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An Analysis of Energy Needs and Seawater Extraction Approaches for the Integration of Primary Magnesium Production with Nuclear Power

The cost of primary magnesium production is highly dominated by the energy required to produce the metal. This, along with the current geographical concentration of its production, makes the magnesium price vulnerable as energy-production paradigms begin to rapidly change. In this work, four nuclear-integration pathways for the efficient supply of heat and electricity to magnesium production from a nuclear power plant are considered. A production route using electrified metallothermic reduction is compared to a nuclear-heat-integrated electrolytic pathway. It is shown that the latter can achieve significantly lower primary energy consumption, 261 MJ/kg Mg compared with 416 MJ/kg Mg. The electrolytic pathway can also reduce emissions down to around 2.1 kgCO2/kg Mg, compared to around 34 kgCO2/kg Mg in the Pidgeon process. Beyond energy and emissions, the four pathways considered are also compared to their ability to reduce locational dependence of the production plant and reduce process complexity. This work also highlights future advances in magnesium production with a potential scheme for the efficient extraction of magnesium from seawater with coproduction of desalinated water and the potential of demand-side response from magnesium electrolysis. This is the first work to consider the full integration of primary magnesium production using nuclear cogeneration, and it is hoped the study paves the way for the analysis of economic nuclear integration within metal extraction and manufacturing.

22 - GENERAL STUDIES OF NUCLEAR REACTORS

BISON analyses of TRISO fuel performance, its dependence on time-at-temperature, and possible implications for fuel design and qualification

The Advanced Gas Reactor Fuel Development and Qualification (AGR) program has established a substantial technical foundation to support private entry into the U.S. high-temperature gas-cooled reactor market. However, emerging tristructural isotropic (TRISO)-fueled reactor applications include small modular reactors and microreactors with longer fuel residence times, which may expose fuels to higher time-at-temperature (TAT) values than were explored by the AGR program. Increased TAT could affect diffusive and thermomechanical behaviors such as Pd penetration, fission gas release, creep, and fission product transport. In this work, we applied multiscale best-estimate BISON fuel performance modeling to assess these effects within a representative design space based on the AGR-5/6/7 experiment and analyzed trends in predicted particle and compact fuel performance metrics with possible implications for near-term fuel design and qualification. BISON unambiguously predicted that TRISO fuel performance is sensitive to TAT. Increasing TAT was not predicted to increase the magnitude of failure-inducing tangential stresses in particle coating layers. Predictions obtained using a mechanistic model for Pd penetration indicated that penetration depth does not depend strongly on TAT. While these observations suggest that AGR testing provides a conservative upper bound for the steady-state operation of TRISO particles at lower powers and higher residence times, BISON also predicted that the release of poorly retained Ag would increase with TAT. Because these analyses applied models to extrapolate beyond the available experimental data, the authors recommend performing targeted experiments to confirm these predictions. Nevertheless, these predictions may provide reactor developers with enough confidence to make near-term design decisions associated with the potential fuel performance trade-offs of increasing TAT.

11 - NUCLEAR FUEL CYCLE AND FUEL MATERIALS

Tailoring Ion Transport in Li 3‐3y Ho 1+y Cl 6‐x Br x via Transition‐Metal Free Structural Planes and Charge Carrier Distribution

Abstract Localized atomistic disorder in halide‐based solid electrolytes (SEs) can be leveraged to boost Li + mobility. In this study, Li + transport in structurally modified Li 3 HoCl 6 , via Br − introduction and Li + deficiency, is explored. The optimized Li 3‐3 y Ho 1+ y Cl 6‐ x Br x achieves an ionic conductivity of 3.8 mS cm −1 at 25 °C, the highest reported for holmium halide materials. 6,7 Li nuclear magnetic resonance and relaxometry investigations unveil enhanced ion dynamics with bromination, attaining a Li + motional rate neighboring 116 MHz. X‐ray diffraction analyses reveal mixed‐anion‐induced phase transitions with disproportionate octahedral expansions and distortions, creating Ho‐free planes with favorable energetics for Li + migration. Bond valence site energy analysis highlights preferred Li + transport pathways, particularly in structural planes devoid of Ho 3+ blocking effects. Molecular dynamics simulations corroborate enhanced Li + diffusion with Br − introduction into Li 3 HoCl 6 . Li‐Ho electrostatic repulsions in the (001) plane presumably drive Li + diffusion into the Ho‐free (002) layer, enabling rapid intraplanar Li + motion and exchange between the 2d and 4h sites. Li 3‐3 y Ho 1+ y Cl 6‐ x Br x also demonstrates good battery cycling stability. These findings offer valuable insights into the intricate correlations between structure and ion transport and will help guide the design of high‐performance fast ion conductors for all‐solid‐state batteries.

Chemistry

Enhancement of the Physical and Mechanical Properties of Cellulose Nanofibril-Reinforced Lignocellulosic Foams for Packaging and Building Applications

Biobased foams have the potential to serve as eco-friendly alternatives to petroleum-based foams, provided they achieve comparable thermomechanical and physical properties. We propose a facile approach to fabricate eco-friendly cellulose nanofibril (CNF)-reinforced thermomechanical pulp (TMP) fiber-based foams via an oven-drying process with thermal conductivity as low as 0.036 W/(m·K) at a 34.4 kg/m3 density. Acrodur®, iron chloride (FeCl3), and cationic polyacrylamide (CPAM) were used to improve the foam properties. Acrodur® did not have any significant effect on the foamability and density of the foams. Mechanical, thermal, cushioning, and water absorption properties of the foams were dependent on the density and interactions of the additives with the fibers. Due to their high density, foams with CPAM and FeCl3 at a 1% additive dosage had significantly higher compressive properties at the expense of slightly higher thermal conductivity. There was slight increase in compressive properties with the addition of Acrodur®. All additives improved the water stability of the foams, rendering them stable even after 24 h of water absorption.

Chemistry

The CP-PAW Code Package for First-Principles Calculations from a User’s Perspective

CP-PAW is a combined electronic structure and ab initio molecular dynamics code to perform mixed quantum and classical simulations of atomistic condensed phase systems, such as solids, liquids, and molecular systems. As the name suggests, the CP-PAW code unifies the all-electron projector augmented-wave (PAW) method with the Car–Parrinello (CP) approach to determine not only the electronic and nuclear ground states of condensed matter but also to study their properties and dynamics. In addition to briefly outlining the underlying theory, the focus will be on the unique aspects of CP-PAW and how to correctly employ them as a user. How to install CP-PAW using the new build system will also be briefly mentioned.

Blöchl, Peter E [Institute for Theoretical Physic

Temperature-Dependent Transport Characteristics of 2D MoS2 Channel FETs Grown Using Salt-Based Precursors

D Transition Metal Dichalcogenides (TMDs), particularly MoS2, are promising candidates for sub-10 nm Gate All Around (GAA) CMOS FETs. Salt-assisted Chemical Vapor Deposition (CVD) enable lateral MoS2 growth at atmospheric pressure and low temperatures. This work analyzes salt-based precursor-driven CVD-grown MoS2 FETs at various temperatures. MoS2 was grown using Ammonium Molybdate salt, sulfurized at 750∘C, and transferred onto p−Si3/SiO2 substrates. At room temperature, threshold voltage (VT) ranged from -35 V to -25 V, with a peak drain current of 1.2μA/μm. As temperature increased above 325K, VT shifted exponentially, and carrier mobility dropped significantly. At 400 K, the gate lost channel control, though gate leakage current remained low. These results are compared with non-salt-based MoS2 growth to assess salt precursor effects.Notice: This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (https://www.energy.gov/doe-public-access-plan).

Jones, Andrew [ORNL] (ORCID:0009000233849687)

Nuclear Safety [Vol. 36, No. 2, July-December 1995]

Nuclear Safety is a journal that covers significant issues in the field of nuclear safety. Its primary scope is safety in the design, construction, operation, and decommissioning of nuclear power reactors worldwide and the research and analysis activities that promote this goal, but it also encompasses the safety aspects of the entire nuclear fuel cycle, including fuel fabrication, spent-fuel processing and handling, and nuclear waste disposal, the handling of fissionable materials and radioisotopes, and the environmental effects of all these activities. Table of Contents for this issue follows. THE CHORNOBYL ACCIDENT: 195 The Chornobyl Accident Revisited, Part III: Chernobyl Source Term Release Dynamics and Reconstruction of Events During the Active Phase, A. R. Sich; GENERAL SAFETY CONSIDERATIONS: 218 Second ANS Workshop on the Safety of Soviet-Designed Nuclear Power Plants, R. A. Bari; 234 Elements of a Nuclear Criticality Safety Program, C. M. Hopper; 243 Rickover, Excellence, and Criticality Safety Programs, R. E. Wilson; ACCIDENT ANALYSIS: 249 Transient Analysis of the PIUS Advanced Reactor Design with the TRAC-PF1/MOD2 Code, B. E. Boyack, J. L Steiner, S. C. Harmony, H. J. Stumpf, and J. F. Lime; 278 The Hierarchy-By-Interval Approach to Identifying Important Models that Need Improvement in Severe-Accident Simulation Codes, T. J. Heames, M. Khatib-Rahbar, J. E. Kelly, R. P. Jenks-Johnson, and Y.-S. Chen; 290 RELAP5/MOD3 Code Coupling Model, R. P. Martin; 299 Missiles Caused by Severe Pressurized-Water Reactor Accidents, R. Krieg; DESIGN FEATURES: 310 Validation of COMMIX with Westinghouse AP-600 PCCS Test Data, J. G. Sun, T. H. Chien, J. Ding, and W T. Sha; ENVIRONMENTAL EFFECTS: 321 Spent Nuclear Fuel Characterization for a Bounding Reference Assembly for the Receiving Basin for Off-Site Fuel, S. D. Kahook, R. L. Garrett, L. R. Canas, and M J. Beckum; OPERATING EXPERIENCES: 332 Reactor Shutdown Experience, Compiled by J. W. Cletcher; U.S. NUCLEAR REGULATORY COMMISSION INFORMATION AND ANALYSES: 335 Reactor Coolant System Blowdown at Wolf Creek on September 17, 1994, J. V. Kauffman and S. L. Israel; RECENT DEVELOPMENTS: 344 Reports, Standards, and Safety Guides, D. S. Oueener; 349 Proposed Rule Changes as of June 30, 1995; ANNOUNCEMENTS: 320 Symposium on Acceptability of Risk From Radiation—Application to Manned Space Flight; 320 24th DOE/NRC Nuclear Air Cleaning and Treatment Conference; 361 Radiation Biology and Radiation Protection— Modern Developments and Tendencies in Radiation Biology; 362 1997 IEEE Sixth Conference on Human Factors and Power Plants; 354 The Authors; 360 Reviewers of Nuclear Safety, Vol. 36.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS

Nuclear Safety [Vol. 37, No. 1, January-March 1996]

Nuclear Safety is a journal that covers significant issues in the field of nuclear safety. Its primary scope is safety in the design, construction, operation, and decommissioning of nuclear power reactors worldwide and the research and analysis activities that promote this goal, but it also encompasses the safety aspects of the entire nuclear fuel cycle, including fuel fabrication, spent-fuel processing and handling, and nuclear waste disposal, the handling of fissionable materials and radioisotopes, and the environmental effects of all these activities. Table of Contents for this issue follows. GENERAL SAFETY CONSIDERATIONS: 1 A Decision Support System for Maintenance Management of a Boiling-Water Reactor Power Plant, J. H. Shen, A. Ray, and S. Levine; ACCIDENT ANALYSIS: 12 On Prediction of the Ignition Potential of Uranium Metal and Hydride, M. Epstein, W. Luangdilok, M. G. Plys, and H. K. Fauske; 26 An Overview of the Primary Parameters and Methods for Determining Condensation Heat Transfer to Containment Structures, J. Green and K. Almenas; DESIGN FEATURES: 49 Modem Tornado Design of Nuclear and Other Potentially Hazardous Facilities, J. D. Stevenson and Y. Zaho; U.S. NUCLEAR REGULATORY COMMISSION INFORMATION AND ANALYSES: 73 1994 Accident Sequence Precursor Program Results, R. J. Belles, J. W. Cletcher, D. A. Copinger, B. W. Dolan, J. W. Minarick, and P. D. O'Reilly; ANNOUNCEMENTS: 93 American Institute of Chemical Engineers (AlChE) Spring 1997 National Meeting; 94 European Safety and Reliability Association International Conference on Safety and Reliability ESREL ’97; 95 Criticality Safety Challenges in the Next Decade; 96 21st International Symposium on the Scientific Basis for Nuclear Waste Management; 84 The Authors; 88 Letter to the Editor; 90 Indexes to Nuclear Safety, Volume 36.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS

MARVEL Reactor Digital Engineering Developments

The MARVEL reactor project has served to introduce a new generation of engineers to the processes required to transform a reactor design from simply an idea on paper into what will be an approved, constructed, and operational nuclear power system. Much as there have been advances in materials, analysis, and evaluation methodologies over the 50 years since the last reactor was built at INL, so too has the technology for managing the engineering process itself advanced. Digital Engineering tools and methods provide improved coordination between previously siloed engineering disciplines, reduced burdens of non-value-added data transcription processes and bring forward insights and improvements that might otherwise fall later in the design stage, where changes are much more costly. While the tools and techniques to support the full digital engineering vision are not yet complete, the MARVEL design processes provide valuable demonstrations and validations of key aspects and illuminate further areas for implementation by subsequent projects.

21 - SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLAN

Probabilistic Resource Adequacy Suite (PRAS) v0.8 Model Documentation

The Probabilistic Resource Adequacy Suite, or PRAS, is a software package for studying power system resource adequacy. It allows the user to simulate power system operations under a wide range of operating conditions, in order to study the system's risk of failing to meet demand due to a resource shortfall, and identify the time periods and regions in which that risk occurs. This reports documents version 0.8 of the tool.

29 ENERGY PLANNING, POLICY, AND ECONOMY

Nuclear Safety [Vol. 37, No. 3, July-September 1996]

Nuclear Safety is a journal that covers significant issues in the field of nuclear safety. Its primary scope is safety in the design, construction, operation, and decommissioning of nuclear power reactors worldwide and the research and analysis activities that promote this goal, but it also encompasses the safety aspects of the entire nuclear fuel cycle, including fuel fabrication, spent-fuel processing and handling, and nuclear waste disposal, the handling of fissionable materials and radioisotopes, and the environmental effects of all these activities. Table of Contents for this issue follows. GENERAL SAFETY CONSIDERATIONS 181 Attitudes and Practices Regarding Disposal of Liquid Nuclear Waste at Clinton Laboratories in the Very Early Years: A Historical Analysis, S. H. Stow; 202 Off-Site Nuclear Emergency Management in Germany Under the Auspices of the Federal Structure, A. Bayer, S. Bittner, and H. Korn; DESIGN FEATURES: 211 Twenty-Fourth DOE/NRC Nuclear Air-Cleaning and Treatment Conference, R. R. Bellamy, J. J. Hayes, R. D. Porco, and M. W. First; OPERATING EXPERIENCES: 222 Explosion in the Tomsk-7 Reprocessing Plant on April 6, 1993, M. L. Hyder, W. G. Lussie, and F. E. Witmer; 234 Reactor Shutdown Experience, Compiled by J. W. Cletcher; U.S. NUCLEAR REGULATORY COMMISSION INFORMATION AND ANALYSES: 237 Assessment of Spent Fuel Cooling, J. G. Ibarra, W. R. Jones, G. F. Lanik, H. L. Ornstein, and S. V. Pullani; RECENT DEVELOPMENTS: 256 Reports, Standards, and Safety Guides, D. S. Queener; 260 Proposed Rule Changes as of June 30, 1996; ANNOUNCEMENTS: 210 Tennessee Industries Week; 221 21st International Symposium on The Scientific Basis for Nuclear Waste Management; 259 IAEA Programme of Scientific Meetings for 1997; 270 PSAM 4 (International Conference on Probabilistic Safety Assessment and Management); 265 The Authors; 268 Errata.

05 NUCLEAR FUELS

Overcoming the Entropy Penalty of Direct Air Capture for Efficient Gigatonne Removal of Carbon Dioxide

Atmospheric carbon poses an existential threat to civilization via global climate change. Hundreds of gigatonnes of carbon dioxide must be removed from earth’s atmosphere in the next three decades, necessitating a low-cost, energy-efficient process to extract low concentrations of carbon dioxide for conversion to a stable material permanently stored for thousands of years. In this work, the challenge of removing gigatonnes of CO 2 is described via the scale of effort and the thermodynamics of collecting and reducing this diffuse chemical, the accumulation of which imparts a substantial entropy penalty on any atmospheric carbon capture process. The methods of CO 2 reduction combined with upstream direct air capture (DAC) including absorption, membrane separation, and adsorption are compared with biomass torrefaction and permanent burial (BTB). A Monte Carlo model assesses the mass, energy, and economics of the full process of biomass torrefaction from biomass collection and transport to stable carbon burial to determine that 95% of scenarios could remove carbon for less than $200 per CO 2 -tonne-equivalent. Torrefied carbon is further discussed for its long-term stability and availability at the scale required to substantially mitigate the threat of climate change.

biomass

The impact of chemistry on anion migration in bixbyite-structured lanthanide oxides

This manuscript describes atomistic calculations of oxygen vacancy and interstitial migration in bixbyite structured lanthanide oxides. We examine two types of compounds, one in which only one type of lanthanide cation is present and a second class in which two lanthanides are present in a 3:1 ratio as dictated by the symmetry of the bixbyite crystal structure. Using temperature accelerated dynamics and the nudged elastic band method, we quantify the role of chemistry on the energy barriers for the most important pathways for both vacancy and interstitial migration. We then analyze the impact of these variations on the overall diffusivity of each defect, quantifying the contribution of each pathway using the theory of kinosons. We find that vacancy mobility can vary by as much as three orders of magnitude through changes in chemistry at 500 K. Changes in interstitial mobility are more modest but can still vary by an order of magnitude. This points to the ability to tune the mass transport characteristics of these compounds through appropriate choices in chemistry. We have also included supplementary information containing the atomic structures of the relevant pathways.

36 MATERIALS SCIENCE

Nuclear Safety [Vol. 37, No. 2, April-June 1996]

Nuclear Safety is a journal that covers significant issues in the field of nuclear safety. Its primary scope is safety in the design, construction, operation, and decommissioning of nuclear power reactors worldwide and the research and analysis activities that promote this goal, but it also encompasses the safety aspects of the entire nuclear fuel cycle, including fuel fabrication, spent-fuel processing and handling, and nuclear waste disposal, the handling of fissionable materials and radioisotopes, and the environmental effects of all these activities. Table of Contents for this issue follows. GENERAL SAFETY CONSIDERATIONS: 97 The Nuclear Community and the Public: Cognitive and Cultural Influences on Thinking About Nuclear Risk, M. A. Meyer; 109 Twenty-Third Water Reactor Safety Information Meeting, D. A. Copinger; ACCIDENT ANALYSIS: 126 Analysis of a PWR LBLOCA Without SCRAM, Trevor N. Tyler, Rafael Macian-Juan and John H. Mahaffy; DESIGN FEATURES: 139 Vulnerability of Multiple-Barrier Systems, N. C. Lind; ENVIRONMENTAL EFFECTS: 149 A Study of Wet Catalytic Oxidation of Radioactive Spent Ion Exchange Resin by Hydrogen Peroxide, Xingchao Jian, Tianbao Wu, and Guichun Yun; 157 A Comparison Study and Resolution of Differences Between Emergency Response and Safety Analysis Codes Used at the Savannah River Site, A. A. Simpkins; OPERATING EXPERIENCES: 164 Reactor Shutdown Experience, Compiled by J. W. Cletcher; RECENT DEVELOPMENTS: 167 Reports, Standards, and Safety Guides, D. S. Queener; 172 Proposed Rule Changes as of Dec. 31,1995; ANNOUNCEMENTS: 178 American Nuclear Society 1997 Annual Meeting; 178 American Nuclear Society Nuclear Criticality and Safety Division Topical Meeting; 176 The Authors.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS

High-Temperature SiC Cladding End Plug Irradiation Design and HFIR Readiness

This report documents the design of a High Flux Isotope Reactor (HFIR) irradiation experiment intended to evaluate irradiation effects on the hermeticity of silicon carbide (SiC) end plug specimens under a radial fast neutron flux gradient at representative light-water reactor (LWR) temperatures of approximately 300 °C. The overarching goal of this work is to statistically evaluate SiC end plug hermeticity and mechanical properties following irradiation using the high-throughput irradiation capability discussed here. Each specimen consists of a short section of SiC fiber–reinforced SiC (SiC/SiC) tube with a single monolithic SiC end plug joined to one end. The experiment allows up to 66 specimens to be irradiated in six different stacks within a dry, sealed irradiation capsule derived from the previously developed high-temperature SiC/SiC cladding bowing experiment. The neutronics basis, thermal analysis, and HFIR readiness of the experiment are discussed in this report for two possible design cases. The first design case is based on existing approval documentation and components that are on hand and approved for use, so the experiment insertion would require only specimen receipt, specimen pre-irradiation characterization, experiment assembly, and final fabrication package approval. The second design case provides improved thermal robustness and the preferred end plug geometry but requires fabrication of a modified holder and revisions to the HFIR approval documentation, in addition to the other activities required for the first design case, before insertion.

Hott, Daniel [Oak Ridge National Laboratory (ORNL)

Ion Transport and Crystal Rotation in Plastic Crystal Electrolytes Under Applied Electric Fields

Organic ionic plastic crystal electrolytes, containing a plastic crystal and lithium salt, offer a potential balance between mechanical and electrochemical properties for solid state lithium-ion battery electrolytes. These electrolytes contain multiple mobile ionic species (three or four), resulting in complex transport mechanisms which have not yet been established. Plastic crystals are defined by long-range positional order and short-range rotational disorder. It is therefore necessary to quantify changes in the local crystal structure of the electrolyte as current flows through it. Herein, we examine the electrochemical properties of pyrrolidinium-based plastic crystal electrolytes containing lithium salt and zwitterion additives, including measurements of current fraction and limiting current. We obtain species-specific insight into electrolyte transport using pulsed-field gradient nuclear magnetic resonance spectroscopy and find that, while the zwitterion additive increases ionic conductivity, it decreases lithium diffusivity with respect to other ionic components. With operando spatiotemporally resolved wide-angle X-ray scattering we observe location-specific crystal rotations due to the passage of ionic current. In conclusion, we posit that reducing energy dissipation due to rotation is essential for using plastic crystal electrolytes in practical applications.

Yap, Kyra M. K. [University of California, Berkele

Nuclear Security Risks for HALEU Fuels

There is growing interest in high-assay low-enriched uranium (HALEU) for use in advanced nuclear reactors as a high-energy fuel source. The primary objectives of this report are to identify the security risks that directly result from HALEU and to identify the gaps and challenges it presents from a theft and sabotage perspective. This study focuses on HALEU security risks for the front end of the fuel cycle and includes a review of the supply chain, fuel fabrication, and transport for terrestrial reactors.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS

CASM Monte Carlo: Calculations of the thermodynamic and kinetic properties of complex multicomponent crystals

Monte Carlo techniques play a central role in statistical mechanics approaches that connect macroscopic thermodynamic and kinetic properties to the electronic structure of a material. This paper describes the implementation of Monte Carlo techniques for the study of multicomponent crystalline materials within the Clusters Approach to Statistical Mechanics (CASM) software suite, and demonstrates their use in model systems to calculate free energies and kinetic coefficients, study phase transitions, and construct phase diagrams from first principles. Many crystal structures are complex, with multiple sublattices occupied by differing sets of chemical species, along with the presence of vacancies or interstitial species. This imposes constraints on concentration variables, the form of thermodynamic potentials, and the values of kinetic transport coefficients. The framework used by CASM to formulate thermodynamic potentials and kinetic transport coefficients accounting for arbitrarily complex crystal structures is presented and demonstrated with examples of increasing complexity. Additionally, an overview of the capabilities of the CASM software specific to Monte Carlo methods is given, and a new CASM software package is introduced, casm-flow, which helps automate the setup, submission, management, and analysis of Monte Carlo simulations.

Cluster expansion