A FINITE ELEMENT METHOD FOR NEUTRON NOISE ANALYSIS IN HEXAGONAL REACTORS

The early detection of anomalies through the analysis of the neutron noise recorded by incore and ex-core instrumentation gives the possibility to take proper actions before such problems lead to safety concerns or impact plant availability. The study of the neutron fluctuations permits to detect and differentiate anomalies depending on their type and possibly to characterize and localize such anomalies. This method is non-intrusive and does not require any external perturbation of the system. To effectively use the neutron noise for reactor diagnostics it is essential to accurately model the effects of the anomalies on the neutron field. This paper deals with the development and validation of a neutron noise simulator for reactors with different geometries. The neutron noise is obtained by solving the frequency-domain two-group neutron diffusion equation in the first order approximation. In order to solve this partial differential equation a code based on a high order finite element method is developed. The novelty of this simulator resides on the possibility of dealing with rectangular meshes in any kind of geometry, thus allowing for complex domains and any location of the perturbation. The finite element method also permits automatic refinements in the cell size (h-adaptability) and in its polynomial degree (p-adaptability) that lead to a fast convergence. In order to show the possibilities of the neutron noise simulator developed a perturbation in a hexagonal two-dimensional reactor is investigated in this paper.


INTRODUCTION
Being able to monitor the state of nuclear reactors while they are running at nominal conditions is a safety requirement. The early detection of anomalies gives the possibility to take proper actions before such problems lead to safety concerns or impact plant availability. The CORTEX project [1], funded by the European Commission in the Euratom 20162017 work program, aims at developing an innovative core monitoring technique that allows detecting anomalies in nuclear reactors, such as excessive vibrations of core internals, flow blockage, coolant inlet perturbations, etc. The technique is based on using the inherent fluctuations in neutron flux recorded by incore and ex-core instrumentation, referred to as neutron noise, from which the anomalies will be detected and differentiated depending on their type, location and characteristics. The method is non-intrusive and does not require any external perturbation of the system.
To be able to detect, localise and quantify a perturbation in real-time, an automatic algorithm based on machine learning has to be provided with a large set of simulation data [2]. As the number of experiments to effectively train the machine learning algorithms is huge, these experiments must be carried out in a time efficient manner, i.e. fast running techniques are required to carry out the simulations. One useful technique to solve the effect of a perturbation in the neutron noise is to resolve the frequency-domain first-order neutron noise equation in the diffusion approximation. This is a partial differential equation with complex numbers.
This work presents a neutron noise simulator developed with the finite element method, called FEMFFUSION. It can deal with any kind of geometry allowing complex domains and any location of the perturbation. In other words, it computes the same quantities as the frequency domain code CORE SIM [3] but allowing any kind of geometry and a more adaptable structure. Also, the finite element method also permits automatic refinements in the cell size (h-adaptability) and in its polynomial degree (p-adaptability) that leads to an exponentially fast convergence.

THE NEUTRON DIFFUSION EQUATION
In the two energy group approximation, the time-dependent neutron diffusion equation with one group of delayed neutrons, where the matrices are denoted by [ ], are defined as [4] [ where the cross sections matrices are defined as The main unknown of the neutron diffusion equation is the space-and time dependent neutron flux, in its usual separation in the fast and thermal energy groups φ = φ 1 ( r, t), φ 2 ( r, t) T , and the neutron precursor concentration C( r, t). All other quantities have their usual meaning [4].

Static problem
For a given transient analysis in a core reactor, usually, a static configuration of the reactor is considered as initial condition. Associated with the time dependent neutron diffusion equation, (1) and (2), the static solution takes the form This problem is known as the Lambda Modes problem for a given configuration of the reactor core.
To solve the problems (3), a spatial discretization of the equations has to be selected. In this work, a high order continuous Galerkin finite element method is used leading to an algebraic eigenvalue problem associated with the discretization of equation (3) with the following block structure, are the algebraic vectors of weights associated with the fast and thermal neutron fluxes. More details on the spatial discretization used can be found in [5].
The resulting algebraic eigenvalue problem is solved using the Block Inverse-Free Preconditioned Arnoldi Method (BIFPAM) [6] or Newton iteration solver [7].

FIRST-ORDER NEUTRON NOISE THEORY
The first-order neutron noise theory is based on splitting every time dependent term, expressed as X( r, t), into their mean value, X 0 ,which is considered as the steady-state solution, and their fluctuation around the mean value, δX as, The fluctuations are assumed to be small compared to the mean values. This allows to neglect second-order terms (δX( r, t) × δX( r, t)) ≈ 0. Also, the fluctuations of the diffusion coefficients are neglected and δD g ≈ 0 is assumed. This approximation was demonstrated to be valid for light water reactor applications [8]. Thus, the first-order neutron noise equation can be written as [9].
The perturbation source term δS( r, ω) is given by the frequency-domain as changes in the cross sections : δS( r, ω) = δS 1 ( r, ω) δS 2 ( r, ω) = φ s δΣ 12 + φ a δΣ a1 δΣ a2 where By comparing with Eqs. (3), it can be seen that the neutron noise equation is an in-homogeneous equation with complex quantities that has to be solved after the steady-state solution is obtained because φ 1 and φ 2 represent the steady state fast and thermal neutron fluxes, respectively. The related static eigenvalue problem must be solved with the same spatial discretization as the frequency domain neutron noise equation to get coherent results.
Applying the continuous Galerkin finite element discretization to Eq. (6) leads to an algebraic linear system of equation with the following block structure where δΦ = δφ 1 , δφ 2 T are the algebraic vectors of weights associated with the fast and thermal neutron noise fluxes.

NUMERICAL RESULTS
To study the possibilities of the FEMFFUSION code, a two-dimensional hexagonal reactor (2D IAEA benchmark) was considered. This benchmark has a 1/12 reflective symmetry but as the inserted perturbation is not symmetrical, the whole reactor is solved. The fuel assembly pitch is 20.0 cm. Table 1 shows the cross section data for the this reactor. Figure 1 shows the materials layout.
A perturbation in the fuel assembly marked with material 5 of 10% of its cross sections reference values needs to be solved inserted to verify the developed noise simulator: δΣ a1 = 0.00095042, δΣ a2 = 0.00750058, δΣ s12 = 0.00177540, δΣ f 1 = 0.00058708, δΣ f 2 = 0.00960670. Table 2 shows the convergence of the solution depending on the polynomial degree used in the FEM shape functions (FED). We have defined the following error indicators where values with * represent reference results extracted from [10] for the steady-state results and a very fine FEM computation with a finite element polynomial degree equals to 7 and each cell refined into 16 cells for the neutron noise results. φ * a,g and δφ * a,g are the mean flux and the average noise flux, respectively, at assembly a. N A is the number of assemblies in the reactor. Figure 2 represents the mean assembly flux values for the steady state solution using FED = 3. Figure 3 presents the neutron noise magnitude and Figure 4 displays the neutron noise phase. The results shows that the fast neutron noise has ab influence over a wider region. On the other hand, the thermal neutron noise is mostly localized. Also, for this perturbation, the phase of the neutron noise is similar throughout the entire reactor.

CONCLUSIONS
This work presents a neutron noise simulator developed with the finite element method. It can deal with different kinds of geometry allowing complex domains as hexagonal reactors and any location of the perturbation. Also, the finite element method permits automatic refinements in the cell size and in its polynomial degree that leads to fast spatial convergence. This code will permit to train machine learning algorithms to detect perturbations in real-time in operating nuclear reactors.