N. Tsoulfanidis

Conformal Mapping and Hexagonal Nodal Methods —I: Mathematical Foundation

The conventional transverse integration method of deriving nodal diffusion equations does not satisfactorily apply to hexagonal nodes. The transversely integrated nodal diffusion equation contains nonphysical singular terms, and the features that appear in the nodal equations for rectangular nodes cannot be retained for hexagonal ones. A method is presented that conform ally maps a hexagonal node to a rectangular node before the transverse integration is applied so that the resulting nodal equations are formally analogous to the ones for rectangular nodes without the appearance of additional singular terms. Utilizing the invariance of the Laplacian diffusion operator under conformal mappings, it is shown that the diffusion equation for a homogeneous hexagonal node can be transformed to the diffusion equation for an inhomogeneous rectangular node. The inhomogeneity comes in through a smoothly varying mapping scale function, which depends only on the geometry. The steps of conformal mapping from a hexagonal node to a rectangular node are given, and the mapping scale function is derived, evaluated, and applied to nodal equation derivations.

An integral form of the variational nodal method

Abstract An integral form of the variational nodal method is formulated, implemented, and tested. The method combines an integral transport treatment of the even-parity flux within the spatial node with an odd-parity spherical harmonics expansion of the Lagrange multipliers at the node interfaces. The response matrices that result from this formulation are compatible with those in the VARIANT code at Argonne National Laboratory. Spatial discretization within each node allows for accurate treatment of homogeneous or heterogeneous node geometries. The integral method is implemented in Cartesian x-y geometry and applied to three benchmark problems. The method’s accuracy is compared to that of the standard spherical harmonic formulation of the variational nodal method, and the CPU and memory requirements of the two approaches are compared and contrasted. In general, for calculations requiring higher-order angular approximations, the integral method yields solutions with comparable accuracy while requiring substantially less CPU time and memory than the spherical harmonics approach.

A Finite Subelement Generalization of the Variational Nodal Method

Abstract The variational nodal method is generalized by dividing each spatial node into a number of triangular finite elements designated as subelements. The finite subelement trial functions allow for explicit geometry representations within each node, thus eliminating the need for nodal homogenization. The method is implemented within the Argonne National Laboratory code VARIANT and applied to two-dimensional multigroup problems. Eigenvalue and pin-power results are presented for a four-assembly Organization for Economic Cooperation and Development/Nuclear Energy Agency benchmark problem containing enriched UO2 and mixed oxide fuel pins. Our seven-group model combines spherical or simplified spherical harmonic approximations in angle with isoparametric linear or quadratic subelement basis functions, thus eliminating the need for fuel-coolant homogenization. Comparisons with reference seven-group Monte Carlo solutions indicate that in the absence of pin-cell homogenization, high-order angular approximations are required to obtain accurate eigenvalues, while the results are substantially less sensitive to the refinement of the finite subelement grids.

Formulas Giving Buildup Factor for Double-Layered Shields

Formulas that give absorbed dose buildup factors for two-layered shields have been developed based on gamma-ray absorption buildup factors computed with the Monte Carlo Neutral Particle Transport Code System (MCNP). The shielding materials considered were water, lead, steel, concrete, and some of their combinations for two-layered shields with thicknesses between 1 to 10 mfp. Gamma energy considered ranged from 0.5 to 6 MeV. The formulas reproduce MCNP results with a difference of <10%, in most cases <3%.

Neutron Fluence at the Pressure Vessel of a Pressurized Water Reactor Determined by the MCNP Code

Pressure vessel fluence and reaction rates for dosimetry foils in the cavity surrounding the pressure vessel of a pressurized water reactor were determined with a Monte Carlo calculation using the MCNP code. Source neutrons were sampled from a position probability distribution derived from the utility-provided normalized assembly segment power output. The MCNP model was based on one-eighth core symmetry. Source segment spatial biasing, energy cutoff spatial importance functions, and weight windows were employed as variance reduction techniques. Computed reaction rates were compared with measured ones and in one case to discrete ordinates transport code calculations. Computed reaction rates matched the measured ones within ±10% for 21 of 33 cases and within +15% for 26 of 33 cases. Neutron flux and fluence >0.1111 and MeV at the pressure vessel location were computed to <10% statistical uncertainty. The estimated maximum fluence per cycle was found to be of the order of 10 17 n/cm 2 .

Spherical harmonics — Finite element treatment of neutron transport in cylindrical geometry

Abstract A variational spherical harmonics (DPN) method is used in conjunction with the finite element method to solve the transport of neutron beams. The method is developed for finite axisymmetric cylindrical geometry. In this model, the angular distribution along the path of neutron travel is separated into two sub-ranges of the forward and the backward moving particles. The system of the model equations is cast in terms of an even and an odd vector analogue to the even-parity transport technique. The finite element method is applied in space to solve the resulting coupled system of equations. A source-free cylinder with specified incident flux on the z = 0 surface is considered in this study. The model is based on the DP1 and DP3 orders of expansion. The study demonstrates the advantages of the DPN method in accounting for the discontinuity of the angular flux in the direction of particles with no approximation. Results for the scalar flux are presented in isodose curve (contour) form for homogeneous and non-homogeneous media with different absorbing and purely scattering media. Numerical results compared favorably with those obtained by the exact solutions.

Neutron Energy Spectrum Calculations in Three PWR

Transport calculations of neutron energy spectra were performed for three PWR: ANO-1, ANO-2 and McGuire-1. The fluxes were calculated In R-θ geometry utilizing the eighth core symmetry. Taking the R-θ flux as a basis, a 3-D flux was synthesized by multiplying the R-θ flux with a “leakage correction” factor obtained from a 2-D R-Z flux divided by a 1-D flux. Using cross sections derived from VITAMIN-C, the neutron spectrum was obtained in terms of 26 energy groups. The calculated neutron spectra were used to compute reaction rates for several threshold reactions which were in turn compared with measured ones obtained by Irradiating foils placed at several in-vessel and ex-vessel locations. Integral fluxes for neutron energy above 100 keV and above 1 Mev, as well as corresponding fluences for the lifetime of the plant, have been obtained for the location in front of the PV.

Exposure Buildup Factors of UO2 Using the Monte Carlo Method

When the gamma dose rate around an irradiated nuclear reactor fuel element is calculated, it is important to know the attenuating characteristics of the fuel element itself, one of them being the buildup factor. Exposure buildup factors of uranium dioxide (UO{sub 2}) for ten gamma-ray energies (0.050 to 10.0 MeV) have been computed for ten material thicknesses (0.5 to 10.0 mean free paths) using the MCNP code. The accuracy of the MCNP model was checked by computing the buildup factors of oxygen and uranium and comparing these results with the data given in the literature for these elements. The results indicate that the UO{sub 2} exposure buildup factors, for the energies and distances studied, are close to those of uranium.

CADMIUM CUTOFF ENERGY FOR MCNP MODELING OF DOSIMETRY REACTIONS

Reactor pressure vessel neutron fluence calculations are verified by comparing calculated reaction rates of dosimetry foils assumed to be placed in the cavity surrounding the pressure vessel with corresponding experimentally determined dosimetry reaction rates at the same location. The irradiated foils used are bare and cadmium covered. The cadmium-covered foils see a different neutron flux than the uncovered foils. Computer models computing reaction rates for the foils must approximate the effect of the cadmium foil cover. The MCNP code modeling of reactor cavity dosimetry reactions to validate the computed neutron flux hitting the pressure vessel of a pressurized water reactor (PWR) requires the choice of an appropriate cadmium cutoff energy. Since the neutron spectrum is harder in the cavity than in the core, the same cadmium cutoff energy may not be universally applied. The correct cutoff energy for the cavity of a PWR was computed by using MCNP4a and a representative cavity spectrum. Four cadmium-covered foil reactions were analyzed, and the appropriate energy cutoff was determined to be [approximately] 0.6 eV for [approximately]0.5-mm-thick cadmium covers.

Nuclear Energy, Introduction

In terms of technical progress of the human species/society, the second half of the twentieth century is marked by two developments: the computer and nuclear energy. And the two are related since progress in the development and applications of nuclear energy owes a lot to the power of computations made possible by the digital computer.