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At least 829 records · Page 46
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Performing a multi-unit level-3 PSA with MACCS

MACCS (MELCOR Accident Consequence Code System), WinMACCS, and MelMACCS now facilitate a multi-unit consequence analysis. MACCS evaluates the consequences of an atmospheric release of radioactive gases and aerosols into the atmosphere and is most commonly used to perform probabilistic safety assessments (PSAs) and related consequence analyses for nuclear power plants (NPPs). WinMACCS is a user-friendly preprocessor for MACCS. MelMACCS extracts source-term information from a MELCOR plot file. The current development can combine an arbitrary number of source terms, representing simultaneous releases from a multi-unit facility, into a single consequence analysis. The development supports different release signatures, fission product inventories, and accident initiation times for each unit. The treatment is completely general except that the model is currently limited to collocated units. A major practical consideration for performing a multi-unit PSA is that a comprehensive treatment for more than two units may involve an intractable number of combinations of source terms. This paper proposes and evaluates an approach for reducing the number of calculations to be tractable, even for sites with eight or ten units. The approximation error introduced by the approach is acceptable and is considerably less than other errors and uncertainties inherent in a Level 3 PSA.

42 ENGINEERING↗
DOE OSTI2024

A Prefire Approach for Probabilistic Assessments of Postfire Debris-Flow Inundation

Increases in wildfire activity and rainfall intensification are driving more postfire debris flows (PFDF) in many regions around the world. PFDFs are most common in the first postfire year and may even occur before a fire is fully controlled. This underscores the importance of assessing postfire hazards before a fire starts. Evaluation of PFDF hazards prior to fire can help strategize interventions lessening the negative effects of future fires. However, debris-flow runout and inundation analyses are not routine in PFDF hazard assessments, partially due to time constraints and substantial uncertainties in boundary conditions. Here, we propose a prefire PFDF inundation assessment framework using a debris-flow runout model based on the Herschel-Bulkley (HB) rheology (HEC-RAS v6.1). We constrain model inputs and parameters using Bayesian posterior analysis, rainfall-runoff simulations, and a debris-flow volume model. We use observations from recent PFDF incidents in northern Arizona, USA, to calibrate model components and then apply our prefire inundation assessment framework in a nearby unburned area. Specifically, we (a) identify yield stress as the most influential factor on inundation extent and arrival time in a HB model, (b) establish posterior distributions for model parameters suitable for forward modeling by leveraging uncertainties in field observations, and (c) implement a predictive forward analysis in an area that has not burned recently to evaluate PFDF inundation under several future fire scenarios. This study improves our ability to assess postfire debris-flow hazards before a fire begins and provides guidance for future applications of single-phase rheological models when assessing PFDF hazards.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2019

A probabilistic assessment of PWR SNF cladding pinching failure considering potential transportation accident conditions

This report presents a probabilistic assessment of pressurized water reactor (PWR) spent nuclear fuel (SNF) rod cladding pinching failure triggered by a transportation accident, after 300 years of dry storage. Three thousand pinching scenarios, which were previously evaluated assuming a baseline 9-m cask drop and a horizontal impact on an unyielding surface, are reassessed assuming more realistic train transportation accidents. Each scenario was defined using Latin Hypercube Sampling (LHS) of SNF rod conditions and loading parameters, and it is simulated with a nonlinear finite element (FE) model. Cladding failure is evaluated under consideration of hydride-related embrittlement effects. The reassessment indicates that cladding pinching failure is not significantly affected by the assumptions on transportation accidents, but is rather controlled by other parameters, mainly the pellet-cladding gap width. The reevaluated, conditional failure probability ranges from 1.62 E-7 to 6.47 E-7, for 5,000 km of transportation by train.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

Machine Learning Derived Dynamic Operating Reserve Requirements in High-Renewable Power Systems

Accurately forecasting wind and solar power output poses challenges for deeply decarbonized electricity systems. Grid operators must commit resources to provide reserves to ensure reliable operations in the face of forecast errors, a process which can increase fuel consumption and emissions. We apply neural network-based machine learning to expand the usefulness of median point forecast data by creating probabilistic distributions of short-term uncertainty in demand, wind, and solar forecasts that adapt to prevailing grid conditions. Machine learning derived estimates of forecast errors compare favorably to estimates based on incumbent methods. Reserves derived from machine learning are usually smaller than values derived using incumbent methods, which enables fuel savings during most hours. Machine learning reserves are generally larger than incumbent reserves during times of higher forecast error, potentially improving system reliability. Performance is tested using multi-stage production simulation modeling of the California Independent System Operator (CAISO) system. Machine learning reserves provide production cost and greenhouse gas (GHG) emission reductions of approximately 0.3% relative to historical 2019 requirements. Savings in the 2030 timeframe are highly dependent on battery storage capacity. At lower levels of battery capacity, savings of 0.4% from machine learning reserves are shown. Significant quantities of battery storage are expected to be added to meet California's resource adequacy needs and GHG reduction targets. Addition of these batteries saturate reserve needs and results in minimal within-hour balancing costs in 2030.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2023

Expansion of Hazards and Probabilistic Risk Assessments of a Light-Water Reactor Coupled with Electrolysis Hydrogen Production Plants

This report builds upon the body of work sponsored by the Department of Energy (DOE) Light-Water Reactor Sustainability (LWRS) Flexible Power Operation and Generation (FPOG) program that presented generic probabilistic risk assessments (PRAs) for the addition of a heat extraction system (HES) to light-water reactors to support the co-location of a high temperature hydrogen electrolysis facility (HTEF). Probabilistic and deterministic hazards assessments and risk analyses are leveraged throughout this report. Several improvements and new analyses are included in this report. First, higher amounts of detail in the specifications of the generic HTEFs are used to produce scaled results for a 100, 500, and 1000 MW nominal hydrogen production facility. An additional hazard assessment of 1000 kg of hydrogen storage is performed. The facility hazards and footprint are assessed to determine the safe distance required for placement near the nuclear power plant (NPP). Second, specific designs for corresponding HESs for the different levels of support required by the HTEFs are analyzed in the PRA model. Third, a hazards analysis of the specified HTEFs leads not only to effects of the quantified risk assessment for the NPP, but also qualitative hazards assessment for the community. Finally, a seismic analysis and a high winds analysis have each been added to the PRA. The results investigate the applicability of the potential licensing approaches which do not require a full United States (U.S.) Nuclear Regulatory Commission (NRC) licensing review. The PRAs are generic and include listed assumptions. The HTEF design built for this project has further eliminated many conservative assumptions from the prior PRAs in this series. The PRA results indicate that the 10 CFR 50.59 licensing approach is justified due to the minimal increase in initiating event frequencies for all design basis accidents, with none exceeding 7.7%. The PRA results for core damage frequency and large early release frequency support the use of NRC Regulation Guide 1.174 as further risk information that supports a change without a full licensing amendment review. The hazard analyses and PRA confirm the need for engineered blast barriers of storage tanks and the common production header leaving the HTEF. The hazards analyses and PRA also confirm with high confidence that using the assumptions of design in this report that the safety case for licensing an HES addition and an HTEF sited with its unprotected high-pressure stage components 187 meters from the NPP’s transmission towers (the most fragile structure, system, and component) is strong.

08 HYDROGEN↗
DOE OSTI2021

A Bayesian and HRA-Aided Method for the Novel Reliability Analysis of Software

Technological advancements and nuclear power plant modernization has inspired considerable research in the areas of safety and reliability, yet there remains a lack of consensus for the reliability assessment of digital instrumentation and control (I&C) systems. Motivated by the lack of consensus for reliability analysis methods, this work employs a novel framework that incorporates Bayesian, human reliability, and common-cause failure (CCF) modeling techniques. The novel framework allows the use of state-of-the-art or classical modeling techniques when accounting for human and CCF effects on system reliability. The Bayesian and HRA-Aided Method for the Reliability Analysis of Software (BAHAMAS) is demonstrated by a case study for the quantification of software hazards found in a previous analysis of a digital reactor trip system. The results demonstrate the ability of BAHAMAS to account for human activities during the software development life cycle and their influence on software reliability. BAHAMAS is a flexible tool for extending the coverage of conventional probabilistic risk assessments to include modernized digital I&C systems.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2022

Probabilistic estimation of depth-resolved profiles of soil thermal diffusivity from temperature time series

Abstract. Improving the quantification of soil thermal and physical properties is key to achieving a better understanding and prediction of soil hydro-biogeochemical processes and their responses to changes in atmospheric forcing. Obtaining such information at numerous locations and/or over time with conventional soil sampling is challenging. The increasing availability of low-cost, vertically resolved temperature sensor arrays offers promise for improving the estimation of soil thermal properties from temperature time series, and the possible indirect estimation of physical properties. Still, the reliability and limitations of such an approach need to be assessed. In the present study, we develop a parameter estimation approach based on a combination of thermal modeling, sliding time windows, Bayesian inference, and Markov chain Monte Carlo simulation to estimate thermal diffusivity and its uncertainty over time, at numerous locations and at an unprecedented vertical spatial resolution (i.e., down to 5 to 10 cm vertical resolution) from soil temperature time series. We provide the necessary framework to assess under which environmental conditions (soil temperature gradient, fluctuations, and trend), temperature sensor characteristics (bias and level of noise), and deployment geometries (sensor number and position) soil thermal diffusivity can be reliably inferred. We validate the method with synthetic experiments and field studies. The synthetic experiments show that in the presence of median diurnal fluctuations ≥ 1.5 ∘C at 5 cm below the ground surface, temperature gradients > 2 ∘C m−1, and a sliding time window of at least 4 d the proposed method provides reliable depth-resolved thermal diffusivity estimates with percentage errors ≤ 10 % and posterior relative standard deviations ≤ 5 % up to 1 m depth. Reliable thermal diffusivity under such environmental conditions also requires temperature sensors to be spaced precisely (with accuracy to a few millimeters), with a level of noise ≤ 0.02 ∘C, and with a bias defined by a standard deviation ≤ 0.01 ∘C. Finally, the application of the developed approach to field data indicates significant repeatability in results and similarity with independent measurements, as well as promise in using a sliding time window to estimate temporal changes in soil thermal diffusivity, as needed to potentially capture changes in bulk density or water content.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

Clusters have edges: the projected phase-space structure of SDSS redMaPPer clusters

ABSTRACT We study the distribution of line-of-sight velocities of galaxies in the vicinity of Sloan Digital Sky Survey (SDSS) red-sequence Matched-filter Probabilistic Percolation (redMaPPer) galaxy clusters. Based on their velocities, galaxies can be split into two categories: galaxies that are dynamically associated with the cluster, and random line-of-sight projections. Both the fraction of galaxies associated with the galaxy clusters, and the velocity dispersion of the same, exhibit a sharp feature as a function of radius. The feature occurs at a radial scale Redge ≈ 2.2Rλ, where Rλ is the cluster radius assigned by redMaPPer. We refer to Redge as the ‘edge radius’. These results are naturally explained by a model that further splits the galaxies dynamically associated with a galaxy cluster into a component of galaxies orbiting the halo and an infalling galaxy component. The edge radius Redge constitutes a true ‘cluster edge’, in the sense that no orbiting structures exist past this radius. A companion paper tests whether the ‘halo edge’ hypothesis holds when investigating the full three-dimensional phase-space distribution of dark matter substructures in numerical simulations, and demonstrates that this radius coincides with a suitably defined splashback radius.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Effects of social structure and management on risk of disease establishment in wild pigs

Contact heterogeneity among hosts determines invasion and spreading dynamics of infectious disease, thus its characterization is essential for identifying effective disease control strategies. Yet, little is known about the factors shaping contact networks in many 28 wildlife species and how wildlife management actions might affect contact networks. Wild pigs in North America are an invasive, socially-structured species that pose a health concern for domestic swine given their ability to transmit numerous devastating diseases 2 such as African swine fever (ASF). Using proximity loggers and GPS data from 48 wild pigs in Florida and South Carolina, USA, we employed a probabilistic framework to estimate weighted contact networks. We determined the effects of sex, social group, and spatial distribution (monthly home range overlap and distance) on wild pig contact. We also estimated the impacts of management-induced perturbations on contact and inferred their effects on ASF establishment in wild pigs with simulation. Social group membership was the primary factor influencing contacts. Between-group contacts depended primarily on space use characteristics, with fewer contacts among groups separated by >2 km and no contacts among groups >4 km apart within a month. Modeling ASF dynamics on the contact network demonstrated that indirect contacts resulting from baiting (a typical method of attracting wild pigs or game species to a site to enhance recreational hunting) increased the risk of disease establishment by ~33% relative to direct contact. Low-intensity population reduction (<5.9% of the population) had no detectable impact on contact structure but reduced predicted ASF establishment risk relative to no population reduction. Here, we demonstrate an approach for understanding the relative role of spatial, social, and individual-level characteristics in shaping contact networks and predicting their effects on disease establishment risk, thus providing insight for optimizing disease control in spatially- and socially-structured wildlife species.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2024

Experimental investigation of probabilistic failure of SiC/SiC composite tubes under multiaxial loading

This paper presents an experimental investigation of the failure behavior of SiC fiber-reinforced SiC matrix (SiC/SiC) composite tubes under multiaxial loading. A new testing apparatus is designed to independently apply axial stress and internal pressure to SiC/SiC tubes. Acoustic emission (AE) is used to monitor the damage growth in the specimen. Based on the measured stress–strain response, a strain-based criterion is proposed to determine the proportional limit stress (PLS). The proposed strain-based criterion is compared with the PLS determined by the AE measurement. The PLS is determined for different loading ratios to form a multiaxial failure surface. By testing multiple replicates, the statistical variations of the PLS are determined. In conclusion, based on the experimental results, a mathematical model is developed to characterize the probability distribution of the PLS of SiC/SiC composites under multiaxial loading.

36 MATERIALS SCIENCE↗
DOE OSTI2024

ASME Design Code Rule Changes for Nuclear Graphite

The American Society of Mechanical Engineers Boiler Pressure and Vessel Code (ASME BPVC) Section III, Division 5, Article HHA-3000 outlines graphite core component and graphite core assembly design guidelines. Graphite core components are defined as ?components manufactured from graphite that are installed to form a graphite core assembly within the reactor pressure vessel of a high temperature, graphite moderated fission reactor.? (p. 413) Graphites? inherent defect distributions do not allow for deterministic material reliability. Rather, graphite has variable strength distributions which change by grade. Article HHA-3000 outlines two semi-probabilistic methods, the full and simplified assessments, which set design load limit targets for each of three component structural reliability classes. The Design Task Group was officially recognized as a specialized task group within ASME November of 2023, though we?ve been collaborating since 2022. The purpose of the Design Task Group is to correct, clarify, and make HHA-3000 function as intended. The Design Task Group will sunset once we?ve achieved our objectives. The Design Task Group was specifically told to not write new Code. While there may be more precise and more accurate methods to determine reliability targets, the current methods are conservative, relatively simple to implement, and have thus far been considered satisfactory for setting design reliability targets. Much of the ground-work to write proposal files and background documents for records to make the changes needed to achieve our objective have been completed. The Design Task Group has documented much of their work through papers, presentations, and memorandums. Three memorandums in which INL team members had substantial contributions are found in the Appendices: FEA Modeling for the Baseline Program, Evaluating the Effects on Margin of Updating the Threshold and Shape Parameters in the Full Assessment, and Interpretations of the Full and Simplified Assessments in ASME BPVC. Most of the on-going work to achieve the Design Task Group?s objective will be addressing comments on existing records and moving records through the balloting process. The Design Task Group met bi-weekly mostly through the end of FY2023. Since February 2024, the Design Task Group has mostly been completed with solving and documenting the technical issues associated with the assessments. Unless new tasks are identified, the remaining work of the Design Task Group will be political and editorial.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Advanced Closures for Nonlinear Fluid Simulations of Plasmas (Final Report)

Advanced Closures for Nonlinear Fluid Simulations of Plasmas E. Held, Utah State University (Principal Investigator) J.-Y. Ji, Utah State University (Co-Investigator) Utah State University (USU) continued to develop and apply closures for hybrid fluid/kinetic simulations of magnetized fusion plasmas. The closures developed by USU's fusion theory and computation group use the drift ordering and are referred to as "continuum" and "general moment" approaches. They differ in their representation for the distribution functions which probabilistically describe the density of particles in 5 dimensional phase space. The two chosen velocity variables, needed in studies of kinetic physics for magnetized plasmas, are the speed and the component of a particle's velocity along the magnetic field. In magnetized plasmas, the charged particle trajectories are tied closely to the magnetic field lines. Both approaches account for binary interactions between particles using the Coulomb collision operator. Theoretical and computational development along both lines provided avenues for verification between the two approaches. This funding allowed USU to further develop physically accurate and numerically efficient models that advance our understanding of kinetic physics in experiments like the International Thermonuclear Experimental Reactor (ITER). The ITER device will likely prove the physical practicality of thermonuclear fusion by producing 500 megawatts of clean, abundant fusion power. This work helped to realize this goal by advancing our simulation capability using hybrid fluid/kinetic models.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Distributionally Robust Decentralized Volt-Var Control With Network Reconfiguration

Here, this paper presents a decentralized volt-var optimization (VVO) and network reconfiguration strategy to address the challenges arising from the growing integration of distributed energy resources, particularly photovoltaic (PV) generation units, in active distribution networks. To reconcile control measures with different time resolutions and empower local control centers to handle intermittency locally, the proposed approach leverages a two-stage distributionally robust optimization; decisions on slow-responding control measures and set points that link neighboring subnetworks are made in advance while considering all plausible distributions of uncertain PV outputs. We present a decomposition algorithm with an acceleration scheme for solving the proposed model. Numerical experiments on the IEEE 123 bus distribution system are given to demonstrate its outstanding out-of-sample performance and computational efficiency, which suggests that the proposed method can effectively localize uncertainty via risk-informed proactive timely decisions.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

Natural Language Processing-Enhanced Nuclear Industry Operating Experience Data Analysis to Support Risk Model Parameter Estimations

This set of slides has been prepared for a talk at the INL AI/ML Symposium held on September 8, 2022. Presentation outline: Background - Nuclear power plant operating experience data sources • Research focus and motivation - Analyzing free-text operating experience data: present and future • Research method - Input - Methodological steps - Output • Conclusions and next steps

99 GENERAL AND MISCELLANEOUS↗
DOE OSTI2024

Central Asia Seismic Hazard Assessment (CASHA): A Probabilistic Seismic Hazard Assessment for Kazakhstan, Kyrgyzstan and Tajikistan

Probabilistic seismic hazard assessments (PSHA) underpin the calculation of earthquake loads in most building codes around the world. In Central Asia, the building codes are slowly being updated to incorporate some of the contemporary concepts of seismic hazard representation. There is also a regional desire to coordinate hazard assessments and building code modernization. However, some challenges remain. Expertise in the region related to seismic hazard assessments is still largely compartmentalised, requiring a significant amount of training and capacity building in seismic hazard assessment related topics. In addition, there are vast amounts of seismic data (bulletin and waveforms), both from analogue and digital eras, that the region’s countries stored but until recently did not use or share among themselves or with the broader seismological community around the world. Finally, after the collapse of the Soviet Union in the 1990s, many of the countries’ seismic networks suffered a major setback with the lack of attention and budget to update existing equipment and installation of new instruments. In order to address these issues, the United States Department of Energy through Lawrence Livermore National Laboratory (LLNL) initiated a project in 2016 to engage and train local scientists in Central Asia to install new equipment, to enhance the quality of seismic monitoring and reporting, to improve and harmonise the regional earthquake catalogue, and to conduct national probabilistic seismic hazard assessments using the new and improved datasets. To achieve the seismic hazard assessment related goals, a series of workshops were held in Almaty, Kazakhstan; Bishkek, Kyrgyzstan; and Dushanbe, Tajikistan from 2016 until 2020. During the time that the COVID-19 pandemic restricted travel, workshops continued online (22 online workshops were hosted in two years). Finally, in May 2022, an in-person workshop in Istanbul, Turkey brought together all project participants along with civil engineers engaged with building code activities in their respective countries, providing a platform to discuss the implementation of the hazard models into updates of building codes in each country, as well as to discuss model parameters, sensitivity analyses and model results in terms of hazard maps, uniform hazard spectra and hazard deaggregations. The workshops were a combination of lectures and hands-on exercises, and included international participation as well as local scientists and engineers. The workshops served several purposes, including training, coordination of data collection, interactions between local earth scientists and engineers, and brainstorming and knowledge exchange among local and international experts. This report outlines the new earthquake catalogue compilation effort and the PSHA project undertaken in Kyrgyzstan, Tajikistan, and Kazakhstan as part of this initiative. The southern part of this region is tectonically active with moderate to high levels of both shallow crustal seismic activity and occurrence of deeper earthquakes under the Hindu Kush and Pamir mountain ranges. Deeper earthquakes also occur near southwestern Kazakhstan, under the eastern Greater Caucasus and Caspian Sea. Large portions of central and northern Kazakhstan, on the other hand, are in stable continental regions with low levels of seismic activity. This study systematically compiles and improves all available data on local seismicity, active faults, and ground motion attenuation characteristics of the region; and builds a framework to enable a contemporary PSHA to be carried out with the engagement of local scientists. While the project was regional, the seismic hazard assessments are primarily driven by the countries’ own national preferences and understanding of data collection, interpretation, and validation of results.

58 GEOSCIENCES↗
DOE OSTI2020

Three-dimensional microstructure-explicit and void-explicit mesoscale simulations of detonation of HMX at millimeter sample size scale

Fully three-dimensional (3D) microstructure-explicit and void-explicit mesoscale simulations of the shock-to-detonation (SDT) process of pressed granular HMX (octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine) are performed. The overall size scale of the models is up to 3 × 3 × 15 mm3, with ∼30 000 grains and 206 265 voids. The models account for the heterogeneous material microstructure, constituent distribution, constituent morphology, and voids. Loading conditions considered involve piston velocities in the range of 600–1200 m/s or pressures in the range of 4–8 GPa. The focus is on analyzing the SDT process and the effects of microstructure and voids on the run-to-detonation distance (RDD). Companion two-dimensional (2D) simulations are also carried out to assess the differences between 2D and 3D. Statistically equivalent microstructure sample sets (SEMSSs) are generated and used for both 2D and 3D, allowing the prediction of the statistical and probabilistic Pop plots (PPs). The predictions are in general agreement with trends in available experimental data in the literature. It is found that both the microstructure (heterogeneous grain size, morphology, and size distribution) and voids significantly affect the RDD and the PPs. These effects are systematically delineated and quantified via the use of SEMSSs with different combinations of attributes. A recently developed probabilistic formulation for the PPs is used to characterize the results, allowing uncertainties in the relations between the shock pressure and RDD arising from material heterogeneities to be quantified. The probabilistic formulation is further used to quantify the confidence levels in the ranked order of influences of different combinations of microstructure and voids on the PPs.

Miller, Christopher (ORCID:0000000206844381)↗
DOE OSTI2022

A Note on the Probability of Initiation Problem

The purpose of this note is to draw attention to Pál for his derivation of the Pál-Bell equation which is used by LLNL and LANL to model Probability of Initiation (POI) problems. Both laboratories credit the equation to Bell and Lee. Although the equation formulated by Bell and Lee and the equation derived by Pál appear to be the same, the equation by Bell and Lee is flawed. Scattering, which was absent in Bell’s original formulation of a POI problem, was later introduced by Bell and Lee by replacing a fission term with a scattering term. Such a replacement, however, leads to an incorrect average number $\overline{v}$ of neutrons that is produced in a fission event. By approaching a POI problem from probabilistic point of view, Pál formulates the problem as a branching process that i) describes the growth of a population of neutrons in a fissile medium, and ii) predicts $\overline{v}$ correctly. There are five reasons for writing this report. The first is to clarify the failing in Bell and Lee’s derivation. The second is to establish the origin of the equation used by the laboratories. The third is to elucidate the fixed point technique formulated by Mokhtar-Kharroubi to solve the Pál-Bell equation. The fourth is to derive the α eigenvalue that is associated with a super-critical situation. The fifth is to describe in the Appendix a second order positivity preserving technique that accelerates the convergence of the first order fixed point method developed by Mokhtar-Kharroubi.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

A Note on the Probability of Initiation Problem

The purpose of this note is to draw attention to Pál’s derivation of the Pál Bell equation which is used by LLNL and LANL to model Probability of Initiation (POI) problems. Both laboratories credit the equation to Bell and Lee. Although the equation formulated by Bell and Lee and the equation derived by Pál appear to be the same, the derivation by Bell and Lee is flawed. Scattering, which was absent in Bell’s original formulation of a POI problem, was later introduced by Bell and Lee by simply replacing a fission term with a scattering term. Such a replacement, however, leads to an inaccurate average number v̅ of neutrons that is produced by a fission event. By approaching a POI problem from probabilistic point of view, Pál formulates the problem as a branching process that determines v̅ correctly. There are six reasons for writing this report. The first is to clarify the failing in Bell and Lee’s derivation. The second is to establish the origin of the equation used by the laboratories. The third is to elucidate the fixed point technique formulated by Mokhtar-Kharroubi for solving the Pál-Bell equation. The fourth is to derive the α eigenvalue that is associated with a time dependent Pál-Bell equation. The fifth is to show in Appendix A that the non-linear fission operator of the Pál-Bell equation is descended from the operator of the Galton-Watson process of stochastic theory. The sixth is to describe in Appendix B a second order positivity preserving technique that accelerates the convergence of the first order fixed point method developed by Mokhtar-Kharroubi.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗