Criticality calculation for cluster fuel bundles using grey Dancoff factor

This paper applies the grey Dancoff factor calculated by Monte Carlo method to the criticality calculation for cluster fuel bundles. Dancoff factors for five symmetrically different pin positions of CANDU37 and CANFLEX fuel bundles in full three-dimensional geometry are calculated by Monte Carlo method. The concept of equivalent Dancoff factor is introduced to use the grey Dancoff factor in the resonance calculation based on equivalence theorem. The equivalent Dancoff factor which is based on the realistic model produces an exact fuel collision probability and can be used in the resonance calculation just as the black Dancoff factor. The infinite multiplication factors based on the black Dancoff factors calculated by collision probability or Monte Carlo method are overestimated by about 2 mk for normal condition and 4 mk for void condition of CANDU37 and CANFLEX fuel bundles in comparison with those based on the equivalent Dancoff factors.     

Improvement on application of Dancoff factor and resonance interference table

An improvement for application of Dancoff factor is developed. It combines Stamm'ler's two-term method for resonance integral calculation with neutron current method for Dancoff factor calculation. Stamm'ler's formulation, which is originally derived for the infinite lattice geometry, can be easily revised to contain the Dancoff factor explicitly, while the neutron current method can easily calculate the Dancoff factor for general irregular assembly geometry. For the resonance interference effects the resonance interference factor table is built in pairs of nuclides, only for the interference between 238 U and other resonance nuclides, spanning over a range of background cross-section and number density ratio of the pairing nuclides. A series of verification calculations have been carried out for problems of infinite lattice and single assembly geometry, with two or multiple resonance absorbers. For these verification calculations, our improvement on Dancoff factor application and resonance interference give good results.     

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.     

Applicability of the enhanced neutron current method for non-convex fuel shapes

The enhanced neutron current method is applied for the evaluation of the background cross-section for fuels with non-convex shape. The evaluated background cross-sections are compared with those obtained by the conventional method (the equivalence theory) commonly used in the resonance calculation. However, the Dancoff factor for a fuel with non-convex shape is not directly obtained by the traditional method currently used in the resonance calculation. Therefore, a new method for the evaluation of the Dancoff factor is also proposed, which utilizes the variation of moderator density and is applicable to the non-convex shapes. The test calculations for fuels with non-convex shape indicate that the background cross-section obtained by the enhanced neutron current method and the conventional method agree well with each other. Since the enhanced neutron current method can be applicable not only for the convex shapes but also for non-convex shapes, it will be useful for the resonance calculations in complicated geometry.     

基于等效几何理论的复杂几何燃料共振计算方法研究

针对各种研究堆、实验堆以及新型反应堆中广泛应用的复杂几何燃料的共振计算难题,本文基于全局 局部耦合策略开展了可处理复杂几何燃料的等效几何共振计算方法研究。针对复杂几何燃料的孤立问题,基于燃料的逃脱概率守恒,建立了复杂几何燃料模型的等效一维圆柱（或平板）燃料模型;基于燃料到外围结构材料区的碰撞概率守恒,获得了燃料外围结构材料的等效尺寸;根据复杂几何燃料的丹可夫因子守恒,建立了等效一维圆柱（或平板）燃料外围的慢化剂尺寸;针对等效一维圆柱（或平板）燃料模型,采用伪核素子群方法进行了有效自屏截面计算。将该方法应用于非棒状几何燃料的共振计算,结果表明,该方法具有很强的几何处理能力,且具有较高的计算精度和计算效率。     

The surface current method in gray Dancoff factor calculation

The Dancoff correction is important in the calculation of the effective cross section of resonant isotopes in a heterogeneous system. Although the neutron current method is a simple and straightforward approach to estimate the Dancoff factor, its use is limited to the black Dancoff factor. In this paper, we expand the current method used to determine both the black and gray Dancoff factors. The method developed also relies on a neutron transport solver, where a fixed source on a fuel rod surface has an outward direction, a cosine distribution, and a constant shape. The detector is located on the surface of the rods to measure incoming and outgoing currents; therefore,there is no need to calculate the chord length, and the development, validation, and verification of the code can be omitted. The mathematical foundation of the suggested method is derived using the integral transport equation. The effects of the moderator and lattice configuration are followed by a sensitivity analysis of the Dancoff factor for several problems, including pressurized water reactor and cluster fuel assemblies. The maximum and average relative errors of the calculated results are approximately 0.3% and0.05%, respectively.     

全陶瓷微密封燃料有效多群截面计算方法研究

全陶瓷微密封（FCM）燃料是一种弥散颗粒燃料。由于弥散颗粒燃料存在双重非均匀性,传统的确定论方法及蒙特卡罗方法皆难以处理这种双重非均匀效应以获得有效多群截面。本文基于超细群方法建立FCM燃料的有效多群截面计算方法。为描述燃料棒内TRISO颗粒的非均匀性,在共振能量段,通过采用超细群方法求解包含TRISO颗粒的一维球模型得到超细群缺陷因子,通过超细群缺陷因子修正所有核素的超细群截面即可将颗粒和基质均匀化。由于TRISO颗粒在热能区也存在较强的自屏效应,在热能区,利用穿透概率及碰撞概率等价得到多群缺陷因子,通过多群缺陷因子修正所有核素的多群截面将燃料和基质均匀化。均匀化后的FCM燃料组件即可视为普通压水堆燃料组件进行共振计算。利用丹可夫修正因子等价得到FCM燃料组件各燃料棒的等效一维棒模型,对一维棒模型求解超细群慢化方程从而得到共振能量段的有效自屏截面。数值结果表明,该方法能有效处理FCM燃料的双重非均匀性,得到精确的有效自屏截面。     

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。     

Development of efficient Krylov preconditioning techniques for multi-dimensional method of characteristics

Two techniques are proposed in the preconditioning for the Krylov sub-space method called the Generalized Minimal RESidual (GMRES) method to accelerate inner iterations based on the method of characteristics (MOC). The GMRES method is an iterative method to solve a linear algebraic system byminimizing the norm of the residual vector. The proposed preconditioning technique is based on the first flight collision probability which is efficiently made by the multi-dimensional MOC code. To simplify the preconditioner, slight couplings among regions are ignored by considering the mean free path. And another proposed technique makes simplified preconditioner by the scaling matrix which can homogenize and de-homogenize the fuel region and the cladding region. The scaling technique reduces the size of the matrix and also reduces the calculation time of inverse matrix. Numerical results show that the preconditioner simplified by the mean free path efficiently reduces the number of iterations for the GMRES algorithm. And the scaling technique keeps the efficiency of preconditioner even in the multi-dimensional geometry. The total calculation time is found to be reduced when these techniques are employed.     

三区非均匀栅元X-Y几何块方法中子输运计算

基于中子积分输运理论,应用综合界面流和碰撞几率技巧的块方法,导出了处理三区非均匀栅元结构的二维（Ｘ－Ｙ）几何多群中子输运问题的数值模型。即对于由若干栅元组成的按Ｘ－Ｙ几何排列的堆芯结构,对每一类栅元剖分为圆柱形元件（如燃料棒、控制棒、可燃毒物棒等）、包壳和慢化剂三个均匀区,用碰撞几率（ＣＰ或ＰＩＪ）方法计算各区的中子通量分布;对于相邻栅元用ＤＰ１近似的中子流来耦合;因此,块方法具有精度高、速度快、能灵活处理各种几何问题的优点,是目前动力堆组件计算最有前途的方法之一。基于块方法基本理论,发展了三区栅元模型,导出了计算方法,编制了ＦＯＲＴＲＡＮ计算机程序。为验证其精度和适用性,对两个例题进行了计算,并与其它程序的计算结果进行了比较,证明功率分布和本征值均符合较好。     

模块化2D/1D耦合特征线法MMOC程序开发及验证

本文基于耦合求解的思想,轴向、径向均采用特征线法（MOC）,通过开展2D/1D耦合MOC理论模型、模块化几何预处理方法研究,开发了2D/1D耦合MOC 3D中子输运求解程序MMOC,并开展了1D/2D/3D C5G7基准题验证。k_eff的相对误差分别为0.082%、0.045%、0.032%,该程序准确有效,计算精度满足中子输运计算的要求。     

基于任意三角形网格的中子扩散变分节块法

传统的基于矩形和六角形几何的堆芯计算程序已不适用于具有复杂几何的新型反应堆堆芯计算,本文开展了基于任意三角形网格的多群中子扩散变分节块方法研究。首先,采用ANSYS软件对计算区域进行三角形网格剖分,并利用坐标变换将任意三角形变换为正三角形;其次,采用Galerkin变分技术建立包含节块中子平衡方程的泛函,将三角形节块内变量利用正三角形内正交基函数进行展开;最后,利用变分原理,获得中子通量密度与节块边界上分中子流的响应关系,并基于传统的源迭代法对其进行求解。基于上述理论模型开发了程序TriVNM,并采用不同几何基准题进行了验证。结果表明,TriVNM计算的堆芯keff和归一化功率分布与参考解吻合较好,该计算方法适用于复杂几何堆芯扩散计算。     

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.     

Improvement to the Calculation of Collision Probabilities in Annular System and Its Application to Cluster Geometry

A new method of calculating first-flight collision probabilities in cluster geometry is developed. The collision probabilities in an annular system as well as the transition probabilities for a neutron escaping from one subcell to reach other subcells are calculated and combined. The former is an improvement on Bonalumi's approximation and the latter is an extension of the Dancoff correction to subcells containing fuel rods and coolant. As applications of the present method, we calculate the collision probabilities in a hexagonal cluster with 7 fuel rods and in a square cluster with 28 fuel rods. Fairly good agreement is obtained by comparing the probabilities obtained from our method with the results obtained by the exact method. One-group calculations of the flux distribution are carried out in these systems.     

基于改进缺陷因子方法的FCM燃料有效多群截面计算方法研究

全陶瓷微胶囊封装（FCM）燃料是重要的候选事故容错燃料,与传统燃料相比,FCM燃料的双重非均匀性使得其有效多群截面计算面临较大的挑战。本文提出一种改进的缺陷因子方法来处理FCM燃料在共振能区和非共振能区的自屏效应,实现FCM燃料的等效均匀化。通过颗粒丹可夫因子守恒来构建新的等效模型以克服传统的体积权重等效模型无法考虑燃料棒间自屏的影响;在共振能量段,基于新的等效一维球模型求解超细群慢化方程获得共振能量段的超细群缺陷因子;在非共振能量段,利用新等效模型的特征值计算获得快群和热群的多群缺陷因子;在此基础上实现FCM燃料棒的等效均匀化。本方法已在高保真中子学程序NECP-X上实现,并在一系列工况下进行了测试,与蒙特卡罗程序的比较表明,本方法能处理不同情况下的双重非均匀性,并可获得准确的有效自屏截面。     

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.     

直接三维中子输运特征线法并行计算软件的实现

作为中子输运问题的一种重要确定论方法,特征线法（MOC）具有强几何适应性、计算流程简洁、易于大规模并行的优点。ANT-MOC是自主开发的中国数值反应堆1.0（CVR1.0）中的三维特征线法中子输运计算程序,主要用于压水堆、快堆的堆芯输运计算。ANT-MOC支持基于构造实体几何（CSG）的复杂几何建模、高效的用户输入方式、面向矩形/六边形网格的射线追踪算法,以及基于轨迹链分解的并行算法和负载平衡策略。在国产超算上,ANT-MOC可以扩展到约10万处理器核,并行效率在50%以上。针对压水堆、快堆计算问题进行验证和参数敏感性分析,结果表明ANT-MOC计算结果具有较好的稳定性和准确度。     

基于OpenMP的中子输运方程特征线法并行计算研究

特征线法是目前求解反应堆中子输运方程的主要计算方法之一。本文开发了基于OpenMP的中子输运方程特征线法并行计算程序,以提高特征线法的计算效率。OpenMP是共享存储体系结构上的一个并行编程模型,采用Fork-Join并行执行方式,适合于SMP共享内存多处理系统和多核处理器体系结构。通过相关基准题测试验证,表明所开发的程序在有效增殖因数以及相对中子通量(归一化栅元功率)分布等参数上都能取得良好的精度,且使用OpenMP能取得良好的加速效果,使计算时间显著减少。     

子群法与特征线法结合的中子共振计算

传统的中子共振自屏计算方法采用了有理近似,局限于处理简单的共振模型,在处理复杂燃料栅元/组件时会引入较大误差。为提高复杂情况下共振计算的精度,将子群法共振模型与特征线方法结合,推导了子群法-特征线法方程。基于WIMSD格式的69群数据库,编制了可用于任意二维几何中子共振计算的SGMOC程序。通过数值验证表明,该程序计算结果与MCNP程序计算结果吻合良好,具有较高的计算精度与几何通用性。