing experiments is summarized in table 1.3.4.3. Note that in such conditions, experiments with high-resolution diagnostics are really scarce. In the literature, a variety of power transients are investigated: ramp, square... [START_REF] Visentini | Experimental investigation of heat transfer in transient boiling[END_REF]- [2] as well as transient pool boiling in water [START_REF] Rosenthal | An experimental study of transient boiling[END_REF]- [START_REF] Sakurai | Transient pool boiling heat transfer Part 1: incipient boiling superheat[END_REF]- [START_REF] Sakurai | Transient pool boiling heat transfer. Part 2: boiling heat transfer and burnout[END_REF]- [START_REF] Sakurai | Mechanisms of transitions to lm boiling at CHFs in subcooled and pressurized liquids due to steady and increasing heat inputs[END_REF]- [START_REF] Park | Transient CHF phenomena due to exponentially increasing heat inputs[END_REF] and ow boiling crisis in other liquids [START_REF] Visentini | Experimental investigation of heat transfer in transient boiling[END_REF]- [START_REF] Schei | Étude expérimentale et modélisation du transfert de chaleur de l'ébullition transitoire[END_REF].

Boiling Regimes

Investigation U bulk [m.s ¡1 ] ∆T sub [K] p [bar] τ [ms]
Heater Geometry Diagnostic (1971 -[39]) Johnson Despite the numerous experimental results available in the literature, generating numerous correlations and LUTs, signicantly fewer mechanistic models exist attempting to predict the critical heat ux under subcooled ow boiling and steady-state conditions. Reviews have been proposed in the literature, for instance by Tong and Tang [START_REF] Tong | Boiling heat transfer and two-phase ow[END_REF], Kandlikar [START_REF] Kandlikar | Critical heat ux in subcooled ow boiling -An assessment of current understanding and future directions for research[END_REF], Celata [START_REF] Celata | Critical Heat Flux, Post-CHF Heat Transfer and Their Augmentation[END_REF] or Ahmed et al. [START_REF] Ahmed | Towards understanding the critical heat ux for industrial applications[END_REF]. It appears that only sparse mechanistic approaches exist which will be presented in the following.

Boundary Layer Separation Model

Kutateladze and Leont'ev [START_REF] Kutateladze | Some applications of the asymptotic theory of the turbulent boundary layer[END_REF] proposed a pure hydrodynamical model disregarding thermal eects. The bubble's generation process is modeled as a gas injection throughout a porous wall and the latter interferes with the dynamic boundary layer. A critical bubble injection mass ux leads to the separation of the boundary layer inducing the disappearance of the ow at the wall. The critical mass ux is expressed in equation 1.17 and described in gure 1.20. Therefore, the heater is no longer cooled which leads to the boiling crisis.

pρ inj v inj q crit = 2f 0 ρ 0 U 0 (1.17) Kutateladze and Leont'ev proposed that the ow boiling crisis is due to two additive contributions: CHF in pool boiling and the heat ux needed to separate the dynamic boundary layer (q P CHF,f low = q P CHF,pool q P BL ). Tong [START_REF] Tong | Boundary-layer analysis of the ow boiling crisis[END_REF] decided to assume only the boundary layer separation. Nevertheless, the quantity ρ inj and v inj are arduous to model, estimate and to link with q P CHF . This diculty leads this approach to use existing experimental data to close the conditions by correlations: the Tong's approach generated the Tong-68 correlation. Celata [START_REF] Celata | Rationalization of existing mechanistic models for the prediction of water subcooled ow boiling critical heat ux[END_REF] improved the correlation with the use of more than 1800 data points leading to the correlation known as the modied-Tong correlation having the following expression: As it can been seen, this approach is dicult to adapt and needs a correlation using a lot of experimental data to be applicable. Then, this approach appears more as a correlation than as a mechanistic model. Also to conclude, the hydrodynamic patterns which should result from this purely hydrodynamic and adiabatic approach were not experimentally observed close to CHF. This is for instance reported in the work of Mattson [START_REF] Mattson | A Photographic Study of Subcooled Flow Boiling and the Boiling Crisis in Freon-113[END_REF].

q P CHF = C h lv ρ l U 0 c Re (1.
Interfacial Lift-o Model velocity promotes the development of the KH instability generating thick dry patches and wetting front. In the latter, heat transfer is signicantly more ecient than in the dry patches. The wetting fronts face an over pressure (P v ¡ P l ) on the liquid side due to surface tension and the curvature direction pushing the liquid towards the wall. The latter is calculated with the KH instability theory and needs one single parameter determined by photographic study. On the wall side, the interface is pushed away from the wall because of the vapor generation due to boiling. Consequently, there is a competition between the two opposite eects, which is dominated by (P v ¡ P l ) at low heat ux. CHF is dened as the heat ux which balances the two term. From an analytic development, Galloway and Mudawar obtain the following expression for the CHF:

q P CHF = ρ v pc p,l ∆T sub h lv q d P v ¡ P l ρ v (1.19)
Sturgis and Mudawar [START_REF] Sturgis | Critical heat ux in a long, rectangular channel subjected to one-sided heating-I. ow visualization[END_REF]- [START_REF] Sturgis | Critical heat ux in a long, rectangular channel subjected to one-sided heating-II. Analysis of critical heat ux data[END_REF] explore the inuence of the subcooling and the ow condition on the model. The experiments were realized on a long plate heater (10 cm) at atmospheric pressure with FC-72. They show that this approach presents a good prediction for near-saturation conditions (∆T sub = 3K) and for dierent values of bulk ow Chapter 1. Nuclear Safety Challenge and State of the Art (0.25 to 6 m.s ¡1 ). However, at higher subcooling (∆T sub = 29K), the model prediction presents inconsistent trends and inaccuracies. The authors suppose that these discrepancies are due to the rising impact of the condensation eect, which is not not considered in the model.

Criteria in the bubbly layer

This kind of approach relies on the existence of a bubbly layer at the vicinity of the wall.

For subcooled conditions, it considers a certain quantity and realizes an interaction budget with its environment. Finally it sets a limiting criterion on the considered quantity which characterizes the boiling crisis. There are two kinds of models using this approach: the critical enthalpy in the bubbly layer [START_REF] Tong | Inuence of axially non-uniform heat ux on DNB[END_REF] and the Bubble Crowding Model [START_REF] Weisman | Theoretical prediction of maximum heat ux in power transients[END_REF].

Tong [START_REF] Tong | Inuence of axially non-uniform heat ux on DNB[END_REF] proposes to study the enthalpy in the bubbly layer. Along the heater, the upstream subcooled liquid is heated by the saturated liquid generated by the heater and progressively increases its enthalpy until reaching a critical value. Beyond this particular value, quenching brings subcooled water to the surface which avoids the development of a vapor lm. When the bubbly layer reaches its critical enthalpy, there is no longer quenching and bubbles swell to large vapor pockets triggering DNB. The simplied energy model is described in gure 1.22. Then, some simplifying hypothesis are made slightly before DNB:

Void fraction is supposed to be uniform along the heater, implying the same values of the averaged thermophysical properties in the bubbly layer (ρ, c p ) the bubbly layer thickness s and the streamwise average velocity in it V are also uniform along the heater. The average temperature is approximately uniform The heat transfer coecient h from the bubbly layer to the bulk ow is approximately uniform.

The energy balance can then be simplied to the following equation:

d(H ¡ H b ) dz C(H ¡ H b ) = C c p q P h , with C = h ρV sc p (1.20)
where:

1

.3. Boiling Regimes

H is the average enthalpy of the bubbly layer H b is the average enthalpy of the bulk

The present equation cannot be closed mechanistically as the terms composing C are not suciently known. The author determines C through a correlation :

C = 0.15 (1 ¡ X DN B ) 4.31 (G ¢ 10 ¡6 ) 0.478 [in. ¡1 ]
Because of the lack of physical description of the key parameters involved in the model, this approach is dicult to adapt and needs a closure correlation. However, this approach takes into account the ow history eect in a simplied way and it also enables to have a non-uniform heat ux distribution.

Introduced by Weisman and Pei [START_REF] Weisman | Theoretical prediction of maximum heat ux in power transients[END_REF], the Bubble Crowding Model also relies on the study of a bubbly layer, interacting with its environment: some bubbles undergo an upward motion along the wall; some bubbles migrate towards the core region due to the turbulent uctuation velocity. As the heat ux increases, more and more bubbles are generated impeding cooling by liquid from the core region. CHF is considered as reached when the void fraction in the bubbly layer reaches a certain critical value. Here, instead of studying the energy balance in the bubbly layer as in the Tong's model presented above, Weisman and Pei study a balance of mass which is described in gure 1.23: The process gives an expression of CHF as follow:

q P CHF = G 3 h lv (x 2,crit ¡ x 1 ) ¢ h l,sat ¡ h l,d h l ¡ h l,d (1.21)
where:

G 3 is the mass ow rate from the core to the bubbly layer x 2,crit is the quality in the bubbly layer, corresponding to the critical void fraction.

x 1 is the quality in the bulk ow. h l is the liquid enthalpy for the bulk conditions.

Chapter 1. Nuclear Safety Challenge and State of the Art h l,sat is the liquid enthalpy at saturation conditions. h l,d is the enthalpy at the point of bubble detachment.

The innovation of this approach relies in the term G 3 where the core water intake is due to the turbulent velocity uctuations, which is expressed as follow:

G 3 = ρ » V v 11 (v I ¡ v 11 )P (v I ) dv I (1.22)
where v 11 is the mean velocity at which vapor is ejected from the wall. G 3 can be seen as the results of an eective transport velocity

P (v I ) = 1 c 2πv I RM S e ¡ 1 2 ¡ v I /v RM S © 2 (1.23)
In order to use quantitatively their model, Weisman and Pei experimentally determined the critical void fraction, assessed equal to 0.82. Despite the fairly good results obtained with this model in the comparison study of Inasaka & Nariai [START_REF] Inasaka | Evaluation of subcooled critical heat ux correlations for tubes with and without internal twisted tapes[END_REF], serious limits also appear. First, no theoretical evidence is given for the void fraction criterion: there is no clear phenomenology reason why there should be one (lower than 1, obviously). Secondly, its accuracy is doubtful as Styrikovich [START_REF] Styrikovich | The eect of two-phase ow pattern on the nature of heat transfer crisis in boiling[END_REF] found that the void fraction in the bubbly layer at CHF can vary from 0.3 to 0.95.

Liquid Sublayer Dryout (LSD)

This model has received the greatest interest during the last three decades. It assumes that departed bubbles form vapor blankets which are then advected by the forced ow (see gure 1.24). These vapor blankets, whose length equals the KH instability critical wavelength, are trapped in a superheated layer (SHL) dened as the region near the wall whose temperature exceeds the saturation value. No contact exists between the vapor blankets and the heating wall, which implies the existence of a liquid sublayer, sometimes called macrolayer, between the blankets and the wall. The boiling crisis is stated to occur when the heat ux at the wall is high enough to vaporize the whole liquid sublayer in a time period corresponding to the passage of a vapor blanket.

Introduced by Lee & Mudawar [START_REF] Lee | A Mechanistic critical heat ux model for subcooled ow boiling based on local bulk ow conditions[END_REF], this model was developed for pool boiling and worked satisfactorily for high pressures up to 50 bars [START_REF] Celata | Assessment of correlations and models for the prediction of CHF in water subcooled ow boiling[END_REF]. Katto modied and extended the model for a pressure range within 1 to 200 bars [START_REF] Katto | A prediction model of subcooled water ow boiling CHF for pressure in the range 0.1-20 MPa[END_REF]. All these approaches rely on experimental constants. Celata [START_REF] Celata | Rationalization of existing mechanistic models for the prediction of water subcooled ow boiling critical heat ux[END_REF], starting from a similar model used physical considerations to get rid of these experimental constants. The overall expression of CHF is given as follows:

q P CHF = ρ l δ h lv L B U B (1.24)
where:

δ is the initial thickness of the liquid sublayer L B is the length of the vapor blanket U B the vapor blanket velocity.

In the approach of Celata [START_REF] Celata | Rationalization of existing mechanistic models for the prediction of water subcooled ow boiling critical heat ux[END_REF], quantities are determined by an iterative process involving physical considerations. In the paper of Inasaka & Nariai [START_REF] Inasaka | Evaluation of subcooled critical heat ux correlations for tubes with and without internal twisted tapes[END_REF], eleven models and correlations were compared and showed that a good prediction may be obtained using the 1.3. Boiling Regimes

Mechanistic approaches under transient heat ux

As seen above, several dierent approaches have been developed for steady-state subcooled ow boiling. Nevertheless, models focusing on boiling crisis for subcooled ow boiling under transient heating conditions are signicantly scarcer.

Based on the LSD approach of Katto [START_REF] Katto | A prediction model of subcooled water ow boiling CHF for pressure in the range 0.1-20 MPa[END_REF], Serizawa [START_REF] Serizawa | Theoretical prediction of maximum heat ux in power transients[END_REF] proposed a description for the transient boiling crisis, valid for pool and ow boiling, with or without subcooling. He proposed that once the heat ux has reached the steady-state CHF value, the transient CHF corresponds to the heat ux increase during the time period needed to vaporize the liquid sublayer. To this end, he realizes a energy balance in the macrolayer and includes a term of liquid supply from the surrounding. This model makes use of several constants which must be determined through experiments.

Pasamehmetoglu [START_REF] Pasamehmetoglu | Critical Heat Flux modeling in forced convection boiling during power transients[END_REF] - [START_REF] Pasamehmetoglu | Critical Heat Flux modeling in pool boiling for steady-state and power transients[END_REF] investigated specically the case of exponential heat ux excursions and permitted to reduce the number of experimental constants by introducing a modeling of the heat transfer due to liquid turbulence. Even though this approach is similar to Serizawa, it studies the macrolayer thining through two cumulative mechanisms: thermal thinning due to liquid evaporation and hydrodynamic thinning due to KH instabilities. From the time assimilated to CHF in steady state t CHF,SS , Pasamehmetoglu attempts to calculate the time needed to dryout the macrolayer (see gure 1.25) and denes it as the time of CHF in transient conditions t CHF,T R .

Pasamehmetoglu suggests that the time-derivative of the macrolayer thickness results

B 0 ¡ 1 ¡ Av Aw © ¡ 1 K cp∆T sub h lv © ¡ Liquid supply hkkkkkkikkkkkkj W f ρ l (A w ¡ A v ) loooooooooooooooooooooooooooomoooooooooooooooooooooooooooon Thermal thinning
] where:

δ is the thickness of the liquid sublayer.

A v is the heater area covered by vapor.

A w is the total heater area. W f is the transient liquid supply rate. K is a correction factor for vapor mass behavior in pool boiling with high subcooling. δ c is the thickness of the liquid sublayer due to KH instability (subscript 0 in gure 1.25 at time of steady state CHF). The latter is given by the following expression:

δ c = π 2 σ ¢ ρ l ρ v ρ l ρ v 1 B 2 o ¢ A v A w 2
The liquid supply rate term takes into account the turbulent velocity uctuations in the case of ow boiling conditions. As Weisman and Pei (see above), Pasamehmetoglu considers the uctuation component perpendicular to the wall and the same characteristic velocity calculation

³ V v 11 (v I ¡ v 11 )P (v I ) dv I .

Nonetheless, where Weisman and Pei use a

Gaussian distribution, which is a good approximation far from the wall, he assumes a hyperbolic distribution as

P (v I )W 1 v I n I 1.3. Boiling Regimes with n I [2,
V] being a tuning constant. The author chose such a questionable distribution for two reasons: computational convenience and similar trend than Gaussian distribution for large v I values. He can then connect the liquid supply rate to the heat ux:

W f = £ q P w,CHF,SS q P w n A w q P w,CHF,SS h lv c p ∆T sub with n = n I ¡2 [0, V].
The author nds that a value of n = 2 gives the best t for the data set he uses. Finally, once the time at transient CHF t CHF,T R is calculated, he can then calculate the ratio between transient and steady state CHF through the exponential nature of the heat ux input:

η = q P w,CHF,T R q P w,CHF,SS = exp ¢ t CHF,T R ¡ t CHF,SS τ (1.25)
with τ the exponential excursion period and concludes with the value of the transient CHF as presented in gure 1.26. One can also notice the sensibility of the model to the exponent n. These two mechanistic approaches calculate transient CHF by assessing the excess of heat ux compared to the steady-state conditions. In this direction, Chang et. al. [START_REF] Chang | Transient-eects modeling of critical heat ux[END_REF] proposed a map classifying the transient CHF according to their assumed limiting phenomena. As presented in gure 1.27, this map represents a four-regime classication relying on two parameters which are the thermodynamic quality and the mass ux. The authors note that the vertical limit between the regime I and the other ones is arbitrary dened by G = 100 kg.m ¡2 .s ¡1 . Besides, their approach is applied to a transient heat ux input and to a transient inlet mass ux.

For each regime, a transient correction factor is dened as above as η = q P w,CHF,T R q P w,CHF,SS . The latter is the result of the corresponding transient eect, as presented in table 1.4. The correction factor η m taking into account the macrolayer depletion is assessed with the approach of Serizawa and Pasamemhetoglu. The correction factor η u taking into account the upstream history eect is assessed with correlations. Focusing on transient heat ux input, it is interesting to note that this classication does not take into account the transient nature of the heat input: the trend of the transient (linear, exponential,...) or the characteristic time of the power escalation. Indeed, it would be expected that the mechanisms leading to boiling crisis would depend on it, especially for fast transients.

One can notice that the presented approaches aim on determining a transient correction factor. It considers that the transient boiling follows the steady-state path on the boiling curve and phenomenology, the specicity from transient cases coming only after reaching the steady-state critical heat ux. Besides, due to the limitations of instrumentation at that time, experimental data used by Serizawa and Pasamehmetoglu to develop their models were intrinsically space-averaged values. For fast transients, these averaged values only provide a limited information. The Liquid Sublayer Dryout phenomenology was then generalized for conditions for which no experimental observations of the mechanism involved in the boiling crisis could be done.

Objectives

In this chapter, we presented the BORAX-type accident, an accidental scenario of key importance to dimension the safety design of experimental reactors. Due to the physics of neutronic feedbacks and heat transfer, if the core reaches the critical heat ux during such a transient, this may lead to core failure. In order to have an overview of the heat transfer mechanisms involved during such a transient, we presented the state of the art 1.4. Objectives for single-phase and two-phase heat transfer description. Particularly, we focused on the ow boiling crisis and the transient ow boiling crisis. We have shown that there is a clear lack of understanding and modeling of the transient ow boiling crisis. Consequently, to ll this gap, particularly in conditions applicable to experimental reactors, this thesis will investigate the transient ow boiling crisis at high subcooling and moderate pressure.

In order to improve the understanding and the modeling of the dierent heat transfer phenomena involved in the BORAX-type accident, a partnership between CEA and MIT has been realized. It aims to experimentally investigate the dierent aspects of the transient heat transfer from single phase to boiling crisis. Dierent experiments were designed and manufactured in order to record and observe the involved mechanisms using high speed visible and infrared cameras with precisely controlled conditions of temperature, ow conditions and power escalation period. The record of such high detailed data implies the generation of a massive amount of information, about several Terabits of high speed videos and IR thermography, and more data is yet to come. It is important then to use them properly in order to build a physical understanding of the boiling processes during such transients. In this context, the objectives of my Ph.D. work are to:

1. Contribute to the production of new experimental data on transient ow boiling crisis at high subcooling in a narrow channel. The novelty of this new experimental campaign is also to investigate the impact of the pressure. This objective includes my contribution to the setting of the new designed apparatus, to adapt the existing post-processing tools and to assess the value of CHF by analyzing the data.

2. Analyze the massive amount of data of transient CHF. This includes parametric trend investigation, high speed video and IR thermography observations as well as the development of the required tools.

3. Propose a model describing the transient ow boiling crisis taking into account the parameters of the problem. These parameters are the pressure, the temperature, the ow conditions and the power escalation period.

In Chapter 2, we present the experimental aspects of this thesis. We describe the experimental apparatus in which the tests are realized with precisely controlled conditions of ow, pressure, temperature and power escalation period. We present the dierent equipments and set up enabling to record the phenomenon with a high space and time resolution and how the raw data are post-processed. Finally I present the test matrices and how I contribute in this experimental campaign and the wider experimental partnership between CEA and MIT.

At the beginning of the thesis, experimental data using high resolution diagnostics had already been produced by the group at MIT for transient ow boiling crisis at high subcooling and atmospheric pressure. In Chapter 3, we present the analysis of these data and outcome of the observation of the numerous visible and thermography videos. We then explain how it leaded us to propose a model describing the transient ow boiling crisis and how this model is mathematically formulated and non-dimensionalized. Finally, we discuss the qualitative and quantitative predictability of this model.

In Chapter 4, we present the new experimental results of transient ow boiling crisis experiments at high subcooling and moderate pressure, which I contributed to produce. We also present an investigation of the relevancy of the usual parameters characterizing the ow on the transient ow boiling crisis. Finally, we present the extension of the model Chapter 1. Nuclear Safety Challenge and State of the Art presented in Chapter 3 to this new wider range of conditions and re-discuss its qualitative and quantitative use for boiling crisis prediction. My Ph.D. work is part of an experimental collaboration between CEA and MIT aiming to a better modeling and understanding of the dierent phenomena involved during a BORAX-type accident. More particularly, the experimental part of my Ph.D., which stood from July 2018 to June 2019, aims to contribute to the achievement of the nal phase of this experimental collaboration. In this Chapter, I present a general description of the experimental apparatus and the methods of data processing (raw data post-processing and DNB determination) used for this work, and my contributions to their extension to the new apparatus. Then, I present the implementation of the Schlieren visualization technique, that I suggested and realized, to the existing apparatus as well as its outcomes. Finally, once the technical background of the experiment presented, I present in details the experimental collaboration between the two institutes and how my work ts in, to better apprehend my contribution and to introduce the test matrices analyzed in chapters 3 and 4.

Chapter 2. Experimental Investigations 2.1 Experimental Apparatus

Flow Loop

The ow loop is the core component for the investigation of ow boiling heat transfer. It is made of stainless steel 316L components and covered with thermal insulator (see photo on gure 2.1). The working uid used is deionized (DI) water and the thermohydraulic conditions are precisely controlled. A variable frequency pump Grundfos CRNE3-12 X-FGJ-G-F-HQQE is used to generate the owrate (up to 0.8 L.s ¡1 ). A preheater and a chiller are used to control the water temperature (from 25 o C to 180 o C). An accumulator permits to control the loop pressure (from 1 to 14 bar) by the means of a compressed nitrogen tank. The main duct is divided in two junctions in order to have the possibility to use two dierent test sections (one at a time) without the need of a time-consuming modication of the ow loop. The ow is directed vertically upwards at the location of the test sections as it is generally the case in experimental nuclear reactors. Located on the left branch of the ow loop (picture of gure 2.1), the rst test section (TS1) had been formerly developed in order to investigate how the mass ux and the subcooling impact transient boiling. This test section is presented in gure 2.2. The results of these investigations can be found in [START_REF] Richenderfer | Investigation of subcooled ow boiling and CHF using high-resolution diagnostics[END_REF]- [START_REF] Kossolapov | The Boiling Crisis of water under exponentially escalating heat inuts in subcooled ow boiling at atmospheric pressure[END_REF]. The test section TS1 is made of stainless steel and includes four openings: three quartz windows for optical access and a fourth one to hold the cartridge. This cartridge is a worked piece of Shapal TM ceramics which is made to place the heater at the channel wall. It is electrically non-conductive. The heater is held on the cartridge by front clamps. The electrical power is supplied from behind by the means of the cooper leads. The test section TS1 holds a 3 ¢ 1 cm 2 rectangular channel, 1 cm being the distance between the heater and the facing wall. The heater is placed roughly 65 hydraulic diameters downstream the beginning of the rectangular channel in order to ensure a fully developed dynamic boundary layer. In order to have such length downstream the heater, the vertical pipe below the test section is also made of a rectangular channel which is smoothly connected to the test section.

Located on the right branch of the ow loop (picture of gure 2.1), the second test section (TS2) was developed in order to investigate the eect of pressure, subcooling and high mass ux on transient boiling crisis in a narrow channel. The test section and the corresponding heater are presented in gure 2.3. The TS2 is mostly similar to the TS1. The test section is also modular in term of heating apparatus: a second heater can be placed instead of the front quartz window. For the present project, the conguration is a rectangular channel with a dimension of 18 ¢ 1.95 mm 2 , which uses a single heating cell. Conversely to TS1, the entrance length is directly integrated within the test section: the heater is placed 20 cm downstream the channel entrance. This location enables the complete development of the dynamic boundary layer in the direction normal to the heater (1.95 mm in width), but not in the transverse direction (18 mm in width) avoiding wall eects. The narrowness of the channel implies the design of a non-intrusive clamping system embracing the shape of the channel wall. Finally, the TS2 cartridge is compatible with the TS1 test section, allowing for the use of this cartridge and the corresponding heater with TS1 and its channel. The opposite is not possible as the clamps of the TS1 cartridge overcome the TS2 width.

Heaters

The heater is the centerpiece of the boiling investigations. Fixed on the cartridge, the function of the heater is to provide the necessary thermal power to boil water while resisting to both mechanical and thermal stresses. It has to be transparent to infrared (IR) radiations since IR thermography is used. The mechanical stress is mostly due to the pressure dierence between the inside and the outside of the duct, up to 14 bars, and the force applied by the clamp to x the heater on the cartridge and avoid any leakage. The thermal stress is due to the temperature unhomogeneity during transient heatings, the energy coming from its surface. The choice was made to use a Sapphire substrate coated with a thin lm. The latter is chosen electrically conductive in order to heat by Joule eect. It is also thin, several hundreds of nanometers, in order to neglect its heat capacity and to assume the temperature as homogeneous in the thickness. The lm material also needs to be opaque to IR radiation for the IR thermography (see section 2.2). Two materials, Indium Tin Oxide (ITO) and Titanium, were used to this end. Metallic pads, made of silver or gold, ensure the electrical connection between the active lm (ITO of Titanium) and the back side of the heater, where electric power is supplied. A scheme of the heater design can be found in gure 2.4.

The TS1 heater is made of a square substrate with a dimension of 20 ¢ 20 ¢ 1 mm 3

Test Section

Cartridge

Assembled view Exploded view This heater is made longer than the TS1 heater in order to investigate signicantly higher mass uxes, a length eect being expected. The active heating area diers accordingly to the heat ux needed: for a given power input, the heat ux is higher for a smaller active area. The active surface ranges from 40 ¢ 6 mm 2 to 10 ¢ 4 mm 2 (see section 2.4 for the history).

Data Acquisition

General information on the working conditions is monitored by the means of sensors located at various places on the ow loop. The main sensor sockets and their location are presented in gure 2.1 (scheme). In practical use, only a few of them are used to control the ow loop conditions. The water temperature is monitored by two type-T thermocouples, located at the inlet and outlet of the test sections. The loop pressure is monitored by three pressure transducers, located at the inlet and outlet of the pump, and at the outlet of the test section. Lastly, the mass ux in the test section is monitored by the means of the turbine owmeter located at the pump outlet. The volumetric ow rate is recorded then converted to a mass ux using the hydraulic diameter of the test section channel and the measured values of temperature and pressure. During an experimental run, the dierent high speed data acquisition systems (electrical input, High Speed Visible (HSV) camera and High Speed InfraRed (HSIR) camera) are simultaneously initiated by a 5 V triggering signal. The trigger enables the synchronization of the dierent systems and also avoids the recording of a massive quantity of dispensable data: the full recording (electric, HSIR, HSV) of a 100 ms-long experiment is about 1 Gb of data. The electrical power, P = U I, is determined by the measurement of the current I in the circuit and the voltage U at the heater terminals. These two time-dependent signals are recorded by a high speed data acquisition system (Agilent U2542A) running with a sample rate of 20 kHz.

As presented in section 2.1.1, the test sections are designed with an optical access to the experiment. Set-up options are illustrated in gure 2.5. The IR thermometry is realized by the means of an IRC806 high speed infrared camera, which is cooled by liquid nitrogen in order to avoid any in-camera IR radiation interfering with the IR sensors. The camera records the radiation emitted in the medium-IR spectrum, mainly in the range 3 ¡ 5 µm, in the form of a time-dependent eld of counts representing the number of photons received by the pixels. The spectrum 3 ¡ 5 µm is of interest in this work as it corresponds to the temperature range involved in experiments with boiling water at atmospheric and moderate pressure. As the HSIR camera does not directly record a temperature information, the data have to be post-processed in order to be used for IR thermography (see section 2.2). The frame rate of the HSIR camera can be as high as 3125 frames per second (FPS) depending on the number of pixels involved. Indeed, high speed cameras (HSV or HSIR) frame rates are mainly limited by the data rate: the smaller is the number of recording pixels, the greater the frame rate can be. The HSIR camera records the radiation emitted from the back of the heater by the means of an IR mirror (see gure 2.5), which perfectly reects IR radiations but is transparent to visible light. The Sapphire substrate being almost perfectly transparent to mid-IR wavelengths, we can visualize the thermal pattern on the coating lm as ITO or Titanium are opaque to IR radiation. Indeed, it means that the lm radiates accordingly to its temperature eld while perfectly shielding the radiation coming from the water behind. The IR video is then post-processed (see section 2.2) in order to extract the time-dependent wall temperature and heat ux elds.

The photographic visualization can be realized throughout the three quartz windows with a Phantom v25.12 high speed camera. The frame rate can be as high as several hundreds of thousands of frames per second according to the number of pixels recorded.

As for high speed recording the integration time is really short, i.e. time during which pixels record the incident photons, objects to be recorded must be heavily lit. The most eective setup is the so-called backlit shadowgraphy (see scheme in gure 2.5): A LED light is placed behind the test area. Objects opaque to this light, such as bubbles, appear as shadows with a really good contrast. This technique is always used for the side view, position HSV 1 in gure 2.5. The front view visualization depends on the coating. If the heater is coated with an ITO lm, backlit shadowgraphy can be used (lightening the heater from behind) since ITO and the IR mirror are transparent to visible light (position HSV 2 in gure 2.5). This conguration is not possible with a Titanium coated heater, as titanium is opaque to visible light. However, this coating well reects incident lights and can then be used for frontlit shadowgraphy. The LED light is placed on the front of the heater with a slight angle and the camera is placed at the maximum reection angle. Yet, this technique has a slightly poorer contrast compared to backlit shadowgraphy.

Experimental Protocol

After having presented the dierent elements of the experimental apparatus, let us describe the conduct of an experimental day in order to have a better apprehension of this apparatus.

At the end of every experiment day, the ow loop is drained of its water in order to avoid any possible corrosion. The rst step of the protocol is then to ll the loop with DI water by trying to avoid as much as possible remaining air bubbles in the pipes. After lling the loop, water must be degassed of unwanted dissolved non-condensable gas. To this end, a slight ow rate is imposed by the pump and the loop is derived towards an exchange-membrane device connected to a vacuum pump. When water ows across the device, dissolved non-condensable gas are pumped out, leaving the liquid depleted of its dissolved gas. The degassing process lasts for about two hours, a time which is usually used to properly place the optical setup: cameras, mirrors and light sources. Then, the HSIR camera calibration process is performed. It is made to identify the relationship between the heater temperature and the signal of the photons counts received by the camera. It takes into account all parasitic IR radiations from the environment by using a radiation model that will be presented in section 2.2. To this end, the temperature of the liquid water is progressively increased by the preheater while a slight owrate is imposed in the loop. The temperature rise is suciently slow to consider that both the temperature of water and the temperature of the heater are equal, at least for a given time. During the process, the water temperature measured by the thermocouples and the signal acquired by the HSIR camera are simultaneously recorded, usually every 5 o C from 25 o C to 110 o C. The result is a calibration curve which gives the HSIR signal counts as a function of the temperature. An example is presented in gure 2.6. This calibration 2.1. Experimental Apparatus curve is used later on in the post-processing procedure (see section 2.2). The calibration process lasts for about 1.5 hour. The uid is placed at the desired conditions of ow, pressure and bulk temperature. Once these conditions are stabilized, the test signal is prepared. The electrical input signal is rst generated by a programmable function generator (Rigol DG1022U). It is next amplied by two programmable power supplies (Chroma 62050P-100-100 DC) connected in parallel, generating a power up to 10 kW. For an exponential transient, the input signal is an exponential function dened by its escalation period and its peak voltage. After having reached the peak voltage, the signal is decreased following an exponential decrease of a period about 5 ms. In order to reach DNB conditions and to measure the CHF for a given working condition and escalation period, we proceed sequentially. Tests are realized, and as long as no manifestation of a dry spot is observed with the HSIR camera, the peak voltage is progressively increased. Dry spots are characterized on the HSIR videos by a localized and rapid rise of the signal counts, translating into a similar rise of the wall temperature. This sequential increase of the peak voltage is realized carefully: if the peak power reaches a value signicantly higher than the power needed to trigger the boiling crisis, the heater could remain in lm boiling during a too long period of time. The wall temperature could be high enough to generate a thermal shock and therefore the failure of the heater.

Once dry spots are clearly observed with the HSIR camera, the test is realized at least two more times in order to record three trials or more for the same working condition (water temperature, pressure, mass ux, escalation period). This ensures the repeatability of the test. A new working condition is then established and tested until all the test matrix is investigated. Nonetheless, it is worth to note that boiling crisis can only be conrmed after the post-processing, which gives the wall temperature and Heat Flux to Water videos. This is not a real-time process but takes place after the experiments.

Chapter 2. Experimental Investigations

Data Processing

Once the day of experiment ends, several steps of post-processing are required to nally obtain the exploitable data: wall temperature distribution, heat ux transferred to water distribution and critical heat uxes. This section describes these dierent steps.

Determination of the Temperature and Heat Fluxes elds

The heating lm (ITO or Ti) is designed thin and opaque to IR radiation. The lm is thin enough to assume that its thermal capacity is negligible and to assume a negligible temperature variation in its thickness. The latter property enables the assimilation of its 2-D temperature eld to the boiling surface one. As it is opaque, it shields the radiation emitted by water. The choice of Sapphire as a substrate was leaded by the constraints of robustness and transparency to IR radiations (section 2.1.2). However, sapphire is not a perfectly transparent material in the spectrum of interest (3 ¡ 5µm). This implies that it emits radiation due to its own temperature and also partially absorbs the incident radiation. Finally the environment, i.e. the experimental room, also emits background IR radiation which has to be taken into account. In order to properly determine the boiling surface temperature and Heat Flux to Water distributions, it is then necessary to explicit these contributions resulting to the HSIR observations: boiling surface radiation, substrate absorption/emission, background radiation and the reections at the dierent interfaces (gure 2.7). To this end, we use the conduction-radiation model inverse method presented by Bucci et al. [START_REF] Bucci | A mechanistic IR calibration technique for boiling heat transfer investigations[END_REF]. This method estimates the 3-D temperature distribution in the whole substrate corresponding to the IR radiation observed by the HSIR camera, considering all the contributions described above. This is made for all the recorded frames. Once the 3-D time-dependent temperature distribution assessed, it is then possible to determine the time-dependent boiling surface temperature and Heat Flux to Water distributions. [START_REF] Bucci | A mechanistic IR calibration technique for boiling heat transfer investigations[END_REF] To this end, the heater is assimilated to its substrate which denes the physical domain for the solution. The latter is discretized using a regular hexahedral meshing 2.2. Data Processing (see gure 2.8). In a x ¡ y plane, the domain underneath the active area are squareshaped cells corresponding to the HSIR camera pixels. The IR light source is assumed to be planar [START_REF] Bucci | A mechanistic IR calibration technique for boiling heat transfer investigations[END_REF], radiating in the z direction. The surrounding domain meshing is less rened, the temperature distribution being more uniform. For the TS1 heater geometry the meshing in the z direction is uniformly distributed as the sapphire is thinner and the side eects are negligible. However, the TS2 heater having a thicker substrate and having active surfaces comparatively smaller to the substrate x-y cross section, it was chosen to realize a better renement near the active surface -substrate interface. To have an order of magnitude of the mesh sizes, for a TS1 heater observed with IR frames having a denition of 84 ¢ 84 pixels, the mesh size is 200,000 cells. For a TS2 heater having a 40 ¢6 mm 2 active area and observed with IR frames having a denition of 452 ¢58 pixels, the mesh size is around 2.8 million cells. Once the physical space is discretized, an algorithm processes each frame of the input HSIR video. A functional diagram is presented in section 2.9. For a given time step, the active heating area temperature (T ¦ ito in scheme g. 2.9) is guessed by the means of the counts observed by the HSIR camera. The heat conduction problem is then solved by a 3-D nite volume implicit scheme using the space discretization presented above. All boundaries are modelled by adiabatic walls except the one representing the active heated area (red mesh on gure 2.8). It results in a 3-D temperature distribution in the substrate (T ¦ (x, y, z, t) in scheme g. 2.9). Using the calibration curve, the radiation model computes the corresponding IR radiation signal that the HSIR camera would receive from the present computed temperature eld (R ¦ (x, y, t) in scheme g. 2.9). This computed signal R ¦ (x, y, t) is then compared with the observed signal dis- tribution R(x, y, t). If the two distributions are not suciently similar, a new guess is established using Newton's method and the radiation model. If the two counts distributions are suciently similar, the 3-D temperature eld is kept as the initial conditions Figure 2.9: Functional diagram of the post-processing algorithm [START_REF] Bucci | A mechanistic IR calibration technique for boiling heat transfer investigations[END_REF] for the following step. A time step usually converges in less than 5 iterations. The time step used for the transient problem is given by the HSIR frame rate, and is about a few tenth of milliseconds. This was assessed to be suciently small for the numeric scheme to be stable and to compute accurate solutions. This results to the determinations of the time-dependent boiling surface temperature distribution T w (x, y, t), hereafter referred to as wall temperature and the time-dependent Heat Flux to Sapphire distribution q P s (x, y, t). The latter is dened with the Fourier's Law as:

q P s (x, y, t) = k s T w (x, y, t) ¡ T (x, y, z c , t) z c
where k s is the thermal conductivity of sapphire, T the temperature eld in the sapphire and z c the distance between the center of the rst cell below the active area and this latter. Knowing the time-dependent electric power supplied to the heater P and the active heating area A h , one can nally deduct the Heat Flux to Water as:

q P w (x, y, t) = P (t) A h ¡ q P s (x, y, t) (2.1)
In the following, we will name the Heat Flux to Water simply as Heat Flux as this is the distribution relevant to investigate the boiling phenomena. The time-dependent mean wall temperature T w (t) and mean Heat ux to water q P w (t) are then determined by computing the mean value of the distribution on the active heating area for each time step.

Determination of DNB and CHF

Once the time-dependent wall temperature and Heat Flux distributions are known, we now have to determine the transient CHF, and therefore to characterize what is DNB in transient conditions. For a steady-state heat input, CHF corresponds to the lowest heat ux implying the transition from nucleate boiling to lm boiling. However, its denition for a transient heat input is more challenging. In our investigation, we dene the moment when DNB occurs as the moment we start to observe the rst irreversible dry spot developing on the boiling surface. A dry spot is dened as a pocket of vapor contacting the boiling surface. An irreversible dry spot is dened as one that never gets rewetted once it appears, as long as the heat ux is increasing. Typically, as the heat ux continuously increases, there is no stable state for such a dry spot: it can only grow. On the IR thermography images, a dry spot is characterized by a zone on the boiling surface presenting a Heat Flux almost null compared to the neigbouring area (see lower line in gure 2.10). This zone also presents a wall temperature signicantly higher than the surrounding area (see upper line in gure 2.10), typically above 200 o C for cases at atmospheric pressure. The dierent types of dry spots are illustrated in gure 2.10. The snapshots of the left row (744 ms) show the wall temperature and Heat Flux distributions after the moment when a few bubbles appear. The bubbles are characterized by small spots with higher Heat Flux. The snapshots in the middle rows (from 797.6 ms to 804.8 ms) show the growth and collapse of a reversible dry spot located downstream the heater. It is worth to note that the collapsing of a dry spot generates an area with higher Heat Flux where it quenches (red spot upstream the dry spot). The developing dry spot in the middle of the heater also collapses a few milliseconds later. Finally the snapshots of the right rows (from 819.2 ms to 824 ms) show the development of irreversible dry spots. These dry spots are irreversible as, once they appear, they continue to grow and eventually coalesce until the entire active heating area is covered by a vapor lm. However, we used to stop the exponential power excursion prior this kind of phenomenon happens in order to save the heater integrity. The time of DNB corresponds to the frame just before the occurrence of the rst irreversible dry spot. CHF is assumed to be the corresponding Heat Flux. The current optical set-up enables a high resolution diagnostic on the wall temperature and heat ux distributions through the HSIR camera, and the visualization of the bubbles dynamics through the HSV camera. However, the knowledge of the thermal distribution in the uid, even qualitatively, is not accessible in the current state. More particularly, we are interested in the behavior dierence between pool and ow boiling. In order to ll the gap, I proposed to set up a visualization apparatus using the Schlieren method. This method is intensively used in the eld of aerospace or ballistics in order to visualize shockwaves or in the eld of thermal engineering in order to visualize the thermal eld variations around a heating body such as a ame, a radiator or even a human body [START_REF] Settles | Schlieren and Shadowgraph Techniques[END_REF].

The key concept of the Schlieren visualization method is to translate a refractive 2.3. Schlieren Visualization index heterogeneity into a light amplitude dierence, and thus to observe it with conventional optical sensors. Consequently, it is possible to observe the spatial distribution causing these index heterogeneities: shockwaves, chemical species, density or thermal gradients. To this end, the test area is lit by a punctual light source. In the other side, a focusing optical system, such as lenses or parabolic mirrors, focuses the source light onto a knife edge (see gure 2.11a). The visualization device is placed behind. If the test area has a homogeneous refraction index, the light rays coming from the punctual light source are not deviated and all the light reaches the visualization device. We then observe a homogeneous enlightened background. However, if the test area has a heterogeneous refraction index distribution, the light rays are slightly deviated. This implies that some of the deviated rays are stopped by the knife edge and some others are superimposed with other light rays. Consequently, the observed background appears with a heterogeneous light amplitude, revealing the refraction index heterogeneity in the test area. The patterns depends on the knife edge orientation and geometry (see gure 2.11b). More quantitative information about the Schlieren technique can be found in the book of Settles [START_REF] Settles | Schlieren and Shadowgraph Techniques[END_REF].

Application to the apparatus

Many specic setups making use of the Schlieren method exist [START_REF] Settles | Schlieren and Shadowgraph Techniques[END_REF], however we chose to use a single parabolic mirror setup similar to the one illustrated with the scheme gure 2.12. We chose this type of setup, because of its compactness (about twice smaller than with lenses) and because of the cost (a satisfactory good quality parabolic mirror is signicantly less expensive than a similar quality lens). The resulting setup is presented h Figure 2.12: Functional scheme of the schlieren visualization method [START_REF] Settles | Schlieren and Shadowgraph Techniques[END_REF] in gure 2.13. The light source is generated by a white LED on which a holed aluminium sheet is placed to make it punctual. We did not use the combination lamp-lens-slit as depicted in gure 2.12 as our lighting setup is suciently powerful. The parabolic mirror, placed behind the test section, has a diameter of 10 cm and a focal length of 30 cm. The test area is the uid surrounding the heater. The light beam crosses the two windows perpendicular to the heater of an eective dimension of 40 mm ¢ 6 mm. The knife edge is a diaphragm in order to observe the temperature gradient in all directions. Finally we use the HSV camera Phantom v25.12 for visualization. The reection angle of the light beam on the mirror is set to be as small as possible in order to visualize the test area as perpendicular as possible. It resulted that the LED and the diaphragm were really close, less than one centimeter (g. 2.13) and that the light crosses twice the test area. As discussed in section 1.3.4.3, the Schlieren method cannot provide quantitative information on the temperature eld in the condition of multiple bubbles, especially under ow conditions. However, the high sensitivity of the Schlieren technique enables a qualitative visualization of the temperature eld in the water in pool and ow boiling.

The tests are realized at atmospheric pressure and with a subcooling of 30 K and we apply a step-function to the electrical heater. The bright background region in the images corresponds to the 1-cm width channel and the superimposed white line corresponds to the location of the channel wall.

The visualization made for pool boiling, i.e. stagnant ow conditions, is shown in gure 2.14. It presents snapshots taken every 0.3 seconds from an arbitrary time close to the beginning of the stepwise electrical input (0 s). From 0 s to 0.9 s we can observe the development of the single-phase thermal boundary layer. The ONB occurs between 0.9 s and 1.2 s. Indeed, on the lower part of the snapshot 1.2 s, one can observe the beginning of the thermal boundary layer destabilization due to bubbles nucleation. The following images show that because of the appearance of bubbles, the thermal boundary layer is progressively destabilized, and the entire channel temperature eld is progressively inuenced. The visualization of the ow boiling experiment is presented in gure 2.15 and shows snapshots taken every 0.6 seconds from an arbitrary time close to the beginning of the electrical input (0 s). The Reynolds number is set at 10000. In the single liquid phase, the thermal boundary layer is too thin to be observed, conversely to the pool boiling situation. In nucleate boiling, presented on the 1.2 s to 2.4 s snapshot images, the region where the temperature eld is inuenced by the boiling process is mainly limited near the wall in a thickness of about a quarter of the channel width. It is also interesting to note that we can observe vortices due to the turbulent nature of the the ow. From these experiments, we can notice that the thermal patterns in the channel are signicantly dierent between pool and ow boiling situations. When in pool boiling, the entire channel is thermally inuenced by the nucleate boiling while only a restrained region near the wall is inuenced in turbulent ow boiling. This connement behavior is expected to be stronger in case of stronger forced convection and higher subcooling. Indeed, at higher subcooling, bubbles are signicantly smaller leading to a smaller destabilization of the boundary layer and the stronger ow squeezes more the thermal boundary layer.

Project Time Line and Test Matrices

As stated above, the present work takes place in a wider experimental partnership between CEA and MIT aiming to build a better modeling of the heat transfer phenomena involved during a BORAX-type accident. This collaboration was divided in three phases with specic goals implying the design of three specic experimental campaigns. In order to have a better insight of the work done, I present in this section an overview of the entire collaboration, how the doctorate work takes place in the dierent phases of the project and the dierent experimental diculties faced. Then, the relevant test matrices are presented with the corresponding contributions.

Overview of the partnership between CEA and MIT

The aim of the partnership is to propose a better modeling and physical understanding of the heat transfer phenomena involved during a BORAX-type accident in order to yield a best-estimate approach for the simulation codes. The partnership includes the investigation of the single phase heat transfer, boiling heat transfer and CHF. The whole project is divided in three phases which were ordered according to the experimental constraints. Each phase enabled to gather experience in order to design the experimental apparatus for the following phase, which faces more extreme conditions. My Ph.D. work took place during the third and nal phase of the project. Figure 2.16 chronologically presents the dierent phases of the partnership along with my Ph.D. milestones, which are detailed in the following.

Started in 2014, i.e. more than three years before the beginning of my Ph.D. work, Phase 1 aimed at the validation and improvement of heat transfer correlations under an exponential power escalation at atmospheric pressure. This phase introduced the use of IR thermography and HSV photographic visualization, proposing high resolution diagnostics. It focused on the single-phase heat transfer, the determination of the ONB and the FDNB heat transfer. The investigation of the CHF was not in the scope of this campaign, involving higher constraints on the materials (electric input, thermo-mechanical stress on the heater...). The work realised during this rst phase of the partnership has been published by Su et al. [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part I: Pool boiling[END_REF]- [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part II: Flow boiling[END_REF]. It made use of two apparatus: a pool boiling apparatus [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part I: Pool boiling[END_REF] and a ow loop [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part II: Flow boiling[END_REF] using a rectangular channel of 1 ¢ 3 cm 2 . In the latter, the bulk velocity went up to 2 m.s ¡1 . The tests were realized in water with subcooling degrees of 10 K, 25 K and 75 K and with power escalation periods ranging from 5 ms to 500 ms.

From early 2016 to late 2017, Phase 2 aimed at the determination of transient subcooled CHF at atmospheric pressure. Also using IR thermography and HSV photographic visualization to get high resolution diagnostics, it aimed to get a better understanding of the mechanisms leading to DNB and to measure the impact of the subcooling, escalation period and mass ow. The experiments were realized in a ow loop constituted of a rectangular channel of 1 ¢ 3 cm 2 with bulk velocities up to 2 m.s ¡1 . It used the Test Section TS1 presented in section 2.1.1. The tests were realized in water with subcooling degrees of 10 K, 25 K, 50 K and 75 K and with power escalation periods ranging from 5 ms to 500 ms. The experimental results were presented in Kossolapov et al. [START_REF] Kossolapov | The Boiling Crisis of water under exponentially escalating heat inuts in subcooled ow boiling at atmospheric pressure[END_REF], on which I am co-author, with some additional experiments made in September 2019. Moreover, as these experimental data were already available at the beginning of my Ph.D., I was able to start the development of a model for highly subcooled ow boiling crisis presented in Chapter 3 of this thesis manuscript.

In the continuity of the rst two phases, Phase 3 started almost simultaneously with my Ph.D. work. This nal phase aims to investigate the transient subcooled ow boiling crisis at moderate pressure in a narrow channel. In addition to the parameters investigated in the previous phase, this phase also investigates the impact of pressure, from the atmospheric pressure up to 12 bar which is a relevant value for experimental reactors. The water temperature for the tests ranges from 25 o C to 100 o C giving a subcooling ranging from 25 K to 163 K depending on the pressure. In order to investigate the phenomena in a narrow channel, the Test Section TS2 with a rectangular channel of 1.95 ¢ 18 mm 2 was designed. Due to the small hydraulic diameter in TS2, the use of the same pump allowed tests with a bulk velocity rising up to 19 m.s ¡1 . In the following, I present the events of Phase 3 and my contribution to this phase. The major objectives were the investigation of transient CHF in a narrow channel with a higher uid velocity and a higher pressure. It required to design a specic test section (TS2) using a longer heater. This development was allocated to the team at MIT. The initial planning expected the design and the manufacture of TS2 and the corresponding heaters to be realized by the rst half of 2018, i.e. by my arrival at MIT. It would then have been followed by the shakedown of the modied ow loop and nally the actual investigations on CHF. This planning would have allowed me to remain available in the laboratory for about nine months with a functional apparatus to conduct the required experiments. However, the project started several months later than initially expected and the complexity of this new design associated with the challenge to set up a new apparatus with such a small channel width (1.95 mm) naturally implied some delays on the expected schedule. The manufactured new test section was available in early September 2018. We were then able to modify the ow loop from one single branch loop to a double loop, making this way two test sections ready to use (as shown in the picture Fig. 2.1). This set up work was realized along with the graduate student responsible of Phase 3 at MIT, Florian Chavagnat.

Another delicate point was the functional and reliable coating of heaters. Indeed, the Sapphire substrates were manufactured by an external supplier, but the entire metallic coating, i.e. everything except ITO, was realized internally at MIT allowing us to be exible and to realize dierent combinations. The planned design was to use ITO as the material for the eective heating area of 40¢6 cm 2 with pads made of Titanium and Gold, the Titanium layer being placed between the ITO and the Gold layers for adherence. ITO was privileged as it is not subject to oxidation under these conditions and as it is transparent, which allows the use of backlit shadowgraphy presenting excellent contrasts (see section 2.1.3). However, during the rst experimental campaigns, we noticed the existence of hot spots at the junction between the active heating area and the pads (see gure 2.17). These spots generated temperature anomalies and severe unhomogeneities of temperature and heat ux on the whole heating area. Consequently, the distribution of bubbles was also unhomogeneous as shows the front view on gure 2.17. With such anomalies, any quantitative investigation was impossible. Dierent trials were realized in order to x this issue. Nothing was successful with an ITO active area. After several combinations, the use of a thicker layer of Titanium (500 µm) with pads made of Gold or Silver made the rst eective and usable heaters. These were operational in early April 2019, i.e. two months before the end of my stay. Nonetheless, these heaters always ended with hot spots or pads failure after several transient heat inputs. In June 2019, after realizing the failure of Titanium coatings on other experiments of the group, a mass spectroscopy showed that the Titanium used to coat the heaters was contaminated by Gallium. The obtention of a new Titanium ingot denitively xed the hot spots problem and other unexpected failures. The heaters manufactured for the experiments of July 2019 and after were perfectly functioning and reliable, i.e. after the end of my visit. This explains why the majority of the test points were realized after my departure, at the beginning of the third year of my PhD.

In parallel with investigating and solving the hot spots issue, I realized dierent tasks along with the development of the model presented in Chapter 3. I worked on the development of a heater based on a Silicon substrate in order to investigate the impact of the substrate eusivity through fast power escalation. Unfortunately by lack of time, this attempt did not end with a fruitful conclusion. More details can be found in Appendix B. Moreover, the development of the model leaded me to assess the thermal inuence of 3). Beyond the experimental development I made during this period, I also worked on the post-processing of data. I adapted the script which post-processes the raw HSIR data to Temperature and Heat Fluxes distributions from the existing script to one applicable in our investigations (see section 2.2.1). The existing script was made to post-process data generated with the TS1 heater, a 20 ¢20¢1 mm 3 Sapphire heater with an ITO active area of 1 ¢ 1 cm 2 as presented in Bucci et al. [START_REF] Bucci | A mechanistic IR calibration technique for boiling heat transfer investigations[END_REF]. I modied the substrate and heating area geometries, the latter being adaptable to the used heater. The mesh was also modied to t the new geometry and the dierences in term of heat conduction in the substrate: from an homogeneous mesh in the z direction before, the mesh is here rened at the vicinity of the active heating area to better take into account the 2D heat conduction in the substrate. Finally, the optical properties of Titanium were measured, with an IR spectrometer, then implemented to the script for the radiation part of the algorithm. I also developed a bubble recognition algorithm in order to quantitatively analyze the HSV videos for high subcooling and transient conditions. Indeed, for such conditions, bubbles are so poorly spatially and temporally resolved that it is complicated to use conventional ways to analyze them quantitatively (manual counting or conventional algorithms). The current state of this work is presented in Appendix B.

With operational heaters, the experiments were realized at dierent dates as indicated in the time line in Fig. 2.16. I contributed to the experiments during the period of my visit, but was not able to take part physically to the experiments conducted after June 2019. Nevertheless, I fully contributed to the post-processing work for the whole set of experiments (Calibration implementation -data preparation -post-processing computation -DNB determination). In gure 2.16, it can be noticed that the experimental runs are not continuous and several weeks or months can happen between two runs. This is due to three reasons. The rst one is that the experimental facilities are shared with all the contributors of the group and the ow loop was shared with about ve dierent projects. Therefore, the latter was not always available for us. The second one is that between two runs, the post-processing and pre-analysis of the data is interesting to improve the next runs and adjust the test matrices. Finally the last reason is the preparation of new heaters. In the following section, we present the test matrices conducted during this experimental campaign.

Test Matrices

We present in this section a summarized version of the test matrices. Extended test matrices can be found in Appendix A. Table 2.1 presents the test matrix of the experiments of Phase 2. Even though I was not involved in the realization of these points, it is relevant to precise them as these were used for the development of the model that will be presented in the following Chapter 3. Additional points were realized in September 2019 to extend the analysis presented in [START_REF] Kossolapov | The Boiling Crisis of water under exponentially escalating heat inuts in subcooled ow boiling at atmospheric pressure[END_REF], using power escalation periods as short as 1.5 ms. I contributed to the post-processing calculations of these new points.

Parameters Values

Pressure [bar] 1

Period [ms] 5 -10 -20 -50 -100 -200 -500 As the rst eective tests, the aim was to assess the apparatus and particularly the reliability of the heaters. To this end, we investigated the impact of the pressure and temperature on the CHF, starting with small values of pressure, subcooling and mass ux. Starting with a low pressure enabled us to ensure the sucient sealing of the section. The small values of subcooling and mass ux enabled us to start with conditions of a low expected CHF in order to test the reliability of the heaters in regards of the heat ux. Unfortunately, the latter were starting to deteriorate when we imposed a little to high heat ux (about 15 M W.m ¡2 ). On April 9 th 2019, only three conditions were realized. For these tests, we used a power escalation period of 5 ms, which is the target period corresponding to reactor conditions (see section 1.4). The following day, we decided to start with a longer escalation period, allowing for the need of lower heat uxes, and then sequentially shortened the period. This unfortunately also ended by the failure of the heater. On April 21 st , we tried to perform experiments in the older test section (TS1) in order to assess the impact of length on CHF with comparable conditions than during Phase 2 (atmospheric pressure, 50 K of subcooling and a Reynolds number of 35,000). Unfortunately, only two periods were realized successfully (50 ms and 100 ms), as the heater failed for a shorter period (20 ms). In May, we decided to decrease the active heating area in order to reach higher heat uxes for a same power input. To do so, the width was reduced from 6 mm to 4 mm and the length remained at 40 mm. (see Fig.

Subcooling [K] (Bulk Temperature [ o C]) Reynolds Number [-] (Mass ux [kg.m -2 .s -1 ]) 35,000 (2073) 35,000 (1274) 35,000 (877) 35,000 (733) 25
A.1). It enabled us to investigate conditions with higher CHF, but the limit was still too low to investigate the nominal values. After these heaters, we realized that the Titanium used to coat the heater was contaminated by Gallium as stated in section 2.4.2. After the change of the Titanium ingot, the following heaters were signicantly more reliable, enabling a proper experimental campaign.

On July 17 th and 21 st 2019 were realized experiments aiming to investigate the impact of the pressure and the water temperature on the transient CHF for a low mass ux (Table 2.2) and a high mass ux (Table 2.3). The points realized in the matrices were selected in order to get the conditions necessary to analyze this impact either at constant bulk temperature or constant subcooling. The heated length was shortened to 20 mm to make the surface smaller in order to ensure the heat ux to be suciently high to reach the boiling crisis. Other tests were also realized during these runs and all details can be found in table A.2.

Parameters Values

Mass ux [10 3 kg.m -2 .s [START_REF] Baudin | Turbulent ow and transient convection in a semi-annular duct[END_REF]. These series intended to investigate the impact of the ow condition on the transient CHF in case the boiling phenomena are restrained at the vicinity of the wall, i.e. when bubbles are really small compared to the channel. Tests with similar ow conditions (mass ux, shear velocity, Reynolds numbers,...) were then realized with two dierent channel widths, i.e. using the two test sections. The tests were realized at high pressure and high subcooling in order to ensure that the bubbles remain small compared to both channel widths. Other tests were also realized during these runs and all details can be found in table A. On September 21 st 2019 was realized a series intending to investigate the impact of the mass ux on the transient CHF at high pressure and high subcooling. The corresponding test matrix is presented in table 2.5. The heated length was shortened to make the surface smaller in order to ensure the heat ux to be suciently high to reach the boiling crisis. Other tests were also realized during these runs and all details can be found in table A. [START_REF] Baudin | Turbulent ow and transient convection in a semi-annular duct[END_REF]. In this Chapter, we described the experimental apparatus used for the investigation, the post-processing methods and the experiments which took place. Before my visit at MIT, the experiments of Phase 2, corresponding to the test matrix table 2.1 were already realized. Therefore, I had the opportunity to analyze the data and the observations, which allowed me to develop a model seeking to describe the transient ow boiling crisis at high subcooling and atmospheric pressure. The following Chapter 3 develops this model. In this chapter, we investigate the occurrence of the highly subcooled transient ow boiling crisis at atmospheric pressure using the experimental data of phase 2. From the analysis of these data, we develop a model describing the boiling crisis under such conditions.

Parameters

Chapter 3. Homogeneous Mantle Model at atmospheric pressure

Experimental observations and interpretation

Among the various experimental working conditions of the phase 2 (see Test Matrix on gure 2.1), we focus on the conditions of high subcooling combined with forced ow, which are particularly relevant situations regarding our objectives. For a high subcooling (∆T sub ¥ 50 K) we observe that the heating wall is homogeneously covered by numerous tiny bubbles (about 100 µm in diameter) of short life-time (less than 100 µs). This behavior is not taken into account by the boiling crisis models of the literature. Besides, we decide to consider a smaller subcooling equal to 25 K in order to assess the validity limit of the phenomenology. To summarize, we investigate the ow boiling crisis of water at atmospheric pressure in the following range:

Power escalation period, from 5 ms to 500 ms.

Subcooling, from 25 K to 75 K.

Reynolds number, from 8500 to 35000.

Transient boiling crisis for high Reynolds numbers and high subcoolings

In gure 3.1, we present a representative transient boiling curve under the conditions of interest. It features the mean heat ux as a function of the mean wall superheat. Pictures of the boiling process, as recorded by the HSV camera, are presented for the dierent stages of the transient boiling curve.

Single phase regime

Single phase heat transfer is realized in the subcooled water and results from the cumulative actions of the advection process and the transient heat conduction as presented in section 1.3.3. Nevertheless, the heat ux is still small at this stage (point 1 in gure 3.1).

Onset of Nucleate Boiling and Overshoot

Boiling incipience occurs once the temperature on the wall is large enough to activate the nucleation sites. Under these conditions of subcooling and for fast transients, one observes a temperature overshoot (OV) (point 2 on gure 3.1) which is described in section 1.3.3.

During the OV, we observe larger bubbles than during FDNB.

Fully Developed Nucleate Boiling

After the overshoot, most of the nucleation sites are activated and the wall appears to be homogeneously covered by bubbles. Our HSV analysis shows that bubbles are small in high subcooling conditions, the order of magnitude of the bubble diameter being 100 µm.

Their lifetime is also short, about 100 µs. Qualitatively, this can be explained considering that bubbles grow up to a small diameter then rapidly collapse due to the cold surrounding water. More interestingly, the HSV reveals a pulsating bubble behavior. Bubbles alternatively grow and collapse with a frequency greater than 10 kHz. Indeed, they hardly remains more than two frames on the HSV video recorded at 20, 000 FPS. In addition, videos show that bubbles hardly lift-o from their nucleation site but remain conned at the vicinity of the wall. This observation is consistent with the Schlieren visualization of ow boiling (g. 2.15) where we observe that the inuence of the ow is highly restrained near the wall. It is worth to note that for ∆T sub = 25 K, condition used to assess the validity limit, this mechanism is less observed as bubbles have a longer lifetime, about a few milliseconds. They also have a width up to 3 mm with the front view. Here, width is specied instead of diameter as bubbles may not remain spherical for such size and temperature unhomogeneities. This process is observed during the entire FDNB regime, the density of bubbles increasing with the heat ux. This evolution is illustrated by HSV images in gure 3.1 where the density of bubbles is seen to increase from point 3a to 3b. The period of the power escalation does not qualitatively change this behavior.

Boiling Crisis

As the heat ux increases, bubbles become more and more numerous on the wall. Local and intermittent dry spots occur due to lateral coalescence of some neighboring bubbles.

These dry spots may be reversible, i.e. the wall can be rewetted. However, as the heat ux transferred from the heating surface to the uid increases, the lateral coalescence of the neighboring bubbles leads to the formation of irreversible dry spots, i.e. to the boiling crisis (point 4 in gure 3.1). According to observations made from both HSV and IR camera, the spatio-temporal dynamics of the dry spots changes with the time period τ . For fast transients, dry spots appear uniformly on the entire heating wall. Conversely, for longer transients, dry spots seem to preferentially appear downstream the heating wall, as illustrated by point 4 in Fig. 3.1. The limit between fast and long transients mainly depends on the imposed ow regime, it starts from about 20 ms for strong forced ow to 100 ms for lower ones.

As stated earlier, the boiling curve corresponds to a time evolution, where each point Chapter 3. Homogeneous Mantle Model at atmospheric pressure is associated to a specic value of time. A more detailed analysis on the time sequence of the transient ow boling crisis under the investigated conditions is presented in Appendix C.

Physical description of the model

Considering the observations reported in the former section, we propose the following mechanism: boiling crisis is triggered when bubbles generated at the wall can no longer condense in the surrounding liquid, then coalesce laterally generating irreversible dryspots. Indeed, under conditions of high subcooling and strong forced convection, the observations revealed that each bubble grows, then collapses due to the surrounding cold liquid and nally disappears. Subsequently, other bubbles reproduce the same pattern constituting Nucleation and Condensation Cycles (NCC) happening at high frequency (about 10 kHz). These NCCs contribute to eciently transfer the heat from to wall to the liquid enhancing its temperature. When the liquid reaches its saturation temperature, nucleated bubbles cannot condensate anymore which triggers the boiling crisis.

The bubbles remaining small and at the vicinity of the wall, let us to consider a small layer of uid at the vicinity of the wall of thickness δ, hereafter called the "mantle". In this mantle, the criterion corresponding to the proposed boiling crisis mechanism is that the local volumetric energy density [J.m ¡3 ] reaches the value corresponding to liquid water at saturation conditions. This scenario is schematically summarized in gure 3.2 and sequentially described hereafter.

The transient process starts with single-phase forced convection until rst bubbles appear on the heated wall (ONB). During FDNB, bubbles have a pulsating behavior. They are small and promptly collapse after their nucleation leading to high frequency Nucleation and Condensation Cycles (NCC). These eects are only perceptible near the wall. The model postulates that the high mixing induced by NCCs leads to homogenize the energy throughout the thickness of the mantle, i.e. in the direction y. This thickness δ is assumed to be constant throughout the transient process. This drastic assumption is statistically reasonable considering that the mantle is dened as the region dominated by the inuence of pulsating bubbles and the size of these bubbles does not signicantly changes during FDNB (see pictures 3a, 3b and 4 in Fig. 3.1). The thickness δ is also assumed to be uniform alongside the heater in a rst approximation to avoid dispensable tting constants.

During FDNB, the energy balance in the mantle results from the competition between the heat input supplied from the wall and the cooling processes due to forced convection (that will be detailed in sections 3.2.3 and 3.2.4). Because of the exponential nature of the heat input, the energy density rises in time. Once it reaches the critical value, i.e. when the local subcooling becomes null, bubbles generated at the wall can no longer condense. They coalesce triggering the formation of irreversible vapor pockets, which next results in a boiling crisis.

The model describes the exchanges between two homogeneous zones presented in gure 3.2:

The mantle of thickness δ, where the energy density is homogenized by the NCCs. This thickness is assumed to be small compared to the channel width. This hypothesis is supported by the observation using the schlieren visualization technique 3.1. Experimental observations and interpretation u bulk u bulk q'' w q'' w u bulk The bulk, where cold liquid water ows at a high Reynolds number and high subcooling. The bulk temperature is considered as being unmodied during the transient heating process. This statement is supported by the schlieren visualization (see section 2.3).

The energy density homogeneity in the mantle is a reasonable approximation. Indeed, during FDNB, the high speed videos revealed the small size of the bubbles, around 100µm, which is two orders of magnitude smaller than the width of the channel. The Schlieren visualization technique has shown that in case of turbulent ow condition with highly subcooled water (section 2.3), the region thermally inuenced by the heating wall and the generated bubbles are also restrained to the close vicinity of the wall. Therefore, we can conservatively assume that the mantle thickness would also be small compared to the channel width. Plus, it is also relevant to assume that the mantle thickness is in the order of magnitude of the bubble diameter while being larger, about a few units larger. By their pulsating behavior, the generated bubbles would then mix the uid in mantle and homogenize its temperature. Besides, energy density is considered to be constant in the direction of the width of the heater. This model relies on some input data which are experimental quantities : i) thermal and hydraulic operating conditions (pressure, subcooling, mass ow rate, time period for the exponential rise), ii) the space-averaged heat ux q P w (t).

This approach is akin to the model of Weisman and Pei, see section 1.3.4.4, who proposed an interaction between two distinct domains due to turbulent velocity uctuations. However, Weisman and Pei considered the exchange of bubbles whereas our model proposes to study an exchange of energy. Moreover, our model is also akin to the critical enthalpy approach proposed by Tong [START_REF] Tong | Inuence of axially non-uniform heat ux on DNB[END_REF] (see section 1.3.4.4) who conducted an enthalpy balance in the close vicinity of the wall and proposed a criterion on the enthalpy value. However, the lack of physical modeling of the various terms of his balance and the stated critical enthalpy led Tong to use numerous correlations on experimental data. Finally, these two approaches are limited to steady-state situations and cannot be used for transient heating conditions.

Mathematical formulation 3.2.1 Criterion for the Boiling Crisis

As stated above, the criterion triggering the boiling crisis under the considered conditions is when energetically, an elementary volume of liquid water (dV ) in the mantle is heated from the initial subcooled temperature to saturation, corresponding to an energy rise of c p ρ l ∆T sub dV . The critical energy density rise E Q cr will then be dened by:

E Q cr = ρ l c p ∆T sub (3.1)
When the local energy density rise in the mantle reaches this critical value E Q cr , the next generation of bubbles will no longer condense. These bubbles will instead laterally merge triggering the boiling crisis. The critical energy density rise E Q cr is thus a suitable criterion to characterize the boiling crisis.

It is worth to note that this criterion does not include the latent heat. We have shown that under conditions we intend to apply the model on, bubbles have a short lifetime: less than 100 µs. This is 2 to 5 orders of magnitude smaller than the period of the exponential power excursion, and so the characteristic time scale of the investigated phenomenon. As stated in section 3.1.2, the model assumes that the energy required to generate a bubble, i.e. involving the latent heat, is then entirely released to the adjacent liquid in the mantle, i.e. converted to sensible heat. In other words, the contribution of the latent heat is mechanistically considered as a intermediate step which eciently transfers the heat from the wall to the liquid phase. This temporary process having a time scale much smaller than the time scale of the investigated phenomenon,i.e. the power escalation period τ , it is consistent not to include the latent heat in this criterion for the proposed model.

In the following, we aim to establish an equation on the local energy density rise describing its evolution during the transient heating.

Governing equations of the mantle

To describe mathematically the evolution of the local energy density, one needs to establish an energy balance in the mantle. Let us consider the mantle region of thickness δ which separates the wall from the bulk uid as depicted in Fig. 3.3. The assumptions discussed in section 3.3 consider that the physical quantities do not depend on the z variable. The successive nucleation-condensation cycles of the bubbles induce an homogenization 3.2. Mathematical formulation in the transverse direction y inside the mantle. The heat ux q P w (t) supplied from the wall to the uid contributes to the bubbles generation. The energy of the uid in the mantle increases gradually due to heat released by condensation. Conversely, part of this energy is lost due to two contributions. The rst one is the advection in the streamwise direction due to the vertical velocity component (u m in Fig. 3.3). The second one is the heat exchange between the mantle and the bulk due to turbulent velocity uctuations (v m in Fig. 3.3). These exchanges will be discussed in section 3.2.3 and 3. Chapter 3. Homogeneous Mantle Model at atmospheric pressure

The considered system σ(x) has a total energy E σ (x, t) (J) carried by the uid within it. By considering the bulk energy density E Q 0 [J.m ¡3 ] and the energy density rise E Q (x, t)

[J.m ¡3 ], one can dene:

E σ (x, t) = ¡ E Q (x, t) E Q 0 © dx δ dz
The variation of energy dE σ is written:

dE σ = E σ (x, t dt) ¡ E σ (x, t) = (E Q (x, t dt) ¡ E Q (x, t)) dx δ dz thus dE σ = ∂E Q ∂t dx δ dz dt
The exchange of energy at the interface is written:

dE σ = Q w ¡ Q σ(x) x Q σ(x) x¡ ¡ Q σ(x) b
where (see gure 3.4):

Q w is the heat coming from the wall.

Q σ(x)
x¡ is the heat coming by advection from σ(x ¡ dx).

Q σ(x) x
is the heat going by advection to σ(x dx).

Q σ(x) b
is the heat transferred at the interface mantle-bulk (at a distance y = δ from the wall). the expressions of the dierent terms are:

Q w = q P w dx dz dt Q σ(x) x¡ = u m (E Q 0 E Q (x, t)) δ dz dt Q σ(x) x = u m (E Q 0 E Q (x dx, t)) δ dz dt Q σ(x) b = q P b dx dz dt = v m E Q (x, t) dx dz dt with q P b = v m E Q (x, t
) the net heat ux transferred at the interface mantle-bulk. The demonstration of its expression will be presented in section 3.2.4. These expressions substituted in equation 3.2.2 gives:

dE σ = q P w dx dz dt ¡ u m (E Q (x dx, t) E Q (x, t)) δ dz dt ¡ v m E Q (x, t)dx dz dt = ¢ q P w ¡ u m ∂E Q ∂x δ ¡ v m E Q (x, t)
dx dz dt

Mathematical formulation

The expression equalized with the expression in partial time derivative gives the equation of the mantle:

∂E Q ∂t u m ∂E Q ∂x v m δ E Q = q P w δ (3.2)
In the present situation with an established forced convection and with high heat uxes supplied from the wall, we have:

-kinetic energy and pressure eects which are small compared to the other terms.

-gravitational potential energy and viscous dissipation eects which are small compared to thermal eects at the vicinity of the wall.

It is then satisfactory to neglect these usual contributions in the present model.

Equation (3.2) can be transformed into a dimensionless expression. The dimensionless volumetric energy density E ¦ is dened with respect to the critical energy:

E ¦ = E Q /E Q cr .
The length of the heater, L, is used to set a dimensionless form of the variable x: x ¦ = x/L. Time period τ of the exponential rise is used to dene a dimensionless time: t ¦ = t/τ . Reporting these three quantities in Eq. 3.2, we obtain :

∂E ¦ ∂t ¦ N x ∂E ¦ ∂x ¦ N y E ¦ = τ E Q cr δ q " w (3.3)
where N x and N y are two non-dimensional groups dened as:

N x = u m τ L N y = v m τ δ
A well posed mathematical problem requires for this rst order partial dierential equation, initial and boundary conditions. At time t ¦ = 0, the temperature of the uid is uniform in the whole channel, equals to the inlet temperature : E ¦ (x ¦ , 0) = 0.

The uid entering the heated section has a temperature equal to the bulk temperature which remains the same during the whole transient heating process. This condition reads E ¦ (0, t) = 0, dt ¡ 0. To fully formulate the mathematical problem, we still need to dene the two velocities u m and v m characterizing the the cooling process induced by the forced ow.

Heat transport induced by the streamwise ow

The velocity in the streamwise direction is u( x, t) = U (y) u I ( x, t). Neglecting the uctuating component u I ( x, t), this velocity will be approximated by its mean component U (y), this latter being given by the law of the wall for a turbulent channel ow. As strong velocity gradients are present near the wall, a spatial average along the mantle thickness will also be considered:

u m (δ) = 1 δ » δ 0 U (y) dy (3.4)
Once the bubbles are generated at the wall, the mean velocity prole is perturbed from the law of the wall. Nonetheless, its use as a rst approximation to compute u m is reasonable. The latter is a space averaged velocity within the thickness δ which is greater than the diameter of the bubbles, implying that most of the area accounted for to compute the Chapter 3. Homogeneous Mantle Model at atmospheric pressure mean velocity is sparsely inuenced by the bubbles.

Moreover, as described above, the bubbles are small, quasi-spherical and pulsate at high frequency without having time to slip over the surface. This leads to think that these ow perturbations do not inuence the streamwise average ow as the growth and collapse would be achieved symmetrically compared to the center of a bubble and as the period of the pulsation is small compared to the characteristic time period of the experiment (τ ). This growth and collapse behavior however inuences the instantaneous ow contributing to the mantle mixing. Finally, bubbles are conned in close vicinity of the wall where the velocity is strongly reduced due to the viscous boundary condition. Therefore, most of the weight U taken into account by the calculation of the mean velocity u m is taken where the uid is in single-phase liquid.

Heat transport induced by the transverse turbulent velocity uctuation 3.2.4.1 Assessment of the expression of the velocity v m

Then, one has to describe the heat exchange between the mantle and the bulk due to turbulence in the y direction. The advection at the interface by the turbulent velocity uctuations is taken into account by a characteristic velocity v m . Let us rst recall the usual denitions for the quantities used in the description of turbulent ows.

u ¦ is the wall shear velocity calculated with the McAdams correlation, δ ν = ν/u ¦ is the viscous sublayer unit, with ν the kinematic viscosity of water.

-Re τ = u ¦ e/ν is the wall Reynolds number where e is the channel half width.

The corresponding values of these quantities under the investigated conditions are presented in table 3.1. Hereafter, we will use the following denitions: U = U/u ¦ for the dimensionless velocity and Y = y/δ ν for the normalized distance to the wall. As featured in section 3.2.2, the heat exchanges between the mantle and the bulk are due to the transverse turbulent velocity uctuations. Let us consider a given transverse turbulent velocity uctuation v I , the elementary transfer of heat at the mantle-bulk interface dq P b is given by :

Re [¡] ∆T sub [K] Re τ [¡] δ ν [µm] u ¦ [cm.s ¡1 ] 8500
dq P b (v I ) = 5 E Q 0 v I P (v I ) dv I dv I 0 (E Q 0 E Q ) v I P (v I ) dv I dv I ¡ 0
where P (v I ) is the probability density function (PDF) of the transverse velocity component v I . As illustrated in gure 3.5, the rst line describes a transport of cold bulk water towards the mantle and the second one describes the heated water leaving the mantle to the bulk. In order to have the net heat transport, all the possible velocity uctuations weighted with their probability must be taken into account. This results to the following integral:

q P b = » V ¡V dq P b (v I ) (3.5) = » 0 ¡V E Q 0 v I P (v I ) dv I » V 0 (E Q 0 E Q ) v I P (v I ) dv I (3.6) = E Q 0 » 0 ¡V v I P (v I ) dv I (E Q 0 E Q ) » V 0 v I P (v I ) dv I (3.7) = E Q 0 » V ¡V v I P (v I ) dv I E Q » V 0 v I P (v I ) dv I (3.8) v I being a turbulent velocity uctuation, » V ¡V v I P (v I ) dv I = xv I y = 0
the rst term of equation 3.8 is then null and the net heat ux at the interface mantle-bulk is given by q P b = v m E Q with:

v m = » V 0 v I P (v I ) dv I (3.9)
With v m the characteristic velocity of the transverse turbulent velocity uctuation. This characteristic velocity is also the expected value of the positive velocity uctuation:

v m = E(v I | R ).
Besides, decomposing v m enables to better understand its two contributions: the cold bulk water inow and the warmed water outow. Indeed, as xv I y = 0,

³ 0 ¡V v I P (v I ) dv I = ¡ ³ V 0 v I P (v I
) dv I = ¡v m and one can rewrite equation 3.7 as:

q P b = Warmed mantle water outow hkkkkkkkikkkkkkkj v m (E Q 0 E Q ) ¡ v m E Q 0 lo omo on
Cold bulk water inow (3.10) which is illustrated in gure 3.5

The formulation of the characteristic velocity v m is akin to the characteristic velocity dened by Weisman & Pei [START_REF] Weisman | Theoretical prediction of maximum heat ux in power transients[END_REF] and Pasamehmetoglu [START_REF] Pasamehmetoglu | Critical Heat Flux modeling in pool boiling for steady-state and power transients[END_REF]:

E(v I ¡ v v | rv v , Vs) = » V vv (v I ¡ v v )P (v I ) dv I (3.11)
with v v the vapor ejection velocity due to boiling. In their case, they aim to determine the velocity of liquid inowing a layer (the bubbly layer for Weisman and Pei and the macrolayer for Pasamehmetoglu) at low subcooling where vapor eectively leaves the denominated zone. In our case, bubbles do not depart from the wall because of the high subcooling. Thus, at a distance greater than the bubble diameter, in particular at a distance δ from the wall, there is no vapor mass ux escaping. Moreover, bubbles have a Postive uctuations transporting hot water from the mantle to the bulk Negative uctuations transporting cold water from the bulk to the mantle pulsating behavior. In our model, we chose to assume that the velocity eld generated by the bubble growth is damped by the bubble collapse and that these velocities have no signicant inuence on the exchanges at the interface mantle-bulk. For these reasons, v v 0 appears to be a quite reasonable approximation in our case and thus our proposed expression of v m does not contradict theirs. However, the present expression establishment and its use in equation 3.3 takes into account the whole distribution of the transverse turbulent velocity uctuations, and not only a part as in [START_REF] Weisman | Theoretical prediction of maximum heat ux in power transients[END_REF] and [START_REF] Pasamehmetoglu | Critical Heat Flux modeling in pool boiling for steady-state and power transients[END_REF], improving the use of this physical quantity. Plus, these authors make use of assumed expressions of the PDF P (v I ), Gaussian for Weisman and Pei and Hyperbolic for Pasamehmetoglu, whose accuracy is not quantitatively assessed.

As a determination of v m being necessary to exploit the model and equation 3.2, we present in the following an analysis of Direct Numerical Simulation (DNS) data which aims to accurately determine the latter quantity. A comparative study is also conducted to assess the validity of the assumptions of Weisman and Pei, and Pasamehmetoglu.

DNS analysis

DNS simulations performed by Moser et al. [START_REF] Moser | Direct numerical simulation of turbulent channel ow up to Re τ = 590[END_REF] investigated turbulent ows inside a rectangular channel. Their data yield PDF as a function of the distance to the wall and for three values of the Re τ parameter (180, 395 and 590). Graham et al. [START_REF] Graham | A Web services accessible database of turbulent channel ow and its use for testing a new integral wall model for LES[END_REF] performed similar DNS simulations with Re τ = 1000 and made available instantaneous velocity elds in steady-state conditions.

For Re τ equals 180, 395 and 590, Moser et al. propose in their data sets, the PDFs of v I for dierent y from the wall to the channel centerline. The use of equation 3.9 at each available y enables then to compute the expected value v m for these value of Re τ (see gure 3.7a).

For Re τ equals 1000, Graham et al. propose in their database the full velocity eld of their simulation. The simulation domain consists of a mesh made of 2048 ¢ 512 ¢ 1436 nodes and the database presents 4000 snapshots of the eld. In the x and z directions, the domain is periodic. The authors provide statistic quantities of the ow such as the mean velocities or the Root Mean Square (RMS) compounds, but not the actual PDF needed to calculate v m . It is then necessary to directly compute the expected value v m from the raw statistics. In order to have enough points to compute the expected value, we assume the ergodic hypothesis: for a given y coordinate, a quantity X veries X = xXy x,z , which are respectively the temporal average for a given position and the space average in a plan

x ¡ z for a given time. By symmetry, we will only investigate half of the channel in the y direction, i.e. between the wall and the channel centerline. A node over two will be taken into account in this direction as this reduction gives a satisfactorily ne information on the eld v m . For each y coordinate analyzed, all the x¡z nodes of the y-constant layer are used to compute the expected value. We also combine the data of three randomly chosen time steps in order to smooth a potential punctual statistic event, such as vortices. Then, each value v m is obtained from a statistic made of 2048 ¢ 1536 ¢ 3 points, i.e. more than nine millions points. The following procedure aims to validate the fact that this sample size is satisfactorily large.

To investigate the statistical convergence, let us dene two quantities: the normalized mean transverse velocity V and the two v m components computed with the positive and negative uctuations. From a nite statistic sample of N observations of normalized transverse velocities v n , V is dened as

V = E(v I | R) = N ņ=1 v n N (3.12)
This quantity theoretically equals zero. We also dene two v m components

v m,v I ¡0 = E(v I | R ) = 1 N v n ¡0 v n v m,v I 0 = E(v I | R ¡ ) = ¡ 1 N v n 0 v n (3.13)
where v m,v I ¡0 is v m computed with the positive uctuations and v m,v I 0 is v m computed with the negative uctuations. These values theoretically compensates each other. In gure 3.6, we present the value taken by these three quantities as a function of the sample size N.

Qualitatively, v m,v I ¡0 and v m,v I 0 uctuate for small values of N and converge to a common value for larger values. Particularly, they have converged for the sample size 2048 ¢ 1536 ¢ 3 (see gure 3.6b) which is represented by the vertical dashed line. More quantitative criteria can be assessed with the mean velocity. First of all, for the sample size 2048 ¢1536¢3, the absolute value taken by the mean velocity is negligible compared to the characteristic velocity of the channel ow, i.e. the shear velocity u ¦ : | V |=| V | /u ¦ 10 ¡3 (see gure 3.6a). Moreover, the mean velocity is negligible compared to our quantity of interest, v m , which is the most relevant criterion for our computations:

| V | v m 10 ¡2 (3.14)
This conrms that for a sample size of 2048 ¢1536¢3 observations, V has converged to zero. Thus this sample size is adequate for the calculation of v m . Lastly, we will take v m as the average of the two denitions, v m = 1 2 (v m,v I ¡0 v m,v I 0 ) which can be rewritten as After having dened the method to compute the expected value v m , let us compare our results with the ones of Weisman and Pei, and of Pasamehmetoglu which need less computing resources. Pasamehmetoglu assumed a hyperbolic distribution, P (v

v m = 1 2N N ņ=1 sgn(v n ) v n (3.15)
I )W(1/v I ) n
with n, an adjustable empirical coecient taken in the interval s2, Vr. In order to t his model with the experimental data set, n is taken equal to 4. In our case, v v is taken null and the computation of the expected value Eq. 3.11 corresponds to the integral ³ V 0 dx/x n¡1 , which diverges. Consequently, the approach of Pasamehmetoglu cannot be used in our case. In their approach, Weisman and Pei assumed that P (v I ) has a Gaussian distribution everywhere in the domain, i.e. dy r0, Re τ s:

P wp (v I ) = e ¡ 1 2 ¡ v I vrms © 2 v rms c 2π
This assumption is relevant in the bulk, but is not true near the wall where anisotropy eects deviate the probability density function from a Gaussian distribution. In order to quantify the discrepancy between this assumption and the actual uctuations extracted from the DNS data, we compare the expected values v m = E(v I | R ) computed with the Gaussian distribution (v m,wp ) with the ones computed with DNS data as above (v m ). The relative discrepancy is dened as

| v m ¡ v m,wp | v m
The comparison is plotted in Fig. 3.7b. A maximum dierence of about 50 %, is observed at y = 5 and reduces as the distance increases, being about 5 % at y = 50. Based on this analysis, we decided to use DNS data to compute v m . This calculation for dierent values of y and for the four Re τ gives the parametric function presented in gure 3.7a.

Finally, with the knowledge of v m for dierent values of y and Re τ , it is now possible to interpolate the value of v m for any kind of ow conditions.

Model Implementation and Output

We will now discuss the various parameters involved in equation 3.2: q P w (t) is the time-dependent mean heat ux, measured from experimental data. In this rst approach, a space-averaged value is used for the heat ux, which is consistent with the observations until boiling crisis. Indeed, an investigation on the space standard deviation of the heat ux has been realized. It shows that for the investigated conditions (high subcooling and strong ow conditions), the standard deviation remains small. Moreover, when the heat ux is not averaged on the whole surface, but depends on the x coordinate (as q P w (x, t)), the results are signicantly similar.

the characteristic velocities u m and v m have been presented in section 3.2.3 and 3.2.4. Both quantities depend on the operating conditions (pressure, bulk temperature, mass ow rate) and on the mantle thickness δ.

The mantle thickness δ is a priori unknown.

The present model assumes that boiling crisis happens when the condition

E Q (x, t CHF ) = E Q cr is reached.
It is used to estimate the a priori unknown quantity δ.

In order to assess the thickness δ, we use the heat ux distribution for the various operating conditions. From the IR thermography, we determine the time corresponding to CHF t CHF , as described in section 2.2.2. Then we consider the mean heat ux q P w (t), with t ranging between zero and t CHF . Using a guess-estimate approach, a rst value of δ is chosen and Eq.( 3.3) is then integrated with an explicit forward Euler method. The right-hand-side term acts as a source term and is computed using the experimental time-function q P w (t) with 0 ¤ t ¤ t CHF . If the volumetric energy density rise at time t CHF diers from the critical value, the δ value is considered wrong. Another value is xed and the process is repeated. The correct value δ is obtained when convergence is achieved. A criterion of convergence, equal to 10 ¡3 has been selected: 3) is solved for the considered set of experimental conditions. The obtained values for the thickness δ are plotted as a function of the time period τ in Fig. 3.9a. The values can also be normalized using the length δ ν = ν/u ¦ which is the relevant turbulent length scale in the transverse direction (Fig. 3.9b):

| max x ¦ [0,1] (E ¦ (x ¦ , t ¦ CHF )) ¡ 1| 10 ¡3
δ = δ δ ν
Each of the eight curves plotted in Fig. 3.9 consists in seven points (τ = 5, 10, 20, 50, 100, 200, 500 ms). For each point, three runs were conducted. Therefore, the results presented in Fig. 3.9 are derived from 168 measurement points. The uncertainty is calculated as ∆δ = (∆δ m ) 2 (∆δ t ) 2 (σ δ ) 2 . ∆δ m is the measurement uncertainty resulting from the propagated uncertainty on the heat ux to water. ∆δ t is the temporal uncertainty due to the boiling crisis frame assessment. Finally, σ δ is the standard deviation on the three tests. To determine the measurement uncertainty ∆δ m , we add (or subtract) to the heat ux input its own uncertainty (q P w (t) ¨∆q P w (t)) in the algorithm and we run it from the beginning to t CHF . This provides a value of δ slightly dierent compared to the original one. This discrepancy gives the measurement uncertainty. We observe that the relative measurement uncertainty follows linearly the relative measurement on the heat ux. To determine the temporal uncertainty ∆δ t , we add (or subtract) to t CHF the number of frames corresponding to the uncertainty on the determination of the moment of DNB. The uncertainty on the frame of DNB is usually due to the visualization of the irreversible dry spots characterizing DNB (see section 2.2.2). This uncertainty is of only one frame for the fast transients, but can be of several frames for longer periods (2-3). The algorithm is then stopped at slightly dierent moments implying slightly dierent values of δ. The discrepancy with the original value gives the temporal uncertainty ∆δ t .

The mantle thickness is presented in Figure 3.9a. Its value ranges from 100 µm for the short periods and high Reynolds numbers, to 2 mm for long periods and small Reynolds numbers. For the high subcooling degrees (50 K and 75 K), these values are slightly larger (short periods) or signicantly larger (long periods) than the bubble diameter. This supports the assumption presented in section 3.2.3. However, for 25 K of subcooling, the mantle thickness would take values smaller than the bubble diameters (discussed in section 3.3.1.2). One can note than the thickness δ increases with the period τ until a plateau which is reached for τ larger than 100 ms. This increasing trend is regular for most of the tests, except for the ones with ∆T sub = 25K. The value taken by this plateau decreases with the increase of the subcooling and the Reynolds number, but no quantitative regularity is observed. However, when the mantle thickness is normalized by the length δ ν (Fig. 3.9b), the value taken at the plateau only depends on the subcooling showing three distinct limit values.

These observations lead to propose a new normalization on the mantle thickness and on the power escalation period. The mantle thickness is qualitatively related to the thermal boundary layer. In order to account for both turbulent dynamic boundary layer and thermal boundary layer, a relevant approach is to dene a normalization quantity as δ T = δ ν /P r = α/u ¦ , with P r and α being respectively the Prandtl number and the thermal diusivity of water at the considered inlet conditions. Therefore we dene the non-dimensional mantle thickness δ as

δ = δ δ T
To normalize the power escalation period, we choose to use the time scale linked with the advection in the mantle in the streamwise direction as it corresponds to the major velocity component. This time scale is also dened for the limit of laminar conditions. Moreover, as each curve δ(τ ) converges towards an upper limit (plateau) associated with steady-state conditions, it seems worth to use this time scale at steady-state in order to assess the transient nature of the escalation process. Therefore, we dene τ x as

τ x = L u m,V (3.16)
with u m,V the value taken by u m in steady-state, i.e. for long τ . This quantity only depends on the steady-state quantities and can be obtained analytically. When τ approaches innity, corresponding to the steady-state condition, Eq.( 3.3) is reduced to the following equation:

dE ¦ ss (x ¦ ) dx ¦ v m,V L u m,V δ V E ¦ ss (x ¦ ) = L δ V q P ss u m,V E Q cr
where both subscripts "ss" and "V" stand for steady-state conditions. Particularly, q P ss equals the steady-state CHF. This dierential equation is solved using the boundary condition E ¦ ss (0) = 0:

E ¦ ss (x ¦ ) = q P ss E Q cr v m,V ¡ 1 ¡ e ¡λV x ¦ © with λ V = v m,V L u m,V δ V (3.17)
As E ¦ ss (x ¦ ) is a increasing function on x ¦ , the maximum volumetric energy density is observed downstream of the heater, i.e. at x ¦ = 1.

Setting E ¦ ss (1) = 1, i.e. E Q (x = L) = E Q
cr , we obtain the following relation:

1 ¡ e ¡λV v m,V = E Q cr q P ss (3.18)
Beyond the thermal hydraulics conditions, the left-hand-side term only depends on δ V and the right-hand-side one only depends on the CHF value at steady-state. The value of τ x can thus be fully determined knowing the value of CHF at steady-state.

The experimental points are then plotted in gure 3.10 using normalized quantities on both axis: It shows that all available data sets are satisfactorily tted by the following equation:

δ (τ ) = δ V 1 ¡ e ¡τ/τx ¨(3.19)
where δ V is the only tuning parameter and corresponds to the normalized mantle thick- ness in steady-state conditions. A single value, equals to 180 in the present case, is suitable for all operating conditions investigated in this chapter (Atmospheric pressure).

Discussion

Results presented in gure 3.10 show that the quality of the t between the experimental results and equation (3.19) depends on the subcooling. For high subcoolings (50 K and 75 K), the t is excellent showing the validity of the model for these high degrees of subcooling. However, some tests, associated with short periods (τ /τ x 1) and a small subcooling (25 K) present some discrepancies which can be explained. To be valid, the present model needs a sucient degree of subcooling so that nucleation-condensation cycles (NCCs) presented in section 3.1.2 can take place. However, for the concerned conditions, the major assumption of the model can reach its limit: bubbles have a longer lifetime, about a few milliseconds, and can be bigger than the computed mantle thickness. The bubble width being about 3 mm, these are not necessarily spherical. It implies that the duration of a growth and collapse cycle may no longer be negligible in front of the power escalation period. Therefore, the mixing induced by NCCs may not be ecient enough to make the mantle homogeneous, giving a rst explanation for the observed discrepancies. Moreover, Kossolapov et al. [START_REF] Kossolapov | The Boiling Crisis of water under exponentially escalating heat inuts in subcooled ow boiling at atmospheric pressure[END_REF] observed that for such conditions (25 K of subcooling and short periods) at the considered ow conditions, the boiling crisis could randomly be triggered by the rst generation of bubbles. Necessarily, the absence of nucleate boiling implies the invalidity of the concept of NCC and the homogeneous mantle in these conditions, also explaining the discrepancies of these points with the general trend of gure 3.10. This shows that a degree of subcooling as low as 25 K may be the limit of validity for the model, especially for short periods. Nonetheless, it is worth to note that for long periods Chapter 3. Homogeneous Mantle Model at atmospheric pressure (τ /τ x ¡ 1), equation (3.19) matches perfectly with all data, including those obtained with 25 K of subcooling. Equation (3.19) proposes a simple relation between the mantle thickness δ through its non-dimensional form δ and the dierent working parameters. This shows that the model is able to accurately handle the dierent working parameters. Equation (3.19) includes a unique non a priori calculable constant δ V which corresponds to the value taken by the normalized mantle thickness at steady-state for all the conditions of this set.

The t with the experimental data gives δ V equals 180 for the current set of conditions. However, this quantity is expected to depend on other parameters such as the pressure or the quantity λ V dened in Equation (3.17). A higher pressure would lead to smaller bubbles then would reduce the constant δ V . The quantity λ V represents the ratio at steady-state of heat removal between the x and the y directions. It linearly depends on the heated length L. For the current set of experimental conditions, λ V takes values close to unity or above, meaning that heat removal in the x direction prevails or is equivalent to that in the y direction. If considering cases where the heat removal in the y direction signicantly prevails, one would expect the constant δ V to decrease since in this case heat transfer takes place along the whole heater.

The present model relies on an energy balance in the uid and does not take explicitly into account the surface condition of the heater. Nonetheless, it is known that boiling phenomena, including boiling crisis, depend on key parameters which are surface conditions: nucleation site density, roughness, wettability, porosity... In our experiments, the exponential electrical heat input is released either in the substrate or to the water. The proportion in time of each part is determined by the eusivity of both media (water and sapphire) and by the surface conditions once the boiling occurs. Therefore, our approach implicitly takes into account the surface conditions as well as the thermal inertia of the substrate via the time function q P w (t), dened as the net heat ux released to the water. Moreover, at such a high degree of subcooling, bubbles present a high-frequency pulsating behavior which establish NCCs, mixing the uid in the mantle. It is then reasonable to assume that the major phenomenon leading to boiling crisis is the transient heating process in the mantle as described in equation 3.2. This would certainly not be the case for lower subcooling.

Energy distribution along the heated wall

Solving the energy balance equation (3.3) allows one to access to the function E ¦ (x ¦ , t ¦ ). The volumetric energy density rise is plotted in Fig. 3.11 as a function of the distance x ¦ , at time t ¦ = t ¦ CHF corresponding to the time of boiling crisis. Results are presented for two conditions of subcooling, 50 K (Fig. 3.11a) and 75 K (Fig. 3.11b). Both cases include seven values of the time-period τ superimposed with the analytic solution for steady-state conditions Eq.(3.17). These gures show that the set of curves are very similar at 50 K and 75 K. This result conrms the relevance of the dimensionless approach. As expected, the volumetric energy density rise is seen to increase with the coordinate x ¦ . Moreover, the boiling crisis is more likely to happen downstream the heated plate for long time-periods, greater than 20 ms, as a regular increase of E ¦ with x ¦ is noticeable in both gures. Indeed, for the long periods, about 20 percent of the plate length (0.8

Results and physical discussion

x ¦ 1) is associated with a volumetric energy density in the mantle greater than 90% of the critical energy. One will also note that curves with a time period longer than 50 ms almost overlap, nearly matching the analytic solution derived for an innite time-period. A dierent behavior should be observed for very fast transients. The curves presented in Fig. 3.11 for the shortest time-periods, i.e. 5 ms and 10 ms, reveal a dierent increase of the volumetric energy density rise with x ¦ . For τ = 10 ms, more than 50 percent of the plate length (0.5

x ¦ 1) has a volumetric energy density in the mantle greater than 90% of the critical energy. This ratio increases up to 80% of the heating length for τ = 5 ms. One expects therefore a large dry spot to occur and to cover most of the plate. The existence of these two regimes of boiling crisis has been conrmed experimentally. A link with the two non dimensional groups N x and N y will be shown in the following section.

Impact of the non dimensional groups N x and N y on boiling crisis

The energy balance equation (3.3) includes two non dimensional groups dened as:

N x = u m τ L N y = v m τ δ

Physical meaning of N x

The group N x is the ratio of two characteristic times: τ which is the power excursion period, and L/u m which is the time needed for a uid particle to travel in the mantle from the inlet (x = 0 i.e. x ¦ = 0) to the outlet section (x = L i.e. x ¦ = 1).

A N x value smaller than unity means that the uid particle is "trapped" in the heated mantle during the τ period. As a result, the temperature of the mantle increases. A N x value greater than unity means that the uid particle has enough time to be advected alongside the heated area during a time period τ . The temperature of the mantle is slightly modied. From the mathematical point of view, the group N x represents the magnitude of the spatial variations in the ow direction. No energy gradient should be observed in the limit of a zero value leading to a homogeneous incipience of boiling crisis all over the heated plate: the ow is "frozen". On the opposite, large values of N x correspond to a strong variation in the vertical direction: the boiling crisis is expected to happen in the form of dry spots localized downstream the plate. These two predicted regimes are conrmed by experiments as presented in Figure 3.13. In terms of heat removal, N x can also be interpreted as the eciency of the cooling process in the streamwise direction. Chapter 3. Homogeneous Mantle Model at atmospheric pressure For the experiments of phase 2 currently investigated, the heating cell is 1 cm in length. For conditions of Re = 35000 and ∆T sub = 75K, the frozen ow regime is observed for a time period τ ¤ 10ms while the ow driven regime happens for τ ¥ 20ms. One expects these values to be modied by a change in the heater length. Other experiments have been conducted using the same channel and similar thermohydraulic conditions but with a longer heater, 4 cm in length. The quantity N x becomes equal to 0.73 for the power excursion period τ = 50 ms, now corresponding to the frozen ow regime. The space-variation of the volumetric energy density rise at t=t CHF and the heat ux eld measured on the heated plate obtained for these conditions are presented in Fig. 3.14.

Results and physical discussion

The energy prole shows a plateau for x ¦ greater than 0.4. The heat ux distribution clearly reveals two distinct areas. At the inlet part of the plate (x ¦ ¤ 0.4) a large amount of heat ux is released in the uid while beyond x ¦ = 0.4, many blue spots corresponding to a small heat ux are visible revealing that dry spots are present in this area. From the previous analysis, N x appears to be a relevant quantity, more than the period τ , to predict the location where the boiling crisis can occur, i.e. homogeneously along the heater or localized downstream the heater. Indeed, considering the same time period as in Fig. 3.14, i.e. τ = 50 ms but the 1 cm-long heater, one observes the boiling crisis occurs, localized only at the downstream extremity of the heater. This feature is not surprising as these experimental conditions correspond to N x = 5. This conclusion may be useful for industrial applications, particularly for nuclear industry. If considering for instance, an accidental scenario of the BORAX-type, the computation of N x would give information on the particular zones where the fuel rods could be damaged. If the thermohydraulic conditions and the power escalation period correspond to a high value of the N x group, the boiling crisis will only occur in the downstream part of the core, whereas a small value of N x would imply a possible occurrence of the boiling crisis all along the entire core.

Results and physical discussion

Interpretation of N y and experimental evidences

The non dimensional group N y represents the ratio between two characteristic times, τ the power excursion period, and δ/v m which is the time needed for a uid particle, located at the boundary bulk-mantle, to travel away from the wall over a length δ at the velocity v m . Qualitatively, N y is the number characterizing the eciency of the system to transfer energy from the mantle to the bulk in the direction normal to the wall via the turbulent uctuations of the normal velocity. N y increases with the Reynolds number. The role of this group N y can be illustrated considering the results obtained in 1964 with the SPERT IV facility [START_REF] Crocker | Reactor Power Excursion Tests In The SPERT IV Facility[END_REF]. The reactor is made of 64 cm tall fuel plates, where power pulses were generated for the experiments. The uid was highly subcooled (∆T sub 90 K) owing between the fuel plates with a bulk velocity up to U b = 3.7 m.s ¡1 . As the characteristic velocity u m is much smaller than U b , the N x group can be highly maximized by U b τ /L. All runs were conducted with a power escalation period equal to 10 ms, leading to a N x value smaller than 0.1 due to the large value of L. This situation corresponds to the "frozen regime". Various tests reported in [START_REF] Crocker | Reactor Power Excursion Tests In The SPERT IV Facility[END_REF] are presented in Fig. 3.15. All recordings were performed with a similar power input and a time-period τ of ten milliseconds. The only change between the four tests was the bulk velocity as indicated in Fig. 3.15. At low velocity (Tests B-23 and B-29), a temperature plateau is observed, 70 K above the saturation temperature. This does not happen for the highest velocities (Tests B-33 and B-39) where smaller temperature excursions are observed. A better cooling is achieved as the bulk velocity increases due to the turbulent uctuations. The velocity v m increases, therefore preventing the occurence of boiling crisis.

Relationship between N x and N y

In gure 3.12, it is worth to note that the quantities N x and N y have a similar behavior. They increase, almost linearly, with the power escalation period τ and their behaviors with the dierent parameters are also similar. It is then interesting to determine the relationship between these two parameters. To this end, we plot in gure 3. In order to have an analytic explanation of the constant 3.75 10 ¡4 , let us rst dene two quantities of interest in the mantle:

F (δ ) = » δ 0 U (y )dy G (δ ) = v m (δ ) F (δ )
F only depends on the mantle thickness as it is solely computed with the universal velocity prole U dened with the law of the wall. G depends also on the ow conditions 94

Conclusion

as the function v m depends on Re τ . With this expressions, we can deduce that:

N y N x P e ¦ = G P r L e (3.21)
After computing the dierent working conditions, the quantity G /P r is seen to be constant for steady-state conditions and when the turbulence is fully developed (Re ¡ 10 4 ).

This explained the excellent agreement observed at large values of N x between the curves and the line 3.75 10 ¡4 N x . One will notice that this is not true for the case Re = 8500 which is really slightly underneath the line, but in this case the turbulence is not fully developed. It is worth to note that the value taken here by G /P r are dependent to the value taken by the constant δ V (here equal to 180).

This property is interesting for the estimation of CHF in steady-state conditions as it proposes a new relation between CHF in steady-state q P SS and the working conditions. Indeed, using equation 3.18 and equation 3.21, we can obtain the following relation

q P SS = E Q cr v m (δ V /P r, Re τ ) 1 ¡ exp ¡Pe ¦ G P r L e ¨(3.22)
which is only function of the conditions of the problem and of the two constants δ V = 180 and G P r = 3.75 10 ¡4 .

In the next chapter, we will examine the impact on the model of higher pressures and longer heating length.

Conclusion

In this chapter, we have developed a new model aiming to describe the ow boiling crisis generated by exponential power transients at high subcooling and atmospheric pressure. The development of this model is based on the visualization (HSV and IR thermography) of the phase 2 experiments. Particularly, this model accounts for the Nucleation and Condensation Cycle (NCC) a mechanism not previously taken into account to model boiling crisis for such conditions of high subcooling. The model includes heat removal contributions in both streamwise and normal directions using turbulent ow characteristics determined by a DNS data analysis.

The present model makes use of a physical a priori unknown quantity which is the mantle thickness δ. Using a non-dimensional approach, we have shown that this quantity obeys to a simple mathematical relation involving the operating conditions and a unique experimental tuning constant δ V common to all investigated tests. The t between the experimental points and this simple mathematical relation is excellent for highly subcooled conditions, i.e. for 50 K of subcooling and more, which are the most relevant for nuclear safety applications. The tting is also excellent at 25 K of subcooling for long periods, i.e. for τ /τ x ¡ 1, which reveals the validity limit in term of subcooling of our approach. The quality of this tting shows that from this simple formulation, the model is able to take into consideration the dierent operating parameters explaining the role of each. Moreover, the two non-dimensional groups N x and N y arising in the model provide physical explanations for experimental observations. The group N x appears as a predicting tool to explain spatial occurrence of the boiling crisis on the heating wall, i.e. homogeneously along or localized downstream the heater. This result can be usefully applied for instance in nuclear safety as it allows to predict the localization of the failed fuel during an accident. The group N y quanties the contribution of heat removal in the transverse direction due to turbulent velocity uctuations. It brings into light eects formerly reported in large-scale reactor tests.

Lastly, the ow conditions to be considered for the prediction of boiling crisis deserve a few comments. Quantities usually mentioned in the literature are either the bulk velocity U b [m.s ¡1 ], the mass ux G [kg.m ¡2 .s ¡1 ] or the Reynolds number Re [-]. Never- theless, these quantities are relevant if considering the ow in the whole cross-section. As discussed in the present work, for high subcooling and strong forced convection, physical phenomena involved in the occurrence of boiling crisis are located in the region near the wall, whose characteristic length scale is the mantle thickness δ. Therefore the relevant quantities to be used for these specic conditions should be the wall quantities, namely the wall Reynolds number Re τ [-], the shear velocity u ¦ [m.s ¡1 ] and the viscous sub-layer

unit δ ν [m].
This chapter investigated exponential power transients and homogeneous heat ux elds. An extension to other types of transients and to heterogeneous heat ux elds is however possible as the current mathematical formulation of the model allows it. Finally, a better understanding of the constant δ V would enable the present model to become fully predictive. The impact of the pressure and of the parameter λ V is also important to determine. The next chapter investigates the impact of these two parameters, and attempts to highlight the physical meaning of the constant δ V in order to allow its determination for any operating conditions.

Chapter 4

The transient boiling crisis at moderate pressure and high mass ux In this chapter, we present a parametric analysis of the new boiling crisis experiment results. The parameters we investigate are the pressure, the temperature, the ow conditions, the exponential power excursion period and the heated length. A discussion concerning the most suitable parameter characterizing the ow is realized using the two dierent channel widths (10 mm and 1.95 mm). Finally, in light of these new results, we will then extend the understanding and the application of the homogeneous mantle model.

Chapter 4. The transient boiling crisis at moderate pressure and high mass ux

Experimental results

In this section, we present the inuence on the transient CHF of the thermodynamic conditions of water (pressure and temperature), the heated length, the ow intensity and the power escalation period. For each parametric study, a photographic description is also presented in order to understand the impact of these parameters on the boiling patterns.

In this section, we use the mass ux G as the ow parameter as this is the raw parameter used for the experimental setting. (Figure 4.1a), the CHF also clearly increases with the subcooling, but with a less strong impact. This impact of the subcooling of the value on transient CHF is consistent with the results reported in the literature for steady-state , e.g. review of Celata [START_REF] Celata | Critical Heat Flux, Post-CHF Heat Transfer and Their Augmentation[END_REF], Bruder [START_REF] Bruder | Critical Heat Flux in Flow Boiling Review of the Current Understanding and Experimental Approaches[END_REF] and Kandlikar [START_REF] Kandlikar | Critical heat ux in subcooled ow boiling -An assessment of current understanding and future directions for research[END_REF], where a linear relation is observed between CHF and the subcooling. This linearity is consistent with the notion of critical energy rise (E Q cr = ρ l c p ∆T sub ) as introduced the model in Chapter 3. This is related with the need of energy -connected somehow to CHF -to heat a certain layer of uid at the vicinity of the wall. Moreover, the authors of the previously mentioned reviews reported similar slopes for bulk velocities of 10 m.s ¡1 and 40 m.s ¡1 , the stronger ows being shifted upwards. This means that in steady-state, the dependency of CHF to the subcooling does not really depend on the velocity. Conversely, we observe for these transient cases a greater enhancement with the subcooling for the highest velocity. This dierence of qualitative behavior might be explained with the model presented in the previous chapter considering the dimensionless group N x . For τ = 5 ms and L = 2 cm, ows with a mass ux of G = 2500 kg.m ¡2 .s ¡1 give values of N x 0.3 whereas G = 15000 kg.m ¡2 .s ¡1 leads to N x 1. It shows that the rst case may be considered as transients with respect to convection inside the mantle, whereas the high ow leads to steady-state behavior. Figure 4.2 presents the inuence of the subcooling on the boiling pattern. As presented for example in [START_REF] Kossolapov | The Boiling Crisis of water under exponentially escalating heat inuts in subcooled ow boiling at atmospheric pressure[END_REF] for atmospheric pressure conditions, the subcooling also has a clear impact on the bubble size for higher pressures: the bubble diameters decrease with the increase of the subcooling. This is due to the colder bulk water which makes the thermal boundary layer to be thinner and so bubbles. Under a pressure of 2.3 bar, the bubble size variation is clearly noticeable between 25 K and 75 K of subcooling. The characteristic bubble diameter goes respectively from about a millimeter to a few hundred of micrometers. It is worth to note that for 25 K, there is almost a direct transition from ONB to DNB, just as observed under atmospheric pressure [START_REF] Kossolapov | The Boiling Crisis of water under exponentially escalating heat inuts in subcooled ow boiling at atmospheric pressure[END_REF], with only a few cycles of nucleation before the boiling crisis. Under a pressure of 8.9 bar, we observe the same qualitative eect of the subcooling but with even smaller bubble sizes (under a hundred micrometers) as the pressure is higher (Fig. 4.2). Finally, with high subcooling degrees, we observe the pulsating behavior presented in the previous chapter, with frequencies depending on the subcooling. For 25 K and 2.3 bar the few observed pulsating bubbles before DNB pulsate with frequencies ranging between one to two kilo Hertz. For 50 K and 75 K and 2.3 bar, the pulsating frequencies are about a few tens of kilo Hertz which is similar to what was reported in the previous chapter for atmospheric pressure. For 75 K and 8.9 bar, the pulsating frequency is roughly comparable to the frequency at 2.3 bar Chapter 4. The transient boiling crisis at moderate pressure and high mass ux stant subcooling, the bulk temperature increases with the pressure implying a variation of the thermophysical properties of water such as viscosity (mostly), thermal conductivity... It is worth to note that in the literature, most investigations only focus on the inuence of pressure at constant subcooling (see reviews [START_REF] Celata | Critical Heat Flux, Post-CHF Heat Transfer and Their Augmentation[END_REF]- [START_REF] Bruder | Critical Heat Flux in Flow Boiling Review of the Current Understanding and Experimental Approaches[END_REF]- [START_REF] Kandlikar | Critical heat ux in subcooled ow boiling -An assessment of current understanding and future directions for research[END_REF]). In ow boiling, neglecting the inuence of the bulk temperature can be misleading as the viscosity highly evolves (a reduction factor of 3 between 25 o C and 100 o C) aecting the size of the boundary layer and thus the impact of the ow. In pool boiling (e.g. Sakurai [START_REF] Sakurai | Mechanisms of transitions to lm boiling at CHFs in subcooled and pressurized liquids due to steady and increasing heat inputs[END_REF]), this impact would be negligible because of the the absence of the ow. Plus, the other thermophysical properties vary less drastically than the viscosity. On the following, we present the analysis of these two parametric approaches.

Inuence of the subcooling

Figure 4.3 presents the inuence on the transient CHF of pressure at constant subcooling for two dierent mass uxes in the test section TS2. For the high mass ux (gure 4.3b), the pressure has a slight inuence on CHF. We can observe a small decrease of the CHF with the increase of the pressure. For a lower mass ux (G = 2500 kg.m ¡2 .s ¡1 ), the pressure has no signicative inuence on CHF (gure 4.3a), as the slight observable decrease of CHF with the pressure is much smaller than the uncertainty. These observations are in agreement with the reviews in the literature [12]- [START_REF] Bruder | Critical Heat Flux in Flow Boiling Review of the Current Understanding and Experimental Approaches[END_REF], which have shown that the direct impact of the pressure on CHF at steady-state is weak. These reviews however also describe a possible decrease of CHF with the increase of pressure at constant subcooling, which is also consistent with our observations. This trend, amplied by the mass ux, may be interpreted in regards of an energy approach describing the heating of a thin layer of uid, such as our model presented in the previous chapter. Indeed, when the pressure increases with a constant subcooling, the bulk temperature rises implying a water viscosity drop. For a given mass ux, this results in an increase of the Reynolds number and thus the thinning of the thermal turbulent boundary layer -related to the thickness of the layer to be heated. A thinner layer requiring less energy to be heated, a smaller heat ux is then necessary to reach the boiling crisis.

Experimental results

On gure 4.4, we present the inuence of the pressure at constant bulk temperature on the transient CHF for two mass uxes in TS2. Similarly to Figure 4.3, we can observe that the pressure has a weak inuence on the transient CHF. Inuence which also seems to be amplied by the mass ux. Conversely to above, CHF slightly increases with the pressure at constant bulk temperature. This is due to the increase of the subcooling which has a signicant impact on the CHF value as presented in section 4.1.1. In the left cell, the subcooling remains constant and in the right cell, the bulk temperature remains constant. At constant subcooling, we can observe that the increase of pressure shrinks the bubbles for all boiling regimes (ONB -FDNB -Post-DNB). Two phenomena explain this shrinking. First of all, a higher pressure increases the vapor density. For a same quantity of vapor generated, the bubbles will be smaller. Secondly, an increase of the bulk temperature implies a drop of the viscosity. This leads to a thinning of the thermal turbulent boundary layer. The characteristic bubble size is then smaller. On the other hand, we can observe at constant bulk temperature that the increase of pressure shrinks the bubble size more intensely. This shrinking is due to the increase of the subcooling which is qualitatively explained in section 4.1.1. It is worth to note that the bubble size is more impacted by an increase of the pressure at constant bulk temperature than at constant subcooling. Finally it is interesting to note the dynamics of coalescence in post-DNB with this condition of pressure. For the two cases at 50 K of subcooling, we observe that the bubbles merge to create only single really large pockets of vapor (see post-DNB images of the upper and bottom-left cells), which probably ll the entire width of the channel, which is 2 mm width. Conversely, in the case with 75 K of subcooling, we observe that the bubbles coalesce in clusters and generate smaller pockets (see post-DNB image of the bottom-right cell). This particularity might be explained by both the high 4.1. Experimental results described by the Homogeneous Mantle Model. When the heated length is more important, the uid, during its streamwise advection, has the possibility to receive more energy for a given heat ux. The uid will then reach the critical energy with a lower heat ux than it would be with a smaller heater, implying a lower CHF value.

Inuence of the mass ux

In gure 4.7, we present the value of the transient CHF as a function of the mass ux for dierent thermodynamic conditions. We can notice that the value of the transient CHF clearly increases with the mass ux independently of the conditions. These observations are in agreement with the literature reviews [START_REF] Celata | Critical Heat Flux, Post-CHF Heat Transfer and Their Augmentation[END_REF]- [START_REF] Bruder | Critical Heat Flux in Flow Boiling Review of the Current Understanding and Experimental Approaches[END_REF]- [START_REF] Kandlikar | Critical heat ux in subcooled ow boiling -An assessment of current understanding and future directions for research[END_REF] investigating steady-state CHF. The increase of the transient CHF with the mass ux is due to the enhancement of the subcooled water supply at the vicinity of the wall. This enhancement improves the cooling process. Consequently, the necessary heat ux to start the nucleation and then to imply DNB is higher. The highest CHF are then found at the higher subcoolings and mass uxes as shows gure 4.7. In order to have a more rened investigation on the impact of mass ux, we present in gure 4.8 the inuence of the mass ux on the transient CHF for a single thermodynamic condition and for two heating lengths. It shows a clear increasing trend with the mass ux, almost linear. This is slightly dierent from the observations reported in the literature for steady-state conditions, where CHF follows a power trend with G, with an exponent below unity (e.g. a law as G 0.4 as proposed by Tong [START_REF] Tong | Boundary-layer analysis of the ow boiling crisis[END_REF]).

Inuence of the transient period and coupling with the ow conditions

The qualitative inuence of the ow in steady-state and the general inuence of the power escalation period are known and were presented previously. In section 4.1.4, we presented the inuence of the ow conditions. The decreasing of CHF with the period is presented in section 1.3.4.5 or by the reference of table 1.3.4.3. However, the coupling of these two parameters is not well reported in the literature. Plus, the previous chapter via the quantity N x , has shown that the concept of transient boiling crisis fundamentally depends on the ow conditions. Figure 4.9 presents the values of CHF as a function of the power escalation period for dierent values of the mass ux. First of all, it is interesting to note how CHF depends on the power escalation period in pool or ow boiling. In pool boiling, see black curve in Figure 4.9, we can clearly observe the usual increase of the transient CHF with the decrease of the period. However, for the three cases in ow boiling shown on this curve, the transient CHF does not depend on the period (for the investigated intervals of mass ux and period). For such ow conditions, as the system responses as it would do in steady-state condition, these runs can thus be considered as in quasi steady-state. This clearly shows the importance of the ow and period coupling to discuss the transient boiling crisis. From the perspective of our model, the calculation of N x for these three cases of ow boiling always gives values larger than 1 (except for τ = 5 ms and G = 2500 kg.m ¡2 .s ¡1 which gives 0.7). It shows that N x is a robust indicator of the transient nature of the experiment for a given couple ow-period. This steady-state behavior can also be observed in the boiling patterns. Indeed, gure 4.10 compares the boiling patterns for experiments with τ = 5 ms and τ = 100 ms and with a mass ux of G = 10000 kg.m ¡2 .s ¡1 using the mean of HSV images around ONB, during FDNB and around DNB. We observe no qualitative dierence of behavior between the two periods. For a given regime, bubble sizes are really similar and do not depend on the power escalation period. At ONB, really tiny bubbles (several tens of micrometers) homogeneously nucleate on the heater with a relatively low density (upper line in gure 4.10). During FDNB, bubble density increases, covering more heating surface (middle line in gure 4.10). They also increase in size but on the same order of magnitude. We can also observe for this mass ux and this subcooling that bubbles have the pulsating behavior described in the previous chapter, but conversely to before, they depart as they are strongly advected by the ow. This shows that under a suciently strong forced ow, the boiling patterns are most likely to be governed by the ow. Bubbles remain in a small layer of uid near the wall whose thickness is related with the turbulent thermal boundary layer. Figure 4.11 supports this statement. It presents for three dierent mass uxes (5000, 10000, 15000 kg.m ¡2 .s ¡1 ), the visualization of DNB with HSV images, Heat Flux and wall temperature distributions under a pressure of 12 bars in a water subcooled at 163 K and with an escalation period of 100 ms. We can observe that the bubble size decreases when the mass ux increases (cf. HSV images), until about tens of micrometer, following the expected thinning of the turbulent thermal boundary layer. It is worth to specify that even in the bubbles, i.e. the vapor phase, the assumption of a continuum medium can still be applied, as the Knudsen number is about 3 10 ¡3 , so Kn 10 ¡2 . The increase of the mass ux also generates more inuence of the turbulence on the heat transfer and bubble patterns. Indeed, we can observe on the IR thermography stripes on the Heat Flux and wall temperature distributions in single phase. When bubbles appear, we can can observe such stripes on the bubble advection. These stripes get thinner with the increase of the mass ux and they oscillate more intensely in the transverse direction, revealing a more intense turbulent velocity uctuation in the z-direction. Finally, we can observe at DNB that the general boiling behavior does not depend on the period τ since it is the same for 5 ms and 100 ms, presenting small and stretched dry spots (see wall temperature and heat ux distributions). It is also worth to note that even though the heat ux at DNB increases with the mass ux, the wall temperature does not fundamentally vary as shown on the last line of gure 4.11. Conversely, the boiling behavior between pool boiling and ow boiling and between two transients in pool boiling are signicantly dierent. Figure 4.12 supports this statement. It features HSV images of two pool boiling transients (5 ms and 100 ms) and one ow boiling transient, as we have shown that, under these conditions, all transients are equivalent to quasi steady-state. The gure features images separated by 0.1 τ from a moment t 0 taken slightly before ONB. In pool boiling, bubbles appear homogeneously on the heater after ONB. The bubble size highly depends on the transient period -see row t 0 0.3τ and t 0 0.4τ . This dierence is due to the development of the superheated layer (SHL), governed by the transient heat conduction normal to the wall. For the slow transient (100 ms), SHL has enough time to grow far from the wall before nucleation starts (see section 1.3.2.2), which results to the progressive nucleation of large bubbles. Conversely, for the fast transient (5 ms), SHL is signicantly smaller and the superheat is signicantly more important: it results in an explosive nucleation of small bubbles. It is worth to note that even with such a high subcooling (∆T sub = 163 K), the bubbles do not pulsate as observed under other conditions. Each bubble is seen to grow up to a certain diameter and then oscillate around this size. Interestingly, this phenomenon is not observed for pool boiling at atmospheric pressure, even for a subcooling of 75 K. Under these thermodynamic conditions, we still observe the pulsating behavior of bubbles. The reason is that for these "low" conditions of pressure and subcooling, bubbles can expand beyond the SHL to reach the subcooled bulk water which implies their immediate collapse. This supports the statement that in the conditions of these new experiments, the bubbles remain in a SHL which governs their size and their dynamics. After ONB, bubble size and density increase until the bubbles coalesce and generate dry spots triggering the boiling crisis. This behavior, depending on the power escalation period, is to be contrasted with the case of ow boiling.

As seen above, when the ow is strong enough, the boiling behavior (bubble size and dynamics) and the CHF values do not depend on the power escalation period, but is mainly governed by the ow. The response of the system to the exponentially escalating heat input is not dependent of the period. Consequently under such conditions, the system can be considered as in quasi steady-state conditions.

Connement and ow characterization

As seen in section 1.3.4, the ow boiling (crisis) has been widely investigated in the literature. Among this signicant amount of studies, the parameter characterizing the ow condition diers: the most usual parameter is the mass ux G, but studies with the bulk velocity U b or the Reynolds number Re are also common. This shows a lack of consensus on how the ow impacts the boiling crisis and boiling heat transfer in general. These quantities are overall quantities and represent the ow characteristics in the whole bulk. However, our investigations at high subcooling in the previous chapter have shown the importance of the ow characteristics close to the wall: the shear velocity u ¦ and the wall Reynolds number Re τ . In order to have a new insight on this problematic, tests have been performed in the two test sections providing two channels with two widths: 10 mm and 1.95 mm. The ow conditions were chosen to enable an overlap of the data obtained in the dierent test sections. The core idea is that the relevant parameter representing the ow should present a continuity when the test section changes. The tests are realized at high subcooling (100 K) and under a high pressure (10 bar) in order to generate bubbles having such small sizes that their impact remains only at the vicinity of the heating wall avoiding the eects of the opposite wall. The tests are realized with three escalation periods (5 ms, 20 ms and 100 ms) and a 2-cm long heater. 4.14 show that for a given thermodynamic condition, representing CHF as a function of Re or as a function of Re τ is equivalent: the relative trends between the two geometries are identical. This gure also shows that a Reynolds number does not seem to be a relevant quantity to represent the ow in the boiling crisis problem under such conditions. Indeed, the values taken by the Reynolds numbers (usual and wall) are superimposed for the two test sections whereas the values taken by CHF are signicantly dierent. Moreover, one can note that the trends, especially for 20 ms and 100 ms, are signicantly dierent between the two test sections (between a Reynolds number of 2000 and 8000 and between a wall Reynolds number of 450 and 1100). show that quantities based on a velocity present a signicantly better overlap than with Reynolds numbers. These two representations present a similar behavior. For 20 ms and 100 ms, CHF presents a similar trend as it is almost continuous, the CHF values in TS1 seem to be slightly higher than in TS2 (about 20%) at the superimposed value of x coordinate. The representation with the bulk velocity seems to be slightly better (smaller jump at the junction). For the fast transient (5 ms), the behavior of CHF with the two test sections seems to be dierent. A continuity seems to appear around u ¦ 10 cm.s ¡1 , but the point at 15 cm.s ¡1 with TS1 seems to exceed and breaks the continuity. It is worth to note that the representation with the mass ux (gure 4.17) is really similar to the velocity representations. Particularly, it is practically identical to the one with the bulk velocity. This is due to the fact that at a given thermodynamic condition and under such a strong turbulent condition, the bulk velocity and the mass ux are practically proportional. These comparative studies show that none of these ve parameters are perfectly suited to characterize the ow conditions on the value of CHF. The best choice would be to use a parameter of velocity or the mass ux, but we have shown that these are not perfectly satisfactory. The previous chapter has risen a characteristic quantity of the boiling crisis : the mantle thickness δ. By making the same comparative study to the mantle thickness that has been realized for CHF (see Appendix D), we observe that the Reynolds numbers (Re and Re τ ) clearly fail to catch a continuity or any rational trend on the mantle thickness such as for CHF. However, a representation of CHF as a function of one of the velocity parameters seems to be signicantly more promising. It also shows that the shear velocity is the most satisfying parameter to study the behavior of the mantle thickness. Figures 4.18 and 4.19 present the mantle thickness (dimensional and non-dimensional forms) as a function of the shear velocity. We can observe that for the dimensional form, both continuity and slope at the junction between the two curves are excellent. Moreover, its evolution was expected when the ow becomes more intense, the thickness decreases accordingly to the thinning of a thermal turbulent boundary layer as seen if gure 4.11. Concerning the non-dimensional form, its trend is even more interesting. We can observe that for τ = 100 ms (gure 4.19a), the continuity of the results between the two test sections is excellent as well and the continuity of the slope at the overlap. For τ = 20 ms (gure 4.19b), the slope at the junction is excellent and the continuity is also really good.

Chapter 4. The transient boiling crisis at moderate pressure and high mass ux

It is interesting to observe that for the points made in TS1, when the shear velocity increases, δ seems to increase until a maximum and decreases to make the overlap with the TS2 points. Finally for τ = 5 ms (gure 4.19c), the continuity at the overlap between the two test sections is also good, even though the slopes are almost orthogonal. This is however not really problematic as the case for τ = 20 ms is similar, but showing a slope change which ensure the continuity. This shows that there is a phenomenological reason underneath and not just the result of a coincidence. These comparative studies lead us to two conclusions: under conditions of high subcooling, the best parameter to characterize the ow conditions in boiling crisis studies seems to be the shear velocity. It also shows that the quantity to investigate is the normalized mantle thickness δ . It is worth to note that the behavior of δ seems to be dierent compared to the one presented in the conditions of the previous chapter. Particularly, we can notice that for τ = 100 ms, δ seems to be constant, nearly equals 85 (which is dierent from the previously known δ V = 180), then regularly decreases with the increase of the shear velocity. The two other periods (20 ms and 5 ms) have a behavior even dierent, δ increases then decreases with the increase of the shear velocity.

Extension of the Homogeneous Mantle Model

In the following section, we extend the Homogeneous Mantle Model to higher values of pressure and λ (i.e. higher values of L and u ¦ ). This will rationalize the a priori inconsistent behavior of δ presented in gure 4.19 and extend the understanding of the model to a wider parameter range.

Extension of the Homogeneous Mantle Model

Chapter 3 has shown that the study of a homogeneous mantle and its dimensionless thickness enables to understand quantitatively the transient boiling crisis. However, it only investigated the conditions of atmospheric pressure, with relatively low mass ux and in a single cross section geometry. This section aims at extending the previously proposed model to a wider range of conditions.

Extended analysis of the Mantle Energy Equation

Let us recall the dimensionless equation of energy in the mantle:

∂E ¦ ∂t ¦ N x ∂E ¦ ∂x ¦ N y E ¦ = τ E Q cr δ q P w (t ¦ )
In order to compare the contributions of the cooling process in each x and y direction, let us divide the latter by N x to obtain

γ ∂E ¦ ∂t ¦ ∂E ¦ ∂x ¦ λ E ¦ = q ¦ (t ¦ ) (4.1)
where

γ = 1 N x = L u m τ λ = N y N x = v m L u m δ q ¦ (t ¦ ) = L q P w (t ¦ ) δ u m E Q cr
In this form, the parameters γ and λ allow to make to following analysis:

Inuence of γ

The parameter γ quanties the magnitude of the transient nature of the phenomenon.

When γ 3 1 the term γ ∂ t ¦E ¦ becomes negligible compared to the other terms. All the remaining terms evolve in time accordingly to the source term q ¦ (t ¦ ), the problem can then be assimilated as in quasi steady-state. Conversely, when γ 4 1, the term γ ∂ t ¦E ¦ is signicant and the problem is transient, i.e depends on the power escalation period τ .

Inuence of λ

The parameter λ compares the magnitudes of the cooling processes in the y and the x direction. The third term of the left-hand-side is not a derivative, but directly proportional to E ¦ . λ E ¦ may then be interpreted as a cooling term eective along the whole heater.

For large values of λ, associated with a strong turbulent ow condition, the turbulent velocity uctuations become a much more ecient advection cooling process than the streamwise mean ow advection.

It is worth to note that the parameter λ is always dened, especially in steady-state conditions where it converges to λ V (see chapter 3), showing that this quantity is an intrinsic characteristic of the phenomenon, in regards of the model. Finally, q ¦ (t ¦ ) is the dimen- sionless heat ux and may be seen as the source term of the equation. Its value taken at

t ¦ CHF is the dimensionless CHF q ¦ chf . ux
According to the values taken by the two characteristic parameters γ and λ, four borderline cases of transient ow boiling crisis congurations permit to discuss how the source term q ¦ (t ¦ ) is transferred or stored in the mantle. These four borderline cases are characterized as follow:

1. γ 3 1 and λ 3 1 :

∂E ¦ ∂x ¦ = q ¦ (t ¦ )
In this borderline case, there is no term in time derivative, which means that the problem can be assumed as in quasi steady-state. In other words, the heating rate is suciently small enough to be integrally balanced by the cooling processes that eciently evacuate the energy. Then, the energy in the mantle evolves as the power escalation does and the value of CHF is expected to be similar to the one in steady-state for the given set of conditions. Moreover, the absence of the term λ E ¦ shows that the dominant cooling process is the streamwise advection. In terms of boiling pattern, due to the term ∂ x ¦E ¦ , we expect the dry spots to occur heterogeneously downstream the heated length.

2. γ 4 1 and λ 3 1:

γ ∂E ¦ ∂t ¦ = q ¦ (t ¦ )
In this borderline case, the absence of the term λ E ¦ reveals that heat transfer by turbu- lent velocity uctuations is negligible and the dominant cooling process is the streamwise advection. The existence of the time derivative term implies that the problem cannot be assumed as quasi steady-state. The heat input increases too fast for the cooling processes to remove eciently the energy in the mantle and a part of the heat input at the wall is stored in the mantle.

3. γ 3 1 and λ 4 1:

λ E ¦ = q ¦ (t ¦ )
In this borderline case, there is no term in time derivative meaning that the mantle can be assumed as in steady-state condition. In other words, the heat input is suciently slow for the cooling processes to evacuate eciently the energy. Then, the energy in the mantle evolves as the power escalation does. The term λ E ¦ cools the mantle along the whole length of the heater via the turbulent velocity uctuations normal to the wall. This cooling process prevails over the streamwise advection and increases with the increase of the ow and with the heated length.

4. γ 4 1 and λ 4 1:

γ ∂E ¦ ∂t ¦ λ E ¦ = q ¦ (t ¦ )
In this borderline case, all the contributions weight on the phenomenon. γ 3 1 implies that the process is transient and some heat is stored in the mantle. Besides, the term λ E ¦ enables an ecient normal cooling process all along the heater.

These four borderline cases bring to light four zones in the space (γ, λ) which permit to describe both the transient nature of the phenomenon and the governing cooling process. These zones, called in the following "quadrants", are presented in Figure 4.20. It is worth to note that as these are borderline cases, the strict analysis presented above has to be qualied for the points near the borders of the quadrant, i.e. close to the dashed lines. It is however useful to discuss parametric trends and behaviors as we present in the following.

Mantle thickness behavior and experimental evidence

After the analytic study of the mantle equation, let us analyze the experimental results in regards of the proposed description.

In order to apply the model to the whole test matrix, the range of Re τ which is limited to 1000 in section 3.2.4.2 has been extended to a value of wall Reynolds number equals to 2500. To do so, we analyzed DNS data with Re τ = 5200 made available by Lee and Moser [START_REF] Lee | Direct numerical simulation of turbulent channel ow up to Re τ = 5200[END_REF]. The data set proposes the full velocity eld of their simulation at steadystate. Their simulation domain consists of a mesh made of 10240 ¢ 1536 ¢ 7680 nodes and the database presents 11 snapshots. This analysis process and the assumptions used are similar to the case of Re τ equals 1000 (see section 3.2.4.2). For reasons of le size and time, we only use a single snapshot, which already gathers almost 80 million points for each investigated y coordinate. Figures 4.21a It is worth to note that for values of y greater than 100, the graph is more noisy. With the experience gained in the previous chapter, we knew that the points beyond this value would probably not be used. It was then chosen to decrease the accuracy of the analysis. ux The space (γ, λ) enables to investigate the impact of the dierent phenomena described by the homogeneous mantle model. However, these are not the parameters which are controlled experimentally. Series of points controlled by the evolution of experimental parameters such as the ow condition or the power escalation period generate paths in the the space (γ, λ).

In gure 4.22, we present series of points showing the impact of the ow parameter (G or Re) and the power escalation period. Figure 4.22a displays the three ow boiling crisis series presented in section 4.1.5. At 12 bar, 163 K of subcooling and in test section TS2, series with variable power escalation periods are made with three mass uxes. For these conditions, we may notice that all the points are located in the quadrant 3 -except for two points at really small periods which are at the boundary of the quadrants 3 ¡ 4 (γ 1.5). This means that even for signicantly small power escalation periods, the problem remains quasi stationary in the sense of the model. This is consistent with the quantitative results (CHF independent of the power escalation period) and photographic observations (similar boiling pattern) presented in section 4.1.5. For a given mass ux, we can see that the increase of the power escalation period moves the points to the low values of γ, which is expected, and without impacting the value of λ. Moreover, it is worth to note that a variation of the mass ux impacts both coordinates in γ and in λ : the points evolve diagonally in the upper right direction with the increase of the mass ux. Figure 4.22b presents the series at atmospheric pressure and at 50 K of subcooling presented in Chapter 3. The ow being signicantly less intense, these series are located in a region where λ is one order of magnitude lower. Qualitatively, the impact of a variation of the mass ux or the power escalation period is similar to above: the increase of the power escalation period makes the points migrate to the lower values of γ and the increase of the mass ux makes them migrate to the higher λ. It is however interesting to note that in this region, a series in power escalation period generates a curved path. When γ 3 1, the path tends to an evolution with the power escalation period which does not impact λ similarly to what was presented above: it follows an horizontal path. When γ 4 1, the path tends to a diagonal direction where both γ and λ are impacted. In order to discuss how the quantities δ and q ¦ chf evolve according to the location of a test in the space (γ, λ), we decide to consider individually the cross sections following the quadrants 1 ¡ 2 and 1 ¡ 3. It enables to illustrate individually the impact of γ and λ independently from the other parameters. We will also present how the system behaves when it follows a transverse path using the series of points presented in section 4.2.

The cross section following the quadrants 1 ¡ 2 is well represented by the points at atmospheric pressure presented in Chapter 3. In gure 4.23, we plot for these points the values of δ and q ¦ chf as a function of γ. We can observe that these quantities have dierent trends depending on the quadrants. In quadrant 1, the quantities are roughly constant. Then, the quantities strongly evolve in quadrant 2: δ decreases with the increase of γ while q ¦ chf increases with it. Moreover, we have seen in section 4.3.1 that the quadrants enable to qualitatively predict the boiling crisis patterns. In gure 4.23, two points, A and B, are highlighted throughout their respective location in the cross sections: point A being characteristic for the quadrant 1 and point B being characteristic for the quadrant 2. Their heat ux distributions at DNB are also presented for illustration. Let us recall that the dry spots appear as low heat ux zones, i.e. blue, on the heat ux distribution. In quadrant 1 (point A), we observe that the dry spots characterizing the boiling crisis occur downstream the heater, which is predicted by the predominance of the space derivative compared to the time derivative (γ 1). Conversely, in quadrant 2 (point B), we observe that the dry spots occur homogeneously along the whole heater, predicted by the model due to the predominance of the time derivative compared to the space derivative (γ ¡ 1). It is worth to note that this analysis is similar with the one presented in Chapter 3 where the transient nature is characterized by τ /τ x , τ x equals L/u m with u m taken at steady-state conditions.

In gure 4.24, we present the cross section following the quadrants 1 ¡ 3 by the mean of the evolution of δ and q ¦ chf as a function of λ in this region. These latter behave faithfully to the theoretical predictions presented in section 4.3.1. In quadrant 1, λ À 1, the quantities remain constant and in quadrant 3, these latter strongly evolve: δ decreases with the increase of λ and q ¦ chf increases with it. In order to illustrate the qualitative predictions of section 4.3.1, gure 4.25 enlightens three points of the series presented in section 4.1.5 and on gure 4.22a. We recall that these points, featuring power escalation period series for dierent mass uxes, are realized in the test section TS2 with a heated length of 1 cm under a pressure of 12 bar and with a subcooling of 163 K. The points A and B are both with a mass ux of 15000 kg.m ¡2 .s ¡1 but with two power escalation periods, respectively 5 ms and 100 ms. The points B and C have the same power escalation period (100 ms) but with two dierent mass uxes, respectively 15000 kg.m ¡2 .s ¡1 and 5000 kg.m ¡2 .s ¡1 . The location of these points in the cross section following the quadrant 1 ¡ 3 is presented in the left-hand side graph with their corresponding values of δ . As we study points with dierent power escalation periods, we also illustrate their locations in the space (γ, λ) in the right-hand side graph. Finally we present the boiling pattern at DNB of these three points with an HSV image and the heat ux distribution. We can note that these three points are located in the quadrant 3, meaning that each of them can be considered as quasi steady-state. This quasi steady-state behavior has been previously mentioned in section 4.1.5 where it was shown that the value of CHF and the boiling pattern do not depend on power escalation period for this range of mass ux and power escalation periods. As the value of λ is higher for the series at 15000 kg.m ¡2 .s ¡1 than for the one at 5000 kg.m ¡2 .s ¡1 , the heat removal is then more important and the CHF is expected to be higher. This enhancement of CHF is conrmed by the results presented in gure 4.9 and is also clearly illustrated here with the points A-B (λ about 20 and CHF around 45 M W.m ¡2 ) and C (λ about 8 and CHF around 25 M W.m ¡2 ). In these series, comparing the dimensional CHF is relevant to bring into light the impact of λ as the thermodynamic conditions are the same. Now let us review the experimental points investigated in section 4.2 in regards of the insight given by this new analysis. On gure 4.26 are presented the dierent experimental points in the space (γ, λ). For the three power escalation periods, the evolution of δ as a function of u ¦ is also recalled. We can observe than for a period of 100 ms, the series of points remain in the quasi steady-state region (quadrants 1 and 3). The series begins almost in the quadrant 1 (λ 1) and reaches the core of the quadrant 3 when u ¦ increases (λ 4 1). This path regularly evolving in the direction λ explains the regular decrease of δ with the increase of u ¦ . Concerning the points at small values of u ¦ , i.e. next to the quadrant 1, δ does not signicantly vary. It shows the plateau presented in gure 4.24. Concerning the series of points with power escalation periods of 20 ms and 5 ms, their hybrid paths in the space (γ, λ) may clarify the non monotonous evolution of δ in such conditions. We may notice the relative curvature of the paths, curvature which increases with the decrease of the period. This makes the path of the series with a period of 20 ms crosses through quadrants. It starts in the quadrant 4, close to quadrant 2, and nishes in quadrant 3. This means that at the beginning of the path, the process is transient with a moderate inuence of the turbulence uctuations (λ 4 ¡ 5) and nishes as a quasi steady-state process with a strong inuence of the turbulent uctuation (λ 60).

The path taken by this series can then be interpreted in two steps, explaining the 2-slope evolution of δ . For a small shear velocity, the series path remains in the low turbulenceinuenced region and switches from quasi steady-state to transient. It means that at this rst step, the increase of the shear velocity mostly reduces the transient nature of the problem. The evolution can then be assimilated to an evolution from the quadrant 1 to the quadrant 2. This means that δ is expected to increase, which is observed. For a greater shear velocity, we can observe that the increase of the shear velocity only aects its position in the map in the direction λ and not signicantly in the direction γ. Therefore in this second step, the evolution can be assimilated to the evolution from the quadrant 1 to the quadrant 3. Consequently, δ is expected to decrease with the increase of the shear Chapter 4. The transient boiling crisis at moderate pressure and high mass ux velocity, which is also observed. The analysis of the 5 ms series is similar. Even though the points remain in quadrant 4, the path curvature is such that the previous analysis remains valid. In the rst step, γ evolves from 50 to 5 whereas λ evolves from 6 to 15.

As the evolution in γ is much more important at this stage, the impact of the transition from a transient problem to a quasi steady-state problem dominates. This explains the increase of δ . Conversely, at the second stage, γ only evolves from 4 to 2 whereas λ evolves from 15 to 70. This explains that δ decreases with the increase of the shear velocity. These two series of points show the competition between the transient process and the impact of the turbulence. It again shows that the transient nature of the problem not only depends on the power escalation period, but also on the ow conditions.

Analytic expression of the transient CHF

In this section, let us analytically consider the equation 4.1. To this end, let us assume that the heat ux transferred to water is exponential q P w (t) W e t/τ . This implies that the dimensionless heat ux can be written as q ¦ (t ¦ ) = q ¦ 0 e t ¦ with q ¦ 0 the dimensionless heat ux at the initial time. It is worth to note that this assumption is not perfectly correct in real experimental conditions as we know that a part of the injected electric heat input, perfectly exponential, is lost in the substrate and that this part varies between the single phase and the boiling heat transfer. It is however a consistent limit situation that deserves a few comments, in regards of the experimental apparatus but also in regards of a nuclear reactor application.

The source term q ¦ (t ¦ ) being proportional to e t ¦ , it is relevant to use the method of separation of variables to solve this equation by assuming that E ¦ can be written as E ¦ (x ¦ , t ¦ ) = f (x ¦ ) e t ¦ . Consequently, equation 4.1 is reduced to an ordinary dierential equation (ODE) on f :

f I (x ¦ ) k f (x ¦ ) = q ¦ 0 with dx ¦ r0, 1s k = γ λ
The solving of this ODE gives

f (x ¦ ) = A e ¡kx ¦ q ¦ 0 k A R The boundary condition of equation 4.1 imposes E ¦ (0, t ¦ ) = 0, dt ¦ R ¦ , implying f (0) = 0 and thus f (x ¦ ) = q ¦ 0 k ¡ 1 ¡ e ¡kx ¦ ©
and nally

E ¦ (x ¦ , t ¦ ) = q ¦ (t ¦ ) k ¡ 1 ¡ e ¡kx ¦ © d(x ¦ , t ¦ ) r0, 1s ¢ r0, t ¦ CHF s
The criterion on the boiling crisis presented in chapter 3 imposes E ¦ (1, t ¦ CHF ) = 1. Using this condition enables us to access the expression of the non-dimensional CHF assuming the heat ux transferred to water follows an exponential trend: This gure presents two interesting characteristics: rst of all, we can notice that there is no point under the line y = x, secondly a signicant number of experimental points is closely represented by the analytic solution. To properly explain these two observations, it is worth to recall the heating conguration of the heater (see Chapter 2). The heat is generated by Joule eect in a thin lm (ITO or Tinanium) coated on an inert substrate (Sapphire). Consequently, the heat input is partitioned to a heat ux transferred to the water (used as an input of our model) and a heat ux transferred to the substrate. For a given set of conditions, this partition depends on the boiling regime The rst result, no point being under the line y = x, is explained by the fact that the part of the electric heat ux transferred to water is signicantly small in single phase, then increases when the boiling starts. This result is illustrated in gure 4.29 where the heat ux transferred to water (input of the model) is below the representative exponential trend. Quantitatively, it implies that the energy deposited to water during the transient will be lower than the one obtained using the exponential as a source term. Therefore, to reach the energy criterion of boiling crisis 3.1 in the experiment, an excess of heat ux is needed compared to the analytic case. Consequently, this analysis enables to estimate a minimum value for CHF:

q ¦ chf = k 1 ¡ e ¡k
min(q ¦ exp ) = q ¦ chf or q P w,CHF ¥ u m E Q cr L δ k 1 ¡ e ¡k (4.3)
Consequently, this expression 4.3 proposes a conservative estimation of the value of CHF which is interesting for nuclear safety design.

The second property, i.e. most of the points are really close to the analytic value of q ¦ This shows that for the set of experimental conditions investigated here, the assumption of an exponential trend is satisfactory. Let us estimate the discrepancy with the latter assumption using the following equation developed in [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part II: Flow boiling[END_REF]: q P w (t) = q P 0 e t/τ h w h s h w where h s and h w are respectively the heat transfer coecient (HTC) to the substrate and to the water. It describes the part of heat ux going to the water q P w (t) from an exponential electrical heat ux input q P 0 e t/τ injected in the heating lm. h s is the HTC to substrate which equals ε s / c τ th(L s / c α s τ ), α s , ε s and L s being respectively the diusivity, eusivity and thickness of the substrate. h w is the HTC to water which signicantly evolves between the single phase and the boiling heat transfer, due to the enhancement of the heat transfer in boiling. Since the HTC presents a jump during the transient, this shows that to have an exponential trend on q P w , the condition h s 3 h w is necessary. To tend to that condition, either we increase h w , particularly in single phase, or we decrease h s . The increase of the mass ux increases h w and the increase of the period τ decreases h s . Figure 4.29 supports this statement. It shows that the increase of the mass ux or the excursion period implies the heat ux to water signal to become closer to the exponential trend. It is worth to note that even with a heat ux to water signal signicantly deviating from the exponential trend (left graph in gure 4.29), the discrepancy on q ¦ chf remains satisfactory (8.0 %). In an industrial context, such as a nuclear fuel plate, the heating conguration is signicantly dierent. A fuel plate is made of a central meat which contains the ssile material. The latter is protected by cladding plates on both sides. Meat and cladding plates thickness are about the same order of magnitude, about 1 mm and less. For example in the SPERT-IV reactor, a meat of 0.5 mm of U-Al is cladded by two 0.5-mm thick plates of Aluminium. Heat is generated in the whole volume of the meat and is transferred in the cladding and nally to the water. In this consideration, it is realistic to think that when the meat undergoes an exponential power excursion, the inertial eect of the whole fuel plate would be minor. Consequently, the heat ux transferred to the water would properly follow the exponential trend during the entire transient until DNB. Plus, in such context, the mass ux is usually high, allowing to have a consequent HTC to water. For these reasons, it is relevant to assume that in industrial applications, the heat ux to water closely follows the exponential escalation generated by the nuclear fuel. The use of equation 4.2 would then be reliable to estimate CHF. Therefore, beyond being conservative, this expression would also propose a relatively accurate estimation of the transient CHF for industrial congurations.

Prediction of the mantle thickness and model closure

We have shown that the theoretical analysis of the equation of the mantle (Eq. 4.1) enables to rise the existence of four quadrants in the space (γ, λ) which describe the dominant phenomena of the transient ow boiling crisis. These predictions are consistent with the experimental observations. Moreover, the assumption of a purely exponential trend of the heat ux transferred to the water, which is realistic and consistent with the industrial applications, enables to derive an analytic value of the non-dimensional CHF. However, in order to properly apply the model to any situation, it is of key importance to know the value of the mantle thickness. The latter being involved in all the calculations and Chapter 4. The transient boiling crisis at moderate pressure and high mass ux quantities of the model. On gure 4.30, we plot in the space (γ, λ) the available points of the test matrix eligible to be analyzed by the model, colorized according to the value taken by δ . We can observe that the non-dimensional mantle thickness qualitatively follows the predicted evolution as formerly presented in section 4.3.1. In the quadrant 1, δ takes its maximum value and is globally independent of the the parameters γ and λ, presenting then a plateau. When only γ or λ increases, i.e. the cross section following quadrants 1 ¡ 2 or 1 ¡ 3, we observe that δ decreases. Yet, some points present irregularities in their positions: some points having the same coordinate (γ, λ) present dierent values of δ and some series seem to follow dierent trends. One can for instance see that in gure 4.25, the cluster of the points A,B-C are above other points which seem to follow another trend. Yet, these irregularities make the quantity δ (γ, λ) complicated to correlate. Chapter 3 has shown that in atmospheric pressure, a simple relation (Eq. 3.18) enables to assess the non-dimensional mantle thickness once the working conditions are known and using a single experimental constant δ V . In order to make the proposed model to be fully predictive and applicable, an extended relation enabling the assessment of the mantle thickness needs to be established in the entire range of parameters. This includes the impact of the pressure and the parameter λ as discussed at the end of Chapter 3. The excellent tting obtained by using equation 3.19 for the atmospheric pressure and moderate velocities in Chapter 3 encourages us to try its use for wider ranges of conditions. However, to apply equation 3.19 to any transient condition, it is needed to know the steady-state CHF at the given working conditions. Yet in the test matrix, many experimental tests have been realized with short periods (usually 5 ms) without the corresponding steady-state tests. Therefore, we chose to assess the value of δ V using an inverse method (see 4.31). Knowing the working conditions, the value of the corresponding δ obtained experimentally and by assuming the validity of Eq. 3.19 we assess, by the mean of an iterative method, the expected value of δ V at this given condition. In gure 4.32, we show the results of the inverse method on the points obtained with 50 K and 75 K of subcooling, presented in Chapter 3 (atmospheric pressure in the test section TS1 with a heated length of 1 cm). More particularly, we plot the corresponding function δ V 1 ¡ e τ /τx ¨using the previously computed δ V and τ x . We can observe a certain dispersion on the computed value of δ V , illustrated by the asymptotic value for τ /τ x ÝÑ V in gure 4.32. This dispersion is connected with the dispersion of the experimental points with the tting function. It is worth to note that points giving the larger discrepancy correspond to the small periods. Besides, this method generates satisfactory values of δ V This method is next applied to all the points. This generates computed values of δ V , representing the expected values of the non-dimensional mantle thickness at steady- state. On gure 4.33, we present the results as a function of the corresponding λ V . We can observe that it seems to follow a certain asymptotic trend: for the small λ V , it seems to be constant and for the large λ V , it seems to evolves as 1/ c λ V . Yet, regarding the ux important dispersion, one has to be careful using this method and future work is expected to permit to improve its accuracy and decreases the dispersion.

Conclusion

In this chapter, we analyzed the impact of the experimental parameters on the transient CHF. The investigated parameters were the pressure, the temperature, the heater length, the mass ux and nally the power escalation period. Particularly, it has been shown that the investigation of the impact of pressure cannot be realized without setting a quantity characterizing the temperature: bulk temperature or subcooling. Plus, we have shown that the ow conditions have a major impact on the transient nature of the transient boiling crisis. If the forced convection is suciently important, heat ux escalating exponentially with a period as short as a few tens of millisecond may be considered as quasi steady-state.

We have discussed the pertinence of the possible parameters describing the ow conditions for the transient ow boiling crisis at high subcooling. The analysis shows that to describe CHF, Reynolds numbers are not the proper indicators, conversely to the velocity quantities which present smaller discontinuity when the channel width changes. Plus, the same exercise applied to the mantle thickness shows that its representation as 4.4. Conclusion a function of the shear velocity presents an excellent match with data. This shows that the mantle thickness is a relevant quantity to describe the transient ow boiling crisis at high subcooling.

We have extended the model developed in Chapter 3 to higher values of the parameter λ. The analysis of equation 4.1 leaded to develop a 4-quadrant diagram describing the impact of the key parameters γ and λ. Each quadrant permits to propose a qualitative description of the boiling crisis behavior. Moreover, the analytic solution of equation 4.1 assuming an exponential form of the heat ux, which is realistic for industrial applications, permits to propose a conservative estimation of CHF. Nonetheless, these analysis can be realized only with the knowledge of the mantle thickness. If its behavior is well understood, a quantitative method to predict the value of the mantle thickness without access to experimental data is yet to be determined. However, with the prior knowledge of the steady-state CHF for a given set of working conditions, the use of the model permits to describe the corresponding transients. Finally, the model is a helpful tool to understand the physical phenomena involved in the transient ow boiling crisis at high subcooling.

Conclusion and Perspectives

In this thesis, we investigated the transient ow boiling crisis of water at high subcooling and moderate pressure. This work is in the continuity of the partnership between CEA and MIT aiming to improve the understanding and the modeling of the dierent heat transfer phenomena involved in the BORAX-type accident.

By the conduct of experimental tests and their analysis, we aimed to understand the impact of the parameters of pressure, temperature, ow condition and power escalation period. We have seen that the impact of pressure cannot be investigated independently of the temperature, either considering constant bulk temperature or subcooling, and both choices permit to bring to light interesting physical phenomenon. The conditions of high subcooling and strong forced convection highly dierentiate the behaviors and the dominant mechanisms from the situations of pool boiling and at low subcooling or saturated. The dependency on the power escalation period of the transient CHF highly depends on ow conditions, relieving a certain coupling. Moreover, the thermal inuence on surrounding water is fundamentally dierent between pool conditions or turbulent forced ow conditions, conditioning the modeling approaches.

We observed and analyzed the space-time highly resolved videos and IR thermography catching the transient ow boiling crisis at high subcooling, highlighting the underlying mechanism. In such condition, the boiling heat transfer is a wall phenomena. Nucleation and Condensation Cycles eciently transfer the heat from the wall to a neighbouring layer of liquid and the boiling crisis is triggered when this layer reaches the saturation temperature. The mathematical formulation of this mechanism enables to describe and predict the phenomenology of the boiling crisis and takes into account two cooling processes. The model brings to light two non-dimensional parameters γ and λ, which respectively describe the transient nature of the process and dominant cooling process. Particularly, the exploitation of the model has shown that the transient behavior of the system cannot be predicted only with the power escalation period, explaining the experimental observations. The non-dimensional quantity γ permits to describe it more accurately by involving the power escalation period, the heated length and the ow conditions.

Using the ndings of the model analysis, the knowledge of the steady-state CHF permits to conservatively estimate the value of CHF for any power escalation period and subcooling. Indeed, from the steady-state CHF, we can determine δ V (Eq. 3.18), which enables to determine the quantity δ (Eq. 3.19) and nally a conservative estimation of CHF (Eq. 4.2 and Eq. 4.3). As this arises from a physical model, this estimation takes physically into account a large number of conditions relevant to the problem: not only the pressure, temperature ow conditions and power escalation period as initially aimed, but also the channel width and the heated length. Therefore, this model can be applied to industrial simulation codes such as CATHARE to predict the transient CHF during nuclear accidental transients. Indeed, in such codes, the steady-state CHF for a given condition is considered as known, using correlations or look-up tables.

Nonetheless, the present model lacks of a physical closure condition. Indeed, the model can be currently used but needs a steady-state CHF to predict extended ranges of conditions. Yet, the model describes the energy evolution in a mantle having a thickness uniform alongside the wall and constant in time. This was chosen for simplicity and to avoid in a rst approach any overtting coming from free parameters. An interesting perspective of improvement would be to mechanistically describe the behavior of the mantle during the transient, for instance by investigating its space-time development similarly to a boundary layer. This would possibly generate a second equation coupled with the rst one describing the model and generate a coupled system of two unknowns, closing the system. It would also be interesting to add more points to the 3D surface δ (γ, λ), which would ensure the existing relations to remain valid. We would suggest to measure additional series of period with conditions exploring other combinations of parameters (γ, λ). It would also be interesting to investigate series in steady-state in order to develop an accurate relation between δ V and λ V , which would also be a closure condition for the model.

Finally it would be interesting to explore the limits of the model and possibly to extend them. The model has been developed for "extreme" conditions. It would be interesting to quantify the minimal conditions to validate the assumptions of the model (currently done by observation of the high speed videos) and how to adapt it otherwise. It would be also interesting to explore its applicability to other geometries such as pipes or annular ducts. This would be most probably the case for geometries where the curvature radii are large. Finally, we suggest to investigate the impact of the surface conditions in a more controlled manner and with totally dierent surface conditions. It would validate the fact that the surface conditions is a second order parameter in the model behavior, correctly taken into account by the use of the experimental heat ux, and if not, how the surface conditions quantitatively impact it.

Bubble detection algorithm

Using backlit shadowgraphy, the videos obtained by the front HSV camera present an excellent contrast. It would then be interesting to quantitatively analyze these frames: bubble size distribution, bubble density, evolution in time... However, in case of high subcooling, bubbles are numerous and the experimental setup allows a relatively small space and time resolutions for such tiny bubbles. The development of a bubble detection algorithm is then necessary. In the following, we present the dierent approaches explored and their results, as well as the possible perspectives.

Threshold lter and bubble footprint

The original frames issued by the backlit shadowgraphy are gray scale images (256 bits) where the heater appears in light gray and any manifestation darker. Most of the dark manifestations appearing are bubbles, but the backlit shadowgraphy is suciently sensitive to also catch the high variations of the refractive index due to the high temperature gradient. Due to the forced convection, these variations appear slightly before ONB, i.e.

when the gradient of temperature near the wall is high, in the form of thin stripes.

The rst approach is the gray scale threshold. First of all, the threshold lter inverses the color to make the background dark and the object light. A pre-lter is also applied to clean any constant impurities on the frame, such as possible dust. A threshold is then chosen to accurately lter the background from the bubbles. If a pixel has a gray scale higher than the threshold, the lter gives to that pixel a value of 1, and 0 otherwise. The application of that lter to the whole bitmap generates a binary frame where detected objects are white (pixel value of 1) and the background is black (pixel value of 0). If the threshold is too small, the lter is too sensitive and too much noise is detected. This increases the number of small "fake" bubbles that are detected. If the threshold is too large, the lter is not sensitive enough. Not all bubbles are detected and the detected ones may appear too small. From the threshold lter applied to the HSV frames, it results successive bubble footprint maps that can be analyzed. On gure B.2, we present on an example frame the dierent images resulting from this process. The rst quantitative information that can be obtained is the proportion of the heating surface covered by bubbles. For example, in Figure B.3, this proportion is plotted as a function of the normalized time for three dierent power escalation periods. This time is normalized with the power escalation period and the origin is taken at ONB. This gure allows us for noticing that the development of boiling takes a longer normalized time for a longer period. For a short period, the nucleation can be considered as "more explosive" whereas for longer period, it is more progressive. This goes in the direction of the analysis of the model and the use of γ (or N x ) and the time analysis which will be presented in Appendix C. The footprint also permits to isolate bubbles in case of relatively low density. It enables to count them and to determine their size (diameter or foot print area). Processing this information permits to draw the bubble size distribution as presented in B.4. Particularly, this distribution can be evaluated for each frame of the transient process between ONB and DNB. It is worth to note that such a distribution is akin to the ones of Zhang et al. [START_REF] Zhang | Percolative scale-free behavior in the boiling crisis[END_REF] who presented the percolative scale-free behavior of steady-state boiling crisis. In their work, they use the heat ux foot print which is possible as they investigated boiling near saturation conditions (subcooling up to 5 K), consequently presenting high spatial resolution. This approach is not applicable with our conditions of high subcooling as bubbles appear too small in the heat ux distribution, but suciently resolved in HSV frames. Moreover, the transient nature may present dierent footprint distributions. One can notice that when the bubbles are too close, their footprint can merge and generate a single bigger footprint. For a more accurate detection of bubbles in such case, and thus for a more accurate identication their size distribution, it is necessary to individually detect them. Geometric methods exist to detect several bubbles from a single footprint. It is for example possible to assume that bubbles in such conditions are spherical or ellipsoidal and then to estimate their centers and dimensions. However in our case, bubbles are so small that they appear only as clusters of several pixels. Consequently, it is not possible to extrapolate their geometric shape. Another approach is to develop an edge detection algorithm.

Edge Detection

Edge detection permits to capture the edge of an object by detecting an important spatial variation of the grey scale bitmap. Two methods were investigated: the Laplacian of Gaussian method (LoG) and the multi-scale wavelet transform.

The LoG uses the Laplacian (second space derivative) to emphasize the spatial variation of the grey level distribution. However, this operator is so sensitive that it captures any noise on the frame. Consequently, we rst apply a Gaussian operator to the frame to smooth the grey level distribution. Then, a judicious threshold permits to catch the bubble edges as presented in Figure B.5. This method has three tuning parameters: the standard deviation σ of the Gaussian lter, the kernel of the lter and the threshold. and

q P ref = q P 0 ε l ε s tanh Ls c αsτ ε l (C.5)
The reference quantities ∆T ref and q P ref depend on thermal properties of the substrate (α s , ε s ) and of the liquid water (ε l ), the power escalation period (τ ) and the initial value of the electrical power input (q P 0 ). They can therefore easily be transposed to other experiments. For this reason, we propose to represent the temperature and the heat ux released in the uid using these specic reference quantities even though their relevance is a priori limited to the heat diusion process.

Time evolution to transient boiling crisis

The normalized wall temperature and heat ux transferred to the water are presented in Each gure shows ve curves which are in close coincidence. Time evolution of the heat ux shows two distinct processes. At early stages, heat transfer develops in the liquid phase. The heat ux increases slowly and regularly. Boiling inception occurs leading to a sharp increase of the heat ux until the triggering of the boiling crisis. This brutal change in the heat ux slope is due to the high eciency of the boiling heat transfer. Indeed, the eusivity of water being signicantly smaller than for sapphire, heat is preferentially transferred to the sapphire in single phase. This preference changes after ONB as boiling heat transfer is really ecient leading to a high value of the heat transfer coecient h w . Note that this slope discontinuity becomes smoother with the increase of τ , it is almost not visible for the cases at 500 ms. This behavior is due to two reasons. First, the term tanh(L s / c α s τ ) decreases with τ which implies a better single-phase heat transfer to the water with the increase of the period. Secondly, the activation of the nucleation sites is signicantly dierent between short and long periods: for long periods, the number of activated nucleation sites progressively increases and so the heat transfer; for short period, nucleation sites are simultaneously activated homogeneously on the heater increasing sharply the heat transfer. This is observed with both the HSV and HSIR cameras.

One notices at rst glance that the normalized time required for boiling crisis to occur slightly depends on the time period τ , the subcooling or the Reynolds number. It ranges, in the present case, between 6.8 and 8.2 τ . Naturally, this value is specic to the present experimental setup and in particular, strongly connected to the quantity q P 0 . The parameter τ mainly impacts the relative duration of the single-phase and two-phase regimes. Boiling onset occurs at a larger normalized time as the time period τ is reduced while the two-phase ow duration is shortened. Now considering a xed value of τ , the critical heat ux is seen to increase with the subcooling and with the Reynolds number. Fixing both the period and the subcooling, one observes the increase of CHF with the Reynolds number as the ow exerts a more ecient cooling in this case. Remembering that the quantity q P ref does not depend on the subcooling neither on the Reynolds number (and slightly on the period), the vertical scale appears to be a relevant indicator to compare the CHF values.

Once is xed the period τ , time evolution of the normalized temperature at the wall (lower row in Fig. C.1) also presents a close superimposition of the ve curves. Similar trends are observed with the heat ux: the rise is fastened as the time period is reduced, time duration associated to heat transfer in the single liquid phase increases and that for two-phase ow is reduced. The last period, just before the occurrence of boiling crisis, deserves a few comments. The curves measured with a subcooling of 75 K are correctly superimposed with those at 50 K in the single-phase regime then diverge after boiling incipience. This is not surprising remembering that the non dimensional temperature Θ w was obtained starting from a heat diusion model in the liquid single-phase. A close superimposition between the curves is clearly perceptible for the escalation period τ = 5 ms. For a large τ associated with long time scales, convection eects are signicant as revealed by the dispersed charts plotted in Time evolution of the normalized wall temperature is changed after boiling inception. The slope drop is due the high increase of the heat transfer coecient. A large period τ leads to a smooth increase : nucleate boiling develops progressively on the heated wall until dry spots occur leading to boiling crisis in a time delay of about two times the period τ . In contrast, a sharp temperature overshoot is observed at boiling inception when the time period is small (5 ms). Nucleation occurs homogeneously on the heater with a high density of activated nucleation sites. Boiling crisis is then triggered rapidly, in about half a period τ This representation reveals that at least three fourths of the total time is spent in single phase, i.e. 5 to 7 τ . This long time duration corresponds to the beginning of the exponential excursion where the temperature conditions are not met for nucleation to start. Then, the time duration corresponding to two-phase ow reduces with the decrease of the power excursion period. For a small value of τ , time at boiling inception (ONB) is large and so the corresponding heat ux at the wall. Boiling occurs in the form of numerous tiny bubbles developing homogeneously along the heated wall in a very short time interval. In the following, a more quantitative insight on these dierent time characteristics is presented

Résumé en français

Les réacteurs expérimentaux de type piscine sont des réacteurs nucléaires notamment utilisés pour des tests de matériaux sous irradiation ou encore pour la production de radioisotopes pour la médecine nucléaire. Le c÷ur est généralement placé directement dans la piscine à une dizaine de mètres de profondeur et un écoulement forcé est réalisé pour son refroidissement. Le combustible nucléaire est sous forme de plaques formant ainsi des canaux dans lesquels circule l'eau, larges de quelques millimètres à quelques centimètres. Lors d'un accident de réactivité par exemple initié par l'éjection d'une barre de contrôle, le réacteur subit une excursion de puissance de forme exponentielle avec un temps caractéristique de l'ordre de la dizaine de millisecondes. Contrairement aux réacteurs commerciaux de type REP, du fait du fort enrichissement du combustible des réacteurs piscines, cette excursion de puissance ne peut être atténuée, avant la chute des barres de contrôle, que par une production conséquente de vide dans le liquide modérateur, i.e. production de vapeur. Du fait des conditions thermohydrauliques dans le c÷ur, la forte excursion de puissance peut induire l'ébullition de l'eau circulant entre les plaques. Si le système reste en ébullition nucléée, la forte production de vapeur permet alors l'arrêt de l'excursion exponentielle. Si le système transite vers l'ébullition en lm, la quantité de vapeur alors présente dans le c÷ur du réacteur peut ne plus être susante pour que les contre-réactions neutroniques arrêtent l'excursion de puissance. De plus, l'isolation thermique engendrée par le lm de vapeur peut porter atteinte à l'intégrité du c÷ur: une fusion du c÷ur est alors possible, voire une explosion de vapeur. Ce scénario accidentel est appelé accident de type BORAX. Pour la sûreté nucléaire, il est alors important de pouvoir prédire la crise d'ébullition en chauage transitoire dans des conditions applicables à ce genre de réacteur.

La problématique de la crise d'ébullition avec chauage stationnaire a été largement étudiée ces cinquante dernières années pour des applications industrielles. De nombreuses valeurs expérimentales, tables ou encore corrélations sont disponibles dans la littérature pour estimer la valeur du ux critique dans une large plage de conditions de fonctionnement. Néanmoins, du fait de la faible compréhension des phénomènes mis en jeu et des faibles résolutions métrologiques des expériences disponibles dans la littérature, il y a peu de modèles mécanistes robustes et validés décrivant le phénomène de crise d'ébullition. Concernant les cas où le chauage en paroi suit une loi exponentielle en temps, applicable aux scénarii accidentels décrit ci-dessus, les connaissances sont encore plus restreintes. Le Commissariat à l'Énergie Atomique et aux Énergies Alternatives (CEA) a alors décidé de développer un partenariat avec le Massachusetts Institut of Technology (MIT) pour combler ces lacunes. Les travaux de cette thèse s'inscrivent dans ce partenariat et ont précisément pour but de: produire de nouvelles données expérimentales à haute résolution dans des conditions applicables aux réacteurs de type piscine, développer une méthode capable de prédire le ux critique avec chauage transitoire, développer la compréhension physique de la crise d'ébullition transitoire. Ce schéma illustre l'élément chauant avec la conguration optique permettant de réaliser la thermographie infrarouge. Les expériences reproduisant le scénario accidentel d'intérêt sont réalisées en injectant une puissance électrique de forme exponentielle en temps dans le lm conducteur électriquement. Cette puissance est alors transférée vers le substrat en saphir et vers l'eau. Durant l'expérience, et grâce au fait que le saphir est transparent aux longueurs d'onde mises en jeu, la caméra infrarouge ultra rapide (HSIR) enregistre un lm infrarouge des rayonnement émis à l'arrière de l'élément chauant. Par un procédé de post-traitement explicité dans Bucci et al. [START_REF] Bucci | A mechanistic IR calibration technique for boiling heat transfer investigations[END_REF], il est alors possible d'extraire la distribution de température sur la paroi chauante (T w (x, y, t)) ainsi que la distribution de densité de ux thermique transmise à l'eau (q P w (x, y, t)). On peut alors étudier le déclenchement de la crise d'ébullition qui est dans ce travail déni comme l'apparition de la première poche de vapeur stable.

Dispositif expérimental et données associées

Comme évoqué précédemment, ce travail s'inclut dans un partenariat entre le CEA et le MIT. Celui-ci a pour but général de proposer de meilleures modélisations des transferts thermiques impliqués dans les accidents de type BORAX. Il a été divisé en trois phases. La phase 1 étudie les transferts thermiques transitoires en régimes monophasique et d'ébullition nucléée sans arriver à la crise d'ébullition. Nous n'en discuterons donc pas ici. La phase 2 étudie la crise d'ébullition transitoire à pression atmosphérique dans un canal de 1 cm de large. Enn la phase 3 étudie la crise d'ébullition transitoire à pression modérée, jusqu'à 12 bar, et dans un canal plus étroit de 1,95 mm. Les données de la phase 2, qui étaient déjà disponibles au début du doctorat, seront exploitées pour développer le modèle de manteau homogène décrit ci-après. Concernant les expériences et les données de la phase 3, j'ai contribué à la mise en place du dispositif durant mon séjour de 1 an au MIT, j'ai réalisé le travail de post-traitement et j'ai exploité ces résultats pour étendre l'application du modèle.

Le modèle de manteau homogène à pression atmosphérique Au début de l'excursion exponentielle, les échanges thermiques sont monophasiques du fait de puissances thermiques encore faibles. Quand le ux thermique en paroi est suisamment important, il y a déclenchement de l'ébullition nucléée (ONB) où les premières bulles apparaissent. Au fur et à mesure que la puissance augmente, le système passe en régime d'ébullition nucléée pleinement développée (FDNB) et la densité de bulles sur la paroi chauée augmente. Durant ce régime, il apparait trois caractéristiques propres aux bulles dans ces conditions de forte sous-saturation et convection forcée. Les bulles restent connées en proche paroi et ont un petit diamètre, de l'ordre de quelques centaines de micromètres. De plus, ces bulles se recondensent immédiatement après leur croissance leur donnant une comportement pulsant à haute fréquence, supérieur à la dizaine de kiloHertz. Ce phénomène dénommé ici Cycles de Nucléation et de Condensation (NCC) transfère alors ecacement l'énergie de la paroi chauante vers le liquide environnant. Au vu de ces observations, la modélisation du phénomène va alors être réalisée au travers de l'étude d'une ne couche d'épaisseur δ en proche paroi et homogène en température dans son épaisseur qui sera appelée manteau. Son homogénéité est causée par le rapide brassage dû aux NCC. Durant le régime de FDNB, au fur et à mesure de l'augmentation du ux en paroi, la température dans le manteau augmente progressivement sous l'eet de la recondensation des bulles. Quand le manteau atteint la température de saturation, les bulles ne peuvent alors plus se recondenser. Elles coalescent et forment des poches de vapeur : c'est la crise d'ébullition. Cette équation décrivant la physique de la crise d'ébullition possède alors des paramètres calculables à partir des conditions expérimentales de fonctionnement (température, écoulement...) mais aussi de l'épaisseur du manteau δ qui est a priori inconnue. Pour surmonter cela, nous allons considérer le problème inverse : à partir des essais expérimentaux compatibles avec l'utilisation du modèle (forte soussaturation et écoulement forcé), nous allons calculer les valeurs de δ associées à ces points dans le cadre de l'approche du modèle. La stratégie de cette approche est de pouvoir prédire la valeur que prend l'épaisseur du manteau en fonction des paramètres du problème. Le résultat de ce processus permet d'avoir des séries de δ en fonction de la période τ et ce pour les diérentes conditions de fonctionnement. Les valeurs dimensionnées de cette épaisseur, de l'ordre de quelques centaines de micromètres à quelques millimètres, sont compatibles avec la taille de bulles observée et les tendances sont cohérentes entre elles. Les valeurs de δ des diérentes séries sont cependant dispersées et un adimensionnement adéquat permet de les regrouper. Nous proposons comme adimensionnement δ = αδ/u ¦ pour l'épaisseur du manteau, avec α la diusivité thermique de l'eau et u ¦ la vitesse de cisaillement. Ce choix d'adimensionnement permet de prendre en compte à la fois les eets de turbulence et les eets thermiques. Pour la période, l'adimensionnement est choisi comme τ = τ /τ x = u m,V τ /L, avec u m,V la vitesse caractéristique u m calculée pour des conditions de chauage quasi-statique. Le résultat de ces adimensionnements est alors présenté sur la gure D.14. Ils permettent de regrouper l'ensemble des points sur une courbe unique (en pointillés) qui a la bonne propriété d'avoir une équation simple δ (τ ) = δ V p1 ¡ e ¡τ q. Cette équation possède un unique paramètre libre δ V non es- timable a priori, caractérisant la valeur prise par δ pour des chauages quasi-statiques et commun à l'ensemble des points utilisés ici. Il est alors possible de déterminer l'épaisseur du manteau en fonction des conditions de fonctionnement, permettant donc d'appliquer le modèle et de prédire la crise d'ébullition.

L'équation (D.1) peut aussi être adimensionnée en introduisant les grandeurs adi-

mensionnées x ¦ = x/L, t ¦ = t/τ et E ¦ = E Q (x ¦ , t ¦ )/E Q crit .
On obtient alors l'équation suivante

∂E ¦ ∂t ¦ N x ∂E ¦ ∂x ¦ N y E ¦ = τ E Q cr δ q " w (D.2)
où N x = u m τ /L caractérise le refroidissement par advection dans le sens de l'écoulement et N y = v m τ /δ caractérise le refroidissement dû aux uctuations turbulentes de vitesse normales à la paroi chauante. Cet adimensionnement permet de faire une analyse physique de la crise d'ébullition en fonction des valeurs que prennent ces deux groupements adimensionnés. Par exemple, la valeur prise par N x permet de prédire la nature transitoire du phénomène et la structure spatiale du déclenchement de la crise d'ébullition, homogène le long de la paroi chauante ou hétérogène en aval de la paroi chauante. Ces prédictions physiques sont vériées expérimentalement.

La crise d'ébullition à pression modérée et haute soussaturation et fort écoulement

La réalisation des expériences de la phase 3 a permis de faire une étude paramétrique du ux critique en transitoire en fonction de la pression (jusqu'à 12 bar), de la température (sous-saturation jusqu'à 163 K), des conditions d'écoulement (jusqu'à 20 m.s ¡1 ) et de la période d'excursion exponentielle (aussi courte que 2,5 ms).

L'impact de la sous-saturation sur la valeur du ux critique transitoire est similaire aux cas de chauage stationnaire : la valeur du ux critique augmente avec la sous-saturation. Cette dépendance à la sous-saturation est cohérente avec une approche énergétique comme celle du modèle de manteau homogène ainsi qu'avec la dénition d'énergie critique alors introduite.

Pour traiter de l'impact de la pression, il faut tout d'abord déterminer le paramètre de température : sous-saturation ou température du c÷ur de l'écoulement. En eet, avec l'évolution de la pression, ces deux paramètres de température évoluent de manière antagonistes. À sous-saturation imposée, une augmentation de pression signie une augmentation de la température du c÷ur de l'écoulement. Inversement, à température du c÷ur de l'écoulement imposée, une augmentation de pression va alors faire augmenter la sous-saturation. La gure D.15 présente les évolutions du ux critique transitoire (τ = 5 ms) en fonction de la pression à sous-saturation constante (a) et à température du c÷ur de l'écoulement constant (b). À sous saturation constante, il apparait que la valeur du ux critique diminue avec la pression comme observé dans la littérature. Inversement, à température de c÷ur constant, la valeur du ux critique augmente avec la pression, montrant l'importance prédominante de la sous-saturation dans le processus de crise d'ébullition. Enn, ces tendances antagonistes montrent que l'impact de la pression sur le ux critique ne peut être discuté qu'en indiquant clairement le paramètre de température retenu. Concernant la vitesse d'écoulement ainsi que la période d'excursion exponentielle, il est plus intéressant d'étudier leur impact de manière couplée. Des résultats à forte sous-saturation et pression modérée sont présentés sur la gure D.16 en fonction de la période τ et pour plusieurs conditions d'écoulement (un sans écoulement en noir, et trois avec ux massique important). Nous pouvons remarquer que pour le cas sans écoulement, la valeur du ux critique dépend de la période d'excursion. Pour les longues périodes, la valeur de ux critique semble être sur un plateau, puis augmente au fur et à mesure que τ diminue. Ce comportement est cohérent avec les résultats de ux critique transitoire de la littérature, autant sans qu'avec un écoulement modéré. Cependant, nous pouvons voir sur la gure D.16 que pour les cas en convection forcée, le ux critique ne dépend pas de la période τ sur la plage étudiée et la courbe présente alors un plateau. Ce comportement indépendant de la période, qui peut être expliqué par la forte convection imposée ici (au delà de 5000 kg.m ¡2 .s ¡1 soit 5 m.s ¡1 ), peut alors être associé à un comportement en chauage quasi-statique. Ce comportement quasi-statique, même pour des périodes très courtes (e.g. 5 ms), peut être prédit par le modèle introduit précédemment via le groupement adimensionné N x qui caractérise la nature transitoire d'un essai ainsi que la structure de l'ébullition lors de la crise. Dans ces cas ci, la valeur prise par N x est toujours au dessus de l'unité ce qui prédit un comportement quasi-statique du point de vue temporel et une structure de déclenchement hétérogène du point de vue spatial. Ces deux prédictions sont vériées expérimentalement par le plateau pour la propriété temporelle, et par les observation des caméras ultra rapides pour la propriété spatiale. Reynolds ne sont pas à même de caractériser l'écoulement pour étudier le phénomène de crise d'ébullition dans ces conditions : le raccordement entre les essais avec les deux entrefers est médiocre. Une caractérisation basée sur une vitesse est bien plus cohérente, i.e. bon raccordement, la vitesse de cisaillement (u ¦ ) étant la plus satisfaisante.

Pour étudier les nouveaux points de la phase 3 à la lueur du modè1e précédemment présenté, le formalisme de ce dernier est repris. Pour cela, l'équation D.2 est divisée par N x pour obtenir l'équation:

γ∂ t ¦E ¦ ∂ x ¦E ¦ λE ¦ = q ¦ (D.3)
avec γ = L/u m τ caractérisant la nature transitoire du phénomène, λ = v m L/u m δ caractérisant le terme de refroidissement prédominant, et enn q ¦ = Lq P w /Lu m E Q crit qui est le terme source. L'étude de cette équation vis à vis des paramètres γ et λ permet de mettre en place un diagramme à quatre quadrants, présenté sur la gure D.17, qui dénissent les cas limites du modèle ainsi que la nature de la crise d'ébullition associée. La position sur la direction horizontale du diagramme, caractérisant la valeur prise par γ, permet de déterminer la nature transitoire ainsi que la structure spatiale de la crise d'ébullition. Si le point de fonctionnement est situé sur la gauche du diagramme, le phénomène est quasistatique et la crise d'ébullition est déclenchée en aval de l'élément chauant, et s'il est sur la droite, le phénomène est transitoire et la crise d'ébullition est déclenchée de manière homogène spatialement. La position sur la direction verticale du diagramme permet de déterminer la source de refroidissement prédominante. Si le point de fonctionnement est situé en bas du diagramme, le processus de refroidissement prédominant est l'advection dans le sens de l'écoulement, et s'il est situé en haut, le processus prédominant est le refroidissement par les uctuations turbulentes de vitesse, normales à la paroi. chf expérimental en fonction de sa valeur prédite analytiquement via l'équation (D.4), il apparait deux choses. La première est que tous les points sont au dessus de la bissectrice, signiant que l'équation (D.4) permet de faire une prédiction conservative du ux critique. La deuxième est que beaucoup de ces points sont au voisinage de la bissectrice, signiant que l'équation (D.4) permet de faire une prédiction précise du ux critique. Cette précision est illustrée avec la gure D.18b qui présente l'écart relatif entre les valeurs analytiques et expérimentales. Il apparait en eet que la dispersion est faible : plus d'un tiers des points sont prédits par l'équation (D.4) avec moins de 5% d'erreur et plus de 60% avec moins de 20% d'erreur. Dans la conguration des expériences (Fig. D.11), q P w (t) suit une loi d'autant plus proche de l'exponentielle quand le ux massique augmente et que la période est grande. Ces points de fonctionnement correspondent aux points ayant le moins de déviation par rapport à la prédiction analytique. Enn, il est intéressant de noter que dans un cadre d'applications industrielles, on s'attend à ce que le ux thermique transmis à l'eau soit quasiment exponentiel. En eet, les applications industrielles fonctionnent souvent avec des ux massiques élevés et le chauage est réalisé dans le volume du combustible, contrairement aux expériences où le chauage est réalisé dans un lm disposé sur un substrat.

Conclusion

Ce travail de thèse a étudié expérimentalement et a modélisé la crise d'ébullition transitoire de l'eau en écoulement forcé, à forte sous-saturation et à pression modérée. La réalisation des expériences a permis de comprendre l'impact des paramètres principaux du problème que sont la pression, la température, les conditions d'écoulement forcé ainsi que la période d'excursion exponentielle. L'analyse des vidéos de la caméra ultra rapide ainsi que de la thermographie infrarouge ont permis de développer un modèle énergétique de la crise d'ébullition dans ces conditions. Ce modèle est simple et ne comporte qu'un seul paramètre libre (δ V ). Néanmoins, il permet de proposer une méthode de prédic- tion de la crise d'ébullition, et avec une hypothèse supplémentaire simplicatrice mais réaliste, il permet de prédire la valeur du ux critique de manière conservative, précise, et applicable à des congurations industrielles. De plus, son adimensionnement permet d'avoir une meilleure compréhension physique de la crise d'ébullition: prédiction de la nature spatio-temporelle du déclenchement et prédiction de la source de refroidissement prédominant du processus. Pour la suite, il serait intéressant de développer mécanistiquement le comportement de l'épaisseur du manteau δ pour pouvoir proposer une relation physique de fermeture du modèle. Il serait aussi intéressant de faire des essais dans des zones de l'espace (γ, λ) où peu de points sont actuellement disponibles, permettant ainsi de consolider nos capacités de prédiction et de compréhension de l'épaisseur adimensionnée δ . Enn il serait intéressant d'étudier les limites paramétriques de validité de ce modèle et de le confronter à des états de surface diérents. Keywords: Critical Heat Flux, Transient Flow Boiling Crisis, Exponential power escalation, Subcooled water, BORAX.

Abstract:

In case of a reactivity insertion accident in an experimental nuclear reactor, heat generation in the core can grow exponentially in time, with a power escalation period ranging from a few to a few hundreds of milliseconds. Due to neutronic and thermohydraulic effects, boiling crisis may arise, possibly leading to an explosive reaction. If the boiling Crisis has been widely investigated in steady-state conditions, this has not been the case for transient heat inputs. The aim of the present work is to understand and to predict the transient flow boiling crisis in the conditions of moderate pressure and high subcooling.

To this end, an experimental campaign has been realized making use of space and time highly resolved videos and IR thermography covering a wide range of experimental parameters. The analysis of the massive amount of data produced by these experiments gives a better insight on the dependency of the transient Critical Heat Flux to the different parameters of interest (power escalation period, flow velocity, subcooling, pressure, channel width, heating length). Moreover, their fine analysis enables to highlight the underlying mechanisms. For conditions of forced flow and high subcooling, the bubbles generated at the wall present a pulsating behavior. This specific process leads to an efficient heat transfer from the wall to the neighboring fluid. Boiling crisis is stated to occur when a thin layer of liquid contacting the wall reaches the saturation temperature. Starting from these observations, a model is developed which brings to light two nondimensional parameters useful to describe the transient nature of the process and the dominant cooling processes.

With the knowledge of the steady-state CHF, the model permits to conservatively estimate the value of the Critical Heat Flux for any power escalation period and subcooling. This model is now ready for implementation into simulation codes to investigate nuclear accidental transients.

Contents Introduction 1

 1 Nuclear Safety Challenge and State of the Art 1.1 The BORAX-Type Accident . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Reactor Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Boiling Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Experimental Investigations 2.1 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Schlieren Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Project Time Line and Test Matrices . . . . . . . . . . . . . . . . . . . . . 3 Homogeneous Mantle Model at atmospheric pressure 3.1 Experimental observations and interpretation . . . . . . . . . . . . . . . . 3.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Results and physical discussion . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The transient boiling crisis at moderate pressure and high mass ux 4.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Connement and ow characterization . . . . . . . . . . . . . . . . . . . . 4.3 Extension of the Homogeneous Mantle Model . . . . . . . . . . . . . . . . 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

BibliographyA

  Detailed Test matrices B Miscellaneous experimental explorations C On the Time Sequence of the Transient Flow Boiling Crisis at High Subcooling D Connement and ow characterization -extended results Résumé en Français vii List of Figures1.1 OSIRIS Research Reactor, CEA Saclay . . . . . . . . . . . . . . . . . . . . 1.2 Cooling Scheme of OSIRIS reactor [11] . . . . . . . . . . . . . . . . . . . . 1.3 Experimental results of SPERT-IV B-38 test replicating a BORAX type transient [19] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 BORAX-1 installation drawing [20] . . . . . . . . . . . . . . . . . . . . . . 1.5 Images of the destructive test of BORAX-1 [20] . . . . . . . . . . . . . . . 1.6 Overview of the SL-1 reactor [32] . . . . . . . . . . . . . . . . . . . . . . . 1.7 Outcome of the nal destructive test of SPERT-I [52] . . . . . . . . . . . . 1.8 Scheme of the ssion process . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 U-235 Neutron-Induced Fission Yield [37] . . . . . . . . . . . . . . . . . . . 1.10 Stability of the ssion products . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Schematic steps of chain reaction . . . . . . . . . . . . . . . . . . . . . . . 1.12 Cross Sections versus incident neutron energy [34] . . . . . . . . . . . . . . 1.13 Criticality types and controllability of the core . . . . . . . . . . . . . . . . 1.14 Boiling curve representative in log-log scale of a steady-state heat transfer . 1.15 Boiling curve representative of the heat transfer under transient heat ux input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.16 Transient heat transfer regimes in turbulent forced convection (Su et al. [76]) 1.17 Boiling regime diagram as a function of the equilibrium quality x and the heat ux. The blue line represents DNB mechanism and the red one represents the dryout mechanism (adapted and reprinted from Todreas & Kazimi [79]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.18 Boiling crisis mechanisms. a) DNB b) dryout (Todreas & Kazimi [79]) . . . 1.19 Sample of Look-up table from Groenveld et al. [27] . . . . . . . . . . . . . 1.20 Dynamic boundary-layer separation leading to ow boiling crisis (Tong and Tang [83]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.21 Flow Boiling crisis mechanism with the interfacial lift-o model (Zhang and Mudawar [87]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.22 Enthalpy balance in the Tong Critical enthalpy model in the bubbly layer (Tong and Tang [83]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.23 Mass balance in the Bubble Crowding Model (Weisman and Pei

  (a) and an overview of the impact of the knife edge orientation and geometry (b)[START_REF] Settles | Schlieren and Shadowgraph Techniques[END_REF] . . . . . . . . . 2.12 Functional scheme of the schlieren visualization method[START_REF] Settles | Schlieren and Shadowgraph Techniques[END_REF] . . . . . . . . 2.13 Picture of the schlieren experimental apparatus . . . . . . . . . . . . . . . 2.14 Schlieren visualization of pool boiling at atmospheric pressure under a heat ux step function. ∆T sub = 30 K . . . . . . . . . . . . . . . . . . . . . . . 2.15 Schlieren visualization of ow boiling at atmospheric pressure under a heat ux step function. ∆T sub = 30 K and Re = 10000 . . . . . . . . . . . . . . 2.16 Time line representing the collaboration between the CEA and the MIT on the transient heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . 2.17 Visualization of Hot spots using the HSV and HSIR cameras . . . . . . . . 3.1 Transient ow boiling curve for p = 1bar, ∆T sub = 75 K, Re = 35000, τ = 50ms: HSV images illustrate the successive regimes from single phase to boiling crisis. The images represent the whole surface of the heater (1cm x 1cm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Interpretation of successive events leading to boiling crisis in conditions of high subcooling and strong forced convection. Front backlit shadowgraphy images correspond to the case p = 1 bar, ∆T sub = 75 K, Re = 35000, τ = 50 ms and represent the entire heating surface (1 cm x 1 cm). The bottom schemas illustrate the model consideration with a side view. . . . . 3.3 Schematic description of the Mantle . . . . . . . . . . . . . . . . . . . . . . 3.4 Description of the entire system considering the total length L of the heater and the exchanges of the elementary system σ(x) used for the energy balance. 3.5 Schematic description of the velocity uctuation contributions at the interface mantle-bulk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Statistic convergence analysis for the set Re τ = 1000 of Graham et al. [26] and y = 41. The quantities are plotted as functions of the sample size N. The dashed line corresponds to the sample size taken for the upcoming calculation, i.e. 2048 ¢ 1536 ¢ 3. . . . . . . . . . . . . . . . . . . . . . . . x List of Figures 3.7 (a) Dimensionless velocity v m as a function of the dimensionless normal coordinate y calculated for Re τ = 180, 395, 590, 1000 with DNS data of Moser et al. [53] and Graham et al. [26]. (b) Discrepancies with the Gaussian distribution assumption. . . . . . . . . . . . . . . . . . . . . . . . 3.8 Schematic description of the guess-estimate algorithm determining the mantle thickness δ for a given experimental run. . . . . . . . . . . . . . . . . . 3.9 Thickness of the mantle δ [m] and its dimensionless form δ [¡] as a function of the excursion period τ [s] for the eight operating conditions. . . . . 3.10 Non-dimensional thickness δ

3. 13

 13 Heat ux maps evolution and E ¦ (x ¦ ) at the time of boiling crisis (t ¦ = 1) for dierent values of N x . The right maps are taken just after the boiling crisis, the left one being 0.4 τ prior. The conditions are ∆T sub = 50 K, Re = 35000 and p = 1bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 Heat ux maps evolution and E ¦ (x ¦ ) at the time of boiling crisis (t ¦ = 1) for a 4x0.6 cm 2 heating cell. The right map is taken just after the boiling crisis, the left one being 0.4 τ prior. The conditions are ∆T sub = 50 K, Re = 35000 and p = 1bar, τ = 50 ms (N x = 0.73). . . . . . . . . . . . . . . 3.15 SPERT IV test results for 10 ms power excursion period with dierent ow conditions [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 N y /P e ¦ as a function of N x . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 CHF as a function of the subcooling at constant pressure and for τ = 5 ms in TS2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 HSV images presenting the characteristic bubble size for dierent subcooling temperatures under 2.3 bar and 8.9 bar, with τ = 5 ms and G = 2500 kg.m ¡2 .s ¡1 in test section TS2. . . . . . . . . . . . . . . . . . . . 4.3 CHF as a function of pressure at constant subcooling and for τ = 5 ms in TS2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 CHF as a function of pressure at constant bulk temperature and for τ = 5 ms in the test section TS2. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 HSV images presenting the impact of the pressure on the characteristic bubble size with a constant bulk temperature and a constant subcooling, with τ = 5 ms and G = 2500 kg.m ¡2 .s ¡1 in the test section TS2. The notation used to name the thermodynamic conditions is p ¡ T b / ∆T sub . The active heating area has a dimension of 20 ¢ 4 mm 2 . . . . . . . . . . . 4.6 CHF as a function of the heated length for τ = 5 ms and G = 2500 kg.m ¡2 .s ¡1 in the test section TS2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Transient CHF (τ = 5 ms) as a function of the mass ux for dierent thermodynamic conditions, with a heated length of 2 cm in the test section TS2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Transient CHF (τ = 5 ms) as a function of the mass ux for dierent heated lengths, under a pressure of 12 bars and 163 K of subcooling in TS2.104 xi List of Figures4.9 CHF as a function of τ for dierent mass uxes, under a pressure of 12 bar and 163 K of subcooling -or Bulk temperature of 25 o C -in the test section TS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 HSV images in frontlit shadowgraphy for power escalation periods of 5 ms and 100 ms around ONB, during FDNB and around DNB. The experiments are realized in the test section TS2 under a pressure of 12 bars and in a water owing at 10000 kg.m ¡2 .s ¡1 at 163 K of subcooling -or Bulk temperature of 25 o C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 HSV and HSIR visualization, 2.6 ms after DNB for dierent mass uxes. The experiments are realized in test section TS2 under a pressure of 12 bars in a water at 163 K of subcooling -or Bulk temperature of 25 o Cwith an escalation period of 100 ms. . . . . . . . . . . . . . . . . . . . . . 4.12 HSV images every 0.1 τ starting from ONB under a pressure of 12 bars and 163 K of subcooling -or Bulk temperature of 25 o C -in TS2. The two rst lines present transient pool boiling with periods of 5 ms and 100 ms. The bottom line presents a transient ow boiling with a period of 5 ms. . . 4.13 CHF as a function of the wall Reynolds number for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . . . . . . . . . . . . . 4.14 CHF as a function of the Reynolds number for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . . . . . . . . . . . . . . . 4.15 CHF as a function of the bulk velocity for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 CHF as a function of the shear velocity for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17CHF as a function of the mass ux for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18 Mantle thickness as a function of the shear velocity for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . . . . . . . . . . 4.19 Non dimensional mantle thickness as a function of the shear velocity for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . 4.20 Schematic representation of the four quadrants as a function of the parameters γ and λ and their associated simplied equations. The location of a quadrant in the horizontal direction permits to determine the time dependency of the problem (Steady-state or Transient). The location of a quadrant in the vertical direction permits to determine the dominating cooling process (Streamwise advection or Turbulent velocity uctuations). . 4.21 Statistical convergence analysis for the set Re τ = 5200 of Lee and Moser [50] for y = 41. The indicators are plotted as functions of the sample size. The dashed line corresponds to the sample size 10240 ¢ 7680 taken to compute v m . Dimensionless velocity v m as a function of the dimensionless normal coordinate y calculated for Re τ = 180, 395, 590, 1000, 5200 with DNS data of Moser et al. [53], Graham et al. [26] and Lee and Moser [50]. 4.22 Path of power escalation period series in the space (γ, λ) for dierent ow conditions. (a) Series with variable period for three mass uxes (5000, 10000, 15000 kg.m ¡2 .s ¡1 ) at 163 K of subcooling, 12 bar in the test section TS2. (b) Series with variable period for three Reynolds numbers (8500, 25000, 35000) at 50 K of subcooling, atmospheric pressure in the test section TS1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.23 Section in the space (γ, λ) focusing on the quadrant 1 ¡ 2 presenting the non-dimensional mantle thickness and non-dimensional CHF. . . . . . . . . xii List of Figures 4.24 Section in the space (γ, λ) focusing on the quadrants 1 ¡ 3 presenting the non-dimensional mantle thickness and non-dimensional CHF. . . . . . . . . 4.25 Location of the investigated points A, B and C in the Cross section following the quadrants 1 ¡ 3 and the space (γ, λ). The HSV image and heat ux distribution at DNB of these points are presented. . . . . . . . . . . . . . . 4.26 Location in the space (γ, λ) of the experimental points investigating the parameter of ow presented in section 4.2 . . . . . . . . . . . . . . . . . . . 4.27 Comparison between the experimental nondimensional CHF and the analytic nondimensional CHF assuming that the heat ux transferred to water follows an exponential trend (Eq. 4.1). The black dashed line represents the plot y = x and the green dashed line represents an excess of 20% with the ploty = 1.2 x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.28 Histogram representing the distribution of the underestimation of the critical heat ux using equation 4.3 among the investigated points. The bin width is 5 % . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.29 Sample of heat uxes transferred to water as a function of time (both normalized) super imposed with an exponential trend. The number written in red presents the relative discrepancy between the experimental and the analytic q ¦ chf . The experiments have been realized at 12 bar, 163 K of subcooling, for heated length of a 1 cm and in the test section TS2. . . . . 4.30 Colorized map presenting the value of δ in the space (γ, λ) . . . . . . . 4.31 Schematic description of the inverse method aiming to assess the value of the steady-state non-dimensional mantle thickness δ V for a given working condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.32 Illustration of the inverse method algorithm results on the experimental series measured with 50 K (left) and 75 K (red) of subcooling at atmospheric pressure in the test section TS1 with a heated length of 1 cm. The black dashed line represents the tting function (Eq. 3.19). The solid lines represent the Eq. 3.19 using the computed values of δ V and τ x of the experimental points (errorbars) . . . . . . . . . . . . . . . . . . . . . . . . 4.33 Computed value of δ V as a function of the corresponding parameter λ V . The black dashed lines represent possible asymptotic trends as 180 when λ V ÝÑ 0 and as 180 c λ V when λ V ÝÑ V . . . . . . . . . . . . . . . . . B.1 Silicon substrate. Failure of the Titanium coating . . . . . . . . . . . . . . B.2 Comparative view of an HSV image: Original view, Pre-Processed Gray Scale and Binary threshold at the normalized time t ¡ t onb τ = 0.78. Case for τ = 50 ms with the condition ∆T sub = 75 K, Re = 35, 000 and p = 1 bar in the test section TS1 with a heating area of 1 ¢ 1 cm 2 . . . . . . . . . . . B.3 Proportion of the heating surface covered by bubbles as a function of the normalized time. Case for τ = 5, 10, 50 ms with the condition ∆T sub = 75 K, Re = 35, 000 and p = 1 bar in the test section TS1 with a heating area of 1 ¢ 1 cm 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Bubble footprint area distribution compared with the Original and Binary threshold frames at t ¡ t onb τ = 0.78. Case for τ = 50 ms with the condition ∆T sub = 75 K, Re = 35, 000 and p = 1 bar in the test section TS1 with a heating area of 1 ¢ 1 cm 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Figures B.5 LoG processing compared with the Original and Binary threshold frames at t ¡ t onb τ = 0.48. Case for τ = 50 ms with the condition ∆T sub = 75 K, Re = 35, 000 and p = 1 bar in the test section TS1 with a heating area of 1 ¢ 1 cm 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Normalized heat ux released to water q ¦ and normalized wall temperature Θ w as a function of the normalized time t ¦ for τ = 5, 50, 500ms plotted from the beginning of the exponential power excursion to CHF. . . . . . . C.2 Illustration of the time sequence of an exponential transient heat input heading to the boiling crisis. The graph represent the mean heat ux transferred to the water as a function of the normalized time. The ONB and DNB are illustrated by the heat ux distribution at the given time. . . C.3 Overall time sequence of the exponential power excursion. The normalized time of the dierent occurrences and regimes is presented as a function of the power escalation period τ . Series with the conditions Re = 25000 and ∆T sub = 75 K at atmospheric pressure in the test section TS1. . . . . . . . C.4 Dimensionless time of ONB and time t ¦ min to reach the saturation temperature at the wall as function of the time period τ for the dierent operating conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.5 Dimensionless time for DNB to occur as function of the time period τ for the dierent operating conditions. . . . . . . . . . . . . . . . . . . . . . . . C.6 Dimensionless durating of the two-phase regime ∆t ¦ = t ¦ CHF ¡ t ¦ ON B as function of the time period τ for the dierent operating conditions. . . . . D.1 CHF as a function of the wall Reynolds number for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . . . . . . . . . . . . . D.2 CHF as a function of the Reynolds number for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . . . . . . . . . . . . . . . D.3 CHF as a function of the shear velocity for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.4 CHF as a function of the mass ux for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.5 CHF as a function of the bulk velocity for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.6 Non dimensional mantle thickness as a function of the wall Reynolds number for three dierent periods at 10 bar and 100 K of subcooling. . . . . . D.7 Non dimensional mantle thickness as a function of the Reynolds number for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . D.8 Non dimensional mantle thickness as a function of the shear velocity for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . D.9 Non dimensional mantle thickness as a function of the bulk velocity for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . D.10 Non dimensional mantle thickness as a function of the mass ux for three dierent periods at 10 bar and 100 K of subcooling. . . . . . . . . . . . . . D.11 Schéma de l'élément chauant et de la thermographie infrarouge . . . . . . D.12 Description de la séquence temporelle d'une excursion exponentielle jusqu'à la crise d'ébullition sous forte sous-saturation et convection forcée. Les images d'ombroscopie présentent l'ensemble de la surface chauante (1 ¢ 1 cm 2 ) dans le cas p = 1 bar, ∆T sub = 75 K, Re = 35000, τ = 50 ms. Les schémas illustrent le modèle, vu de côté. . . . . . . . . . . . . . . . . . . . xiv List of Figures D.13 Schéma représentant les contributions thermiques considérées par le modèle pour le bilan d'énergie. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.14 Épaisseur du manteau adimensionnée δ [-] en fonction de la période adimensionnée τ /τ x [-] pour huit conditions de fonctionnement (∆T sub , Re) représentées par les barres d'erreur. La courbe de régression est représentée en pointillés. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.15 Flux critique transitoire (τ = 5 ms) en fonction de la pression à soussaturation constante (a) et à température du c÷ur de l'écoulement constante (b) avec un ux massique de 15000 kg.m ¡2 .s ¡1 . . . . . . . . . . . . D.16 Flux critique en fonction de la période τ pour diérentes valeurs de ux massique sous une pression de 12 bar and 163 K de sous-saturation (Température de bulk de 25 o C) . . . . . . . . . . . . . . . . . . . . . . . . . . . D.17 Diagramme à quatre quadrants en fonction des paramètres γ and λ et des équations simpliées associées. . . . . . . . . . . . . . . . . . . . . . . . . . D.18 (a) Comparaison entre les ux critiques adimensionnés expérimentaux et analytiques (Eq. (D.4)). La droite en pointillés noirs représente la bissectrice y = x et la droite en pointillés verts représente un excès de 20%. (b) Statistique de la sous estimation relative des résultats analytiques par rapport aux valeurs expérimentales. . . . . . . . . . . . . . . . . . . . . . . xv List of Tables1.1 Orders of magnitude for PWR and pool-type RTR . . . . . . . . . . . . . . 1.2 Value of coecient C of the correlation 1.16 at dierent subcooling degrees [76] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Overview of ow boiling crisis experimental investigations at moderate pressure under exponential power escalations with water. L represents the streamwise heating length, l represents the heated width, d is the diameter of the wet perimeter (diameter of the wire or inner diameter of the tube). . 1.4 Regime description (Chang et al. [16]) . . . . . . . . . . . . . . . . . . . . 2.1 Test Matrix of Phase 2 experiments. The tests were realized with the test section TS1 using an ITO active area of 10 ¢ 10 mm 2 . . . . . . . . . . . . 2.2 Test Matrix investigating the inuence of pressure and water temperature on the transient CHF at G = 2500 kg.m ¡2 .s ¡1 . The tests were realized with the test section TS2 using a Ti active area of 20 ¢ 4 mm 2 . The values in the table correspond to the subcooling (in bold and italics) at the corresponding pressure and bulk temperature. . . . . . . . . . . . . . . . . 2.3 Test Matrix investigating the inuence of pressure and water temperature on the transient CHF at G = 15000 kg.m ¡2 .s ¡1 . The tests were realized with the test section TS2 using a Ti active area of 20 ¢4 mm 2 . The values in the table correspond to the subcooling at the corresponding pressure and bulk temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Test Matrix investigating the inuence of the ow intensity and of the channel width. The tests were realized using a Ti active area of 20 ¢ 4 mm 22.5 Test Matrix investigating the inuence of the mass ux on the transient CHF at high pressure and subcooling. The tests were realized with the test section TS2 using a Ti active area of 10 ¢ 4 mm 2 . . . . . . . . . . . . . . 2.6 Test Matrix investigating the inuence of the mass ux and of the power escalation period on transient CHF at high pressure and subcooling. The tests were realized using the test section TS2 using a Ti active area of 10 ¢ 4 mm 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Turbulent ow characteristics of the investigated conditions . . . . . . . . . A.1 Extended test matrix of the experimental points realized in April and May 2019 investigating on the reliability of the experimental apparatus . . . . . A.2 Extended test matrix of the experimental points realized in July 2019 investigating the impact of the pressure and the temperature . . . . . . . . . A.3 Extended test matrix of the experimental points realized in September 2019 investigating the connement and the ow conditions . . . . . . . . . . . . A.4 Extended test matrix of the experimental points realized in September 2019 investigating the impact of the mass ux at high pressure and high subcooling142 xvii List of Tables

  Overall non-dimensional characteristic number of the mantle [¡] Superscripts ¦ Non-dimensional quantities relative to the mantle model Non-dimensional quantities relative to turbulence

Chapter 1 .

 1 Nuclear Safety Challenge and State of the Art 1.1 The BORAX-Type Accident Through the understanding of the transient boiling crisis, the aim of the present work is to improve the safety of research reactors regarding the BORAX-type accident. This section will introduce the research reactors and their relevant characteristics regarding the problem. It will also describe the BORAX-type accident and give a brief historical overview of the thematics. 1.1.1 The Pool-Type Research Reactors Among the various types of nuclear reactors, Research and Test Reactors (RTR), such as the former OSIRIS reactor at CEA Saclay (see Fig 1.1) are designed to: Provide a neutron ux for various research studies such as material durability under irradiation conditions ; Product radioisotopes for nuclear medicine, such as Technetium 99m, and other various industrial purposes ; Organize training sessions on real nuclear reactors.

Figure 1 . 1 :

 11 Figure 1.1: OSIRIS Research Reactor, CEA Saclay

Figure 1 . 3 :

 13 Figure 1.3: Experimental results of SPERT-IV B-38 test replicating a BORAX type transient[START_REF] Crocker | Reactor Power Excursion Tests In The SPERT IV Facility[END_REF]

Figure 1 . 4 :Figure 1 . 5 :

 1415 Figure 1.4: BORAX-1 installation drawing[START_REF] Dietrich | Experimental investigation of the self-limitation of power during reactivity transients in a subcooled, water-moderated reactor[END_REF]

1. 1 .Figure 1 . 6 :

 116 Figure 1.6: Overview of the SL-1 reactor[START_REF] Holield | Sl-1 accident: Atomic energy commission investigation board report[END_REF]

1. 1 . 3 . 3

 133 The SPERT program Started in 1954, the SPERT program, which stands for Special Power Excursion Reactor Test program, was designed to study the safety of water reactors under high and fast reactivity insertion. Non-destructive and destructive tests were leaded in order to gather data and to understand the underlying physics.

Figure 1 . 7 :

 17 Figure 1.7: Outcome of the nal destructive test of SPERT-I [52]

Figure 1 . 8 :

 18 Figure 1.8: Scheme of the ssion process

Figure 1 . 9 :

 19 Figure1.9: U-235 Neutron-Induced Fission Yield[START_REF]U-235 neutron-induced ssion yield[END_REF]

FissionFigure 1 . 10 :

 110 Figure 1.10: Stability of the ssion products

Chapter 1 .

 1 Nuclear Safety Challenge and State of the Art

Figure 1 .

 1 Figure1.12: Cross Sections versus incident neutron energy[START_REF]Neutron-nuclei interaction spectrum[END_REF]

For

  this reason the unit of reactivity called the dollar and written $, is introduced. It is dened as 1$ = β. Therefore, comparing the reactivity ρ to the unit of reactivity $ provides the state of criticity of the nuclear core. The criticality types are summarized in gure 1.13.

Figure 1 . 13 :

 113 Figure 1.13: Criticality types and controllability of the core

Figure 1 . 14 :

 114 Figure 1.14: Boiling curve representative in log-log scale of a steady-state heat transfer

Chapter 1 .Figure 1 . 15 :

 1115 Figure 1.15: Boiling curve representative of the heat transfer under transient heat ux input

  The latter expression reveals three possible regimes: a forced convection regime when τ 4 1, a transient conduction regime when τ 3 1 and a transition regime when τ 1 (τ [10 ¡1 , 1]). It is experimentally proved in gure 1.16.

Figure 1 . 16 :

 116 Figure 1.16: Transient heat transfer regimes in turbulent forced convection (Su et al. [76])

Figure 1 . 17 :

 117 Figure 1.17: Boiling regime diagram as a function of the equilibrium quality x and the heat ux. The blue line represents DNB mechanism and the red one represents the dryout mechanism (adapted and reprinted from Todreas & Kazimi [79])

Figure 1 . 18 :

 118 Figure 1.18: Boiling crisis mechanisms. a) DNB b) dryout (Todreas & Kazimi [79])

Figure 1 . 19 :

 119 Figure 1.19: Sample of Look-up table from Groenveld et al. [27]

Figure 1 . 20 :

 120 Figure 1.20: Dynamic boundary-layer separation leading to ow boiling crisis (Tong and Tang [83])

  Figure 1.21: Flow Boiling crisis mechanism with the interfacial lift-o model (Zhang and Mudawar [87])

Figure 1 . 22 :

 122 Figure 1.22: Enthalpy balance in the Tong Critical enthalpy model in the bubbly layer (Tong and Tang [83])

Figure 1 . 23 :

 123 Figure 1.23: Mass balance in the Bubble Crowding Model (Weisman and Pei [85])

³ V v 11

 11 (v I ¡ v 11 )P (v I ) dv I . By assuming that the probability density function P of the turbulent velocity uctuations is a gaussian distribution, the standard deviation becomes the Root-Mean-Square (RMS) of v I (σ v I = v I RM S ) and:

Figure 1 . 24 :

 124 Figure 1.24: Mechanism scheme of Liquid Sublayer Models (Celata et al. [15])

Figure 1 . 25 :

 125 Figure 1.25: Schematic descritiption of the thinning mechanisms involved in the model of Pasamehmetoglu. The blue line represents the case when the hydrodynamic mechanism dominates and the red line the case when the thermal mechanism dominates (Adapted from Pasamemhetoglu al. [61])

Figure 1 . 26 :

 126 Figure 1.26: Application of the model to experimental data of Kataoka et al. [41] and its sensibility to the exponent n (Pasamemhetoglu al. [61])

Figure 1 .

 1 Figure 1.27: Transient CHF classication proposed by Chang et al. (Ahmed et al. [1] adapted from Chang et al. [16])

Figure 2 . 1 :

 21 Figure 2.1: Functional scheme (left) and picture (right) of the ow loop

Figure 2 . 2 :

 22 Figure 2.2: View of the Test Section 1 (TS1) and the corresponding cartridge

Figure 2 . 3 :Figure 2 . 4 :

 2324 Figure 2.3: View of the Test Section 2 (TS2) and the corresponding cartridge

Figure 2 . 5 :

 25 Figure 2.5: Scheme (left) and Picture (right) of the test section TS2 optical setting for IR thermography, side and front view backlit shadowgraphy. The setting for test section TS1 is similar. On the picture, the HSV camera is located to perform a front view backlit shadowgraphy.

Figure 2 . 6 :

 26 Figure 2.6: Calibration curve showing the signal received by HSIR as a function of the heater temperature

Figure 2 . 7 :

 27 Figure 2.7: IR radiation contributions observed by the HSIR camera[START_REF] Bucci | A mechanistic IR calibration technique for boiling heat transfer investigations[END_REF]

Figure 2 . 8 :

 28 Figure 2.8: Meshing for a TS2 heater with 40 ¢ 6 mm 2 active heating area observed with IR frames having a denition of 452 ¢ 58 pixels.

Figure 2 . 10 :

 210 Figure 2.10: Escalation to DNB on TS1 with a 40 ¢ 6 mm active area heater at 50 K of subcooling, 35000 Reynolds Number, 100 ms of power escalation period and at atmospheric pressure

Figure 2 . 11 :

 211 Figure 2.11: Functioning scheme of the Schlieren method (a) and an overview of the impact of the knife edge orientation and geometry (b)[START_REF] Settles | Schlieren and Shadowgraph Techniques[END_REF]

Figure 2 . 13 :

 213 Figure 2.13: Picture of the schlieren experimental apparatus

Figure 2 . 14 :

 214 Figure 2.14: Schlieren visualization of pool boiling at atmospheric pressure under a heat ux step function. ∆T sub = 30 K

Figure 2 . 15 :

 215 Figure 2.15: Schlieren visualization of ow boiling at atmospheric pressure under a heat ux step function. ∆T sub = 30 K and Re = 10000

2 ) 2 and 1 .95x18 mm 2 )Figure 2 . 16 :

 2212216 Figure 2.16: Time line representing the collaboration between the CEA and the MIT on the transient heat transfer

Figure 2 . 17 :

 217 Figure 2.17: Visualization of Hot spots using the HSV and HSIR cameras

For the rst eective

  tests made in April 2019 (see time line Fig. 2.16), the tests were realized with a heater having a Ti active area of 40 ¢ 6mm 2 in the narrow channel (TS2) (see Fig. A.1).

1

 1 Experimental observations and interpretation . . . . . . . . . 3.1.1 Transient boiling crisis for high Reynolds numbers and high subcoolings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Physical description of the model . . . . . . . . . . . . . . . . . 3.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . 3.2.1 Criterion for the Boiling Crisis . . . . . . . . . . . . . . . . . . 3.2.2 Governing equations of the mantle . . . . . . . . . . . . . . . . 3.2.3 Heat transport induced by the streamwise ow . . . . . . . . . 3.2.4 Heat transport induced by the transverse turbulent velocity uctuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Model Implementation and Output . . . . . . . . . . . . . . . . 3.3 Results and physical discussion . . . . . . . . . . . . . . . . . . 3.3.1 Evaluation of the mantle thickness . . . . . . . . . . . . . . . . 3.3.2 Energy distribution along the heated wall . . . . . . . . . . . . 3.3.3 Impact of the non dimensional groups N x and N y on boiling crisis 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Figure 3 . 1 :

 431 Figure 3.1: Transient ow boiling curve for p = 1bar, ∆T sub = 75 K, Re = 35000, τ = 50ms: HSV images illustrate the successive regimes from single phase to boiling crisis. The images represent the whole surface of the heater (1cm x 1cm).

Figure 3 . 2 :

 32 Figure 3.2: Interpretation of successive events leading to boiling crisis in conditions of high subcooling and strong forced convection. Front backlit shadowgraphy images correspond to the case p = 1 bar, ∆T sub = 75 K, Re = 35000, τ = 50 ms and represent the entire heating surface (1 cm x 1 cm). The bottom schemas illustrate the model consideration with a side view.

Figure 3 . 3 :Figure 3 . 4 :

 3334 Figure 3.3: Schematic description of the Mantle

Figure 3 . 5 :

 35 Figure 3.5: Schematic description of the velocity uctuation contributions at the interface mantle-bulk.

Figure 3 . 6 :

 36 Figure 3.6: Statistic convergence analysis for the set Re τ = 1000 of Graham et al. [26] and y = 41. The quantities are plotted as functions of the sample size N. The dashed line corresponds to the sample size taken for the upcoming calculation, i.e. 2048 ¢ 1536 ¢ 3.

3. 2 .

 2 Figure 3.7: (a) Dimensionless velocity v m as a function of the dimensionless normal co- ordinate y calculated for Re τ = 180, 395, 590, 1000 with DNS data of Moser et al. [53] and Graham et al. [26]. (b) Discrepancies with the Gaussian distribution assumption.

Figure 3 . 8 :

 38 Figure 3.8: Schematic description of the guess-estimate algorithm determining the mantle thickness δ for a given experimental run.

3. 3

 3 Results and physical discussion 3.3.1 Evaluation of the mantle thickness 3.3.1.1 Experimental results and correlation Equation (3.

3. 3 .Figure 3 . 9 :

 339 Figure 3.9: Thickness of the mantle δ [m] and its dimensionless form δ [¡] as a function of the excursion period τ [s] for the eight operating conditions.

Figure 3 .

 3 Figure 3.10: Non-dimensional thickness δ [-] as a function of the normalized period τ /τ x [-] for the eight operating conditions (∆T sub , Re), represented by the points with error bars. The tting function (dashed line) corresponds to δ V = 180 (see legend).

  Figure 3.11: E ¦ (x ¦ , t ¦ chf ) [¡] for 50 K and 75 K of subcooling, for Reynolds number of 35000 and a time-period ranging within 5 ms to 500 ms. The analytic solution corresponds to the steady-state case, given by Eq. (3.17).

  These quantities are intrinsically connected with the mantle thickness δ as they involve the characteristic velocities u m and v m . The iterative solving procedure of Equation (3.2) yields a single couple (N x , N y ) for each experimental run. These quantities are plotted as a function of the time period τ in Figs. 3.12.

Figure 3 . 12 :

 312 Figure 3.12: Dimensionless groups N x and N y [¡] versus the excursion period τ [s]

Figure 3 . 13 :

 313 Figure 3.13: Heat ux maps evolution and E ¦ (x ¦ ) at the time of boiling crisis (t ¦ = 1) for dierent values of N x . The right maps are taken just after the boiling crisis, the left one being 0.4 τ prior. The conditions are ∆T sub = 50 K, Re = 35000 and p = 1bar. 91

Figure 3 . 14 :

 314 Figure 3.14: Heat ux maps evolution and E ¦ (x ¦ ) at the time of boiling crisis (t ¦ = 1) for a 4x0.6 cm 2 heating cell. The right map is taken just after the boiling crisis, the left one being 0.4 τ prior. The conditions are ∆T sub = 50 K, Re = 35000 and p = 1bar, τ = 50 ms (N x = 0.73).

Figure 3 . 15 :

 315 Figure 3.15: SPERT IV test results for 10 ms power excursion period with dierent ow conditions [19].

  16 N y /P e ¦ as a function of N x , with P e ¦ dened as P e ¦ = Re τ P r = u ¦ e α (3.20) P e ¦ is the Peclet number comparing the heat diusion and the turbulent heat advection in the normal direction from the wall to the channel centerline.

Figure 3 . 16 :

 316 Figure 3.16: N y /P e ¦ as a function of N x

Contents 4 . 1

 41 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . 98 4.1.1 Inuence of the subcooling . . . . . . . . . . . . . . . . . . . . . 98 4.1.2 Inuence of the pressure . . . . . . . . . . . . . . . . . . . . . . 99 4.1.3 Inuence of the heated length . . . . . . . . . . . . . . . . . . . 102 4.1.4 Inuence of the mass ux . . . . . . . . . . . . . . . . . . . . . 103 4.1.5 Inuence of the transient period and coupling with the ow conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.2 Connement and ow characterization . . . . . . . . . . . . . . 108 4.3 Extension of the Homogeneous Mantle Model . . . . . . . . . 113 4.3.1 Extended analysis of the Mantle Energy Equation . . . . . . . 113 4.3.2 Mantle thickness behavior and experimental evidence . . . . . . 115 4.3.3 Analytic expression of the transient CHF . . . . . . . . . . . . 122 4.3.4 Prediction of the mantle thickness and model closure . . . . . . 125 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Figure 4 .

 4 Figure 4.1 presents the transient CHF as a function of the subcooling at constant pressure for two mass uxes in the test section TS2. For G = 15000 kg.m ¡2 .s ¡1 (Figure 4.1b), we can observe that CHF highly increases with the subcooling. For G = 2500 kg.m ¡2 .s ¡1

Figure 4 . 3 :

 43 Figure 4.3: CHF as a function of pressure at constant subcooling and for τ = 5 ms in TS2.

Figure 4 . 4 :

 44 Figure 4.4: CHF as a function of pressure at constant bulk temperature and for τ = 5 ms in the test section TS2.

Figure 4 .

 4 Figure 4.5 presents the inuence of the pressure on the boiling patterns, at constant subcooling and at constant bulk temperature The illustrated boiling patterns are ONB, FDNB and slightly after DNB. The upper cell illustrates the boiling pattern at a pressure of 2.3 bar with a bulk temperature of 75 o C or a subcooling of 50 K. The lower cells illustrates the same boiling patterns when the pressure increases up to 4.8 bar.In the left cell, the subcooling remains constant and in the right cell, the bulk temperature remains constant. At constant subcooling, we can observe that the increase of pressure shrinks the bubbles for all boiling regimes (ONB -FDNB -Post-DNB). Two phenomena explain this shrinking. First of all, a higher pressure increases the vapor density. For a same quantity of vapor generated, the bubbles will be smaller. Secondly, an increase of the bulk temperature implies a drop of the viscosity. This leads to a thinning of the thermal turbulent boundary layer. The characteristic bubble size is then smaller. On the other hand, we can observe at constant bulk temperature that the increase of pressure shrinks the bubble size more intensely. This shrinking is due to the increase of the subcooling which is qualitatively explained in section 4.1.1. It is worth to note that the bubble size is more impacted by an increase of the pressure at constant bulk temperature than at constant subcooling. Finally it is interesting to note the dynamics of coalescence in post-DNB with this condition of pressure. For the two cases at 50 K of subcooling, we observe that the bubbles merge to create only single really large pockets of vapor (see post-DNB images of the upper and bottom-left cells), which probably ll the entire width of the channel, which is 2 mm width. Conversely, in the case with 75 K of subcooling, we observe that the bubbles coalesce in clusters and generate smaller pockets (see post-DNB image of the bottom-right cell). This particularity might be explained by both the high

Figure 4 . 6 :

 46 Figure 4.6: CHF as a function of the heated length for τ = 5 ms and G = 2500 kg.m ¡2 .s ¡1 in the test section TS2.

Chapter 4 .

 4 The transient boiling crisis at moderate pressure and high mass ux ΔT sub = 150 K, p = 8.9 bar ΔT sub = 130 K, p = 5.4 bar ΔT sub = 110 K, p = 6.2 bar ΔT sub = 110 K, p = 11 bar ΔT sub = 90 K, p = 3.6 bar ΔT sub = 75 K, p = 2.3 bar ΔT sub = 90 K, p = 7 bar ΔT sub = 75 K, p = 4.8 bar ΔT sub = 50 K, p = 4.8 bar

Figure 4 . 7 :

 47 Figure 4.7: Transient CHF (τ = 5 ms) as a function of the mass ux for dierent thermodynamic conditions, with a heated length of 2 cm in the test section TS2.

Figure 4 . 8 :

 48 Figure 4.8: Transient CHF (τ = 5 ms) as a function of the mass ux for dierent heated lengths, under a pressure of 12 bars and 163 K of subcooling in TS2.

4. 1 .

 1 Experimental results

Figure 4 . 9 :

 49 Figure 4.9: CHF as a function of τ for dierent mass uxes, under a pressure of 12 bar and 163 K of subcooling -or Bulk temperature of 25 o C -in the test section TS2

Chapter 4 .Figure 4 . 10 :

 4410 Figure 4.10: HSV images in frontlit shadowgraphy for power escalation periods of 5 ms and 100 ms around ONB, during FDNB and around DNB. The experiments are realized in the test section TS2 under a pressure of 12 bars and in a water owing at 10000 kg.m ¡2 .s ¡1 at 163 K of subcooling -or Bulk temperature of 25 o C.

Figure 4 . 11 :

 411 Figure 4.11: HSV and HSIR visualization, 2.6 ms after DNB for dierent mass uxes. The experiments are realized in test section TS2 under a pressure of 12 bars in a water at 163 K of subcooling -or Bulk temperature of 25 o C -with an escalation period of 100 ms.

τ = 5 2 .

 52 ms, G = 5000 kg.m

Figure 4 .

 4 Figure 4.12: HSV images every 0.1 τ starting from ONB under a pressure of 12 bars and 163 K of subcooling -or Bulk temperature of 25 o C -in TS2. The two rst lines present transient pool boiling with periods of 5 ms and 100 ms. The bottom line presents a transient ow boiling with a period of 5 ms.

Figures 4. 13

 13

and 4 .

 4 14 respectively present CHF as a function of the wall Reynolds number Re τ and the usual Reynolds number Re. Then are presented the values of CHF as a function of two characteristic velocities, i.e. bulk velocity U b (gure 4.15) and shear velocity u ¦ (gure 4.16). Finally, gure 4.17 presents CHF as a function of the mass ux G.

Figures 4. 13

 13 Figures 4.13 and 4.14 show that for a given thermodynamic condition, representing CHF as a function of Re or as a function of Re τ is equivalent: the relative trends between the two geometries are identical. This gure also shows that a Reynolds number does not seem to be a relevant quantity to represent the ow in the boiling crisis problem under such conditions. Indeed, the values taken by the Reynolds numbers (usual and wall) are superimposed for the two test sections whereas the values taken by CHF are signicantly dierent. Moreover, one can note that the trends, especially for 20 ms and 100 ms, are signicantly dierent between the two test sections (between a Reynolds number of 2000 and 8000 and between a wall Reynolds number of 450 and 1100).

  Figures 4.13 and 4.14 show that for a given thermodynamic condition, representing CHF as a function of Re or as a function of Re τ is equivalent: the relative trends between the two geometries are identical. This gure also shows that a Reynolds number does not seem to be a relevant quantity to represent the ow in the boiling crisis problem under such conditions. Indeed, the values taken by the Reynolds numbers (usual and wall) are superimposed for the two test sections whereas the values taken by CHF are signicantly dierent. Moreover, one can note that the trends, especially for 20 ms and 100 ms, are signicantly dierent between the two test sections (between a Reynolds number of 2000 and 8000 and between a wall Reynolds number of 450 and 1100).

  (a) τ = 100 ms (b) τ = 20 ms (c) τ = 5 ms

Figure 4 .

 4 Figure 4.13: CHF as a function of the wall Reynolds number for three dierent periods at 10 bar and 100 K of subcooling.

Figure 4 . 14 :

 414 Figure 4.14: CHF as a function of the Reynolds number for three dierent periods at 10 bar and 100 K of subcooling.

Figure 4 .

 4 Figure 4.15 and 4.[START_REF] Chang | Transient-eects modeling of critical heat ux[END_REF] show that quantities based on a velocity present a signicantly better overlap than with Reynolds numbers. These two representations present a similar behavior. For 20 ms and 100 ms, CHF presents a similar trend as it is almost continuous, the CHF values in TS1 seem to be slightly higher than in TS2 (about 20%) at the superimposed value of x coordinate. The representation with the bulk velocity seems to be slightly better (smaller jump at the junction). For the fast transient (5 ms), the behavior of CHF with the two test sections seems to be dierent. A continuity seems to appear

  (a) τ = 100 ms (b) τ = 20 ms (c) τ = 5 ms

Figure 4 . 15 :

 415 Figure 4.15: CHF as a function of the bulk velocity for three dierent periods at 10 bar and 100 K of subcooling.

  (a) τ = 100 ms (b) τ = 20 ms (c) τ = 5 ms

Figure 4 . 16 :

 416 Figure 4.16: CHF as a function of the shear velocity for three dierent periods at 10 bar and 100 K of subcooling.

Figure 4 . 17 :

 417 Figure 4.17: CHF as a function of the mass ux for three dierent periods at 10 bar and 100 K of subcooling.

  (a) τ = 100 ms (b) τ = 20 ms (c) τ = 5 ms

Figure 4 . 18 :

 418 Figure 4.18: Mantle thickness as a function of the shear velocity for three dierent periods at 10 bar and 100 K of subcooling.

  (a) τ = 100 ms (b) τ = 20 ms (c) τ = 5 ms

Figure 4 . 19 :

 419 Figure 4.19: Non dimensional mantle thickness as a function of the shear velocity for three dierent periods at 10 bar and 100 K of subcooling.

Figure 4 . 20 :

 420 Figure 4.20: Schematic representation of the four quadrants as a function of the parameters γ and λ and their associated simplied equations. The location of a quadrant in the horizontal direction permits to determine the time dependency of the problem (Steady-state or Transient). The location of a quadrant in the vertical direction permits to determine the dominating cooling process (Streamwise advection or Turbulent velocity uctuations).

  and 4.21b present the convergence study and gure 4.21c presents v m as a function of y for Re τ = 5200 along with the other sets.

  (a) V as a function of N (b) v m as a function of N (c) v m as a function of y

Figure 4 . 21 :

 421 Figure 4.21: Statistical convergence analysis for the set Re τ = 5200 of Lee and Moser [50] for y = 41. The indicators are plotted as functions of the sample size. The dashed line corresponds to the sample size 10240 ¢ 7680 taken to compute v m . Dimensionless velocity v m as a function of the dimensionless normal coordinate y calculated for Re τ = 180, 395, 590, 1000, 5200 with DNS data of Moser et al. [53], Graham et al. [26] and Lee and Moser [50].

Figure 4 . 22 :

 422 Figure 4.22: Path of power escalation period series in the space (γ, λ) for dierent ow conditions. (a) Series with variable period for three mass uxes (5000, 10000, 15000 kg.m ¡2 .s ¡1 ) at 163 K of subcooling, 12 bar in the test section TS2. (b) Series with variable period for three Reynolds numbers (8500, 25000, 35000) at 50 K of subcooling, atmospheric pressure in the test section TS1

Figure 4 . 23 :

 423 Figure 4.23: Section in the space (γ, λ) focusing on the quadrant 1 ¡ 2 presenting the non-dimensional mantle thickness and non-dimensional CHF.

Figure 4 . 24 :

 424 Figure 4.24: Section in the space (γ, λ) focusing on the quadrants 1 ¡ 3 presenting the non-dimensional mantle thickness and non-dimensional CHF.

Chapter 4 .

 4 The transient boiling crisis at moderate pressure and high mass ux

Figure 4 . 25 :

 425 Figure 4.25: Location of the investigated points A, B and C in the Cross section following the quadrants 1 ¡ 3 and the space (γ, λ). The HSV image and heat ux distribution at DNB of these points are presented.

Figure 4 . 26 :

 426 Figure 4.26: Location in the space (γ, λ) of the experimental points investigating the parameter of ow presented in section 4.2

(4. 2 )

 2 On gure 4.27, we compare the non dimensional CHF obtain experimentally compared with the analytic non-dimensional CHF assuming that the heat ux transferred to water perfectly follows an exponential trend.

Figure 4 . 27 :

 427 Figure 4.27: Comparison between the experimental nondimensional CHF and the analytic nondimensional CHF assuming that the heat ux transferred to water follows an exponential trend (Eq. 4.1). The black dashed line represents the plot y = x and the green dashed line represents an excess of 20% with the ploty = 1.2 x.

Figure 4 . 28 :

 428 Figure 4.28: Histogram representing the distribution of the underestimation of the critical heat ux using equation 4.3 among the investigated points. The bin width is 5 %

Figure 4 . 29 :

 429 Figure 4.29: Sample of heat uxes transferred to water as a function of time (both normalized) super imposed with an exponential trend. The number written in red presents the relative discrepancy between the experimental and the analytic q ¦ chf . The experiments have been realized at 12 bar, 163 K of subcooling, for heated length of a 1 cm and in the test section TS2.

Figure 4 . 30 :

 430 Figure 4.30: Colorized map presenting the value of δ in the space (γ, λ)

4. 3 .Figure 4 . 31 :

 3431 Figure 4.31: Schematic description of the inverse method aiming to assess the value of the steady-state non-dimensional mantle thickness δ V for a given working condition.

Figure 4 . 32 :

 432 Figure 4.32: Illustration of the inverse method algorithm results on the experimental series measured with 50 K (left) and 75 K (red) of subcooling at atmospheric pressure in the test section TS1 with a heated length of 1 cm. The black dashed line represents the tting function (Eq. 3.19). The solid lines represent the Eq. 3.19 using the computed values of δ V and τ x of the experimental points (errorbars)

Figure 4 . 33 :

 433 Figure 4.33: Computed value of δ V as a function of the corresponding parameter λ V . The black dashed lines represent possible asymptotic trends as 180 when λ V ÝÑ 0 and as 180 cλ V when λ V ÝÑ V

1 : 2 : 3 : 4 : 5 :

 12345 ) ΔTsub (K) Tbulk (oC) Re Retau u* (cm/s) Vb (m/s) G (kg/m2/s) L (mm) w (mm) Extended test matrix of the experimental points realized in April and May 2019 investigating on the reliability of the experimental apparatus Transient Date P (bar) ΔTsub (K) Tbulk (oC) Re Retau u* (cm/s) Vb (m/s) G (kg/m2/s) L (mm) w (mm) Extended test matrix of the experimental points realized in July 2019 investigating the impact of the pressure and the temperature Transient Date P (bar) ΔTsub (K) Tbulk (oC) Re Retau u* (cm/s) Vb (m/s) G (kg/m2/s) L (mm) w (mm) Extended test matrix of the experimental points realized in September 2019 investigating the connement and the ow conditions Transient Date P (bar) ΔTsub (K) Tbulk (oC) Re Retau u* (cm/s) Vb (m/s) G (kg/m2/s) L (mm) w (mm) Extended test matrix of the experimental points realized in September 2019 investigating the impact of the mass ux at high pressure and high subcooling Transient Date P (bar) ΔTsub (K) Tbulk (oC) Re Retau u* (cm/s) Vb (m/s) G (kg/m2/s) L (mm) w (mm) Extended test matrix of the experimental points realized in January 2020 investigating the impact of the mass ux and the power escalation period at high pressure and high subcooling

Figure B. 2 :

 2 Figure B.2: Comparative view of an HSV image: Original view, Pre-Processed Gray Scale and Binary threshold at the normalized time t ¡ t onb τ = 0.78. Case for τ = 50 ms with the condition ∆T sub = 75 K, Re = 35, 000 and p = 1 bar in the test section TS1 with a heating area of 1 ¢ 1 cm 2

Figure B. 3 :

 3 Figure B.3: Proportion of the heating surface covered by bubbles as a function of the normalized time. Case for τ = 5, 10, 50 ms with the condition ∆T sub = 75 K, Re = 35, 000 and p = 1 bar in the test section TS1 with a heating area of 1 ¢ 1 cm 2

Figure B. 4 :

 4 Figure B.4: Bubble footprint area distribution compared with the Original and Binary threshold frames at t ¡ t onb τ = 0.78. Case for τ = 50 ms with the condition ∆T sub = 75 K, Re = 35, 000 and p = 1 bar in the test section TS1 with a heating area of 1 ¢ 1 cm 2

  Fig.C.1 for three values of the escalation period τ = 5, 50, 500 ms and ve sets of experimental conditions combining the subcooling ∆T sub = 50, 75 K and the Reynolds number Re = 8500, 25000, 35000. These quantities are plotted from the beginning of the exponential excursion to the boiling crisis.

Figure C. 1 :

 1 Figure C.1: Normalized heat ux released to water q ¦ and normalized wall temperature Θ w as a function of the normalized time t ¦ for τ = 5, 50, 500ms plotted from the beginning of the exponential power excursion to CHF.

  Fig.C.1-lower right.

  . The time period τ = 50 ms corresponds to an intermediate behavior as illustrated by Fig. C.1 (middle column).

Figure C. 3 :

 3 Figure C.3: Overall time sequence of the exponential power excursion. The normalized time of the dierent occurrences and regimes is presented as a function of the power escalation period τ . Series with the conditions Re = 25000 and ∆T sub = 75 K at atmospheric pressure in the test section TS1.

  . The time measured at ONB and at CHF are normalized and presented in gures C.4 and C.5. The normalized times t ¦ ON B and t ¦ CHF are associated with their uncertainty δt ¦ dened asδt ¦ = (δt ¦ t ) 2 σ 2 (C.6) δt ¦t is the uncertainty due to the time resolution of the high-speed IR camera. The frame rate of the camera being 2500, δt ¦ t = 0.4ms/τ . σ is the standard deviation of the three repetitions of each experiment.

  Figure D.4: CHF as a function of the mass ux for three dierent periods at 10 bar and 100 K of subcooling.

[

  Les expériences ont été développées et réalisées au MIT. Le dispositif expérimental est constitué de sections d'essais sur lesquelles raccordées à une boucle hydraulique permettant d'imposer précisément les conditions de pression, de débit et de température. Une section d'essai est une pièce en acier inoxydable dans laquelle passe un canal rectangulaire vertical et qui possède quatre ouvertures. Trois de ces ouvertures sont des hublots optiques permettant d'observer les phénomènes à l'intérieur du canal. La dernière ouverture permet de placer l'élément chauant au niveau du canal. Cet élément chauant est constitué d'un substrat en saphir sur lequel est déposée une ne couche de conducteur électrique permettant un chauage par eet Joule. Des caméras ultra rapides permettent d'observer les phénomènes menant à la crise d'ébullition avec une grande résolution spatio-temporelle. Une caméra ultra rapide visible (> 20 000 fps) permet d'observer la dynamique des bulles avec une vue de face ou de côté. Une caméra ultra rapide infrarouge ( 3 000 fps) permet de réaliser une thermographie infrarouge dont le principe est décrit sur la gure D.11.[START_REF] Bucci | A mechanistic IR calibration technique for boiling heat transfer investigations[END_REF]

Figure D. 11 :

 11 Figure D.11: Schéma de l'élément chauant et de la thermographie infrarouge

L

  'analyse et l'observation des données de la phase 2 à pression atmosphérique des expériences de crise d'ébullition transitoire à forte sous-saturation et convection forcée ont mené à développer un modèle original basé sur une approche énergétique. Les conditions de fonctionnement sélectionnées pour l'étude de cette partie sont ∆T sub = 25 K, 50 K, 75 K, Re [8500, 35000], τ [5 ms, 500 ms]. Sur la gure D.12 ci-après, la séquence temporelle d'une excursion exponentielle est illustrée jusqu'à la crise d'ébullition.

Figure D. 12 :

 12 Figure D.12: Description de la séquence temporelle d'une excursion exponentielle jusqu'à la crise d'ébullition sous forte sous-saturation et convection forcée. Les images d'ombroscopie présentent l'ensemble de la surface chauante (1 ¢ 1 cm 2 ) dans le cas p = 1 bar, ∆T sub = 75 K, Re = 35000, τ = 50 ms. Les schémas illustrent le modèle, vu de côté.

Figure D. 13 : 1 )

 131 Figure D.13: Schéma représentant les contributions thermiques considérées par le modèle pour le bilan d'énergie.

Figure D. 14 :

 14 Figure D.14: Épaisseur du manteau adimensionnée δ [-] en fonction de la période adimensionnée τ /τ x [-] pour huit conditions de fonctionnement (∆T sub , Re) représentées par les barres d'erreur. La courbe de régression est représentée en pointillés.

  Figure D.15: Flux critique transitoire (τ = 5 ms) en fonction de la pression à soussaturation constante (a) et à température du c÷ur de l'écoulement constante (b) avec un ux massique de 15000 kg.m ¡2 .s ¡1 .

Figure D. 16 :

 16 Figure D.16: Flux critique en fonction de la période τ pour diérentes valeurs de ux massique sous une pression de 12 bar and 163 K de sous-saturation (Température de bulk de 25 o C)

Figure D. 17 :

 17 Figure D.17: Diagramme à quatre quadrants en fonction des paramètres γ and λ et des équations simpliées associées.

  Figure D.18: (a) Comparaison entre les ux critiques adimensionnés expérimentaux et analytiques (Eq. (D.4)). La droite en pointillés noirs représente la bissectrice y = x et la droite en pointillés verts représente un excès de 20%. (b) Statistique de la sous estimation relative des résultats analytiques par rapport aux valeurs expérimentales.

Titre:

  Étude expérimentale et modélisation de la crise d'ébullition en transitoire rapide de l'eau à fort sous-refroidissement et à pression modérée Mots clés : Flux Critique, Chauffage transitoire, convection forcée, sous-saturation, excursion exponentielle, BORAX.Résumé : Lors d'une insertion accidentelle de réactivité dans un réacteur nucléaire expérimental, la puissance du coeur peut augmenter de manière exponentielle, avec un temps caractéristique allant de quelques millisecondes à quelques centaines de millisecondes. À cause des effets neutroniques et thermohydrauliques, le système peut atteindre les conditions de crise d'ébullition à même d'engendrer une réaction explosive. Bien que la crise d'ébullition ait été largement étudiée en conditions de chauffage stationnaires, ce n'est pas le cas pour les transitoires notamment de type excursions de puissance. Le but de ce travail est donc de comprendre et de prédire la crise d'ébullition sous l'effet d'un chauffage transitoire rapide de l'eau sous fortes sous-saturations à pression modérée. Des campagnes expérimentales ont été réalisées pour étudier la crise d'ébullition dans de telles conditions au moyen de vidéos et de thermographie IR hautement résolues en temps et en espace. L'analyse de ces données a permis de déterminer la dépendance du flux critique en transitoire rapide en fonction des différents paramètres d'intérêt (temps caractéristique d'excursion de puissance, vitesse d'écoulement, sous-saturation, pression, largeur du canal, longueur de chauffe). De plus, une analyse approfondie de ces données a permis de mettre en évidence les mécanismes sous-jacents à la crise d'ébullition dans ces conditions. En convection forcée et avec de fortes sous-saturations, les bulles générées en paroi présentent un comportement pulsant. Ce phénomène assure un transfert de chaleur efficace depuis la paroi vers le fluide environnant. Le déclenchement de la crise d'ébullition se produit lorsqu'une fine couche de fluide adjacente à la paroi atteint les conditions de saturation. Un modèle développé à partir de ces observations met en évidence deux paramètres adimensionnés utiles pour décrire la nature transitoire du processus ainsi que pour identifier le mode de refroidissement dominant. Grâce à la connaissance du flux critique en régime permanent, le modèle permet d'estimer de manière conservative le flux critique en fonction de la période d'excursion de puissance et du sousrefroidissement. Ce modèle est maintenant prêt à être implémenté dans des codes de simulation pour l'étude des transitoires accidentels.

  Functional scheme (left) and picture (right) of the ow loop . . . . . . . . 2.2 View of the Test Section 1 (TS1) and the corresponding cartridge . . . . . 2.3 View of the Test Section 2 (TS2) and the corresponding cartridge . . . . . 2.4 View of the heater design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Scheme (left) and Picture (right) of the test section TS2 optical setting for IR thermography, side and front view backlit shadowgraphy. The setting for test section TS1 is similar. On the picture, the HSV camera is located to perform a front view backlit shadowgraphy. . . . . . . . . . . . . . . . . 2.6 Calibration curve showing the signal received by HSIR as a function of the heater temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 IR radiation contributions observed by the HSIR camera [8] . . . . . . . . 2.8 Meshing for a TS2 heater with 40 ¢ 6 mm 2 active heating area observed with IR frames having a denition of 452 ¢ 58 pixels. . . . . . . . . . . . . 2.9 Functional diagram of the post-processing algorithm [8] . . . . . . . . . . . 2.10 Escalation to DNB on TS1 with a 40 ¢ 6 mm active area heater at 50 K of subcooling, 35000 Reynolds Number, 100 ms of power escalation period and at atmospheric pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Functioning scheme of the Schlieren method

. . . . . . . . . . . . . . . . . . . . ix List of Figures 2.1

  Extended test matrix of the experimental points realized in January 2020 investigating the impact of the mass ux and the power escalation period at high pressure and high subcooling . . . . . . . . . . . . . . . . . . . . . 143

					Nomenclature
	v v				Transverse vapor ejection velocity	[m.s ¡1 ]
	x				Streamwise coordinate	[m]
	y				Transverse coordinate	[m]
	Greek letters Nomenclature α Diusivity	[m 2 .s ¡1 ]
	∆t ¦	Latin letters = t CHF ¡t ON B τ	Normalized duration of the two-phase regime	[¡]
		c p	Specic heat	[J.kg ¡1 .K ¡1 ]
		D h	Hydraulic diameter	[m]
		e		Channel half width	[m]
		E	Q	Energy density rise	[J.m ¡3 ]
		E	Q cr	Critical energy density rise	[J.m ¡3 ]
		G	Mass ux	[kg.m ¡2 .s ¡1 ]
		h		Heat Transfer Coecient	[W.m ¡2 .K ¡1 ]
		h lv	Liquid-vapor latent heat	[J.kg ¡1 ]
		k		Thermal conductivity	[W.m ¡1 .K ¡1 ]
		L		Heater length	[m]
		L s	Heater thickness	[m]
		N	Sample size	[¡]
		P		Probability density function of velocity uctuation	[s.m ¡1 ]
		p		Pressure	[bar]
		q	P w	Heat ux transferred to the water	[W.m ¡2 ]
		t			Time	[s]
		T w	Wall temperature	[K]
		T b	Bulk Temperature	[ o C]
		T sat	Saturation temperature	[K]
		U		Mean velocity in the streamwise direction	[m.s ¡1 ]
		u		velocity in the streamwise direction	[m.s ¡1 ]
					xviii	xix

A.5 u ¦ Wall shear velocity [m.s ¡1 ] V Mean velocity in the normal direction to the wall [m.s ¡1 ] v velocity in the normal direction to the wall [m.s ¡1 ]

  The BORAX-Type Accident . . . . . . . . . . . . . . . . . . . . 6 1.1.1 The Pool-Type Research Reactors . . . . . . . . . . . . . . . .

	Chapter 1
	Nuclear Safety Challenge and State of
	the Art
	Contents
	1.1

6 1.1.2 Phenomenology of the BORAX type accident . . . . . . . . . . 7 1.1.3 History of the BORAX type accident . . . . . . . . . . . . . . . 9 1.2 Reactor Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.1 Elements on neutrons matter interaction . . . . . . . . . . . . . 12 1.2.2 Core Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.3 Reactivity Eects . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Boiling Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3.1 Boiling regimes and boiling curve . . . . . . . . . . . . . . . . . 20 1.3.2 General impact of the thermohydraulic parameters on the boiling curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3.3 Impact of the transient nature of the heat input on the boiling curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3.4 Flow Boiling Crisis . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Table 1 .

 1

	Core Temperature ( o C)	300	30
	Core Pressure (bar) Mass ow (kg.s ¡1 )	150 2.10 4	1-15 0-10 3
	Thermal Power (MW)	3500	50
	Core Dimension [diameter x height] (m) 5 x 4	0.6 x 0.6
	Fuel Enrichment (%)	3-4	20-95

1.1.1, orders of magnitude of the main reactor characteristics are given to propose a comparative overview of PWRs and pool-type RTRs.

PWR Pool-type RTR 1: Orders of magnitude for PWR and pool-type RTR

Table 1 .

 1

3: Overview of ow boiling crisis experimental investigations at moderate pressure under exponential power escalations with water. L represents the streamwise heating length, l represents the heated width, d is the diameter of the wet perimeter (diameter of the wire or inner diameter of the tube). Chapter 1. Nuclear Safety Challenge and State of the Art 1.3.4.4 Mechanistic approaches for steady-state heat ux

Table 1 . 4

 14

	Regime	Boiling Crisis mechanism	Transient Eect	η
	I	(quasi)	Hydrodynamic	Macrolayer	η m
		Pool Boiling	instability	depletion
	II	Subcooled DNB	Bubble	Macrolayer	η m
	III	Low quality DNB	crowding (local) Bubble Crowding	depletion Macrolayer layer	η m ¢ η u
		& Transition	(bubbly layer)	& upstream eect
	IV	Annular dryout	Liquid lm	Upstream eect	η u
			dryout

: Regime description (Chang et al.

[START_REF] Chang | Transient-eects modeling of critical heat ux[END_REF]

)

Table 2 .

 2

1: Test Matrix of Phase 2 experiments. The tests were realized with the test section TS1 using an ITO active area of 10 ¢ 10 mm 2

Table 2 .

 2 2: Test Matrix investigating the inuence of pressure and water temperature on the transient CHF at G = 2500 kg.m ¡2 .s ¡1 . The tests were realized with the test section TS2 using a Ti active area of 20 ¢ 4 mm 2 . The values in the table correspond to the subcooling (in bold and italics) at the corresponding pressure and bulk temperature.

	Parameters	Values
	Mass ux [10 3 kg.m -2 .s -1 ]	15,000
	Period [ms]	5
	Bulk Temperature [ o C]	25	50	75	100
	2.3		75	50
	3.6		90
	4.8			75	50
	5.4	130
	Pressure [bar]
	6.2		110
	7			90
	8.9				75
	12	163

Table 2 .

 2 3: Test Matrix investigating the inuence of pressure and water temperature on the transient CHF at G = 15000 kg.m ¡2 .s ¡1 . The tests were realized with the test section TS2 using a Ti active area of 20 ¢ 4 mm 2 . The values in the table correspond to the subcooling at the corresponding pressure and bulk temperature. On September 9 th and 19 th 2019 were realized a set of experiments in the two dierent test sections, i.e. investigating two values of channel width. The corresponding test matrix is presented in table 2.

Table 2 .

 2 4: Test Matrix investigating the inuence of the ow intensity and of the channel width. The tests were realized using a Ti active area of 20 ¢ 4 mm 2

Table 2 .

 2 5: Test Matrix investigating the inuence of the mass ux on the transient CHF at high pressure and subcooling. The tests were realized with the test section TS2 using a Ti active area of 10 ¢ 4 mm 2On January 12 th 2020 were realized series to investigate the impact of the power escalation period at dierent mass uxes at high pressure and high subcooling. The corresponding test matrix is presented in table 2.6.

	Parameters	Values
	Pressure [bar]	12
		25 / 163
	Mass ux [kg.m -2 .s -1 ]	0 -2,500 -10,000 -15,000
	Period [ms]

*

15,000 kg.m -2 .s -1 ** 0 and 15,000 kg.m -2 .s -1 2.5 * -5 -7.5 ** -10 -15 ** -20 -30 ** -50 -100 -200 Bulk Temperature [ o C] / Subcooling [K]

Table 2 .

 2 6: Test Matrix investigating the inuence of the mass ux and of the power escalation period on transient CHF at high pressure and subcooling. The tests were realized using the test section TS2 using a Ti active area of 10 ¢ 4 mm 2

chf , is more quantitatively illustrated by the fact that 61% of the investigated points remain under a discrepancy of 20% from the analytic solution as presented in gure 4.28.

Acknowledgements

One thesis. One journey that drove me during the last three years in three labs, two continents and to so many incredible people. In term of working environment, I had the chance to be welcomed in the Laboratoire de Modélisation et Simulation en mécanique des Fluides (LMSF) in CEA Saclay (CEA/DES/ISAS/DM2S/STMF) and the Laboratoire d'Informatique pour la Mécanique et les Sciences de l'Ingénieur (LIMSI) in Université Paris-Saclay and CNRS, both in France, and in the Red Lab in the Department of Nuclear Science and Engineering at MIT in the USA. This variety of environment was a signicant plus for me, enabling me to expand the horizon of my scientic and human experience throughout these three years.

One thesis. One jury that I would like to thank. First of all, I would like to thank Prof. Catherine Colin and Dr. Philippe Lafon for accepting to review my manuscript. I also would like to thank Prof. Michel Gradeck, Dr. François Luseyran and Dr. Vadim Nikolayev for accepting to be examiners of my Ph.D. defense. All your valuable questions and comments will improve the future work in the continuity of this thesis.

at the same subcooling. Finally for the case at 125 K and 8.9 bar the frequency rises to 100 kHz and above, as a signicant number of bubbles are only observed during a single frame on the HSV video recorded at 89, 000 F P S.

Inuence of the pressure

The specic aspect of a parametric study on pressure is that other parameters are coupled with the pressure and the bulk temperature. Indeed, on a rst hand, when the pressure varies with a constant bulk temperature, the subcooling increases with the pressure as the saturation temperature increases. On the other hand, when the pressure varies with a con-Chapter 4. The transient boiling crisis at moderate pressure and high mass ux quantity of energy deposited and the bubbles size dierence when DNB is reached. For the cases at 50 K of subcooling, the bubbles dimension are about the order of magnitude of the channel width. When the bubbles pulsate, they transfer more energy into the bulk, which warm up. When the pockets of vapor appear at the boiling crisis, the surrounding water is relatively warm enabling their expansion to the whole channel. This does not happen for higher subcooling and pressure, where the pockets cannot expend as much.

Then, it is worth to mention that for such conditions, i.e. narrow channel, low subcooling and pressure, the model presented in Chapter 3 is expected not to be applicable.

Inuence of the heated length

In the literature, e.g. [START_REF] Celata | Critical Heat Flux, Post-CHF Heat Transfer and Their Augmentation[END_REF]- [START_REF] Kandlikar | Critical heat ux in subcooled ow boiling -An assessment of current understanding and future directions for research[END_REF], the steady-state CHF in tubes decreases with the increase of the ratio L/d and becomes almost constant beyond a threshold value (between 20 and 40 [START_REF] Celata | Critical Heat Flux, Post-CHF Heat Transfer and Their Augmentation[END_REF]) depending on the ow conditions. This trend is conrmed in Figure 4.6, which presents the inuence of the heated length on the transient CHF for dierent pressures and subcooling temperatures (for TS2, L = 2 cm gives L/d = 10 and L = 4 cm gives L/d = 21). This supports the importance of the system history on the transient boiling crisis process, as for example mentioned in the literature the development of boundary layers or the variation of the local subcooling. This is also consistent with the physics Appendix A

Detailed Test matrices

In this appendix, we present the detailed test matrices of Phase 3 presented in section 2.4. The following parameters are given in the tables:

The pressure (control parameter)

The temperature condition: bulk temperature (control parameter) and corresponding subcooling

The ow condition: the mass ux (control parameter) and the corresponding Reynolds number, wall Reynolds number, shear velocity and the bulk velocity

The geometric conditions: the test section used, the length and width of the heating area.

The exponential power escalation period

The number of trials for the given thermal hydraulic conditions.

Appendix B

Miscellaneous experimental explorations

Silicon substrate

As presented in section 2.2.1, the exponential electrical heat input is released to both the substrate and the water generating two heat uxes being respectively q P s and q P w as dened in section 2.2. The proportion of each part and its evolution in time are determined by the eusivity of both media and by the escalation period following these expressions developed in [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part II: Flow boiling[END_REF]: q P s (t) = q P 0 e t/τ h s h s h w q P w (t) = q P 0 e t/τ h w h s h w with h s and h w being respectively the HTC between the heating lm and the substrate, and the heating lm and the water. Particularly,

In case of single phase heat transfer, h w can be estimated using for example the Sieder-Tate correlation and equation 1.3.3 (see section 1.3.3 or Su et al. [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part II: Flow boiling[END_REF]). In case of boiling heat transfer, h w is more complicated to assess.

In order to investigate the impact of the substrate thermal inertia, it is interesting to compare the boiling behavior during a transient heating with substrates having dierent eusivities. I decided to try crystalline Silicon for dierent reasons. Silicon has as dierent eusivity than Sapphire, it is partially transparent to IR radiations, and nally some silicon wafers were available to conduct fast exploratory tests. From a 1 mm-thick silicon wafer, several 2 ¢ 2 cm 2 squares were cut in order to be used with the test section TS1.

The sides were also manually rounded for the gold pads to ensure an electrical connection with the back of the heater. The silicon substrates were then coated to make gold pads and a 1 ¢ 1 cm 2 titanium active heating area. IR spectroscopy was realized in order to assess the silicon absorption, air-silicon reection and titanium-silicon reection spectra in order to use the conduction-radiation model presented before (section 2.2.1).

Several trials have been realized, but I faced two main issues. The rst is the fragility of the silicon wafer. A sapphire substrate is extremely robust but a heater made with silicon is easily breakable. This issue implies to be careful during manipulation but can be overcame, which is not the case with the second issue. I noticed that the titanium coating does not remain on the silicon once the heater is in contact with water. The gure B.1

shows the results of a single droplet of water on the coating. Retrospectively, such severe reaction of the Titanium may be due to the contamination by Gallium (see section 2.4.2). The new Titanium ingot could possibly improve the adherence of the Titanium on the Silicon substrate. Unfortunately by lack of time, no more investigations have been tried in this direction. Yet, it would be interesting in the future to do so. We also investigated the multi-scale wavelet transform method developed by Guo et al. [START_REF] Guo | A novel multi-scale edge detection technique based on wavelet analysis with application in multiphase ows[END_REF]. This method is particularly interesting as it considers dierent lter resolution and allows us for choosing the weight of each contribution. This large number of tuning parameters permits to be highly accurate, but consequently its tuning may be a long process. Currently, promising preliminary results are ongoing, but more tuning should be rened.

Perspectives

To nalize the bubble detection process, the coupling of the two modules, footprint assessment and edge detection, should be implemented. The next improving method of this work would be not to process the frames individually, but to also include previous frames and following frames. This would have two impacts. The rst one would be to catch the bubble dynamics, their evolution in size and in space. It would also enable to consider a bubble as such and maybe increase the accuracy of the bubble recognition. Finally, a validation process should be realized in order to quantify the accuracy and the uncertainty of the detection method.

Appendix C

On the Time Sequence of the Transient Flow Boiling Crisis at High Subcooling

In this section, we analyze the time characteristics of the dierent regimes involved during a transient boiling crisis. The experimental points involved in this analysis are the experiments of Phase 2 in ow boiling (Re = 8500, 25000 and 35000) and at high subcooling (50 K and 75 K). This Appendix is adapted from the conference paper we submitted to HEFAT 2020 Amsterdam. This paper was accepted, but the conference had to be canceled due to the worldwide pandemic situation.

Time dynamics of the wall temperature and the heat ux

Normalized temperature and heat ux

As the temperature of the liquid entering the test section is well below the saturation temperature, the heat transfer at the wall is rst performed in the liquid by conduction and convection. When the heating process becomes fast -i.e. for small values of τ , heat transfer in the liquid is limited to a short distance from the wall. Heat diusion is expected to be the dominating process. Disregarding end eects, the heat ux is mainly transmitted in the single transverse direction. Accounting for both the liquid medium and the substrate, the heat diusion problem may be solved analytically ( [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part I: Pool boiling[END_REF], Eq.( 12)).

For an electrical heat ux input q P 0 e t/τ released by the ITO between the substrate and the water, we have:

This analysis suggests to plot the curves using non dimensional quantities dened as :

with

Time sequence of the transient CHF As stated above, from the beginning of the process to the boiling crisis, one may split the transient path in two parts: the single-phase regime and the two-phase regime as illustrated in gure C.2. The single phase regime is limited between the beginning of the power escalation period and ONB. The two-phase regime is limited between ONB and DNB. A quantitative minoration of t ¦ ON B can be determined through a simple criterion. Indeed, since boiling can not be observed as long as the liquid temperature is below saturation, a minimum time value can be computed considering the following criterion:

Single

and the use of Eq. ( 12) proposed by Su et al. [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part I: Pool boiling[END_REF] gives the following analytic expression:

Plotted on Fig.C.4, this quantity cannot precisely catch t ¦

ON B as a certain wall superheat is needed to trigger nucleation. The precise determination of this wall superheat needs the knowledge of the surface condition of the heating surface and also depends on the operating conditions and on the escalation period [START_REF] Su | Transient boiling of water under exponentially escalating heat inputs. Part I: Pool boiling[END_REF]. However, it is interesting to note that this very simple criterion satisfactorily catches the parametric trends of the period τ and of the subcooling. We observe a similar slope of t ¦ min and t ¦ ON B for small periods (i.e. less than 20 ms). An equal dierence of 0.5τ between the two levels of subcooling is also observed showing that this time t ¦ min yields a valuable quantitative information on the subcooling eect. This theoretical time also indicates that convection eects start to have a signicant impact for τ greater than 20 ms where the discrepancy increases for higher period showing the increase of the ow condition inuence. To summarize, time t ¦ min , easily accessible from a theoretical point of view, overviews the impact of some major operating parameters (τ , ∆T sub , Re) and proposes a quantitative lower value of t ¦ ON B .

Regarding CHF, the normalized time to CHF t ¦ CHF presented in Fig. C.5 follows a similar qualitative trend than the normalized time of ONB. For this set of experiments, it is interesting to remarkthe non-dimensional time for DNB to occur varies in a narrow range between 6.7 and 8.2. However a more quantitative feature is not determined at this stage. (C.10)

Consequently, one expects this quantity to vanish when the power escalation period takes values smaller than 5 ms : CHF would then become concomitant with ONB. In other words, at signicantly shorter escalation periods, the large delay to nucleation, synonym of a large wall superheat, would generate such an explosive ONB that it would lead directly to boiling crisis.

This analysis brings to light some time characteristics of the transient ow boiling crisis at high subcooling. Particularly, it shows that the normalized duration of the two-phase regime has a regular dependency on the power escalation period. Plus, this dependency is weakly impacted by the subcooling and the ow for these investigated ranges. It would be interesting to extend this analysis to the other available experimental points to conrm these results, and to investigate the inuence of the other parameters (higher thermohydraulic conditions, pressure, lengths,...).

Appendix D Connement and ow characterization -extended results

In this appendix, we detail the impact of the connement and the ow characterization discussed in section 4.2. First of all, we present the value of CHF as a function of the wall Reynolds number, Reynolds number, shear velocity, bulk velocity and mass ux. Secondly, we present the value of the non-dimensional mantle thickness as a function of the same parameters.

Impact on CHF