Two-dimensional nodal transport method for triangular geometry

The advanced nodal method for solving the multi-group neutron transport equation in two-dimensional triangular geometry is developed. To apply the transverse integration procedure, an arbitrary triangular node is transformed into a regular triangular node using coordinate transformation. The angular distributions of intra-node neutron fluxes and its transverse-leakage are represented by the S_N quadrature set. The spatial distributions of neutron flux and source in the regular triangle are given approximately by an orthogonal quadratic polynomial, and the spatial expansion of transverse-leakage is approximated by a second-order polynomial. To establish a stable and efficient iterative scheme, the improved nodal-equivalent finite difference algorithm is used. The results for several benchmark problems demonstrate the higher capability of the method to yield the accurate results in significantly smaller computing times than those required by the standard finite difference method and the finite element spherical-harmonics method.     

Transmission probability method for solving neutron transport equation in three-dimensional triangular-z geometry

This paper presents a transmission probability method (TPM) to solve the neutron transport equation in three-dimensional triangular-z geometry. The source within the mesh is assumed to be spatially uniform and isotropic. At the mesh surface, the constant and the simplified P₁ approximation are invoked for the anisotropic angular flux distribution. Based on this model, a code TPMTDT is encoded. It was verified by three 3D Takeda benchmark problems, in which the first two problems are in XYZ geometry and the last one is in hexagonal-z geometry, and an unstructured geometry problem. The results of the present method agree well with those of Monte-Carlo calculation method and Spherical Harmonics (P_N) method.     

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.     

A new spectral nodal method based on Larsen’s extended diamond approach

In this article, we describe a new spectral nodal method for solving discrete ordinates (S_N) neutron transport problems with anisotropic scattering for arbitrary order N of angular quadrature. The key to our new spectral nodal method is a consistent derivation of nonstandard auxiliary equations that relate angular neutron fluxes only in the upwind directions. These nonstandard equations are angularly coupled extensions of very basic auxiliary equations proposed by Edward W. Larsen in his extended diamond scheme of solving S₂ problems in the presence of scattering and free from spatial truncation error. The resulting method here is also free from spatial truncation error and, in contrast to previously developed spectral nodal methods, it is compatible with an efficient use of iteration on the scattering source and is free from the storage of cell-edge angular fluxes.     

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.     

Spherical harmonics moments of neutron angular flux for spherically symmetric systems

Spherical harmonics moments of the neutron angular flux are calculated for spherically symmetric systems. Neutron angular fluxes are determined by employing the S_N method. Spherical harmonics moments of S_N method calculations are compared to the singular eigen function expansion computations. Total flux and current determined by both of these methods show a good agreement, so the higher order moments are also compared. This study was undertaken to investigate accuracy in calculation of the spherical harmonics moments by the purely numerical S_N 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 closed-form solution for the two-dimensional transport equation by the LTSN nodal method in the energy range of Compton effect

In the present work we report on a closed-form solution for the two-dimensional Compton transport equation by the LTS_N nodal method in the energy range of Compton effect. The solution is determined using the LTS_N nodal approach for homogeneous and heterogeneous rectangular domains, assuming the Klein–Nishina scattering kernel and a multi-group model. The solution is obtained by two one-dimensional S_N equation systems resulting from integrating out one of the orthogonal variables of the S_N equations in the rectangular domain. The leakage angular fluxes are approximated by exponential forms, which allows to determine a closed-form solution for the photons transport equation. The angular flux and the parameters of the medium are used for the calculation of the absorbed energy in rectangular domains with different dimensions and compositions. In this study, only the absorbed energy by Compton effect is considered. We present numerical simulations and comparisons with results obtained by using the simulation platform GEANT4 (version 9.1) with its low energy libraries.     

An exponential spectral nodal method for one-speed X,Y-geometry deep penetration discrete ordinates problems

Presented here is an exponential spectral nodal method applied to deep penetration X,Y-geometry heterogeneous neutron transport problems in the discrete ordinates (S_N) formulation. This numerical method uses the spectral Green's function (SGF) scheme for solving the one-dimensional transverse-integrated S_N exponential nodal equations with no spatial truncation error. Based on the physics of deep penetration problems, we approximate the transverse leakage terms by exponential functions. We show in two numerical experiments that the SGF-exponential nodal method (SGF-ExpN) generates very accurate results when compared to the conventional transport nodal methods for coarse-mesh deep penetration S_N problems, specially in highly absorbing media.     

Application of the PN method to the matrix form transport equation

In this note, the neutron group fluxes are calculated through the implementation of the matrix form transport equation solution by the P_N method and the main objective is to obtain accurate numerical results. Some results for two sample problems are graphically presented. A neutron flux to dose conversion is also proposed to simulate irradiation planning.     

Wavelet-based angular dependence analysis of the heterogeneous calculation on MOX fuel lattice

A wavelet-based transport method is developed to satisfy the high order angular approximation, which has been proved to be necessary in the heterogeneous calculation of MOX fuel lattice. Based on the new angular discretization scheme, the angular dependence of flux is analysed to find out the origin of complicated angular anisotropy and its effects on the heterogeneous calculation. Both of the geometric and neutronic effects are investigated quantitatively to find out the angular dependence in heterogeneous calculations. Comparisons between the traditional S_N angular discretization scheme and wavelet-based scheme are analysed to indicate the challenges brought from the MOX fuel lattice heterogeneous calculation. An effective solution is given by using wavelets in the angular discretization of neutron transport equation. Improvements of high order angular approximation are suggested.     

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.     

Thermal Neutron Spectra in Light Water Poisoned with Cadmium and/or Indium

The thermal neutron spectra in light water of slab geometry poisoned with Cd and/or In were measured by the time of flight method, using a 20-MeV electron linear accelerator. The thermal spectra were simulated to those in the Pu built-up core of a commercial light water reactor corresponding to a fuel burnup of about 15,000 MWD/T. The results of measurements were compared with calculations based on the S ₄ method using the Haywood scattering law. Fairly good agreement was obtained between the calculated and measured results except in a limited range of energy above the 0.176 eV resonance of Cd. It is concluded that the P ₁ components of the source neutrons as well as the neutron scattering kernel play a significant role in the calculation of the thermal neutron spectra with large flux gradients, and that the scattering kernel of light water based on the Haywood model will be accurate enough to evaluate the infinite multiplication constant k_∞ of light water reactor cores with high fuel burnup within an error of about — 0.17%, as estimated from the uncertainty in the spectrum calculation in the region above the Cd resonance. It is also emphasised from the two- dimensional S ₄ calculations that the effect of reentrant hole perturbation should be evaluated quantitatively in the interpretation of the measured angular neutron spectra produced within finite media.     

Hydrogen Permeation through Incoloy-800

The poloidal distribution of the first wall 14 MeV neutron flux and the tritium breeding ratio in a Tokamak fusion reactor were calculated using Monte Carlo method. The poloidal distribution of the 14 MeV neutron flux in the first wall was found to be quite different from that of the primary incident flux. The tritium breeding ratio calculated by the Monte Carlo method became about 5% larger than the value obtained from S_N transport calculations.     

Natural convection heat transfer from horizontal rod bundles in liquid sodium. Part 2: Correlations for horizontal rod bundles based on theoretical results

Natural convection heat transfer from horizontal rod bundles in N_xm × N_ym arrays (N_xm, N_ym = 5–9) in liquid sodium was numerically analyzed for three types of the bundle geometry (in-line rows, staggered rows I and II). The unsteady laminar two-dimensional basic equations for natural convection heat transfer caused by a step heat flux were numerically solved until the solution reaches a steady state. The PHOENICS code was used for the calculation considering the temperature dependence of thermophysical properties concerned. The surface heat fluxes for each cylinder were equally given for a modified Rayleigh number, R_f, ranging from 0.0637 to 63.1 (q = 1×10⁴ to 7×10⁶ W/m²). S_x/D and S_y/D for the rod bundle, which are the ratios of the distance between center axes on the abscissa and the ordinate to the rod diameter, respectively, were ranged from 1.6 to 2.5 on each bundle geometry. The spatial distribution of Nusselt numbers, Nu, on horizontal rods of a bundle was clarified. The average value of Nusselt number, Nu_av, for three types of bundle geometry with various values of S_x/D and S_y/D were calculated to examine the effect of the array size, S/D and R_f on heat transfer. The bundle geometry for the higher Nu_av value under the condition of S_x/D×S_y/D = 4 was examined by changing the ratio of S_x/S_y. A correlation for Nu_av for the three types of bundle geometry above mentioned including the effects of S_x/D and S_y/D was developed. The correlation can describe the theoretical values of Nu_av for the three types of bundle geometry in N_xm × N_ym arrays (N_xm, N_ym = 5–9) for S_x/D and S_y/D ranging from 1.6 to 2.5 within 10% difference.     

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.     

Flux expansion nodal method for solving multigroup neutron diffusion equations in hexagonal-z geometry

A flux expansion nodal method (FENM) has been developed to solve multigroup neutron diffusion equations in hexagonal-z geometry. In this method, the intranodal fluxes are expanded into a set of analytic basis functions for each group. In order to improve the nodal coupling relations, a new type of nodal boundary conditions is proposed, which requires the continuity of both the zero- and first-order moments of partial currents across the nodal surfaces. The response matrix technique is used for the iterative solution of the nodal diffusion equations, which greatly improves the computational efficiency. The numerical results for a series of benchmark problems show that FENM is a very accurate and efficient method for the prediction of criticality and nodal power distributions in the reactors with hexagonal assemblies.     

An Improvement of the Transverse Leakage Treatment for the Nodal SN Transport Calculation Method in Hexagonal-Z Geometry

To overcome the divergent behavior of the NSHEX code, a nodal S_N code for hexagonal geometry, for some transport calculations, an improvement has been made in the calculation of the axial leakage. The axial leakage, previously calculated by using the quadratic transverse leakage approximation (QLA), is calculated by a new method of analytically treating the spatial distribution within a node, based on the axial homogeneity of the ordinary core. The verification tests were performed for the KNK-II model geometry of the NEACRP 3-D Neutron Transport Benchmarks and the large assembly-size KNK-II model. It is found that k_ett values obtained by introducing the new method agree with the reference Monte Carlo calculation results within 0.1% Δk/k for the KNK-II model, although the QLA method did not converge for two cases. Furthemore the new method succeeded in calculations for the large assembly-size model, in which the QLA method failed for all cases. Thus the new method has been found accurate and convergence achieved for the cases in which the QLA method failed.     

On the integration scheme along a trajectory for the characteristics method

The issue of the integration scheme along a trajectory which appears for all tracking-based transport methods is discussed from the point of view of the method of characteristics. The analogy with the discrete ordinates method in slab geometry is highlighted along with the practical limitation in transposing high-order S_N schemes to a trajectory-based method. We derived an example of such a transposition starting from the linear characteristic scheme. This new scheme is compared with the standard flat-source approximation of the step characteristic scheme and with the diamond differencing scheme. The numerical study covers a 1D analytical case, 2D one-group critical and fixed-source benchmarks and finally a realistic multigroup calculation on a BWR-MOX assembly.