Evaluation of coupling coefficients in nodal/modal analysis

A procedure for developing neutron coupling coefficients for power-distribution calculations in LWRs using nodal analysis is described. The coefficients account for the fast and thermal flux variation within each node or fuel assembly by utilizing a 1-D diffusion-theory analysis of the flux ‘modes’ within the nodal volume. The procedure utilizes the fact that the fuel assemblies in most LWRs are arranged in such a manner that the reactor volume consists of a large number of neutronically self-sustaining zones. The paper provides both a physical and an analytical justification for the evaluation of neutron coupling coefficients in two energy groups.     

A nodal modal method for the neutron diffusion equation. Application to BWR instabilities analysis

Fast codes, capable of dealing with three-dimensional geometries, are needed to be able to simulate spatially complicated transients in a nuclear power reactor. In this paper, we propose a modal method to integrate the neutron diffusion equation in which the spatial part has been previously dicretized using a nodal collocation method. For the time integration of the resulting system of differential equations it is supposed that the solution can be expanded as a linear combination of the dominant Lambda modes associated with a static configuration of the reactor core and, using the eigenfunctions of the adjoint problem, a system of differential equations of lower dimension is obtained. This system is integrated using a variable time step implicit method. Furthermore, for realistic transients, it would be necessary to calculate a large amount of modes. To avoid this, the modal method has been implemented making use of an updating process for the modes at each certain time step. Five transients have been studied: a homogeneous reactor, a non-homogeneous reactor, the 3D Langenbuch reactor and two transients related with in-phase and out-of-phase oscillations of Leibstadt NPP. The obtained results have been compared with the ones provided by a method based on a one-step backward discretization formula.     

The analytic nodal diffusion solver ANDES in multigroups for 3D rectangular geometry: Development and performance analysis

In this work we address the development and implementation of the analytic coarse-mesh finite-difference (ACMFD) method in a nodal neutron diffusion solver called ANDES. The first version of the solver is implemented in any number of neutron energy groups, and in 3D Cartesian geometries; thus it mainly addresses PWR and BWR core simulations.     

A modal multidimensional kinetics method using pseudo-harmonics

This work proposes to use the pseudo-harmonics technique as a modal method in spatial kinetics. The pseudo-harmonics are the eigenfunctions associated with the leakage + removal operators in the multigroup steady-state diffusion equation. In this work, the pseudo-harmonics are obtained from the diffusion equation discretized with the Coarse Mesh Finite Difference Method (CMFD) coupled to the Nodal Expansion Method (NEM).     

A polynomial nodal model using Legendre expansions

A polynomial nodal model which uses Legendre expansions was developed for the multi group diffusion equation in 1-D. The development depends upon a least-squares minimization of the approximate functions over the node. Analytical expressions were developed for the polynomial coefficients. Diffusion theory interface and boundary conditions have been applied. Sample problems with analytical solutions and fine-mesh finite-difference solutions have been compared with the method. For most reactor-type problems, fourth-order polynomial expansions appear to be adequate.     

Steady state thermal analysis of PWRs by a single-pass procedure using a simplified nodal layout

A single-pass procedure using a prescribed nodal layout has been developed for steady state PWR core analysis. Recommended nodal patterns for modelling the reactor core by the one stage analysis method have been proposed. By using this one-stage simplified method, DNBR analyses of cores having either one hot subchannel or two equally hot subchannels can be accomplished by a thermal-hydraulic computer code with a capacity of seven or seventeen radial nodes, respectively. The advantages of the simplified method over the other multistage approaches also are discussed and compared.     

A Conformal mapped nodal SP3 method for hexagonal core analysis

Conformal mapped nodal simplified P₃ equations are derived and implemented for the two-dimensional neutronics analysis of fast reactor cores where hexagonal assemblies are loaded. The one-dimensional simplified P₃ equations are solved by using the nodal expansion method. The partial currents response matrices are constructed for the coupled simplified P₃ equations by applying the relationship between the partial currents and the surface-averaged fluxes. These matrices are then solved non-linearly.     

Simplified method for the seismic qualification using measured modal data

The seismic response estimation by the response spectrum method using only the experimental modal data are presented here. The modal participation factors (MPFs) used for the response estimation are calculated using the experimental mode shapes only. This response is compared with the response estimated using the conventional MPFs but the experimentally extracted mode shapes are used along with the mass matrix estimated corresponding to the measured degree of freedoms (dofs) using physical dimensions of the structure. The presented study eliminates the uncertainty associated with analytical modelling for evaluating mass matrix.     

Combined analysis and evaluation of piping systems using the computer

The organization and theoretical aspects of a computer program designed primarily for the analysis and evaluation of power piping with particular emphasis on nuclear class I and II piping are described. The program includes a one-dimensional (radial coordinate in a cylindrical system) thermal analysis which computes the required entries in eqs (10), (11) and (13) of section NB 3600. Six types of loadings may be considered simultaneously: deadweight, sustained design mechanical loads, thermal expansion, thermal anchor movement, earthquake-response spectra or time history, and other anchor movement. For dynamic analysis the generalized Rayleigh-Ritz method using unit loads to generate the trial vectors is utilized. This method provides a consistent reduction of both the stiffness matrix and the mass matrix. The Ritz method is not without its problems, however, in the selection of the so-called ‘master nodes’. While getting enough master nodes is generally not a problem (easier than lumping masses) it is easy to over-specify and therefore introduce ill conditioning. To overcome this shortcoming a method is described for checking the trial vectors for their degree of parallelism in kinetic energy space, and for automatically removing those degrees of freedom which may lead to difficulties. Sample solutions with the program are presented.     

Study on modal characteristics of perforated shell using effective Young's modulus

For the perforated cylindrical shell submerged in fluid, it is almost impossible to develop a finite element model due to the necessity of the fine meshing of the shell and the fluid at the same time. This necessitates the use of solid shell with effective material properties. Unfortunately the effective elastic constants are not found in any references even though the ASME code is suggesting those for perforated plate. Therefore in this study the effective material properties of perforated shell are suggested by performing several finite element analyses with respect to the ligament efficiencies.     

Sensitivity analysis of fuel pin failure performance under slow-ramp type transient overpower condition by using a fuel performance analysis code FEMAXI-FBR

The FEMAXI-FBR is a fuel performance analysis code and has been developed as one module of core disruptive evaluation system, the ASTERIA-FBR. The FEMAXI-FBR has reproduced the failure pin behavior during slow transient overpower. The axial location of pin failure affects the power and reactivity behavior during core disruptive accident, and failure model of which pin failure occurs at upper part of pin is used by reflecting the results of the CABRI-2 test. By using the FEMAXI-FBR, sensitivity analysis of uncertainty of design parameters such as irradiation conditions and fuel fabrication tolerances was performed to clarify the effect on axial location of pin failure during slow transient overpower. The sensitivity analysis showed that the uncertainty of design parameters does not affect the failure location. It suggests that the failure model with which locations of failure occur at upper part of pin can be adopted for core disruptive calculation by taking into consideration of design uncertainties.     

Linear modal analysis of out-of-phase instability in boiling water reactor cores

An eigenvalue problem governing BWR core nuclear thermal-hydraulic modes which result in out-of-phase power oscillations is formulated. This formulation is based on the linearization approximation to nonlinear feedback terms and the very simple models for neutronics and thermal-hydraulics. The eigenvalue problem in 5 × 5 matrix formulation can be easily solved without using a computer. A series of the calculations are carried out, at a high-power and low-core-flow condition, to investigate the dependence of the eigenvalues and eigenfunctions on the void reactivity coefficient and the subcriticality of spatial neutronic modes, where the latter parameter is identical to the eigenvalue separation of the higher-harmonic neutronic mode. These results show that the threshold value of the void coefficient for initiating the unstable out-of-phase oscillation strongly depends on the subcriticality. The oscillation mode becomes more unstable with an increase in the absolute value of the negative void coefficient, whereas the mode becomes more stable, almost linearly, with increasing subcriticality. The resonant frequency of the oscillation and the phase shifts between the nuclear thermal-hydraulic variables are consistent with previous measured or calculated values.     

A lattice homogenization procedure using the nodal expansion method and equivalence theory

In this paper we give the generalization of the nodal expansion method to incorporate the flux discontinuity factors arising in the equivalence theory formulation of lattice homogenization. The application of equivalence theory to a 2-D uniform lattice in the multigroup diffusion theory approach is shown to yield (in the lowest-order approximation) Benoist-type diffusion coefficients, cell discontinuity factors and flux-weighted cross-sections. The nodal expansion method is then shown to provide a convenient formalism to calculate the Benoist diffusion coefficient. Numerical results for two LWR benchmark problems are also given.     

A nodal SN transport method for three-dimensional triangular-z geometry

A new nodal S_N transport method has been developed to perform accurate transport calculation in three-dimensional triangular-z geometry, where arbitrary triangles are transformed into regular triangles via a coordinate transformation. The transverse integration procedure is applied to treat the neutron transport equation in the regular triangle. The neutron angular distributions of intra-node fluxes are represented using the S_N quadrature set, and the spatial distributions of neutron fluxes and sources are approximated by a quadratic polynomial. The nodal-equivalent finite difference algorithm for 3D triangular geometry is applied to establish a stable and efficient iterative scheme. The present method was tested on four 3D Takeda benchmark problems published by the nuclear data agency (NEACRP), in which the first three problems are in XYZ geometry and the last one is in hexagonal-z geometry. The results of the present method agree well with those of the reference Monte-Carlo calculation method, the difference in k_eff being less than 0.1%. This shows that multi-group reactor core/criticality problems can be accurately and effectively solved using the present method.     

Extension of the analytic nodal diffusion solver ANDES to triangular-Z geometry and coupling with COBRA-IIIc for hexagonal core analysis

In this paper the extension of the multigroup nodal diffusion code ANDES, based on the Analytic Coarse Mesh Finite Difference (ACMFD) method, from Cartesian to hexagonal geometry is presented, as well as its coupling with the thermal–hydraulic (TH) code COBRA-IIIc for hexagonal core analysis.     

Residual load method for modal analysis of piping systems subjected to seismic excitation

Due to the fact that piping systems normally possess a large number of unknowns, the computation of eigenvectors and frequencies is limited. It does not seem to be reasonable to calculate all modes because of the wellknown phenomenon that the accuracy of the higher modes is deteriorating the more the higher the frequency of these eigenforms. When the modal-analysis method is used, this means, that the complete response of piping systems remains unknown.In this paper two methods which pretend to take the neglected higher modes into account are discussed. These are the residual-load-method and the modal-acceleration-method. Error investigations on both methods reveal that the residual-load-method might be given preference.Two examples are presented in which piping systems are dealt with the residual-load-method. It seems surprising, that even for branched systems it is sufficient to compute a few eigenvectors and then apply the residual load.     

Characteristic evaluation of electromagnetic flow couplers using a liquid metal MHD analysis code

It is known that electromagnetic flow couplers have high efficiency. In fact, an experiment using a small-scale annulus model shows that it has the maximum efficiency of 60%. However, another experiment using a middle-scale sector model exhibits that the maximum efficiency is less than 20%. In order to analyze this discrepancy, a computer code for solving magnetohydrodynamic flows is developed and is applied to the evaluation of these experiments. The obtained results show that the efficiency of the small-scale annulus model is high as observed in the experiment and the low efficiency observed in the middle-scale sector model is ascribed to the energy loss due to the eddy current arising in the ends of the magnetic region and the electric contact resistance between the stainless steel duct and the copper bus bar.     

Fuel performance evaluation of rock-like oxide fuels

The concept of the rock-like oxide (ROX) fuel has been developed for the annihilation of excess plutonium in light water reactors. Irradiation tests and post-irradiation examinations were carried out on candidate ROX fuels. The ternary fuel of YSZ–spinel–corundum system, the single-phase fuels of YSZ, the particle-dispersed fuels of YSZ in spinel or corundum matrix, and the blended fuels of YSZ and spinel or corundum matrix were fabricated and submitted to irradiation testings. The fuels containing spinel showed chemical instabilities with the vaporization of MgO component, which caused fuel restructuring. The swelling behavior was improved with the particle-dispersed fuels. However, the particle-dispersed fuels showed higher fractional gas release (FGR) than blended type fuels. The FGR of YSZ single-phase fuels were comparable to what would be expected for UO₂ fuel at the similar fuel temperatures. The YSZ single-phase fuel showed the best irradiation performance among the ROX fuels investigated.     

A modal analysis of free-surface and internal gravity waves in a fast-reactor hot pool

This paper follows the classical approach and presents a linear, inviscid, free-mode analysis of a stratified fluid in a prismatic tank consisting of an eccentric above-core structure (ACS) as well as four pump pods and eight intermediate heat exchanger (IHX) standpipes. The prismatic geometry (i.e. the horizontal cross-section is independent of the vertical position) implies a decoupling of the mathematics governing variations in the horizontal and vertical planes; the latter depends only on the vertical density profile and provides the dispersion relation between frequency and eigenwavenumber. After exploiting the analogy between the horizontal-plane problem and acoustic vibrations in a rigid cavity, the modal shapes and eigenwavenumbers of the first seven modes are determined by employing acoustic elements in the BERDYNE finite-element structural dynamics code.