Multigroup Formulation Research Articles

Neutron angular flux reconstruction in slab geometry using multigroup discrete ordinates transport models

In this article, we present an application of the coarse-mesh Deterministic Spectral Method (SDM) to generate multigroup angular fluxes in one-dimensional spatial domains using the neutron transport stationary equation, in the formulation of discrete ordinates (SN), considering isotropic scattering source. After obtaining the analytical solution of the SN equations, we replace the integral term of the scattering source in the original neutron transport equation. Thus, we obtain analytically two expressions for angular fluxes in the multigroup formulation, considering the neutron propagation in the positive ( ) and negative ( ) directions, presenting a meaningful reduction in the computational time of simulations of typical neutron shielding problems.

Convergence study of CMFD based acceleration schemes for multi-group transport calculations with fission source using fourier analysis

Application of the coarse mesh finite difference (CMFD) method and its variants to accelerate the convergence of criticality calculations has been extremely popular for neutron transport simulations. Despite their widespread use in the multi-group form, the convergence performance of these acceleration schemes is usually studied in the one-group domain and results extended to multi-group problems. Theoretical investigation of the convergence behavior of multi-group CMFD and related methods has not been previously performed and is an important work to complement existing studies. In this paper, we present the Fourier convergence analysis of multi-group CMFD, partial current CMFD (pCMFD), optimally diffusive CMFD (odCMFD) and linear prolongation CMFD (lpCMFD) schemes for acceleration of the fission source neutron transport calculations. The multi-group formulation of the error iteration matrix has been found to be analogous to the results obtained from one-group analysis. It has also been shown that for certain simplified cases, the spectral radius of multi-group CMFD is independent of the group coupling terms and can be determined as the maximum among individual energy groups. The results extend current understanding and provide deeper insight into the application of CMFD based acceleration techniques for more realistic multi-group neutron transport.

Multigroup neutron diffusion equation with the finite volume method in reactors using MOX fuels

ABSTRACTThe use of mixed oxide (MOX) fuel to partially fill the cores of commercial light water reactors (LWRs) gives rise to a reduction of the radioactive waste and production of more energy. However, the use of MOX fuels in LWRs changes the physics characteristics of the reactor core, since the variation with energy of the cross sections for the plutonium isotopes is more complex than for the uranium isotopes. Although the neutron diffusion theory could be applied to reactors using MOX fuels, more emphasis on treatment of the energy discretization should be placed. This energy discretization could be typically 4–8 energy groups, instead of the standard 2-energy group approach. In this work, the authors developed a finite volume method for discretizing the general multigroup neutron diffusion equation. This method solves the eigenvalue problem by using Krylov projection methods, in which the size of the vectors used for building the Krylov subspace does not depend on the number of energy groups, but it can solve the multigroup formulation with upscattering and fission production terms in several energy groups. The method was applied to MOX reactors for its validation.

A new multigroup method for cross-sections that vary rapidly in energy

We present a numerical method for solving the time-independent thermal radiative transfer (TRT) equation or the neutron transport (NT) equation when the opacity (cross-section) varies rapidly in frequency (energy) on the microscale ε; ε corresponds to the characteristic spacing between absorption lines or resonances, and is much smaller than the macroscopic frequency (energy) variation of interest. The approach is based on a rigorous homogenization of the TRT/NT equation in the frequency (energy) variable. Discretization of the homogenized TRT/NT equation results in a multigroup-type system, and can therefore be solved by standard methods.We demonstrate the accuracy and efficiency of the approach on three model problems. First we consider the Elsasser band model with constant temperature and a line spacing ε=10−4. Second, we consider a neutron transport application for fast neutrons incident on iron, where the characteristic resonance spacing ε necessitates ≈16,000 energy discretization parameters if Planck-weighted cross sections are used. Third, we consider an atmospheric TRT problem for an opacity corresponding to water vapor over a frequency range 1000–2000cm−1, where we take 12 homogeneous layers between 1–15km, and temperature/pressure values in each layer from the standard US atmosphere. For all three problems, we demonstrate that we can achieve between 0.1 and 1 percent relative error in the solution, and with several orders of magnitude fewer parameters than a standard multigroup formulation using Planck-weighted (source-weighted) opacities for a comparable accuracy.

Multi-group formulation of the temperature-dependent resonance scattering model and its impact on reactor core parameters

A multi-group formulation for the exact neutron elastic scattering kernel is developed. It incorporates the neutron up-scattering effects stemming from lattice atoms thermal motion and it accounts for them within the resulting effective nuclear cross-section data. The effects pertain essentially to resonant scattering off of heavy nuclei. The formulation, implemented into a standalone code, produces effective nuclear scattering data that are then supplied directly into the DRAGON lattice physics code where the effects on Doppler reactivity and neutron flux are demonstrated. The correct accounting for the crystal lattice effects influences the estimated values for the probability of neutron absorption and scattering, which in turn affect the estimation of core reactivity and burnup characteristics. The results show an increase in values of Doppler temperature feedback coefficients up to −10% for UOX and MOX LWR fuels compared to the corresponding values derived using the traditional asymptotic elastic scattering kernel. This paper also summarizes research performed to date on this topic.

Analytic Function Expansion Nodal Method for Multigroup Diffusion Equations in Cylindrical (r,θ,z) Geometry

A coarse-mesh nodal method in cylindrical (r, θ,z) geometry, e.g., of pebble bed reactors, based on the analytic function expansion nodal (AFEN) methodology, is described in this paper. Two unique features are (a) no use of transverse integration—allowing a nodal scheme in (r, θ,z) geometry—and (b) nodal solution expressed in terms of analytic basis functions—leading to high accuracy and readily available reconstruction of homogeneous flux distributions. Additional features of multigroup formulation, two methods of void region treatment, and coarse-group-rebalance acceleration are implemented in the TOPS code and tested on several benchmark problems, including the Organisation for Economic Co-operation and Development/Nuclear Energy Agency PBMR-400 Benchmark Problem. The TOPS results are in excellent agreement with those of the VENTURE code, using significantly less computer time.

Mathematical properties of a kinetic transport model for carriers and phonons in semiconductors

We present studies on the mathematical properties of a multigroup formulation of the Bloch–Boltzmann–Peierls equations. The considered model equations are based on a general carrier dispersion law and contain the full quantum statistics of both the carriers and the phonons. Moreover, the transport model allows the investigation of particle distributions with arbitrary anisotropy with respect to the main direction. We prove the boundedness of the solution according to the Pauli principle and study the conservational properties of the multigroup equations. In addition, the existence of a Lyapounov functional to the proposed model equations is proved and expressions for the equilibrium solution are given. Numerical results are presented for the stationary state distributions of a coupled system of electrons and longitudinal optical phonons in GaAs.

Multigroup Discriminant Analysis Using Linear Programming

In this paper we introduce a nonparametric linear programming formulation for the general multigroup classification problem. Previous research using linear programming formulations has either been limited to the two-group case, or required complicated constraints and many zero-one variables. We develop general properties of our multigroup formulation and illustrate its use with several small example problems and previously published real data sets. A comparative analysis on the real data sets shows that our formulation may offer an interesting robust alternative to parametric statistical formulations for the multigroup discriminant problem.

Boundary condition perturbation technique for use in spatial homogenizatton method in transport theory

An expression is derived for the first-order change in the ratio of linear functionals of the fundamental solution of the neutron transport equation due to a perturbation in the boundary condition of the system. The perturbation formula is derived in the context of energy dependent transport theory. The multigroup formulation is also given. Two numerical examples are used to verify the correctness of the formalism. As an application, use of this method for improving spatial homogenization techniques is also discussed

Variational Principles and Convergence Acceleration Strategies for the Neutron Diffusion Equation

New extrapolation techniques are presented based on the inverse power method to facilitate solution of the multigroup neutron diffusion equation. Unlike the usual acceleration approaches, no estimate of the dominance ratio is required to calculate optimal extrapolation factors. At each outer iteration, the extrapolation factors that correspond to a stationary point of an appropriate functional have been calculated. This technique has been used successfully in the calculation of direct, direct/adjoint, and fixed-source eigenvalue problems for a multigroup formulation of the neutron diffusion equation discretized by finite elements. Numerical tests allow the performance of the variational method to be compared with that of the Chebyshev method.

Calculation of the transport of neutral atoms in highly ionized plasmas using neutron transport methods

The linear Boltzmann equation for the uansport of neuual atoms in highly ionized plasmas is compared with the neutron transport equation. It is shown that, when the plasma is isotropic, the neutral transport equation can be solved with neutron uansport methods. Two formulations of the neutral transport equation that are amenable to solution by existing multigroup neutron uansport codes are given – the multigroup formulation and the multispecies approximation. In the first, the solution provides the neuttal energy (as well as spatial and angular) disuibution, whereas in the latter, the specuum of each neutral species is assumed to be known and fixed. As an illustration, a problem of neutral penetration to and interaction with the plasma of the ST Tokamak is solved. The solution is obtained for a two-species model using the discrete ordinates code ANISN.

Two-Dimensional (r,z) Multigroup Calculation for an Intermediate Spectrum Critical Assembly

The results of two-dimensional (r,z) multigroup calculations of an experimental low-power reactor (PPA-19) are described. Comparison is made between calculated results and experimental measurements of reactivity, power distribution, and sodium activation. Generally good agreement is observed between calculation and experiment. This calculation was performed in 1954 and represents the first application of Roe's two-dimensional multigroup formulation to the analysis of an operating critical assembly. The over-all objective of such calculations is to ascertain the accuracy of two-dimensional multigroup methods in order to facilitate their application to future problems in reactor design. (auth)