Application of correction technique using leakage index combined with SPH or discontinuity factors for energy collapsing on pin-by-pin BWR core analysis

A correction technique to capture the spectral interference effect on collapsed cross sections is combined with the superhomogenization (SPH) factor or the discontinuity factor (DF) and is applied to the pin-by-pin core analysis for boiling water reactors (BWRs). The spectral interference effect has relationship with variations of neutron leakage in each pin-cell from the viewpoint of neutron balance. In order to correct collapsed cross sections, a new correction technique, in which the neutron leakage in each pin-cell is used as a correction index, was proposed in the previous study. By this correction technique, the reference coarse group cross sections are well reproduced and the calculation accuracies are improved. However, the reference fine group calculation results could not be reproduced since the correction technique cannot reduce energy collapsing errors. Thus, we combine the correction technique with the SPH factor or the DF to reduce energy collapsing errors. In order to verify and discuss the applicability of the correction technique with the SPH factor or the DF, two-dimensional benchmark calculations considering typical characteristics of BWR cores are carried out. The correction technique with the DF more accurately reproduces the reference fine group calculation results than that with the SPH factor.     

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.     

Reduction of the Spatial Discretization Error in the Method of Characteristics using the Diamond-difference Scheme

In this paper, the diamond-difference (DD) scheme, which is commonly used in discrete-ordinate codes, is applied to the method of characteristics (MOC) to reduce the spatial discretization error of the flat flux approximation. Smaller spatial discretization error allows coarser background mesh division, which leads to smaller computational burden. Some theoretical considerations on the DD scheme are discussed to clarify the strength of this method. An absorption cross section weighted DD scheme (AWDD), which utilizes macroscopic absorption cross section to set the weight, is also discussed. The DD and AWDD schemes are implemented to AEGIS, which is a lattice physics code based on MOC. Then the AEGIS code is applied to two different benchmark problems whose spatial discretization errors are large. The calculation results indicate that from the viewpoint of spatial discretization error, the AWDD scheme is superior to the conventional MOC in which the step characteristics approximation is commonly used. Since incorporation of the AWDD scheme to current MOC codes is very simple, it will be a good candidate of spatial discretization method for MOC codes.     

Application of the multigrid amplitude function method for time-dependent MOC based on the linear source approximation

ABSTRACT

An efficient numerical scheme for time-dependent MOC calculations is proposed. In the present scheme, one of the most successful factorization method, the multigrid amplitude function (MAF) method, is employed to achieve faster computation with the minimum degradation for the temporal integration of the scalar flux. In addition, the MAF method is re-derived based on the linear source approximation, which is not applied for time-dependent MOC calculations in the past studies as far as the authors’ knowledge, to reduce the spatial discretization error with the coarser flux region separation. The accuracy and computational time of the present scheme are evaluated through the calculation of the TWIGL and the C5G7-TD 2D benchmark problems. The present calculation results show that the present scheme is 6.2 times faster than the conventional method while achieving the same accuracy in the C5G7-TD benchmark problem.     

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.     

Neutron Transport Calculations by Using Double-Differential Cross Sections

Some test calculations were carried out to demonstrate the usefulness of double-differential cross sections for neutron transport calculations including anisotropic scattering. A transport code system NITRAN was applied for the purpose. In NITRAN, the anisotropy of elastic and inelastic scattering can be treated in a general form by double-differential total neutron-emission cross sections, which are generated from single-differential and/or original double-differential cross section data base.

The test calculations were performed for neutron flux spectra in aluminum and lead slabs, and also for tritium production rates in a natural lithium sphere. Since the treatment free from collision kinematics is possible by using the double-differential cross sections in the Sⁿcalculations, the discretization of secondary neutron energy distribution becomes independent of the segmentation of angular distribution. A significant improvement due to this independence can be seen in calculating the anisotropy of general inelastic scattering and the extreme anisotropy of elastic scattering by heavy nuclei. For precise anisotropic transport calculations, it is therefore concluded that the nuclear data of double-differential type are more suitable than those of single-differential type.     

Wavelet-based angular dependence analysis of the heterogeneous calculation on MOX fuel lattice

A wavelet-based transport method is developed to satisfy the high order angular approximation, which has been proved to be necessary in the heterogeneous calculation of MOX fuel lattice. Based on the new angular discretization scheme, the angular dependence of flux is analysed to find out the origin of complicated angular anisotropy and its effects on the heterogeneous calculation. Both of the geometric and neutronic effects are investigated quantitatively to find out the angular dependence in heterogeneous calculations. Comparisons between the traditional S_N angular discretization scheme and wavelet-based scheme are analysed to indicate the challenges brought from the MOX fuel lattice heterogeneous calculation. An effective solution is given by using wavelets in the angular discretization of neutron transport equation. Improvements of high order angular approximation are suggested.     

Optimization treatment of point-wise nuclear data in Monte Carlo criticality and burnup calculations

The Monte Carlo codes used for neutron transport calculations are always time consuming, a large proportion of which is possessed by the treatment of continuous-energy cross sections. In this paper, two companion methods are developed for the optimization treatment of point-wise nuclear data, the first of which is called Computational-Expense Oriented (CEO) method based on the unionized energy grid approach and reconstructs only the computationally expensive cross sections in neutron transport simulation, and the other of which is called energy bin (EB) method, a companion of CEO method when the reaction rate tallies for MC-coupling burnup calculation are performed. These two methods are implemented in the code RMC, a Monte Carlo (MC) code used for reactor analysis, and tested on fast reactor core and BWR assembly problems. The numerical results show that CEO method, in comparison with reconstructing all cross sections under the unionized grid, requires the sharply decreased computer memory while achieving almost the same computational efficiency, and EB method can optimize the processing of nuclide-specific energy grid search and thus effectively reduce the total search number while requiring very small computer memory.     

Substep methods for burnup calculations with Bateman solutions

When material changes in burnup calculations are solved by evaluating an explicit solution to the Bateman equations with constant microscopic reaction rates, one has to first predict the development of the reaction rates during the step and then further approximate these predictions with their averages in the depletion calculation. Representing the continuously changing reaction rates with their averages results in some error regardless of how accurately their development was predicted. Since neutronics solutions tend to be computationally expensive, steps in typical calculations are long and the resulting discretization errors significant.In this paper we present a simple solution to reducing these errors: the depletion steps are divided to substeps that are solved sequentially, allowing finer discretization of the reaction rates without additional neutronics solutions. This greatly reduces the discretization errors and, at least when combined with Monte Carlo neutronics, causes only minor slowdown as neutronics dominates the total running time.     

Fast Neutron Spectra from Natural Uranium Target Bombarded with High Energy Electrons

The numerical solution of the transport equation has the errors caused by the approximations used in the computational method. In the past estimations of these errors have been performed experimentally. In the present study, formulas to estimate the errors have been derived on the basis of the perturbation theory. This method enables us to deterministically estimate the numerical errors due to the iteration, spatial discretization and Legendre polynomial expansion of scattering transfer cross sections.

Using the error estimation method developed in the present study, two examples of error analyses were carried out to confirm its validity and applicability to error estimation for a practical purpose. The errors of the calculated tritium breeding ratio for ⁷Li in a infinite slab geometry were estimated, and they agreed well with the values predicted from direct calculation. As the second example, error analysis was carried out for one-dimensional nuclear calculations on two types of commercial fusion reactor blankets. In this analysis the tritium breeding ratio and the fast neutron leakage flux from the inboard shield were investigated, and the errors from different causes were quantitatively compared.     

Reduction in spatial discretization error in the method of characteristics by using the mobile-chord ray tracing method

The mobile-chord method is applied to the method of characteristics (MOC) to reduce the spatial discretization error in ray traces. In the mobile-chord method, the offset of a ray trace in a strip depends on the azimuthal angle and this variation cancels the spatial discretization error of each ray trace. Although the mobile-chord method has been employed in the evaluation of the collision probability, it has not yet been applied to the MOC. The mobile-chord method is implemented in the AEGIS code, which is a lattice physics code based on the MOC. Verification calculations are carried out for the pin-cell and whole core geometries of the C5G7 benchmark problem by using UO₂ and mixed oxide (MOX) fuels. The calculation results indicate that the spatial discretization error in the mobile-chord method is smaller than that in the equidistant ray tracing method, which is commonly used in conventional MOC codes. Since the mobile-chord method can be used along with the cyclic ray tracing method, it is expected to be an attractive candidate for conventional MOC codes.     

氘核中子反应截面的法捷耶夫方程理论计算

中子引起的轻核反应是核数据研究的重要内容。当前我国核数据库中氘核中子反应截面的计算结果局限于采用s 波可分离势,且入射能量在20 MeV以下。需要发展三体核反应的法捷耶夫方程理论方法,采用超出s 波的核子 核子相互作用,从而对更高能量范围内氘核全套中子反应截面做出准确的描述。本文介绍了利用法捷耶夫方程计算n+d三核子反应体系的弹性散射微分截面、破裂反应、破裂反应出射中子和质子的双微分截面的理论框架及数值计算结果,同时计算了弹性散射总截面和破裂反应总截面的激发函数。计算结果与实验数据及CENDL 32、ENDF/B Ⅷ.0、JENDL 5、JEFF 33等数据库中的评价数据符合较好。     

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%。     

Resonance Calculation for Large and Complicated Geometry Using Tone's Method by Incorporating the Method of Characteristics

A new efficient approach for evaluating the background cross section, which is based on Tone's method, is presented. Though the collision probability method is used in the conventional Tone's method, the method of characteristics (MOC) is used in the present method. Since the computation time of MOC is shorter than that of the collision probability method in a large and complicated geometry, the present method will be useful not only for lattice physics calculation, but also for analyses of advanced reactors with complicated geometry. Verification calculations are carried out in two configurations, i.e., a PWR fuel assembly geometry and a multiassembly geometry adjacent to the baffle-reflector region. The validity of the present method has been confirmed through the results of verification calculations.     

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.     

Evaluation of Dancoff Factors in Complicated Geometry using the Method of Characteristics

A simple and efficient method to estimate the Dancoff factor in a complicated geometry, named “the Neutron current method,” is presented in this paper. In this method, Dancoff factors are evaluated from the flux values obtained by the method of characteristics (MOC). By setting appropriate neutron sources in the non-fuel regions of target geometry and then executing fixed source calculation by MOC, the neutron current method can evaluate Dancoff factors for complicated geometry. It was demonstrated that the neutron current method can easily be adopted for complicated geometries, such as a PWR fuel assembly or large-scale geometry that is difficult to handle by the traditional collision probability method. By utilizing the neutron current method instead of a traditional collision probability method, the calculation time of Dancoff factors in complicated large geometry is drastically reduced.     

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。     

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.     

一维特征线法中子输运程序开发及验证

为探讨两维/一维综合法堆芯分析方法,本文基于特征线法研制了一维中子输运程序--PEACH-1D.不同于通常的平源近似特征线方法,PEACH-1D可对子区的中子源项作线性近似;程序运用指数函数插值表和渐近源外推技术来加速计算过程.相关数值结果表明,PEACH-1D具有很高的计算精度和效率,线性源近似的特征线法具备处理较粗网格的能力,值得推广.     

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。