STUDY OF CAVITATION IN LIQUID SODIUM AND SIMULATION OF DYNAMIC CORE DEFORMATIONS

Due to the high confinement of the closed fuel assemblies in Sodium-cooled Fast Reactors (SFR), the liquid sodium filling the space between these assemblies may vaporize in case of an important mechanical excitation (cavitation process). That phenomenon would therefore induce a specific force feedback to the assemblies leading to a specific neutronic response of the core. Neutronic of SFR is indeed sensible to mechanical deformation. Hence the necessity to have a good prediction of the displacement of these mechanical structures in the framework of Fluid-Structure Interaction (FSI). To study this kind of transient at a reactor scale, we propose in this paper a coupling procedure between the fluid and the structures. The fluid behaviour is investigated by Merkle’s three-equations model and simulated within a coarse mesh. The structures are modelled by a mass-spring system subjected to the fluid forces. In order to stabilize the numerical code coupling, a relaxation process is added and some results of this computational work are presented.


INTRODUCTION
Mechanical excitation of SFR fuel assemblies may appear for example in case of external events like earthquakes or internal initiators due to a mechanical default. Recent developments at CEA have proposed a homogeneized core modelling [1] to investigate such transients at the reactor scale. In [2] and [3], this modelling is applied to a core subjected to excitation of the assemblies and the neutronic effect (which can be important in case of mechanical deformation of SFR core) is investigated using the CNTT (Cast3M Neutronics Transient Tool) solver [4]. However the procedure to model cavitation in [2] and [3] requires an additional physical study and a numerical modelling has to demonstrate how vaporization of the fluid may appear during a mechanical transient of the core. We also show that the model in [1] has to be completed to take into account the effects of fluid convection which are not negligible regarding the pressure forces exerted on the mechanical structures. The challenge in the context of our study is then to build a numerical tool able to: • Reproduce the main effects of the cavitation phenomenon • Couple Fluid and Structure behaviour • Model a whole reactor core for a reasonable computational cost The choice has been made to consider the flow model of the Navier-Stokes equation for the fluid flow. In order to model the two-phase flow behaviour, a mixture velocity field is used where the dynamics of each phase is governed by a mass transfer coefficient. This approach has been used using Merkle's model [5] which consists in a three-equations modelling. Within the design of a SFR, two recent works [2] and [6] have used different softwares and methodologies in order to investigate the neutronic behaviour of a deformed core. In this context, the work [3] presents different kind of mechanical transients and hence neutronic transients impacted by different thermal-hydraulic initiators. Such sensitivity of fast reactor cores to deformations can be explained by the relative high free mean path λ of the neutrons regarding the core radius R: a very simple calculation indeed shows (see [2]) that λ varies relatively faster than R. Let us denote N the homogenized atomic density of an isolated core and V the volume. Then we have in case of deformation:

GENERAL
Since R evolves like V 1/3 , Eq. (1) implies: And then: These results show with very simple considerations that growing the size of a core exacerbates the neutron leakage.

An academic test case
In order to study mechanical effects on neutronics, specific displacements are assumed. Such academic displacements are described in [2]: the displacement of the assemblies does not depend on their position (see Fig. 1). The local Navier-Stokes equations (4) are numerically assumed to solve the fluid Pressure field P . This one corresponds to an unstationary convection-diffusion equation-type with a constraint on the mass balance driven by P . To simulate the fluid in a domain with moving boundaries, the Arbitrary Lagrangian-Euler (ALE) method is used. A description of this technique can be found in [7].

Modelling by means of domain homogenization
Modelling and simulating feedbacks of hydraulic effects on movements of assemblies are quite challenging: the convection term in the momentum equation is indeed non linear and may then lead to complex phenomena which would require a very fine mesh size. This is of course a high limitation in order to perform simulations at the whole reactor scale. In Broc et al. [1] the so-called "UPφ" model has been applied. It consists in homogenizing the whole core: the fluid part is not simulated but completely modelled.

Results of the UPφ model in several transients
Results of chaining FSI calculations to neutronics within an homogeneized core in different scenarios leading to a core mechanical excitation are provided in [3]. In some cases, the FSI model can lead to negative pressure values. This suggests that the fluid may vaporize through a cavitation process. This one can not be taken into account by the model (4). This finding is crucial: vapour apparition in the inter-assembly space may lead to different pressure forces exerted on the fuel assemblies. This vapour is characterized (in a two-phase equilibrium) by a constant pressure value P sat . The linear fluid model in the U P φ model in such a case would lead to a wrong pressure estimation and then to a wrong FSI coupling.

Phase change and the UPφ model
A "made-by-hand" modelling of vaporization or gas presence in the inter-assembly space has been performed in [2]. If one wants to assess the behaviour of the assemblies, all the inter-assembly liquid sodium is removed from a certain time step during a simulation. However in such an approach, the gas dynamics is not simulated.

Case description and analytical model
In order to upgrade the modelling of the fluid behaviour in case of movement of the assemblies, a local case has been studied: it consists in the study of the fluid behaviour in a domain where two parallel plates are moving and driven by a constant pressure reference at the left/right borders. The configuration of such a case is presented in Fig. 2. Solving Eq. (4) in the configuration of Fig. 2 leads us to formulate the analytical model Eq. (5) for the pressure term P . This has enabled us to predict which set of movement frequency and minimal gap distance h 0 can possibly lead to reach P sat . The results of this analysis at the local scale has been presented in [8].

1D-Homogenization in 2D geometries
In order to reduce the computational cost as mentioned in Sec. 1, the 2D whole core domain (interassembly space) is homogenized into a 1D domain: the meshed domain between too assembly edges (inter-assembly space) is then created by a N x × 1 mesh. With this technique only a mean effect on pressure is simulated and a correlation has to be used in order to re-build a correct head loss term (see Eq. (7)).

Modelling of the vaporization process under core excitation
In order to model the cavitation process, a set of equations has to be capable of reproducing mass exchanges between the liquid and gaseous phases in a physical manner. A large variety of models used to model cavitation processes can be found in [9]. In our case the mass exchanges are managed by the term Γ v . To close the set of equations Eq. (7), this term is splitted into the vaporization term . m + and the condensation term . m − (Eq. (8)). The mixture equation model implemented in the CFD code Code Saturne is used (see [10]). The mixture density ρ m , velocity u m and dynamic viscosity µ m are then written as a function of the void fraction α such as: By injecting Eq. (6) into Eq. (4) and by averaging (spatially) the variables according to the 1Dhomogenization, an equation for < α > (void fraction transport) is then obtained to build the following averaged 3-equation model: The exchange term m − results from a local analysis for boiling sodium. Its application to cavitating sodium is justified due to the similitude between the two processes: with t ∞ = 1 ms corresponding to a physical relaxation term.

Modelling of the assemblies displacement
The displacement of each assembly (indexed by i) in 2D is modelled by a damped mass-spring system submitted to an external force from the fluid part.
This displacement leads to a fluid volume update which requires a new calculation of the fluid variables in the framework of a FSI code coupling.

Numerical scheme for the FSI code coupling
A native coupling consisting at each step in calculating the pressure field and then the displacement field is numerically unstable due both to the high confinement of the assemblies and the important density of the liquid medium. Consequently an implicit time scheme with under-relaxation has been implemented and is presented in a nutshell in Eq. (10), n + 1 depicting the current time step and k the local iteration step of the relaxation process.
The relaxation term λ is empirically determined when calculations become stable.

Comparison of The Pressure Fields in a 1-Ring Configuration
In the cases described here, the assemblies are initially moved outwards (Fig. 1). Results within a 2D configuration made of one ring of assemblies are depicted in Fig. 3 with or without cavitation modelling. Effects of cavitation are observable on the right picture: In the one hand the region around the central assembly where cavitation occurs is maintained at P sat . In the other hand the distance of the ring assemblies to the central assembly is also different in the two cases.
By the way no experimental data are available related to the cavitation process under two moving boundaries. However models similar to Eq. (7) are compared to experimental data in [11] in case of cavitation on 2D foil sections and show a good agreement with them.

Transient Reactivity Effects of Assemblies Displacement
The CNTT solver [4] has been used to compute the reactivity effect from deformation of the 1-Ring 3D core. The assemblies made of fissile fuel are assumed to be straight (without bending) and their movement is taken from the 2D FSI calculations described in Subsection 4.1. Influence of cavitation on the assembly displacement can be seen on the right side in Fig. 4. This behaviour is also observable on the curve depicting the reactivity effect. The high value of hte initial reactivity effect is due to the small size of the 1-Ring configuration.

CONCLUSIONS
A numerical model enabling code coupling between thermal-hydraulics and mechanics with chaining to neutronics in the context of a SFR basic design has been presented here. This model is made of a mixture modelling of the cavitation process within a 1D-homogenized geometry in order to perform calculations at the scale of a reactor. In order to take into account the top-bottom fluid flowering in the inter-assembly space, an upcoming step consists in performing the same FSI coupling in 3D.