Neutron Diffusion Theory Research Articles

Subdiffusive processes in BWRs

The objective of this work is the analysis of subdiffusive processes in BWR reactors. Nuclear reactors are highly heterogeneous systems where the phenomena at the reactor scale are not possible to describe with constitutive standard laws (normal diffusion) of energy, momentum, and mass. However, there are methods to achieve it such as the volumetric average to upscale the reactor and incorporate up-scaled constitutive laws. However, it can also be proposed constitutive laws of fractional order to consider anomalous diffusion. For the neutronic processes, the neutron diffusion theory is non-appropriated to the reactor scale, because this theory is applied to the homogeneous reactor. Given the heterogeneity of these systems in describing the dynamic behaviour of BWRs with point models, it represents a challenge to the incorporation of the heterogeneous effects whose main characteristic is that the transport phenomena are subdiffusive with relaxation times (i.e., non-instantaneous) by nature. The point models called reducer-order models are important for stability analysis and to develop control strategies. In linear or nonlinear stability analysis and system control, they are carried out with reduced order models that are zero dimension, i.e., point models that depend exclusively on time. This is because existing methodologies propose mathematical expressions in the time or frequency domain for stability analysis or to establish control strategies. The challenge is for ROM models to be able to consider subdiffusive effects on neutron motion. This is achieved by various routes from the P1 theory that includes the temporal term with relaxation effects (relaxation time) or by proposing a more general constitutive equation of fractional order for the neutron current vector, as is presented in this work. Moreover, the Fractional Order Models behavior must be as a transport theory based model, as well as in those nuclear and thermohydraulic codes which are very useful but complex, for which is sometimes it is impossible to obtain analytical expressions. To incorporate these effects in this work we consider point models of fractional order, which represent the subdiffusive phenomenon when the fractional order (known as anomalous diffusion coefficient) is greater than 0 and less than 1.

Analytical solution of the fractional neutron point kinetic equations using the Mittag-Leffler function

A new analytical solution of the fractional point kinetic equations is developed in the present work, which considers one group of delayed neutrons as well as a constant reactivity. One of the main novelties and theoretical contributions of the developed solution is that it does not require approximating the Inverse Laplace transform by numerical means, as other solutions do, and instead it is obtained analytically using the Green and the Mittag-Leffler functions. A set of algorithms and MATLAB codes are developed with the purpose to implement and compute the developed solution in an adequate way, which represents the major computational contribution. Numerical experiments show that the developed solution is very accurate solving short-time dynamics phenomena, being able to reproduce, in a precise way, data that is reported in literature. The development of the proposed analytic solution represents an important contribution to validate numerical methods that are currently used to solve the neutron fractional point kinetic equations. Program SummaryProgram Title: Mittag-Leffler FNPKECPC Library link to program files:https://doi.org/10.17632/v4kfhzw7sf.1Developer's repository link:https://github.com/Cruz-Lopez-Carlos-Antonio/Mittag-Leffler-FNPKE/tree/mainLicensing provisions: Creative Commons by 4.0Programming language: MATLAB (2021a) / Python 3Nature of problem:Fractional Neutron Point Kinetic Equations describe the neutron population in a nuclear reactor, considering the changes on transport and thermohydraulic properties of the system. For such task, a mass balance approach based on fractional and integral derivatives is used, which improves and generalizes the integer case that uses the classical definitions of these mathematical operators. From a physical point of view, this new approach takes into account memory and non-local effects that are omitted in the standard neutron diffusion theory and which are relevant in scenarios where the current neutron density undergoes strongly variations on time. From a mathematical point of view, using fractional derivatives leads to a new and more general type of differential equations of arbitrary order that requires different methods of solution and approximation, and which must include the integer approach as particular case. In such context, the development of analytical solutions is a fundamental problem because it provides a framework to carry out verifications of numerical methods, as well as an insight to perform theoretical comparisons with the classical model. The developed Mittag-Leffler FNPK codes analytically solve the Fractional Neutron Point Kinetic Equations defined in terms of the Caputo's derivatives and subject to classical initial conditions.Solution method:The developed codes are written in MATLAB and Python programming languages, and solve the mentioned fractional differential equation system using a power series expansion as well as the Laplace transform. Such solution considers a single group of precursors of delayed neutrons and it is expressed in terms of the 2-parameter Mittag-Leffler function, being accurately computed considering a small number of terms of the infinite sum. Unlike other approaches, in the present case it is not necessary to use a numerical procedure to find the inverse Laplace transform, because it is analytically found using the Green function. Even when the developed solution is found for a constant reactivity case, the Mittag-Leffler FNPKE codes can extended it to more general scenarios where the reactivity is a function of time or where it depends on the neutron density itself. In these cases, it is necessary to apply a subinterval methodology where the time is divided in small steps, assuming a constant reactivity in each of them. The smaller of the intervals, the more accurate the approximation. With the purpose to overcome the numerical issues related to the computation of the 2-parameter Mittag-Leffler function, the developed codes integrate the most advanced algorithm reported in literature, which is useful also to estimate the high-order derivatives of such function.Additional comments including restrictions and unusual features:The developed codes do not face issues related to arithmetical precision digits, as other solutions do, because it is not expressed in terms of high order polynomials and instead it is written as a linear combination of the Green function. Additionally, several steps can be precomputed without regard the neutron parameters and initial conditions, which represents an important advantage in terms of the computational time.

A 3D coupled thermo-mechanical and neutron diffusion numerical model for irradiated concrete

This paper presents a Finite Element-based formulation for assessing the behavior of irradiated cementitious materials under operational conditions, including both thermo-mechanical and neutron diffusion processes. Specifically, the mathematical model employs a two-group neutron diffusion theory in conjunction with heat conduction theory. This methodology accounts for the transition of fast neutrons to thermal neutrons and the associated heat generation resulting from the absorption of thermal neutrons during radiation exposure. The 3D coupled model is solved monolithically, as a single, integrated system. This proposed formulation addresses a significant gap in commercial software by enabling the analysis of multiphysics scenarios using a single tool. Additionally, it currently represents a unique fully coupled 3D model based on the Finite Element Method for tackling this specific multiphysics problem. Through validation against existing literature results, this approach demonstrates accurate evaluation of the radiation field, temperature variations, and stress/strain states in irradiated concrete operating under service conditions. Furthermore, to investigate the material’s durability, the proposed numerical formulation is extended to include mesomechanical studies.

Physics-constrained neural network for solving discontinuous interface K-eigenvalue problem with application to reactor physics

Machine learning-based modeling of reactor physics problems has attracted increasing interest in recent years. Despite some progress in one-dimensional problems, there is still a paucity of benchmark studies that are easy to solve using traditional numerical methods albeit still challenging using neural networks for a wide range of practical problems. We present two networks, namely the Generalized Inverse Power Method Neural Network (GIPMNN) and Physics-Constrained GIPMNN (PC-GIPIMNN) to solve K-eigenvalue problems in neutron diffusion theory. GIPMNN follows the main idea of the inverse power method and determines the lowest eigenvalue using an iterative method. The PC-GIPMNN additionally enforces conservative interface conditions for the neutron flux. Meanwhile, Deep Ritz Method (DRM) directly solves the smallest eigenvalue by minimizing the eigenvalue in Rayleigh quotient form. A comprehensive study was conducted using GIPMNN, PC-GIPMNN, and DRM to solve problems of complex spatial geometry with variant material domains from the field of nuclear reactor physics. The methods were compared with the standard finite element method. The applicability and accuracy of the methods are reported and indicate that PC-GIPMNN outperforms GIPMNN and DRM.

A new neutron-gamma porosity measurement method for pulsed neutron logging tools

As an alternative method for the compensated neutron porosity measurement method, the neutron-gamma porosity measurement method is widely used in cased holes. However, the neutron-gamma porosity measurement method has some problems, such as the lack of detailed theoretical analysis and low sensitivity in high-porosity formations. The purpose of this paper is to clarify the theory of neutron-gamma porosity measurement, improve the measurement sensitivity, and eliminate the effects of environmental factors. Based on the neutron diffusion theory and the gamma transport theory, the distribution of capture gamma rays is described. A new neutron-gamma porosity measurement method is developed based on inelastic and capture gamma-ray distribution. The applicability of the new method in different environments is analyzed using the Monte Carlo method, and the effectiveness of this method is verified using the synthetic model. The results indicate that the capture gamma-ray ratio is only related to the source-detector distance, gamma-ray attenuation length, and neutron migration length. By combining density, different neutron cross sections, inelastic gamma-ray count ratio, and capture gamma-ray count ratio, the parameter secondary gamma-ray hydrogen index (SGH) related to hydrogen index can be extracted. The dynamic range of SGH is much higher than thermal neutron count ratio and the capture gamma-ray count ratio. The errors in porosity calculation by this method under different borehole and lithologic conditions are generally less than 2 pu. The calculation error of this method is close to that of the compensated neutron porosity measurement method. In addition, the applicability of the new method is analyzed without density data, and the results indicate that the calculation results are basically consistent with the results of calculation with density data. The new method has great potential for extensive applications in cased holes or open holes as an alternative for the compensated neutron porosity logging measurement method.

Solution of Neutron Diffusion Problems by Discontinuous Galerkin Finite Element Method With Consideration of Discontinuity Factors

Abstract Neutronics calculations are the basis of reactor analysis and design. Finite element methods (FEM) have gained increasing attention in solving neutron transport problems for the rigorous mathematical formulation and the flexibility of handling complex geometric domains and boundary conditions. In order to reduce the computational errors caused by the homogenization of cross sections, this paper adopts the discontinuous Galerkin finite element method (DG-FEM) to solve the generalized eigenvalue problem formulated by the neutron diffusion theory and compensates the homogenization error by incorporating discontinuity factors. The results show that the discontinuous Galerkin finite element method can introduce the discontinuity factors with clear mathematical and physical meanings. The computational results of the discontinuous Galerkin finite element method are slightly better than those of the continuous Galerkin finite element method. However, the computation cost of the former is higher than that of the latter. Although good parallel efficiency can be achieved, the discontinuous Galerkin finite element method is not preferable for large-scale problems unless the effect of the discontinuity factors is significant.

Reactor space-dependent transfer function at low frequency approximated by neutron diffusion theory perturbation

The space-dependent transfer function defines the response of the neutron flux in reactor core to a periodic perturbation at a point in the core. To compute it at low excitation frequency for the ZED-2 zero power reactor, we introduce a perturbation method. The ZED-2 reactor is used to investigate the properties of CANDU® lattices. In CANDU®, the low frequency noise (0.1–5 Hz) allows one to study mechanical oscillations of structures in the core, the reactor control systems, as well as delayed neutron effects. The method for computing the ZED-2 transfer function was inspired by the results of measurements in the ZED-2 reactor. They showed that, at low excitation frequency, the imaginary part of the space-dependent transfer function is, up to normalization by the steady-state neutron flux, the same over all the core, while the real part varies significantly with the position of the flux detectors or the perturber in the core. This method was then found useful to compute the response of the ZED-2 reactor core to low frequency perturbations where other methods were found hard to apply. This opens the path to generalization of this method to CANDU® reactors. A rough calculation shows for which types of thermal reactors this perturbation theory is not worth applying. Also, the perturbation theory introduced here can be applied to other problems described by linear operators having a second smallest eigenvalue larger in absolute value compared to the perturbation. A further development, using the perturber induced reactivity, was found to make the core response normalization unnecessary. The theoretical basis of that method is outlined.

A study on the solution of the spatial kinetics equations in the neutron diffusion theory

In this work, recent studies on the solution of the spatial kinetics equations in the neutron diffusion theory are presented. A nodal technique for the treatment of spatial variables is applied, resulting in the definition of average quantities with respect to neutron fluxes, delayed neutron precursor concentrations, and current densities. The current densities are expressed in terms of the average neutron fluxes to adjust the number of additional unknowns that originate in the nodal scheme. Thus, two systems of coupled differential equations, dependent on the temporal variable and involving the neutron fluxes and delayed neutron precursor concentrations must be solved. A proposal for an iterative scheme with independent solutions of these two systems is developed. Analytical solutions are derived for the delayed neutron precursor concentrations equations. For neutron fluxes equations, numerical and analytical solutions are discussed. The latter use Padé approximations and the Schur–Parlett algorithm to evaluate matrix exponentials. Numerical results obtained through the proposed techniques are compared with results available in the literature, showing satisfactory agreement and contributing to establishing benchmark results. The proposed approach of solving the two systems independently lead to a reduction of the stiffness of the system, a relevant issue for the global approach, and required lower computational costs.

New exact solitary wave solutions for fractional model

<abstract><p>This manuscript involves the new exact solitary wave solutions of fractional reaction-diffusion model using the exp $ \mathrm{(-\ }\varphi \left(\eta \right) \mathrm{)} $-expansion method. The spatial model of fractional form is applied in modeling super-diffusive systems in the field of engineering, biology, physics (neutron diffusion theory), ecology, finance, and chemistry. The findings of miscellaneous studies showed that presented method is efficient for exploring new exact solutions to solve the complexities arising in mathematical physics and applied sciences. The new solutions which are obtained in the form of the rational, exponential, hyperbolic and trigonometric functions have a wide range in physics and engineering fields. Several results would be obtained under various parameters which shows good agreement with the previous published results of different papers. The proposed method can be extended to solve further problems arising in the engineering fields. My main contribution is programming and comparisons.</p></abstract>

New basis functions for reactor core calculations using the nodal expansion method

In 1977 Finnemann, Bennewitz and Wagner introduced the Nodal Expansion Method (NEM), by combining neutron diffusion theory, polynomial expansions, interface currents continuity, and weighted residual techniques. Extensive testing clearly indicated that it was consistent, computationally efficient, and accurate for meshes up to the typical size of assemblies. Two classical two-dimensional benchmarks were simulated: IAEA and LRABWR. Moments weighting was suggested to be preferable to Galerkin weighting. However, considering nodes of the size of assemblies, the fifth-order expansion using Galerkin weighting provided more accurate results than the fourth-order expansion using moments weighting (G3 and M2 on Finnemann's notation, respectively). The quadratic approximation has been considered appropriate to represent the transverse leakage terms (B2 on Finnemann's notation). Several studies seeking more accurate representations for the transverse leakage have been published since then. However, fourth-order expansion using moments weighting remains widely used as the most appropriate choice to approximate the average transverse fluxes, which may restrict the potentiality of the method.A detailed analysis of the NEM's basis functions has been performed, which allowed the development of formalism capable of generating polynomial expansions of any order. The new basis functions should introduce the additional terms required to describe the average transverse flux more properly, considering its distortions at fuel-reflector interfaces. In the same way that increasing the number of terms in convergent series leads to more accurate results, increasing the expansion order should bring the result closer to the expected solution. Comparisons of two- and three-dimensional LWR static benchmarks simulations are presented using up to tenth-order polynomial expansions, using both weight functions: moments and Galerkin. The introduced basis functions have led to more accurate results for keff and normalized assembly power distribution, including nodes at fuel-reflector interfaces. The increases in computing time using up to tenth-order expansions, for both weight functions, have not exceeded 40% for the simulated two-dimensional benchmarks and 90% for three-dimensional ones.

Variational nodal method for three-dimensional multigroup neutron diffusion equation based on arbitrary triangular prism

With the development of nuclear technology, new concepts of reactor core have been continuously proposed, and new types of geometric assembly and reactor core configuration would be adopted to achieve better characteristics. Traditional nodal method based on rectangle or hexagonal nodal is no longer applicable for complex geometry modeling. Given that triangles have the advantage of constructing arbitrarily complex geometry, in this study, variational nodal method of three-dimensional multigroup neutron diffusion theory based on arbitrary triangular prism was developed. ANSYS code is used to divide calculation region into arbitrary triangle mesh, which is then transformed into regular triangle mesh by coordinate transformation, and a set of orthogonal polynomial basis function is built. Variables such as nodal neutron scalar flux and neutron current on nodal surface are expanded based on those polynomial basis functions. The response relationship between neutron partial neutron current and neutron flux can be obtained by variational principle, and traditional power method is employed to solve those response matrix system. Based on above, TriVNM code was developed, and 4 types of typical benchmarks including rectangle, hexagonal assembly reactor and reactor with curved boundary were employed to verify TriVNM. Both power distributions and keff calculated by TriVNM code agree well with reference results, and the maximum relative error of all the benchmarks’ power doesn’t exceed 1% and error of keff doesn’t exceed 50 pcm. TriVNM code achieves the same or higher level of calculation accuracy compared with other diffusion codes. Mesh sensitivity analysis shows that for large mesh size TriVNM code still has good accuracy.

Three dimensional steady‐state neutronics/thermal‐hydraulics coupled simulation for a molten salt reactor moderated by zirconium hydride rods

Zirconium hydride (ZrH) is considered as an alternative moderator in the molten salt reactor (MSR) due to its excellent slowing-down ability, good thermal stability, and high radiation resistance. In the MSR, ZrH moderator is inserted in the form of rod into the fuel salt of the reactor core, and an axial and a transverse fuel salt flow occur between adjacent fuel salt channels. This inter channel flow mixing effect of fuel salt makes the molten salt reactor moderated by zirconium hydride rods (ZrH-MSR) substantially different from other type of reactors in both neutronics and thermal hydraulics. Based on the multi-group neutron diffusion theory and the subchannel thermal-hydraulics model, a steady-state analysis code for ZrH-MSR was developed and verified in this work. Then, a 25 MWth ZrH-MSR proposed by Transatomic Power Corporation was selected to evaluate its steady-state characteristics due to its cost-effective and simplified design. By applying this coupled code, the effects of control rod position, inlet fuel salt flow velocity, and residence time of fuel salt in the external loop on the distributions of the neutron fluxes, delayed neutron precursors (DNPs), and temperature for the 25 MWth ZrH-MSR were analyzed in detail. The simulation results show that the control rod position influences the neutron fluxes greatly, which indirectly affects the distributions of DNPs and core temperature. The inlet fuel salt flow velocity and the residence time of fuel salt in the external loop have little influence on the neutron fluxes of the core but affect the distribution of DNPs significantly. These numerical results can provide valuable information for the research and design of ZrH-MSR.

Cálculo de criticalidade pela teoria de difusão de nêutrons: uma análise comparativa de aproximação da densidade de corrente

Este trabalho apresenta uma análise comparativa entre algumas formas de aproximação das densidades de corrente em cálculos de criticalidade usando a teoria da difusão de nêutrons. Como os reatores nucleares são compostos por diversos materiais, definindo regiões heterogêneas onde os parâmetros nucleares variam de forma significativa, é fundamental que as correntes sejam expressas de tal forma a preservar continuidade nas interfaces das regiões. A partir de uma integração nodal na equação da difusão de nêutrons estacionária, apresentamos quatro propostas de aproximação para as densidades de corrente nas interfaces. Uma vez construído o modelo, o cálculo do parâmetro que define a criticalidade depende da determinação do autovalor dominante. Aqui apresentamos e discutimos três métodos do cálculo deste autovalor. A comparação dos resultados numéricos é realizada a partir de três problemas teste, em meios heterogêneos, disponíveis na literatura. Os resultados obtidos indicam que as aproximações mais efetivas para as densidades de corrente nas interfaces, para cálculos do autovalor e dos fluxos, são aquelas que relacionam os coeficientes de difusão dos dois nós comuns à interface (propostas 3 e 4). Além disso, o método da secante se mostrou mais eficiente para determinação do parâmetro da criticalidade.

Safety Analysis of the TRIGA 2000 U3Si2-Al Fuel Core Under Reactivity Insertion Accidents

TheTRIGA 2000 reactor in Bandung is planned to change its fuel type from the TRIGA fuel rod type to the U 3 Si 2 -Al plate type of low enriched uranium of 19.75 % with uranium density of 2.96 gU/cc. A study on the neutronic parameters from the equilibrium core has been done. To ensure safe operation of the new fuel, thermodynamic evaluation of the core needs to be done. The purpose of this study is to conduct a reactor safety analysis of reactivity insertion during withdrawal of the control rod and to study the effect of this reactivity insertion on the power and the maximum temperature of the fuel and the cladding. Reactivity insertion accident is the main factor of the design basis accidents in nuclear reactor design. A simulation of transient for reactivity insertion has been carried out using a coupled neutronic and thermal-hydraulic MTR-DYN code. The code was developed based on three-dimensional multigroup neutron diffusion theory. The coupled space and time-dependent problem were solved by adiabatic model. Transient analysis was performed for a reactivity insertion of 32.33 pcm/s with the assumption that all of the control rods were rapidly withdrawn. For the insertion at a low power of 100 W, the maximum power achieved was 2.74 MW while a maximum power of 2.3 MW was achieved for the power transient of 1 MW. The maximum temperature of the coolant, the cladding, and the fuel for TRIGA 2000 core does not exceed the allowable safety limit for reactivity insertions . Normal 0 false false false EN-US X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif"; mso-ansi-language:EN-US; mso-fareast-language:EN-US;}

Numerical calculation of spectral problems in SP3 approximation by FEM

The SP3 approximation of the neutron transport equation allows improving the accuracy for both static and transient simulations for reactor core analysis compared with the neutron diffusion theory. Besides, the SP3 calculation costs are much less than higher order transport methods (SN or PN). Another advantage of the SP3 approximation is a similar structure of equations that is used in the diffusion method. Therefore, there is no difficulty to implement the SP3 solution option to the multi-group neutron diffusion codes. In this work, the application of the SP3 methodology based on solution of the - and -spectral problems has been tested for reactor benchmarks. The FEM is chosen to achieve the geometrical generality. The results calculated with the diffusion and SP3 methods are compared with the reference calculation results.

Analytical Solution for Multi-Energy Groups of Neutron Diffusion Equations by a Residual Power Series Method

In this paper, a multi-energy groups of a neutron diffusion equations system is analytically solved by a residual power series method. The solution is generalized to consider three different geometries: slab, cylinder and sphere. Diffusion of two and four energy groups of neutrons is specifically analyzed through numerical calculation at certain boundary conditions. This study revels sufficient analytical description for radial flux distribution of multi-energy groups of neutron diffusion theory as well as determination of each nuclear reactor dimension in criticality case. The generated results are compatible with other different methods data. The generated results are practically efficient for neutron reactors dimension.

Pin-by-pin power reconstruction method using expansion in pseudo-harmonics

A pin-by-pin power reconstruction method was developed using the expansion in eigenfunctions for the reconstruction of neutron flux distribution in a homogeneous fuel assembly. The eigenfunctions used were the so-called pseudo-harmonics, which are eigenfunctions associated with the leakage plus removal operator of the neutron diffusion theory. The modulation technique was used to obtain the heterogeneous distribution of power density. In this technique, the power density homogeneous distribution, calculated with the reconstructed neutron flux is multiplied by a form function. This form function is obtained in the calculation of assembly homogenization. The results obtained by this method show good agreement between the reconstructed pin-by-pin power and that of the reference heterogeneous calculation, for the benchmark used. The largest deviation found was of the order of 4.7%, in a peripheral cell of a fuel element with the faces towards the region of the baffle/reflector.

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

ABSTRACTThe 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.

Neutronic and Thermal-Hydraulic Safety Analysis for the Optimization of the Uranium Foil Target in the RSG-GAS Reactor

The G. A. Siwabessy Multipurpose Reactor ( Reaktor Serba Guna G.A. Siwabessy , RSG-GAS) has an average thermal neutron flux of 2×10 14 neutron/(cm 2 sec) at the nominal power of 30 MW. With such a high thermal neutron flux, the reactor is suitable for the production of Mo-99 which is widely used as a medical diagnostic radioisotope. This paper describes a safety analysis to determine the optimum LEU foil target by using a coupled neutronic and thermal-hydraulic code, MTR-DYN. The code has been developed based on the three-dimensional multigroup neutron diffusion theory. The best estimated results can be achieved by using a coupled neutronic and thermal-hydraulic code. The calculation results show that the optimum LEU foil target is 54 g corresponding to the reactivity change of less than the limit value of 500 pcm. From the safety analysis for the case when the primary flow rate decreased by 15% from its nominal value, it was found that the peak temperatures of the coolant and cladding are 69.5°C and 127.9°C, respectively. It can be concluded that the optimum LEU foil target can be irradiated safely without exceeding the limit value. Received: 10 December 2015; Revised: 2 August 2016; Accepted: 4 August 2016

HPC implementation in the time-dependent neutron diffusion code AZKIND

This article presents a summary of the development of the computer code AZKIND. This code is based on multi-group neutron diffusion theory, where the kinetics equations include delayed neutron precursors. For space discretization a Galerkin process is applied using a nodal finite elements method, and the well-known θ-method is used for time discretization. A high performance computing methodology was implemented in AZKIND to solve the resulting linear algebraic system Av→=b→ where numerical solution of large algebraic systems representing full nuclear reactor cores is achieved with an acceleration technique based on the open source linear algebra PARALUTION library. This implementation allows AZKIND threading into a GPU thousands of arithmetic operations for parallel processing. The acceleration is demonstrated with the use of different nuclear fuel arrays resulting in extremely large matrices, getting a speedup ratio up to 48. Finally, the acceleration capabilities of the HPC implementation are presented in the simulation of a power transient in an actual boiling water reactor core.