Simplified P3 Research Articles

Application of the hierarchical domain decomposition boundary element method to the simplified P3 equation

In this paper, the hierarchical domain decomposition boundary element method (HDD-BEM), which has been developed to solve the diffusion equation, is applied to the simplified P3 (SP3) equation. The HDD-BEM solution for the SP3 equation is provided in the present paper. A computer program, ABEMIE, based on the HDD-BEM is developed, and a two-dimensional one-group anisotropic-scattering benchmark problem is solved with it to verify the present HDD-BEM for the SP3 equation.Through numerical benchmarking, it is shown that the present method results in good agreement with the solution obtained using the existing SPN solver based on the finite element method for both eigenvalue and neutron flux distribution. This benchmark result suggests that the HDD-BEM is suitable for application to the SPN equation.

Treatment of Staggered Mesh for BWR Pin-by-Pin Core Analysis

In this paper, the neutron current interpolation (NCI) method is proposed to treat the staggered mesh on the basis of the Simplified P3 (SP3) theory code. In the NCI method, the response matrix formulation is applied to the SP3 theory code, and neutron partial currents at assembly boundary are interpolated using low-order polynomial functions, i.e., flat, linear or quadratic function. The pin-by-pin fine-mesh core analysis method is considered as a candidate next-generation core calculation method for BWR. Since the number of fuel rods in BWR assembly is increased with the higher burn-up, the different types of assembly may be adjacent in a BWR core. Therefore, when the cell-homogenized pin-by-pin analysis method is applied to the BWR core analysis, it must have a capability to treat the staggered mesh. Our previous study indicates that the SP3 theory is appropriate for pin-by-pin BWR calculations. However, the SP3 theory code that can treat the staggered mesh has not been developed so far. For these reasons, the treatment method of the staggered mesh is investigated to develop such SP3 theory code. The performances of the NCI method are evaluated through comparison with the cell-heterogeneous detailed transport calculations by the method of characteristics (MOC). Two-dimensional, 2 × 2 multi-assembly geometries are used to compare the prediction accuracy of the NCI method. The 2 × 2 multi-assembly geometries consist of 9 × 9 UO2 and 10 × 10 MOX fuel assemblies. The calculation results indicate that the prediction accuracy of the NCI method is similar to that of the detailed transport calculation method.

Treatment of Staggered Mesh for BWR Pin-by-Pin Core Analysis

In this paper, the neutron current interpolation (NCI) method is proposed to treat the staggered mesh on the basis of the Simplified P3 (SP3) theory code. In the NCI method, the response matrix formulation is applied to the SP3 theory code, and neutron partial currents at assembly boundary are interpolated using low-order polynomial functions, i.e., flat, linear or quadratic function. The pin-by-pin fine-mesh core analysis method is considered as a candidate next-generation core calculation method for BWR. Since the number of fuel rods in BWR assembly is increased with the higher burn-up, the different types of assembly may be adjacent in a BWR core. Therefore, when the cell-homogenized pin-by-pin analysis method is applied to the BWR core analysis, it must have a capability to treat the staggered mesh. Our previous study indicates that the SP3 theory is appropriate for pin-by-pin BWR calculations. However, the SP3 theory code that can treat the staggered mesh has not been developed so far. For these reas...

Applicability of the Diffusion and Simplified P3 Theories for Pin-by-Pin Geometry of BWR

The pin-by-pin fine-mesh core calculation method is considered as a candidate next-generation core calculation method for BWR. In this study, the diffusion and simplified P3 (SP3) theories are applied to the BWR pin-by-pin fine-mesh calculation. The performances of the diffusion and SP3 theories for cell-homogeneous pin-by-pin fine-mesh calculation for BWR are evaluated through comparison with a cell-heterogeneous detailed transport calculation by the method of characteristics (MOC). Two-dimensional, 2 × 2 multi-assemblies geometry is used to compare the prediction accuracies of the diffusion and SP3 theories. The 2 × 2 multi-assemblies geometry consists of 9 × 9 UO2 fuel assemblies that have two different enrichment splittings. To minimize the cell-homogenization error, the SPH method is applied for the pin-by-pin fine-mesh calculation. The SPH method is a technique that reproduces a result of heterogeneous calculation using that of homogeneous calculation. The calculation results indicated that the diffusion theory shows a discrepancy larger than that of the SP3 theory on the pin-wise fission rate distribution. In contrast to the diffusion theory, the SP3 theory shows a much better accuracy on the pin-wise fission rate distribution. The computation time using the SP3 theory is about 1.5 times longer than that using the diffusion theory. The BWR core analysis consists of various calculations, e.g., the cross section interpolation, neutron flux calculation, thermal hydraulic calculation, and burn-up calculation. The function of the calculation time for the neutron flux calculation is usually less than half in the typical BWR core analysis. Therefore, the difference in the calculation time between the diffusion and SP3 theories would have no significant impact on the calculation time of the BWR core analysis. For these reasons, the SP3 theory is more suitable than the diffusion theory and is expected to have sufficient accuracy for the 2 × 2 multi-assemblies geometry used in this study, which simulates a typical situation of the actual BWR core.

Applicability of the Diffusion and Simplified P3 Theories for Pin-by-Pin Geometry of BWR

The pin-by-pin fine-mesh core calculation method is considered as a candidate next-generation core calculation method for BWR. In this study, the diffusion and simplified P3 (SP3) theories are applied to the BWR pin-by-pin fine-mesh calculation. The performances of the diffusion and SP3 theories for cell-homogeneous pin-by-pin fine-mesh calculation for BWR are evaluated through comparison with a cell-heterogeneous detailed transport calculation by the method of characteristics (MOC). Two-dimensional, 2 × 2 multi-assemblies geometry is used to compare the prediction accuracies of the diffusion and SP3 theories. The 2 × 2 multi-assemblies geometry consists of 9 × 9 UO2 fuel assemblies that have two different enrichment splittings. To minimize the cell-homogenization error, the SPH method is applied for the pin-by-pin fine-mesh calculation. The SPH method is a technique that reproduces a result of heterogeneous calculation using that of homogeneous calculation. The calculation results indicated that the diff...

Fast Tree Multigrid Transport Application for the Simplified P3 Approximation

Calculation of neutron flux in three-dimensions is a complex problem. A novel approach for solving complicated neutron transport problems is presented, based on the tree multigrid technique and the Simplified P3 (SP3) approximation. Discretization of the second-order elliptic SP3 equations is performed for a regular three-dimensional octree grid by using an integrated scheme. The octree grid is generated directly from STL files, which can be exported from practically all computer-aided design-systems. Marshak-like boundary conditions are utilized. Iterative algorithms are constructed for SP3 approximation with simple coarse-to-fine prolongation and fine-to-coarse restriction operations of the tree multigrid technique. Numerical results are presented for a simple cylindrical homogeneous one-group test case and for a simplistic two-group pressurized water reactor pressure vessel fluence calculation benchmark. In the former homogeneous test case, a very good agreement with 1.6% maximal deviation compared with DORT results was obtained. In the latter test case, however, notable discrepancies were observed. These comparisons show that the tree multigrid technique is capable of solving three-dimensional neutron transport problems with a very low computational cost, but that the SP3 approximation itself is not satisfactory for all problems. However, the tree multigrid technique is a very promising new method for neutron transport.

The impact of the harmonic crystal model on the neutron resonance parameters of 237Np

DYN3D is a nodal diffusion code for 3D steady-state and transient analysis of Light Water Reactor (LWR) cores with hexagonal or square fuel element geometry. In addition to the neutron kinetics, it comprises of a thermal-hydraulics model for flow in parallel coolant channels. Macroscopic cross section data libraries generated with variation of burn-up, reactor poisons concentrations and thermal-hydraulic feedback parameters are linked to the code. Two-group and multi-groups versions of the code are available.Currently, at the Helmholtz-Zentrum Dresden-Rossendorf (HZDR), the DYN3D code is being extended and adopted for the application to block-type High Temperature Gas-Cooled Reactors (HTGRs). In this paper, we give an overview of the latest developments of DYN3D concerning block-type HTGR.The simplified P3 (SP3) transport approximation is implemented into the multi-group DYN3D code to take anisotropy of the neutron flux and heterogeneity of the core more precisely into account. The SP3 method previously implemented into DYN3D for square fuel element geometry of LWR is being extended for hexagonal geometry of the graphite blocks, where the hexagons are subdivided into triangular nodes to be able to perform a systematic mesh refinement.One of the main challenges in cross section generation for the HTGR core calculations is the treatment of the so-called “double heterogeneity”. The modified Reactivity-Equivalent Physical Transformation (RPT) approach is applied in order to eliminate the double-heterogeneity of HTGR fuel elements in the deterministic lattice calculations. The main steps of the RPT method are described. The use of the method for the cross section generation of a simplified HTGR core including its verification is presented.A 3D heat conduction module coupled with a channel-type coolant flow model is implemented to take the temperature reactivity feedback to neutronics physically correctly into account. It is shown that there is significant redistribution of the produced heat by heat conduction between the graphite blocks.

The Simplified P3 Approximation

The simplified P3 (SP3) approximation to the multigroup neutron transport equation in arbitrary geometries is derived using a variational analysis. This derivation yields the SP3 equations along with material interface and Marshak-like boundary conditions. The multigroup SP3 approximation is reformulated as a system of within-group problems that can be solved iteratively. An “explicit” iterative algorithm for solving the within-group problem is described, Fourier analyzed, and shown to be more efficient than the traditional FLIP implicit algorithm. Numerical results compare diffusion (P1), simplified P2 (SP2), and simplified P3 calculations of a mixed-oxide (MOX) fuel benchmark problem to a reference transport calculation. The SP3 approximation can eliminate much of the inaccuracy in the diffusion and SP2 calculations of MOX fuel problems.