An application of a higher order finite difference method to a natural convection experiment in the hot plenum of an LMFBR

A two dimensional thermal-hydraulic analysis of a natural circulation experiment has been performed to evaluate the effectiveness of a higher order finite difference method for solving the Navier-Stokes and the energy equations. In the method, the convection terms appearing in each equation are solved by the Method of Characteristics using the third order Lagrange type polynomial as the interpolation function, and an iterative procedure is applied to solve the time derivative terms of each equation stably with second order accuracy. The analytical results have been compared with an experiment in which the temperature and the velocity distributions in the plenum region were measured with their fluctuations, and it was shown that the higher order finite difference method could simulate natural convection phenomena involving fluctuations well.     

Effects of dimensional change strain in nuclear graphite component stress analysis

Shrinkage and thermal stresses are induced into graphite components when they are irradiated in nuclear reactor cores. These stresses have to be taken into account in the reactor design and subsequent safety case assessments. This is usually done using graphite irradiation constitutive models programmed into a finite element code. The models use empirical data for the irradiation induced property and dimensional change, which are obtained from graphite material test reactor programmes. The dimensional change in nuclear graphite is one of the most important strains induced by the irradiation fluence. In this paper the effect of two different numerical methods to calculate the dimensional change strain is examined. Then the effect on the predicted stress using two different empirical models for dimensional change is studied. The solutions show that although the difference between two models is small, there are considerable differences in the stress profile.     

571.2 MHz次谐波聚束腔结构的物理设计

文章给出了北京正负电子对撞机二期工程(BEPCⅡ)未来预注入器中571.2 MHz次谐波聚束腔结构的物理设计步骤和方法。分别使用二维有限差分法分析软件Superfish和三维有限积分法分析软件MAFIA对优化后的聚束腔结构进行了计算,得到了比较一致的结果。另外,对Superfish和MAFIA在设计聚束腔结构上的优缺点进行了比较。     

A finite element method for neutron transport. Part IV: A comparison of some finite element solutions of two group benchmark problems with conventional solutions

The one group finite element formulation of the neutron transport equation is adapted to treat multigroup shielding problems. The method is extended to eigenvalue problems by using a source iteration technique. The results for one-dimensional two group shielding problems show that the finite element method is fast and accurate; in the case of transport problems the solutions are free from the ray defects often found with discrete ordinate methods. Results for eigenvalue problems show that the method, when compared with discrete ordinate and collision probability methods and with diffusion theory in appropriate cases, is again fast and accurate. In the majority of cases, approximate values of the lowest eigenvalue when plotted against the reciprocal of the square number of nodes, lie very close to a straight line; consequently a very good estimate of the benchmark eigenvalue can often be found with coarse finite element methods. The results for two group problems have shown that the accuracy and speed achieved for the corresponding one group benchmark problems are maintained. These results and those of Part III, for the two-dimensional two region one group problems indicate that the finite element method is promising for the multigroup two dimensional problems. For both sets of results the finite element representation is used for the spatial dependence of the angular flux. The directional dependence of the angular flux is treated by expansions: either the Spherical Harmonics, the continuous representation; or Walsh Functions, the discrete representation. Walsh Functions do not appear to have any particular advantage over Spherical Harmonics. In the case of one dimension when the Spherical Harmonics reduce to Legendre functions, they are superior to Walsh Functions.     

蒸汽发生器一维传热的适定性研究

使用了适定性的数学理论———特征线方法 ,对蒸汽发生器中的一维瞬态两相流传热进行了分析研究 ,目的是为了了解其数学物理本质。根据分析结果 ,蒸汽发生器二次侧的两相流传热是适定的。同时 ,根据这个结果分析得到了求解蒸汽发生器一维传热方程的逆风有限差分格式是稳定的。     

三门核电站1号核岛底板混凝土浇注温度场分析方法

对三门核电站1号核岛底板混凝土浇注实测温度结果进行了分析,并运用Matlab编程,采用一维及二维差分法对混凝土浇注及养护过程温度场进行模拟。通过结果的对比表明,差分法在预测混凝土温升方面具有较好的准确性和良好的工程实用性,二维差分法计算结果与实测结果较为一致且具有更高的精度,可以用于模拟混凝土温度场及应力场。     

Development of the finite element method of body fit nodalization for mixed convection analysis in rod bundles

In the reactor rod bundle analysis, mixed convection phenomena are very important after the reactor shutdown. In this paper, the finite element method based on the body fit nodalization are developed to analyze the mixed convection phenomena in a complex geometry. The velocity distribution and the temperature distribution in the reactor rod bundles are obtained using the above two methods. To validate the developed methods, a comparison of the present results with the analytic solutions for a concentric tube is taken. The results show that the mixed convection in a complex geometry can be treated very well with these two methods, and that the finite element method with the body fit nodalization is more efficient than the finite difference method with the body-fitted coordinate system.     

Exact Error Estimation for Solutions of Nuclide Chain Equations

The exact solution of nuclide chain equations within arbitrary figures is obtained for a linear chain by employing the Bateman method in the multiple-precision arithmetic. The exact error estimation of major calculation methods for a nuclide chain equation is done by using this exact solution as a standard. The Bateman, finite difference, Runge-Kutta and matrix exponential methods are investigated.

The present study confirms the following. The original Bateman method has very low accuracy in some cases, because of large-scale cancellations. The revised Bateman method by Siewers reduces the occurrence of cancellations and thereby shows high accuracy. In the time difference method as the finite difference and Runge-Kutta methods, the solutions are mainly affected by the truncation errors in the early decay time, and afterward by the round-off errors. Even though the variable time mesh is employed to suppress the accumulation of round-off errors, it appears to be nonpractical. Judging from these estimations, the matrix exponential method is the best among all the methods except the Bateman method whose calculation process for a linear chain is not identical with that for a general one.     

Monte-Carlo simulation of a stochastic differential equation

For solving higher dimensional diffusion equations with an inhomogeneous diffusion coefficient,Monte Carlo (MC) techniques are considered to be more effective than other algorithms,such as finite element method or finite difference method.The inhomogeneity of diffusion coefficient strongly limits the use of different numerical techniques.For better convergence,methods with higher orders have been kept forward to allow MC codes with large step size.The main focus of this work is to look for operators that can produce converging results for large step sizes.As a first step,our comparative analysis has been applied to a general stochastic problem.Subsequently,our formulization is applied to the problem of pitch angle scattering resulting from Coulomb collisions of charge particles in the toroidal devices.     

Radiation effects on heat and mass transfer in MHD stagnation-point flow over a permeable flat plate with thermal convective surface boundary condition,temperature dependent viscosity and thermal conductivity

We study a steady laminar 2-D MHD viscous incompressible flow over a permeable flat plate with thermal convective boundary condition and radiation effects. The viscosity and thermal conductivity of fluid are assumed to vary linearly with temperature. Similarity representation of the governing partial differential equations is obtained via group method. Similarity equations are then solved numerically by implicit finite difference technique. Effects of convective heat transfer parameter (b), radiation parameter (R,) magnetic field parameter (M), the thermal conductivity parameter (S), suction parameter (fw), Prandtl number (Pr) and Schmidt number (Sc) on the dimensionless axial velocity, temperature, concentration, wall temperature, the rate of heat transfer and the rate of mass transfer are investigated. Good agreement is found between the numerical results of the present paper with published result for special case.     

A Compensator Design Controlling Neutron Flux Distribution via Observer Theory

To suppress the spatial xenon oscillations in a nuclear reactor, an implementable stabilization scheme is proposed based on the finite dimen- sional compensator theory in control theory for the distributed parameter systems. The method is applied to a one-dimensional reactor whose dynamics is governed by one-group diffusion equation with its associated iodine and xenon dynamics. The modal decomposition of the state variables enables us to use the pole assignment algorithms developed in finite dimensional systems to obtain the stabilizing compensator gains. This allows us to estimate the states of a reactor in a transient using output measurement data and arbitrary initial conditions, and control the states using the estimated values. The resulting compensator is tested by using model-based data for measurement output through numerical simulations. The results show that unstable spatial xenon oscillations initiated by perturbations can be controlled by the finite dimensional compensator.     

有限差分方法下组件通量精细分布重构方法

为准确处理中子探测器的响应,对于扩散差分计算得出的通量分布也有必要进行组件通量精细分布重构,二维双二次样条函数是最理想、方便的选择。给出了实现的具体方法和公式。此方法可适应任何通量分布形状、任意网格划分,并在组件边缘保证中子流连续。     

Solution of group diffusion equation for X-Y geometry by finite Fourier transformation

A new difference equation to the two dimensional diffusion equation for x-y geometry is derived by using the finite Fourier transformation. This difference equation has a form of a coupled equation of the 3 point difference equations for each coordinate, and can be easily solved by the iterative method of the alternative direction implicit method. Group diffusion equations are solved using this difference equation and sample calculations show that accurate results can be obtained with less mesh points than the usual 5 points difference equation.     

密度锁内无扰动时稳态温度场分析

实验模拟了密度锁内无扰动时稳态温度场分布。结果发现,稳态温度分布曲线上存在一个温度分层结束点;它是导热层与恒温层的分界,只有当温度分层结束点在密度锁内才能有效地抑制传热。应用半无限大平板导热模型、一维等截面直肋稳态导热模型和Fluent流体计算软件对无扰动时稳态温度场分布进行了理论计算。结果表明,半无限大平板导热模型是计算密度锁内无扰动时稳态温度场分布和温度分层结束点位置的最佳方法。     

Spherical and Cylindrical Ion Acoustic Shock Structures in Plasmas with q-Nonextensive Electron Velocity Distribution

The propagation of ion acoustic shock waves in cylindrical and spherical geometries has been investigated. The plasma system consists of cold ions, nonextensive electrons and thermal positrons. Spherical and cylindrical Korteweg–de Vries–Burger equations have been derived by reductive perturbation method and their shock behavior is studied by employing finite difference method. It is found that shock waves can be produced in this medium. The important effects of the q-nonextensive electron on the properties of ion acoustic waves are discussed. Furthermore, it is observed that the positron concentration, ratio of electron to positron temperature, geometry parameter and the plasma kinematic viscosity significantly modifies the shock structure.     

Estimation of maximum coated particle fuel compact packing fraction

This work develops an analytic fuel fraction packing model for a high temperature gas cooled reactor fuel compact fabricated from overcoated particles of a single size. The model includes the effects of one dimensional compression and finite matrix grain size. One dimensional compression limits the maximum fuel packing fraction to about 48% for the pressed compact in this single sized particle system. This limit is due to two effects. The first is that the process of die loading limits the pre-compression packing configuration to one that is stable under gravity, which is not the most space efficient one. The second effect is due to the one dimensional compression which reduces only the axial dimension of the particle lattice rather than uniformly compressing the lattice. The die wall can also limit the maximum packing fraction by preventing the nearby particles from moving into a more space efficient configuration.     

单位瞬态法在核电厂疲劳监测系统中的应用

核电厂疲劳监测系统中的非稳态导热反演计算是关键步骤,本文提出的单位瞬态法目的就是解决反演计算问题。首先研究了一维瞬态热传导内外壁温度的线性关系,其次建立了单位瞬态热传导有限元模型,然后通过数值计算实现了由内壁温度计算外壁温度或由外壁温度计算内壁问题双向计算过程,最后通过1组试验数据验证单位瞬态法的正确性,为疲劳监测系统的推广应用奠定了基础。     

Application of finite fourier transformation for the solution of the diffusion equation

Application of the finite Fourier transformation is discussed for the solution of the diffusion equation in one dimension, two dimensional x-y and triangular geometries. It can be shown that the equation by the Nodal Green's function method in Cartesian coordinate can be derived as a special case of the finite Fourier transformation method.     

三维带控制棒反应性温度系数的计算分析

介绍了三维带控制棒计算反应性温度系数的计算方法，对于２００ＭＷ供热堆，在不同棒位下作了三维反应性温度系数的计算，结果表明，慢化剂温度系数的大小和控制棒插入的深度有一定关系。以运行工况临界棒位下的慢化剂温度系数为参考，对二维计算结果作了分析，结果表明第二维无控制棒计算是保守的近似。     

Optimization studies of fuel loading pattern for a typical Pressurized Water Reactor (PWR) using particle swarm method

A three dimensional multi-energy group computer model PRISHA, which solves the neutron diffusion equations using finite difference method is developed for Pressurized Water Reactor (PWR). This computer code can find an optimum loading of a group of fresh fuel assemblies along with fuel assemblies of different exposures. The successive line over relaxation (SLOR) method is used to solve neutron diffusion equations. After validation of this part of computer code against an IAEA – PWR benchmark problem with 177 fuel assemblies in the core, particle swarm optimization (PSO) method is incorporated in the code for finding the optimum fuel loading pattern. A typical PWR core with 157 fuel assemblies, where 289 fuel pins are arranged in 17 × 17 rectangular arrays in a fuel assembly, was analyzed using this computer model for two cycles using PSO method. Different numbers of particles and iterations were used in PSO method. The results are found to be not very sensitive to either the number of particles or the number of iterations used in PSO method for considered case. However, a number of experiments have to be performed to arrive at the best global fitness parameter. Reasonably low power peaking factors were obtained for both the cycles.