Science and Technology of Nuclear Installations

Volume 2015, Article ID 140979, 14 pages

http://dx.doi.org/10.1155/2015/140979

## Properties of Neutron Noise Induced by Localized Perturbations in an SFR

Institute of Research and Development, Duy Tan University, K7/25 Quang Trung, Da Nang 550000, Vietnam

Received 31 March 2015; Revised 6 June 2015; Accepted 10 June 2015

Academic Editor: Keith E. Holbert

Copyright © 2015 Hoai-Nam Tran. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Investigation of the properties of neutron noise induced by localized perturbations in a sodium-cooled fast reactor has been performed using a multigroup neutron noise simulator. Three representations of the noise source associated with the perturbations of absorption, fission, and scattering cross sections, respectively, were assumed to be located at the first fuel ring around the central assembly. The energy- and space-dependent noise, that is, the amplitude and the phase, was calculated in a wide range of frequencies, for example, 0.1–100 Hz. The results show that in the important energy range (>1.0 keV) where the noise amplitude is significant the phase is almost constant with energy at the calculated frequencies despite the source types. At low frequencies, the variation of the phase is negligibly small at a large distance from the source. The perturbation in several fast groups has a significant contribution and dominates the amplitude and the phase of the induced noise.

#### 1. Introduction

Online diagnostics for monitoring the operating status of light water reactors (LWRs) based on analyzing the detector signals of neutron noise were deployed widely in various countries [1–5]. Measurement of the neutron noise in fast reactors, one of the next generation nuclear systems, and a test facility has also been conducted [3, 6, 7]. However, the knowledge and experience of the neutron noise in fast reactors in both measurement and simulation are very little compared to that of LWRs. Most neutron detectors used in LWRs, such as ionization chamber and fission chamber, are sensitive to thermal neutron. In many cases for simplicity one can investigate the neutron noise in the thermal group to represent the behaviour of the detector noise. Thus, recent numerical development for simulating the neutron noise in LWRs is based on two-group diffusion theory [8–11]. Nevertheless, numerical simulation still remains a challenge to reproduce and interpret the measurement data for improving core surveillance. It is a more difficult task for a fast reactor with less knowledge and experience.

Similar to static calculations in deterministic method, the neutron noise in a fast reactor should be calculated based on a multigroup model. In measurement, the detector signal is a combination of the energy-dependent noise with the cross section of the detector as a weighting function. In a sodium-cooled fast reactor (SFR), a fission chamber consisting of fissionable material, for example, or , coated on the inner wall of the chamber has a major potential for in-core fast neutron detection [12]. Thus, the energy-dependent cross section of the detector is complicated. This is also one of the difficulties for numerical simulation in predicting or interpreting the measurement phenomena. To simulate the neutron noise in fast reactors with hexagonal fuel assemblies, a neutron noise simulator was developed based on multigroup diffusion theory [13, 14]. The tool consists of two modules: a static module for solving the eigenvalue problem of a static state and a noise module for solving the neutron noise equation with a given source in a frequency domain. An application was performed for investigating the neutron noise behaviour induced by periodic core deformation effect in a large SFR [15].

In a realistic fluctuation of a system, the noise source can be modelled via a linear combination of the fluctuations of all cross section types in the first order approximation. The contribution of the perturbation of each cross section type in the total source depends on a specific scenario. For instance, the vibration of a control rod in an LWR can be modelled as the vibration of the absorption cross section, while to simulate the noise induced by fuel vibration, it is necessary to consider all cross section types including the fission cross section. The problem in an SFR is even more complicated since the fluctuation of cross sections is strongly energy-dependent and is considered in a multigroup theory. Therefore, prior to assessing realistic scenarios of fluctuations in an SFR, it is worth investigating the properties of the neutron noise induced by the fluctuations of absorption, scattering, and fission cross sections separately. This is because these fluctuations lead to different properties of the noise sources, respectively, and, as a result, different properties of the induced noise.

The present paper aims at investigating the properties of neutron noise induced by localized perturbations in a large SFR core using the noise simulator. Three representations of the noise sources associated with the localized perturbations of absorption, fission, and scattering cross sections, respectively, were assumed to be located at the center of the core. The space- and energy-dependent neutron noise has been calculated in a wide range of frequencies, for example, 0.1–100 Hz.

The paper is organized as follows. Section 2 presents briefly the principles of the neutron noise equation in multigroup diffusion theory which was solved in a frequency domain in the noise simulator. Section 3 describes the core model of a large SFR and the assumption of local perturbations as the noise source. Results and discussion on the properties of the energy- and space-dependent noise at the calculated frequencies are also presented. Finally, some concluding remarks are given in Section 4.

#### 2. Principles of the Neutron Noise Simulator

The basis of the neutron noise equation is the assumption of small stationary fluctuations of the system; that is, the averaged value of a time-dependent quantity over time is equal to the static value. Assume that all time-dependent terms can be split into a stationary component, , which corresponds to the value at the steady state, plus a small fluctuation, , as

By assuming the small fluctuations, the first order noise is taken into account, products of fluctuation terms can be neglected from time-dependent diffusion equations, and the result is a linear equation for the fluctuation of the flux. Subtracting the static equation and after performing a Fourier transform of all time-dependent terms, the first order space- and frequency-dependent neutron noise equation in multigroup diffusion theory is written as follows:where denotes the energy group, is the neutron noise in group , is the diffusion coefficient in group , and is the production cross section in group . is defined aswithwhere is the absorption cross section in group . is the scattering cross section from group to group , is the velocity of neutron in group , and is the frequency-dependent fission energy spectrum, which is obtained from the equation of delayed neutron as

The last term in (2), , denotes the noise source in group , which is calculated via the fluctuations of the macroscopic cross sections and the static flux, , as follows:where with represents the fluctuation of macroscopic cross sections. In (6), the fluctuation of the diffusion coefficient is neglected. The neutron noise equation (2) is an inhomogeneous equation with an external source, of which all quantities are frequency-dependent, that is, complex quantities. In order to solve the noise equation, it is necessary to define a noise source. Therefore, one needs to know the characteristics of the static state such as the and the static flux, , to calculate the noise source according to (6). This means that the solution of the static equation is also required. The simulator was implemented with two modules: a static module solving the static equation and a noise module solving the noise equation. Finite difference approach is used for the spatial discretization of the system with hexagonal fuel assemblies, where a hexagonal assembly is radially divided into triangular right prisms. Therefore, in a 3D model, each fundamental node has five interfaces including two equilateral triangular bases and three rectangular sides. More detailed description about the noise simulator can be seen in [14].

Figure 1 displays the spatial discretization of a hexagonal system in a 60-degree domain. In this figure, the () coordinates are used to handle the triangular fine meshes and the () coordinates are used to handle the hexagonal coarse meshes. A power iterative solution procedure is implemented for solving the balance equations of both the static and the noise equations. A coarse mesh finite difference (CMFD) method is employed for accelerating the convergence of both the static and the noise solutions, in which a coarse mesh is radially defined as a hexagonal assembly. In previous works, benchmarking calculations for the static state of the ESFR core were performed and had a good agreement with ERANOS [14]. Noise calculations in a two-group model had also a good agreement with analytical solutions [13]. These results give certain assurance for the noise calculations and further investigation of the noise behaviour in fast reactors.