Neutron Transport Research Articles

Compact fusion blanket using plasma facing liquid Li-LiH walls and Pb pebbles

Liquid plasma facing walls allow for increased neutron-wall loading expanding the design space of fusion power-plants and experimental devices towards compact high-field reactors. This study presents the design of a compact radial build blanket for fusion devices composed of variable quantities of Lead (Pb) and Lithium-Lithium Hydride (Li-LiH). A tank-like cylindrical neutronic model of the early design of the stellarator reactor proposed by Renaissance Fusion is implemented in OpenMC (neutron transport and dose rate analyses). The reactor's radial build composition and blanket layer thicknesses are varied to fulfill the requirements on tritium breeding ratio (TBR), nuclear heat extraction, radiation shielding (for the coils, internal structures and external environment) for a stellarator-based power-plant. The analyses suggest that a radial build lower than a meter thick between the plasma and coils would be sufficient to allow for a TBR ∼ 1.60, an energy multiplication factor of ∼ 1.07, to capture ≥ 90% of the nuclear heat, limit the neutron fluence at the coils below 1019 n/cm2, and limit the structural damage on the liquid metal vessel and magnet structure. In particular, a blanket composed of 32 cm of Pb and Li-LiH, 54 cm of a heavy metal hydride such as vanadium hydride (VH2), along with a 1.3 m of concrete bioshield, would minimize the radial build of the stellarator reactor while fulfilling tritium breeding, shielding and heat extraction requirements.

Tool Developments in the OpenMC Code: Correlated Sampling and Transient Fission Matrix Approach Coupled to OpenFOAM

This article outlines the advancements made in broadening the application scope of the OpenMC neutron transport code to include thermohydraulic coupling and nuclear data uncertainty propagation. These developments primarily involve the incorporation of the correlated sampling (CS) technique, facilitating the propagation of thermal feedback or cross-section sampling on neutronic calculations through neutron weight adjustments. The CS technique is integrated with computer-aided-design (CAD)–based meshing and the Transient Fission Matrix (TFM) approach. Together, these components enable comprehensive handling of neutronics-thermohydraulic coupling: The TFM approach addresses neutron kinetics via precalculated neutron transport matrices, the CS technique accounts for thermal feedback impacts on the matrices, and CAD meshing defines volumes corresponding to each matrix bin to align results with computational fluid dynamics codes, specifically OpenFOAM. Implementation details and verification procedures are elaborated, alongside an analysis on existing limitations and possible perspectives.

Symmetry Determining Equations and Lie Groups of the Neutron Transport Equation

The Grigoriev-Meleshko Method, an indirect Lie group theory method, is used to derive the symmetry determining equations (SDEs) of the neutron transport equation (NTE) and the coupled delayed neutron precursor equations (DNPEs). A solution to the SDEs is a Lie group of transformations that can be used to reduce the order of the NTE and DNPEs or outright solve the equations. We found several solutions of the SDEs and worked through the mathematical algorithm to demonstrate relationships of instantiations of the NTE and its known solutions with the Lie groups. Examples of solutions include the Lie group that allows for the transformation of the differential form of the NTE to the integral form of the NTE; the Lie groups that permit Case’s solution; and the Lie group used to transform from the NTE to the α -eigenvalue form of the NTE.

An OpenFOAM solver for multiphysics modeling of fusion reactor design: The nemoFoam code

This paper introduces a multiphysics code, named nemoFoam, designed within the OpenFOAM environment to address the coupling of neutron and photon transport with thermal-hydraulics in nuclear reactor simulations. The code, conceived both for fusion and fission applications, employs a modular approach where the neutronic module is currently based on multi-group neutron diffusion equations, including a mono-kinetic treatment for photons. The thermal-hydraulic module is built on the standard OpenFOAM solver. The paper focuses on presenting the key features of nemoFoam and discussing the structure of the code, the nuclear and thermal-hydraulic models, and the coupling strategy between them.In order to assess the performance of the neutronic module, the code is applied to a fusion case study, indeed a benchmark against the Serpent Monte Carlo code for neutron and photon transport is performed, applying nemoFoam to the Affordable, Robust and Compact (ARC) fusion reactor design. Simulation results demonstrate good agreement with Monte Carlo benchmarks, emphasizing the potential of the code for multiphysics simulations. The modular design of nemoFoam allows users to extend the implemented model equations to study additional phenomena, focus only on selected aspects independently and, potentially, to add new solvers in each module.

Development of a diffusive simplified double P[formula omitted] approximation to the neutron transport equation

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 (PN 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 (SPN 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 (SDPN equations). Numerical results show that the SDPN approximation provides faster and more accurate results than the SPN equations for some problems with strong variations of the angular neutron flux.

Numerical solution of static and spatial kinetics self-adjoint angular flux neutron transport equation

This paper elucidates a comprehensive derivation of the variational formulation pertaining to the static and spatial kinetics self-adjoint angular flux (SAAF) neutron transport equation. The methodology employed for discretization of the spatial variable is the finite element method, while the energy group discretization is executed via the group method, and the directional discretization is conducted using the discrete ordinates method. Analytic expressions for the discretization to variable separation under both vacuum and reflective boundary conditions are furnished. Constructed upon the MOOSE framework, a code designated as SAAFCGSN has been developed for the resolution of the SAAF neutron transport equation. The zero-order scattering matrix within this computational framework is managed through an innovative "decoupling" method, thereby enhancing the computational efficiency significantly. The functionality and robustness of the SAAFCGSN code are corroborated through meticulous evaluation involving seven distinct steady-state scenarios as well as two transient states. Empirical outcomes verify the compatibility of the SAAFCGSN code with both structured and unstructured mesh, inclusive of their amalgamations, thus facilitating maintenance and ensuring elevated computational accuracy. In addition, a performance analysis benchmarked against IAEA standards reveals that the adoption of the scattering matrix "decoupling" method propels a computational speed increase exceeding 30 %, signifying a notable advancement in calculation efficiency.

Efficient and Portable Vectorized Sweep Kernels for the Transport Equation on 3D Cartesian Grids Using the Kokkos Framework

This paper describes the implementation of efficient and portable vectorized sweep kernels as part of the resolution of the neutron transport equation on three-dimensional Cartesian grids using the discrete ordinates ( S n ) method for the angular variable and the diamond differencing (DD) scheme for the spatial discretization. Vectorization is set up along the directions within the same octant and is independent of the spatial discretization order; therefore, the extension of this technique to high-order DD or discontinuous Galerkin schemes is immediate. Our implementation is written in C++17 and relies on the Kokkos performance portability framework. This library allows one to express shared-memory parallelism (including vectorization) in a machine-independent way and supports many backends including CUDA and OpenMP. Our vectorization procedure relies on the portable single instruction multiple data types provided by Kokkos. The method has been implemented for DD schemes up to order 2 and yields promising results on CPUs supporting standard vector instructions.

Proof-of-principle of parametric stellarator neutronics modeling using Serpent2

This contribution presents neutron transport studies for the 5-period helical-axis advanced stellarator stellarator using the Serpent2 code. These studies utilize a parametric geometry model, enabling scans in neutronics modeling by varying the thickness of the reactor layers. For example, the tritium breeding ratio (TBR) can be determined by exploring various blanket material options and thicknesses to identify the threshold configuration that meets the TBR design criterion of 1.15. We found out that with the helium-cooled pebble ped candidate option, the TBR criterion is met with a breeding zone thickness of 26 cm, while with the dual-coolant lithium lead the threshold is exceeded at a thickness of 46 cm. Furthermore, the geometry includes non-planar field coils, allowing to study the fast neutron flux in these superconducting coils with a technological limit of 1×1091/cm2s . It is shown that the neutron fast flux is not constant at the coils, necessitating a neutron transport simulation to determine the distribution of the fast-flux at the coils. We show that the peak fast flux can be more than a factor of 2 higher than the average flux, and that the peak flux location rotates helically.

Utilization of Magnesium Chloride (Mg-Cl) cycle for Hydrogen Production of SOMBRERO Fusion Reactor

The aim of this study is to investigate the hydrogen production potential of the Mg-Cl (magnesium chloride) cycle in the SOMBRERO fusion reactor. Three different percentages of UO2 nuclear fuel, namely 2%, 6% and 10%, have been used while keeping the fuel zone thickness constant. The performance of the SOMBRERO fusion reactor has been considered statically. The neutronic calculations have performed with Monte Carlo neutron transport code. Firstly, it has been determined that the tritium breeding ratio (TBR) and energy multiplication factor (M) values depends on neutronically. Secondly, the amount of hydrogen production with by using energy multiplication factor (M) have been performed. The highest hydrogen production has been obtained as 8,12687 kg/s for 10% UO2 fuel ratio.

Coupling Nuclear Predictions into Damage Simulations with SPECTRA-PKA

Modern nuclear physics software is well validated, providing advanced capabilities to support the engineering of the future generations of fission and fusion reactors. Transport simulators can model the transport of neutrons through reactor geometries, and inventory codes can accurately predict transmutation and activation. Meanwhile, material modeling applies a variety of techniques to understand how structural damage and composition changes will alter the properties of materials, ultimately limiting their usable lifetime in a reactor. Bridging the gap between nuclear simulation tools and materials modeling is a necessary step if these lifetimes are to be accurately predicted, which, for fusion, is critical to provide the necessary assurance of commercial viability. SPECTRA-PKA is a tool developed to compute the rates of structural damage source events, i.e. the primary knock-on atoms (PKAs), using the same nuclear data as used by transport and inventory simulations. Now, it has been interfaced with the binary collision approximation code SDTrimSP, allowing those PKA events, distributed spatially and temporally in an atomic system, to be converted into damage cascades. This computational infrastructure provides insight into the variation in damage distributions between different materials under the same nuclear environment. Example simulations for materials under fusion reactor conditions demonstrate how the rich detail of the nuclear environment can be applied directly to modeling, without the need for integral-average measures that omit those details.

Nuclear analysis for the test cell lithium system interface cell during operation and shutdown

This paper presents the shielding analyses, which will be provided for supporting the design of the Test Cell Lithium System Interface Cell (TLIC) and safety assessment. The study considers the radiation decay source from IFMIF-DONES operation with beam-on. Radiation shielding simulations have been performed for the safety concerns. Detailed simulation geometry was created from CAD drawing for the TLIC. Neutron transport calculation was performed for all TLIC components concerned. The particle flux, biological dose and absorbed dose maps are provided in the study. A series of neutron transport are performed using the MCNP Monte Carlo McDeLicious neutron source code coupled with FISPACT for inventory calculations for TLIC design.

Neutron irradiation-induced shutdown dose rate estimation in EU DEMO divertor

The EU DEMO fusion reactor, which is part of the EUROfusion PPPT program and the successor to ITER, aims to facilitate the transition of fusion from the science laboratory to industrial use. Its key objective is net energy production. Its critical component, the divertor, manages power exhaust, ash, and helium removal. The key elements of the divertor include Plasma Facing Components (PFCs), shielding liners, reflector plates, divertor cassettes, and a vacuum vessel locking system. It is imperative that the shutdown dose rates are assessed to ensure workers' radiation protection during maintenance, plan the disposal of radioactive material, and ensure the overall safety of the plant. The shutdown dose rate tool employed in the present study is R2Smesh, which couples MCNP for radiation transport and FISPACT for neutron activation to calculate neutron-induced shutdown dose rates, utilizing an advanced, semi-heterogeneous EU DEMO divertor model, presenting a challenging task in neutron transport calculations.

On the Application of the Nyström Method to Two-Dimensional Neutral Particle Transport Problems in Heterogeneous Media

Over time, several methods were developed to deal with neutral particle transport problems. The interest in these problems is related to their wide range of applications, from neutron transport and heat transfer in nuclear reactors to radiative transfer in atmospheric clouds. Unlike the discrete ordinates or discrete ordinates–like methods, integral methods do not require discretization of angular variables. Instead, angular variables are completely eliminated by an integration procedure over the solid angle, which allows elimination of the ray effect. That said, this paper presents a new approach to estimate the scalar flux in two-dimensional fixed-source neutron transport problems in a heterogeneous medium, considering isotropic scattering and vacuum and reflective boundary conditions. Here, the Nyström method is combined with the singularity-subtraction technique to present an integral formulation for the scalar flux in a mesh grid over all regions of the domain. The iterative method of the Neumann series is used as an alternative to direct methods to solve the resulting system of equations generated from the domain discretization. Numerical results are given to verify the offered method.

A numerical validation between the neutron transport and diffusion theories for a spatial kinetics problem

In this paper, a comparative analysis of numerical results of the neutron transport and diffusion theories for steady-state and transient multigroup problems is presented. The neutron transport equation is known as the one that best describes the behavior of the neutron population in a nuclear reactor. However, due to the difficulty of working with its complete form, other models are considered as approximations to this equation. One such approximation is the neutron diffusion equation, which uses the Fick's Law. It is well known, however, that the diffusion model may not work well under specific conditions. The objective of this work is to establish a quantitative comparison of numerical results obtained for the K dominant eigenvalue and for the scalar fluxes from the two theories and to analyze the influence of the model on the results.

Reduced-order methods for neutron transport kinetics problem based on proper orthogonal decomposition and dynamic mode decomposition

This work proposes two reduced-order methods to address the high computing resource consumption of neutron transport kinetics problems. The first method is based on proper orthogonal decomposition (POD), which integrates the POD basis with the governing equation. The second method is based on dynamic mode decomposition (DMD), which constructs an approximate linear relationship to predict the time-dependent physical quantities. Numerical results demonstrate that both two reduced-order methods can achieve high accuracy and efficient prediction. For the selected cases, POD can obtain a speed-up ratio of 105 – 429, while DMD can obtain a speed-up ratio of 2300 – 12500. For step transient situations, POD and DMD predict power errors do not exceed 0.42% and 0.05%. For ramp transient situations, POD and DMD predict power errors do not exceed 0.69% and 1.9%. This work can provide some useful suggestions for applications and further development in neutron transport kinetics reduced-order model.

Kinetics Parameters Estimation for the Coupled Neutron-Photon Point Kinetics Model in Monte Carlo Eigenvalue Calculations

The delayed photoneutron effect should be considered in the point kinetics equation (PKE) applied to D2O and beryllium-moderated or -reflected systems because of its importance on reactor kinetics. This paper proposes neutron-photon-coupled PKE with a clear definition of adjoint-weighted kinetics parameters after providing the mathematical derivation from the neutron and photon transport and precursor equations. The Monte Carlo algorithms to estimate the kinetics parameters are induced from their definition and implemented in the eigenvalue calculation module of McCARD. They are verified through homogeneous infinite-medium problems with few-group neutron and photon cross sections by comparing with analytic solutions. The proposed PKE and kinetics parameters estimation methods are applied for the kinetics analyses of the artificial control rod drop transient problem in the RB reactor. Photonuclear physics is also implemented in a time-dependent Monte Carlo simulation to calculate the reference data of neutron amplitude in the core to be compared with the PKE results. As good agreement is shown in the neutron amplitude, it is validated that the developed PKE is capable of predicting the reactor kinetics considering photoneutrons.

The Second-Order Features Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (2nd-FASAM-L): Mathematical Framework and Illustrative Application to an Energy System

The Second-Order Features Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (abbreviated as “2nd-FASAM-L”), presented in this work, enables the most efficient computation of exactly obtained mathematical expressions of first- and second-order sensitivities of a generic system response with respect to the functions (“features”) of model parameters. Subsequently, the first- and second-order sensitivities with respect to the model’s uncertain parameters, boundaries, and internal interfaces are obtained analytically and exactly, without needing large-scale computations. Within the 2nd-FASAM-L methodology, the number of large-scale computations is proportional to the number of model features (defined as functions of model parameters), as opposed to being proportional to the number of model parameters. This characteristic enables the 2nd-FASAM-L methodology to maximize the efficiency and accuracy of any other method for computing exact expressions of first- and second-order response sensitivities with respect to the model’s features and/or primary uncertain parameters. The application of the 2nd-FASAM-L methodology is illustrated using a simplified energy-dependent neutron transport model of fundamental significance in nuclear reactor physics.

Mitigating Spatial Error in the Iterative Quasi–Monte Carlo (iQMC) Method for Neutron Transport Simulations with Linear Discontinuous Source Tilting and Effective Scattering and Fission Rate Tallies

The iterative Quasi–Monte Carlo (iQMC) method is a recently proposed method for neutron transport simulations. iQMC can be viewed as a hybrid between deterministic iterative techniques, Monte Carlo simulation, and Quasi–Monte Carlo techniques. iQMC holds several algorithmic characteristics that make it desirable for high-performance computing environments, including an O ( N − 1 ) convergence scheme, a ray-tracing transport sweep, and a highly parallelizable nature similar to analog Monte Carlo. While there are many potential advantages of using iQMC, there are also inherent disadvantages, namely, the spatial discretization error introduced from the use of a mesh across the domain. This work introduces two significant modifications to iQMC to help reduce the spatial discretization error. The first is an effective source transport sweep, whereby the source strength is updated on the fly via an additional tally. This version of the transport sweep is essentially agnostic to the mesh, material, and geometry. The second is the addition of a history-based linear discontinuous source tilting method. Traditionally, iQMC utilizes a piecewise constant source in each cell of the mesh. However, through the proposed source tilting technique, iQMC can utilize a piecewise linear source in each cell and reduce spatial error without refining the mesh. Numerical results are presented from the two-dimensional (2-D) C5G7 and Takeda-1 k-eigenvalue benchmark problems. The results show that the history-based source tilting significantly reduces error in global tallies and the eigenvalue solution in both benchmarks. Through the effective source transport sweep and linear source tilting, iQMC was able to converge the eigenvalue from the 2-D C5G7 problem to less than 0.04% error on a uniform Cartesian mesh with only 204 × 204 cells.

Coupling between LOTUS and CTF with DYN3D within a multiscale and multiphysics software development

Coupling of nuclear codes can be performed at several scale levels and is necessary to improve their reliability and sustainability. Mainly, the coupling of nuclear codes that use nodal expansion neutronics with channel thermal hydraulics has been carried out at the fuel assembly level while, only recently, the coupling of nuclear codes that use advanced neutronics, thermal hydraulics, and thermo-mechanics has been carried out at either the fuel pin or materials level. Meanwhile, in the UK, a multiscale and multi-physics software development between NURESIM and CASL is being developed, which includes a coupling software environment that enables the coupling of nuclear codes at several scale levels. Full coupled reactor physics at either the fuel pin or materials level can be obtained by coupling the transport code LOTUS, the subchannel code CTF, and the nodal code DYN3D. In this journal article, a multi ways coupling between LOTUS and CTF at either the fuel pin or materials levels with DYN3D at the fuel assembly level is compared to a multi ways coupling between DYN3D and CTF at the fuel pin level with DYN3D at the fuel assembly level and a multi ways coupling between Open MC and CTF at the materials level with DYN3D at the fuel assembly level. These comparisons have been carried out to present the coupled reactor physics verifications at either the fuel pin or materials levels. These show that the multi ways coupling between LOTUS and CTF at either the fuel pin or materials levels with DYN3D at the fuel assembly level outperforms the multi ways coupling between DYN3D and CTF at the fuel pin level with DYN3D at the fuel assembly level due to the application in the former of full neutron transport. Also, these show that the multi ways coupling between LOTUS and CTF at the materials level with DYN3D at the fuel assembly level agrees with the multi ways coupling between Open MC and CTF at the materials level with DYN3D at the fuel assembly level due to the application in both of full neutron transport.

Energy Dissipation of Fast Electrons in Polymethylmethacrylate: Toward a Universal Curve for Electron-Beam Attenuation in Solids between ∼0 eV and Relativistic Energies.

Spectroscopy of correlated electron pairs was employed to investigate the energy dissipation process, as well as the transport and the emission of low-energy electrons on a polymethylmethacrylate surface, providing secondary electron spectra causally related to the energy loss of the primary. Two groups are identified in the cascade of slow electrons, corresponding to different stages in the energy dissipation process. The characteristic lengths for attenuation due to collective excitations and momentum relaxation are quantified for both groups and are found to be distinctly different: λ_{1}=(12±2) Å and λ_{2}=(62±11) Å. The results strongly contradict the commonly employed model of exponential attenuation with the electron inelastic mean free path as characteristic length, but they essentially agree with a theory used for decades in astrophysics and neutron transport, albeit with characteristic lengths expressed in units of angstroms rather than light-years.