A Mouly
email: aude.mouly@cea.fr

N Seiler

F Bertrand

M Gradeck Lemta

MODELLING OF CORE MOLTEN MATERIAL FAST VAPORISATION AND EXPANSION FOR THE EVALUATION OF SFR SEVERE ACCIDENT CONSEQUENCES

Keywords: SFR, severe accidents, expansion phase, dimensional analysis, MOREINa

The fourth generation of sodium fast reactors (SFR) is developed taking into account severe accidents as soon as the pre-conceptual phase is under way in order to increase safety margins. In this context, parametric and fast-running tools are used in addition to mechanistic tools during the reactor design phase for probabilistic margin evaluations.

The present paper is dedicated to preliminary transient modelling and studies of the expansion phase using a new parametric tool named MOREINa. Potentially occuring after a secondary power excursion in a SFR with low void worth core (non energetic primary phase), the expansion phase implies a vaporisation of superheated materials, their expansion and their interaction with the coolant, potentially jeopardizing the vessel mechanical integrity. MOREINa has a new expansion phase model considering both fuel and steel in order to evaluate the energy released during this secondary phase and its consequences on the reactor vessel structures. This modelling, based on dimensional analysis and involving mass, momentum and energy balances, is done to simulate the vaporisation of two overheated liquids whose vapors supply a bubble of vapour and fission gases, the bubble expansion due to pressure difference between the bubble and the cover gas located at the top of the vessel and finally the FCI induced by thermohydraulic instabilities and contributing to the further fragmentation of the molten materials. The energy released is evaluated to assess order of magnitudes of the mechanical loading applied on the reactor vessel. Comparison with a CEA validated parametric tool DETONa are finally presented and analyzed.

INTRODUCTION

Improving reactors technology in order to excel safety and reliability at industrial scale is one of the current objectives of the fourth generation of nuclear reactors (GEN IV). Among the six nuclear concepts of this generation, Sodium Fast Reactor (SFR) has been selected for its ability to sustainably manage radioactive waste and nuclear fuel ressource [START_REF]Les reacteurs nucleaires caloporteur sodium[END_REF]. This neutron fast reactor concept is well known in France because of previous fast reactors feedback (Rapsodie, Phenix and Superphenix). At CEA, further developments of SFR are conducted in the scope of safety standard of the GEN IV. Thus, taking into account hypothetical severe accidents involving the melt down of the reactor's core is required in the safety analyses during the preconceptual phase.

Severe accidents are studied in the safety analysis of design phase by using two different approaches [START_REF] Bertrand | Methodologies d'etudes de surete et applications pour la pre-conception de differents types de reacteurs nucleaires de quatrieme generation[END_REF] : the deterministic approach and the probabilistic approach. The first approach uses unfavourable scenario to demonstrate the compliance of the reactors to safety requirement and verify the margins regarding the safety criteria. Complex mechanistic codes are used such as SIMMER and EUROPLEXUS during this approach.

The second one studies the behaviour of the reactor with different scenarios using the occurrence probabilities of phenomena (Best Estimate Plus Uncertainties method). This approach is more appropriate to calculate safety margin probabilities. Other codes are needed in this second approach owing to the high CPU time of the complex mechanistic codes which prevent their use from such probabilistic studies. In this context, the R&D progress in CEA provided fast-running tools in order to evaluate safety margins, uncertainties studies and parametric studies. The present paper is dedicated to the presentation of one of them, named MOREINa * . This tool models a part of the expansion phase which is a specific phase of severe accidents in SFR.

In this paper, we detail the modelling and we also present first results. At first, the expansion phase will be described. Then, the physical modelling will be detailed. In a second time preliminary results and preliminary sensitivity studies will be shown and analyse. Finally, a comparison with a validated parametric tool will be presented.

THE EXPANSION PHASE

The expansion phase during a SFR severe accident is a highly hypothetical situation studied in the reactor preconceptual phase in order to achieve safety requirements. It can potentially occur in SFR severe accident when molten materials are forming pools in the internal structures of the vessel. This phase begins when the pool composed by the core molten materials undergoes a recriticality induced by its compaction. This pool recriticality involves a secondary power excursion, implying an increase of the pool temperature. If the pressure of the pool is lower than the increasing saturation pressure obtained with a correlation on the saturated temperature of the liquid materials, the molten materials vaporisation begins by flashing and creates a bubble of vapour. The pressure difference between the bubble vapour and the reactor's cover gaz initiates an expansion of the vapour towards the cover gas in the mechanism.

Because of the thermal equilibrium break up during the expansion of the vapour bubble by vapour pressure decrease, the pool vaporises and supplies the bubble to reach another thermal equilibrium between the vapour and the liquid. The bubble expansion can also potentially induce instabilities which can promote fragmentation between the different fluid layers. These instabilities are driven by the density difference between the fluid layers and the velocity of the vapour expansion. Furthermore, if the expansion leads the liquid molten core materials into the sodium, a fuel coolant * MOdel of vapouR Expansion and fuel coolant Interaction in Natrium interaction can also occur and induce Sodium vaporisation and the resulting bubble expansion. The mechanical energy released by the vaporisations, expansions and interaction will potentially jeopardize the reactor vessel structure (see figure 1).

Figure 1. Phenomenological tree of the expansion phase

DESCRIPTION OF THE PHYSICAL MODELLING

The models implemented in MOREINa describe the vaporisation of superheated molten materials in the internal structures of the core and the expansion of the generated vapours forming a bubble because of the pressure difference with the upper cover gas.

Main assumptions

In MOREINa, the studied system is considered adiabatic. This is justified because four order of magnitude are separating the lost energy from lateral wall to the thermal energy of the pool during the transient. Moreover, the pool is supposed to be non compressible. The initial superheated liquid materials are assumed to be composed of liquid steel, liquid fuel and fission gases. The non missibility of the liquid steel in the liquid fuel involves liquid steel bulks homogeneously distributed in the liquid fuel.

The distribution of temperature is parabolic along the z axis with the hotest part at the center of the liquid pool materials because the power excursion is driven by the neutron flux in the pool. At the uppermost location, the non liquid part of the core is full of Argon gas (see figure 2). This modelling is an academic case and has been chosen to study the behaviour of the molten materials vaporisation and vapour bubble expansion. The sodium will be included in MOREINa, between the pool and the argon gas, in further developments.

At the initial time, the metastable state of the pool is assumed to break and vapour production is beginning to reach a stable state. Argon gas and the fission gases are supposed to be perfect gases and their evolution is assumed to be isentropic.

Tool characteristics

MOREINa is a 0D multi-area explicit sheme calculation tool whose equations are solved using the fourth order of Runge-Kutta calcultation method (RK4).

In the calculation tool, the core modelling is developped considering a cylindrical geometry in which the vapour bubble evolution is supposed to be planar. This configuration has been chosen to fit with the geometry of the reactor core. A mesh grid is implemented along the height of the core in order to discretize the temperature and the mass in the core (see figure 2).

Figure 2. Mesh grid scheme implemented in MOREINa

This calculation tool is considered as a 0D multi-area tool because the balance equations constituting the modelling are applied on different zones of the pool. One zone is defined by one or more meshes, depending on the transient evolution. The first zone is the molten materials above the vapour bubble also cited as slug (see the blue part in the figure 2). The second one is named the vaporisation zone (see the red part in the figure 2) and corresponds to the mesh(es) with the higher temperature at initial time. Finally the last one in the system covers the molten materials below the vapour bubble (the green part in the figure 2).

The different areas defined at initial time evolve during the transient because of the molten pool materials vaporisation (see figure 2,b). Indeed, the temperature in this zone decreases as the pool vaporises. The adjacent meshes are included in the vaporized area when the temperatures of the meshes become equal (see figure 2,b).

The vapour formed during the transient supplies one unique bubble in which the fission gases, the steel vapour and the fuel vapour are mixed. This bubble is considered to be at the hotest part of the mesh, in the middle of the vaporisation zone throughout the transient.

Major phenomena

The bubble expansion of the modelling is decribed by a momentum balance equation applied on the liquid above the vapour, the slug.

∂ ż1 (t) ∂t = 1 (z pool (t) -z 1 (t)) P bubble -P Ar (t) ρ l pool -g (1)
Where z 1 is the upper vapour/liquid interface elevation and z pool the liquid/Argon interface elevation. g is the gravity acceleration and ρ l b is the density of the pool above the bubble. The Argon pressure P Ar (t) and the fission gasses pressure P GF are calculated using Laplace's law for diatomic gases. The global vapour pressure of the bubble P bubble is calculated using Dalton's law and Raoult's law by adding partial pressures obtained with the saturation pressure and the steel mole fraction in the pool for the liquids (y s ). P GF is calculated with Laplace's law for fission gases.

P bubble = y s P sat s (T sat ) + (1 -y s )P sat f (T sat )) + P GF (2)
The molten materials vaporisation has been separately modelled between the two superheated molten materials, fuel and steel, to treat the non-miscibility of the dedicated liquids. For each liquid, a system to solve has been expressed using two balance equations.

First, the enthalpy balance equations have been expressed using thermal properties and vapour mass fraction for the dedicated liquids (see the first equation line in the system 3). The second equation is the derivated mass balance expressing the two-phase flow volume with the vapour density ρ v , the liquid density ρ l and the vapour mass fraction x (see the second equation line in the system 3). The liquid volume change has been neglected because of the low order of magnitude difference with the vapour volume change (n is representing the studied liquid, either fuel or steel).

         m n x n dh n v dT sat n + (1 -x n ) dh n l dT sat n - V tot n m n dP sat n dT sat n dT sat n dt + ∆h n l,v dx n dt = dQ n dt m n dx n dt 1 ρ n v - 1 ρ n l + x n ∂(1/ρ n v ) ∂T sat n + (1 -x n ) ∂(1/ρ n l ) ∂T sat n dT sat n dt = dV v dt (3)
Being included in one unique pool, a conducto-convective heat transfert (Q s = -Q f ) has been implemented between the liquid steel and the liquid fuel. Having unknown bulks of steel in the fuel vapour, the heat exchange surface has been maximised resulting from stable droplets of steel in the fuel whose limit size is given by the critical Weber number (W e c = 12.).

dQ s dt = - dQ f dt = hS ech ∆T = hS ech (T sat s -T sat f ). (4)
In order to verify the consistency of the mass and energy exchanges during the transient, the mechanical energy and a mass balance are calculated. The mechanical energy balance is described by the four energies involved during the transient ; the expansion energy, the compression energy, the kinetic energy and finally the potential energy (Equation 5).

E expansion = E compression + E kinetic + E potential (5)
The expansion energy, E expansion , is the energy released by the vapour bubble expansion, especially the pressure and volume vapour bubble evolution. The compression energy, E compression , represents the energy of the Argon compression, is also calculated with the pressure and volume evolution of the Argon gas. The kinetic energy, E kinetic , is the energy induced by the slug motion and finally, the potential energy, E potential , is also the energy calculated by the slug evolution inside the core.

Instabilities located at the upper vapour/liquid interface

The vapours bubble expansion and the density difference between the liquids and the vapours involves Rayleigh-Taylor (RT) instabilities at the liquid/vapour interface during the transient (see figure 3). Due to these instabilities, molten materials jets from the slug are created and liquid droplets may be included in the vapour according to jet break up during the transient. To take into account this phenomenon, a model dealing with droplets production in the vapour has been implemented in MOREINa.

Rayleigh-Taylor instabilities in MOREINa's modelling are described in figure 3. The slug is represented in orange and the vapour driving the liquid during the expansion is represented in blue. The perturbations, induced by RT instabilities, are sinusoidal perturbations and only the most unstables ones leading to droplet formation are modelled. The most unstable wavelength is calculated using the cut-off wavelength, defined as the wavelength above which the perturbations becomes unstables and start to growth (Equation 6).

λ M RT (t) = √ 3 × λ c RT (t) = √ 3 × 2π σ pool (ρ l pool (t) -ρ v bubble (t))z 1 (t) . ( 6
)
Jets induced by the perturbations growth are supposed to be homogeneously distributed at the bubble interface. Experiences have showed an hexagonal distribution pattern [START_REF] Fermigier | Two-dimensional pattern in Rayleigh-Taylor instability of a thin layer[END_REF] with the most probable wavelength of the RT instabilities as the period of the hexagon. Thus, the jet density per unit surface is calculated as a function of the wavelength and the mean number of jets in an hexagonal mesh.

Figure 3. Droplet creation and inclusion scheme in MOREINa

The jets created by RT instabilities are growing during the transient. The rate of the jet growth has been studied with experiments [START_REF] Lewis | The instability of liquid surface when accelerated in a direction perpendicular to their planes, II[END_REF] showing that it reaches an asymptotical speed. Two different correlations from different works [5] [6] have been implemented into the model at user's convenience. The first one, also named Corradini's velocity, expresses the velocity with the local scale of the system, the most unstable wavelength. The second one, named Lainault's velocity, expresses the velocity using the local and the global scale of the system, the vessel diameter and also the most unstable wavelength.

v e = K λ C RT z1 (7
)

v e = α 1 (z 1 λ c RT ) n + α 2 (z 1 D T ) m (8)
Where K = 4.65, α 1 = 2.3, α 2 = 0.35, n = 0.28 and m = 0.4.

Jets breaks up and droplet creation are well known as the fragmentation process. In MOREINa, the main fragmentation regime is a transient turbulent regime involving a combination of Kelvin-Helmholtz instabilities, Rayleigh-Taylor instabilities and capillarity forces to create the droplets [START_REF] Marmottant | Atomisation d'un liquide par un courant gazeux[END_REF]. Due to the modelling choices in MOREINa, the flow evolution cannot be locally described around the perturbations. Then, the jet breakup implying spherical droplets creation has been postulated to occur when the length of the jet is above λ M RT (t). The spherical droplets diameter is considered equal to d drop = λ M RT (t)/2 [START_REF] Manchon | Modeling and analysis of molten fuel vaporization and expansion for a sodium fast reactor severe accident[END_REF] (see figure 3). Droplets' families are created each time jets are fragmented during the transient. The families are divided into two parts, fuel droplet and steel droplets features, each one depending on the fragmentation time.

A second type of fragmentation is also considered in the modelling in order to take into account the fragmentation of the created droplet in the vapour flow due to the competition between the flow inertia force and the droplet surface tension force. A simple model has been implemented involving the comparison of the droplet Weber number and the critical Weber number.

Thermal evolution of the droplets in the vapour

Because of the axial distribution of the pool temperature, droplets created by fragmentation process and included in the vapour bubble have a different temperature than the vapour in the vaporisation area. Thus, heat transfers between the two phases must be modelled because it can potentially involve phase changes : droplet partial or total vaporisation and vapour condensation on the droplet surface.

Thermal evolutions of created droplets have been implemented in MOREINa by evaluating two total heat transfers, one for each liquid fluid. This total heat transfer is calculated by summing each transfers between the droplets of corresponding part of the family (steel or fuel) and the vapour bubble. For each part of the family, the conducto-convective and radiative heat transfers are considered and calculated in order to evaluate the temperature evolution of the droplets integrated in the vapour.

dQ i,j dt = dQ rad i,j dt + dQ conv i,j dt = S i,j ϕ i,j (T vapour -T i,j ) + S i,j i,j ν(T 4 vapour -T 4 i,j ). ( 9
)
The index i corresponds to the family's number and the index j represents the steel or fuel part in the family. Furthermore, S i,j is the heat exchange surface calculated by summing all the droplets surfaces and h i,j , the heat exchange coefficient between the vapour bubble and the droplets. The averaged temperature of the vapour is named T vapour and T i,j corresponds to the droplet's temperature. ν is the Boltzmann constant and finally, i,j is the liquid absorbtivity.

The temperature of the droplets is updated using the calculated heat exchange and compared to the saturated temperature of the corresponding liquid. If the obtained temperature is higher than the saturated one, vaporisation is simulated. The vaporised mass is calculated using an enthalpy balance equation and droplets are partialy or totally vaporised. If the updated temperature is smaller than the saturated temperature, no vaporisation is considered and the heat exchange heats up the droplets.

The total heat exchanges are finally added to the vaporisation systems as a source in the enthalpy equation to integrate the potential vapour condensation on the droplets surface.

PRELIMINARY RESULTS

Preliminary results have been obtained with MOREINa. This academic case has been developped to study and verify this first part of the expansion phase modelling : pool vaporisation and vapour expansion in the core.

Reference case

This case has been computed without integrating instabilities at the upper vapour-liquid interface. The different parameters used for the calculation of this reference case are summarised in the followed table.

The meshgrid chosen for the computation is composed of 1000 meshes and the temperature difference between the hotest part of the meshgrid and the lowest is ∆T = 5446 -3700 = 746K. Meshgrid sensitivity studies on the time of discretisation have highlighted the value 10 -6 s as the most adapted time of discretisation for this transient. The results (see figure 4) show the evolution of the relevant variables during the transient. 52 kg of liquids are vaporised during the transient. This molten materials vaporisation induces a 200 K decrease of the temperature. The fuel temperature decrease is higher than the steel temperature decrease (around 50 K) because of the smaller value of fuel latent heat of vaporisation than steel latent heat of vaporisation. This difference involves an higher fuel mass vaporisation than steel one.

The pressure difference between the vapour bubble pressure and the Argon pressure generates an increase of the upper vapour/liquid interface velocity until 44 m/s and an increase of the corresponding height z 1 (t). The acceleration evolves due to the pressure difference and the liquid mass above the vapour. The more the vapour pression decreases, the more the acceleration decreases. When it reaches negative values, the velocity decreases until the system finds a stable state and the transient ends. The vapour pressure has an high value (17 MPa) at initial time because of the fission gases and decreases with the expansion to reach 9.8 MPa. The Argon pressure is increasing during the transient due to the expansion of the vapour volume and the decrease of the argon volume.

During the transient, the mechanical energies are highlighting the explained phenomenon : the expansion energy grows proportionnaly to the vapour volume evolution due to the vaporisation of the liquids. The compression energy increases because of the argon pressure increase induced by the expansion of the vapour and finally the kinetic energy follows the upper vapour/liquid interface velocity during the transient. Furthermore, the expansion energy is at the final state equal to the Argon compression energy. This demonstrates no energy loss during the transient and validates the numerical implementation in MOREINa. The results presented in this part are performed with the instability models implemented in MOREINa considering the jets growth rate expressed with the local and the global scale of the system, the Lainault velocity [START_REF] Lainault | Modelisation de la liberation d'energie lie aux accidents graves dans les reacteurs nucleaires a neutrons rapides[END_REF] (Equation 8).

The results (see figure 5) are similar as the reference case where no instability is taken into account until 0.004 s, the beginning of the droplets inclusion inside the vapour. 470 kg of fuel droplets and 170 kg of steel droplets are included inside the vapour bubble during the transient. The vaporised mass of the droplet is almost totally coming from the fuel droplets because of the fuel latent heat vaporisation smaller than the steel one. 33 kg of fuel under the form of droplets are vaporised during the transient which impacts the vapour pressure and the fuel temperature. The vapour pressure increases around 0.005 s because of the higher amount of vapour formed during the transient. High thermal exchanges are implemented between the vapour and the droplets created by fragmentation which induce an important decrease of the difference between fuel and steel temperature in this case.

Furthermore, the upper vapour/interface velocity reaches an higher maximun (3 %) due to the loss of mass in the slug by droplets inclusions. Because of this higher interface velocity and the more vaporised mass during the transient, the Argon pressure is also higher (16 %).

Finally, the mechanical energies implies in this transient are 5 % higher than the mechanical energies calculated in the reference case because of the implied vaporisation and the more important induced global bubble vapor during the transient. The DETONa parametrical tool was developped by Xavier Manchon [START_REF] Manchon | Modeling and analysis of molten fuel vaporization and expansion for a sodium fast reactor severe accident[END_REF] to model the fuel vaporization and a global bubble expansion in a SFR. DETONa focuses on the mechanical energy released by the fuel vapour expansion pushing away the sodium phase towards the vessel structures. This code is computed in a spherical geometry and implies the presence of a sodium liquid column around the fuel pool (see figure 6).

No interaction between the fuel and the sodium are modelled and the vapour formed is supposed to be between the fuel pool and the sodium around. An Argon phase is also simulated in order to represent the core cover gas. This parametric tool has been validated on experiments. Comparing MOREINa's results to DETONa's results is interesting to assess relevancy of MOREINa's results. For the comparison, some adaptations have been performed in MOREINa :

• A sodium column has been implemented above the melted pool ;

• Steel and fissions products have been eliminated from the pool for the comparison calculation ;

• The vessel volume has been adapted to cope with DETONa's vessel volume.

Because of the two different geometries, spherical geometrie in DETONa modelling and cylindrical geometrie in MOREINa modelling, the results are expected to be the same order of magnitude but small discrepancies are expected. The parameters used for this comparison are resumed in the followed table (Table II).

The 1 √ 3 in the expression of DETONa's characteristic time (Equation 10) originate from the difference between the volume and the surface in a spherical geometry.

The two different expressions of the expansion time underscore the time-gap between the two modelling. To conclude, the comparison of the fuel vapour expansion results between MOREINa and DETONa allows to validate the order of magnitude of the model and the gap obtained between the two geometries were expected.

CONCLUSION

In the scope of the safety standards of the GEN IV, a modelling of the vaporisation and the expansion of the formed vapour have been developped and implemented in a fast-running tool. The preliminary results obtained from a transient study of this first part of the expansion phase modelling have been presented. Two differents results have been analyzed, one without taking account instabilities and another one taking account this phenomenon. Finally a comparison with a validated tool, DETONa have also been conducted for the fuel vaporisation and expansion modelling. Further developments will focus on the sodium interaction phenomenon after the expansion phase to evaluate the total loadings applied on the second barrier (i.e the vessel).

Figure 4 .

 4 Figure 4. Relevant values during the MOREINa's transient

Figure 5 .

 5 Figure 5. Relevant values during the MOREINa's transient with instabilities

Figure 6 .

 6 Figure 6. Modeled geometry in DETONa[START_REF] Manchon | Modeling and analysis of molten fuel vaporization and expansion for a sodium fast reactor severe accident[END_REF]

  evolution and the acceleration is approximated by R/δt 2 exp . bubble -P Ar ) = 0.048s(10)By applying the same equation to MOREINa's system, the expansion time expression is obtained.

Table I .

 I Calculation initial parameters of the reference case

	Parameters	Unit	Value
	Core internal structures height	m	1.5
	Core internal structures radius	m	1.7
	Total pool mass	t	40000
	Steel mass fraction in the pool	-	0.3
	Argon pressure	Pa	1.12 × 10 5
	Argon volume	m 3	6.1
	Fission gases pressure	P a 33.44 × 10 5
	Fission gases volume	m 3 5.15 × 10 -3

ACKNOWLEDGMENTS

The autors would like to thanks CEA's Generation IV SFRAG project, which suports this study.

Table II. Calculation parameters of the reference case

The energies involved in the two transient have the same order of magniture (figure 7). The expansion energy and the compression one calculated at the end of MOREINa's transient admit a 7 % deviation compared with DETONa final energies. This gap, also found on the vaporised fuel mass (7%) and the temperature (2%) due to the geometry difference between the two models. In the spherical geometry, the pressure is applied on a section evolving with the bubble expansion during the transient. On the contrary, in the cylindrical geometry the pressure section is constant. This geometrical feature impacts the momentum equation and the pool vaporisation.

Furthermore, the two transients have an important temporal gap : MOREINa's transient is around 0.07 s while DETONa's transient is around 0.042 s. This gap can be explained by the expansion time.

To obtain the expansion time δt 2 exp , the momentum equation is applied to DETONa's spherical volume. The sodium radius evolution is assumed to be enough small to be neglected face to the bubble radius Vapour bubble

Vapour and liquid ∆h l,v [J.kg -1 ] Phase change specific enthalpy exch[-]

Exchange

Area Radius in spherical geometry


Juan Esteban Luna

Tony Glantz
email: tony.glantz@irsn.fr

Arthur V S Oliveira
email: arthur.oliveira@univ-lorraine.fr

Alexandre Labergue
email: alexandre.labergue@univ-lorraine.fr

M Gradeck
email: michel.gradeck@univ-lorraine.fr

Simulation of dispersed flow film boiling in LOCA conditions considering different fuel rod blockage ratios

Keywords: LOCA, Nuclear reactor, Clad ballooning, Thermal-hydraulics, Mechanistic model, NECTAR

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

After a LOCA, the safety systems inject water into the nuclear core plenum with the objective to prevent nuclear rods from reaching safety limit temperatures and thus mitigate the accident consequences. During this phase called "re-flooding", the injected water comes into contact with the nuclear rods at high temperatures producing a complex flow of steam and dispersed droplets downstream the quenching front. At this phase, this dispersed flow serves as a coolant for the nuclear rod section downstream the quenching front. At the same time, some nuclear rods can undergo large deformations, leading to a decrease in the fluid passage area of some sub-channels. As a result, the effectiveness of security systems can be compromised. Furthermore, these blocked sub-channels cause the steam to deviate to the less blocked sub-channels [START_REF] Oliveira | Parametric effects on the flow redistribution in ballooned bundles evaluated by magnetic resonance velocimetry[END_REF][START_REF] Oliveira | Velocity field and flow redistribution in a ballooned 7×7 fuel bundle measured by magnetic resonance velocimetry[END_REF]. Consequently, the cooling of blocked sub-channels can be compromised because forced convection with steam is supposed to be the dominant phenomenon in the cooling of the assembly [START_REF] Oliveira | Mechanistic modeling of the thermal-hydraulics in polydispersed flow film boiling in LOCA conditions[END_REF]. On the other hand, Ruyer et al. [START_REF] Ruyer | Two-phase flow across a partially damaged core during the reflood phase of a loca[END_REF] found that droplets show an inertial behavior and do not deviate with the steam. Subsequently, the droplets could improve the cooling of these blocked sub-channels. Thus, understanding this phenomenon is a major issue for nuclear safety design. However, the physical characterization of this flow involves complex and intricate phenomena such as droplets fragmentation and coalescence, turbulence, droplets interactions or impacts onto the wall, and also thermodynamic non-equilibrium between phases. Figure 1 represents the main thermal-hydraulic phenomena downstream of the quenching front. Additionally, Table 1 shows the range of some thermal-hydraulic parameters considered in this two-phase flow [START_REF] Carrillo | Experimental thermal hydraulics study of the blockage ratio effect during the cooling of a vertical tube with an internal steamdroplets flow[END_REF][START_REF] Jin | Investigation of the thermal-hydraulic non-equilibrium in a 7x7 rod bundle during reflood[END_REF][START_REF] Jin | Experimental study of droplet sizes across a spacer grid location under various reflood conditions[END_REF]. Furthermore, previous studies indicate that the liquid droplet size distribution in this two-phase flow can be adequately represented by a log-normal distribution [START_REF] Carrillo | Experimental thermal hydraulics study of the blockage ratio effect during the cooling of a vertical tube with an internal steamdroplets flow[END_REF][START_REF] Jin | Investigation of the thermal-hydraulic non-equilibrium in a 7x7 rod bundle during reflood[END_REF][START_REF] Jin | Experimental study of droplet sizes across a spacer grid location under various reflood conditions[END_REF][START_REF] Jin | Development of a droplet breakup model for dry spacer grid in the dispersed flow film boiling regime during reflood transients[END_REF]. Steam Temperature (T s )

< 800 • C Wall Temperature (T w ) 300 • C -1200 • C
For this reason, and intending to support safety analysis of pressurized water reactors, the IRSN has developed a code, name DRACCAR * , capable of describing the fuel assemblies' behavior during a LOCA. To do so, DRACCAR calculates the thermal-hydraulics in all the reactor circuits and the thermo-mechanical be-havior of fuel assemblies [START_REF] Glantz | DRACCAR: A multi-physics code for computational analysis of multi-rod ballooning, coolability and fuel relocation during LOCA transients Part one: General modeling description[END_REF]. Moreover, the IRSN has launched several experimental campaigns that serve to validate DRACCAR code. In the thermal-hydraulics axis, the main experiment (named COAL † [START_REF] Repetto | Core coolability in Loss Of Coolant Accident: the COAL experiments investigating the thermal hydraulics of a rod bundle with blocked area during the reflooding[END_REF][START_REF] Repetto | Core coolability in loss of coolant accident: the COAL experiments[END_REF]) consists of a 7 x 7 bundle of electrically heated rods at a full-length scale, with a blocked region by the presence of clad ballooning in the bundle. In this way, COAL experiments give useful information about the thermal-hydraulics phenomena during the re-flooding phase in a bundle scale. However, information on the heat transfer between the rods and the two-phase flow at the sub-channel scale is not well characterized. Consequently, another experiment, named COLIBRI ‡ , was developed to accurately characterize the thermal-hydraulic phenomena downstream of the re-wetting front at a sub-channel scale. The COLIBRI experimental bench consists of a test section of a single heated tube and a supply system of droplets and super-heated steam [START_REF] Carrillo | Experimental thermal hydraulics study of the blockage ratio effect during the cooling of a vertical tube with an internal steamdroplets flow[END_REF].

Additionally to COLIBRI bench, a mechanistic model in a numerical code was also developed, named NECTAR § , in order to calculate the heat and mass transfers involved between the polydisperse flow and the high-temperature nuclear rods at the scale of a subchannel. Additionally, NECTAR calculates as well the droplets dynamics. One of the characteristics of this 1D code is that it considers the droplets' distribution, the droplets' dynamics in the steam core, and the droplets' fragmentation. NECTAR has been first validated by experimental measurements carried out on the COLIBRI experimental bench [START_REF] Carrillo | Experimental thermal hydraulics study of the blockage ratio effect during the cooling of a vertical tube with an internal steamdroplets flow[END_REF], for three different geometries representing the cladding swelling at the sub-channel scale and different thermal-hydraulic conditions [START_REF] Oliveira | Mechanistic modeling of the thermal-hydraulics in polydispersed flow film boiling in LOCA conditions[END_REF].

In addition, the IRSN carried out measurements of the redistribution flow rate in a 7 × 7 assembly with 16 ballooned rods for different geometric conditions (blockage ratio, blockage length, and coplanarity of the deformation) as well as for different flow rates and without heat transfer between the flow and the wall.

The results showed that the blockage ratio plays a major role in the flow redistribution and as a result, the amount of deviated flow is proportional to the blockage rate. This blockage ratio (τ b ), defined in Eq. 1, is a parameter used to characterize the reduction in the cross-section due to the deformation of the cladding in a LOCA; where S b and S 0 are, the cross-sectional area of a blocked and an intact sub-channel respectively.

τ b = 1 - S b S 0 , τ b(90%) = 0.9, τ b(61%) = 0.61 (1)
The experimental results obtained with COLIBRI have shown an improvement in heat transfer in the case of a blocked sub-channel, owing to the steam acceleration (i.e. not deviated) in the blocked sub-channels. As the steam mass flow rate entering the test section was kept equal in all cases, the steam velocity, therefore, increased due to a Venturi effect. For this reason, the COLIBRI bench has been modified to account for the effect of flow redistribution within a more realistic assembly. With the aim of estimate the thermal-hydraulic behavior of these new experiments, we present a simulation of the dispersed flow film boiling by considering this steam flow deviation using NECTAR for two blockage ratios (τ b = 61% and 90%) and lengths (100 mm and 300 mm) to be used in COLIBRI bench.

NECTAR CODE

The code is based on the heat balance between the nuclear rod (hot wall to be cooled) and the internal flow of superheated steam and dispersed droplets. NECTAR uses an 1D axial model of steam and droplets in thermal and dynamic non-equilibrium. The droplets are considered to be spherical and at saturation. The dissipated heat flux at the wall due to the internal flow (Φ int ) is therefore the sum of four different contributions:

Φ int = Φ c,ws + Φ r,ws + Φ imp,wd + Φ r.wd (2)
For the calculation of the different heat transfer mechanisms NECTAR uses the following correlations:

• Gnielinski correlation [START_REF] Gnielinski | New equations for heat and mass transfer in the turbulent flow in pipes and channels[END_REF], for the forced wall/steam convection (Φ c,ws ) :

N u s = f /8 (Re s -1000) P r 1 + 12.7(f /8) 1/2 P r 2/3 -1 (3)
• For the steam/droplets convection (Φ c,sd ), the Ranz and Marshall correlation [START_REF] Ranz | Evaporation from drops[END_REF] with the modification of Yuen and Chen [START_REF] Yuen | Heat-transfer measurements of evaporating liquid droplets[END_REF]:

N u sd (1 + (h s -h d ) /h f g ) = 2 + 0.6Re 1/2 M P r 1/3 f (4)
• Impacts of droplets onto the wall (Φ imp,wd ). The code uses the data (for the estimation of h i ) and the model from Gradeck et al. [START_REF] Gradeck | Heat transfer for Leidenfrost drops bouncing onto a hot surface[END_REF] for the calculation of the energy taken by a single drop:

E d = ts 0 h i (T w -T s ) + ε w σ SB T 4 w -T 4 d πd (t) 2 4 dt (5)
Where h i , ε w , d(t) and σ SB represent the transfer coefficient by the impact of a drop, the emissivity of the wall, the droplet diameter, and the Stefan Boltzmann's constant respectively. The Biance et al. model [START_REF] Biance | On the elasticity of an inertial liquid shock[END_REF] was used to calculate the droplet diameter during impact (d(t)). Additionally, the total heat flux extracted by droplets impacts q imp,wd can be estimated as:

q imp,wd = 6 ṁ πρ d d 3 E d ( 6
)
Where ṁ is the mass flux density of droplets impinging the wall. Moreover, NECTAR code estimates this mass flux with Hewitt et Govan [START_REF] Hewitt | Phenomenological modelling of non-equilibrium flows with phase change[END_REF] correlation for droplet deposition rate, with the correlation used by Guo et Mishima [START_REF] Guo | A non-equilibrium mechanistic heat transfer model for post-dryout dispersed flow regime[END_REF] to estimate the concentration of the droplets in the steam core:

ṁ = 0.18 ρ s D h /σ t αρ d if αρ d /ρ s < 3, 0.083(αρ d /ρ s ) -0.65 ρ s D h /σ t αρ d if αρ d /ρ s > 3 (7)
• For the radiation heat transfer calculation between the steam, the droplets and the wall, the code uses the thermal resistance model of Sun et al. [START_REF] Sun | Calculation of Combined Radiation and Convection Heat Transfer in Rod Bundles Under Emergency Cooling Conditions[END_REF]. The radiation heat flux between each phase can be expressed as:

q r,ij = F r,ij σ B (T 4 i -T 4 j ) (8)
Where F r,ij is the form factor, defined as

F r,ij = 1/ (R i + R j + R i R j /R k ) (9)
Where the indices i, j and k represent each phase (steam, droplets or the wall). Finally, the thermal resistance R of each phase is expressed as

R s = (1 -ε s ) / [ε s (1 -ε s ε d )] (10)
R d = (1 -ε d )/ [ε d (1 -ε s ε d ] (11)
R w = 1/(1 -ε s ε d ) + (1 -ε w )/ε w (12)
Table 1 summarizes the conservation equations used in NECTAR. Furthermore, the volumetric evolution of the dispersed phase (i.e. the droplets) is determined according to the moment density equations [START_REF] Carrillo | Experimental thermal hydraulics study of the blockage ratio effect during the cooling of a vertical tube with an internal steamdroplets flow[END_REF], [START_REF] Repetto | Core coolability in loss of coolant accident: the COAL experiments[END_REF] described along the vertical z-axis. The model does not consider the coalescence of the droplets; however, fragmentation of the droplets is possible. For this, the Chou and Faeth model [START_REF] Chou | Temporal properties of secondary drop breakup in the bag breakup regime[END_REF] is used to calculate the new distribution after fragmentation. Moreover, the wall is considered to be thermally "thin" (Bi ≪ 1) and the simulations are only valid in the Leidenfrost regime.

Table II. Conservation equations of the NECTAR code [START_REF] Oliveira | Mechanistic modeling of the thermal-hydraulics in polydispersed flow film boiling in LOCA conditions[END_REF].

Equations

Mass conservation (steam)

dms dt = ṁev Momentum balance (droplet) ρ d u d du d dz = 3 4d v.C D 1+B ρ s (u d -u s ) |u d -u s | + (ρ s -ρ d ) g C D = 24 Re(1+B) , B = Cps(Ts-Tsat) h f g+(Q L / ṁd ) , v = 1 1-6.55α d

Energy balance

Cp s T s (z + dz) ṁs (z) + d ṁs(z) dt -ṁs (z)T s (z) = Φ ev Φ ev = Φ imp,wd + Φ r,wd + Φ sd + Φ r,sw

SIMULATION DOMAIN AND BOUNDARY CONDITIONS

3 shows the simulation domain and the boundary conditions of the NECTAR simulation. The code uses the same geometry of the COLIBRI experimental bench, described by Peña Carrillo et al. [START_REF] Carrillo | Experimental thermal hydraulics study of the blockage ratio effect during the cooling of a vertical tube with an internal steamdroplets flow[END_REF], which represents a blocked fluid sub-channel as a circular section venturi. As explained by Peña Carrillo et al. [START_REF] Carrillo | Experimental thermal hydraulics study of the blockage ratio effect during the cooling of a vertical tube with an internal steamdroplets flow[END_REF], this choice of geometry allows to accurately assess the heat transfers paths in this mist flow at the subchannel scale, providing valuable information for DRACCAR's code validation. Oliveira et al. [START_REF] Oliveira | Parametric effects on the flow redistribution in ballooned bundles evaluated by magnetic resonance velocimetry[END_REF][START_REF] Oliveira | Velocity field and flow redistribution in a ballooned 7×7 fuel bundle measured by magnetic resonance velocimetry[END_REF] found that the blockage ratio plays a major role on the flow redistribution in a bundle with blocked sub-channels. Additionally, they found that the amount of deviated flow is proportional to the blockage ratio. Therefore, for the present work, the amount of deviated steam is taken as proportional to the blockage ratio (according to Eq. 1).

ṁe = (1 -τ b ) ṁ0 (13)
According to the results of Oliveira et al. [START_REF] Oliveira | Parametric effects on the flow redistribution in ballooned bundles evaluated by magnetic resonance velocimetry[END_REF][START_REF] Oliveira | Velocity field and flow redistribution in a ballooned 7×7 fuel bundle measured by magnetic resonance velocimetry[END_REF] and Ruyer et al. [START_REF] Ruyer | Two-phase flow across a partially damaged core during the reflood phase of a loca[END_REF], the steam deviation is carried out up to the transition zone between the intact and blocked section, however, in the COLIBRI experimental bench, the deviation is carried out in a single point immediately before the transition zone (out of NECTAR simulation domain). Consequently, the steam mass flow entering the COLIBRI test section has been already deviated (reduced). For this reason, before the thermal calculation, the NECTAR code determines the steam mass flow rate according to Eq. 13.

Finally, in order to simulate the new experimental conditions of the COLIBRI experimental bench, the following input conditions have been set : [START_REF] Oliveira | Parametric effects on the flow redistribution in ballooned bundles evaluated by magnetic resonance velocimetry[END_REF] wall temperatures between 450°C and 800°C, (2) initial mass flow rate of steam ( ṁ0 ) of 4 kg/h, at 1 bar (absolute pressure) and at 400°C of temperature and (3) droplets mass flow rate of 20 kg/h, which corresponds to a volume fraction of 5.10 -3 . The initial droplets velocity is 10 m/s. The droplets distribution is log-normal, as shown in Fig. 4. The average diameter d 10 is 250 µm, for a maximum diameter of 700 µm and a minimum diameter of 5 µm. In this study, we chose two different blockage ratios with two different blockage lengths, named 61%/100mm, 61%/300mm, 90%/100mm, and 90%/300mm; with this notation, the first value (before the slash) of the notation corresponds to the simulated blockage ratio and the second one (after the slash) correspond to the simulated blocking length. Finally, the droplets' distribution used is typical in LOCA case. Nevertheless, this parameter depends on other factors, such as the mass flow of steam. Therefore, it is not possible to exactly fix a droplet distribution for the COLIBRI experimental bench. However, although there is an increase in the heat flux removed from the wall with an increase in the droplets' diameter, this is not significant (as is the increase in the mass flow of droplets or steam) [START_REF] Oliveira | Etude du refroidissement d'une zone représentative d'un assemblage de coeur de REP / Study of the cooling of a PWR's core assembly representative zone[END_REF].

RESULTS

We present herein the results of internal heat flux per unit of blocked length for the chosen blockage ratios (τ b = 61% and τ b = 90%) and lengths (100 mm and 300 mm) with and without flux deviation (bypass).

As expected, there is a decrease in heat transfer when there is a steam deviation for both blockage lengths simulated (Fig. 5 and Fig. 6). On the other hand, there is a small difference in the heat flux released by the wall between the two blockage ratios when the steam deviation is simulated. For the cases with a more severe blockage length (300 mm), this difference is even smaller (Fig. 6). This is due to an slight increase in heat transfer by droplets impacts (Φ imp,wd ), in the most blockage case.

Table III summarizes the heat flux per unitary blockage length for each mechanism of heat transfer. In general, the results show a slight increase in the internal heat flux (Φ int ) per blocked length for the cases of 300 mm of blockage length. This is due to an improvement in convective heat transfer between the steam and the wall (Φ c,ws ). Also, a decrease in droplets impacts heat transfer (c) is evidenced while increasing blockage length. On the other hand, when comparing the results with and without steam deviation, a reduction in heat flow is highlighted. However, this reduction is cushioned by the increase in heat transfer due to droplets impacts (Φ imp,wd ) and by the decrease in the steam temperature that favors wall/steam convection (Φ c,ws ) (Fig. 7). This means that the decrease in heat transfer between the steam and the wall is damped because even though the heat transfer coefficient was reduced (by reducing the mass flow), the temperature difference increased. Consequently, the decrease in heat is not proportional to the decrease in the steam mass flow. Furthermore, this reduction in steam temperature is due to convective heat transfer between the steam and the droplets (Φ c,sd ). A discontinuity in the steam temperature is evidenced in Fig. 7 for the cases without steam deviation. This is due to the droplets' fragmentation in the blocked zone. In this way, the importance of droplets is seen in the wall cooling. On the other hand, the influence of the droplets can also be evidenced on the heat transfer with the wall, when even for the cases without blockage (τ b = 0%), the droplets' impact heat transfer mechanism (Φ imp,wd ) is the most important for both blockage length ratio simulated. Furthermore, this same behavior is reflected for blocked sub-channels when there is steam deviation, where (Φ imp,wd ) represents more than half of the total heat flow contribution. These results will be compared with experimental results that will be obtained soon in the new campaign with the experimental bench COLIBRI.

CONCLUSIONS

The simulations carried out with NECTAR code allows the analysis of the influence of steam deviation on the heat transfer which take place in a representative tube of a PWR sub-channel cooled by a steam flow with dispersed droplets during a LOCA. This analysis yielded the following results:

• The steam-to-wall convection (Φ c,ws ) was reduced with the reduction of the mass flow. However, the heat transfer between the droplets and the steam (Φ c,sd ) caused the steam temperature to be lower than in the non-deviation cases. Consequently, the reduction in heat exchange between the wall and the steam was not proportional to the steam mass flow reduction.

• Increasing the blockage length slightly improves heat transfer in the presence of steam deviation. However, in the simulations carried out the blockage length is not an influencing factor in the subchannels cooling.

• The droplets have an important role in the sub-channel cooling for the droplets fraction volume studied. This is reflected in the importance of heat transfer by droplets impact onto the wall (Φ imp,wd ) for the cases with steam deviation and the cases without blockage. Additionally, the droplets decrease the steam temperature, thus improving the heat transfer with the wall.

These numerical results obtained with NECTAR should be validated experimentally with the new COLIBRI experimental campaign. In addition, given the great influence of droplets on cooling, it is necessary to take into account for the possible effect of droplets' interactions on the different heat transfers.

Figure 1 .

 1 Figure 1. Heat and mass transfer phenomena in LOCA. [3].

Figure. 2

 2 shows the different transfers: forced wall/steam convection (Φ c,ws ), wall/steam radiation (Φ r,ws ), steam/droplets convection (Φ c,sd ), steam/droplets radiation (Φ r,sd ), wall/droplets radiation (Φ r,wd ), wall/droplets impacts (Φ i,wd ) and evaporation of droplets (m ev ).

Figure 2 .

 2 Figure 2. Heat and mass transfer paths considered in NECTAR [3].

Figure 3 .

 3 Figure 3. Simulation Domain of NECTAR simulation.

Figure 4 .

 4 Figure 4. Log-normal droplets distribution (µ=0.45, σ= ln(d 10 ), d 10 =250µm).

Figure 5 .

 5 Figure 5. Internal heat flow as a function of the average wall temperature for the cases with a 100mm blockage length.

Figure 6 .

 6 Figure 6. Internal heat flow as a function of the average wall temperature for the cases with a 300mm blockage length.

Figure 7 .

 7 Figure 7. Steam temperature in the blocked zone for 61%/100mm and 90%/100mm (Tw=607°C).

Table III .

 III Internal heat flux per blockage length for the studied cases (W/o BYP: without bypass and W/ BYP: with bypass).

				Internal heat flux per blockage length [kW/m]
	Heat	τ b = 0%		τ b = 61%			τ b = 90%
	Transfer 100mm 300mm	100mm	300mm	100mm	300mm
	Mech.	W/o	W/o	W/	W/o	W/	W/o	W/	W/o	W/	W/o
		BYP	BYP	BYP	BYP	BYP	BYP	BYP	BYP	BYP	BYP
	Φ imp,wd	1.25	1.21	1.4	1,01	1,35	0,88	1,63	0,26	1,32	0,205
		49%	46,5% 56,2% 34,8% 52,1% 31,3% 60.83% 5,6% 50,6% 4,3%
	Φ c,ws	1.02	1.14	0.85	1.64	1.02	1.74	0.84	3.97	1.09	4.4
		40%	43.4% 33.9% 56.6% 39.2% 61.6% 31.2% 87.4% 41.9% 94.5%
	Φ r,wd	0,19	0,17	0,2	0,20	0,18	0,15	0,19	0,3	0,17	0,187
		7.57%	6.71% 8.05% 7.04% 6.83% 5.46% 7.22% 6.55% 6.67% 3.9%
	Φ r,ws	0,09	0,09	0,05	0,05	0,05	0,05	0,02	0,02	0,02	0,02
		3.38%	3.42% 1.91% 1.54% 1.91% 1.59% 0.72% 0.37% 0.74% 0.35%
	Φ int	2,56	2,61	2,5	2,89	2,60	2,83	2,68	4,53	2,6	4,81

† COAL: COolability of a fuel Assembly during LOCA ‡ COLIBRI: COoLIng of Blockage Region Inside a PWR Reactor. § NECTAR: New Experimental Code for Thermal-hydraulic Analysis in a Representative geometry.

ACKNOWLEDGMENTS

This work is carried out as part of a PhD thesis funded by the Institut de Radioprotection et de Sûreté Nucléaire (IRSN) in Cadarache, France.

NOMENCLATURE

Latin letters


B Jourdy

D Guenadou
email: david.guenadou@cea.fr

N Seiler

A Labergue

M Gradeck
email: michel.gradeck@univ-lorraine.fr

SCALE EFFECTS METHODOLOGY APPLIED TO UPRISING JETS IN ASTRID

Keywords: SFR, Scale effects, Impinging jets, Buoyant jets

The CEA is involved in the development of the 4 th generation of nuclear reactors, among which are the Sodium Cooled Fast Reactors (SFR). To support their design and safety, specific codes were developed and should be validated using experimental results from relevant mock-ups. Due to the complexity of building a full-sized prototype in the nuclear field, most of the experiments are performed on reduced-sized models but it may lead to scale effects. Such scale effect is under study in this paper, focusing on a critical issue in SFR reactor, which is the rising of the jet outgoing the core at low power. On this issue, a scale effects methodology is detailed, using the SFR's reduced scale mock-up MICAS as the reference scale. To achieve that, dimensionless Navier-Stokes equations under Boussinesq's approximation are considered and the Vaschy-Buckingham theorem is applied to determine the most relevant dimensionless numbers: the densimetric Froude number. Experimental campaigns have been performed to measure velocity fields. Their analysis clearly shown dependency of mean pathway of the jet on this dimensionless number. Particularly, plotting the jet angle as a function of the normalized densimetric Froude number, a change of behaviour for a threshold value of 0.45 has been observed. This result also shows negligible effects on the jet rise of the phenomena happening before jet impingement on the Upper Core Structure (UCS).

INTRODUCTION

In order to close the nuclear fuel cycle and use the wide spread uranium 238 isotope instead of the uranium 235 currently being used in Gen II and Gen III nuclear reactors, CEA is involved in the development of Gen IV sodium-cooled fast neutron reactors (SFR). As building a scale 1 experiment, for characterizing the reactor prototype behaviour, is both complex and expensive, most experiments are performed on smallscale mock-ups to assess new design options and validate associated simulation software. Due to sodium opacity and its reactivity with water, sodium experiments are very complex to carry out, even at small scale.

That is why a majority of such experiments are performed with a simulating fluid which is water, owing to its close properties to the sodium ones.

In 2015, in order to support the French experimental reactor project called ASTRID [START_REF] Saez | Status of ASTRID nuclear island pre-conceptual design[END_REF], the 1:6th scale mock-up MICAS was commissioned to study the thermal-hydraulics behaviours inside the upper plenum of this one. This mock-up allows investigations of different thermo-hydraulics issues as the gas entrainment at free surface [START_REF] Guenadou | Analysis of the Gas Entrainment by Vortex in the Upper Plenum of the ASTRID Reactor Mock-up[END_REF] or the behaviour of the jets coming out from the core [START_REF] Guenadou | Flow analysis in the upper plenum of the micas model in support of the astrid reactor program[END_REF]. For this specific issue, these jets can be subjected to thermo-hydraulics oscillations at low power operation. And, after impinging the Upper Core Structure (UCS), they may raise due to buoyancy forces. This leads to temperature oscillating variations of some submerged vessel components and thermal fatigue, which impacts on the lifetime of the reactor.

This study focuses on the thermal-hydraulic behaviour of the hot impinging jets out of the core, especially on the transition from inertia-dominated to buoyancy-dominated jets. The first part briefly introduces both ASTRID prototype and MICAS mock-up, and provides details about the phenomenon under investigation. A theoretical study is also provided. Then, a description of the experiments conducted on MICAS and their experimental results are discussed. These results will be used as a reference for a later scale effect analysis by comparison with results at lower scales.

THE ASTRID PROTOTYPE AND MICAS MOCK-UP

General Overview

From 2011 to 2019, the CEA has been involved in the development of the 4 th generation ASTRID (Advanced Sodium Technological Reactor for Industrial Demonstration) prototype [START_REF] Tenchine | Some thermal hydraulic challenges in sodium cooled fast reactors[END_REF]. This project aimed at demonstrating the technical options chosen and at giving an idea of the operating costs. Its power was 600MWe. However, even if this project has stopped in 2019, the CEA still carries on studies on sodium reactors and technologies. Especially, efforts head to build a numerical reactor, implying a strong validation of numerical codes.

The MICAS facility is a 1:6 th sized mock-up representing the hot plenum of the ASTRID recator, based in the CEA Cadarache [START_REF] Guenadou | Plateau Facility in Support to ASTRID and the SFR Program: an Overview of the First Mock-up of the ASTRID Upper Plenum, MICAS[END_REF]. The scale was chosen to be a compromise between the overall size and the detail of the geometry of the vessel as MICAS geometry is homothetic to ASTRID's one, but some geometrical simplifications were necessary. Its dimensions are about 2.5 m in diameter and 1.7 m in height. Cut views of both ASTRID and MICAS are shown in Figure 1.

Core catcher

Primary pumps

Phenomenon of Interest and Problematic

Figure 2 represents a simplified schema of the flow and main components above the core. Upstream of the core, 288 hot jets are first impinging and going through a porous plate, creating loss of pressure. This porous plate deflects a small part of the flow horizontally. A part of the water (around 15% [START_REF] Guenadou | Flow analysis in the upper plenum of the micas model in support of the astrid reactor program[END_REF]) flows through the Upper Core Structure (UCS) and exit around the barrels. The bottom plate of the UCS deflects the entire part of the flow after impingement, resulting in a unique almost radial jet spreading into the upper plenum of vessel where the fluid temperature is lower than the jet's one.

At low power operating conditions, the inertia of this jet decreases and buoyancy effects become predominant, causing the jet to rise. If so, the flow pattern in the vessel is modified and leads to thermal fluctuations, causing thermal fatigue of the components of the vessel. As this phenomenon is crucial for the life expectancy of the reactor, a deep investigation is necessary.

Experimental results on MICAS are used to validate numerical codes to simulate ASTRID's thermalhydraulics behavior. However, as we are using a downscaled mock-up to validate codes, we need to ensure that the experimental conditions are representative of the reactor ones and that the phenomenon under investigation (i.e the raise of the impinging jets) can be transposed from a scale to another without distortions. So the main point of this study is to find the most relevant dimensionless numbers to ensure the similarity of the flow, and study the dependence of the rise of the jet on these dimensionless numbers. A parameter of interest for the later scale effect's study is the critical value of this dimensionless numbers, for which the jet rises.

THEORETICAL STUDY

Similarity of the Flow

In order to be representative of the ASTRID's flow behavior in MICAS, the Vaschy-Buckingham theorem [START_REF] Debongnie | Sur le Théorème de Vaschy-Buckingham[END_REF] has been applied to MICAS as shown in the Table 1. This theorem allows to select dimensionless numbers whose conservation ensures the similarity of the flow between two different scales. The nomenclature is given at the end of the document. This theorem also gives geometrical dimensionless parameters such as 𝜋 1 or 𝜋 9 : these parameters do not interfere from ASTRID to MICAS as the transformation is linear from a scale to another, but can be important for the design of mock-up that need geometrical distortion.

𝜋 1 = 𝐻 𝐿 𝜋 2 = 𝑝 𝜌. 𝑢 2 = 𝐸𝑢 𝜋 3 = 𝜂 𝜌. 𝑢. 𝐿 = 1 𝑅𝑒 𝜋 4 = 𝑔. 𝐿 𝑢 2 = 1 𝐹𝑟 2 𝜋 5 = 𝑇 -𝑇 ∞ 𝑇 𝑐 -𝑇 ∞ 𝜋 6 = 𝛽(𝑇 𝑐 -𝑇 ∞ ) 𝜋 7 = 𝛼 𝑢. 𝐿 = 1 𝑃𝑒 𝜋 8 = 𝜌 ∞ -𝜌 𝜌 ∞ 𝜋 9 = 𝑠 𝐿
However, as the conservation of every 𝜋 groups is not possible from a scale to another, we use the dimensionless governing equations known as the conservation equations in order to determine the most relevant dimensionless numbers to ensure similarity from a scale to another and calculate our experimental conditions. The study of the MICAS mock-up is divided in two different regions: first, the under-UCS zone where 288 hot jets exit the core. Then the vessel zone, where a unique jet (from the merge of the 288 core jets) is diffusing into the vessel.

Just above the core, we assume axisymmetric incompressible steady jets in cylindrical coordinates where the velocity vector 𝑢 in the (𝑟, 𝑧) coordinate system is defined by 𝑈, its axial component, and V its radial component. The notation 𝑋 * means the dimensionless variable 𝑋/𝑋 0 , 𝑋 0 being the characteristic parameter associated to the variable 𝑋. As (𝜌 ∞ -𝜌)/ 𝜌 ∞ ~0.2%, we assume the Boussinesq approximation to be valid. The scaling parameters are chosen as 𝑈 0 , the average axial velocity at the jet exit, 𝐿 0 the half-width of the jet, 𝑇 𝑐 the jet's temperature out of the core and 𝜌 0 the jet's density at its exit out of the core. This leads to the following dimensionless conservation equations:

Mass conservation:

𝜕𝑈 * 𝜕𝑧 * + 1 𝑟 * 𝜕 𝜕𝑟 * (𝑟 * 𝑉 * ) = 0 (1)
Energy conservation: Between the porous plate and the UCS, the overall fluid temperature is homogeneous at the hot jets' temperature, around 55°C. The dimensionless momentum equation for a jet in this zone can be reduced to its axial one, as 𝑉 ≪ 𝑈, and is given by equation ( 3):

𝑈 * 𝜕𝑈 * 𝜕𝑧 * + 𝑉 * 𝜕𝑈 * 𝜕𝑟 * = - 1 𝜌 * 𝜕𝐸𝑢 𝜕𝑧 * + 1 𝑅𝑒 . 1 𝑟 * . 𝜕 𝜕𝑟 * { 𝑟 * 𝜕𝑈 * 𝜕𝑟 * } (3)
In this zone, the similarity is ensured by the conservation of the dimensionless numbers from equation ( 2) and ( 3), and the conservation of the interaction between jets. Important parameter to ensure that the dynamic interaction between the 288 jets is preserved, and thus we can consider only one single jet, are the dimensionless numbers 𝜋 1 and 𝜋 9 (see Table 1) with 𝐻 the nozzle-to-plate distance, 𝑑 the jet diameter and 𝑠 the inter-jet distance [START_REF] Lai | Dynamic Interaction of Multiple Buoyant Jets[END_REF]. As MICAS is a linear downscaled mock-up from ASTRID, these ratios are automatically preserved. Thus, the main dimensionless numbers of interest in this zone are the Euler number and the Reynolds number. As this zone has a constant temperature, the Prandtl number is not important under the UCS.

After impingement of the UCS, the jets are merging and a single jet is spreading in the vessel. This unique jet can be studied as a planar jet in the (𝑥 0 , 𝑦 0 , 𝑧 0 ) Cartesian system of coordinates (see Figure 2): as the axisymmetric hypothesis is still valid for planar jets, the 𝑦 0 component can be neglected.

In the vessel, the jet is not on the 𝑥 0 axis but bend with an angle 𝜃 0 that we define as the angle between the horizontal axis 𝑥 0 and the jet axis 𝑥 (see Figure 2). This angle, by convention, is positive if the jet is going downward (nominal conditions) and negative if the jet is going upward. This definition implies the relation given equation ( 4):

{ 𝑥 ሬ ሬԦ = cos(𝜃 0 ) 𝑥 0 ሬሬሬሬԦ -sin(𝜃 0 ) 𝑧 0 ሬሬሬሬԦ 𝑧 ሬԦ = cos(𝜃 0 ) 𝑧 0 ሬሬሬሬԦ + sin(𝜃 0 ) 𝑥 0 ሬሬሬሬԦ (4)
By keeping 𝑈 as the axial velocity (on the 𝑥 axis) and 𝑉 the velocity component perpendicular to the jet's axis (on the 𝑧 component), the dimensionless momentum equations on these axis become (equation ( 5)):

{ 𝑈 * 𝜕𝑈 * 𝜕𝑥 * + 𝑉 * 𝜕𝑈 * 𝜕𝑧 * = 1 𝑅𝑒 𝜕 𝜕𝑧 * { 𝜕𝑈 * 𝜕𝑧 * } -( 1 𝐹𝑟 2 - 1 𝐹𝑟 𝐷 2 ) sin(𝜃 0 ) 𝑈 * 𝜕𝑉 * 𝜕𝑥 * + 𝑉 * 𝜕𝑉 * 𝜕𝑧 * = 1 𝑅𝑒 𝜕 𝜕𝑧 * { 𝜕𝑉 * 𝜕𝑧 * } -( 1 𝐹𝑟 2 - 1 𝐹𝑟 𝐷 2 ) cos(𝜃 𝑂 ) (5)
As the flow is fully turbulent in MICAS (𝑅𝑒 > 10 4 ), the Reynolds number may have a small influence on the jet's behavior as jets tends to be Reynolds invariant [START_REF] Heller | Self-Similarity and Reynolds Number Invariance in Froude Modelling[END_REF]. The main parameters to achieve similarity from ASTRID to MICAS (as long as the flow is fully turbulent) are the Euler's number for the jets under the UCS, and the densimetric Froude number for the jet spreading in the vessel.

This dimensionless number is defined by equation ( 6) where Δ𝜌 = 𝜌 ∞ -𝜌 is the density difference between the hot jets and the surrounding environment, 𝑢 is the velocity of the jet, 𝑔 the gravitational constant and 𝐿 the characteristic length scale of the jet.

𝐹𝑟 𝐷 = 𝑢 √(∆𝜌 𝜌 ∞ ⁄ ) 𝑔 𝐿 (6)
This dimensionless number is frequently used for buoyant jet studies [START_REF] Lai | Dynamic Interaction of Multiple Buoyant Jets[END_REF] [9] [START_REF] Papakonstantis | Inclined Negatively Buoyant Jets 1: Geometrical Characteristics[END_REF], giving credit to its choice as a similarity parameter.

Buoyancy in Jets

For fully turbulent submerged free jets under Boussinesq's approximation, the buoyancy problem only depends on the discharge angle 𝜃 0 , and the initial jets parameters such as the flow rate 𝑄 0 , buoyancy 𝐵 0 and momentum 𝑀 0 [START_REF] Papakonstantis | Inclined Negatively Buoyant Jets 1: Geometrical Characteristics[END_REF]. Here, two characteristics length scales can be defined as 𝑙 𝑀 = 𝑀 0 3/4 / 𝐵 0 1/2 and 𝑙 𝑄 = 𝑄 0 / 𝑀 0 1/2 . 𝑙 𝑀 represents the characteristic length from momentum-driven to buoyancy-driven jet, and 𝑙 𝑞 represents the characteristic length at which the jet's exit still influence the flow.

These two characteristics length scales are related to the densimetric Froude number (equation ( 7)) as

𝑙 𝑄 𝑙 𝑀 = 𝐾 0 𝐹𝑟 𝐷 ( 7
)
𝐾 0 is a constant that depends of the jet's nozzle. For a round jet, 𝐾 0 = (𝜋/4) 1/4 . For 𝐹𝑟 𝐷 ≫ 1, it has been demonstrated [START_REF] Papakonstantis | Inclined Negatively Buoyant Jets 1: Geometrical Characteristics[END_REF] that the geometrical parameters such as the distance from the nozzle exit to the onset of its rise 𝑋 𝑍 as shown in Figure 2 only depends on the characteristic length scale, the densimetric Froude number and the initial discharge angle (equation ( 8)) :

𝑋𝑧 𝐿 𝐹𝑟 𝐷 = 𝐾(𝜃 0 ) (8)
In our case, the characteristic length scale is the jet diameter. 𝐾(𝜃 0 ) is a constant that only depends on the initial discharge angle. This relation shows the dependency of the jet's behaviour on the densimetric Froude number. In this study, we characterize the rise of the jet by two parameters: the distance before its raise 𝑋𝑧 and the asymptotic angle 𝜃 𝑓 as presented in Figure 3 below. This study will only focus on the evolution of this asymptotic angle with the densimetric Froude number.

Experimental Conditions

The inlet conditions of MICAS taken to be representative of the ASTRID reactor are calculated based to the similitude in densimetric Froude number at the core exit. The similitude in this zone allows the control of all parameters (velocity and temperature) and a relation between the jet's behaviour and the inlet conditions can then be determined. The influence of the Euler number on the behaviour of the jet will be studied later. In order to observe the rise of the jet, we decrease the inlet velocity by decreasing the mass flow and we keep the temperature differences between the jets and the environment, leading to a decrease of the densimetric Froude number with the decrease of the mass flow. The flow rate is measured with Coriolis flowmeters (±0.1% of measurement range).

Most of the components in MICAS are made of optical grade transparent polymer to allow laser measurements. The maximal temperature conditions are 60°C and the minimal temperature which can be reached is around 10°C for the cold jets in the vessel. As we the repartition of the flow in MICAS as in ASTRID (95% of hot jets, 5% of cold ones), the operating conditions are given in Table 2. Hot jets have a temperature of 55°C and cold ones are at 10°C [START_REF] Guenadou | Plateau Facility in Support to ASTRID and the SFR Program: an Overview of the First Mock-up of the ASTRID Upper Plenum, MICAS[END_REF].

The Velocity Measurement Setup

The velocity fields are measured thanks to Particle Image Velocimetry (PIV) as shown Figure 4, allowing us to record 2D large velocity fields of the jets after impingement. In our setup, the laser is set on a 3-axis motion table to get a positioning as accurate as possible. Nylon particle with a density of 1000 kg.m -3 are mixed with the water. These particles with a 4µm diameter are enlightened by the laser sheet and scatter the light. Then, their motion is recorded by a 4 Mpixels CCD camera, perpendicular to the laser sheet. Its acquisition rate is 15 Hz and results are averaged over 150 images, i.e. 10s of acquisition. PIV results averaged on a higher number of images (over 300) are giving identical results, validating the stationary nature of the flow for a 10s acquisition.

Temperature Measurements

The inlet temperature for both hot and cold jets are measured with PT100 probes (±0.1°𝐶). We also put PT100 probes in the UCS to ensure that we have no thermal exchanges between the hot jets and its environment before their impingement on the UCS and spreading into the vessel.

In the vessel, we use PT100 probes to measure the ambient temperature. These probes are disposed in the middle of the vessel and next to the pump pit (green dots in the Figure 5), to measure a mixing temperature.

We also use an array of thermocouples (±1°𝐶) to have informations about the stratification and the establishment of the flow's temperature. These 15 thermocouples are disposed on a pole with spacing of 1cm.

EXPERIMENTAL RESULTS AND COMMENTS

Velocity Measurement Results

The time-averaged PIV results show the raise of the jet in the vessel (displayed in Figure 6). For typical cases. For high mass flow rate (i.e. momentum-driven flow), we notice an initial angle around 20° (picture 1 of the Figure 6). This angle seems to be constant when buoyancy becomes unimportant and only depends on the geometry of the UCS. When the mass flow rate decrease, we notice in picture 2 of the Figure 6 that the jet start to rise and so in pictures 3 and 4, with an increasing asymptotic angle.

Because of some difficulties to achieve thermal establishment at the nominal temperature (i.e 53°C for ambient temperature and 55°C for hot jets), some results have higher temperature differences leading to a bigger asymptotic angle even if the mass flow rate is higher, as we can see in pictures 3 and 4 on the Figure 6.

Temperature Results

As expected, the ambient temperature in the upper plenum of the vessel is subjected to thermal fluctuations. We start measurements when the flow is stabilized, i.e. when the temperature in the vessel is fluctuating around its mean value as shown Figure 7. When the flow is steady, we process to PIV measurements. The fluctuations around the mean value in the vessel are increasing with the mass flow. Maximum fluctuations observed go up to 0.4°𝐶 around the mean value for 𝑄 = 151 𝑚 3 . ℎ -1 (see the Figure 7). We define the asymptotic angle of the jet by the angle between horizontal (referred as 0°) and jet centreline at the point where the jet start to rise. The asymptotic angle is calculated by plotting the maximum velocity magnitudes in the jet (representing the jet centreline), and defining a window in which we make a linear regression on these maximal magnitudes. This window is user-defined, meaning that it can differ from a test to another: a wrongly-positioned window could lead to aberrant points and modify the calculated angle. Hence, the window dimensions only needs to catch the jet centreline.

Asymptotic angle results

In order to show the jet's behaviour dependency on the densimetric Froude number, we plot the evolution of the asymptotic angle with this dimensionless number. The densimetric Froude number is calculated in two different zones: in the entrance of the vessel after impingement of the UCS, and at the core's exit. In the vessel, equations ( 5) and ( 8) shown a dependency of the jet's behaviour only on the densimetric Froude number, result we want to confirm in our experiment. But, as we only control inlet conditions out of the core, we need to study the evolution of this angle with the densimetric Froude number calculated with our inlet conditions. In this zone, the jet's behaviour is supposed to be dependent on the Euler number as shown by equation ( 3), before diffusing in the vessel and be dependent on the densimetric Froude number.

First, we want to ensure that the behaviour of the free submerged jet in the vessel only depends on the densimetric Froude number. To do so, we calculate the experimental densimetric Froude number regarding the vessel features: the characteristic length is the planar jet's exit, calculated thanks to PIV results. The velocity is an output velocity, i.e. an average of the velocity vectors on the jet exit (equation ( 9)). The density differences are given thanks to the PT100 probes' measurements and tabulated values of density's evolution with temperature.

𝑢 𝑉𝑒𝑠𝑠𝑒𝑙 = 1 𝑛 𝑣𝑒𝑐 ∑ 𝑢 𝑖 ( 9
)
Figure 9 shows that the evolution of 𝜃 𝑓 seems to be linear with the 𝐹𝑟 𝐷 on a range from 0.5 to 4.7, beyond which the angle becomes constant (between 17° and 21°). After this threshold value of 𝐹𝑟 𝐷 = 4.7, buoyancy effects become negligible and the jet behave like a momentum-driven jet only. This saturation may depends on the UCS geometry, hypothesis that will be studied later. A regression on the linear zone gives a critical densimetric Froude number of 2.5, defined as the transition from positive to negative angle. This figure shows the dependence on the angle with the densimetric Froude number as expected by equations ( 5) and ( 8).

Figure 10 represents the evolution of the asymptotic angle 𝜃 𝑓 with the initial densimetric Froude number 𝐹𝑟 𝐷 using as parameters the inlet conditions at the exit of the core. This 𝐹𝑟 𝐷 is calculated with the same differences of density as previously (the jet's and ambient temperature being respectively the same in both cases), but the characteristic length is the diameter of the jets in the core, and the velocity is an averaged velocity of the jets given equation [START_REF] Papakonstantis | Inclined Negatively Buoyant Jets 1: Geometrical Characteristics[END_REF].

𝑢 𝐶𝑜𝑟𝑒 = 𝑄 𝑛 𝑗𝑒𝑡 𝑆 𝑗𝑒𝑡 [START_REF] Papakonstantis | Inclined Negatively Buoyant Jets 1: Geometrical Characteristics[END_REF] The evolution of 𝜃 𝑓 seems again to be linear with the 𝐹𝑟 𝐷 on a range from 3 to 17 before reaching the asymptotic value around 20°. Buoyancy become negligible for 𝐹𝑟 𝐷 = 17. A regression on the linear zone give a critical densimetric Froude number of 9.5.

As the jets undergo non-linear phenomena such as loss of pressure induced by the porous plate, deviation from the barrels or impingement on the UCS, the evolution of the asymptotic angle was not expected to be similar to the evolution observed in Figure 9. To enlighten the differences due to non-linear phenomena happening between the core and the UCS, we normalize these results with the densimetric Froude number obtained at nominal mass flow in both cases.

CONCLUSIONS

In order to be representative of the flow in ASTRID, experimental inlet conditions on the 1:6 scaled mockup MICAS are derived from densimetric Froude number conservation, applied to the inlet conditions out of the core. The phenomenon under study being the rise of a hot jet in a colder environment, experiments have been conducted in MICAS to characterize this phenomenon. Theoretical study imply that the behavior of the jet depends on the Euler number under the UCS and on the densimetric Froude number in the vessel. Thanks to PIV measurements for velocities and jet's angle, and temperature measurements thanks to PT100 probes, we have been able to plot the evolution of the asymptotic angle with the densimetric Froude number. This number has been calculated in two different zones, i.e. out of the core and in the vessel after impingement of the UCS. In both cases, the evolution of the asymptotic angle seems to be linear with the densimetric Froude number until the jet become momentum-dominated only. When buoyancy effects become negligible, the jet's angle has a constant value around 20°, that is assumed to be dependent on the mock-up geometry. Differences were expected between these two results, as non-linear phenomena appear between the exit of the core and the spreading in the vessel. But normalized results shown that these nonlinear phenomena were negligible on the evolution of the asymptotic angle. This result imply that the Euler number has negligible effects on the rise of the jet. The critical normalized densimetric Froude number for which we consider that the jet rises has a value of 0.45. A comparison of this result with the critical densimetric Froude number at other scales will provide information about scale effects on this phenomenon.

NOMENCLATURE

Figure 1 .

 1 Figure 1. Cut Views of ASTRID (Left) and MICAS (Right)

Figure 2 .

 2 Figure 2. Schema of the flow path above the core in ASTRID and MICAS in two different cases of mass flow rate and associated experimental results (velocity vectors colored according to velocity scale)

Figure 3 .

 3 Figure 3. Characteristics features of the jet's rise

Figure 4 Figure 5 .

 45 Figure 4. PIV setup on MICAS

Figure 6 . 5 . 3 . 1 .

 6531 Figure 6. PIV results with decreasing mass flow

Figure 7 .XzFigure 8

 78 Figure 7. Stabilization of the temperature in the vessel with time for Q = 151 m 3 /h measured with PT100 probe

Figure 9 .

 9 Figure 9. Evolution of the asymptotic angle with the densimetric Froude number in the upper plenum of the vessel

Figure 11

 11 Figure 11 gather the experimental data from Figure 9 and Figure 10 normalized by the densimetric Froude number at nominal mass flow rate. This figure shows that the results from both experiments are identical, leading to the conclusion that the non-linear phenomena have no influence on the raise of the jet in the upper plenum at MICAS's scale.The normalized critical densimetric Froude number for which the jet's angle is 𝜃 𝑓 = 0° is 𝐹𝑟 𝐷 = 0.45. The transition from buoyant to momentum-driven jet is reached for 𝐹𝑟 𝐷 = 0.75.

Figure 10 .

 10 Figure 10. Evolution of the asymptotic angle with the densimetric Froude number at the outlet of the core

Table 1 . Dimensionless numbers obtain with Vaschy-Buckingham theorem

 1

	(𝑈 * 𝜕𝑇 * 𝜕𝑧 * +	𝑉 * 𝑟 *	𝜕𝑇 * 𝜕𝑟 * ) =	1 𝑅𝑒	1 𝑃𝑟	1 𝑟 *	𝜕 𝜕𝑟 * (𝑟 * 𝜕𝑇 * 𝜕𝑟 * )	(2)

Table 2 . Experimental conditions on MICAS compared to ASTRID's ones

 2

		Experimental conditions
		ASTRID (nominal)	MICAS (study range)
	𝑇 𝑐 (°C)	~570	~55
	𝑇 𝑐 -𝑇 ∞ (°𝐶)	~150	1,2 𝑡𝑜 4,4
	𝑄 𝑐 (𝑚 3 . ℎ -1 )	~32 214	32 𝑡𝑜 165
	𝐹𝑟 𝐷	~ 3 𝑡𝑜 27

ACKNOWLEDGMENTS

The authors acknowledge the support from the CEA and its collaboration with the Lorraine University, especially the Energy & Theoretical and Applied Mechanics Laboratory. The authors also acknowledge the team of the PLATEAU platform for their support during the experimental campaigns.


Formalisms Theoretical questions Numerics and summary

Keywords:

Polymerisation with Attachement

dc i dt " J i´1 ´Ji , i ě 2 , J i " a i c 1 c i ´bi`1 c i`1 , i ě 1 , dc 1 dt " ´J1 ´ř8 i"1 J i .
Ball, Carr, Penrose, Comm. Math. Phys 104(4), 1986

Equilibrium is given by J i " J " 0, which implies

c i " Q i z i , Q i " a 1 a 2 ¨¨¨a i´1 b 2 b 3 ¨¨¨b i , i ě 1
for some z. Looking at the mass at equilibrium, F pzq :"

ÿ iě1 iQ i z i
It is natural to look for a solution of

F pzq ? " ρ :" ÿ iě1 ic i p0q " ÿ iě1 ic i ptq
Equilibrium of the BD model

$ ' ' ' & ' ' ' % dc i dt " J i´1 ´Ji , i ě 2 , J i " a i c 1 c i ´bi`1 c i`1 , i ě 1 , dc 1 dt " ´J1 ´ř8 i"1 J i .
Ball, Carr, Penrose, Comm. Math. Phys 104(4), 1986

Q i " a 1 a 2 ¨¨¨a i´1 b 2 b 3 ¨¨¨b i F pzq " ÿ iě1 iQ i z i ? " ρ
Equilibrium of the BD model

$ ' ' ' & ' ' ' % dc i dt " J i´1 ´Ji , i ě 2 , J i " a i c 1 c i ´bi`1 c i`1 , i ě 1 , dc 1 dt " ´J1 ´ř8 i"1 J i .
Ball, Carr, Penrose, Comm. Math. Phys 104(4), 1986

If the serie F pzq " ř iě1 iQ i z i has a finite radius of convergence z s and if suptF pzq , z ă z s u ": ρ s ă 8 , then there is a critical mass such that there is no equilibrium with mass ρ ą ρ s .

Formalisms Theoretical questions Numerics and summary

Equilibrium Metastability Coarsening

Equilibrium of the BD model

$ ' ' ' & ' ' ' % dc i dt " J i´1 ´Ji , i ě 2 , J i " a i c 1 c i ´bi`1 c i`1 , i ě 1 , dc 1 dt " ´J1 ´ř8 i"1 J i .
Ball, Carr, Penrose, Comm. Math. Phys 104(4), 1986

If the serie F pzq " ř iě1 iQ i z i has a finite radius of convergence z s and if suptF pzq , z ă z s u ": ρ s ă 8 , then there is a critical mass such that there is no equilibrium with mass ρ ą ρ s .

Remark

We may consider that lim iÑ8 b i {a i " z s Formalisms Theoretical questions Numerics and summary

Equilibrium Metastability Coarsening

Equilibrium of the BD model Equilibrium of the BD model

$ ' ' ' & ' ' ' % dc i dt " J i´1 ´Ji , i ě 2 , J i " a i c 1 c i ´bi`1 c i`1 , i ě 1 , dc 1 dt " ´J1 ´ř8 i"1 J i . Ball,
$ ' ' ' & ' ' ' % dc i dt " J i´1 ´Ji , i ě 2 , J i " a i c 1 c i ´bi`1 c i`1 , i ě 1 , dc 1 dt " ´J1 ´ř8 i"1 J i .
Ball, Carr, Penrose, Comm. Math. Phys 104(4), 1986

If ρ ď ρ s , then (with strong convergence)

lim tÑ8 c i ptq " Q i z i , F pzq " ρ If ρ ą ρ s , then (with weak convergence) lim tÑ8 c i ptq " Q i z i s , ρ ´ρs " "loss of mass to 8"

Remark

There is a Lyapounov function (or relative entropy), given by

H z pcq " ÿ iě1 " c i ˆln ˆci Q i z i ˙´1 ˙`Q i z i * . SBD model SDE $ ' ' ' ' ' ' & ' ' ' ' ' ' % C1ptq " C in 1 ´2J1ptq ´ÿ iě2 J i ptq , C i ptq " C in i `Ji´1 ptq ´Ji ptq , J i ptq " Y ì ´şt 0 a i C1psqC i psqds Ȳ í `1´ş t 0 b i`1 C i`1 psqds ¯CTMC Xn :" # C " pC i q iě1 P N N : n ÿ i"1 iC i " n + . " qpC , R ì C q " a i C1pC i ´δ1,i q , qpC , R í C q " b i C i , R ì C " C ´e1 ´ei `ei`1 R í C " C `e1 `ei´1 ´ei
Formalisms Theoretical questions Numerics and summary

Equilibrium Metastability Coarsening

Equilibrium of the SBD model

SDE $ ' ' ' ' ' ' & ' ' ' ' ' ' % C1ptq " C in 1 ´2J1ptq ´ÿ iě2 J i ptq , C i ptq " C in i `Ji´1 ptq ´Ji ptq , J i ptq " Y ì ´şt 0 a i C1psqC i psqds Ȳ í `1´ş t 0 b i`1 C i`1 psqds ¯CTMC Xn :" # C " pC i q iě1 P N N : n ÿ i"1 iC i " n + . " qpC , R ì C q " a i C1pC i ´δ1,i q , qpC , R í C q " b i C i , R ì C " C ´e1 ´ei `ei`1 R í C " C `e1 `ei´1 ´ei
Equilibrium, for any pa i q, pb i q, n :

ΠpC q " B n,z n ź i"1 pQ i z i q C i C i ! , Equilibrium of the SBD model SDE $ ' ' ' ' ' ' & ' ' ' ' ' ' % C1ptq " C in 1 ´2J1ptq ´ÿ iě2 J i ptq , C i ptq " C in i `Ji´1 ptq ´Ji ptq , J i ptq " Y ì ´şt 0 a i C1psqC i psqds Ȳ í `1´ş t 0 b i`1 C i`1 psqds ¯CTMC Xn :" # C " pC i q iě1 P N N : n ÿ i"1 iC i " n + . " qpC , R ì C q " a i C1pC i ´δ1,i q , qpC , R í C q " b i C i , R ì C " C ´e1 ´ei `ei`1 R í C " C `e1 `ei´1 ´ei
Equilibrium, for any pa i q, pb i q, n :

ΠpC q " B n,z n ź i"1 pQ i z i q C i C i ! , Detailed balance property : ΠpC qqpC , R ì C q " ΠpR ì C qqpR ì C , C q
Formalisms Theoretical questions Numerics and summary

Equilibrium Metastability Coarsening

Rescaled SBD model, n Ñ 8

SDE $ ' ' ' ' ' ' & ' ' ' ' ' ' % c1ptq " c in 1 ´2 ρ n J1ptq ´ÿ iě2 ρ n J i ptq , c i ptq " c in i `ρ n J i´1 ptq ´ρ n J i ptq , J i ptq " Y ì ´şt 0 n ρ a i c1psqc i psqds Ȳ í `1´ş t 0 n ρ b i`1 c i`1 psqds ¯CTMC X ρ n :" # c P R N : n ρ c i P N , n ÿ i"1 ic i " ρ + . # qpc, r ì cq " n ρ a i c1pc i ´δ1,i q , qpc, r í cq " n ρ b i c i , r ì c " c ´ρ n e1 ´ρ n e i `ρ n e i`1 r í c " c `ρ n e1 `ρ n e i´1 ´ρ n e i Rescaled SBD model, n Ñ 8 SDE $ ' ' ' ' ' ' & ' ' ' ' ' ' % c1ptq " c in 1 ´2 ρ n J1ptq ´ÿ iě2 ρ n J i ptq , c i ptq " c in i `ρ n J i´1 ptq ´ρ n J i ptq , J i ptq " Y ì ´şt 0 n ρ a i c1psqc i psqds Ȳ í `1´ş t 0 n ρ b i`1 c i`1 psqds ¯CTMC X ρ n :" # c P R N : n ρ c i P N , n ÿ i"1 ic i " ρ + . # qpc, r ì cq " n ρ a i c1pc i ´δ1,i q , qpc, r í cq " n ρ b i c i , r ì c " c ´ρ n e1 ´ρ n e i `ρ n e i`1 r í c " c `ρ n e1 `ρ n e i´1
´ρ n e i

Large volume limit : convergence towards the BD model (for a wide class of "reasonable" coefficients) on finite time intervals

n ÿ i"1 " ´ci ln ˆn ρ Q i z i ˙`ρ n ln n ρ c i ! `Qi z i * `ρ n ln B z n " n ÿ i"1 " c i ˆln c i Q i z i ´1˙`Q i z i * `Rn pcq `ρ n ln B z n " H z pcq ´8 ÿ i"n`1 Q i z i `

Remark

For ρ ą ρ s , we believe that a single giant cluster emerges, of size « np1 ´ρs {ρq (see work on limiting shapes of random combinatorial structures)

Metastability BD $ ' ' ' & ' ' % dc dt J ´Ji , ě 2 , i " i c c i ´bi`1 c i`1 , ě 1 , dc 1 dt " ´J1 ´ř8 i"1 J i .
' For an z ą z , there exists admissible configuration f " f i pzq such that J i " J ‰ 0 and 1 pzq " z. We start with c in " f and consider z OE z s :

piq For algebraically large time t, cptq ´f is exponentially small piiq lim tÑ8 cptq ´f ptq is not exponentially small piiiq ř iąn ˚ci ptq ď ř iąn ˚ci p0q `J˚t with J ˚exponentially small, Metastability SBD for c 1 ptq " z

Taking the monomer number as a constant allows to view the SBD process as a superposition of (independent) Birth-Death process on N

˚. i ´1 i i `1 api ´1qz bpiq apiqz bpi `1q
Formalisms Theoretical questions Numerics and summary

Equilibrium Metastability Coarsening

Metastability SBD for c 1 ptq " z Taking the monomer number as a constant allows to view the SBD process as a superposition of (independent) Birth-Death process on N

˚. i ´1 i i `1 api ´1qz bpiq apiqz bpi `1q
' For z ă z s : sub-critical, absorption at 1 is almost sure.

' For z ą z s : super-critical, absorption at 1 is NOT almost sure.

Metastability for the SBD ?

Numerical simulation "shows" metastability with sharp transition between "metastable state" and stationary state

Metastability SBD for c 1 ptq " z nucleation : we look for the first time a cluster of size greater than n appears :

τ n :" inftt ě 0, ÿ iěn C i ptq ą 0u i ´1 i i `1 api ´1qz bpiq apiqz bpi `1q
Formalisms Theoretical questions Numerics and summary

Equilibrium Metastability Coarsening

Metastability SBD for c 1 ptq " z nucleation : we look for the first time a cluster of size greater than n appears :

τ n :" inftt ě 0, ÿ iěn C i ptq ą 0u i ´1 i i `1 api ´1qz bpiq apiqz bpi `1q
There exists a quasi-stationary distribution,

P Π qsd n tCptq P ¨| τ n ą tu " Π qsd n and P Π qsd n tτ n ą tu " exp p´J n pzqtq
where Π qsd n is given by, for some (explicit) J n pzq, f n pzq

Π qsd n pC q " n ź i"2 pf n i q C i C i ! e ´f n i ,
Formalisms Theoretical questions Numerics and summary

Equilibrium Metastability Coarsening

Metastability SBD for c 1 ptq " z nucleation : we look for the first time a cluster of size greater than n appears :

τ n :" inftt ě 0, ÿ iěn C i ptq ą 0u i ´1 i i `1 api ´1qz bpiq apiqz bpi `1q
Theorem (Hingant, Y. 2021)

(for a class of initial condition Π in ), for any ε, and z close enough to z s , there exists K ˚, γ n ˚, J n ˚ą 0 such that }P Π in pC ptq P ¨| τ n ˚ą tq ´Πqsd n ˚} ď K ˚epJ n ˚´γnqt , P Π in pτ n ą tq ě p1 ´εqe ´Jn ˚t , where § K ˚, 1{γ n ˚are at most algebraically large § J n ˚is exponentially small Metastability SBD for c 1 ptq " z nucleation : we look for the first time a cluster of size greater than n appears :

τ n :" inftt ě 0, ÿ iěn C i ptq ą 0u i ´1 i i `1 api ´1qz bpiq apiqz bpi `1q
Method of proof : (i) Coupling arguments exploiting independence of particles (ii) Known probability of absorption for birth-death process

Metastability SBD for c 1 ptq " z nucleation : we look for the first time a cluster of size greater than n appears :

τ n :" inftt ě 0, ÿ iěn C i ptq ą 0u i ´1 i i `1 api ´1qz bpiq apiqz bpi `1q

Remark

Whether similar results holds true for the original SBD is an open question.

Nucleation + Coarsening dynamics

We start from a rescaled model

$ ' & ' % dc ε i dt " 1 ε " J ε i´1 ´Jε i ‰ , i ě 2 , ρ " c ε 1 ptq `ε2 ÿ iě2 ic ε i ptq . 1 ε a ε 1 C ε 1 C ε 1 Ý ÝÝÝÝÝ á â ÝÝÝÝÝ Ý 1 ε b ε 2 C ε 2 C ε 2 C ε i´1 1 ε a ε pεpi´1qqC ε 1 C ε i´1 Ý ÝÝÝÝÝÝÝÝÝÝÝ á â ÝÝÝÝÝÝÝÝÝÝÝ Ý 1 ε b ε pεiqC ε i C ε i 1 ε a ε pεiqC ε 1 C ε i Ý ÝÝÝÝÝÝÝÝÝ á â ÝÝÝÝÝÝÝÝÝ Ý 1 ε b ε pεpi`1qqC ε i`1 C ε i`1 ,
and we look for f ε pt, xq "

ř iě2 c ε i ptq1 rpi´1{2qε,pi`1{2qεq pxq

SBD

One sample path simulation of the "nonlinear" SBD, with a i " i 2{3 , b i " a i pz s `q{i 1{3 q, n " 500, ρ " 1 ą ρ s " 0.1056 dynamics : the "three eras" hypothesis § Appearance of large clusters. § Coarsening dynamics that leads to a single cluster, in a competition-like process. § Steady-state convergence of small cluster-size. piq Jpx, tq " ax r c 1 ptq ´bx, r ă 1 : f Ñ δ 0 (nucleation keeps creating "infinitesimal" clusters, no coarsening)

Nucleation + Coarsening dynamics

piiq Jpx, tq " pac 1 ptq ´bqx r , r ă 1 : f Ñ f 8 , c 1 ptq Ñ b a many large clusters persist, no coarsening) § case piq § case piiq

  Hingant, Y. (2016))we have f ε Ñ f (in C pr0, T s; w ´˚´Mpr0, 8qqq) ´bpxqs ϕ 1 pxqf pt, xq dx , for all ϕ P C 0 r0, 8q, which is the weak form ofBf Bt `BpJpx, tqf pt, xqq Bx " 0 , limxÑ0Jpx, tqf pt, xq " Nptq .

  tqf pt, xqq Bx " 0 , limxÑ0Jpx, tqf pt, xq " Nptq .

  Equilibrium of the rescaled SBD model, n Ñ 8If ρ ď ρ s , then for c n Ñ c (strongly), and z " F ´1pρq lim nÑ8 ´ρ n lnpΠ n pc n qq " H

z pcq If ρ ą ρ s , then for c n Ñ c (weak-*), and z s " F ´1pρ s q lim nÑ8 ´ρ n lnpΠ n pc n qq " H zs pcq Method of proof : Same as Anderson et al. 2015 + continuity property of H z pcq. ´ρ n ln Π n pcq "

One sample path simulation of the "nonlinear" SBD, with a i " i 2{3 , b i " a i pz s `q{i 1{3 q, n " 10000, ρ s " 0.1056

Metastability SBD

One sample path simulation of the "nonlinear" SBD, with a i " i 2{3 , b i " a i pz s `q{i 1{3 q, n " 500, ρ " 1 ą

Rescaled SBD model, n Ñ 8

Large volume limit : convergence towards the BD model (for a wide class of "reasonable" coefficients) on finite time intervals 3 Methods of proof :

(1) Tightness and identification of the limit (convergence in law) (2) Contraction of }c n ´c} (pathwise convergence) (3) Contraction of } ř jěi c n j ´řjěi c j } (pathwise convergence) Formalisms Theoretical questions Numerics and summary

Equilibrium Metastability Coarsening

Equilibrium of the rescaled SBD model, n Ñ 8

Theorem (Hingant, Y. ( 2019))

If ρ ď ρ s , then for c n Ñ c (strongly), and z " F ´1pρq lim nÑ8 ´ρ n lnpΠ n pc n qq " H z pcq Formalisms Theoretical questions Numerics and summary

Equilibrium Metastability Coarsening

Equilibrium of the rescaled SBD model, n Ñ 8

Theorem (Hingant, Y. ( 2019))

If ρ ą ρ s , then for c n Ñ c (weak-*), and z s " F ´1pρ s q lim nÑ8 ´ρ n lnpΠ n pc n qq " H zs pcq

Coarsening dynamics

For large density ρ ą ρ s , many super-critical clusters form in a small time. § How many super-critical clusters are they and how fast do they grow ? § How long does it takes for a single large cluster to take over the other ones ? § Are there situations where many large clusters persist ?

Nucleation + Coarsening dynamics Theorem (Deschamps, Hingant, Y. ( 2016))

Nptq is an explicit function of c 1 ptq, and is given by a quasi steady-state approximation of c ε 2 " f ε pt, 2εq, given by the solution of $ ' & ' % 0 " rJ i´1 pc 1 q ´Ji pc 1 qs , i ě 2 , c 1 ptq " c 1 .

When c 1 ą lim xÑ0 bpxq apxq , the solution of J i " J ‰ 0 is linked to the loss of mass in the classical BD theory. The aggregation dynamic is linked to the disease 'onset' Hence studying quantitatively the properties of the aggregation dynamic is relevant to understand some mechanisms of the Proteopathies. This can be done by reproducing the aggregation process in vitro.

Protein aggregation diseases : Working hypothesis

Does a Mathematical model reproduce the data ?

How does that help to understand the mechanistic phenomenon of the aggregation process ?