Measurements of Doppler Effect of Coated Particle Fuel Rod in SHE-14 Core Using Sample Heating Device

Reactivity decrease due to temperature rise of a single fuel rod sample was measured in the SHE-14 core using a sample heating device with purpose to verify the calculation accuracy of the Doppler effect for resonance neutron absorption in a Very High Temperature Reactor. The fuel rod sample was a stuck of coated particle fuel compacts containing 4% enriched UO₂ kernels, and it was heated up to about 700°C in a sample heating tube which was placed along the axis of the core.

The difference of reactivity decrease between the two same size samples of fuel rod and graphite rod due to temperature rise can be interpreted as the increased resonance neutron capture of ²³⁸U, i.e. Doppler effect. The SRAC code system was applied to the Doppler effect calculation where the collision probability method was used in the cell calculation and the one-dimensional, multi-group diffusion approximations were adopted in the core calculation. The results gave a ratio of the calculated to the measured Doppler effect of 0.93.     

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。     

A New Mixed Method with Finite Difference and Finite Element Method for Neutron Diffusion Calculation

A new method for obtaining three-dimensional neutron flux distribution in a reactor has been developed by taking into account the fact that the X-Y planar geometry is generally complex but the geometry along Z-axis is simple. In this method, the finite element method is applied to the X-Y plane calculation and the finite difference method to the Z-axis. For solving a three-dimensional neutron diffusion equation, these two methods are iterated successively until a consistency of the leakage coefficients is attained between the two. The present method is embodied as a computer program FEDM for FACOM M200 computer. With this program, a three-dimensional diffusion calculation was performed for comparing some numerical results with those by a conventional standard computer code ADC. The comparison has shown that they agree well with each other. Computing time required for this problem by the FEDM was shorter than that by the ADC for obtaining same accuracy on the eigenvalue. To indicate usefulness of this method, a demonstration calculation for a reactor with a complex geometry was performed, which was a difficult case to calculate with a conventional finite difference code.     

HARMONY程序计算中子扩散方程高阶λ本征值问题的基准验证

高阶λ谐波在反应堆堆芯功率重构、换料优化、ADS次临界反应堆物理特性研究等领域有着重要应用价值。为进行高阶λ谐波的计算,本文基于隐式重启动Arnoldi方法(IRAM)编制了可用于一维、二维、三维笛卡尔坐标系中子扩散方程的任意阶λ谐波及本征值计算的HARMONY程序,并进行了基准题的数值验证。结果表明,HARMONY程序能实现高阶λ本征值问题计算,具有较高的精度,为未来基于λ谐波的ADS次临界反应堆物理特性研究奠定了基础。     

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.     

Approach to Knowledge Based Man-Machine Communication for BWR Start-Up Guidance

A novel direct calculation method of response matrices on heterogeneous lattices by using the Monte Carlo method is proposed. These direct response matrices (DRMs) can be used in core calculations in place of the conventional homogenized lattice constants. The DRMs are formalized by four sub response matrices (sub-RMs) in order to respond to a core eigenvalue, k; thus the DRMs can be re-evaluated on each outer iteration in the core calculations. The sub-RMs can be evaluated by analyzing each neutron's trajectory from ordinary lattice calculations with the Monte Carlo code. Since these sub-RMs are calculated directly under an actual complex assembly geometry, i.e., without a homogenization process, intra-assembly heterogeneous effects can be reflected on global partial current balance calculations. With using two of the sub-RMs, which deal with neutron production probabilities for each fuel pin, and the obtained partial current balance, pin-wise neutron production distributions can also be reconstructed. The DRM method is applied to infinite lattice calculations and multi-assembly calculations. As a result, it is shown that the derivation of the method is valid and intra-assembly heterogeneity is reflected in the results correctly.     

基于OpenMC的瞬发中子衰减常数计算模块开发与验证

瞬发中子基波衰减常数α可定量描述反应堆内中子随时间的变化,是计算绝对反应性所需的中子动力学参数之一,对次临界（特别是较深次临界）绝对反应性的精确测量具有重要意义。本文在开源程序OpenMC基础上,基于k α迭代方法,以中子径迹长度上的平均时间吸收权重修正作为k α迭代参数因子,在输运过程中对瞬发、缓发中子分别考虑,开发了具有瞬发α本征值问题计算功能的OpenMC PA模块。以Godiva衍生基准题和MUSE 4次临界实验装置为计算对象,对程序计算瞬发α本征值问题能力进行验证。结果表明,该计算模块有优于MCNP4C的计算速度与计算范围,计算值与参考值的相对误差小于05%。OpenMC PA能满足次临界系统瞬发α本征值和中子动力学参数计算需求。     

Analysis of Heavy-Water-Moderated,Cluster-Type Fuel Lattices by Cluster Physics Code ‘MESSIAH’

This paper presents a comprehensive analysis performed by a new cluster analysis code ‘MESSIAH’ on reactor physics constants measured in the critical facility for a pressure-tube-type, heavy-water-moderated reactor. The MESSIAH code system utilizes the method of the collision probability to solve the neutron transport equation. The effective space dependent cross sections are calculated in the thermal and resonance energy range before the eigenvalue calculation for the whole energy range. With use of these cross sections, the multi-group, space dependent transport equation is solved, and the flux distribution, spectrum and k _eff are obtained to the input bucking. In the above three steps the method of the collision probability is used consistently and extensively. The treatment of leakage neutrons from lattices in MESSIAH is also confirmed by an independent method using a Monte Carlo calculation. The calculated reactor physics constants, especially the micro-parameters and the activation traverse of Dy, agreed fairly well with the experiment. The diffusion calculation with use of the group constants calculated by MESSIAH predicts the reactivity of 0% void core excellently (<0.12%). However, for a 100% void core, the calculated reactivity was slightly lower than the experiment (~0.74%), which was attributed to over prediction of the diffusion constants.     

Reactivity Worth of Fuel Elements

Reactivity worths of fuel elements were measured in the Ozenji Critical Facility (OCF) and analyzed with three group perturbation method. The result shows that the worth of one single fuel pin can be well predicted by calculation over a very wide range of the core spectrum, namely from a lattice of 2.5% enrichment and 0.43 volume ratio to that of 1.5% enrichment and 3.5 volume ratio.

The analysis indicates the importance of thermal neutron flux peaking remaining after the removal of a fuel pin. Only by incorporating this effect can the reactivity worth of a fuel pin be correctly evaluated. In the present study, the neutron spectrum in the water hole where the peaking occurred was assumed to be the same as in the reflector. The reflector spectrum seems to provide better agreement with experiment than the core spectrum. Validity of the analysis was extended to a bundle of sixteen fuel pins by measuring the reactivity worths of bundles of fuel pins as well as the thermal neutron flux distributions. One dimensional diffusion calculations were employed throughout the analysis.     

Influence of the external neutron sources in the criticality prediction using 1/M curve

The influence of external neutron sources in the process to obtain the criticality condition is estimated. To reach this objective, the three-dimensional neutron diffusion equation in two groups of energy is solved, for a subcritical PWR reactor core with external neutron sources. The results are compared with the solution of the corresponding problem without external neutron sources, that is an eigenvalue problem. The method developed for this purposes it makes use of both the nodal method (for calculation of the neutron flux) and the finite differences method (for calculation of the adjoint flux). A coarse mesh finite difference method was developed for the adjoint flux calculation, which uses the output of the nodal expansion method. The results regarding the influence of the external neutron source presence for attaining criticality have shown that far from criticality it is necessary to calculate the reactivity values of the system.     

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.     

JFNK求解高温气冷堆中子扩散问题的预处理方法研究

预处理方法是非线性求解方法JFNK中效率和数值收敛特性的关键。基于包括活性区、石墨反射层和外围含硼碳砖在内的高温气冷堆堆芯中子扩散本征值问题,对JFNK求解过程中的预处理环节进行了研究。根据矩阵元素的物理性质简化得到不同的近似雅克比矩阵,使用线性预处理方案ILU、SIPLU进行了预处理,并对其预处理效果、矩阵稀疏性、预处理时间等参数进行了分析。结果表明,块雅克比型近似阵是对原雅克比矩阵的较好近似,其能够在保留各中子能群内部耦合关系的前提下,构造结构简单、适用性强的近似雅克比矩阵。对于此类高温气冷堆中子扩散问题,选取SIPLU方法能获得性能良好的预处理矩阵,达到高效JFNK计算。     

Discrete formulation for two-dimensional multigroup neutron diffusion equations

The objective of this paper is to introduce a new numerical method for neutronic calculation in a reactor core. This method can produce the final finite form of the neutron diffusion equation by classifying the neutronic variables and using two kinds of cell complexes without starting from the conventional differential form of the neutron diffusion equation. The method with linear interpolation produces the same convergence as the linear continuous finite element method. The quadratic interpolation is proven; the convergence order depends on the shape of the dual cell. The maximum convergence order is achieved by choosing the dual cell based on two Gauss' points. The accuracy of the method was examined with a well-known IAEA two-dimensional benchmark problem. The numerical results demonstrate the effectiveness of the new method.     

Development and Verification of an Efficient Spatial Neutron Kinetics Method for Reactivity-Initiated Event Analyses

A space/time nodal diffusion code based on the nodal expansion method (NEM), EPISODE, was developed in order to evaluate transient neutron behavior in light water reactor cores. The present code employs the improved quasi-static (IQS) method for spatial neutron kinetics, and neutron flux distribution is numerically obtained by solving the neutron diffusion equation with the nonlinear iteration scheme to achieve fast computation. A predictor-corrector (PC) method developed in the present study enabled to apply a coarse time mesh to the transient spatial neutron calculation than that applicable in the conventional IQS model, which improved computational efficiency further. Its computational advantage was demonstrated by applying to the numerical benchmark problems that simulate reactivity-initiated events, showing reduction of computational times up to a factor of three than the conventional IQS. The thermohydraulics model was also incorporated in EPISODE, and the capability of realistic reactivity event analyses was verified using the SPERT-III/E-Core experimental data.     

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。     

High-order cross-section homogenization method

A high-order cross-section homogenization method based on boundary condition perturbation theory is developed to improve the accuracy of nodal methods for coarse-mesh eigenvalue calculations. The method expands the homogenized parameters such as the cross-sections and the neutron flux discontinuity factor in terms of the node surface current-to-flux ratio. The expansion coefficients are evaluated during the nodal calculations using additional precomputed homogenization parameters. As a result, it is possible to correct (update) the homogenized parameters to arbitrary order of accuracy for the effect of reactor core environment (fuel assembly neutron leakage) with very little computational effort in the core calculation. The reconstructed fine-mesh flux (fuel-pin power) is a natural byproduct of the new method. A benchmark problem typical of a BWR core is analyzed in one dimension, monoenergetic diffusion theory by modifying a nodal method based on a bilinear, flat as well as a fine-mesh intranodal flux shape. The homogenized parameters are first computed using exact (fine-mesh) albedos and compared to those determined from a fine-mesh core calculation. Two nodal (coarse-mesh) examples are given to show how well this approach works as a higher-order perturbation method is utilized. The paper concludes by showing that this method succeeds in giving excellent results for cores that may be difficult to model using standard nodal methods.     

Development of Zircaloy Clad Hafnium Rods for BWR Long Life Neutron Absorbers

Control rod columns of the High Temperature Engineering Test Reactor (HTTR) core have each three cylindrical void holes for insertion of the control rods. The reactivity worths of the simulated void holes were measured in the Very High Temperature Reactor Critical Assembly (VHTRC) by the pulsed neutron method with the purpose of evaluating the neutron streaming effect on the reactivity worths of the void holes. The reactivity worths were determined using the revised King-Simmons' formula. The measured results are compared with the calculated ones based on the nuclear data of ENDFIB-IV, using the Benoist's anisotropic neutron diffusion coefficients. It is concluded that the reactivity worths are enlarged up to about twice as large as that obtained using the conventional isotropic neutron diffusion coefficients, and it is also found that the neutron streaming effect that is defined as the relative fraction components exceeding the reactivity worths obtained by the conventional isotropic neutron diffusion coefficients are likely overestimated about 11 and 32% with the void holes being in the core and reflector regions, respectively. Then, the reactivity decrease due to the neutron streaming effect is able to be evaluated around 1%Δk for the initial critical core of the HTTR.     

Validation of neutron current formulations for the response matrix method based on the SP3 theory

The pin-by-pin fine mesh BWR core analysis code SUBARU has been developed as a next-generation BWR core analysis code. SUBARU is based on the SP3 theory and the response matrix method is used for flux calculations. The SP3 theory consists of the 0th and 2nd order neutron fluxes. Therefore, the relations among the 0th and 2nd order partial neutron currents and the fluxes are required to apply the response matrix method. SUBARU is approximated the relations among the partial neutron currents and the fluxes are similar to that the diffusion theory. Our previous study revealed that the prediction accuracy of SUBARU is much higher than that of conventional core analysis codes. However, validity of the above approximation is not directly investigated so far. Therefore, relations among the partial neutron currents and the fluxes are theoretically derived and calculation results with the rigorous and the conventional formulations are compared. The calculation results indicate that the approximation of the conventional formulation is appropriate for the BWR core analysis.     

A Krylov–Schur solution of the eigenvalue problem for the neutron diffusion equation discretized with the Raviart–Thomas method

Mixed-dual formulations of the finite element method were successfully applied to the neutron diffusion equation, such as the Raviart–Thomas method in Cartesian geometry and the Raviart–Thomas–Schneider in hexagonal geometry. Both methods obtain system matrices which are suitable for solving the eigenvalue problem with the preconditioned power method. This method is very fast and optimized, but only for the calculation of the fundamental mode. However, the determination of non-fundamental modes is important for modal analysis, instabilities, and fluctuations of nuclear reactors. So, effective and fast methods are required for solving eigenvalue problems. The most effective methods are those based on Krylov subspaces projection combined with restart, such as Krylov–Schur. In this work, a Krylov–Schur method has been applied to the neutron diffusion equation, discretized with the Raviart–Thomas and Raviart–Thomas–Schneider methods.     

定态中子输运方程本征值计算方法研究

散射源项各向异性展开阶数较大或者离散纵标方法的角度离散方向较多时,中子输运方程本征值的计算迭代容易失败。为了克服该问题,本文通过数学上的推导,构造了中子输运方程α本征值迭代求解的一种新方法,数值算例表明该方法提高了收敛速度,不收敛的问题也得到明显改善。对keff本征值的计算进行改进,改进后的方法不依赖迭代初值,数值算例表明改进方法的计算结果较好。