Multigroup Transport Theory Research Articles

Universal modification of vector weighted method of dependent trials

Vector weighted Monte Carlo algorithms for the estimation of linear functionals of set of solutions of systems of the 2nd kind integral equations are studied. A universal modification of the vector weighted method of dependent trials with branching of the Markov chain trajectory relative to the elements of matrix weight is constructed. It is shown that the computational cost of the constructed algorithms is finite in the case when the solutions of systems of the 2nd kind are bounded. The application of constructed method of dependent trials is presented for some problems of the radiation transfer theory with allowance for polarization and multigroup transport theory.

Coarse Mesh Rebalance Acceleration of Fission Source Iteration for Subcritical Source-Driven Systems

Coarse mesh rebalance (CMR) method is formulated for the acceleration of fission source iteration (FSI) in the multigroup transport theory analysis of subcritical source-driven systems. The within-group equations are solved by diamond-differenced SN method and CMR has been employed also for the acceleration of scattering source iterations. By numerical experiments, carried out in spherical geometry, stability and performance characteristics of the proposed acceleration are determined and assessed. Systems containing a central nonmultiplying source region and an outer multiplying blanket region are considered. CMR is stable provided that the number of coarse mesh regions in the blanket is in a stability interval between a minimum and a maximum value, irrespective of the fine mesh used in SN calculations. Maximum acceleration is obtained for an optimum number of blanket coarse mesh regions which is also independent of the fine mesh. Both the width of the stability interval and the optimum speed-up ratio (execution time ratio of the unaccelerated run to the accelerated one) increases as the system becomes optically thicker. If we let the system approach criticality by increasing the blanket size, the number of FSIs increases dramatically if no acceleration is employed. For such slightly subcritical systems, very high speed-up ratios are obtained with CMR acceleration whose performance is independent of the criticality level.

Treatment of Double Heterogeneity in the Resonance and Thermal Energy Regions in High-Temperature Reactors

The fuel in high-temperature reactors (HTRs) consists of a large number of tiny (a few hundred microns in size) TRISO-coated particles dispersed randomly in a graphite matrix. At the resonances of the major nuclide (238U or 232Th) of the fuel, the neutron mean free path in the fuel is often comparable to the kernel dimensions, and hence the dispersion of particles in the graphite matrix must be treated as a heterogeneous medium for obtaining self-shielded cross sections in the resonance (epithermal) groups. HTRs containing only plutonium as fuel (as may be the case in reactors designed for burning plutonium) have high concentrations of the isotopes 239Pu and 240Pu in the kernels. This fact together with their rather large resonance cross section in the thermal groups results in a very short neutron mean free path in the fuel that is comparable to the kernel dimensions. In such cases the fuel zone must be treated as a heterogeneous medium in the thermal energy region as well. However, the resonance treatment method in libraries such as the WIMS library does not cover the two large resonances of plutonium lying in the thermal region. Instead, a large number of groups are used to cover the details of cross-section variation in this region.The heterogeneity of the fuel region together with the heterogeneous distribution of fuel region, graphite moderator, and coolant is referred to as the double heterogeneity of HTRs. The paper describes work we carried out for addressing these problems. A new method for generating a random distribution of TRISO particles in the fuel zone of a pebble or fuel compact and Monte Carlo calculation of the Dancoff factor in the heterogeneous random medium, required for calculating the self-shielded resonance group cross sections, is presented. Dancoff factors obtained by our method are compared with values available in published literature on the subject. The paper also discusses a new methodology developed to solve the double-heterogeneity problem at the stage of the multigroup transport theory solution, which is particularly important in the thermal region occurring in high-content Pu-fueled HTRs. These features have been incorporated in the WIMS library–based lattice code BOXER3. An option for handling the spherical geometry of the lattice cell of a pebble bed reactor has been added in the code. Results of analysis of a number of HTR lattice cell benchmark problems are presented.

SIMMER extension for multigroup energy structure search using genetic algorithm with different fitness functions

Abstract The multigroup transport theory is the basis for many neutronics modules. A significant point of the cross-section (XS) generation procedure is the choice of the energy groups' boundaries in the XS libraries, which must be carefully selected as an unsuitable energy meshing can easily lead to inaccurate results. This decision can require considerable effort and is particularly difficult for the common user, especially if not well-versed in reactor physics. This work investigates a genetic algorithm-based tool which selects an appropriate XS energy structure (ES) specific for the considered problem, to be used for the condensation of a fine multigroup library. The procedure is accelerated by results storage and fitness calculation speed-up and can be easily parallelized. The extension is applied to the coupled code SIMMER and tested on the European Sustainable Nuclear Industrial Initiative (ESNII+) Advanced Sodium Technological Reactor for Industrial Demonstration (ASTRID)-like reactor system with different fitness functions. The results show that, when the libraries are condensed based on the ESs suggested by the algorithm, the code actually returns the correct multiplication factor, in both reference and voided conditions. The computational effort reduction obtained by using the condensed library rather than the fine one is assessed and is much higher than the time required for the ES search.

A probabilistic study of the influence of parameter uncertainty on solutions of the neutron transport equation

The influence of uncertainty in the scattering and absorption cross sections on the solution of the neutron transport equation is studied using polynomial chaos theory. The uncertainty is defined by means of uniform and log-uniform probability distributions. By expanding the neutron flux in a series of polynomial chaos functions we may reduce the stochastic transport equation to a set of coupled deterministic equations, analogous to those that arise in multi-group neutron transport theory, with the effective multi-group transfer scattering coefficients containing information about the uncertainty. This procedure enables existing transport theory computer codes to be used, with little modification, to solve the problem. Applications are made to a transmission problem and a constant source problem in a slab. In addition, we also study the rod model for which exact analytical solutions are readily available. In all cases, numerical results in the form of mean, variance and sensitivity are given which illustrate how absorption and scattering cross section uncertainty influences the solution of the transport equation.

Multigroup multi-layer models of neutron reflection and transmission for reactor transport calculations with anisotropic scattering

In this article, we extend the one-speed multi-layer models to neutron reflection and transmission developed in our earlier work (de Abreu, M.P., 2005. Multi-layer models to neutron reflection and transmission for whole-core transport calculations, Annals of Nuclear Energy 32, 215) to multigroup transport theory. We begin by considering a two-layer boundary region, and we develop for such a region discrete ordinates models to the diffuse reflection and transmission of neutrons for multigroup nuclear reactor core problems with anisotropic scattering. We perform numerical experiments to show that our models to neutron reflection and transmission can be used to replace efficiently and accurately two nonactive boundary layers in whole-core transport calculations. We conclude this article with an inductive extension of our two-layer results to a boundary region with an arbitrary number of layers.

A Spectral Representation of the Linear Multigroup Transport Problem

In 1981, Kanal and Davies made the transformation of the fundamental equation of the one‐velocity linear transport theory by expanding the scattering function for the problem to be solved as a spectral integral over the complete set Case's eigenfunctions for a previously solved transport problem. The obtained equation represented a singular integral equation containing a spectral integral over the spectrum of the solved problem, whose kernel depends on the difference between the scattering function of the problem to be solved and that of the solved problem. As a matter of fact, the above authors derived mathematical reformulation of the well‐known Case's approach to the solution of the one‐dimensional equation of transport. Moreover, the above authors considered also several examples illustrating the validity of such reformulation. In this paper we generalize these results to the problems of the one‐dimensional linear multigroup transport theory.

Inverse problem of spectral analysis of linear multigroup neutron transport theory in plane geometry

Gelfand-Levitan method of differential equation restoration by its spectral function was developed for one velocity linear transport theory. Here this method is applied to a mulligroup transport theory. The systems of algebraic equations connecting the kernel and the spectral function for the characteristic operator of the linear multigroup transport theory are obtained. Decision problem of these systems is investigated.

A discrete-ordinates solution for multigroup transport theory with upscattering

A recently developed version of the discrete-ordinates method is used along with elementary numerical linear-algebra techniques to establish an efficient and especially accurate solution to a class of multigroup transport problems for which upscattering is an important aspect of the model. The problems considered are defined for finite plane-parallel media, and anisotropic scattering from any group to any group is included in the formulation. Computational details of the solution are discussed, and accurate numerical results for two previously defined test problems are established.

Numerical Validation of the Theory of Coupled Reactors for the Heavy Water Critical Assembly DCA

It has been numerically confirmed that the theory of multi-point coupled reactors developed recently holds exactly in the framework of multi-group transport theory. The validation was performed by calculating the coupling coefficients for a two-region reactor constructed in the DCA, where the inner core is the test fuel region and the outer core is the driver region moderated by heavy water. The change of the multiplication factors, namely the coupling coefficients, due to the change of the water levels of the inner and outer cores are investigated, and the resulting change of the nuclear properties of each core is clarified, by which the usefulness of the coupled reactor theory has been shown.

Numerical Validation of the Theory of Coupled Reactors for the Heavy Water Critical Assembly DCA.

It has been numerically confirmed that the theory of multi-point coupled reactors developed recently holds exactly in the framework of multi-group transport theory. The validation was performed by calculating the coupling coefficients for a two-region reactor constructed in the DCA, where the inner core is the test fuel region and the outer core is the driver region moderated by heavy water. The change of the multiplication factors, namely the coupling coefficients, due to the change of the water levels of the inner and outer cores are investigated, and the resulting change of the nuclear properties of each core is clarified, by which the usefulness of the coupled reactor theory has been shown.

The FN Method for Multigroup Transport Theory with Upscattering

An integral transform technique and the FN method are used to develop solutions to a class of multigroup radiation-transport problems. The multigroup model considered allows an anisotropic scattering law and transfer from any group to any group. Computational aspects of the developed solution are discussed, and especially accurate numerical results are reported for two test cases.

97/00421 Some reciprocity-like relations in multi-group neutron diffusion and transport theory over bare homogeneous regions

97/00421 Some reciprocity-like relations in multi-group neutron diffusion and transport theory over bare homogeneous regions

Some reciprocity-like relations in multi-group neutron diffusion and transport theory over bare homogeneous regions

Some simple reciprocity-like relations that exist in multi-group neutron diffusion and transport theory over bare homogeneous regions are presented. These relations do not involve the adjoint solutions and are directly related to numerical schemes based on an explicit evaluation of the fission matrix.

Three-Dimensional Nodal Diffusion and Transport Theory Methods for Hexagonal-zGeometry

Advanced nodal methods for the solution of the multigroup neutron diffusion and transport theory equations in three-dimensional hexagonal-z geometry are described. The code HEXNOD allows an accurat...

The time-dependent multigroup transport equations in reactor kinetics

A rigorous description of nuclear reactor kinetics usually invokes neutron transport theory. If the multigroup transport theory is considered, the description of space-time nuclear reactor kinetics consists of partial integro-differential equations coupled with ordinary differential equations. The purpose of this paper is to give an iterative scheme for the construction of the solution of the coupled systems and to establish some qualitative analysis of the coupled system. The iterative scheme consists of two monotone sequences, which converge monotonically from above and below to a unique solution, respectively. The qualitative analysis includes the existence and uniqueness of a positive solution, explicit upper and lower bounds of the solution, and sufficient conditions in terms of reactor parameters for subcriticality and supercriticality of the time-dependent system.

Development of a Legendre polynomial nodal model for the multigroup transport theory PN approximation

A polynomial nodal model which uses Legendre expansions for the spatial component of the angular neutron flux was developed for the multigroup P N equations. The development depends upon the least-squares minimization of the residuals using the trial functions over the node. Anlytical expressions were developed for the polynomial coefficients. The odd moments of the angular neutron flux over the half ranges were used at the internal interfaces, and the Marshak boundary condition was used at the external boundaries. Sample problems with fine mesh finite-difference solutions of the diffusion and transport equations were used for comparison with the model. The results compared favorably.

A method for computing the discrete spectrum basic to multi-group transport theory

Elementary considerations are used to develop a computational method for establishing the discrete spectrum for a general multi-group model basic to radiation transport.

Multislab multigroup transport theory with L th the order anisotropic scattering

The special case of a triangular transfer matrix relevant to multigroup transport theory in multislabs with L th order anisotropic scattering is discussed, and the F N method is used to establish accurate numerical results for a test problem.

Multigroup transport theory with anisotropic scattering

The special case of a triangular transfer matrix relevant to multigroup transport theory with L th order anisotropic scattering is discussed. The developed theory reduces the calculation of the reflected and transmitted angular fluxes to a sequence of one-group problems involving only angular fluxes at the boundaries. The theory is then extended to yield similar results at any location within a finite slab. The F N method is used to establish particularly accurate numerical results for a test problem.