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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 397 records · Page 22

Sulfur Metabolites Play Key System-Level Roles in Modulating Denitrification

Competition between nitrate-reducing bacteria (NRB) and sulfate-reducing bacteria (SRB) for resources in anoxic environments is generally thought to be governed largely by thermodynamics. It is now recognized that intermediates of nitrogen and sulfur cycling (e.g., hydrogen sulfide, nitrite, etc.) can also directly impact NRB and SRB activities in freshwater, wastewater, and sediment and therefore may play important roles in competitive interactions. Here, through comparative transcriptomic and metabolomic analyses, we have uncovered mechanisms of hydrogen sulfide- and cysteine-mediated inhibition of nitrate respiratory growth for the NRB Intrasporangium calvum C5. Specifically, the systems analysis predicted that cysteine and hydrogen sulfide inhibit growth of I. calvum C5 by disrupting distinct steps across multiple pathways, including branched-chain amino acid (BCAA) biosynthesis, utilization of specific carbon sources, and cofactor metabolism. Here, we have validated these predictions by demonstrating that complementation with BCAAs and specific carbon sources relieves the growth inhibitory effects of cysteine and hydrogen sulfide. We discuss how these mechanistic insights give new context to the interplay and stratification of NRB and SRB in diverse environments.

59 BASIC BIOLOGICAL SCIENCES↗

Practical Techniques for Modeling Gas Turbine Engine Performance

The cost and risk associated with the design and operation of gas turbine engine systems has led to an increasing dependence on mathematical models. In this paper, the fundamentals of engine simulation will be reviewed, an example performance analysis will be performed, and relationships useful for engine control system development will be highlighted. The focus will be on thermodynamic modeling utilizing techniques common in industry, such as: the Brayton cycle, component performance maps, map scaling, and design point criteria generation. In general, these topics will be viewed from the standpoint of an example turbojet engine model; however, demonstrated concepts may be adapted to other gas turbine systems, such as gas generators, marine engines, or high bypass aircraft engines. The purpose of this paper is to provide an example of gas turbine model generation and system performance analysis for educational uses, such as curriculum creation or student reference.

aerothermodynamics↗

Practical Techniques for Modeling Gas Turbine Engine Performance

The cost and risk associated with the design and operation of gas turbine engine systems has led to an increasing dependence on mathematical models. In this paper, the fundamentals of engine simulation will be reviewed, an example performance analysis will be performed, and relationships useful for engine control system development will be highlighted. The focus will be on thermodynamic modeling utilizing techniques common in industry, such as: the Brayton cycle, component performance maps, map scaling, and design point criteria generation. In general, these topics will be viewed from the standpoint of an example turbojet engine model; however, demonstrated concepts may be adapted to other gas turbine systems, such as gas generators, marine engines, or high bypass aircraft engines. The purpose of this paper is to provide an example of gas turbine model generation and system performance analysis for educational uses, such as curriculum creation or student reference.

aerothermodynamics↗

Hybrid NEP-Chemical Vehicle and Propulsion Technology Study for Crewed Mars Missions

Nuclear Electric Propulsion (NEP) is a suite of technologies with the potential to enable crewed opposition-class missions to Mars. Each subsystem comprising an NEP system has multiple technology options that present tradeoffs in vehicle sizing and technology development risk. Multiple vehicle concepts have been studied in the past and are currently being studied at NASA. Three models have been developed to illustrate the impact of key technology parameters on overall sizing for a crewed Mars transportation vehicle. Vehicle mass required to close the mission as a function of NEP system alpha (kWe/kg) and specific impulse (Isp) is estimated with a detailed trajectory model. A power system mass model estimates alpha as a function of power and radiator mass assumption. A power conversion system thermodynamic model predicts the radiator area required to close a Brayton cycle power conversion system. Combined with the architecture mass required for mission closure, the radiator area and alpha model provide insight into how technology development may impact the mission. Our focus is on parametric sweeps of the whole design space rather than any particular point design. Results from the mission model comparing electric propulsion technology (Hall, Ion, Magnetoplasmadynamic) and both 2039 and 2042 opposition-class mission launch windows are included. The results of the study illustrate the technology parameters that can result in mission closure and illustrate where the architecture is sensitive to variations in technology performance or requirements.

Matthew Duchek↗

Environmental Thermodynamic Effects on Future Changes of Mesoscale Convective Systems and Isolated Deep Convection Over the US: A Case Study

Mesoscale convective systems (MCSs) and isolated deep convection (IDC) significantly influence local weather conditions and the hydrological cycle across the United States. Their future changes under warming is critical to the society. However, the exact impacts of thermodynamic environmental changes on them are difficult to resolve using climate model simulations which integrate both dynamic and thermodynamic factors. This study explores a theoretical modeling approach to isolate environmental thermodynamic effects on their future changes through a case study. For a 50-day period during the 2020 summer, a pair of convection-permitting model simulations indicate contrasting changes in MCSs and IDC between inland and coastal regions under pseudo global warming. Driven by the thermodynamic environments of these simulations, a single-column parcel model indicates a decrease in the frequency of IDC occurrences, along with increases in duration and precipitation amount under warming, attributable to rising most unstable convective available potential energy (MUCAPE), convective inhibition (MUCIN), and precipitable water (PW). A multi-column parcel model reveals contrasting changes in the frequency and mean area of MCSs between inland and coastal regions, underscoring the increase in mean MUCIN over inland regions and increases in mean MUCAPE and PW over coastal regions. The increase in mean MCS area over inland regions is linked to the interplay between accelerated gust fronts and enhanced subsidence strength, which is further traced to the unchanged mean MUCIN but increased mean MUCAPE in large-scale environments. In conclusion, these case study results suggest a major role of thermodynamic environmental changes in controlling characteristics of MCSs and IDC under warming.

Mesoscale convective system↗

Environmental sensitivities of shallow-cumulus dilution – Part 1: Selected thermodynamic conditions

Cumulus entrainment, and its consequent dilution of buoyant cloud cores, strongly regulates the life cycle of shallow cumuli yet remains poorly understood. Herein, new insights into this problem are obtained through large-eddy simulations that systematically investigate the sensitivity of shallow-cumulus dilution to cloud-layer relative humidity (RH), cloud- and subcloud-layer depths, and continentality (i.e., the land–ocean contrast). The simulated cloud-core dilution is found to be strongly sensitive to continentality, with fractional dilution rates twice as large over the ocean as over land. Using a similarity theory based on the turbulent-kinetic-energy (TKE) budget, the reduced cloud-core dilution over land is attributed to larger cloud-base mass flux (mb), driven by stronger surface heating and subcloud turbulence. As mb increases, the fractional dilution rate must decrease to maintain energetic equilibrium. A positive sensitivity is also found to cloud-layer RH, with the core dilution increasing by 25 %–50 % for a 10 % enhancement in RH. This sensitivity is interpreted using the buoyancy-sorting hypothesis, in that mixtures of cloud and environmental air are more likely to become negatively buoyant and detrain (rather than diluting the cloud core) in drier cloud layers. By contrast, the sensitivities of (marine) shallow-cumulus dilution to cloud- and subcloud-layer depths are weak, with a 3 % decrease for a doubling for the former and a 4 % reduction in dilution for a 50 % deeper subcloud layer. These surprisingly weak sensitivities are readily explained by offsetting effects in the TKE similarity theory. Altogether, these experimental findings provide useful, though still incomplete, guidance for flow-dependent shallow-cumulus entrainment formulations in large-scale models.

54 ENVIRONMENTAL SCIENCES↗

Vanadyl Phosphates A x VOPO 4 (A = Li, Na, K) as Multielectron Cathodes for Alkali-Ion Batteries

Vanadyl phosphates comprise a class of multielectron cathode materials capable of cycling two Li + , about 1.66 Na + , and some K + ions per redox center. In this review, structures, thermodynamic stabilities, and ion diffusion kinetics of various A x VOPO 4 (A = Li, Na, K, NH 4 ) polymorphs are discussed. Both the experimental data and first-principle calculations indicate kinetic limitations for alkali metal ions cycling, especially between for 0 ≤ x ≤ 1, and metastability of phases with x > 1. This creates challenges for multiple-ion cycling, as the slow kinetics call for nanosized particles, which being metastable and reactive with organic electrolytes are prone to side reactions. Thus, various synthesis approaches, surface coating, and transition metal ion substitution strategies are discussed here as possible ways to stabilize A x VOPO 4 structures and improve alkali metal ion diffusion. Finally, the role of advanced characterization techniques, such as X-ray absorption spectroscopy, diffraction, pair distribution function analysis and 7 Li and 31 P NMR, in understanding the reaction mechanism from both structural and electronic points of view is emphasized.

25 ENERGY STORAGE↗

An Active Broad Area Cooling Model of a Cryogenic Propellant Tank with a Single Stage Reverse Turbo-Brayton Cycle Cryocooler

As focus shifts towards long-duration space exploration missions, an increased interest in active thermal control of cryogenic propellants to achieve zero boil-off of cryogens has emerged. An active thermal control concept of considerable merit is the integration of a broad area cooling system for a cryogenic propellant tank with a combined cryocooler and circulator system that can be used to reduce or even eliminate liquid cryogen boil-off. One prospective cryocooler and circulator combination is the reverse turbo-Brayton cycle cryocooler. This system is unique in that it has the ability to both cool and circulate the coolant gas efficiently in the same loop as the broad area cooling lines, allowing for a single cooling gas loop, with the primary heat rejection occurring by way of a radiator and/or aftercooler. Currently few modeling tools exist that can size and characterize an integrated reverse turbo-Brayton cycle cryocooler in combination with a broad area cooling design. This paper addresses efforts to create such a tool to assist in gaining a broader understanding of these systems, and investigate their performance in potential space missions. The model uses conventional engineering and thermodynamic relationships to predict the preliminary design parameters, including input power requirements, pressure drops, flow rate, cycle performance, cooling lift, broad area cooler line sizing, and component operating temperatures and pressures given the cooling load operating temperature, heat rejection temperature, compressor inlet pressure, compressor rotational speed, and cryogenic tank geometry. In addition, the model allows for the preliminary design analysis of the broad area cooling tubing, to determine the effect of tube sizing on the reverse turbo-Brayton cycle system performance. At the time this paper was written, the model was verified to match existing theoretical documentation within a reasonable margin. While further experimental data is needed for full validation, this tool has already made significant steps towards giving a clearer understanding of the performance of a reverse turbo-Brayton cycle cryocooler integrated with broad area cooling technology for zero boil-off active thermal control.

Guzik, Monica C.↗

The Role of Oxidation Resistance in High Temperature Alloy Selection for a Future with Green Hydrogen

Hydrogen is being considered integral to the future energy landscape but there is limited mechanistic understanding and a lack of predictive models to describe the combined effects of alloy and gas composition, temperature, thermal cycling and water vapor contents on the oxidation behavior of high-temperature materials. Experimental evaluations were combined with coupled thermodynamic-kinetic modeling to investigate the oxidation behavior of five representative Ni-based superalloys in two water vapor contents (10%, 60% H 2 O) under thermal cycling (1-h, 100-h cyles) conditions at 800°C and 1000°C. The alloy with the highest Ti content demonstrated the poorest cyclic oxidation behavior while the alloy with highest Cr and Al contents was expected to continue to support protective formation of a compact Al 2 O 3 scale. Finally, accelerated degradation of the chromia-forming alloys was observed in the higher water vapor content but the impact on transient oxidation of the alumina-forming alloys needs further investigations.

36 MATERIALS SCIENCE↗

4-D Cloud Water Content Fields Derived from Operational Satellite Data

In order to improve operational safety and efficiency, the transportation industry, including aviation, has an urgent need for accurate diagnoses and predictions of clouds and associated weather conditions. Adverse weather accounts for 70% of all air traffic delays within the U.S. National Airspace System. The Federal Aviation Administration has determined that as much as two thirds of weather-related delays are potentially avoidable with better weather information and roughly 20% of all aviation accidents are weather related. Thus, it is recognized that an important factor in meeting the goals of the Next Generation Transportation System (NexGen) vision is the improved integration of weather information. The concept of a 4-D weather cube is being developed to address that need by integrating observed and forecasted weather information into a shared 4-D database, providing an integrated and nationally consistent weather picture for a variety of users and to support operational decision support systems. Weather analyses and forecasts derived using Numerical Weather Prediction (NWP) models are a critical tool that forecasters rely on for guidance and also an important element in current and future decision support systems. For example, the Rapid Update Cycle (RUC) and the recently implemented Rapid Refresh (RR) Weather Research and Forecast (WRF) models provide high frequency forecasts and are key elements of the FAA Aviation Weather Research Program. Because clouds play a crucial role in the dynamics and thermodynamics of the atmosphere, they must be adequately accounted for in NWP models. The RUC, for example, cycles at full resolution five cloud microphysical species (cloud water, cloud ice, rain, snow, and graupel) and has the capability of updating these fields from observations. In order to improve the models initial state and subsequent forecasts, cloud top altitude (or temperature, T(sub c)) derived from operational satellite data, surface observations of cloud base altitude, radar reflectivity, and lightning data are used to help build and remove clouds in the models assimilation system. Despite this advance and the many recent advances made in our understanding of cloud physical processes and radiative effects, many problems remain in adequately representing clouds in models. While the assimilation of cloud top information derived from operational satellite data has merit, other information is available that has not yet been exploited. For example, the vertically integrated cloud water content (CWC) or cloud water path (CWP) and cloud geometric thickness (delta Z) are standard products being derived routinely from operational satellite data. These and other cloud products have been validated under a variety of conditions. Since the uncertainties have generally been found to be less than those found in model analyses and forecasts, the satellite products should be suitable for data assimilation, provided an appropriate strategy can be developed that links the satellite-derived cloud parameters with cloud parameters specified in the model. In this paper, we briefly outline such a strategy and describe a methodology to retrieve cloud water content profiles from operational satellite data. Initial results and future plans are presented. It is expected that the direct assimilation of this new product will provide the most accurate depiction of the vertical distribution of cloud water ever produced at the high spatial and temporal resolution needed for short term weather analyses and forecasts.

Smith, William L., Jr.↗

Understanding Dynamics and Thermodynamics of ENSO and Its Complexity Simulated by E3SM and Other Climate Models

Despite the seeming success of most state-of-the-art climate models in simulating the El Niño-Southern Oscillation (ENSO), there is strong evidence that models achieve realistic levels of ENSO activity do so often for wrong reasons. This is owing to an often occurred near cancelation of large errors in terms of contributions to ENSO growth rate from coupled dynamic and thermodynamic feedback processes. Climate models remain deficient in simulating the observed ENSO’s spatial and temporal complexity that involves interplays of coupled dynamic and thermodynamic feedbacks, interactions across multiple scales, nonlinear processes in the tropical atmosphere and ocean system, biases in mean sate and physical processes, and influences external to equatorial Pacific coupled ENSO dynamics. Our proposed research aims at advancing predictive and process-level understandings of ENSO simulated in E3SM and other climate models under current and future climate conditions with two main objectives: (i) better understanding the aforementioned broad range interactive processes and sources that control fundamental properties of ENSO in E3SM and CIMP6 outputs as well as in observational (reanalysis) data sets, using a hierarchical of coupled dynamical frameworks consisting of theoretical analysis, intermediate complexity modeling, and coupled dynamic diagnostics; (ii) to use this understanding to explore pathways towards improving E3SM’s capability of simulating ENSO and its complexity. More specifically, we will focus on four main thrusts of research: (1) ENSO’s dynamic and thermodynamic feedbacks; (2) the across-scale interactions of ENSO with annual cycle and MJO/WWB/TIW (Madden Julian Oscillation/Westerly Wind Burst/Tropical Instability Wave) activity; (3) key nonlinear processes of ENSO involving atmospheric convective thresholds, nonlinear ocean dynamic heating, and thermocline outcropping; and (4) the impacts of climate mean-state biases/changes and perturbed physical processes on simulated ENSO and its complexity.

54 ENVIRONMENTAL SCIENCES↗

New High-Entropy Perovskite Oxides with Increased Reducibility and Stability for Thermochemical Hydrogen Generation

This project aims to design, synthesize, and test a transformative class of High-Entropy Perovskite Oxides (HEPOs) as redox oxides to enable thermochemical hydrogen generation with improved stability, kinetics, and efficiency. These developed HEPOs are expected to demonstrate improved kinetics with oxygen surface exchange coefficient ( k > 7.5×10 -4 cm/s) in Budget Period (BP) 1, retain its structural stability in a broad range of oxygen non-stoichiometry (Δδ > 0.15) at a low operating reduction temperature of T red < 1400°C in BP 2, and deliver a H 2 yield of over 400 µmol per gram of oxide and high stability with less than 20% degradation after at least 50 cycles in BP 3. This project is feasible due to the unique thermodynamic properties (simultaneously increased reducibility and phase stability) and kinetic characters (stability against particle coarsening and potentially enhanced oxygen transport and surface reaction kinetics) of such HEPOs, and it is enabled by a unique active learning computational design approach. Computational studies have been conducted to investigate the oxygen vacancy formation in complex perovskite systems. * Accurate prediction of V O .. concentration with disordered A-site cations in Fe-based high-entropy perovskite oxides * Combined MC/DFT computation elucidates the mechanism of Co preference on the redox due to the strain introduced by local distortion. In this project, we explored a large number (~150) of perovskite compositions, which are listed in Tables 2 – 4). * All perovskite specimens have been synthesized through a high-throughput high-energy ball milling process, followed by the conventional sintering process. * XRD, SEM/EDS and TGA were performed to confirm the crystal structure, phase homogeneity and oxygen non-stoichiometry for compositionally complex perovskite oxides (CCPOs). * 110 compositions show single-phase from XRD * Unusual aliovalent doping effects in medium-entropy perovskite compositions. * V-shape relation between Δδ vs. x (La 1-x Sr x )(Mn 1/3 Fe 1/3 Ti 1/3 )O 3 * The highest reported hydrogen production for the CCPOs made in this project ( T re = 1350 ºC 30 min, T Ox = 1100 ºC 30 min) * B-site mixing (La 0.8 Sr 0.2 )(Mn 0.2 Fe 0.2 Co 0.4 Al 0.2 )O 3 : 89.97 ± 2.73 mmol H2 /mol oxide (395 ± 10 μmol/g oxide ) (i) No phase transformation during reactions when Co molar ratio is less than 61% (ii) Balance between intrinsic kinetics (oxygen surface exchange) and thermodynamics (oxygen non-stoichiometry) (iii) Preference of Co identified by in-situ XPS * A-site mixing (La 1/6 Pr 1/6 Nd 1/6 Gd 1/6 Ba 1/6 Sr1/6)MnO 3 : 98.48 mmol H2 /mol oxid e (~415 μmol/g oxide )

08 HYDROGEN↗

Flexible Oxy-Fuel Combustion for High-Penetration Variable Renewables

A thermodynamic model was developed for the oxy-combustion Allam-Fetvedt cycle. This information was then used to develop an optimized dispatch strategy using price strips supplied by the modeling teams. The price strips represent future possible grid configurations that include a high penetration of variable renewables and a carbon tax. Multiple optimization strategies and tools were used to maximize the net present value (NPV) of the plant on these potential future grids. The optimization varied the size of the air separation unit, the size of oxygen storage tanks, the size of carbon dioxide storage tanks and the flow rate of the carbon dioxide pipeline. The team was able to determine a dispatch strategy that resulted in a positive NPV for all price strips. This indicates that an oxy-combustion plant with oxygen storage would be economically viable on a future grid with a high degree of variable renewables.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

An experimental database of cell performance for vanadium redox flow battery

The continual growth in energy demand has resulted in the deployment of renewable energy generators to reduce the impact of fossil fuel dependence. However, these generators often suffer from intermittency and require energy storage when there is over-generation and the subsequent release of this stored energy at high demand. One promising energy storage technology which can provide a solution to improve energy management and grid stability, is the redox flow battery. Among the numerous flow battery systems, vanadium redox flow battery is the most iconic solution to large scale energy storage, giving a more efficient link between energy production, especially from renewables, and energy demand. The aim of the current database is to characterize the performance of the cell design and to provide training/validation data for physical model or data-driven model. The database includes hundreds of experimental cell performance data of vanadium redox flow battery with various current densities for multiple charge-discharge cycles. All the cell parameters, chemical parameters, material parameters, operation parameters and thermodynamic parameters of the cell system are listed. Coulomb, voltaic and energy efficiencies are also provided. The database will be helpful for researchers in the field of redox flow batteries.

Gao, Peiyuan↗

Tripropellant engine study

Engine performance data, combustion gas thermodynamic properties, and turbine gas parameters were determined for various high power cycle engine configurations derived from the space shuttle main engine that will allow sequential burning of LOX/hydrocarbon and LOX/hydrogen fuels. Both stage combustion and gas generator pump power cycles were considered. Engine concepts were formulated for LOX/RP-1, LOX/CH4, and LOX/C3H8 propellants. Flowrates and operating conditions were established for this initial set of engine systems, and the adaptability of the major components of shuttle main engine was investigated.

Wheeler, D. B.↗

Dynamic Simulation of a Periodic 10 K Sorption Cryocooler

A transient thermal simulation model has been developed to simulate the dynamic performance of a multiple-stage 10 K sorption cryocooler for spacecraft sensor cooling applications that require periodic quick-cooldown (under 2 minutes) , negligible vibration, low power consumption, and long life (5 to 10 years). The model was specifically designed to represent the Brilliant Eyes Ten-Kelvin Sorption Cryocooler Experiment (BETSCE), but it can be adapted to represent other sorption cryocooler systems as well. The model simulates the heat transfer, mass transfer, and thermodynamic processes in the cryostat and the sorbent beds for the entire refrigeration cycle, and includes the transient effects of variable hydrogen supply pressures due to expansion and overflow of hydrogen during the cooldown operation. The paper describes model limitations and simplifying assumptions, with estimates of errors induced by them, and presents comparisons of performance predictions with ground experiments. An important benefit of the model is its ability to predict performance sensitivities to variations of key design and operational parameters. The insights thus obtained are expected to lead to higher efficiencies and lower weights for future designs.

cryocooler design performance sorption simulation ↗

Experiment and Modeling of Simultaneous Creep, Plasticity and Transformation of High Temperature Shape Memory Alloys During Cyclic Actuation

The present work is focused on studying the cycling actuation behavior of HTSMAs undergoing simultaneous creep and transformation. For the thermomechanical testing, a high temperature test setup was assembled on a MTS frame with the capability to test up to temperatures of 600 C. Constant stress thermal cycling tests were conducted to establish the actuation characteristics and the phase diagram for the chosen HTSMA. Additionally, creep tests were conducted at constant stress levels at different test temperatures to characterize the creep behavior of the alloy over the operational range. A thermodynamic constitutive model is developed and extended to take into account a) the effect of multiple thermal cycling on the generation of plastic strains due to transformation (TRIP strains) and b) both primary and secondary creep effects. The model calibration is based on the test results. The creep tests and the uniaxial tests are used to identify the viscoplastic behavior of the material. The parameters for the SMA properties, regarding the transformation and transformation induced plastic strain evolutions, are obtained from the material phase diagram and the thermomechanical tests. The model is validated by predicting the material behavior at different thermomechanical test conditions.

Kumar, Parikshith K.↗

Power System Optimization Codes Modified

A major modification of and addition to existing Closed Brayton Cycle (CBC) space power system optimization codes was completed. These modifications relate to the global minimum mass search driver programs containing three nested iteration loops comprising iterations on cycle temperature ratio, and three separate pressure ratio iteration loops--one loop for maximizing thermodynamic efficiency, one for minimizing radiator area, and a final loop for minimizing overall power system mass. Using the method of steepest ascent, the code sweeps through the pressure ratio space repeatedly, each time with smaller iteration step sizes, so that the three optimum pressure ratios can be obtained to any desired accuracy for each of the objective functions referred to above (i.e., maximum thermodynamic efficiency, minimum radiator area, and minimum system mass). Two separate options for the power system heat source are available: 1. A nuclear fission reactor can be used. It is provided with a radiation shield 1. (composed of a lithium hydride (LiH) neutron shield and tungsten (W) gamma shield). Suboptions can be used to select the type of reactor (i.e., fast spectrum liquid metal cooled or epithermal high-temperature gas reactor (HTGR)). 2. A solar heat source can be used. This option includes a parabolic concentrator and heat receiver for raising the temperature of the recirculating working fluid. A useful feature of the code modifications is that key cycle parameters are displayed, including the overall system specific mass in kilograms per kilowatt and the system specific power in watts per kilogram, as the results for each temperature ratio are computed. As the minimum mass temperature ratio is encountered, a message is printed out. Several levels of detailed information on cycle state points, subsystem mass results, and radiator temperature profiles are stored for this temperature ratio condition and can be displayed or printed by users.

Juhasz, Albert J.↗