基于改进变分节块法的共轭中子通量密度计算技术

共轭中子通量密度对于核安全和压水堆（PWR）中的探测器计算有着重要的意义,为了消除现有节块方法在处理由于控制棒移动带来的非均匀节块（包括非均匀的截面和不连续因子）时所造成的较大误差,本文提出一种改进的变分节块法（VNM）。确定了不同于前向方程的共轭节块方法的连续条件,不同于传统VNM在全局建立泛函,本文方法为每一个节块建立泛函;构建了含非均匀不连续因子的乘子项,以显式处理表面不连续的共轭中子通量密度;除共轭体中子通量密度、截面和表面分中子流密度外,将表面不连续因子展开为分段正交多项式来构造响应矩阵。含有非均匀节块的BEAVRS基准题数值结果证明,同传统VNM相比,改进的VNM可以将非均匀问题的有效共轭增殖系数和燃料区共轭中子通量密度偏差降低2个量级,有利于实现前向与共轭中子通量密度的高精度内积计算。     

Development of a 2-D 2-group neutron noise simulator for hexagonal geometries

In this paper, the development of a neutron noise simulator for hexagonal-structured reactor cores using both the forward and the adjoint methods is reported. The spatial discretisation of both 2-D 2-group static and dynamic equations is based on a developed box-scheme finite difference method for hexagonal mesh boxes. Using the power iteration method for the static calculations, the 2-group neutron flux and its adjoint with the corresponding eigenvalues are obtained by the developed static simulator. The results are then benchmarked against the well-known CITATION computer code. The dynamic calculations are performed in the frequency domain which leads to discarding of the time discretisation. Then, the developed 2-D 2-group neutron noise simulator calculates both the discretised forward and the adjoint reactor transfer function between a point source and its induced neutron noise, by assuming the neutron noise source as an “absorber of variable strength” type. The neutron noise induced by a “vibrating absorber” type of noise source may also be modeled using the calculated transfer function. The viability of the simulator is verified for different benchmark cases.     

Convergence problems associated with the iteration of adjoint equations in nuclear reactor theory

Convergence problems associated with the iteration of adjoint equations based on two-group neutron diffusion theory approximations in slab geometry are considered. For this purpose first-order variational techniques are adopted to minimise numerical errors involved. The importance of deriving the adjoint source from a breeding ratio is illustrated. The results obtained are consistent with the expected improvement in accuracy.     

Calculation of Coupling Coefficients of Coupled Reactors Theory Using the Generalized Perturbation Theory

Recently, rigorous multi-point equations are derived using the region-wise importance functions to produce fission neutrons. Since the coupling coefficients used in these multi-point equations are calculated with the weight of these importance functions but not the adjoint function used in the conventional perturbation theory, errors due to the change of the flux is introduced in the coupling coefficients for a perturbed system if the unperturbed flux is used.

It is shown that using the generalized perturbation theory, the coupling coefficients using the unperturbed flux can be obtained taking into account the first order change of the flux due to the perturbation, and the same accuracy as the conventional perturbation theory in which the adjoint function is used can be obtained in the case of one-point reactor.     

A response-based time-dependent neutron transport method

An efficient response-based solution to the time-dependent neutron transport equation in a semi-infinite slab is derived. The solution is based on polynomial expansions of the source terms and neutron flux in the time domain. The expansion coefficients of the flux solution are computed in terms of response functions, which are special cases of Green’s functions for arbitrary in-volume and surface sources. The resulting response equation, which is a convolution integral equation in time, is reduced to a linear algebraic system of equations in the expansion coefficients. Two example problems are solved using the response-based method, and the extension of the method to general (finite, heterogeneous) problems is discussed.     

Application of the method of implicit non-stationary iteration (MINI) to 3D neutron diffusion problems

This paper extends an earlier one on the method of implicit non-stationary iteration (MINI). The extension is to the solution of symmetric and non-symmetric sets of linear equations arising in a three-dimensional Cartesian finite difference neutron diffusion representation of a reactor. MINI is used to solve the symmetric equations for neutron flux of a particular energy group, as well as the non-symmetric equations for neutron upscatter and region rebalance. Calculations are reported for some real two- and three-dimensional reactors, and the performance of MINI is compared with those of other methods. MINI is found to be particularly beneficial for problems involving extensive upscatter.     

Adjoint flux calculation of natural mode equation by time dependent neutron transport

ABSTRACT

The physical implication of the eigenfunction of the adjoint of the natural mode equation is studied. The eigenfunction at a phase space position is theoretically derived to be proportional to the amplitude of the neutron flux sufficiently long time after a source is placed at the position. Meanwhile, the time change rate and distribution of the neutron flux originated in the source must converge to those of the fundamental mode of the natural mode equation. Based on the proportional relation, the adjoint flux is calculated for a MOX- and UO₂-fueled lattice by a continuous-energy Monte Carlo time-dependent neutron transport calculation. The calculated distributions of the adjoint flux agree well with those of the iterated fission probability both in the prompt and delayed critical states.     

Galerkin and Generalized Least Squares finite element: A comparative study for multi-group diffusion solvers

In this paper, the solution of multi-group neutron/adjoint equation using Finite Element Method (FEM) for hexagonal and rectangular reactor cores is reported. The spatial discretization of the neutron diffusion equation is performed based on two different Finite Element Methods (FEMs) using unstructured triangular elements generated by Gambit software. Calculations are performed using Galerkin and Generalized Least Squares FEMs; based on which results are compared. Using the power iteration method for the neutron and adjoint calculations, the neutron and adjoint flux distributions with the corresponding eigenvalues are obtained. The results are then validated against the valid results for the IAEA-2D andBIBLIS-2D benchmark problems. The results of GFEM-2D (developed based on Galerkin FEM) and GELES-2D (developed based on Generalized Least Squares FEM) computer codes are also compared with results obtained from DONJON4 computer code. To investigate the validation of developed computer codes for the calculation with more than two energy groups, the calculations are performed for a benchmark problem with seven energy groups. To investigate the dependency of the results to the number of elements, a sensitivity analysis of the calculations to the number of elements is performed.     

Development of a finite volume inter-cell polynomial expansion method for the neutron diffusion equation

Heterogeneous nuclear reactors require numerical methods to solve the neutron diffusion equation (NDE) to obtain the neutron flux distribution inside them, by discretizing the heterogeneous geometry in a set of homogeneous regions. This discretization requires additional equations at the inner faces of two adjacent cells: neutron flux and current continuity, which imply an excess of equations. The finite volume method (FVM) is suitable to be applied to NDE, because it can be easily applied to any mesh and it is typically used in the transport equations due to the conservation of the transported quantity within the volume. However, the gradient and face-averaged values in the FVM are typically calculated as a function of the cell-averaged values of adjacent cells. So, if the materials of the adjacent cells are different, the neutron current condition could not be accomplished. Therefore, a polynomial expansion of the neutron flux is developed in each cell for assuring the accomplishment of the flux and current continuity and calculating both analytically. In this polynomial expansion, the polynomial terms for each cell were assigned previously and the constant coefficients are determined by solving the eigenvalue problem with SLEPc. A sensitivity analysis for determining the best set of polynomial terms is performed.     

A rigorous weight function for neutron kinetics equations of the quasi-static method for subcritical systems

The purpose of the present work is to develop an efficient solution method to solve the time dependent multi-group diffusion equations for subcritical systems with external sources using the quasi-static method.Usually, the k-eigenfunction for an adjoint criticality equation is used as a weight function to derive a one-point neutron kinetics equation for the amplitude function in the quasi-static method. It is shown that the use of this k-eigenfunction introduces a first order error due to the change of the flux, when the systems are not close to the critical state. It is shown also that the use of the ω-eigenfunction for the adjoint time dependent equation as the weight function can eliminate such first order error resulting from ignoring the term of first order change of the shape function to solve subcriticality problems, and it gives more accurate results than the use of conventional k-eigenfunctions of the critical adjoint equation.     

ADS次临界反应堆的点堆中子动力学方程

加速器驱动的次临界系统（ADS）中的次临界反应堆与临界反应堆相比,中子注量率的空间分布具有严重的不均匀性,同时中子平均能量较高且中子能量变化复杂,中子价值变化大,因而传统的点堆动力学方程不能较为真实地模拟ADS次临界反应堆。本文从含多群中子多组缓发中子先驱核的动力学方程出发,给出其共轭方程。然后利用稳态扩散方程及其共轭方程的共轭关系,推导得出含有归一化功率的动力学方程表达式。进而定义多个特征算子,导出了含有源中子价值的点堆中子动力学方程,并对几种简单情况进行了初步验证,为进一步分析ADS次临界反应堆的动态过程奠定了基础。     

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.     

Kinetics models for safety studies of accelerator driven systems

Safety analyses of Accelerator Driven Systems (ADSs) are mainly performed by codes developed in the past for critical reactors, a point-kinetics model for computing the transient power being often employed. It is shown that this model – that assumes time-independent neutron direct and adjoint flux shapes – may be inaccurate in the ADS case even if it is acceptable for a similar “critical” case. Although the material distribution remains almost unchanged, flux (power) shape variations may be significant in the first case due to external source related effects. An option for a more refined modelling of the neutron adjoint flux in ADS analyses is discussed. This option is shown to be rather complicated in the general case: it may give rise to involvement of several adjoint flux shapes and several sets of related point-kinetics parameters (reactivity, etc.). To improve the accuracy of the point-kinetics treatment, an extended point-kinetics model, which employs several flux (or power) shapes precomputed at steady-state conditions, is proposed in the paper. This approach may help to avoid the spatial kinetics treatment if no strong material movement occurs. The reactivity and other point-kinetics parameters are defined similarly to a critical reactor.     

Two-group sparse-grid nodal neutron balance equation of the BIPR-8 computer program

The transition from finite-difference diffusion equations for solving problems of neutron transport in VVER to the nodal sparse-grid neutron balance equations used in the BIPR-8 computer program is examined. The solution error is decreased by using trial functions to describe the neutron field inside cells in addition to the condition of continuity of the neutron flux and current on the common boundaries between the cells. The corresponding nodal equations for solving the problems of three-dimensional neutron kinetics in the BIPR-8KN program are presented. __________ Translated from Atomnaya énergiya, Vol. 105, No. 1, pp. 8–14, July, 2008.     

三维圆柱几何格林函数节块法

发展了中子扩散计算三维圆柱几何格林函数节块法。通过横向积分将中子扩散方程化为３个互相耦合的一维偏通量方程。对于径向偏通量方程，将径向扩散微分算符分解为平板几何的扩散微分算符和１个附加项之和，将附加项移到方程右端作为１个附加源项，这样，３个方程都化为平板几何一维方程的形式。再借助平板几何第二类边界条件格林函数，建立偏通量积分方程。方程推导中，对圆柱形曲面几何的线积分和横向积分均需对相应的广义线元作积分，对于修正源项，通过分部积分方法将偏通量导数项转化为对格林函数的求导，通过源迭代法求解方程。基准计算表明，该计算精度高、速度快、成为三维圆柱几何堆芯计算的有效方法。     

ASYNT — An adjoint synthesis method for neutron irradiation estimation on the vessels of VVER/PWR reactors

The new ASYNT method is developed and proposed for neutron fluence calculations. This method uses the solution of adjoint neutron transport equation for flux/responses evaluation. The evaluation of flux/responses is reduced to the space and energy integration of the product of 3D adjoint solution and the neutron source distribution, determined by realized loading patterns and operational regimes. The adjoint solution does not depend on the neutron source distribution and is obtained only once for every surveillance site, response and type of reactor. The application of this method results in separability of azimuthal and axial dependence in the 3D adjoint solution. That is why the 3D adjoint solution could be synthesised from the 2D and 1D adjoint solutions. The circular cylinder reactor core presentation of the solution axial dependence is the only approximation used in the ASYNT method.     

三维多群六角形几何中子扩散程序开发

本文基于解析基函数展开方法求解中子扩散方程的原理,利用满足中子扩散方程的解析基函数,将节块内的各群中子注量率近似展开。为提高该方法的计算精度,节块间耦合条件采用面中子注量率和面中子净流连续。节块间耦合条件的选取需利用源迭代法来求解中子扩散方程。源迭代中的内迭代选用加速的高斯 塞德尔方法,外迭代采用Lyusternik-Wagner外推加速收敛技术。针对中子注量率收敛慢、有效增殖因数收敛快、内迭代方程组系数矩阵更新耗时的特点,采用一种新的加速方法--一次外迭代多次内迭代的方法。基于以上理论模型,发展了三维多群六角形几何中子扩散程序HANDF-D,对三维二群vver440基准题、高通量堆临界实验2、三维四群热堆问题、三维七群快堆问题计算的结果表明,该方法能准确快速地给出堆芯有效增殖因数和功率。     

Determination of Large Negative Reactivity by Integral Versions of Various Experimental Methods

The large negative reactivity is measured in Semi-Homogeneous Experimental facility (SHE). Experimental methods are Sjöstrand's pulsed neutron, source multiplication and rod drop methods beside revised King-Simmons' pulsed neutron methods. Neutron detectors are placed at various points in the core region for multi-points measurement.

Usual one-point reactor model analysis resulted in the reactivity values, strongly dependent on the detector position with the increase of subcriticality. In addition, disagreements between the used experimental methods are also pointed out.

In order to overcome these difficulties due to the spatial higher harmonics and the kinetic distortion in the neutron flux distribution, an integral version analysis is applied, in which use is made of multi-points reactor model. In the analysis, space integration of the neutron counts obtained throughout the core region is made with weights of the adjoint function of fast neutrons, calculated using the two- or three-dimensional diffusion code. The negative reactivity values determined by the integral version analysis agreed well with each other within the uncertainty of ～5% in the reactivity range down to ～50 dollars.

It is concluded that all the experimental methods are adequate for precise determination of the large negative reactivity of reactor provided that the integral version analysis is utilized or that correction is made for the change of the neutron generation time using precise calculation.     

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。     

三维六角形节块多群中子扩散程序NDHEX

本文介绍用DIF3D (NOD)求解二、三维六角形几何系统下中子扩散方程的理论模型及数值计算方法。六角形节块内的中子通量密度分布采用高次多项式近似表示,最后导出通量矩方程及偏流的响应矩阵方程。应用粗网再平衡和渐近源外推方法加速收敛。参考此方法编制了计算程序NDHEX,并对一些六角形基准问题进行了计算。结果表明:NDHEX的计算结果与DIF3D(NOD)的计算结果符合很好;与差分程序相比,具有更高的精度与计算效率。它可用于快堆计算。