A new technique for spectral interference correction on pin-by-pin BWR core analysis

A new correction technique to capture the spectral interference effect on collapsed cross sections, which focuses on application to the pin-by-pin boiling water reactor (BWR) core analysis, is proposed. The spectral interference effect, which is caused by adjacent loadings of different types of fuel assemblies, has relationship with variations of neutron leakage in each pin-cell from the viewpoint of neutron balance. Variation of neutron leakage affects neutron spectrum and thus the neutron leakage is considered to be important to correct coarse-group cross sections used in core calculations. We focus on the neutron leakage in each pin-cell and use it as a correction index (i.e., a leakage index (LI)), which is defined as the volume-averaged neutron leakage in a pin-cell. By utilizing the leakage index, we represent the variations of coarse-group cross sections as the linear combination of LIs. In order to verify and discuss the applicability of the present correction technique, two-dimensional benchmark calculations considering typical characteristics of BWR cores are carried out. From the calculation results, the present correction technique well reproduces the reference coarse-group cross sections and improves the calculation accuracies.     

超级均匀化方法用于球床氟盐冷却高温堆扩散计算

球床氟盐冷却高温堆的控制棒位于侧反应射层内,存在无裂变中子源且受堆芯泄漏谱强烈影响的强吸收体区域扩散计算难题。超级均匀化方法(Super Homogenization,SPH)被用于对氟盐球冷却床堆侧反射层中控制棒区域的强吸收体进行等效均匀化处理,同时堆芯除控制棒区域外采用谱修正方法(Spectra Modification,SM),将输运计算的结果作为基准进行验算。结果表明,SM-SPH模型能有效地计算球床氟盐冷却高温堆反射层控制棒价值及通量分布,并且较常规的SPH方法能更好地处理棒间干涉效应。     

Reduction of MOC discretization errors through a minimization of source ratio variances

A new technique to reduce discretization errors for ray tracing in the method of characteristics (MOC) is proposed focusing on depletion calculations of single and multi-assembly geometries. In order to efficiently carry out depletion calculations, a calculation scheme using the superhomogenization (SPH) method can be used. However, the discretization errors are caused by changes of neutron sources and total cross sections according to a depletion. This fact means that improvement of accuracy cannot be expected by the calculation scheme with the SPH method when changes of the above parameters are significant. In order to mitigate this problem, a new approach is developed. In the new approach, the discretization errors are reduced by minimizing a variance of a certain parameter which is composed of a ratio of neutron source to total cross section. The verification results suggest that accuracy is degraded by the SPH method as expected especially in a geometry where neutron sources and total cross sections are drastically changing through a depletion. On the other hand, the new approach gives more accurate results compared to the conventional MOC in all calculation cases. Consequently, improvement of calculation efficiency by the new approach is confirmed.     

基于特征线法计算的超细群慢化方程求解方法

为实现对复杂几何、复杂能谱组件的精细计算,提出了一种基于特征线的超细群慢化方程求解方法。通过耦合特征线法中的固定源计算,在共振能量范围内建立超细群慢化方程,通过精细能谱获得复杂结构下的共振自屏截面。对典型压水堆栅元问题、带有温度分布的栅元问题、燃料内部存在不均匀性的栅元问题以及板状燃料组件问题进行了计算。结果表明,基于特征线的超细群慢化方程求解方法可精确计算复杂几何、复杂能谱问题,为共振计算提供基准。     

SARAX-FXS程序针对非均匀快堆堆芯计算中的改进及应用

SARAX-FXS程序是基于确定论方法,适用于快谱堆芯组件能谱、均匀化参数计算的程序。由于快堆中组件空间自屏的非均匀效应不可忽视,本文将基于一维圆柱、平板几何的碰撞概率方法加入SARAX-FXS模块,并以等效一维模型计算组件的均匀化参数。为保证能群归并前后的核反应率守恒,在组件计算中引入超级均匀化(SPH)因子修正截面。采用快堆基准题MET-1000对程序的计算结果进行验证,结果表明,与参考解相比,SARAX-FXS的一维计算模块具有较高的精度,特征值计算相对偏差在100~200pcm之间。堆芯计算结果显示,引入SPH因子可提高特征值计算的精度约300pcm,功率分布的均方根误差可从约3%下降至约1%。     

A Pebble Bed Reactor cross section methodology

A method is presented for the evaluation of microscopic cross sections for the Pebble Bed Reactor (PBR) neutron diffusion computational models during convergence to an equilibrium (asymptotic) fuel cycle. This method considers the isotopics within a core spectral zone and the leakages from such a zone as they arise during reactor operation. The randomness of the spatial distribution of fuel grains within the fuel pebbles and that of the fuel and moderator pebbles within the core, the double heterogeneity of the fuel, and the indeterminate burnup of the spectral zones all pose a unique challenge for the computation of the local microscopic cross sections. As prior knowledge of the equilibrium composition and leakage is not available, it is necessary to repeatedly re-compute the group constants with updated zone information. A method is presented to account for local spectral zone composition and leakage effects without resorting to frequent spectrum code calls. Fine group data are pre-computed for a range of isotopic states. Microscopic cross sections and zone nuclide number densities are used to construct fine group macroscopic cross sections, which, together with fission spectra, flux modulation factors, and zone buckling, are used in the solution of the slowing down balance to generate a new or updated spectrum. The microscopic cross-sections are then re-collapsed with the new spectrum for the local spectral zone. This technique is named the Spectral History Correction (SHC) method. It is found that this method accurately recalculates local broad group microscopic cross sections. Significant improvement in the core eigenvalue, flux, and power peaking factor is observed when the local cross sections are corrected for the effects of the spectral zone composition and leakage in two-dimensional PBR test problems.     

Application and Research of Super Homogenization Method for PWR Core Pin-by-pin Calculation

Super homogenization (SPH) method was used as the homogenization technique of PWR core Pin-by-pin calculation. For the fuel assembly, the traditional SPH method was used to generate the group constants. For the reflector-assembly, the spatial leakage dependent SPH method was evaluated. The problem of non-convergence in the iteration of SPH factors calculation was solved by improved SPH method which can keep the neutron leakage and reaction activity conserved. Based on KAIST benchmark, the performance of the SPH method applied in PWR core Pin-by-pin calculation was evaluated. Numerical results demonstrate that compared with the results of traditional assembly-homogenized calculation, the Pin-by-pin calculations have the higher accuracy.     

超级均匀化方法在压水堆堆芯Pin-by-pin计算中的应用与研究

在压水堆堆芯Pin-by-pin计算中,采用超级均匀化（SPH）方法作为均匀化技术,对燃料组件传统SPH因子进行计算,生成了Pin-by-pin等效均匀化参数。针对存在中子泄漏现象的反射层组件,研究了与空间泄漏相关的SPH方法,在保证反应率守恒的基础上,同时保证各栅元各能群的中子泄漏率守恒,解决了存在中子泄漏时SPH因子迭代计算的不收敛问题,生成了反射层组件的等效均匀化参数。基于KAIST基准题,分析了压水堆堆芯Pin-by-pin计算中应用SPH因子的堆芯计算精度。数值结果表明,与传统组件均匀化计算方法相比,应用SPH方法的压水堆堆芯Pin-by-pin计算的计算精度更高。     

A note on application of superhomogénéisation factors to integro-differential neutron transport equations

The present article focuses on the application of the SPH factor method to the integro-differential neutron transport equation. While leakage-related parameters are arbitrarily corrected by the SPH factors, the correction procedure for these parameters affects the calculation accuracy. We treat two correction procedures named the simultaneous correction and the direct correction, and compare them with each other in one-dimensional colorset assembly problems. Through numerical testing, we find that the simultaneous SPH correction gives better accuracy than the direct SPH correction, and the higher-order SPH-corrected calculations show better accuracy than the low-order ones. Furthermore, to consider the flux discontinuity between different types of assemblies, the improved SPH method proposed by Yamamoto and the SPH method with the Selengut normalization condition are also tested. Numerical results reveal that the both methods significantly improve the calculation accuracy and that the latter method is more robust than the former method.     

Measurement and Analysis of Cadmium Ratios of Activation Foils in Thermal Critical Assembly (KUCA B 3/8″P36EU-NU-EU)

Abstract

To examine the applicability of foil activation technique for the estimation of neutron spectrum in a thermal reactor, Cd ratios of 8 activation foils (Au, Th, Dy, In, Mn, W, D.U. and E. U.) were measured in the void at the core center of KUCA B3/8″P36 EU-NU-EU assembly. The Cd ratios were analyzed with SRAC code system using 107 group cross sections based on ENDF/B4. To make the correction for polyethylene plates facing to the void to the calculated spectrum with 2-dimensional (r-z) diffusion model, softening factor calculated with 1-dimensional infinite slab model was introduced. This model gave almost same neutron spectrum as that without this correction. For the model which distributes atoms of Al sheath and support cylinder homogeneously into simulating materials, and using pointwise (fine group) cross sections for Au, Th, W and D.U., the calculated values except for W and D.U. almost agreed with the experimental ones. For W and D.U. C/E values were–1.1. Since Cd ratios are sensitive to the change in neutron spectrum except for D.U., this method is useful to judge the appropriateness of calculated neutron spectrum.     

Resonance Treatment Based on Ultra-fine-group Spectrum Calculation in the AEGIS Code

The resonance calculation method using the ultra-fine-group spectrum calculations in the AEGIS code is explained in detail. By a simple benchmark problem, it is verified that the effect of anisotropic scattering on effective cross-sections is not very large and the isotropic scattering source approximation is adequate in practical resonance calculations in LWRs. Furthermore, some efficient numerical algorithms in the ultra-fine-group calculations to reduce the computation time without large degeneration of accuracy are presented. In addition, the SPH method for energy collapsing of cross-sections is adopted in the AEGIS code to reduce the error of energy collapsing.

Through the comparison with continuous-energy Monte-Calro calculation in the pin-cell geometry, the validity of the resonance treatment in the AEGIS code is verified.     

Applicability of angular flux discontinuity factor preserving region-wise leakage for integro-differential transport equation

In the current core analysis, spatial homogenization is utilized to reduce the computational time. The discontinuity factor (DF) is one of the effective correction factors to reduce spatial homogenization error. The DF in diffusion equation is widely used; on the other hand the DF in transport equation has not been put to practical use although several efforts have been carried out. In this paper, the angular flux discontinuity factor (AFDF) as the DF for the integro-differential transport equation (e.g., the discrete-ordinate method, the method of characteristics) is theoretically described and its applicability is discussed. The AFDF is used to preserve the region-wise neutron leakage at each spatial mesh and defined as a ratio of heterogeneous and homogeneous angular fluxes at the homogenized region surface. In a homogeneous calculation with the AFDF, the angular flux is discontinuous at the region surface. In this paper the applicability of the AFDF to fuel pin cell homogenization is verified for one-dimensional slab geometry. As a result of this verification, it is confirmed that the AFDF has the capability to reduce the spatial homogenization error of fuel pin cell homogenization.     

Development of a Fine and Ultra-Fine Group Cell Calculation Code SLAROM-UF for Fast Reactor Analyses

A cell calculation code SLAROM-UF has been developed for fast reactor analyses to produce effective cross sections with high accuracy in practical computing time, taking full advantage of fine and ultra-fine group calculation schemes.

The fine group calculation covers the whole energy range in a maximum of 900-group structure. The structure is finer above 52.5 keV with a minimum lethargy width of 0.008. The ultra-fine group calculation solves the slowing down equation below 52.5 keV to treat resonance structures directly and precisely including resonance interference effects. Effective cross sections obtained in the two calculations are combined to produce effective cross sections over the entire energy range.

Calculation accuracy and improvements from conventional 70-group cell calculation results were investigated through comparisons with reference values obtained with continuous energy Monte Carlo calculations. It was confirmed that SLAROM-UF reduces the difference in k-infinity from 0.15 to 0.01% for a JOYO MK-I fuel subassembly lattice cell calculation, and from ?0.21 % to less than a statistical uncertainty of the reference calculation of 0.03% for a ZPPR-10A core criticality calculation.     

The pseudo-resonant-nuclide subgroup method based global–local self-shielding calculation scheme

The pseudo-resonant-nuclide subgroup method (PRNSM) based global–local self-shielding calculation scheme is proposed to simultaneously resolve the local self-shielding effects (including spatial self-shielding effect and the resonance interference effect) for large-scale problems in reactor physics calculations. This method splits self-shielding calculation into global calculations and local calculations. The global calculations obtain the Dancoff correction factor for each pin cell by neutron current method. Then an equivalent one-dimensional (1D) cylindrical problem for each pin cell is isolated from the lattice system by preserving Dancoff correction factor. The local calculation is to perform self-shielding calculations of the equivalent 1D cylindrical problem by the PRNSM. The numerical results show that PRNSM obtains accurate spatial dependent self-shielded cross sections and improves the accuracy of dealing with the resonance interference over the conventional Bondarenko iteration method and the resonance interference factor method. Furthermore, because both global and local calculation is linearly proportional to the size of problems, the global–local calculation scheme could be applied to large-scale problems.     

压水堆堆芯Pin-by-pin均匀化计算中非均匀泄漏修正模型研究

在压水堆堆芯Pin-by-pin均匀化计算中采用均匀泄漏修正模型及非均匀泄漏修正模型对组件计算的中子能谱进行修正,本文研究了Pin-by-pin均匀化计算中均匀泄漏修正模型及非均匀泄漏修正模型的实现方式,提出了非均匀泄漏修正模型和栅元均匀化方法的联合实现方式,并分析比较了不同栅元均匀化扩散系数产生方式的计算效果。数值结果表明,非均匀泄漏修正模型及由其产生的中子泄漏系数能有效提高压水堆堆芯Pin-by-pin计算的精度。     

Applicability of Constant Flux Approximation in Method of Characteristics with Filtering to Tiny Regions

For the method of characteristic (MOC), a system with large-gradient neutron flux caused by a strong absorber or neutron leakage is reported to entail large errors in spatial mesh discretization and require very fine mesh spacing. To apply the MOC to such fine models, the ray-trace path width has to be fine in order to make many paths cross each region. Our new method intends to obtain good accuracy with a coarse path width. With a coarse path width on the MOC, some tiny regions have less ray-tracing paths. The reliability of flux calculation for the regions can be evaluated with the calculation volumes that are estimated in ray tracing. If the discrepancy between the calculation and true volumes becomes large, the accuracy cannot be expected. In this study, the discrepancies were numerically evaluated, and it is found that the discrepancies occur on a very tiny region. To make the flux calculation of such tiny regions more reliable, an approximation, in which the outgoing flux is equal to the incoming neutron, is applied instead of the usual MOC equation. The criteria switching on the approximation, which is called filtering, was numerically evaluated for PWR assemblies. The method is validated with numerical benchmark calculations.     

Effective Homogenization Method for Control Rod Channels

A cell calculation method has been developed for accurately treating neutron transport and heterogeneity effects of control rods (CRs) within the bounds of homogeneous neutron diffusion theory. In this method, CR cell-averaged homogeneous neutron cross sections are calculated by a heterogeneous neutron transport calculation with an annular supercell model, in which a CR assembly is surrounded by a homogeneous fuel region. Then, a neutron diffusion calculation is carried out using the homogeneous neutron cross sections in the same supercell, and the CR cell-averaged radial neutron diffusion coefficients are modified in an iterative manner such that the CR cell-interface neutron current which is obtained by the heterogeneous transport calculation can be reproduced by the homogeneous diffusion calculation. In the case of a 1,000-MWeclass FBR, the center CR worth, which was calculated by an RZ diffusion calculation using the cross sections obtained by the above method, agreed within 1% with that obtained by a heterogeneous transport calculation, proving the validity of the method.     

Approximate Treatment of Thermal Expansion Effect in Lattice Transport Calculations

In this paper, thermal expansion effect on neutronics characteristics is approximately taken into account by a correction on cross sections. Dimensions of reactor core components depend on their temperatures due to the thermal expansion phenomena. However, the variation of calculation geometry requires considerable computational load for trajectory based lattice transport calculations such as the characteristics method since ray tracing must be re-executed. Therefore, if a correction on cross sections can accurately capture the effect of geometrical variation due to the thermal expansion, computation time of a lattice transport calculation that treat temperature variation can be reduced. Three different corrections on cross sections were tested in PWR fuel assembly geometry using UO₂ and MOX fuels. It was found that the correction of cross sections that preserves neutron attenuation in a region almost reproduce the reference calculation that explicitly considers geometrical variation due to the thermal expansion. The result of this paper will be useful for lattice calculations in production analysis since material temperatures are frequently changed in such analysis to cover various reactor conditions.     

控制棒尖齿效应中非均匀不连续因子处理技术

To handle the control rod cusping effect in pressurized water reactor (PWR) fuel management calculation, the variational nodal method (VNM) in the fuel management calculation code system NECP-Bamboo has been extended to tread the heterogeneous cross section distribution by expanding the volumetric cross sections into piece-wise polynomials in the early work. However, the partially inserted control rods also introduces heterogeneous discontinuity factor (DF) on nodal interface. Thus, in this paper, an ultimate solution is proposed to fully handle this problem. Firstly, the surface integral in the VNM is modified to contain the discontinuity of neutron flux, incorporating a continuous discontinuity factor in that term. Secondly, the surface DF is expanded into the sum of pieces-wise orthogonal polynomials to construct the nodal response matrixes. Comparing with current representative re-homogenization methods, the application numerical results of the BEAVRS benchmark problem demonstrate the effectiveness of the heterogeneous VNM with heterogeneous DF. It can eliminate the cusping effect by providing more accurate differential control rode worth curves and pin power distributions.     

控制棒尖齿效应中非均匀不连续因子处理技术

为了在压水堆(PWR)中限制控制棒尖齿效应,燃料管理计算程序系统NECP-Bamboo中的变分节块法(VNM)在之前的工作中通过将截面用分片多项式展开,具备了处理非均匀截面的能力。但半插的控制棒同时在节块表面带来了非均匀的不连续因子(DF)。本文提出了一个可以完全解决这个问题的方案。首先,通过包含非均匀不连续因子的表面积分将非均匀不连续因子显式地表达在变分节块法的泛函中,然后用分片多项式展开不连续因子,使其出现在响应矩阵的构建中。与现有的再均匀化方法相比,BEAVRS基准题的数值结果表明,含非均匀不连续因子的非均匀变分节块法可以消除控制棒尖齿效应,同时获得更准确的功率分布。