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.     

Cellular neural network to the spherical harmonics approximation of neutron transport equation in x–y geometry. Part I: Modeling and verification for time-independent solution

This paper describes a novel method based on using cellular neural networks (CNN) coupled with spherical harmonics method (P_N) to solve the time-independent neutron transport equation in x–y geometry. To achieve this, an equivalent electrical circuit based on second-order form of neutron transport equation and relevant boundary conditions is obtained using CNN method. We use the CNN model to simulate spatial response of scalar flux distribution in the steady state condition for different order of spherical harmonics approximations. The accuracy, stability, and capabilities of CNN model are examined in 2D Cartesian geometry for fixed source and criticality problems.     

A multigroup finite-element solution of the neutron transport equation—I: X-Y geometry

The result of extending a variational finite element method of solving the neutron transport equation, to energy dependence, is reported. Detailed results are given, in the form of tables and graphs, of P₁ and higher-order transport solutions to a number of benchmark problems in X-Y geometry. The accuracy and flexibility of the method are demonstrated. Some suggestions are made for the future development of the computer implementation of the method.     

Numerical solution of neutron kinetics equations using A-stable algorithms

This paper gives a detailed account of both theoretical and numerical investigations which have been conducted in the application of A-stable algorithms to neutron kinetics problems. It is broadly divided into three sections. General considerations on desirable features of a reactor dynamics code are followed by the theoretical background. In order to be self-contained, the stability properties of one-step methods are recalled with emphasis on the A-stability concept introduced by Dahlquist. An algorithm is described, based on the interpolation of exp(z) in the unit disc of the complex plane, which generates A-stable schemes w_nn(z), (n= 1,…) with so-called ‘spectral matching’ properties. Practical reasons limit to w₁₁ (z) its use for the integration of the kinetics equations and the analytical properties of this first order rational approximation to the exponential function are studied. A second class of suitable integration schemes is made of the implicit Runge-Kutta (IRK) family, particularly the subclass of diagonally implicit Runge-Kutta (DIRK) methods which are factorizable. Finally, the numerical results obtained with these algorithms are discussed on a set of four point kinetics problems for both fast and thermal-type reactors.     

Finite element spherical harmonics (PN) solutions of the three-dimensional Takeda benchmark problems

A set of multi-group eigenvalue (K_eff) benchmark problems in three-dimensional homogenised reactor core configurations have been solved using the deterministic finite element transport theory code EVENT and the Monte Carlo code MCNP4C. The principal aim of this work is to qualify numerical methods and algorithms implemented in EVENT. The benchmark problems were compiled and published by the Nuclear Data Agency (OECD/NEACRP) and represent three-dimensional realistic reactor cores which provide a framework in which computer codes employing different numerical methods can be tested. This is an important step that ought to be taken (in our view) before any code system can be confidently applied to sensitive problems in nuclear criticality and reactor core calculations. This paper presents EVENT diffusion theory (P₁) approximation to the neutron transport equation and spherical harmonics transport theory solutions (P₃–P₉) to three benchmark problems with comparison against the widely used and accepted Monte Carlo code MCNP4C. In most cases, discrete ordinates transport theory (S_N) solutions which are already available and published have also been presented. The effective multiplication factors (K_eff) obtained from transport theory EVENT calculations using an adequate spatial mesh and spherical harmonics approximation to represent the angular flux for all benchmark problems have been estimated within 0.1% (100 pcm) of the MCNP4C predictions. All EVENT predictions were within the three standard deviation uncertainty of the MCNP4C predictions. Regionwise and pointwise multi-group neutron scalar fluxes have also been calculated using the EVENT code and compared against MCNP4C predictions with satisfactory agreements. As a result of this study, it is shown that multi-group reactor core/criticality problems can be accurately solved using the three-dimensional deterministic finite element spherical harmonics code EVENT.     

Applications of the multidimensional equations to complex fuel assembly problems

For the analysis of reactors with complex fuel assemblies or fine mesh applications as pin by pin neutron flux reconstruction, the usual approximation of the neutron transport equation by the multigroup diffusion equation does not provide good results. A classical approach to solve the neutron transport equation is to apply the spherical harmonics method obtaining a finite approximation known as the P_L equations. In this line, a nodal collocation method for the discretization of these equations on a rectangular mesh is used in this paper to analyse reactors with MOX fuel assemblies. Although the 3D P_L nodal collocation method becomes feasible due to the improvements in computer hardware, a complete treatment of the detailed structure of the fuel assemblies in actual three-dimensional geometry is still prohibitive, thus, an assembly homogenization method is necessary. A homogenization method compatible with our multidimensional P_L code is proposed and tested performing heterogeneous and homogenized calculations. In this work, we apply the method to 2D complex fuel assembly configurations.     

Sub-cell balanced nodal expansion methods using S4 eigenfunctions for multi-group SN transport problems in slab geometry

A highly accurate S₄ eigenfunction-based nodal method has been developed to solve multi-group discrete ordinate neutral particle transport problems with a linearly anisotropic scattering in slab geometry. The new method solves the even-parity form of discrete ordinates transport equation with an arbitrary S_N order angular quadrature using two sub-cell balance equations and the S₄ eigenfunctions of within-group transport equation. The four eigenfunctions from S₄ approximation have been chosen as basis functions for the spatial expansion of the angular flux in each mesh. The constant and cubic polynomial approximations are adopted for the scattering source terms from other energy groups and fission source. A nodal method using the conventional polynomial expansion and the sub-cell balances was also developed to be used for demonstrating the high accuracy of the new methods. Using the new methods, a multi-group eigenvalue problem has been solved as well as fixed source problems. The numerical test results of one-group problem show that the new method has third-order accuracy as mesh size is finely refined and it has much higher accuracies for large meshes than the diamond differencing method and the nodal method using sub-cell balances and polynomial expansion of angular flux. For multi-group problems including eigenvalue problem, it was demonstrated that the new method using the cubic polynomial approximation of the sources could produce very accurate solutions even with large mesh sizes.     

The finite-element method for multigroup neutron transport: Anisotropic scattering in 1-D slab geometry

Proof-tests on 1-D multigroup neutron transport problems are reported for strong anisotropic scattering. These tests have been undertaken as part of the validation of the 3-D multigroup finite-element transport code fel tran for ansisotropic scattering media. To illustrate the treatment of within-group and intergroup anisotropic scattering in the finite-element method the relevant theory is outlined. Ingroup scattering is checked using the backward-forward-isotropic (BFI) scattering law for source and eigenvalue problems. With this law anisotropic scattering problems can be transformed into equivalent isotropic scattering problems. In this way the well-validated isotropic scattering version of fel tran is used to validate the anisotropic version. Intergroup scattering effects are checked by solving few-group source problems for P₁ and P₃ scattering and the BFI scattering law. For P₁ and P₃ scattering checks are made with the discrete-ordinate finite-difference code anisn and the spherical harmonics finite-difference code marc/pn. For the BFI scattering law comparison is made with two-group exact solutions of Williams (1985) for 1-D systems.     

A mixed P1–DP0 diffusion theory for planar geometry

A variational analysis is used to derive a mixed P₁–DP₀ (P₁ spherical harmonics–double P₀ spherical harmonics) angular approximation to the time-independent monoenergetic neutron transport equation in one-dimensional planar geometry. This mixed angular approximation contains a space-dependent weight factor α(x) that controls the local angular approximation used at a spatial point x: α(x) = 1 yields the standard P₁ (diffusion) approximation, α(x) = 0 gives the standard DP₀ approximation, and 0 < α(x) < 1 produces a mixed P₁–DP₀ angular approximation. The diffusion equation obtained differs from the standard P₁ diffusion equation only in the definition of the diffusion coefficient. Standard Marshak incident angular flux boundary conditions are also obtained via the variational analysis. We examine the use of this mixed angular approximation coupled with the standard P₁ approximation to more accurately treat material interfaces and vacuum boundaries. We propose a simple but effective functional form for the weight factor α(x) that removes the need for the user to specify the value. Numerical results from several test problems are presented to demonstrate that significant improvements in accuracy can be obtained using this method with essentially no computational penalty.     

Transport calculations of neutron energy spectra in a graphite cylinder with a D-T source

Neutron energy spectra resulting from the transport of 14.7 MeV neutrons from a collimated D-T source through a graphite cylinder, have been calculated with the discrete-ordinates 2-D transport dot 4.2 code, with multigroup cross-sections generated using the njoy code from the ENDF/B (IV & V) libraries. The results confirm the conclusion of Goldfeld et al. (1985), that energy spectra at mesh points close to the axis of the system, in front of the collimated beam, consist mainly of one-collision contributions of elastically or inelastically scattered neutrons. Investigation of the dependence of the calculated spectra on the order of truncation of the Legendre polynomials expansion of the flux and of the cross-sections (i.e. the order of scattering) leads to the following observations:

• 1.(a) the P₆ or P₇ approximations seem to be adequate enough for flux calculations, with less than 3% error, in spite of the high degree of the source and the cross-section's anisotropy;
• 2.(b) the calculation error is reduced significantly by increasing the order of scattering from P₄ to P₇, mainly in mesh points close to the axis and of those energies in which the anisotropy of the elastic and discrete level inelastic scattering processes is most pronounced.

Finally, the dot 4.2 calculations are compared with Monte Carlo mcnp calculations; both calculated spectra are in a good agreement.     

Application of Finite Element Method to Two-Dimensional Multi-Group Neutron Transport Equation in Cylindrical Geometry

The finite element method is applied to the spatial variables of multi-group neutron transport equation in the two-dimensional cylindrical (r, z) geometry. The equation is discretized using regular rectangular subregions in the (r, z) plane. The discontinuous method with bilinear or biquadratic Lagrange's interpolating polynomials as basis functions is incorporated into a computer code FEMRZ. Here, the angular fluxes are allowed to be discontinuous across the subregion boundaries.

Some numerical calculations have been performed and the results indicated that, in the case of biquadratic approximation, the solutions are sufficiently accurate and numerically stable even for coarse meshes. The results are also compared with those obtained by a diamond difference S _n code TWOTRAN-II. The merits of the discontinuous method are demonstrated through the numerical studies.     

The integral-transform method for solving neutron transport problems with general anisotropic scattering in a cylinder of finite height

In this paper, we present a mathematical technique for solving the integral transport equation for the criticality of a homogeneous cylinder of finite height. The purpose of the present paper is two-fold : firstly, to show that our earlier formalism can be generalized to any order of anisotropy, and secondly to generate the numerical results, which could serve as benchmarks when scattering is linearly anisotropic. We expand the scattering function in spherical harmonics to retain the Lth order of anisotropy. Thereafter, we write the integral transport equations for the Fourier-transformed spherical harmonic moments of the angular flux. In conformity with the integral-transform method for multidimensional geometry, the kernels of these integral equations are represented in their respective factorized form, which consists of a series of products of suitable spherical Bessel functions. The Fourier-transformed spherical harmonic moments are also represented in their separable form by expanding them in a series of products of spherical Bessel functions, commensurate with the symmetry of finite cylindrical geometry. The criticality problem for the cylinder of finite height is then posed as a matrix eigen value problem whose eigen vector is composed of the expansion coefficients mentioned above. The general matrix element is expressed as a product of certain integrals of Bessel functions, which can be evaluated by recursion relations derived in this paper. Finally, a comparison between the present benchmark results and S_N results (twotran) in (r–z) goemetry is presented when scattering is linearly anisotropic.     

Enrichment of 6Li by Ion Exchange

Elementary solutions to the energy-dependent Boltzmann equation with a one-term degenerate scattering kernel are derived in plane geometry, and the weight function W (z) is obtained which makes these solutions mutually orthogonal over the half-range interval of the continuum. The weight function greatly facilitates determination of the expansion coefficients in general solutions and is applied to the problems in infinite half space.

The diffusion length (discrete space eigenvalue) υ₀ is exactly expressed by using the halfrange characteristic function X(z). In a 1/υ-absorbing medium, as the absorption concentration q increases from zero to a critical value g^*, the diffusion length decreases from infinity to the end of the continuum 1/σ_min. For q≥q^*, v₀ vanishes and the neutron density can be represented by the transient term alone, whose exact expression is obtained.     

Calculation of extrapolation distances by low order Chebyshev polynomial approximation of transport equation in slab geometry

In this study, the problem of extrapolated end point has been studied in one-speed neutron transport equation with isotropic scattering by using the Chebyshev polynomial approximation which is called T_N method. Assuming neutrons of one speed, extrapolated end point are calculated for the uniform finite slab using Mark and Marshak type vacuum boundary conditions. It is shown that low order T_N method gives very good results of low order spherical harmonics approximation and diffusion theory for extrapolation of the flux of neutrons leaking from the medium. We present an alternative method which is similar to P₁ method to calculate the extrapolation distances z₀. Moreover, we prefer new solution of transport equation in one-dimensional slab geometry.     

Development of Discrete Ordinates SN Code in Three-Dimensional (X,Y, Z) Geometry for Shielding Design

A discrete ordinates transport code ENSEMBLE in (X, Y, Z) geometry has been developed for the purpose of shielding calculations in three-dimensional geometry. The code has some superior features, compared with THREETRAN which is the only code of the same kind so far developed. That is, the code can treat higher order anisotropic scattering and employs a coarse mesh rebalancing method. Moreover it has a negative flux fix-up routine using a variable weight diamond difference equation scheme and has a ray-effect fix-up option using a fictitious source based on S_N→P_N-1 conversion technique. Formulations for these advanced features in three-dimensional space have been derived.

As the demonstration of the capabilities of the code, several numerical analyses and an analysis of an annular duct streaming experiment in JRR-4 at Japan Atomic Energy Research Institute, have been performed.

As a result of these analyses, confirmation has been obtained for the prospect of applicability of ENSEMBLE to practical shielding design.     

Application of the TN method to critical slab problem for one-speed neutrons with forward and backward scattering

The critical slab problem which includes isotropic forward and backward scattering has been studied in one-speed neutron transport equation using first kind of Chebyshev polynomials. The critical half-thicknesses are computed for different degrees of c and forward and backward scattering with Mark and Marshak boundary conditions in the uniform finite slab. It is shown that T_N method gives accurate results in one-dimensional geometry and the results are agreement P_N approximation.     

Thermal conductivity of (U,Pu,Np)O2 solid solutions

The thermal conductivities of (U_,Pu_,Np)O₂ solid solutions were studied at temperatures from 900 to 1770 K. Thermal conductivities were obtained from the thermal diffusivity measured by the laser flash method. The thermal conductivities obtained below 1400 K were analyzed with the data of (U,Pu,Am)O₂ obtained previously, assuming that the B-value was constant, and could be expressed by a classical phonon transport model, λ = (A + BT)⁻¹, A(z₁, z₂) = 3.583 × 10⁻¹ × z₁ + 6.317 × 10⁻² × z₂ + 1.595 × 10⁻² (m K/W) and B = 2.493 × 10⁻⁴ (m/W), where z₁ and z₂ are the contents of Am- and Np-oxides. It was found that the A-values increased linearly with increasing Np- and Am-oxide contents slightly, and the effect of Np-oxide content on A-values was smaller than that of Am-oxide content. The results obtained from the theoretical calculation based on the classical phonon transport model showed good agreement with the experimental results.