Neutron Diffusion Calculations Research Articles

PWR core calculation based on pin-cell homogenization in three-dimensional pin-by-pin geometry

For the pressurized water reactor two-step calculation, the traditional assembly homogenization and two-group neutron diffusion calculation have been widely used. When it comes to the core pin-by-pin simulation, many models and techniques are different and unsettled. In this paper, the homogenization methods based on the pin discontinuity factors and super homogenization factors are used to get the pin-cell homogenized parameters. The heterogeneous leakage model is applied to modify the infinite flux spectrum of the single assembly with reflective boundary condition and to determine the diffusion coefficients for the SP3 solver which is used in the core simulation. To reduce the environment effect of the single-assembly reflective boundary condition, the on-line method for the SPH factors updating is applied in this paper, and the functionalization of SPH factors based on the least-squares method will be pre-made alone with the table of the group constants. The fitting function will be used to update the thermal-group SPH factors with a whole-core pin-by-pin homogeneous solution on-line. The three-dimensional Watts Bar Nuclear Unit 1 (WBN1) problem was utilized to test the performance of pin-by-pin calculation. And numerical results have demonstrated that PWR pin-by-pin core calculation has more accurate results compared with the traditional assembly-homogenization scheme.

Strengths and weaknesses of the modal expansion method for perturbations calculations in nuclear reactor physics

Reactor design and uncertainties propagation request repeated execution of neutron diffusion calculations. To address this, one can rely on the Generalized Perturbation Theory which is restricted to single integral values such as reaction rate ratios. To overpass such limitations, the perturbed flux could be calculated with the modal expansion method. The difficulty to calculate more than one eigenvector has limited its use. Hence, modal expansion lacks a clear explanation of its strengths and weaknesses. Using recent advances in eigenvalues solvers, we have successfully applied modal expansion to the 2D-BIBLIS and IAEA-3D benchmarks. Our results are promising for fission and capture cross sections perturbations. Still, modal expansion is faulty for local perturbations or when it concerns the reflector. In terms of computational cost, modal expansion is more efficient than solving the direct problem with a calculation time divided by 8 for a basis of 20 eigenvectors.

Evaluating the Diffusion Approximation Capability on the Integral Pressurized Water Reactor (IPWR) Core Calculation

Diffusion approximation is an important approximation used to model a nuclear reactor core with a quite good accuracy and less computational cost. This approximation has been used widely around the globe for various kinds of nuclear reactors. This diffusion approximation is applied in a two-step method, a method combining a high fidelity transport code and low fidelity diffusion code. Meanwhile, innovations in the nuclear core model continue to make the nuclear reactor core safer, more robust, and smaller. The trend of creating smaller and more modular reactor core is emerging in the last ten years. These innovations will affect the core modeling system. Consequently, for smaller reactors, it is important to evaluate the capability of diffusion approximation if one wants to use a computationally cheaper method to model the reactor core. In this paper, neutron diffusion calculation for 160 MWth integral pressurized water reactor (IPWR) core was conducted using the PARCS nodal diffusion code employing the few-group spatially homogenized cross-sections generated by the Serpent Monte Carlo code. Due to its capability to model any reactor geometry in the high-resolution calculation, the results from Serpent were also used as a reference. Two important parameters are compared between PARCS and Serpent: effective neutron multiplication factor and core power distribution. For the full IPWR core model, a discrepancy of 564 pcm between PARCS and Serpent k eff was observed, while the radial power distribution had a maximum error of 4.71 %. It can be said, to some extent, that the diffusion approximation can be applied to IPWR core analysis. However, further improvement is indeed required if one wants more accurate results with low computational costs.

High performance 3D pin-by-pin neutron diffusion calculation based on 2D/1D decoupling method for accurate pin power estimation

The methods and performance of a 3D pin-by-pin neutronics code based on the 2D/1D decoupling method are presented. The code was newly developed as an effort to achieve enhanced accuracy and high calculation performance that are sufficient for the use in practical nuclear design analyses. From the 3D diffusion-based finite difference method (FDM) formulation, decoupled planar formulations are established by treating pre-determined axial leakage as a source term. The decoupled axial problems are formulated with the radial leakage source term. To accelerate the pin-by-pin calculation, the two-level coarse mesh finite difference (CMFD) formulation, which consists of the multigroup node-wise CMFD and the two-group assembly-wise CMFD is implemented. To enhance the accuracy, both the discontinuity factor method and the super-homogenization (SPH) factor method are examined for pin-wise cross-section homogenization. The parallelization is achieved with the OpenMP package. The accuracy and performance of the pin-by-pin calculations are assessed with the VERA and APR1400 benchmark problems. It is demonstrated that pin-by-pin 2D/1D alternating calculations within the two-level 3D CMFD framework yield accurate solutions in about 30 s for the typical commercial core problems, on a parallel platform employing 32 threads.

High-accuracy neutron diffusion calculations based on integral transport theory

In this paper, the Ronen method is developed, implemented, and applied to resolve the neutron flux and the criticality eigenvalue in simple one-dimensional homogeneous and heterogeneous problems. The Ronen method is based on iterative calculations of correction factors to use in a multigroup diffusion model, where the factors are actually given by the integral transport equation. In particular, spatially dependent diffusion constants are modified locally in order to reproduce new estimates of the surface currents obtained by the integral transport operator. The diffusion solver employed in this study uses finite differences, and the transport-corrected currents are introduced into the numerical scheme as drift terms. The corrected solutions are compared against reference results from a discrete ordinate code. The results match well with the reference solutions, especially in the limit of fine meshes, but slow convergence of the scalar flux is reported.

Development and validation of a coupled neutron diffusion-thermal hydraulic calculation procedure for fast reactor applications

The neutron diffusion equation is defined based on the User Defined Function (UDF) and the User Defined Scalar (UDS) functions of the FLUENT. The neutron diffusion equation is solved iteratively by using the solver of the FLUENT with the Finite Volume Method (FVM). At the same time, the mass, momentum and energy equations are solved iteratively. At each iteration, the power distribution (flux distribution) obtained by the iteration of the neutron diffusion equation is transferred to the thermal-hydraulics calculation and is used as the heat source term. At the same time, the temperature distribution obtained from the thermal-hydraulics calculation is transferred to the neutron diffusion calculation and the macroscopic cross sections of the materials are corrected to realize the coupling calculation of the neutron diffusion and the thermal-hydraulics under the same solver of the FLUENT without needing to develop the interface program and the computational cost is saved. 2D-TWIGL benchmark problem is calculated by the FLUENT solver to verify the feasibility of this method to solve neutron diffusion equation. Through the modeling and calculation of the 5 × 5 PWR assembly model, the calculation results are compared with the results of other programs to verify the feasibility of the coupling method and the correctness of data transfer. Then this coupling method is applied to calculate the hot assembly of a modular lead-cooled fast reactor (M2LFR-1000) to verify that the thermal-hydraulics characteristics (the maximum fuel temperature and the maximum cladding outer surface temperature) are all within the corresponding thermal-hydraulics design limits.

Rod Ejection and Drop Accident Analysis of Aqueous Homogeneous Solution Reactor

Aqueous Homogeneous Solution Reactor concept has been proposed for producing medical isotopes(Medical Isotope Production Reactor-MIPR). However, there are several difficulties in transient calculation of aqueous homogeneous solution reactors. First, there are no assemblies in the core which is different from the traditional reactor core. Second, the operation of aqueous solution reactor at a power of 200kW will generate radiolytic-gas bubbles. The void volume created by these bubbles in the solution core will introduce a strong negative reactivity feedback. Third, the complex structure of the coolant pipes immersed in fuel solution requires unstructured neutron diffusion calculation methods. Therefore, analytic basis functions expansion nodal method for arbitrary triangular-z node is established to solve the complex structure geometry neutron diffusion equation. Based on this, a software named TABFEN-K has been developed to solve the three-dimensional space-time neutron kinetic equations. Then，TABFEN-K code is used for typical accident analysis of a solution reactor. A simplified geometry model, bubbles generation model , thermal conduction model and and cross section feedback model are given in this paper. A software called TABFEN-MIPR is developed and used for the simulations of the control rod ejection and drop. The same characteristics in the transient process with the results from literatures are obtained.

Introduction to the non-orthogonal gradient-based finite volume method for three-dimensional neutron diffusion calculations with application to the rectangular reactor core configurations

The objective of this research is to introduce a new adaptive non-orthogonal gradient-based Finite Volume Method (FVM) for the accurate solution of the multi-group three-dimensional neutron diffusion equations. This method of calculation is developed for the first time, based on a new method of mesh movement and Control Volume (CV) deformation using the spatial gradient variations of the reactor power. The mesh movement and CV deformation is developed for rectangular three-dimensional reactor core configurations and different boundary conditions.For verification and evaluation of the proposed method, five benchmark problems have been considered and the results of calculations compared with the reference solutions. The results of solutions involved in obtaining reactor power as well as the corresponding effective multiplication factor showed that the proposed non-orthogonal gradient-based FVM is more accurate than the conventional FVM.

Convex shaped FBR core with graded size pins to prevent re-criticality during Core Disruptive Accident

The core geometry of sodium-cooled fast spectrum reactor (SFR) is not generally in its most reactive configuration. Therefore, the study of postulated Core Disruptive Accident (CDA) has been recognized as one of the most important themes for SFR. There are several analyses and experiments to investigate CDA phenomena. On the basis of these knowledge, some special devices are designed to control the relocation of molten material. This study focuses on the core configuration to maximize neutron leakage effect during its geometrical deforming, instead of introduction of the devices. The authors investigate the convex shaped core design that is closed to spherical shape. It is because the oval spherical core has the least neutron leakage in all practical core geometries. The neutron leakage of the convex shaped core increases by geometry change and it causes negative compaction reactivity insertion. In this paper, the feasibility of the convex shaped core design is investigated based on the parametric survey of compaction reactivity using the neutron diffusion calculation. The convex shaped core with uniform size pins and less neutron leakage shows negative compaction reactivity but the radial power distribution is not flattened. The introduction of graded size pins enables convex core to accomplish negative compaction reactivity and leveled power, simultaneously.

A solution to the singularity problem in the meshless method for neutron diffusion equation

In recent years, the moving least square (MLS) approximation has been widely used for numerical analysis of scattered data. Many numerical methods for solving partial differential equations are developed based on the MLS approximation.The MLS method is ideal for problems where the geometry of the domain is complex. An example of this is the neutron diffusion calculation. The collocation method is known that it has fast convergence speed, however, it is unstable and non-robust in many cases. In particular, it is the reason resulting in the fluctuation of the solution at the position that is on overlapping boundary region. This drawback can be dealt with the meshless local Petrov-Galerkin weak form for nodes on the interface boundary. In this paper, a combination of the weak form of the meshless local Petrov-Galerkin (MLPG) and the collocation method are applied to solve the neutron diffusion equation. Trial functions employed in the weak form of MLPG are constructed via the MLS method. In most MLPG methods numerical integration is required. The exception to this is the case of the collocation method in which the test functions are Dirac delta functions. Therefore, if the shape function is not constructed precisely enough, within an acceptable error range, the second order derivative of the shape function will lack accuracy. One of the reasons for the inaccuracy of the shape function is singularity problems occurring in the process of constructing the shape function. In this study, a solution is introduced to eliminate the singularity problem, allowing us to obtain the derivative of the shape function with sufficient precision. Finally, neutron diffusion problems are implemented in a combination of the weak-form of the meshless local Petrov-Galerkin (MLPG) and the collocation method is used to evaluate the efficiency and accuracy.

Jacobian-Free Newton-Krylov Nodal Expansion Methods with Physics-Based Preconditioner and Local Elimination for Three-Dimensional and Multigroup k-Eigenvalue Problems

Motivated by the high accuracy and efficiency of nodal methods on the coarse meshes and the superlinear convergence and high efficiency of Jacobian-free Newton-Krylov (JFNK) methods for large-scale nonlinear problems, a new JFNK nodal expansion method (NEM) with the physics-based preconditioner and local elimination NEM_JFNK is successfully developed to solve three-dimensional (3D) and multigroup k-eigenvalue problems by combining and integrating the NEM discrete systems into the framework of JFNK methods. A local elimination technique of NEM_JFNK is developed to eliminate some intermediate variables, expansion coefficients, and transverse leakage terms through equivalent transformation as much as possible in order to reduce the computational cost and the number of final-solving variables and residual equations constructed in NEM_JFNK. Then efficient physics-based preconditioners are successfully developed by approximating the matrices of the diffusion and removal terms, transverse leakage terms using the three-adjacent-node quadratic fitting methods, and scatter source terms, which make full use of the traditional power iteration. In addition, the Eisenstat-Walker forcing terms are used in the developed NEM_JFNK method to adaptively choose the convergence criterion of linear Krylov iteration within each Newton iteration based on the Newton residuals and to improve computational efficiency further. Finally, the NEM_JFNK code is developed for 3D and multigroup k-eigenvalue problems in neutron diffusion calculations and the detailed study of convergence, computational cost, and efficiency is carried out for several 3D problems. Numerical results show that the developed NEM_JFNK methods have faster convergence speed and are more efficient than the traditional NEM using power iteration, and the speedup ratio is greater for the higher convergence criterion.

Core geometry for recriticality prevention against CDA in sodium-cooled fast reactor

In core disruptive accident (CDA) of sodium-cooled fast reactors (SFRs), fuel and steel can melt to form a molten pool. The objective of the present work is to investigate design measures for core of SFR to prevent recriticality. This paper studied core geometry of convex cross-section shape to increase neutron leakage from core surface in case of core compaction. In order to evaluate effective neutron multiplication factor, 70-group neutron diffusion calculations were performed by changing parameters related to the convex core geometry (i.e. ratio of inner height to inner diameter). Neutron diffusion code and nuclear data library employed in those calculations are CITATION and JENDL-3.3, respectively. By optimizing core configuration, the compaction reactivity of the convex shaped core showed drastic reduction from 2.07 % of reference cylindrical core (Japan Sodium-cooled Fast Reactor) to -3.28 %.

Acceleration of within group iteration for pin-by-pin calculations

In thermal reactor physics fine mesh neutron diffusion calculations in which homogenization is only at the pin cell level are of considerable interest. However, the slow convergence of iterations on fine-mesh spatial grids limits their applicability to practical problems. We present here an efficient way to accelerate the within-group iterations for pin-by-pin calculations based on a two-dimensional heterogeneous variational nodal method (VNM). Response matrix (RM) equations are formulated that incorporate multiple pins within each node. Within the nodes, finite elements in the x-y plane are employed to describe the piecewise constant heterogeneous geometry. On the nodal interfaces orthogonal polynomials are employed to approximate current distributions. The RM equations are solved using the Red-Black Gauss-Seidel (RBGS) iteration. Investigations on the coarse nodes acceleration (CNA) by combining homogenized pin cell nodes into larger heterogeneous nodes are performed. A series of meshing schemes are examined with a 2D small modular reactor core problem. With sufficient interface expansion order, CNA do not cause significant errors to either eigenvalues or fission rate distributions compared with fine mesh calculations. It is demonstrated that CNA accelerate the RBGS iteration significantly and achieves favorable accuracy-efficiency trade-off.

Fast solution of neutron diffusion problem by reduced basis finite element method

For the important applications which need carry out many times of neutron diffusion calculations such as the fuel depletion analysis and the neutronics-thermohydraulics coupling analysis, fast and accurate solutions of the neutron diffusion equation are demanding but necessary. In the present work, the certified reduced basis finite element method is proposed and implemented to solve the generalized eigenvalue problems of neutron diffusion with variable cross sections. The order reduced model is built upon high-fidelity finite element approximations during the offline stage. During the online stage, both the keff and the spatical distribution of neutron flux can be obtained very efficiently for any given set of cross sections. Numerical tests show that a speedup of around 1100 is achieved for the IAEA two-dimensional PWR benchmark problem and a speedup of around 3400 is achieved for the three-dimensional counterpart with the fission cross-sections, the absorption cross-sections and the scattering cross-sections treated as parameters.

Evaluation of Pin-Cell Homogenization Techniques for PWR Pin-by-Pin Calculation

In general, spatial homogenization, energy group condensation, and angular approximation are all included in the homogenization process. For the traditional pressurized water reactor (PWR) two-step calculation, the assembly homogenization with assembly discontinuity factors plus two-group (2G) neutron diffusion calculation have been proved to be a very efficient combination. However, this changes and becomes unsettled for the pin-by-pin calculation. Thus, this paper evaluates pin-cell homogenization techniques by comparison with the two-dimensional one-step whole-core transport calculation. For the homogenization, both the generalized equivalence theory (GET) and the superhomogenization (SPH) methods are studied. Considering the spectrum interference effect between different types of fuel pin cells, both 2G and 7-group (7G) structures are condensed from the 69-group WIMS-D4 library structure. For practical reactor core applications, the low-order angular approximations, including the diffusion and the SP3 methods, are compared with each other to determine which one is accurate enough for the PWR pin-by-pin calculation. Numerical results have demonstrated that both the GET and the SPH methods work effectively in pin-cell homogenization. In consideration of the spectrum interference effect, the 7G structure is sufficient for the pin-by-pin calculation. Compared with the diffusion method, the SP3 method can decrease the errors dramatically.

A three-dimensional nodal neutron kinetics code with a higher-accuracy algorithm for reactor core in hexagonal-z geometry

A nodal neutron kinetics model with a higher-accuracy algorithm for three-dimensional multi-group neutron diffusion calculation is developed for transient analysis of reactor core in hexagonal-z geometry. The spatial discretization technique is based on a high order nodal expansion method for hexagonal-z geometry. In this method, one dimensional radial and axial spatially flux of each node and energy group are defined as quadratic polynomial expansion and four order polynomial expansion respectively. The partially-integrated radial and axial leakages are both approximated by the quadratic polynomial. The time discretization technique is based on a fully implicit scheme combined with a Runge-Kutta method for solving the neutron flux and delayed neutron precursor equations. A flux weighting method is adopted to calculate homogenized multi-group cross sections for transient analysis of control rod ejection accident. The described numerical methods are implemented into the computer code HNHEXK. This code is tested with one static benchmark problem and several kinetic benchmark problems. The numerical results show that the current method has a good agreement with the reference.

Enhanced finite difference scheme for the neutron diffusion equation using the importance function

Mesh point positions in Finite Difference Method (FDM) of discretization for the neutron diffusion equation can remarkably affect the averaged neutron fluxes as well as the effective multiplication factor. In this study, by aid of improving the mesh point positions, an enhanced finite difference scheme for the neutron diffusion equation is proposed based on the neutron importance function.In order to determine the neutron importance function, the adjoint (backward) neutron diffusion calculations are performed in the same procedure as for the forward calculations. Considering the neutron importance function, the mesh points can be improved through the entire reactor core. Accordingly, in regions with greater neutron importance, density of mesh elements is higher than that in regions with less importance. The forward calculations are then performed for both of the uniform and improved non-uniform mesh point distributions and the results (the neutron fluxes along with the corresponding eigenvalues) for the two cases are compared with each other. The results are benchmarked against the reference values (with fine meshes) for Kang and Rod Bundle BWR benchmark problems. These benchmark cases revealed that the improved non-uniform mesh point distribution is highly efficient.

On an improved box-scheme finite difference method based on the relative event probabilities

The accuracy of the neutron diffusion calculations can be significantly affected by changing mesh point positions in the Box-Scheme Finite Difference Method (BSFDM). Accordingly, optimizing mesh point positions with respect to the relative region's importance for the 2-D rectangular geometry, light water reactors are addressed in this paper.In this study, we present a new algorithm approximation form based on the relative event probabilities (REPs), which can be directly used to evaluate the relative region's importance for optimizing mesh points positions in BSFDM. In accordance with this method, in regions with greater importance, the number of meshes is more than that in regions with less importance. So, the implementation of this method makes an adaptive nonuniform mesh point distribution through the entire reactor core unlike the usual methods. The results are benchmarked against two well-known solutions, namely the TWIGL seed-blanket and the Rod Bundle BWR.The results all suggest that the proposed method has a better accuracy in all benchmarking cases compared to the usual method for any arbitrary mesh point numbers.

Partitioned-Matrix acceleration to the Fission-Source iteration of the Variational Nodal Method

The Variational Nodal Method (VNM) expands the nodal volumetric flux and surface partial current into the sums of orthogonal basis functions without using the transverse integration technique. The exclusion of the transverse integration provides a number of advantages for employing the VNM in Pressurized Water Reactor (PWR) core simulation. The orthogonality of those basis functions guarantees the conservation of neutron balance regardless of the expansion orders, providing an opportunity to accelerate the computationally expensive full-order iteration by using cheap low-order sweeping with high-order moments fixed. This was named as the Partitioned-Matrix (PM) technique in the legacy VNM code VARIANT, and was applied to the within-group (WG) iteration. It is very effective for neutron-transport calculation, but less effective for neutron-diffusion mainly due to the reduced number of high-order partial current moments. In this paper, we extend the PM technique to the Fission-Source (FS) iteration to accelerate the flux convergence by using low-order flux moments also. From the macroscopic acceleration point of view, it converges the fission- and scattering-source distributions by using computationally cheap low-order iteration faster than the original full-order sweeping. Based on our new VNM code VIOLET, considering the fact that the discontinuity factor used for preserving neutron leakage rates during spatial homogenization slows down the nodal iteration convergence, numerical tests were carried out for two typical PWR problems respectively without and with discontinuity factors. By analyzing both the computational effort in terms of FLOP (FLoating-point OPeration) and computing time, the following conclusions have been demonstrated. The legacy PM technique for WG iteration can provide an acceleration ratio of about 2 for the PWR core neutron-diffusion calculation with or without using discontinuity factors, while the one for FS iteration itself can accelerate by a factor of about 3 which is higher. By accelerating both the WG and FS iteration simultaneously, the acceleration ratio is about 4 for both the two PWR problems. In addition, by extending the PM technique from the WG iteration to the FS iteration, the neutron-diffusion calculation of the VNM can be accelerated very effectively with almost no extra storage or implementation cost to the existing computer code.

Exact-to-precision generalized perturbation theory for neutron transport calculation

This manuscript extends the exact-to-precision generalized perturbation theory (EPGPT), introduced previously, to neutron transport calculation whereby previous developments focused on neutron diffusion calculation only. The EPGPT collectively denotes new developments in generalized perturbation theory (GPT) that place premium on computational efficiency and defendable accuracy in order to render GPT a standard analysis tool in routine design and safety reactor calculations. EPGPT constructs a surrogate model with quantifiable accuracy that can replace the original neutron transport model for subsequent engineering analysis. This is achieved by reducing the effective dimensionality of the state variable (i.e. neutron angular flux) via projection onto an active subspace determined using reduced order modeling techniques. Confining the state variations to the active subspace allows one to recast the problem in terms of a small number of ‘pseudo’ responses which are solely dependent on the physics model rather than on the nominal number of responses, the input parameters, or the number of points in the state phase space. This renders a computationally efficient construction of the surrogate whose form is determined using a recursive relationship from the solution of the transport equation. Unlike conventional surrogate techniques, the EPGPT can upper-bound the errors resulting from its predictions with high probability, which can be preset by the user.