G.Th. Analytis

Theoretical investigation of the neutron noise diagnostics of two-dimensional control rod vibrations in a PWR

Abstract In order to develop a method for monitoring control rod vibrations by neutron noise measurements, the noise induced by two-dimensional vibrations of control elements is investigated. The two-dimensional Greens function relating the small stochastic cross-section fluctuations to the neutron noise is determined for a rectangular slab reactor in the modified one-group theory, and subsequently, the neutron response to two-dimensional vibrating noise sources is investigated. Two possible diagnostical applications are considered: (a) the reconstruction of the mechanical trajectory of the vibrating element by neutron noise measurements, and (b) the possibility of locating the vibrating element in the core.

A three-dimensional theoretical investigation of the local and global component of neutron noise in bare homogeneous water moderated reactors and applications

Abstract Starting from the time-dependent three-dimensional two-group diffusion equations for a bare homogeneous critical reactor, it is shown that the fluctuations of the neutron population can be uniquely separated into a local and a global component with each component satisfying a second-order differential equation. It is shown that under certain limitations, the two-group treatment of the neutron noise and the subsequent derivation of the two components, is equivalent to the one-group theory in which the slowing down of the fast neutrons is taken into account through an appropriately chosen slowing down kernel. The theory so developed, is applied in order to investigate the local component of the neutron noise induced by a randomly vibrating infinitely thin absorber in a two-dimensional cylindrical reactor and the neutron noise due to axially propagating perturbations of the moderator density, in a multi-channel model of a three-dimensional slab reactor.

Investigation of the space-dependent global component of the neutron-to-inlet coolant temperature fluctuations transfer function

Abstract By using the one-dimensional linearized stochastic neutronic equation in the diffusion approximation for an axially finite, homogeneous unreflected core with the slowing down of fast neutrons taken into account by an appropriately chosen slowing-down kernel, we have managed to split the thermal neutron noise field into a global and local component. By applying our theory to water-cooled and moderated cores and by disregarding feedback effects, we derive explicit expressions for the global component of the neutron-to-inlet coolant temperature fluctuations transfer function, both in the exact, space-dependent model and in the point model approximation. By numerically evaluating the gains and phases of the transfer functions for different core heights and coolant velocities, we conclude that considerable differences arise between the space-dependent and the point model, even for small cores. In particular, in the case of relatively large cores, we show that the phase-to-frequency relationship of the space-dependent transfer function can be used to calculate the velocity of the coolant through the core.

Analysis of the neutron response to axially-propagating perturbations in a heterogeneous lattice of cylindrical fuel elements via the two-group slowing-down model

Abstract The problem of the neutron response to an infinite frequency-dependent plane unit source in the moderator of an infinite regular heterogeneous lattice of cylindrical fuel elements is analyzed in the one-group diffusion approximation with a two-group slowing-down kernel, within the framework of the source-sink method of Feinberg and Galanin. It is shown that the neutron response divides naturally into a component with a long spatial relaxation length and one with a short relaxation length, which is equal to the thermal diffusion length in the moderator. The latter spatial decay mode is solely due to attenuation effects in the pure moderator and in contrast to the corresponding homogeneous diffusion theory analysis of the same problem, no other localized spatial decay mode exists. Finally, a simple void-propagation model is used for calculating the neutron spectra, due to axially-propagating perturbations of the moderator density in the heterogeneous and the corresponding homogenized system. Although the shapes of the spectra are similar, it is shown that both physically and mathematically, the analysis of this problem by homogeneous theories can lead to erroneous interpretations.

‘Local’ sensitivity volume of an incore neutron detector in a heterogeneous lattice of cylindrical fuel elements via the Feinberg-Galanin model

Abstract By employing a previously-developed frequency-dependent source-sink formalism of Feinberg and Galanin in one-group diffusion theory with a two-group slowing-down kernel, we investigate the ‘local sensitivity volume’ of a neutron detector located in a 2-D heterogeneous lattice of cylindrical fuel elements. It is shown that the local component of the detector response has a spatial relaxation length equal to the thermal diffusion length in the pure moderator and by considering cases with neutronic data typical of the lower and upper parts of a BWR core, it is explicitly shown that while in the lower part the local field of view of the detector extends over approximately half of the cross-sectional area of the four surrounding bundles, in the upper part, it extends over the whole cross section of the four bundles. In both cases though, the perturbations occurring at the four corner subchannels are more heavily weighted.

A semi-analytical solution of the two-group Feinberg-Galanin kinetic adjoint equations for a heterogeneous lattice of cylindrical fuel elements

Abstract A solution of the two-group Feinberg-Galanin kinetic adjoint equations for a 2-D heterogeneous lattice of a finite number of cylindrical fuel rods is presented and the terms corresponding to the local component of the detector response—which are solely due to attenuation effects in the pure moderator—are identified in the solution. By considering two specific numerical examples, it is shown that the dominant contribution to the local component of the detector response arises from the term representing attenuation effects in the thermal kinetic adjoint, which has a spatial relaxation length equal to the thermal diffusion length in the pure moderator, and which is identical to the corresponding term derived if one pursues a one-group, with a two-group slowing-down kernel, analysis.

Formulation of the two-group stochastic Feinberg-Galanin equations for heterogeneous lattices

Abstract The general problem of neutron noise in an infinite heterogeneous reactor consisting of plate-type fuel elements of identical nuclear properties embedded in a moderator is formulated in the two-group diffusion theory by using the source-sink method of Feinberg and Galanin. After linearizing the neutronic stochastic differential equations, the heterogeneous kinetic adjoint equations are formulated and solved by assuming that the detectors are only sensitive to thermal neutrons. It is shown that in a heterogeneous analysis, a localized noise source will give rise to additional local noise components in the response of a near-by detector. These components have very similar properties to the conventional local component of the homogeneous two-group theory but their existence is due to the attenuation of the signal in the pure moderator and is not directly related to the number of energy groups used in the analysis.

Frequency response of the neutron field to localized perturbations in a 3-D heterogeneous lattice and applications

Abstract By using the frequency-dependent source-sink formalism of Feinberg and Galanin in one-group diffusion theory with a two-group slowing-down kernel, a model is developed for the investigation of the frequency response of the neutron field to a localized frequency-dependent perturbation in a 3-D heterogeneous lattice of a finite number of fuel and control rods embedded in an infinite moderator with no upper or lower reflectors. A numerical method for solving the equations of the semi-analytical model is indicated and it is shown that the well-known local component of the neutron-noise field is due to attenuation effects in the pure moderator. Finally, with some simplifications, the formalism is applied to the problem of the ‘field of view’ of an incore neutron detector in a BWR with moderator data typical of the lower and the upper part of a BWR core. Within the framework of the present model, it is shown that in the upper part of the core, although with different weighting for different distances from the detector, the detector ‘sees’ at least the whole cross section of the surrounding bundles.

Three-group three-dimensional analysis of the random neutron field in bare homogeneous reactors and the concepts of local and global neutron noise

Abstract A three-group theory of neutron noise is developed and it is shown that the neutron-noise field in homogeneous multiplying media can be separated into a component with a long attenuation length (global component) and two local components whose attenuation lengths in water-moderated reactors are very small and of the order of the diffusion length of thermal neutrons in the medium. The theory is subsequently applied to the investigation of the neutron noise induced by axially propagating stochastic fluctuations of the moderator density in a simple 1-D BWR model and the evaluated neutron cross-spectra and phases are compared to the corresponding ones obtained by employing the two-group model with two-group data derived by condensing the three-group data in the reactor spectrum. In this respect, it is shown that the results obtained by using the three- and the two-group models are almost identical.

Analysis of some pseudo-discrete spatial decay modes of the two- and three-group neutron transport equations by analytic continuation

Abstract The nature and location of the poles of the dispersion function of the frequency-dependent two- and three-group neutron transport equations for an infinite 3-D homogeneous multiplying medium with constant cross sections and isotropic scattering is investigated. It is shown that on the right half of the complex plane, only one pole exists on the physical sheet of the Riemann surface, which is almost identical to the one obtained in two- and three-group frequency-dependent diffusion theory, and gives rise to the well-known, in neutron noise, long attenuation length global component of the detector response. In contrast to diffusion theory, in transport theory no other pole exists on the physical sheet; the poles associated with the short spatial relaxation length ‘local component’ in homogeneous diffusion theories approaches are bifurcated and are to be found on the adjacent ‘unphysical’ sheets of the Riemann surface of the generic multivalued dispersion function. Although not on the physical sheet, the bifurcated poles are located very close to it and give rise to pseudo-discrete spatial decay modes with very similar characteristics to the local component of the detector response in homogeneous diffusion theories.