Development of the nodal collocation method for solving the neutron diffusion equation

The nodal collocation method is a new technique for the discretization of the multidimensional neutron diffusion equation where the solution sought is expressed in the form of tensorial expansions of Legendre polynomials defined over homogeneous parallelepipeds. In this study, we have truncated the tensorial expansions using the serendipity approximation in an attempt to reduce the total number of unknowns and improve the effectiveness of the discretization. The remaining Legendre coefficients are then determined in order to preserve selected moments of the neutron conservation equation in each parallelepiped. This approach allows a variable order of convergence without sacrificing the consistency peculiar to full tensorial expansions and generates matrix systems which may be resolved by an Alternating Direction Implicit algorithm. Furthermore, we have proved that the linear nodal collocation method and the mesh centered finite difference method are equivalent. Validation results are given for the IAEA 2-D and 3-D benchmarks and for a 2-D representation of a pressurized water reactor (PWR).     

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.     

Averaging the neutron diffusion equation

This paper presents a general theoretical analysis of the neutron motion problem in a nuclear reactor, where large variations on neutron cross-sections normally preclude the use of the classical neutron diffusion equation. A volume-averaged neutron diffusion equation (VANDE) is derived which includes correction terms to diffusion and nuclear reaction effects. A method is presented to determine closure-relationships for the VANDE (e.g., effective diffusivity). In order to describe the distribution of neutrons in a highly heterogeneous configuration, it was necessary to extend the classical neutron diffusion equation. Thus, the averaged diffusion equation includes two correction factors: the first correction is related with the absorption process of the neutron and the second correction is a contribution to the neutron diffusion, both parameters are related to neutron effects on the interface of a heterogeneous configuration. As an example of the VANDE, the plane source in an infinite medium was considered to study the effects of the correction factors on the neutron flux, and the results were compared with classic solution.     

Application of the response matrix method to BWR lattice analysis

A lattice calculation code RESPLA has been developed for light-water reactor lattices on the basis of the response matrix method treating the heterogeneity in pin cells. The spatial dependency of neutron flux distribution along each cell boundary is taken into account by dividing the cell boundary into several subsurfaces and the anisotropy of neutron angular distribution is considered up to the P₁ component by using a relation between the P₀ and P₁ components. The RESPLA code has been applied to BWR lattice calculations and the calculational results have been compared with those obtained by the S_n method and the collision probability method. It has been found that the present response matrix method has the same accuracy as the collision probability method with fine spatial meshes and the error caused by the use of coarse meshes is much smaller than that by the collision probability method. Furthermore, the required computing time is smaller by about a factor of five than that in the collision probability method.     

求解中子输运SP3方程的半解析节块方法

中子输运简化P3 (SP3)方法是对中子输运方程PN的一种近似,可以转换为与中子扩散方法相似的形式.采用节块方法中有效的半解析方法求解中子输运SP3方程,同时也基于粗网有限差分(CMFD)方法采用细网有限差分(FMFD)形式同样求解该方程.通过对NEACRP-L-336基准题(修改)的数值计算,验证了通过Pin-By-Pin的节块计算能够获得与FMFD几乎相同的结果,而Pin-By-Pin的CMFD计算结果与FMFD计算结果有一定的偏差.     

A solution of the neutron diffusion equation in hemispherical symmetry using the homotopy perturbation method

The homotopy perturbation method is used to formulate a new analytic solution of the neutron diffusion equation both for a sphere and a hemisphere of fissile material. Different boundary conditions are investigated; including zero flux on boundary, zero flux on extrapolated boundary, and radiation boundary condition. The interaction between two hemispheres with opposite flat faces is also presented. Numerical results are provided for one-speed fast neutrons in ²³⁵U. A comparison with Bessel function based solutions demonstrates that the homotopy perturbation method can exactly reproduce the results. The computational implementation of the analytic solutions was found to improve the numeric results when compared to finite element calculations.     

Flux expansion nodal method for solving multigroup neutron diffusion equations in hexagonal-z geometry

A flux expansion nodal method (FENM) has been developed to solve multigroup neutron diffusion equations in hexagonal-z geometry. In this method, the intranodal fluxes are expanded into a set of analytic basis functions for each group. In order to improve the nodal coupling relations, a new type of nodal boundary conditions is proposed, which requires the continuity of both the zero- and first-order moments of partial currents across the nodal surfaces. The response matrix technique is used for the iterative solution of the nodal diffusion equations, which greatly improves the computational efficiency. The numerical results for a series of benchmark problems show that FENM is a very accurate and efficient method for the prediction of criticality and nodal power distributions in the reactors with hexagonal assemblies.     

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.     

The implicit restarted Arnoldi method,an efficient alternative to solve the neutron diffusion equation

To calculate the neutronic steady state of a nuclear power reactor core and its subcritical modes, it is necessary to solve a partial eigenvalue problem. In this paper, an implicit restarted Arnoldi method is presented as an advantageous alternative to classical methods as the Power Iteration method and the Subspace Iteration method. The efficiency of these methods, has been compared calculating the dominant Lambda modes of several configurations of the Three Mile Island reactor core.     

Convergence of the iteration scheme in the nodal expansion method for the solution of the diffusion equation

Convergence of the iteration scheme in the nodal expansion method for the solution of the diffusion equation has been established. The proof is applicable to 1-D, 2-D and 3-D problems with commonly occurring boundary conditions. It is restricted to square and cubic nodes and parabolic expansion of the flux over a node.     

Exact solution to the time-dependent one-speed multiregion neutron diffusion equation

Exact solutions are obtained to the time-dependent one-speed neutron diffusion equation in one-dimensional multiregion Cartesian and spherical geometries with multiplication and without delayed neutrons. These solutions enable the study of the one-speed space-time behavior of prompt neutrons in an arbitatry number of neutronically dissimilar material regions. Parametric benchmark calculations are presented.     

Application of linear-extended neutron diffusion equation in a semi-infinite homogeneous medium

The linear-extended neutron diffusion equation (LENDE) is the volume-averaged neutron diffusion equation (VANDE) which includes two correction terms: the first correction is related with the absorption process of the neutron and the second is a contribution to the neutron diffusion, both parameters are related to neutron effects on the interface of a heterogeneous configuration. In this work an analysis of a plane source in a semi-infinite homogeneous medium was considered to study the effects of the correction terms and the results obtained with the linear-extended neutron diffusion equation were compared against a semi-analytical benchmark for the same case. The comparison of the results demonstrate the excellent approach between the linear-extended diffusion theory and the selected benchmark, which means that the correction terms of the VANDE are physically acceptable.     

Assessment of the Prony's method for BWR stability analysis

It is known that Boiling Water Reactors are susceptible to present power oscillations in regions of high power and low coolant flow, in the power-flow operational map. It is possible to fall in one of such instability regions during reactor startup, since both power and coolant flow are being increased but not proportionally. One other possibility for falling into those areas is the occurrence of a trip of recirculation pumps. Stability monitoring in such cases can be difficult, because the amount or quality of power signal data required for calculation of the stability key parameters may not be enough to provide reliable results in an adequate time range. In this work, the Prony's Method is presented as one complementary alternative to determine the degree of stability of a BWR, through time series data. This analysis method can provide information about decay ratio and oscillation frequency from power signals obtained during transient events. However, so far not many applications in Boiling Water Reactors operation have been reported and supported to establish the scope of using such analysis for actual transient events. This work presents first a comparison of decay ratio and frequency oscillation results obtained by Prony's method and those results obtained by the participants of the Forsmark 1 & 2 Boiling Water Reactor Stability Benchmark using diverse techniques. Then, a comparison of decay ratio and frequency oscillation results is performed for four real BWR transient event data, using Prony's method and two other techniques based on an autoregressive modeling. The four different transient signals correspond to BWR conditions from quasi-steady to power oscillations. Power signals from such transients present a challenge for stability analysis, either because of the low number of data points or need of much iteration, and thus reducing their capability for real time analysis. The results show that Prony's method can be a complementary reliable tool in determining BWR's stability degree.     

Progress in nodal methods for the solution of the neutron diffusion and transport equations

Recent progress in the development of coarse-mesh nodal methods for the numerical solution of the neutron diffusion and transport equations is reviewed. In contrast with earlier nodal simulators, more recent nodal diffusion methods are characterized by the systematic derivation of spatial coupling relationships that are entirely consistent with the multigroup diffusion equation. These relationships most often are derived by developing approximations to the one-dimensional equations obtained by integrating the multidimensional diffusion equation over directions transverse to each coordinate axis. Both polynomial and analytic approaches to the solution of the transverse-integrated equations are discussed, and the Cartesian-geometry polynomial approach is derived in a manner which motivates the extension of this formulation to the solution of the diffusion equation in hexagonal geometry. Iterative procedures developed for the solution of the nodal equations are discussed briefly, and numerical comparisons for representative three-dimensional benchmark problems are given.

The application of similar ideas to the neutron transport equation has led to the development of coarse-mesh transport schemes that combine nodal spatial approximations with angular representations based on either the standard discrete-ordinate approximation or double P_n expansions of the angular dependence of the fluxes on the surfaces of the nodes. The former methods yield improved difference approximations to the multidimensional discrete-ordinates equations, while the latter approach leads to equations similar to those obtained in interface-current nodal-diffusion formulations. The relative efficiencies of these two approaches are discussed, and directions for future work are indicated.     

Iterative schemes for obtaining dominant alpha-modes of the neutron diffusion equation

Two new methods of obtaining dominant prompt alpha-modes (sometimes referred to as time-eigenfunctions) of the multigroup neutron diffusion equation are discussed. In the first of these, we initially compute the dominant K-eigenfunctions and K-eigenvalues (denoted by λ₁, λ₂, λ₃ … etc.; λ₁ being equal to the K_eff) for the given nuclear reactor model, by existing method based on sub-space iteration (SSI) which is an improved version of power iteration method. Subsequently, a uniformly distributed (positive or negative) 1/v absorber of sufficient concentration is added so as to make a particular eigenvalue λ_i equal to unity. This gives ith alpha-mode. This procedure is repeated to find all the required alpha-modes. In the second method, we solve the alpha-eigenvalue problem directly by SSI method. This is clearly possible for a sub-critical reactor for which the inverse of the dominant alpha-eigenvalues are also the largest in magnitude as required by the SSI method. Here, the procedure is made applicable even to a super-critical reactor by making the reactor model sub-critical by the addition of a 1/v absorber. Results of these calculations for a 3-D two group PHWR test-case are given. These results are validated against the results as obtained by a completely different approach based on Orthomin(1) algorithm published earlier. The direct method based on the sub-space iteration strategy is found to be a simple and reliable method for obtaining any number of alpha-modes. Also comments have been made on the relationship between fundamental α and k values.     

A multi-region boundary element method for multigroup neutron diffusion calculations

For the analysis of a two-dimensional nuclear system consisting of a number of homogeneous regions (termed cells), first the cell matrices which depend solely on the material composition and geometrical dimension of the cell (hence on the cell type) are constructed using a boundary element formulation based on the multigroup boundary integral equation. For a particular nuclear system, the cell matrices are utilized in the assembly of the global system matrix in block-banded form using the newly introduced concept of virtual side. For criticality calculations, the classical fission source iteration is employed and linear system solutions are by the block Gaussian-elimination algorithm. The numerical applications show the validity of the proposed formulation both through comparison with analytical solutions and assessment of benchmark problem results against alternative methods.     

A solution of the neutron diffusion equation for a hemisphere containing a uniform source

An analytic solution of the diffusion equation for a hemisphere of fissile or non-fissile material is presented which contains a spatially uniform neutron source. Numerical results are given for the flux distribution for one-speed fast neutrons in ²³⁵U and also for a non-fissile element of similar scattering properties. We use these results to check the accuracy of the finite element code EVENT. The procedure is also developed for multigroup calculations. In an Appendix we outline the procedure required when the hemisphere contains a source and is also irradiated by an external current of neutrons.     

Multigroup neutron diffusion equation with the finite volume method in reactors using MOX fuels

The use of mixed oxide (MOX) fuel to partially fill the cores of commercial light water reactors (LWRs) gives rise to a reduction of the radioactive waste and production of more energy. However, the use of MOX fuels in LWRs changes the physics characteristics of the reactor core, since the variation with energy of the cross sections for the plutonium isotopes is more complex than for the uranium isotopes. Although the neutron diffusion theory could be applied to reactors using MOX fuels, more emphasis on treatment of the energy discretization should be placed. This energy discretization could be typically 4–8 energy groups, instead of the standard 2-energy group approach. In this work, the authors developed a finite volume method for discretizing the general multigroup neutron diffusion equation. This method solves the eigenvalue problem by using Krylov projection methods, in which the size of the vectors used for building the Krylov subspace does not depend on the number of energy groups, but it can solve the multigroup formulation with upscattering and fission production terms in several energy groups. The method was applied to MOX reactors for its validation.