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The neutron transport equation models the distribution of the neutron power inside a reactor core. The complexity of integrating this equation implies the use of some angular approximations. Discrete ordinates and spherical harmonics equations (P N equations) are commonly used to this purpose, but even when these approaches are used, the resulting set of equations requires high computational demands. An alternative is to develop simplified formulations of them, such as the simplified spherical harmonics equations (SP N equations), which provide accurate results. The spherical harmonic equations are based on the regularity of solution in terms of the angular dependence. However, in some reactor configurations, the angular fluxes can be discontinuous. For that, the double spherical harmonics equations, where angular fluxes are expanded in terms of two sets of Legendre polynomials, were studied. In this work, a simplified treatment of them is developed that leads to a diffusive set of equations, the simplified double spherical harmonics equation (SDP N equations). Numerical results show that the SDP N approximation provides faster and more accurate results than the SP N equations for some problems with strong variations of the angular neutron flux. • The double spherical harmonics equations approximate the neutron transport. • The simplified double PN equations extend the neutron diffusion theory. • This formulation improves the SPN results if strong angular flux variations appear. [ABSTRACT FROM AUTHOR]

The steady state neutron transport equation can be studied resorting to different eigenvalue formulations, useful to investigate the criticality state of a nuclear system. In this respect, the knowledge of the eigenvalue spectra featuring the main eigenproblems is fundamental, from a practical standpoint, to guide the eigenvalue solvers searching for the dominant eigenpairs. The study of the eigenvalue spectrum shape on the complex plane is also relevant to better understand the physico-mathematical aspects of the neutron transport numerical approximations. This paper investigates the features of the eigenvalue spectra with respect to various parameters like modelling approximations, departure from criticality and spatial heterogeneity in a one-dimensional slab geometry framework. The sensitivity of the energy spectrum characterising the fundamental eigenfunction is then evaluated with respect to the model adopted and to perturbations in the input parameters. • The Neutron transport equation can be arranged into different eigenvalue problems for approaching the critical state. • The knowledge of the eigenvalue spectra featuring the main eigenproblems is fundamental for both theoretical and practical evaluations. • This paper investigates the features of the eigenvalue spectra with respect to various parameters like modelling approximations, departure from criticality and spatial heterogeneity in a one-dimensional slab geometry. • The sensitivity of the energy spectrum characterising the fundamental eigenfunction is evaluated with respect to the model adopted and to perturbations in the input parameters. [ABSTRACT FROM AUTHOR]

• A scenario of SGTR + SLB was simulated in a three-loop Western 1000 MWe PWR. • A comparison is carried out between MELCOR 2.2 and CATHARE 2 codes. • The analysis is focused on thermal–hydraulic and FP releases. • The results are compared, discussed and the differences highlighted. The European project denominated Reduction of Radiological Consequences of design basis and design extension Accidents (R2CA) was launched in September 2019, with a very broad participation: 11 countries, 17 participating organisations, including international organisations, utilities, regulators, technical support organisations, researchers and developers and was coordinated by IRSN. The main goal of the project was to assess the conservatisms in the radiological releases calculations in nuclear power plant (NPP) studies. The work here presented is focused on developing new calculation methodologies and updating computer code models to carry out more detailed assessments of source terms resulting from a steam generator tube rupture (SGTR) accident in the Design Extension conditions A (DEC-A) domain. For this purpose, participants in Work Package 2 (WP2) of R2CA developed and implemented simplified models in the Severe Accident (SA) and Thermal-Hydraulic (TH) codes that take into account iodine spiking and iodine transport phenomena within primary and secondary circuit. These methodologies will not only provide a better estimation of the radiological consequences, but should also serve to improve accident management procedures, innovative instrumentation development and early detection tools. The new approaches are tested in a SGTR + Steam Line Break Outside Containment scenario without significant fuel degradation in a three-loop Western 1000 MW e PWR. During the transient, operators are also assumed to implement emergency operating procedures (EOPs) in line with Westinghouse EOPs to limit the release of radioactivity and control the evolution of plant parameters. The calculations are performed by Tractebel and CIEMAT with the American SA tool MELCOR and Bel-V with the French TH code CATHARE. The results highlighted some limitations of implemented models in predicting iodine behaviour as well as small discrepancies in the TH evolution of the transient, both of which were analysed and discussed in detail. [ABSTRACT FROM AUTHOR]

• The system of GPU accelerated neutron kinetics calculation is established. • The spatial parallel and space-angle parallel modes are used for neutron transport. • Both single- and multi-GPU techniques are used to speed up the GPU-NTLBM. The simulation of neutron transport equation (NTE) has become one of the main trends of nuclear reactor engineering and design owing to its high accuracy, wherein one of its limitations is the low computational speed. To improve this condition, this paper provides a GPU accelerated neutron transport lattice Boltzmann method (NTLBM) for solving neutron transport problem. The simplified finite Boltzmann scheme and the GPU CUDA technique are combined to achieve the spatial parallel calculation and the space-angle full parallel calculation. Due to the ease of implementation and high parallelism, the GPU accelerated NTLBM (GPU-NTLBM) can be very easily and efficiently to realize. Besides, by using the multi-GPU acceleration, the large-scale NTLBM engineering calculation can be efficiently speeded up, and the properties of NTLBM makes the internal boundary treatments convenient. In considering the properties of the NTE, the space-angle full parallel acceleration is applied to further speed up the GPU-NTLBM for the first time, in which all the spatial and the angular dimensions are calculated in parallel. Simulation results show that the NTLBM can be applied to solve the multi-group neutron transport problems accurately and the GPU acceleration can be applied to effectively speed up the NTLBM calculation. The space-angle full parallel acceleration can further improve the GPU-NTLBM acceleration. This paper could provide some parallel perspectives for solving the large-scale neutron transport engineering calculation. [ABSTRACT FROM AUTHOR]

• A new method combining immersed body projections and adaptive mesh resolution. • Provides solutions using optimized meshes specifically to resolve neutron transport. • Overcomes difficulties relating to large geometry conforming meshes. • Particularly beneficial for efficiently resolving complex domains and structures. This article describes a new adaptive immersed body method for the efficient and accurate modelling of neutron transport within geometrically complex domains. It combines two techniques of immersed body projections and self-adaptive mesh refinement to form a unique method that can efficiently resolve problems that contain complex internal structures. The approach allows complete freedom of where mesh resolution is placed and the meshes, which do not need to conform to the problem structure, are optimised for resolving the physics of the problem. Importantly the benefits can be found for problems with many internal structures such as fuel pins, control rods or cooling pipes and channels. Standard finite element meshes typically become large as they capture the geometrical detail and as a result solving times typically increase. This method overcomes this issue and allows for smaller meshes that are optimised to increase solution accuracy, as demonstrated in the numerical examples. [ABSTRACT FROM AUTHOR]

CORTEX is a EU H2020 project (2017-2021) devoted to the analysis of 'reactor neutron noise' in nuclear reactors, i.e. the small fluctuations occurring around the stationary state due to external or internal disturbances in the core. One important aspect of CORTEX is the development of neutron noise simulation codes capable of modeling the spatial variations of the noise distribution in a reactor. In this paper we illustrate the validation activities concerning the comparison of the simulation results obtained by several noise simulation codes with respect to experimental data produced at the zero-power reactors AKR-2 (operated at TUD, Germany) and CROCUS (operated at EPFL, Switzerland). Both research reactors are modeled in the time and frequency domains, using transport or diffusion theory. Overall, the noise simulators managed to capture the main features of the neutron noise behavior observed in the experimental campaigns carried out in CROCUS and AKR-2, even though computational biases exist close to the region where the noise-inducing mechanical vibration was located (the so-called "noise source"). In some of the experiments, it was possible to observe the spatial variation of the relative neutron noise, even relatively far from the noise source. This was achieved through reduced uncertainties using long measurements, the installation of numerous, robust and efficient detectors at a variety of positions in the near vicinity or inside the core, as well as new post-processing methods. For the numerical simulation tools, modeling the spatial variations of the neutron noise behavior in zero-power research reactors is an extremely challenging problem, because of the small magnitude of the noise field; and because deviations from a point-kinetics behavior are most visible in portions of the core that are especially difficult to be precisely represented by simulation codes, such as experimental channels. Nonetheless the limitations of the simulation tools reported in the paper were not an issue for the CORTEX project, as most of the computational biases are found close to the noise source. • Validation activities with dedicated experimental data from zero-power reactors. • Modeling in the time and frequency domain using transport or diffusion theory. • The noise simulators capture the main features of the neutron noise behavior. • Computational biases exist close to the noise source. [ABSTRACT FROM AUTHOR]

• Evaluate the suitability of a site for nuclear installations. • A Meteorological Pressure Index (MPI) is developed to identify sites with resilience capacity in terms of atmospheric transport phenomena. • Natural wind actions on a regional scale is taken into account. • MPI is able to aggregate meteorological data on a regional level and create maps. Today, the evaluation of a site for innovative nuclear installations represents a fundamental step in supporting the implementation of a sustainable nuclear power program. One major concern is the potential for radioactive releases and their effects on the biosphere, particularly on human health. To address this issue, a new Meteorological Pressure Index (MPI) has been developed to identify sites with resilience capacity in terms of transport phenomena correlated to natural wind actions on a regional scale. The MPI utilizes a fuzzy approach that combines data on 3D wind speed, direction, and frequency, providing maps that are useful in the site selection process. The application of MPI in the Sicily region of Italy has allowed to identify bordering areas that that are more affected by efficient air mass transport mechanisms. Consistency analyses have been conducted among MPI maps, observed wind distributions using wind rose charts, and stagnation S and recirculation R factors. The results confirm the MPI's ability to aggregate meteorological data on a regional level, making it a suitable tool when compared to disaggregated data that may not always provide a comprehensive viewpoint. [ABSTRACT FROM AUTHOR]

• A new class of non-linear acceleration methods has been created (SVRM). • The stability of SVRM is shown through a spectral radius and Fourier analysis. • The simplified numerical benchmark indicates a significant effectiveness advantage. In this paper we propose a new non-linear technique for accelerating the source iterations of the discrete-ordinates transport equation. The acceleration method, called Spatially Variant Rebalancing Method (SVRM), is based on the computation of the zeroth and first order spatial variation of the neutron balance equation. The non-linear acceleration is applied to the method of characteristics (MOC) with a step-approximation of the source. The new acceleration is meant to catch the high-order variation of the neutron flux within the spatial mesh. The paper proposes a numerical analysis of the technique based on the explicit computation of the Jacobian. The latter is analyzed with both spectral and Fourier analysis (Hong et al., 2010). Also, a comparison of the new method against CMFD (Smith, 1983), DSA (Larsen, 1982), and BPA (Adams et al., 1988) has been done for a parametrized heterogeneous problem, in order to study the performance of SVRM in different transport regimes. The analysis of SVRM has been constrained to plane geometries. [ABSTRACT FROM AUTHOR]

• The Method of Characteristics is applied as a new transport code called PerMOC. • Both forward and adjoint CMFD acceleration techniques are applied to the MOC kernel. • A newly proposed adjoint solver reduces about 30% the number of transport sweeps. • More speed-ups are achieved using a parallel transport sweeping in the MOC kernel. • The calculated reactivity of 2D C5G7 benchmark has only 51pcm error with the reference. In this research, the Coarse Mesh Finite Difference (CMFD) acceleration technique in both adjoint and forward mode is applied to the MOC kernel, to develop a new code called PerMOC. A new approach in the adjoint MOC kernel which is called thermal-up-scattering-like iteration scheme reduces the total number of adjoint iteration sweeps. In additions, more speed-up is achieved using a parallel transport sweeping in MOC kernel. Validation test for PerMOC C-Sharp code is carried out for four distinct criticality benchmarks in which C5G7 is one of those. In the case of C5G7 benchmark, the best-calculated multiplication factor has 51 pcm differences with the reference case. Furthermore, the newly proposed adjoint solver reduces about 30% the number of adjoint transport sweeps in comparison with the normal adjoint solver. The present article describes the PerMOC methodologies and illustrates its ability to model large problems with acceptable accuracy. [ABSTRACT FROM AUTHOR]

• A GPU-based parallel MOC algorithm is implemented, which includes several solving kernels. • A performance analysis model is applied to analyze the performance of the code and identify the limitation. • The corresponding optimizations according to the analysis are applied and the significant speedup ratio is obtained. The method of characteristics (MOC) is one of the most common methods for solving the neutron transport equation in practical application. Researches have been focused on the acceleration techniques and the parallel algorithm for improving the efficiency of MOC. The Graphics Processing Unit (GPU) provides an alternative method of parallelizing the MOC neutron transport sweep. In this work, a GPU-paralleled 2D MOC code is implemented, which employs the diamond difference (DD) scheme and the step characteristics (SC) scheme. Different parallel schemes which are ray-level, energy-group-level, and polar-angle-level, are analyzed to choose the proper parallel scheme. The C5G7 2D benchmark is calculated to verify the accuracy and efficiency of the code in different schemes with single precision and double precision. The bottlenecks of the GPU code are identified and the code is classified into three categories, which are compute-bound, memory-bound, and latency-bound, according to the performance analysis model introduced in this paper. In addition, corresponding optimization strategies are applied to improve the performance according to the analysis. Moreover, the speed, power efficiency, and hardware cost are compared for CPU and GPU based on a fictitious quarter core PWR problem. Numerical results demonstrate that the energy group-level parallelization can obtain the optimal performance on GPU. Optimization strategies are effective to improve the efficiency of the calculation on GPU, which indicates that the performance analysis model is useful and effective to locate the limitation of the code. Moreover, the GPU-version code is about 30 times faster than the CPU-version code with double precision and about 100 times faster with single precision, while the desired accuracy is kept. And the GPU delivers superior performance in both speed, energy efficiency, and hardware cost. [ABSTRACT FROM AUTHOR]

• The method can be utilized for neutronics modelling of hexagonal geometry reactors with high accuracies and efficiencies. • The new method reduces the eigenvalue errors by an order of magnitude and the error of fluxes by 2% with low order angular approximations. • With P 7 , the proposed methods yields a computational gain of 33.0 compared with the traditional method. An 11-core parallelization results in another 88% reduction in the total computation time. This paper presents recent progress of the development and implementation of the integral transport variational nodal method (VNM) to the neutronics modelling of hexagonal geometry nuclear reactors. In this study, the three-dimensional multi-group neutron transport equation is solved with the variational nodal method utilizing an integral form of the with-in node angular flux distribution to generate response matrices more efficiently. The angular flux distributions on hexagonal nodal interfaces are approximated with even-parity spherical harmonics. The quasi-reflected interface condition (QRIC) method is employed to reduce the number of interfacial angular terms to save computational costs. Parallelization algorithms are accounted to accelerate the response matrix formation process. The newly-implemented methods are examined against the TAKEDA-4 benchmark cases, presenting superior accuracy and efficiency than the standard VNM approach. With the P 7 angular approximation, the joint performance of the integral method and the QRIC method yields a remarkable computational time gain of 33.0. Furthermore, an 11-core parallelization to the response matrix formation process results in another 88% reduction in the total computation time. [ABSTRACT FROM AUTHOR]

• Approximations of small-scale interface and boundary perturbations. • Homotopy and homogenisation approximations were applied to cylindrical pincells. • Perturbations additionally applied to the Feinberg-Galanin method. • Perturbation methods provided a reasonable estimate at a lower computational cost. Rough and perturbed surfaces are examined in the field of reactor physics using homotopy, homogenisation and the Feinberg-Galanin method. The homotopy method allows a problem with a perturbed interface to be represented as an unperturbed problem with a modified boundary condition. Perturbation theory was applied to this method and the results were studied for the neutron transport equation, the neutron diffusion approximation and a hybrid ABH-diffusion approximation. The homogenisation of surface roughness in the fuel-moderator interface was also examined and compared to the homotopy approach. Finally, the effect of larger-scale geometric uncertainties was studied using the Feinberg-Galanin approach. The effectiveness of the methods is determined by examining the change in effective multiplication factor k eff examined both directly and via the six-factor formula. Analytic solutions to approximations of perturbation theory differentials were capable of accurately predicting the change in k eff when compared to numerical solutions. Examining a change in fuel pin radius of 20 µm, we find that for water moderated systems the overall change in k eff is close to 70 pcm and for graphite systems it is 18 pcm. This shows that water and graphite moderated cores are sensitive to small changes in geometry. [ABSTRACT FROM AUTHOR]

• A method to reduce k eff bias without changes to the Monte Carlo simulation process. • Calculation of k eff reliable value and its statistical uncertainty. • Method approval by calculation of the weakly coupled system. • The use of the method for estimation of neutron generation time and β eff. The work describes the realization of the method that allows reducing bias of the effective multiplication factor calculation in criticality problems without any changes in simulation and receiving reliable evaluations of its statistical uncertainty. The method is based on the calculation of a discrete adjoint function of a matrix transport equation. It is demonstrated that the use of the discrete adjoint function may improve estimations of neutron generation time and the effective delayed neutron fraction. Calculations have been performed by means of the MCU (Monte Carlo Universal) code. The «Whiteside problem» weakly coupled system and ZR-6 critical benchmark experiment have been used to show the effectiveness of the method. [ABSTRACT FROM AUTHOR]

Highlights • Review of SPN theoretical literature showing progress in understanding of SPN method. • Generalization of Pomraning's variational analysis of SPN equations. • New SPN paradigm based on a set of coupled integral equations for N + 1 scalar fluxes. • New equations provide an expression for the SPN angular flux as subset of PN flux. • Interface conditions assure regularity of SPN solution through the interface. Abstract We have generalized Pomraning's variational derivation of the Simplified P N (SP N) method and apply it to the first-order transport equation with anisotropic scattering and sources without restriction on the parity of N. The result is a new system of coupled linear integral equations for N + 1 scalar functions, which are shown to be equivalent to the traditional first-order and second-order (even or odd) SP N equations. The integral operator in these equations is the square root of the Laplacian. We view these integral equations as a natural paradigm for SP N in that they do not introduce the artificial current vectors which add two artificial degrees of freedom (per current) to the traditional first-order SP N equations. Interface conditions are derived by requiring the solution to behave smoothly across the interface between homogeneous regions. Surprisingly, these conditions are identical to the traditional ones obtained assuming a slab-like behavior for the angular flux. In an infinite homogeneous medium the integral SP N equations give the exact P N solution and here we join the work of Ackroyd et al. and Chao. We have also derived a simple recurrence relation to obtain the well-known PDE for the scalar flux. By virtue of the equivalence SP N -P N in an infinite homogeneous medium, this technique gives the scalar-flux PDE for P N , which was obtained for N = 1 , 2 , 3 by Davison in a much more involved way. We include also a comparative review of most of the derivations of the SP N equations in the literature, including the recent work by Chao, which shows the progress in the understanding of these equations which culminates with Chao's work. Our integral equations are fully equivalent to those of Chao, in that they predict the same equations and associated SP N angular flux. [ABSTRACT FROM AUTHOR]

Highlights • The kinetics transport equation was reformulated and implemented in the MOC solver of FEM-based high-fidelity neutron transport code PROTEUS. • The transient fixed source problem (TFSP) kinetics solver with the coarse mesh finite difference (CMFD) acceleration was implemented in PROTEUS-MOC. • The TFSP transient solver of PROTEUS were verified using the C5G7-TD 2D and 3D transient benchmark problems, showing very good agreement in power changes with time with the MPACT solutions. Abstract A spatial kinetics transport method based on the transient fixed-source problem has been implemented as the first step in the development of kinetics capabilities for the neutron transport code PROTEUS-MOC. The 2-D and 3-D cases of the C5G7-TD transient benchmark problems were used to verify the implementation. Benchmark results obtained by the new kinetics solver in PROTEUS-MOC are in good agreement with the results of other transport kinetics codes. [ABSTRACT FROM AUTHOR]

Highlights • Second-order sensitivities for uncollided particles are obtained by ray-tracing. • Responses are uncollided angular flux integrals on the system boundary. • Inputs are densities, cross sections, and source emission rates. • In a homogenous spherical test problem, results agree with analytic results. • In a two-region spherical test problem, results agree with those from PARTISN. Abstract The Second-Level Adjoint Sensitivity System (2nd-LASS) that yields the second-order sensitivities of a response of uncollided particles with respect to isotope densities, cross sections, and source emission rates has previously been derived and solved using the off-the-shelf discrete ordinates code PARTISN. In this paper, the 2nd-LASS equations are solved using ray-tracing for the case when the response is evaluated at a point on the boundary (the partial current density or scalar flux). The resulting second derivatives, including all mixed partial derivatives as well as derivatives involving material mass densities, are compared with analytic and PARTISN results for a homogeneous sphere and PARTISN results for a two-region sphere. Second derivatives computed using ray-tracing are within 0.003% of those computed using PARTISN and are over 100 times faster to compute for the spherical test problems. [ABSTRACT FROM AUTHOR]

Most deterministic neutron transport codes implement the multi-group approximation theory to solve the linear Boltzmann equation by which the neutron flux and the interaction cross-sections are averaged over discretized energy groups. In order to further reduce the computational cost for large scale and/or time dependent problems, this multi-group information undergoes additional energy group condensation into the so-called few energy group structure which may be used in a broad range of diffusion based full-core analyses. The accuracy and efficiency of few-group based computations is dependent on the number and the boundaries of the discrete energy group structure. Since the flux spectrum used in the homogenization process may not be known a priori and indeed may evolve in space, with burnup, and during transients, the optimal energy group structure depends on reactor type, design, operating conditions, fuel type, and composition. The Canadian Pressure Tube Super-Critical Water-cooled Reactor (PT-SCWR) is a Generation IV advanced reactor system that uses light water above its thermodynamic critical point as coolant and a plutonium-driven thorium fuel mixture. Considering that the anticipated flux spectrum for such a design deviates significantly from the thermal-neutron dominated CANDU designs, there may be a need for improvements in the number and boundaries of the few-group nuclear data. This paper presents a systematic methodology for the delineation of few-group energy structure for the Canadian PT-SCWR. The methodology used the SCALE (Standardized Computer Analysis for Licensing Evaluation) code package to examine the effect of energy group homogenization and determine the structure which minimizes the sensitivity of the results to energy group partitioning. As such, it utilizes normal and off-normal operating conditions to determine the effect of energy homogenization and determines the minimal group of energy boundaries which can accurately capture the lattice physics phenomena within the lattice cell over a wide range of operating and accident conditions. [ABSTRACT FROM AUTHOR]

In this paper, the neutron noise based on transport theory and diffusion noise theory using Green’s function technique is calculated. As the neutron noise is used for core diagnostic, surveillance and monitoring, calculation of neutron noise precisely can play an important role in monitoring and safety. We compare the accuracy of Green’s function based on transport and diffusion theory in order to survey the differences between these theories. In this study some deviation between results obtained two theories are observed, and the impact of dimensions, cross sections and frequency on the results investigated. [ABSTRACT FROM AUTHOR]

The Tone’s and embedded self shielding methods are based on a heterogeneous-homogeneous equivalence principle. They both provide a formula for an equivalent dilution parameter that can be used to interpolate pretabulated effective cross sections for the infinite homogeneous medium. These two methods have been traditionally applied to fast reactor analysis with fine or ultrafine group energy mesh. We are investigating the accuracy of these two methods for the generation of self-shielded cross sections for typical pressurized water reactor assemblies. [ABSTRACT FROM AUTHOR]

Pyroprocessing is a method involving electrochemistry, thermodynamics, and transport theory. Driven by the current, nuclear materials redistribute between anode, electrolyte, and cathode to realize the desired separation. In the present study, a model for predicting the mass transport for the electrochemical process was developed. The model took into account the following key parameters: diffusion and mass transfer in both the electrode and electrolyte, the anodic dissolution and cathodic precipitation kinetics, the molten salt flow condition, and the solubility of species in molten LiCl-KCl and liquid cadmium. It was validated by available transport data between different pairs of the electrodes, namely liquid cadmium anode to solid cathode, solid anode to solid cathode, and solid anode to liquid cadmium cathode. Time for Pu depositing on the solid cathode as well as Zr dissolving from U-Pu-Zr alloy anode was precisely predicted. The potential and composition changes in various experiments were also well reproduced. The plausible explanation was given to explain the minor disagreement and future work was suggested to improve its capability. The model is useful to understand the reaction mechanisms and safeguard a pyroprocessing facility by monitoring the separation process and tracking the material flow. [ABSTRACT FROM AUTHOR]

A massively parallel implementation of a recently developed technique for numerically integrating the transport equation, The Random Ray Method (TRRM) (Tramm et al., 2017), is applied to several large reactor benchmark problems. The implementation, which is part of a new development called The Advanced Random Ray Code (ARRC), is one of the first parallel implementations of TRRM. Our goal is to better understand the accuracy and performance characteristics of TRRM on massive scale problems, and to provide community software that facilitates further algorithmic development and potentially its application to a broader class of problems. Key features of ARRC include extreme memory efficiency, domain decomposition, a task based parallel structure, and the ability to efficiently utilize Single Instruction Multiple Data (SIMD) vector units. These attributes lead to efficient performance on modern high performance computer (HPC) architectures, enabling the detailed simulation of reactor cores in three dimensions. [ABSTRACT FROM AUTHOR]

Two generalized adjoint solution modules, the generalized adjoint transport equation solver and the generalized adjoint B 1 equation solver, were implemented into DeCART for the general response uncertainty analysis on the doubly heterogeneous system. The generalized adjoint transport equation solver for the doubly heterogeneous system was implemented based on the Sanchez-Pomraning method. Moreover, the generalized adjoint B 1 equation solution scheme was newly set up using the adjoint B 1 equation for an efficient calculation. From the benchmark analysis on the MHTGR-350 Ex. I-1, it was confirmed that the two generalized adjoint solver can provide reliable results for the doubly heterogeneous region. Then, the uncertainties for the k inf and the microscopic cross sections on the same problem were evaluated using DeCART/MUSAD code system. The results reveal that the double heterogeneity does not have a significant effect on the k inf uncertainty but slightly increases the uncertainty of 238 U capture compared with the homogeneous case. [ABSTRACT FROM AUTHOR]

Noise experiments, and the Feynman- α method in particular, are considered a standard in-pile experiment, often used in zero power reactors in a sub critical configuration to estimate the reactivity level. In the Feynman- α method, the variance to mean ratio of event triggered neutron count is fitted on the so called Feynman-Y function, resulting with an experimental estimation of the α decay mode. In a recent study, analytic formulas for the expected statistical uncertainty in the Feynman- α method were derived, using the backward stochastic transport equation. The outline of the present study is to experimentally validate these analytic formulas. The formulas were implemented on 4 different signals, obtained from the MINERVE reactor at CEA Cadarache. For each of the four measurements, the signal was truncated in increasing measurement lengths (ranging from 1 to 60 min), in order to observe the functional behavior of the statistical error through time. Then, the measured statistical error was compared with the analytic estimation. Results indicate a very high correspondence between the expected statistical error and the measured one. [ABSTRACT FROM AUTHOR]

The advances in computer processing power have made it possible to perform a detailed pin-by-pin calculation of the whole reactor core. A transport theory code TRANPIN has been developed to perform a whole reactor core pin-by-pin calculation in 2D hexagonal geometry. This code employs the interface current method based on 2D collision probability for reactor core calculation. The present code divides each lattice cell location in the fuel assembly (FA) into finer regions. The subdivided regions inside the lattice cell are connected using the 2D collision probabilities. The coupling of lattice cells within the assembly and assembly to assembly coupling is achieved using interface currents. The interface currents are obtained by expanding the angular flux leaving or entering the lattice cell surface into double orthonormal P2 polynomials. The interface current method is traditionally used to perform lattice calculations and its application to core calculations in hexagonal geometry is not reported in literature to the best knowledge of authors. The present code can perform the calculation in any multigroup energy structure. The code TRANPIN has been validated against two benchmark problems i) a simplified high temperature test reactor (HTTR) benchmark problem and ii) OECD VVER-1000 MOX Core Computational Benchmark. It may be mentioned that the OECD benchmark is analyzed using ultra fine WIMS library in 172 energy groups. This approach may be considered as unprecedented since the full reactor core calculations are normally performed with cross sections prescribed in few energy groups by some prior transport simulations. The present paper describes the interface current methodology incorporated in TRANPIN. The comparison of results of benchmark analyses with published results is also presented. [ABSTRACT FROM AUTHOR]

The SEM approach is an advanced numerical technique based on combination of high-order Finite Element Method (FEM) and spectral methods, which benefits both from geometric flexibility of Finite Element Method and high accuracy of spectral methods. In this study, the neutron transport equation has been converted into weak form obtained by Generalized Least Squares Method (GLSM) and implementation K + variational principle. In order to analyze the spatial-wise and the angular-wise integrals of K + variational method, the angular part of flux has been developed via spherical harmonics expansion while Spectral Element Method has been applied in spatial part of flux. Spectral nodal discretization and trial function expansion has been implemented via different orthogonal polynomials. In present work, a comparison among different basis shape functions (trial functions) applied in SEM approach has been illustrated. Therefore, a general comparison between the capability of nodal discretization and solution accuracy of different polynomials can be represented. The incorporation of SEM with variety of nodal discretization of polynomials would give an opportunity to study the behavior of each of polynomials in numerical calculations. [ABSTRACT FROM AUTHOR]

This article presents a new efficient space-angle subgrid scale formulation (SGS) for the space-angle phase-space discretisation of the Boltzmann transport equation (BTE). It applies both coarse and fine scale discretisations to the space-angle phase-spaces, for which the fine scale solutions yield more accurate but more expensive solutions compared to those of the coarse scale approach. Accuracy and efficiency are maintained by integrating the fine scale components within the coarse scale formulation. This results in a method that generates linear systems the same size as those that would be generated by the coarse scale system. The formulation is based on using finite elements (FEs) to represent the spatial variable of the BTE, whilst spherical harmonics are employed to represent the particles’ directions. In space, the fine and coarse scales are represented through discontinuous and continuous linear finite elements. In angle, the scales are represented by the full spherical harmonics (coarse scale), and a partitioning of these basis functions into subsets that couple only between themselves (fine scale). Whilst the spatial discretisation has been previously documented, it is the angular treatment that is critical for efficient solutions. This is particularly the case for large angular expansion sizes involving angular discretisations that couple moments through the streaming term (i.e. not discrete ordinate schemes), where it is shown here that the computing costs are reduced to O ( M ) from O ( M 3 ) , where M denotes the angular expansion size. It is shown how the method increases efficiency of the solving process through the improved efficiency of matrix vector multiplication which forms the most expensive component of a Krylov based solver. This matrix vector product is also computed without the matrix being stored in memory which would be a requirement for large scale radiation transport calculations. The method’s capabilities are demonstrated by solving fixed source problems involving a range of material types – from optically thin to opaque. In all cases the new method yields accurate and stable solutions. [ABSTRACT FROM AUTHOR]

The relation between the Rossi alpha and Feynman alpha formulae is well known. The second factorial moment of counts in an interval, which is related to the variance, can be obtained by a double integration of the Rossi alpha expression. However in a recent paper, Munoz Cobo et al. have cast doubts on the generality of this relation stating that this relation is valid only for point models and is not true for space dependent models. The present paper revisits the issue. We examine the multi mode formulae obtained by Munoz Cobo et al. in order to assess the validity of the above relationship. We show that indeed it is possible to derive the Feynman alpha formula by a double integration of the Rossi alpha formula followed by some elementary algebraic manipulation. To demonstrate the generality of this relationship we use two distinct approaches. In the first of these, we show, using the (backward) stochastic transport equation and the space dependent Bartlett formula, that the Rossi alpha and Feynman alpha functions obey the above relationship. In the second, we start from the theory of stochastic point processes and show that the relationship is quite general. [ABSTRACT FROM AUTHOR]

The impact of the approximations in the “two-steps” procedure used in the current generation of nodal simulators for core transient calculations is assessed by using a higher order solution obtained from a direct, whole core, transient transport calculation. A control rod ejection accident in an idealized minicore is analyzed with PARCS, which uses the two-steps procedure and DeCART which provides the higher order solution. DeCART is used as lattice code to provide the homogenized cross sections and kinetics parameters to PARCS. The approximations made by using (1) the homogenized few-group cross sections and kinetic parameters generated at the assembly level, (2) an effective delayed neutrons fraction, (3) an effective fuel temperature and (4) the few-group formulation are investigated in terms of global and local core power behavior. The results presented in the paper show that the current two-steps procedure produces sufficiently accurate transient results with respect to the direct whole core calculation solution, provided that its parameters are carefully generated using the prescriptions described in the present article. [ABSTRACT FROM AUTHOR]

Monte Carlo transport codes are extensively used in neutronic analysis, especially in criticality safety analysis and shielding analysis. Super Monte Carlo Simulation Program for Nuclear and Radiation Process (SuperMC) is a CAD based Monte Carlo program for integrated simulation of nuclear systems. The aim of this paper was to show the capability of SuperMC on criticality calculation with different models. In this study 119 representative benchmarks from the International Handbook of Evaluated Criticality Safety Benchmark Experiments (ICSBEP) were used for validating the code. All the benchmark models and input files were built and completed by employing the models’ conversion and construction function of SuperMC. The results showed that there was a good agreement between SuperMC and experimental data, and the discrepancies were mainly included in the statistical uncertainty. [ABSTRACT FROM AUTHOR]

Monte Carlo method plays an important role in accurate simulation of random media, owing to its advantages of the flexible geometry modeling and the use of continuous-energy nuclear cross sections. Three stochastic geometry modeling methods including Random Lattice Method, Chord Length Sampling and explicit modeling approach with mesh acceleration technique, have been implemented in RMC to simulate the particle transport in the dispersed fuels, in which the explicit modeling method is regarded as the best choice. In this paper, the explicit modeling method is applied to the depletion benchmark for HTGR fuel element, and the method of combination of adjacent burnup regions has been proposed and investigated. The results show that the explicit modeling can provide detailed burnup distribution of individual TRISO particles, and this work would serve as a supplement for the HTGR fuel depletion benchmark calculations. The combination of adjacent burnup regions can effectively reduce the memory footprint while keeping the computational accuracy. [ABSTRACT FROM AUTHOR]

MCNP6 can be described as the merger of MCNP5 and MCNPX capabilities, but it is much more than the sum of these two computer codes. MCNP6 is the result of six years of effort by the MCNP5 and MCNPX code development teams. These groups of people, residing in Los Alamos National Laboratory’s X Computational Physics Division, Monte Carlo Codes Group (XCP-3) and Nuclear Engineering and Nonproliferation Division, Radiation Transport Modeling Team (NEN-5) respectively, have combined their code development efforts to produce the next evolution of MCNP. While maintenance and major bug fixes will continue for MCNP5 1.60 and MCNPX 2.7.0 for upcoming years, new code development capabilities only will be developed and released in MCNP6. In fact, the initial release of MCNP6 contains numerous new features not previously found in either code. These new features are summarized in this document. Packaged with MCNP6 is also the new production release of the ENDF/B-VII.1 nuclear data files usable by MCNP. The high quality of the overall merged code, usefulness of these new features, along with the desire in the user community to start using the merged code, have led us to make the first MCNP6 production release: MCNP6 version 1. High confidence in the MCNP6 code is based on its performance with the verification and validation test suites, comparisons to its predecessor codes, our automated nightly software debugger tests, the underlying high quality nuclear and atomic databases, and significant testing by many beta testers. [ABSTRACT FROM AUTHOR]

Three-dimensional, full core transport modeling with pin-resolved detail for reactor dynamic simulation is important for some multi-physics reactor applications. However, it can be computationally intensive due to the difficulty in maintaining accuracy while minimizing the number of time steps. A recently proposed Transient Multi-Level (TML) methodology overcomes this difficulty by use multi-level transient solvers to capture the physical phenomenal in different time domains and thus maximize the numerical accuracy and computational efficiency. One major problem with the TML method is the negative flux/precursor number density generated using large time steps for the MOC solver, which is due to the Backward Euler discretization scheme. In this paper, the stability issue of Backward Euler discretization is first investigated using the Point Kinetics Equations (PKEs), and the predicted maximum allowed time step for SPERT test 60 case is shown to be less than 10 ms. To overcome this difficulty, linear and exponential transformations are investigated using the PKEs. The linear transformation is shown to increase the maximum time step by a factor of 2, and the exponential transformation is shown to increase the maximum time step by a factor of 5, as well as provide unconditionally stability above a specified threshold. The two sets of transformations are then applied to TML scheme in the MPACT code, and the numerical results presented show good agreement for standard, linear transformed, and exponential transformed maximum time step between the PKEs model and the MPACT whole core transport solution for three different cases, including a pin cell case, a 3D SPERT assembly case and a row of assemblies (“striped assembly case”) from the SPERT model. Finally, the successful whole transient execution of the stripe assembly case shows the ability of the exponential transformation method to use 10 ms and 20 ms time steps, which all failed using the standard method. [ABSTRACT FROM AUTHOR]

The aim of the paper is to provide a contribution for the extension to heterogeneous media of the A ∞ and A N formulations for the linear transport equation. A new approach is proposed, in which the exponential of the optical length is decomposed in a sum of pure exponential terms, as in a homogeneous medium; in this way, a set of coupled diffusion-like equations is obtained. Different approximations are presented and their validity is discussed. [ABSTRACT FROM AUTHOR]

An analytical discrete-ordinates solution is developed for the problem of particle transport in ducts, as described by a one-dimensional model constructed with two basis functions. Two types of particle incidence are considered: isotropic incidence and incidence described by the Dirac delta distribution. Accurate numerical results are tabulated for the reflection probabilities of semi-infinite ducts and the reflection and transmission probabilities of finite ducts. It is concluded that the developed solution is more efficient than commonly used numerical implementations of the discrete-ordinates method. [ABSTRACT FROM AUTHOR]

We consider transport of light, neutrons, or any uncharged particles in a straight duct of circular cross section. This problem first came to fashion some 30 years ago when Pomraning and Prinja formulated their so called “pipe problem”. In the years to follow, investigators applied essentially every known method of numerical solution, including MMRW’s Wiener–Hopf – except possibly one. This presentation concerns that particular numerical solution, which arguably seems to be the most efficient of all. [ABSTRACT FROM AUTHOR]

We consider time-dependent transport in the 2-stream or “rod” model via an attractive matrix formalism. After reviewing some classical problems in homogeneous media we discuss transport in materials whose density may vary. There we achieve a significant contraction of the underlying Telegrapher’s equation. We conclude with a discussion of stochastics, treated by the “first-order smoothing approximation.” [ABSTRACT FROM AUTHOR]

We present a study of two dimensional, fixed-source neutron transport problems on a heterogeneous medium. It is an extension of a methodology used for solving neutron transport problems on an homogeneous medium, that combines nodal schemes with explicit solutions of the transversal integrated equations via the Analytical Discrete Ordinates method (ADO). We consider closed-form solutions of the integrated discrete ordinates transport equation on a two dimensional cartesian geometry, together with a level symmetric quadrature scheme, on each region of interest, in the domain, possibly characterised by different materials. In this work, each solution in a region is coupled with that of its neighbouring regions to provide the whole solution, without resorting to using iterative schemes. The terms involving unknown angular fluxes that arise using nodal schemes, assumed as constant function approximations, are added to the source term. The model proposed leads to a considerable reduction of the order of the associated eigenvalue problems written as perturbations of diagonal matrices, and the solutions obtained are explicit in terms of the spatial variables. Analytical expressions are also derived for the elementary solutions. The numerical results obtained are shown to be in good agreement with other results available in the literature. [ABSTRACT FROM AUTHOR]

Two classic problems of radiative transfer and neutron transport are solved for a spatially-uniform semi-infinite medium with isotropic scattering. General analytical equations are derived for (1) angular moments of the outward current and (2) spatial moments of the total flux within the half-space. Such moments, for example, can provide analytically explicit equations for determining the surface albedo of the medium as well as the mean depth and mean square distance of travel within the medium. The analysis is done with the Case-style eigenmode method as expressed in terms of the Chandrasekhar H -function and its moments. [ABSTRACT FROM AUTHOR]

“Three-dimensional transport theory: an analytical solution of an internal beam searchlight problem, I”, Annals of Nuclear Energy, 36(8), 1256–1261 (2009) by Williams extends the range of analytical solutions, and the associated development of techniques, numerical results and analysis near singularities. The final integrals are not easy to evaluate as the integrands are highly oscillatory, singular and also on infinite range. We report here some further numerical evaluations of expressions of Williams, and also compare these with those of Williams and Ganapol and Kornreich. The numerical results compare very well. The disagreements are very rare, and even then in the fifth decimal place. We are also able to explore the nature of the results near singularities in conformity with the results of Williams. We also verify MCNP-5, the widely used Monte Carlo code against these analytical results. We have found that MCNP is easily able to provide results within 0.1% deviation from the “exact” results for most cases, and within 1% for almost all cases. It is challenged near the singularities, however, where the deviations are larger. [ABSTRACT FROM AUTHOR]

• A new 3-D MOC code called ARCHER is developed for pebble-bed HTRs. • The CMFD acceleration is constructed under complex pebble-bed problem. • The two-level MPI-OpenMP parallel is implemented in the ARCHER code. • The performance of the ARCHER code is verified using several cases. A new deterministic code ARCHER using Three-Dimensional (3-D) Method of Characteristics (MOC) has been developed by Institute of Nuclear and new Energy Technology (INET), Tsinghua University, aiming to obtain high-fidelity transport solution of pebble-bed High Temperature gas-cooled Reactor (HTR) with explicit pebble-bed geometry. However, 3-D MOC usually suffers from the slow convergence rate without efficient acceleration methods. In this work, the Coarse Mesh Finite Difference (CMFD) based on regular mesh is implemented to accelerate 3-D MOC calculation, which is available for the problems with cylinder and cuboid geometry. The complex geometry of reactor core with randomly stacked spherical fuel elements and irregular helium flow area, as well as the topological relationships between fine Flat Source Region (FSR) mesh for MOC and coarse CMFD mesh, are the key but challenging tasks for high-fidelity transport solution. Constructive Solid Geometry (CSG) and Boolean operation are constructed to describe problem geometry, which has the ability to deal with complex topological relationships between meshes. In addition, tree grid structure is utilized to reduce the time complexity of ray tracing and obtain the grid mapping relationship between coarse mesh and fine mesh. Spatial domain decomposition parallel based on MPI is utilized to alleviate the memory pressure and pursue high computational performance. Ray parallel is further achieved through OpenMP with shared memory. Several problems have been used to verify the geometric modeling capability and computational performance of this new 3-D MOC code for pebble-bed HTR. [ABSTRACT FROM AUTHOR]

In this work we solve the transport equation with strongly anisotropic scattering, i.e., with a forward–backward-anisotropic kernel. We treat the problem by finding an integral representation to the solution, which we then project to a finite dimensional space. We verify numerically the robustness of the techniques we develop by performing calculations for several cases found in the literature. Then we obtain new numerical results for the transport equation with strongly anisotropic scattering when the kernel has more than two terms. Our simulations allows us to obtain the total intensity, the total flux, the dominant eigenvalue and the critical thickness with precision to at least five digits. These simulations indicate that adding third and fourth kernel terms contributes to about 1% in the studied cases. [ABSTRACT FROM AUTHOR]

Discrete ordinates method relies on approximating the integral term of the transport equation with the aid of quadrature summation rules. These quadratures are usually based on certain assumptions which assure specific symmetry rules and transport/diffusion limits. Generally, these assumptions are not problem-dependent which results in inaccuracies in some instances. Here, various methods have been developed for more accurate estimation of the independent angle in S 2 approximation, as it is tightly related to valid estimation of the diffusion coefficient/length. We proposed and examined a method to reduce a complicated problem that usually is consisting many energy groups and discrete directions (S N ) to an equivalent one-group S 2 problem while it mostly preserves general features of the original model. Some numerical results are demonstrated to show the accuracy of proposed method. [ABSTRACT FROM AUTHOR]

A computational fluid dynamics (CFD) model of annular two-phase flow with evaporating liquid film has been developed based on the Eulerian–Lagrangian approach, with the objective to predict the dryout occurrence. Due to the fact that the liquid film is sufficiently thin in the diabatic annular flow and at the pre-dryout conditions, it is assumed that the flow in the wall normal direction can be neglected, and the spatial gradients of the dependent variables tangential to the wall are negligible compared to those in the wall normal direction. Subsequently the transport equations of mass, momentum and energy for liquid film are integrated in the wall normal direction to obtain two-dimensional equations, with all the liquid film properties depth-averaged. The liquid film model is coupled to the gas core flow, which currently is represented using the Eulerian–Lagrangian technique. The mass, momentum and energy transfers between the liquid film, gas, and entrained droplets have been taken into account. The resultant unified model for annular flow has been applied to the steam–water flow with conditions typical for a Boiling Water Reactor (BWR). The simulation results for the liquid film flow rate show favorable agreement with the experimental data, with the potential to predict the dryout occurrence based on criteria of critical film thickness or critical film flow rate. [ABSTRACT FROM AUTHOR]

The method of characteristic direction probabilities (CDP) combines the benefits of the collision probability method (CPM) and the method of characteristics (MOC) for the solution of the integral form of the Botlzmann Transport Equation. By coupling only the fine regions traversed by the characteristic rays in a particular direction, the computational effort required to calculate the probability matrices and to solve the matrix system is considerably reduced compared to the CPM. Furthermore, boundary averaging is performed to reduce the storage and computation but the capability of dealing with complicated geometries is preserved since the same ray tracing information is used as in MOC. An analysis model for the outgoing angular flux is used to analyze a variety of outgoing angular flux averaging methods for the boundary and to justify the choice of optimize averaging strategy. The boundary average CDP method was then implemented in the Michigan PArallel Characteristic based Transport (MPACT) code to perform 2-D and 3-D transport calculations. The numerical results are given for different cases to show the effect of averaging on the outgoing angular flux, region scalar flux and the eigenvalue. Comparison of the results with the case with no averaging demonstrates that an angular dependent averaging strategy is possible for the CDP to improve its computational performance without compromising the achievable accuracy. [ABSTRACT FROM AUTHOR]

The introduction of interfacial area transport equation (IATE) has greatly improved the overall performance of two-fluid model due to the fact that the IATE can predict the interfacial area concentration (IAC) more accurately. However, models in the source and sink terms of IATE that are developed for predicting bubble coalescence and breakup are mainly based on small diameter pipe flows. Since the two-phase flow in a large diameter pipe is characterized by its multi-dimensional nature, the IATE should be confirmed before being applied to large pipe flows. From this point of view, local measurements in air–water bubbly flow systems in a large pipe with 101.6 mm diameter were performed by using four-sensor optical probes. The radial profiles of the void fraction, IAC, bubble Sauter mean diameter and interfacial velocity were obtained at two axial locations of z / D = 2 and 29. The simplified one-dimensional, steady-state, adiabatic one-group IATE with eight sets of bubble coalescence and breakup models was evaluated against the present experimental data and that from the literatures. The evaluation results showed that the Sun et al. (2004a) model is the best one for large diameter pipes. This model can well reflect the interfacial transfer mechanisms and can provide a reasonable prediction in general. Besides, the models of Smith et al. (2012a) and Ishii and Kim (2001) are also recommended due to their relative good performance in the prediction of IAC. Further work should be undertaken to develop a new bubble coalescence and breakup model with extra consideration of the multi-dimensional flow behavior in large pipes, if necessary. [ABSTRACT FROM AUTHOR]

A new hybrid diffusion-transport homogenization method has been developed by extending the consistent spatial homogenization (CSH) transport method to include diffusion theory. As in the CSH method, an "auxiliary cross section" term is introduced into the source term, making the resulting homogenized diffusion equation consistent with its heterogeneous counterpart. The method then utilizes an on-the-fly re-homogenization in transport theory at the assembly level in order to correct for core environment effects on the homogenized cross sections and the auxiliary cross section. The method has been derived in general geometry and tested in a 1-D boiling water reactor (BWR) core benchmark problem for both controlled and uncontrolled configurations. The method has been shown to converge to the reference solution with less than 1.7% average flux error in less than one third the computational time as the CSH method - 6 to 8 times faster than fine-mesh transport. [ABSTRACT FROM AUTHOR]

Highlights: [•] Two-group interfacial area transport equation was implemented into a three-field two-fluid model in Fluent. [•] Numerical model was developed for cap-bubbly flows in a narrow rectangular flow channel. [•] Numerical simulations were performed for cap-bubbly flows with uniform void inlets and with central peaked void inlets. [•] Code simulations showed a significant improve over the conventional two-fluid model. [Copyright &y& Elsevier]

Highlights: [•] An acceleration technique for solving fine-group eigenvalue transport problems. [•] Coarse-group quasi transport theory to solve coarse-group eigenvalue transport problems. [•] Consistent and inconsistent formulations for coarse-group quasi transport theory. [•] Computational efficiency amplified by a factor of 2 using hybrid SGD for 1D BWR problem. [Copyright &y& Elsevier]

Highlights: [•] Simulation of boron slug transport in a core with coupled codes. [•] Spatial and temporal discretization to describe boron dilution scenarios. [•] Code validation by code-to-code comparison. [•] Boron dilution simulation including cross flow between fuel assemblies. [Copyright &y& Elsevier]

Highlights: [•] Surface perturbation theory is used to transform a spherical radiation shield to an ellipsoid. [•] The surface transformation function is derived. [•] The accuracy of perturbation estimates depends on the detector location and problem physics. [ABSTRACT FROM AUTHOR]