Generalized Equivalence Theory Research Articles

Advancements to generalized equivalence theory for preserving cross-sections of whole core simulations

The conventional two-step approach, without the use of an equivalence method, can introduce significant error for the simulation of non-LWRs (Light Water Reactors), especially fast reactors, due to the large leakage and high anisotropic neutron distribution. To improve upon the accuracy of the two-step approach, a 3D whole core model is simulated with a transport method to generate region-wise homogenized cross-sections (XS). These XS can then be used in a diffusion whole core solver during the second step to extend the application to cycle and transient analysis. Discontinuity factors (DFs) are then introduced to improve the accuracy during the simulation of the second step. With properly generated DFs from Generalized Equivalence Theory (GET), the region-averaged solutions from the first transport step can then be reproduced by the second step diffusion solver. The Monte Carlo method was selected to perform the whole core transport simulation to generate region-wise XS. However, simulating whole core problems with Monte Carlo may result in poor statistics near the peripheral region especially for partial current tallies. This paper introduces an advancement to GET to reproduce region-wise solutions for select regions when the reaction rates and surface currents have good statistics in these regions but poor statistics in other regions. A reference high-fidelity model was constructed using the Serpent 2 Monte Carlo code based on the EBR-II benchmark evaluation and verification was carried out using the TriPEN-4 method in PARCS. The results show that it is possible to reproduce the exact eigenvalue and power distributions of whole core problems in a feasible manner.

A study on the applicability of simplified few-group GET (Generalized Equivalence Theory) to cylindrical molten salt fast reactor

One of the essential requirements for molten salt reactor (MSR) design is methodology for analyzing multi-physics phenomena, such as the behavior of liquid fuel. In the research of Molten Salt Fast Reactors (MSFRs), the Neutron Diffusion Equation (NDE) is widely employed. This study introduces a method to enhance the accuracy of neutronic analysis of MSFR using the NDE. Using the simple structure of MSFR and the characteristics of liquid nuclear fuel, it is intended to enable the application of the simple equivalence method, which is difficult to perform on the existing fast reactor. A straightforward yet effective approach named the simplified Generalized Equivalence Theory (simplified GET) is proposed for applying flux-volume-weighted homogenized cross-sections and representative Discontinuity Factor (DF) values obtained at material. This approach, while similar to the General Equivalence Theory (GET) method, significantly simplifies the enhancement of accuracy in reactor analysis, minimizing computational efforts. Our investigation spans from the initial core to the burned core, ensuring the applicability of this simple method throughout the reactor's operational life. The proposed method demonstrates promising results, offering a viable solution to improve the accuracy of NDE-based calculations in MSFRs.

A leakage correction with burnup-dependent GPS method in two-step core depletion analysis

The aim of this study is to introduce a burnup-dependent GPS† method in the pin-resolved two-step reactor depletion analysis and to verify how effectively leakage correction works in terms of reactor eigenvalue (keff) and power distribution. The burnup-dependent GPS function sets were generated based on the different color-set models with various conditions, considering the adjacent non-target fuels with fresh or burned fuels at each burnup step. In the functionalization stage, the cross-section-dependent SPH factors were parameterized considering the current-to-flux ratio (CFR) indicating the integrated leakage information and the spectral index (SI) representing the fluctuations in the neutron spectrum. The computational performance for the burnup-dependent GPS method was evaluated with a modified PWR benchmark problem for a UOX-loaded small reactor. The lattice calculations for the group constants generation and the reference core calculations were performed using the DeCART2D transport code. The nodal depletion calculations were performed with an in-house nodal expansion method (NEM) code implemented with a hybrid coarse mesh finite difference (HCMFD) solver. The two-step macroscopic depletion calculations were performed based on 4 types of conditions named; (1) w/o GPS, (2) w/BOC-GPS, (3) w/BU-GPS(F), and (4) w/BU-GPS(F&B), where letters ‘F’ and ‘B’ within the parenthesis indicate ‘Fresh’ and ‘Burned’ respectively. As a result, by individually applying two different GPS functions in the boundaries and internal area of the assembly (i.e., w/BU-GPS(F&B)), it shows a good performance with low errors at a sufficiently acceptable level in terms of reactivity and the pin-wise power distribution.†GPS: Generalized equivalence theory (GET) Plus Super-homogenization (SPH).

Implementation and optimization of hybrid parallel strategy in HNET

For the self-developed three-dimensional whole-core High-fidelity NEutron Transport calculation program HNET, although the numerical acceleration algorithms can improve the computation performance of high-fidelity neutron transport in terms of algorithms and models, it still faces some critical issues such as long computing time and enormous memory requirement. The rapid development of high-performance clusters provides a foundation for the application of massively parallel computing. Most current MOC programs are based on a single-type variable to achieve efficient parallelism, and spatial domain decomposition methods are the most common parallel schemes. However, its parallelism is limited and cannot fully utilize the current state-of-the-art computer resources. To solve this problem, hybrid parallel strategies are implemented in HNET to further expand the parallel degree, improve the speed of computation, and reduce the memory requirement. A hybrid MPI/OpenMP method based on domain decomposition and characteristic rays is proposed for the method of characteristics (MOC). For domain decomposition, the simulation domain is divided into spatial subdomains, with each subdomain handled by different processes. On this basis, the characteristic ray parallelism is implemented taking advantage of the inherent parallelism of the characteristic rays. Meanwhile, the optimization of the hybrid parallel strategy further improves the computation speed by eliminating atomic operations and using private pointer arrays and other techniques. In addition, in the framework of generalized equivalence theory (GET) based two-level coarse mesh finite difference method (CMFD), there is also a certain time consumption in solving CMFD linear system. Hence, for CMFD, a hybrid MPI/OpenMP method based on domain decomposition and secondary domain decomposition can be used. Using the secondary domain decomposition method, each subdomain is divided into sub-subdomains, with each sub-subdomain handled by threads, which allows CMFD to utilize the resources of characteristic ray parallelism in MOC, and also increases the speed of CMFD computation. Numerical results show that for both steady-state and transient calculations, the hybrid MPI/OpenMP in HNET can further expand the parallelism and accelerate the computation. It can take full use of parallel resources and achieve large-scale parallelism.

An improved homogenization method for the treatment of strong absorbers in pebble-bed HTGR

In pebble-bed high temperature gas-cooled reactors (HTGRs), strong absorbers including control rods and absorber balls are placed in the reflector. The strong absorber regions need to be homogenized for the convenience of downstream whole-core neutronics diffusion calculation. Previously, a homogenization method based on the generalized equivalence theory (GET) was studied. This paper proposes an improved homogenization method by employing a more rigorous discontinuity factor (DF) model, for more accurate treatment of the strong absorbers in the HTGR reflector. The proposed method has been implemented in the PANGU code developed by Institute of Nuclear and New Energy Technology (INET), Tsinghua University. Numerical results suggest that the newly proposed homogenization method achieves good agreement with the heterogeneous Monte Carlo transport solution and experimental result.

Methodology for Discontinuity Factors Generation for Simplified P3 Solver Based on Nodal Expansion Formulation

The Simplified Spherical Harmonic (SPN) approximation was first introduced as a three-dimensional (3D) extension of the plane-geometry Spherical Harmonic (PN) equations. A third order SPN (SP3) solver, recently implemented in the Nodal Expansion Method (NEM), has shown promising performance in the reactor core neutronics simulations. This work is focused on the development and implementation of the transport-corrected interface and boundary conditions in an NEM SP3 solver, following recent published work on the rigorous SPN theory for piecewise homogeneous regions. A streamlined procedure has been developed to generate the flux zero and second order/moment discontinuity factors (DFs) of the generalized equivalence theory to minimize the error introduced by pin-wise homogenization. Moreover, several colorset models with varying sizes and configurations are later explored for their capability of generating DFs that can produce results equivalent to that using the whole-core homogenization model for more practical implementations. The new developments are tested and demonstrated on the C5G7 benchmark. The results show that the transport-corrected SP3 solver shows general improvements to power distribution prediction compared to the basic SP3 solver with no DFs or with only the zeroth moment DF. The complete equivalent calculations using the DFs can almost reproduce transport solutions with high accuracy. The use of equivalent parameters from larger size colorset models show a slightly reduced prediction error than that using smaller colorset models in the whole-core calculations.

Fine Mesh Finite Difference acceleration method based on the Generalized Equivalence Theory for the 2-D MOC transport calculation

Normally, the pin-cell-based Coarse Mesh Finite Difference (CMFD) method is directly used to accelerate the high-fidelity neutron transport calculations. However, the acceleration effect still has a bottleneck due to the fact that only the pin-cell homogenized flux can be used to update the fine-mesh flux, such as the flat-source-region flux for MOC calculation. In this paper, a Fine Mesh Finite Difference (FMFD) method based on the Flat Source Region (FSR) level is proposed and implemented in a two-dimensional transport code using Method Of Characteristics (MOC). FMFD could avoid the mapping error from pin-cell-wise mesh to fine FSR mesh. However, the solution of the FMFD system is time-consuming since the scale of the FMFD system is much larger than that of the pin-cell-based CMFD system. Therefore, Two-Level (TL) acceleration scheme is performed where the pin-cell-based CMFD is used to accelerate the solution of FMFD system. Both of these two acceleration methods are based on the Generalized Equivalence Theory (GET). Fourier analysis and numerical results show that FMFD with TL is more efficiency than conventional CMFD.

A STUDY OF LEAKAGE-CORRECTED TWO-STEP METHOD BASED ON THE NODAL EQUIVALENCE THEORY FOR FAST REACTOR ANALYSIS

The conventional two-step method based on the generalized equivalence theory (GET) cannot be directly applied to the fast reactor analysis since the assumption of the space-energy separability is not very valid due to a relatively long neutron mean free path. This study aims to develop a leakage-corrected two-step method for the fast reactor analysis with the aid of the albedo-corrected parameterized equivalence constants (APEC) method. The critical idea of the APEC method is to correct the homogenized group constants (HGCs) including discontinuity factors (DFs) during the nodal calculation through predetermined APEC functions. The APEC functions are functionalized in terms of the normalized leakage parameters such as a current-to-flux (CFR) ratio so that they can correct the cross-sections (XSs) and discontinuity factors by reflecting the in-situ neutron leakage information of the nodal analysis. The feasibility of the APEC-corrected two-step method was investigated by solving 5-group diffusion equations for a two-dimensional sodium-cooled fast reactor with a 6-triangle finite difference method. The 5-group HGCs for fuel assemblies were determined by using a continuous-energy Monte Carl code, and the conventional assembly discontinuity factors are also introduced for each hexagonal fuel assembly. First of all, it was demonstrated that the simple FDM scheme could reproduce the reference nodal quantities with the GET. And the APEC functions are formulated using the reference solutions to evaluate the feasibility of simple APEC functional for both XSs and DFs. Then, a smaller color-set problem was defined to determine practical APEC functions for the original benchmark, and various numerical evaluations are performed in terms of the k-eff value and nodal power distribution.

IMPROVEMENT TO NEM SP3 MODELLING AND SIMULATION

Accurate reactor core steady state safety analysis requires coupling between thermal-hydraulics and three dimensional multigroup pin by pin neutronics. Concerning the neutronics modeling, the Nodal Expansion Method (NEM) code is developed at North Carolina State University in the framework of high fidelity multiphysics coupling with CTF. NEM includes a simplified third-order Spherical Harmonic (SP3) solver. In this work, the solver has been improved by incorporating higher order scattering matrix library. The boundary conditions were corrected with one dimensionalP3theory and a consistent coupling coupling between zeroth- and second-order flux moments was established. Two methods for generating second order discontinuity factors (DFs) has ben developed, one based on the Generalized Equivalence Theory (GET) and one based on Parial Current Equivalence Theory (PCET). DFs were generated with three lattice sizes: single pin, 2 pins and assembly level. These developments were tested using the C5G7 benchmark. The results of the SP3 solver improvement, by usingP2andP3scattering cross sections, show a 50% decrease in the eigenvalue (keff) prediction error compared to the reference transport solution. The GET DFs are applied in the C5G7 core pin by pin calculation and are compared with PCET DFs. The results show that PCET have a better performance in global results (eigenvalue). Concerning the different lattice sizes studies, the results show that DFs generated in smalll colorsets can improve local solutions. However, in order to reveal strong global trends, DFs should be generated in a larger corloset representative of the whole core. For the core calculations, DFs generated with the three colorsets together with an additional mixed type DFs were tested. For the mixed type, DFs generated from assembly size lattice were used for the internal interfaces and DFs generated from 2 pins size lattice were used for the assemblies boundary interfaces. These mixed DFs outperformed all the other configurations indicating that they manage to accomplish a satisfying compromise between global and local trends.

TIME-DEPENDENT HOMOGENIZATION FOR PRESSURIZED HEAVY-WATER REACTORS

A new time-dependent homogenization approach that accounts for inter-assembly leakage has recently been proposed. The new technique extends Generalized Equivalence Theory (GET) to transient simulations through the use of time-dependent, leakage-corrected discontinuity factors that are calculated at each time step by means of a global-local iterative approach to account for the effect of neighbouring nodes so that highly heterogeneous cores are more accurately modelled than when employing single-node, zero-node-boundary-current Assembly Discontinuity Factors (ADFs). The technique has been previously tested for a one-dimensional, two-energy-group, BWR-like benchmark. The present work expands the analysis to a one-dimensional, two-energy-group, Pressurized Heavy-Water Reactor (PHWR) configuration. The PHWR configuration consists of 22 fuel nodes bounded on either side by two nodes of heavy-water (D2O) reflector. Each fuel node spans 28.575 cm and is a one-dimensional stylized representation of a 37-element, natural uranium fuel bundle with D2O coolant residing in a pressure tube that in turn resides in a calandria tube surrounded by D2O moderator. A simple transient induced by instantaneous half-core voiding of the D2O coolant is studied. Three types of calculations are performed: A reference, heterogeneous-node, fine-mesh calculation, a standardly-homogenized-node calculation and a GET-homogenized-node (using ADFs) calculation. The root-mean-square percent errors introduced by standard homogenization and ADF-based homogenization for kinetics calculations in PHWR cores are found to be 4% and 5%, respectively, after 0.5 s. This suggests that the use of a time-dependent homogenization method is desirable, and its use is shown to reduce the RMS errors to a maximum of 0.003% over the course of the transient. The conclusion is that although PHWR cores are not extremely heterogeneous, the accuracy of transient modelling for PHWRs is improved when using time-dependent homogenization over conventional ADFs and that the newly-developed time-dependent homogenization method promises to offer substantial improvements in accuracy for transient results with particular relevance to safety analyses.

Multi-group equivalence in subgroup method based on generalized equivalence theory

Multi-group heterogeneous reaction rates calculated by the subgroup method in resonance calculations are not reproduced, when subgroup-collapsed one-group homogeneous calculations are conducted for the same problem, giving rise to a multi-group equivalence effect. In this paper, a new non-iterative equivalence method introducing partial current discontinuity factors (PCDFs) is proposed to resolve this effect and employed into the Bamboo-Lattice code. Its validity was tested and verified through several fixed-source as well as eigenvalue problems for both single pin-cell and assembly geometries. The numerical results show that preservation of scalar flux, partial currents, neutron leakage and reaction rates is guaranteed by implementing this new method. Moreover, the computational time comparison for different equivalence methods shows that the newly proposed non-iterative equivalence method promises a significantly less computational cost compared to the traditional iterative super-homogenization (SPH) method in treating multi-group equivalence effect.

Advanced two-level CMFD acceleration method for the 3D whole-core high-fidelity neutron adjoint transport calculation

In the 2D/1D method, a global adjoint CMFD based on the generalized equivalence theory is built to synthesize the 2D radial MOC adjoint and 1D axial NEM adjoint calculation and also to accelerate the iteration convergence of 3D whole-core adjoint transport calculation. Even more important, an advanced yet accurate two-level (TL) CMFD acceleration technique is proposed, in which an equivalent one-group adjoint CMFD is established to accelerate the multi-group adjoint CMFD and then to accelerate the 3D whole-core adjoint transport calculation efficiently. Based on these method, a new code is developed to perform 3D adjoint neutron flux calculation. Then a set of VERA and C5G7 benchmark problems are chosen to verify the capability of the 3D adjoint calculations and the effectiveness of TL CMFD acceleration. The numerical results demonstrate that acceptable accuracy of 2D/1D adjoint calculations and superior acceleration of TL CMFD are achievable.

Generalized Equivalence Theory Used with Spatially Linear Sources in the Method of Characteristics for Neutron Transport

A recent hybrid stochastic-deterministic calculation scheme using Monte Carlo–tallied group cross sections in a deterministic solver uses the best of both worlds for accurate and fast reactor agnostic transport simulations. However, neglecting the angular dependence of group cross sections induces large self-shielding errors in resonance groups, causing a large reactivity bias up to 300 pcm in light water reactors. To recover this error, we introduce a two-scale assembly transport calculation scheme: cross sections are tallied at the assembly level, while equivalence parameters are computed in a two-dimensional (2-D) pin cell system. We validate a novel equivalence method based on jump conditions on angular fluxes by comparing to the well-established superhomogenization method for 2-D and three-dimensional (3-D) linear source method of characteristics calculations. Test cases include 2-D and 3-D assemblies of two different enrichments with homogeneous and discretized cross-section discretizations. The linear source approximation enables using coarse source-region discretization for these hot zero-power problems. Both equivalence techniques perform similarly, recover the reactivity bias, and achieve near preservation of reaction rates, supporting this multiscale approach to equivalence.

Two-level time-dependent GET based CMFD acceleration for 3D whole core transient transport simulation using 2D/1D method

A pin-resolved kinetics transport simulation based on the transient fixed-source problem (TFSP) using 2D/1D method has been implemented in this work. The 3D transient fixed source problem is transformed into 2D TFSP and 1D TFSP coupled by the leakage term, while the MOC method and NEM method are used to solve the 2D and 1D TFSP respectively. And a global pin-level time-dependent generalized equivalence theory (GET) based coarse mesh finite difference method (CMFD) is used to synthesize the 2D radial MOC and 1D axial NEM transient calculations, and also to accelerate the iteration convergence of 3D whole-core transient transport calculation. Even more important, an advanced yet accurate two-level time-dependent CMFD acceleration technique is proposed, in which an equivalent one-group CMFD is established to accelerate the multi-group CMFD TFSP solutions, and then to accelerate the 3D whole-core transient transport calculation efficiently. 2D TWIGL and 3D C5G7-TD benchmarks are selected to verify the accuracy and the acceleration effect of the 2D/1D method and the two-level CMFD acceleration. Numerical results indicate that acceptable accuracy of 2D/1D transient calculations and superior acceleration of TL CMFD are achieved.

GET-based time-dependent lattice homogenization using a global-local approach

To reduce computational effort, production-type full-core neutronic calculations are usually performed using few-group diffusion theory and a node-homogenized core model, whereby each homogenized node is assumed to have uniform neutronic properties. Homogenized-node properties are determined such that the flux and reaction rates obtained from few-group diffusion calculations using the node-homogenized model are as close as possible to their counterparts obtained using a detailed-geometry, many-group, full-core transport calculation; what constitutes the heterogeneous, reference results. Generalized Equivalence Theory (GET) is widely used to determine the node-homogenized neutronic properties, namely macroscopic cross sections and discontinuity factors. For time-independent calculations, if the reference results are available, then GET provides a way to generate node-homogenized few-group macroscopic cross sections and discontinuity factors such that the node-homogenized diffusion model produces node fluxes and reaction rates identical to the reference node-integrated fluxes and reaction rates. Since full-core reference results are not usually available, node-homogenized parameters are usually generated from single-node detailed-geometry many-group transport calculations using reflective boundary conditions. The corresponding (approximate) discontinuity factors are known as “assembly” discontinuity factors (ADFs). Methods have also been proposed to account for the true node-boundary conditions without resorting to full-core reference calculations. This work extends the equivalence between the heterogeneous model and the node-homogenized model to time-dependent problems and investigates, specifically, the effect of accounting for the time dependence of discontinuity factors compared to the case of using ADFs. A simple, one-dimensional, two-group diffusion model is used for this preliminary investigation. The heterogeneous reference solution is found using fine-mesh finite differences, whereas the node-homogenized solution is found using coarse-mesh (one mesh per node) finite differences. Results show that accounting for the time dependence of the discontinuity factors can reduce node-flux errors from ∼30% to ∼0.025%.

Pinwise Diffusion Solution of Partially MOX-Loaded PWRs with the GPS (GET PLUS SPH) Method

The generalized equivalence theory (GET) plus superhomogenization (SPH) [GET Plus SPH (GPS)] method, which is a new leakage correction method for the pin-by-pin reactor analysis of light water reactors, has been applied to benchmarks for partial loading of mixed oxide (MOX) fuel in pressurized water reactor (PWR) cores. In the GPS method, the pinwise, cross section–dependent SPH factors are parameterized as a function of normalized leakage, i.e., current-to-flux ratio. As partially MOX-loaded PWRs usually have a stiff gradient of neutron flux on nodal interfaces, the original GPS functions for UO2 cores are slightly modified to take into account the strong spectral interaction. To determine the coefficients of the GPS function, several colorset models are considered to obtain fitting data. In this work, the two-dimensional method of characteristics–based DeCART2D code is used for both colorsets and reference core calculations. The GPS method is implemented in an in-house, pin-by-pin diffusion solver with the pinwise coarse mesh finite difference method. To evaluate the performance of the GPS method on partially MOX-loaded PWRs, the Korea Advanced Institute of Science and Technology (KAIST) 1A benchmark is analyzed in this work. In addition, various small and large variants of the KAIST 1A benchmark are also analyzed using the same GPS functions to demonstrate the general applicability of the predetermined GPS functions. Based on the comprehensive results of this work, it is concluded that the GPS method can clearly improve the accuracy of the conventional GET-based, two-step, pin-by-pin core analyses.

3D Real-Time Simulation of Reactor Core based on Discontinuity Factor Ratio

During the development of full scope simulator (FSS) in nuclear power plant, it is difficult to satisfy with high precision and efficiency of the 3D distribution calculation of reactor power. The accuracy of conventional coarse mesh finite difference method (CMFD) cannot meet the requirement if the nodal size is large. According to the generalized equivalence theory (GET), the finite difference solution of diffusion equation based on discontinuity factor correction was deduced, the calculation method of the discontinuity factor and boundary albedo are presented through the process of assembly homogenization. Through the calculation of typical LMW benchmark problem, the simulation results showed that using CMFD with discontinuity factor ratio (CMFD-DFR) to solve the time-dependent neutron kinetic equation, can not only improve the simulation precision, and can keep the calculation speed, and can meet the requirement of reactor core real-time model of FSS.

3D whole-core neutron transport simulation using 2D/1D method via multi-level generalized equivalence theory based CMFD acceleration

A new 2D/1D method with multi-level generalized equivalence theory (GET) based coarse mesh finite difference (CMFD) acceleration was proposed for the whole core transport calculation in this paper. Fouressential features of this new 2D/1D methods are as follows: (1) Two new defined factors, nodal discontinuity factor (NDF) and modified diffusion coefficient factor (MDF), were defined in this new GET based CMFD method to achieve equivalence between CMFD and the transport solutions and especially insure that only positive coupling coefficients would occur in the CMFD linear system; (2) For accelerating the convergence of the CMFD, a multi-level acceleration scheme was implemented and an innovative self-developed RSILU preconditioned GMRES solver was developed to solve the CMFD linear system; (3) Within the new CMFD framework, 2D MOC and 1D two-node nodal expansion solvers were developed for 3D whole core transport calculation and the sub-plane technique was applied to minimize the nodal error successfully while maintain a reasonable computing time; (4) For implementation, transverse leakage splitting technique was applied to avoid the total source to become negative, and domain decomposition method based on MPI was implemented to take advantages of high performance clusters. The accuracy of this 2D/1D method via multi-level GET based CMFD and the effectiveness and acceleration performance of the multi-level CMFD method were examined for the well-known C5G7 benchmarks problems. The numerical results demonstrate that superior accuracy is achievable and the multi-level acceleration schemeis efficient and enhances the converge speed of both the GET based CMFD acceleration and the MOC calculation.

Pin-wise homogenization for SPN neutron transport approximation using the finite element method

The neutron transport equation describes the distribution of neutrons inside a nuclear reactor core. Homogenization strategies have been used for decades to reduce the spatial and angular domain complexity of a nuclear reactor by replacing previously calculated heterogeneous subdomains by homogeneous ones and using a low order transport approximation to solve the new problem. The generalized equivalence theory for homogenization looks for discontinuous solutions through the introduction of discontinuity factors at the boundaries of the homogenized subdomains. In this work, the generalized equivalence theory is extended to the Simplified P N equations using the finite element method. This extension proposes pin discontinuity factors instead of the usual assembly discontinuity factors and the use of the simplified spherical harmonics approximation rather than diffusion theory. An interior penalty finite element method is used to discretize and solve the problem using discontinuity factors. One dimensional numerical results show that the proposed pin discontinuity factors produce more accurate results than the usual assembly discontinuity factors. The proposed pin discontinuity factors produce precise results for both pin and assembly averaged values without using advanced reconstruction methods. Also, the homogenization methodology is verified against the calculation performed with reference discontinuity factors.

Evaluation of Pin-Cell Homogenization Techniques for PWR Pin-by-Pin Calculation

In general, spatial homogenization, energy group condensation, and angular approximation are all included in the homogenization process. For the traditional pressurized water reactor (PWR) two-step calculation, the assembly homogenization with assembly discontinuity factors plus two-group (2G) neutron diffusion calculation have been proved to be a very efficient combination. However, this changes and becomes unsettled for the pin-by-pin calculation. Thus, this paper evaluates pin-cell homogenization techniques by comparison with the two-dimensional one-step whole-core transport calculation. For the homogenization, both the generalized equivalence theory (GET) and the superhomogenization (SPH) methods are studied. Considering the spectrum interference effect between different types of fuel pin cells, both 2G and 7-group (7G) structures are condensed from the 69-group WIMS-D4 library structure. For practical reactor core applications, the low-order angular approximations, including the diffusion and the SP3 methods, are compared with each other to determine which one is accurate enough for the PWR pin-by-pin calculation. Numerical results have demonstrated that both the GET and the SPH methods work effectively in pin-cell homogenization. In consideration of the spectrum interference effect, the 7G structure is sufficient for the pin-by-pin calculation. Compared with the diffusion method, the SP3 method can decrease the errors dramatically.