Transport Calculations of MOX and UO2 Pin Cells by the Method of Characteristics

A new transport theory code for two-dimensional calculations of both square and hexagonal fuel lattices by the method of characteristics has been developed. The ray tracing procedure is based on the macroband method, which permits more accurate spatial integration in comparison to the equidistant method of tracing. The neutron source within each region is approximated by a linear function and linearly anisotropic scattering can be optionally accounted for. Efficient new techniques for both azimuthal and polar integration are presented. The spatial discretization problem in case of P ₁-scattering has been studied. Detailed analyses show that the P ¹-scattering in case of regular infinite array of fuel cells is significant, especially for MOX fuel, while the transport correction is inadequate in case of real geometry multi-group calculations. Finally, the complicated nature of the angular flux in MOX and UO₂ fuel cells is demonstrated.     

An Approximate Calculation Method of Flux for Spherical and Cylindrical Sources with a Slab Shield

An approximate calculation method for unscattered flux from spherical and cylindrical sources with a slab shield is presented. The method is based on modifying the source geometry to a form convenient for calculation: The sphere is approximated by a conical fragment of spherical shell, and the cylinder by a columnar fragment of cylindrical shell. The procedure for manual calculation is relatively simple, and the difference between results of appoximate and exact calculations is less than +5% for large sources (μ ^s R⁰≦ 4), and±15% for small sources (μ_sR⁰≈0.25). Graphs of the functions for approximate calculation are presented. The method has several advantages over Rockwell's method.     

Riemann boundary conditions for the Boltzmann transport equation using arbitrary angular approximations

In this paper a method for resolving the various boundary conditions (BCs) for the first order Boltzmann transport equation (BTE) is described. The approach has been formulated to resolve general BCs using an arbitrary angular approximation method within any weighted residual finite element formulation. The method is based on a Riemann decomposition which is used to decompose the particles’ angular dependence into in-coming and out-going information through a surface. This operation recasts the flux into a Riemann space which is used directly to remove any incoming information, and thus satisfy void boundary conditions. The method is then extended by its coupling with a set of mapping operators that redirect the outgoing flux to form incoming images resembling other specified boundary conditions. These operators are based on Galerkin projections and are defined to enable reflective and diffusive (white) BCs to be resolved. A small number of numerical examples are then presented to demonstrate the method’s ability in resolving void, reflective and white BCs. These examples have been chosen in order to show the method working for arbitrary angled surfaces. Furthermore, as the method has been designed for an arbitrary angular approximation, both _SN$S_N$ and _PN$P_N$ calculations are presented.     

Unscattered Flux from Spherical and Cylindrical Sources with Shell-Shaped Shields

The effect of shell-shaped shield surrounding a spherical or cylindrical source is studied. The shapes of different shields with a common attenuation parameter μt is classified into six types according to the positions of the inner and outer surfaces with respect to the source and the detector. The exact equations and their approximate formulas for unscattered fluxes from sources with shell-shaped shields are derived. The difference between the results from the approximate and the exact calculations is less than ±2% for dense and/or large sources (μ₈R₀?4), and ±20% for small and/or thin sources (μR_o?4). Functions expressing the effect of shield shape are also derived. This effect is very large, and sometimes reaches a factor of from 2 to 10. The results of calculation for spherical sources are compared with those by Ascoli, and it is found that his approximate method has a large error for large sources and/or dense shields (μR₀?4). The results of calculation for cylindrical sources agree well with those by Taylor and Obenshain, and also with Osano v 8 experiments.     

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 study of the stress intensity factors for single or multiple cracks in thick-walled cylinders

The variation of stress intensity factors of a single semi-elliptical crack and multiple semi-elliptical cracks which are radial symmetric or unsymmetric array in an internal pressurized thick-walled cylinder is studied by use of the “frozen-stress” photo-elastic method. The method of determining mixed-mode stress intensity factors K_I and K_II is given. By means of experimental results and the relative results of other authors, the approximate expression for evaluating stress intensity factors of straight border, semi-circular, semi-elliptical internal surface cracks in thick-walled cylinders are presented.     

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.     

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.     

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.     

Iterative schemes for obtaining dominant alpha-modes of the neutron diffusion equation

Two new methods of obtaining dominant prompt alpha-modes (sometimes referred to as time-eigenfunctions) of the multigroup neutron diffusion equation are discussed. In the first of these, we initially compute the dominant K-eigenfunctions and K-eigenvalues (denoted by λ₁, λ₂, λ₃ … etc.; λ₁ being equal to the K_eff) for the given nuclear reactor model, by existing method based on sub-space iteration (SSI) which is an improved version of power iteration method. Subsequently, a uniformly distributed (positive or negative) 1/v absorber of sufficient concentration is added so as to make a particular eigenvalue λ_i equal to unity. This gives ith alpha-mode. This procedure is repeated to find all the required alpha-modes. In the second method, we solve the alpha-eigenvalue problem directly by SSI method. This is clearly possible for a sub-critical reactor for which the inverse of the dominant alpha-eigenvalues are also the largest in magnitude as required by the SSI method. Here, the procedure is made applicable even to a super-critical reactor by making the reactor model sub-critical by the addition of a 1/v absorber. Results of these calculations for a 3-D two group PHWR test-case are given. These results are validated against the results as obtained by a completely different approach based on Orthomin(1) algorithm published earlier. The direct method based on the sub-space iteration strategy is found to be a simple and reliable method for obtaining any number of alpha-modes. Also comments have been made on the relationship between fundamental α and k values.     

International Symposium on Isotope Separation and Chemical Exchange Uranium Enrichment

From the viewpoint of nuclear criticality safety, it is important to comprehend the reactivity of fuel solutions induced by oscillatory movements such as earthquakes. This paper intends to figure out the reactivity of a fuel solution system with a free surface formed by oscillation by evaluating the fluctuation of the neutron multiplication factor (k _eff ) obtained from a static calculation. To fulfill this intension, criticality calculations with reflecting fluid calculation results have been carried out. In the fluid calculations, the finite volume method and the volume of fluid (VOF) method have been applied in tracking the free surface formed by oscillation. The continuous energy Monte Carlo calculation method has been applied in the criticality calculations. As a result, it has been found that the variation patterns of the k _eff and those of the shape of fuel solutions are classified according to oscillation frequency and the ratio of solution height to the width of the tank (H/L). If a sloshing motion is generated, the k _eff fluctuates widely and has a threshold, with which we can classify the fluctuation type of the k _eff , despite the kind of reflector. If H/L is above the threshold, i.e., H/L =0.4, the k _eff fluctuates to a value below that obtained in the resting state. On the contrary, if H/L is below the threshold, the k _eff fluctuates to a value above that obtained in the resting state. This result implies the criticality calculation for a fuel solution with a free surface using the Monte Carlo method may give a slightly smaller threshold than using other approaches.     

Measurements of k-Infinity on Zero-Leakage Zoned System (FCA IV)

The infinite multiplication factor k _∞ of the test zone composition has been measured for a series of FCA IV assemblies with a central test zone composed to produce a k _∞ value close to unity.

For the experimental determination of k _∞, in which some calculations as well as the reactivity worths of the cell and standard sample measured at the center are necessary, a new method of analysis is presented, and is compared with the current method. The new method provides the theoretical basis for k _∞-experiments. It also simplifies the calculations for analysis.

The FCA IV comprises a series of three assemblies of different test zone compositions representing pure uranium, uranium-sodium and uranium-graphite systems, all of which contain a portion of structural stainless steel. From the analyses made with different cross section sets, it was confirmed that the experimental value of k _∞ could be obtained with an accuracy of ±0.01, and in certain cases, ±0.005, for a zoned system with a central test zone with diameter larger than fifteen times the mean free path.     

Investigation on Criticality and Infinite Multiplication Factor of High Conversion Light Water Reactor Using Zone-Type FCA-HCLWR Core Fueled with Enriched Uranium

—A series of reactor physics experiments have been carried out at the FCA to examine the availability of the nuclear data and computational method currently employed to evaluate the nuclear characteristics of the High Conversion Light Water Reactor. Experimental results of the effective and infinite multiplication factors k_eff and k_∞ are compared with the calculated ones for three zone-type FCA-HCLWR cores fueled with enriched uranium. The calculated k_eff and k_∞ values with use of the SRAC system and the cross section set based on the JENDL-2 data file show a good agreement with the measured ones. The calculated-to-experimental (C/E) values for k_eff and k_∞ do not depend on the cell parameters such as the fuel enrichment, the moderator voidage state and the moderator-to-fuel volume ratio, and these values are similar with each other There is also no inconsistency between the C/E values for k_eff and k_∞ : The average C/E values are 989 and 0 988 for k_eff and k_∞ respectively     

parafish: A parallel FE–PN neutron transport solver based on domain decomposition

A new core solver named parafish is presented for the solution of large neutron transport core calculations. The second-order even-parity form of the time-independent Boltzmann transport equation is solved using an innovative algebraic domain-decomposition method. The spatio-angular discretization is performed using non-conforming finite elements and spherical harmonic expansions (P_N method). The parafish code allows one processor to handle more than one domain. This enables proper evaluations of the speed-up. Also, this enables to show that the domain-decomposition method not only performs well in parallel calculations, but also has an inherent acceleration potential. That is, it yields acceleration even without increasing the number of processors.     

Axial SPN and Radial MOC Coupled Whole Core Transport Calculation

The Simplified P_N (SP_N) method is applied to the axial solution of the two-dimensional (2-D) method of characteristics (MOC) solution based whole core transport calculation. A sub-plane scheme and the nodal expansion method (NEM) are employed for the solution of the one-dimensional (1-D) SP_N equations involving a radial transverse leakage. The SP_N solver replaces the axial diffusion solver of the DeCART direct whole core transport code to provide more accurate, transport theory based axial solutions. In the sub-plane scheme, the radial equivalent homogenization parameters generated by the local MOC for a thick plane are assigned to the multiple finer planes in the subsequent global three-dimensional (3-D) coarse mesh finite difference (CMFD) calculation in which the NEM is employed for the axial solution. The sub-plane scheme induces a much less nodal error while having little impact on the axial leakage representation of the radial MOC calculation. The performance of the sub-plane scheme and SP_N nodal transport solver is examined by solving a set of demonstrative problems and the C5G7MOX 3-D extension benchmark problems. It is shown in the demonstrative problems that the nodal error reaching upto 1,400 pcm in a rodded case is reduced to 10pcm by introducing 10 sub-planes per MOC plane and the transport error is reduced from about 150pcm to 10pcm by using SP₃. Also it is observed, in the C5G7MOX rodded configuration B problem, that the eigenvalues and pin power errors of 180 pcm and 2.2% of the 10 sub-planes diffusion case are reduced to 40 pcm and 1.4%, respectively, for SP₃ with only about a 15% increase in the computing time. It is shown that the SP₅ case gives very similar results to the SP₃ case.     

Experimental Determination of Spectral Indices in Low Enriched UO2-H2O Lattices

To get information about the neutron spectrum in low enriched UO₂-H₂O lattices, the spectral indices SI(U^8c/Dy) and SI(U^8c/U^5f) were measured on the basis of the parallel irradiation technique, which basically irradiates activation foils both in a neutron field to be investigated and in a reference field of thermal neutrons. In the present study, a fuel pellet of UO₂ was used for the measurement of activities caused by the neutron capture of ²³⁸U and the fission of ²³⁵U. Besides the technical details of the measurements, the origins of experimental errors are listed with the method how to eliminate them. The measurements were carried out in lattices of different fuel enrichment to demonstrate the capability of the present method, and the experimental results were compared with the calculated ones. It was found that the results of the present measurements are useful to assess the validity of the cell calculations.     

3-D Neutron Transport Benchmarks

Three-dimensional (3-D) neutron transport benchmark problems proposed from Osaka University to NEACRP in 1988 have been calculated by many participants and the results have been summarized. The results of k _eff, control rod worth, and region-averaged group fluxes for proposed four core models calculated by various 3-D transport codes have been compared. There was consistency among the results, when necessary corrections were made. The solutions of the four core models are quite useful as benchmarks for checking the validity of 3-D neutron transport codes.     

Miscellaneous remarks on choice of moments, moment reduction, local- and global-error bounds

Some observations are made on the merits of Legendre, half-range Legendre, Walsh and E_N representations for the angular dependence of neutron transport in finite element calculations. The splicing of transport and diffusion theory calculations is advocated to reduce the complexity of finite element calculations. To assess the accuracy of the splicing method it is necessary to know the error made by a finite element calculation in a local region of the system. This can be done with some precision by the method of bivariational bounds, which gives in a test example an error of for the captures in a locality.     

Coarse-Mesh Three-Dimensional Transport Calculation Method

A coarse-mesh 3-D (X-Y-Z, Hexagonal-Z) discrete ordinates transport calculation method has been developed. This method employs an weighted diamond difference approximation, the weight in which is a function of neutron direction and scalar flux, and can be easily incorpolated into conventional discrete ordinates transport codes. Results obtained in four-group S₄P₀ calculations on simple fast reactors indicate that, though the computer time of this method has increased by 30–50% compared with that of the conventional finite difference method using the same mesh, the errors of k_eff and the power distribution are reduced remarkably.