Solution of Multi-Group Diffusion Equation in x-y-z Geometry by Finite Fourier Transformation

The multi-group diffusion equation in three-dimensional x-y-z geometry is solved by finite Fourier transformation. Applying the Fourier transformation to a finite region with constant nuclear cross sections, the fluxes and currents at the material boundaries are obtained in terms of the Fourier series. Truncating the series after the first term, and assuming that the source term is piecewise linear within each mesh box, a set of coupled equations is obtained in the form of three-point equations for each coordinate. These equations can be easily solved by the alternative direction implicit method. Thus a practical procedure is established that could be applied to replace the currently used difference equation.

This equation is used to solve the multi-group diffusion equation by means of the source iteration method; and sample calculations for thermal and fast reactors show that the present method yields accurate results with a smaller number of mesh points than the usual finite difference equations.     

Solution of Diffusion Equation in r-z Geometry by Finite Fourier Transformation

It is shown that the monoenergetic diffusion equation in multi-region r-z geometry can be solved by the finite Fourier transformation method which has successfully been applied to x-ygeometry. In this method, a system of linear algebraic equations is derived for Fourier coefficients of fluxes and currents at the material boundaries between regions of constant cross sections, and all the boundary values are determined by solving this equation.

Numerical examples are presented for a problem featuring a fixed source and multiple regions, and the results are compared with those obtained from the current difference method. It is shown that the present method yields a better result with relatively few terms of expansion.     

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。     

Comments on “Stability of the Gas Bubbles in Solids,by G.P. Tiwari”

A new difference equation for the numerical solution of the one-dimensional diffusion equation was obtained by using a semi-analytic method, in which the only approximation employed is that the source distribution within a mesh-region is represented by a linear function.

A test program, EXX-1, was prepared and run for BWR calculations, and comparisons were made with the conventional method. The results show that by using the new difference equation, a very coarse mesh model (1 mesh point per material region) can be applied without seriously impairing the computational accuracy.

It is also shown that the conventional difference equation becomes identical to the new expression if group-constants are multiplied by correction factors and the treatment of the source term appropriately modified.

New difference equations for spherical and cylindrical geometries are also given, in an appendix.     

Iterative method for obtaining the prompt and delayed alpha-modes of the diffusion equation

Higher modes of the neutron diffusion equation are required in some applications such as second order perturbation theory, and modal kinetics. In an earlier paper we had discussed a method for computing the α-modes of the diffusion equation. The discussion assumed that all neutrons are prompt. The present paper describes an extension of the method for finding the α-modes of diffusion equation with the inclusion of delayed neutrons. Such modes are particularly suitable for expanding the time dependent flux in a reactor for describing transients in a reactor. The method is illustrated by applying it to a three dimensional heavy water reactor model problem. The problem is solved in two and three neutron energy groups and with one and six delayed neutron groups. The results show that while the delayed α-modes are similar to λ-modes they are quite different from prompt modes. The difference gets progressively larger as we go to higher modes.     

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。     

Generalization of New Finite Element Method Using ‘Imaginary’ Nodal Points

A method for improving the accuracy of finite element solutions to diffusion equations has been developed. The author previously suggested a method for improving the accuracy of finite element solutions to neutron diffusion equations, a kind of Helmholtz equations, within a short computing time. The method has been generalized so that it can be applied to problems described by the Laplace equation, too, such as temperature distributions and electric fields. In this generalized method, 3 ‘imaginary’ nodal points are added at the midsides of each data-given triangular element and the element is subdivided into 4 triangular subelements of the same dimension to improve accuracy. Then, approximate expressions, which express solutions at the ‘imaginary’ nodal points using those at ‘real’ nodal points, are derived by Jacobi's iteration method. These approximate expressions are used to reduce the number of unknowns in the final linear equations. The computing time required for the method described here is much shorter than that required for the straightforward method of increasing the number of elements 4 times under the same accuracy.     

A Study of the von Neumann Stability Analysis of a Semi-implicit Numerical Method Applied Over the Radial Impurity Transport Equation in Tokamak Plasma

The radial impurity transport equation for tokamak plasma is a form of diffusion–convection–reaction equation. The impurity transport equation is solved to determine the distribution of impurity (non-fuel) ion species with different ionization states perpendicular to magnetic surfaces of tokamak plasma. The equation for each charge (ionization) state Z is a non-linear, second-order in space, first-order in time, parabolic partial differential equation coupled to the previous Z???1 and the next Z?+?1 charge states of the impurity species through its reaction term. The number of differential equations to be solved simultaneously is hence determined by the number of ionization states of the impurity species studied. The solution to the set of these coupled equations can be obtained using a semi-implicit numerical method applied on it. The present study describes the application of von Neumann stability analysis over the semi-implicit numerical method applied over the radial impurity transport equation and determines a generic stability criterion for the method. The stability analysis is further illustrated using the geometry of Aditya tokamak installed at the Institute for Plasma Research Gandhinagar, India as an example. The impurity species considered is oxygen (Atomic number?=?8). This leads to a set of eight coupled equations for charge states Z?=?1 to 8 over which von Neumann analysis is illustrated in present study.     

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.     

Advanced PWR Core Calculation Based on Multi-group Nodal-transport Method in Three-dimensional Pin-by-Pin Geometry

A parallel production code, SCOPE2, has been developed for advanced calculations in the reactor core design of PWRs. In SCOPE2, the multi-group diffusion and/or SP3 transport equations are solved by the Red/Black iterative method within the framework of the finite difference method or the advanced nodal method without non-linear iterations. The effects due to pin-cell homogenization are taken into account by using the SPH factors.

In this paper, calculation methods needed for fast computation are derived including efficient response matrix formulation of the nodal-SP3 method, an analytic solution of the flux moments in the nodal-SP3 transport equations, and coarse-group coarse-mesh diffusion acceleration method. It was found that the present pin-by-pin nodal-SP3 method was more accurate than the finite difference SP3 method with a small additional computational cost in the same meshing scheme.

Tracking calculations of a commercial PWR plant by SCOPE2 revealed that the present model accurately predicted the power distribution and critical boron concentration. A set of depletion calculations in a typical design scheme can be completed within a few hours running on a PC-cluster (16 processors) for the full-core geometry of a 3-loop PWR with 340×3407times;26 meshes based on the 9-group pin-by-pin nodal-SP3 method.     

三维六角形粗网格中子扩散计算方法

本文介绍对于由六角形栅元组件组成的核反应堆堆芯,采用三维粗网格展开法求解中子扩散方程,推导出九点高阶近似差分方程。该方程形式上与低阶有限差分方程类似,可以采用低阶有限差分法的求解技巧求解,所建立的方法适用于两维或三维中子扩散计算,对堆芯设计和燃耗分析很有意义。     

Theoretical Research on Gas-Centrifugal Separation for Uranium Enrichment

Studies are reviewed on the uranium hexafluoride gas isotope separation by centrifugation. A flow field in a centrifuge rotor is numerically determined by solving flow equations for compressible gas in strong rotation. The diffusion equation is also solved to determine the molar concentration distribution of the lighter isotope corresponding to the obtained flow field. A modified version of the Newton method is adopted to get the full solution in static state for these nonlinear equations.

Some numerical results are shown and compared with the experimental data made by Beams et al. and Zippe, and found to be in good agreement.     

Effects of Boundary Conditions on Thermal Response of a Cellulose Acetate Layer Using Hottel’s Zonal Method

Energy can transfer internally by radiation in addition to conduction in translucent polymers. Since radiant propagation is very rapid, it can provide energy within the layer more quickly than diffusion by heat conduction. Thus, the transient thermal response of a layer for combined radiative and conduction may be extremely different from that of conduction alone. In this paper, the behavior of a heat conducting, absorbing, and emitting layer of Cellulose Acetate layer is investigated during the transient interval when both conductive and radiative heat transfer are considered. Specifically, the effects of boundary conditions on the response of the layer are considered here. These boundary conditions include both conductive boundary conditions, such as convection coefficient and convective fluid temperature, and radiation boundary conditions, like radiation surrounding temperature and specular reflectivity. To this end, the governing differential equations including the equation of radiative heat transfer within the material coupled to the transient energy equation with radiative terms are presented. The solution procedure is based on nodal analysis and Hottel’s zonal method extended by the ray tracing method. The transient energy equation including the radiative internal energy source is solved using a time marching finite difference procedure with variable space and time increments.     

A method for solving a stochastic eigenvalue problem applied to criticality

We present a novel approach to calculate stochastic eigenvalues of differential and integral equations using polynomial chaos theory. The method is applied to a criticality problem using the diffusion equation. This technique has the advantage of avoiding the non-linear terms in the conventional method of stochastic eigenvalue calculation but it does require an additional, ‘pseudo-time’, independent variable t.     

Improved nodal expansion method for solving neutron diffusion equation in cylindrical geometry

The challenges encountered in the development of nodal expansion method (NEM) in cylindrical geometry and the method to circumvent these difficulties are introduced and discussed in this paper. Due to the fact that the azimuthal term contains a factor 1/r², the traditional transverse integration fails to produce a 1D transverse integrated equation in θ-direction; a simple but effective approach is employed to obtain the θ-directional transverse integration equation. When the traditional polynomials are used to solve the 1D transverse integral equation in r-direction, some additional approximations, which may undermine the precision of the method, are required in the derivation of the moment equations; in order to preserve the accuracy of calculations, the special polynomial approximation is used to solve the 1D transverse integrated equations in r-direction. Moreover, the Row-Column iterative scheme, which is considered to be the more efficient and convenient schemes in cylindrical geometry, is used to solve the partial currents equations. An improved NEM for solving the multidimensional diffusion equation in cylindrical geometry is implemented and tested. And its accuracy and efficiency are demonstrated through several benchmark problems.     

Solution of the isotopic depletion equations using decomposition method and analytical solutions

In this paper an analytical calculation of the isotopic depletion equations is proposed, featuring a chain of major isotopes founding a typical PWR reactor. Part of this chain allows feedback reactions of (n, 2n) type. The method is based on decoupling the equations describing feedback from the rest of the chain by using the decomposition method, with analytical solutions for the other isotopes present in the chain. The method was implemented in a PWR reactor simulation code that makes use of the nodal expansion method (NEM) to solve the neutron diffusion equation, describing the spatial distribution of neutron flux inside the reactor core. Because isotopic depletion calculation module is the most computationally intensive process within simulation systems of nuclear reactor core, it is justified to look for a method that is both efficient and fast, with the objective of evaluating a larger number of core configurations in a short amount of time.     

A solution of the multigroup transport equation using a weighted residual technique

The paper is in two parts. First a solution of the multigroup diffusion equation using a weighted residual technique is described. The implementation enables high-order polynomial approximations to be made to the flux. Secondly the neutron transport equation is solved by expanding the flux in a series of unnormalized spherical harmonics, obtaining second-order diffusion-like equations for the coefficients in that series and applying the algorithm previously outlined.     

Acid Digestion of Radioactive Combustible Wastes

An iteration method using the Pade approximation is described to accelerate the inner iteration of finite difference equations of two-dimensional diffusion equation In the case of void problems where the system includes regions of very low density, the convergence rate of the usual Iteration method becomes extremely slow, and it becomes often impossible to obtain a converged solution. It is found that the present method using the Pade approximation can give the converged solution for such void problems.     

二维中子输运方程的非结构网格离散纵标数值解法

从一阶中子输运方程出发,对方向变量采用离散纵标法展开,得到一系列关于空间变量的偏微分方程,对这些方程采用最小二乘有限元方法进行离散.编制了二维中子输运方程的非结构网格离散纵标计算程序.对一系列基准问题做了验算,计算结果表明,该方法能用于非结构网格,并具有较高的计算精度.