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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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147 records · Page 6
All sourcesNASA NTRSDOE OSTI
NASA NTRS1999

Dynamics and Structure of Weakly-Strained Flames In Normal- and Micro-Gravity

Strained laminar flames have been systematically studied, as the understanding of their structure and dynamic behavior is of relevance to turbulent combustion. Most of these studies have been conducted in opposed-jet, stagnation-type flow configurations. Flame studies in stagnation flows also allow for the determination of fundamental flame properties such as laminar flame speeds and extinction states that can be also conveniently modeled in detail. Studies at high strain rates are important in quantifying and understanding the response of vigorously-burning flames under conditions in which the transport time scales become comparable to the chemical time scales. Studies of weakly-strained flames can be of particular interest for all stoichiometries. For example, the laminar flame speeds for any equivalence ratio, phi, can be accurately determined by using the counterflow technique only if measurements are obtained at very low strain rates. Furthermore, near-limit flames can be only stabilized by weak strain rates. Previous studies have shown that weakly-burning flames are particularly sensitive to chain mechanisms, thermal radiation, and unsteadiness. The stabilization and study of weakly-strained flames is complicated by the presence of buoyancy that can render the flames unstable to the point of extinction. Such instabilities are caused either by the induced natural convection or the fact that higher density fluid can find itself on top of a lower density fluid. Thus, the use of microgravity (mu-g) becomes essential in order to provide meaningful insight into this important combustion regime. In view of the foregoing considerations, the main objectives of the program are to: (1) Experimentally determine the laminar flame speed at near-zero strain rates; (2) Experimentally determine the extinction limits of near-limit flames; (3) Experimentally determine the response of near-limit flames to unsteadiness and heat loss; (4) Introduce Digital Particle Image Velocimetry (DPIV) in microgravity; (5) Conduct detailed numerical simulations of the experiments; and (6) Provide physical insight into the underlying physico-chemical mechanisms.

Hai Zhang↗
NASA NTRSDate not supplied

Evaluation of Long-Term Storage Stability and Operational Efficiency of Rapid Cycle Amine Adsorbents

With the recent incorporation of Mars Extravehicular Activity (EVA) into the 2025 NASA roadmap, Rapid Cycle Amine (RCA) technology advancement within the Exploration Extravehicular Mobility Unit (xEMU) applies to both the upcoming Lunar Artemis missions and future human exploration of Mars. The incorporation of commercial spacesuit vendors to service NASA’s crewed-space programs since 2021 has increased the demand for regenerable, RCA-based carbon dioxide (CO2) and humidity control adsorbents. The regenerable RCA subunit is the most advanced, long-term solution for CO2 removal and humidity control, enabling both lunar and Martian EVAs. The adsorbent selected for use within the RCA unit must exhibit sufficient storage stability and operational efficiency for prolonged missions. As XploSafe worked to simultaneously develop both a recirculating sub-atmospheric test rig and adsorbents for the RCA system within the xEMU, several novel adsorbents were evaluated for long-term storage stability in support of future extended-duration missions. The chosen materials were periodically evaluated by nuclear magnetic resonance (NMR) spectroscopy, thermal desorption-coupled with gas chromatography/mass spectrometry (TD-GC/MS), and CO2 adsorption to assess long-term storage efficacy across various environmental storage conditions. NMR spectroscopy was utilized by extracting the active CO2 adsorbing chemical from the solid support with deuterated solvent and comparing the spectra over time for degradation and change. The solid adsorbents were also analyzed via TD-GC/MS to reveal any potential off-gassing chemicals over time. Additionally, XploSafe’s breakthrough test rig was utilized to dose the solid adsorbents with a 585 BTU/h metabolic rate flow-equivalent CO2 stream and monitored for 0–99% CO2 breakthrough. The performance metrics from the breakthrough analysis were compared over the storage study duration for CO2 removal and humidity control. Xplo-SA9T was evaluated for 24 months, whereas two additional adsorbent variants, MMPA-Sorbent and LPEI-Sorbent, were each evaluated for 12 months.

John R Tidwell↗
NASA NTRS2026

Spacecraft Water Impurity Monitor, a System for Water Quality Analysis on Exploration Missions Beyond Low Earth Orbit

Exploration missions beyond low-earth-orbit (LEO) will require advanced instrumentation to monitor water quality. Traveling beyond LEO means the transfer of water samples to an Earth-based laboratory for detailed analysis is not feasible. Detailed analysis of water composition during exploration is still necessary, because having the capability to determine the specific organic chemical causing a change in total organic carbon (TOC) or the specific metal or ionic species causing a change in conductivity has the potential to inform the crew health and system management decisions. On a new vehicle such as a lunar or Mars surface habitat or a Mars transit vehicle, finding “new” impurities not seen on ISS should be expected. The key is to identify the impurity so the correct action can be taken. On ISS we can measure TOC, conductivity, and other physical properties but do not have the capability for detailed analysis using vehicle instrumentation. This is acceptable because ISS can send samples down to Earth for further analysis and obtain the detailed composition. The Spacecraft Water Impurity Monitor (SWIM) will provide detailed water quality analysis for missions in which sample down mass is not available. SWIM is a system comprising organic and inorganic analysis modules. For organic chemicals, a gas chromatograph mass spectrometer (GCMS) detects and identifies organic impurities. For inorganic species, a capillary electrophoresis capacitively coupled contactless conductivity detection (CE-C4D) system as well as ion specific electrodes detect and identify metal ions and other inorganic salts / acids. The SWIM technology demonstration project has completed a System Requirements Review (SRR) to finalize detection requirements and is currently working to refine vehicle interface requirements for a future technology demonstration. The project also has begun preliminary flight design activities for the core analyzers in the instrument suite.

Water Monitoring↗
NASA NTRS1989

Overview of the Center for Advanced Space Propulsion

The mission of the Center for Advanced Space Propulsion (CASP), a Center for the Commercial Development of Space (CCDS), is to strengthen U.S. competitiveness in space technology, foster cooperative research with industry and government, assist industry in developing commercial products and services, and perform research in advanced space propulsion. Current areas of focus are (1) advanced chemical propulsion, including high area ratio nozzle performance, variable thrust engine performance, and spray combustion stability; (2) artificial intelligence propulsion applications, including health monitoring of rocket engines, fault pattern detection and diagnosis, and intelligent hypertext applications; (3) microgravity fluid management, including liquid storage and transfer, a subscale orbital fluid transfer experiment, and helical, two-phase flow; (4) electric propulsion, including magnetic annular arc thruster, ion thrustor, and electrostatic plasma accelerator; and (5) laser materials processing.

George W Garrison↗
NASA NTRSDate not supplied

Preservation of Previously Unsampled Primordial Isotopic Components in Ryugu and Bennu Samples

Ivuna-type (CI) carbonaceous chondrites are primitive meteorites whose relative elemental abundances closely match the solar photosphere 1 , making them critical benchmarks for the bulk chemical and isotopic composition of the planet-forming disk. Recent sample-return missions to Ryugu 2 and Bennu 3 have enabled direct laboratory analysis of these rubble-pile CI-like asteroids 4 , formed from the reaccumulated fragments of once-larger parent bodies. As such, they may retain a broader spectrum of primordial nucleosynthetic components than is preserved in known meteorites. Here, we report the nucleosynthetic compositions of silicon (μ 30 Si), magnesium (μ 26 Mg*), and iron (μ 54 Fe), three of the four most abundant planet-building elements, in samples from Ryugu and Bennu, along with the newly discovered Oued Chebeika 002 CI chondrite 5 . In contrast to the isotopic homogeneity of CI chondrites, Ryugu and Bennu preserve resolvable μ 26 Mg* and μ 30 Si heterogeneity, revealing μ 26 Mg*- and μ 30 Si-rich nucleosynthetic components not sampled in meteorites. These components may reflect the preservation of previously unsampled primordial isotopic signals, prior to thermal processing 6,7 and late infall of primordial molecular cloud material to the outer disk 8,9 . Additionally, Ryugu particle C0002 exhibits a μ 26 Mg*-poor and μ 54 Cr-rich signature, consistent with incorporation of presolar material from massive supernovae 10,11 . Ryugu and Bennu samples define one end of a μ 54 Fe–μ 30 Si array among carbonaceous asteroid materials 9 , suggesting that the isotopic compositions of these asteroidal materials reflect mixing between early disk isotopic components identified in this study and material derived from the outermost disk 9 . This observation demonstrates that materials with solar chemical abundances can nonetheless exhibit markedly distinct nucleosynthetic signatures. Thus, Ryugu and Bennu samples are critical records of nucleosynthetic diversity among carbonaceous asteroids and underscore the value of sample-return missions in accessing primitive isotopic signatures absent from meteorite collections.

Martin Bizzarro↗
NASA NTRS1994

New Trends in Coatings Developments for Turbine Blades: Materials Processing and Repair

Turbine engines for aeronautic applications now have to face strenuous requirements concerning not only operating performances but also reliability and repairability. This evolution of the specifications induces important consequences regarding the choice and the design of turbine blade protective coatings. This paper presents several key features concerning the deposition techniques and the real-life behavior of such coatings: (1) complex aluminides well suited to the protection against high temperature oxidation and hot corrosion of directionally solidified nickel-base superalloy turbine blades; (2) ceramic coatings used as thermal barriers - they have to exhibit both high resistance to thermomechanical fatigue and adequate smoothness; (3) processing techniques for thermal barriers, including plasma spraying, electron beam physical vapor deposition and plasma enhanced chemical vapor deposition; and (4) blades and coatings repair techniques.

S Alpérine↗
NASA NTRS2026

On-Demand Production of Hypergols: Steps Towards Scalability

During FY 2025, the team at KSC focused on taking the 2024 proof-of-concept demonstration of hypergol production using novel methods and began scaling up production and increasing purity. The team was able to more than triple the production of hydrazine using a novel method with a drastic increase in the purity. The current method, albeit still limited to production rates on the order of mg/hr, provides a cost savings of approximately 14x when compared to market value pricing through the Defense Logistics Agency (DLA) website. The benefit of this technology is the ability to produce some hypergolic fuels using constituents in Earth’s atmosphere, eliminating most chemical commodity logistics. This technology also allows for the return of domestic production of hypergolic propellant.

Kenneth Engeling↗
NASA NTRS2026

Barium Fluoride as a Standard Reference Material for Infrared Refractometry

This work proposes a standard reference material (SRM) for measurements of refractive index in the infrared spectral region (1–12 μm). The purpose of this material is to enable cross-method comparison of refractive index data obtained through minimum deviation, ellipsometry, interferometry, and prism coupling techniques. This will allow vendors and metrology laboratories, regardless of which technique they employ, to certify that their measurements meet required levels of accuracy, precision, and repeatability. Drawing on foundational work by Tilton at the National Bureau of Standards, a figure of merit for candidate SRM materials is outlined, and three materials are selected for further evaluation. Ultimately, barium fluoride is proposed as the most suitable SRM for the infrared spectral region, based on its broad transmission range, chemical and mechanical stability, low temperature dependence of refractive index, and well-characterized dispersion behavior.

Andrew Howe↗
NASA NTRS1993

A SIMULINK Environment for Flight Dynamics and Control Analysis - Application to the DHC-2 'Beaver', Part 1. Implementation of a Model Library in SIMULINK

The design of advanced Automatic Aircraft Control Systems (AACS's) can be improved upon considerably if the designer can access all models and tools required for control system design and analysis through a graphical user-interface, from within one software environment. This MSc-thesis presents the first step in the development of such an environment, which is currently being done at the Section for Stability and Control of Delft University of Technology, Faculty of Aerospace Engineering. The environment is implemented within the commercially available software package MATLAB/SIMULINK. The report consists of two parts. Part I gives a detailed description of the AACS design environment. The heart of this environment is formed by the SIMULINK implementation of a nonlinear aircraft model in block-diagram format. The model has been worked out for the old laboratory aircraft of the Faculty, the De Havilland DHC-2 'Beaver', but due to its modular structure, it can easily be adapted for other aircraft. Part I also describes MATLAB programs which can be applied for finding steady-state trimmed-flight conditions and for linearization of the aircraft model, and it shows how the built-in simulation routines of SIMULINK have been used for open-loop analysis of the aircraft dynamics. Apart from the implementation of the models and tools, a thorough treatment of the theoretical backgrounds is presented. Part II of this report presents a part of an autopilot design process for the 'Beaver' aircraft, which clearly demonstrates the power and flexibility of the AACS design environment from part I. Evaluations of all longitudinal and lateral control laws by means of nonlinear simulations are treated in detail. The AACS design environment from part I proved to be a very useful tool for designing the control laws of the 'Beaver' autopilot within a very tight time-schedule. The autopilot design process itself will be used as a guideline for future AACS research at the Faculty of Aerospace Engineering. Flight tests of the 'Beaver' autopilot, done after evaluating the control laws in the SIMULINK package, proved to be quite successful. In the future, the AACS design package will evolve into a standardized, integrated design environment which can be applied to virtually any type of aircraft. The AACS design cycle will be shortened further by developing tools for automatically porting control laws from the MATLAB/SIMULINK environment to a piloted real-time flight simulator and the Flight Control Computers of the aircraft.

Aircraft Modelling↗
NASA NTRS2003

Analysis of Surface and Bulk Behavior in Ni-Pd Alloys

The most salient features of the surface structure and bulk behavior of Ni-Pd alloys have been studied using the BFS method for alloys. Large-scale atomistic simulations were performed to investigate surface segregation profiles as a function of temperature, crystal face, and composition. Pd enrichment of the first layer was observed in (111) and (100) surfaces, and enrichment of the top two layers occurred for (110) surfaces. In all cases, the segregation profile shows alternate planes enriched and depleted in Pd. In addition, the phase structure of bulk Ni-Pd alloys as a function of temperature and composition was studied. A weak ordering tendency was observed at low temperatures, which helps explain the compositional oscillations in the segregation profiles. Finally, based on atom-by-atom static energy calculations, a comprehensive explanation for the observed surface and bulk features will be presented in terms of competing chemical and strain energy effects.

Ordering↗
NASA NTRS1977

Lunar Elemental Analysis Obtained From the Apollo Gamma-Ray and X-Ray Remote Sensing Experiment

Gamma-ray and X-ray spectrometers carried in the Service Modules of the Apollo 15 and Apollo 16 spacecraft were employed for compositional mapping of the lunar surface. The measurements involved the observation of the intensity and characteristic energy distribution of gamma rays and X-rays emitted from the lunar surface. A large-scale compositional map of over 10 percent of the lunar surface was obtained from an analysis of the observed spectra. The Apollo 15 flight was at a lunar orbital inclination of 29 degrees as compared with a 9-degree inclination of the Apollo 16 flight; thus, the projected ground track of the Apollo 15 flight covered a larger projected surface area than that of the Apollo 16 flight. The objective of the X-ray experiment was to measure the K spectral lines from Mg, Al, and Si. Spectra were obtained and the data were reduced to Al/Si and Mg/Si intensity ratios and ultimately to chemical ratios. Analyses of the results have indicated (1) that the Al/Si ratios are highest in the lunar highlands and considerably lower in the maria, and (2) that the Mg/Si concentrations generally show the opposite relationship. There is a tendency for the Al/Si values to increase from the western mare areas to the eastern limb highlands. There are distinct chemical contrasts between such features: as the small mare basins and the highland rims. The objective of the gamma-ray experiment was to measure the natural and cosmic-ray-induced activity emission spectrum. At this time, the elemental abundances for Th, U, K, Fe, Ti, Si, and O have been determined over a number of major lunar regions. Regions of relatively high natural radioactivity were found in the Mare Imbrium and Oceanus Procellarum regions. High spots of natural radioactivity were also found south of Fra Mauro; somewhat southwest of Archimedes, and south of Aristarchus. An enhanced region of natural radioactivity was found around Van de Graaff on the far side of tile Moon. In regions other than Mare Imbrium and Oceanus Procellarum, an anticorrelation between natural radioactivity and lunar elevation, as determined from the Apollo laser, altimeter, has been found. From the combined results of the gamma-ray and X-ray spectrometer experiments almost complete information concerning the major element composition of over 10 percent of the lunar surface has been obtained. Distributions have also been mapped for K, Th, and Ti. Interesting correlations between lunar topography and magnetic and gravitational properties have been found.

J I Trombka↗
NASA NTRS2026

Mathematical Model of a Regenerative Fuel Cell for System Optimization

This thesis developed a system-level optimization model of a regenerative fuel cell (RFC) system for long-duration, off-world energy storage applications. Prior RFC design studies have typically been limited to reduced parameter sets and simplified constraints due to computational limitations relative to the number of relevant degrees of freedom. As a result, important nonlinear interactions between subsystems have not been fully captured. This work began to address that gap by developing a higher-fidelity, nonlinear optimization framework that incorporates a broader set of design variables and coupled constraints, enabling a multidimensional model that captures the coupled behavior of RFC subsystems and demonstrates the feasibility of applying optimization to such systems. An expanded system-level optimization approach was established that captures interactions between electrochemical performance, structural requirements, and storage design. This enabled a more comprehensive evaluation of trade-offs than conventional formulations. The model integrates four coupled subsystems: a fuel cell, an electrolyzer, reactant gas, and high-pressure storage tanks, and was formulated to accommodate a wide range of mission parameters, including operational time and required output power. It incorporates constraints on available solar array power, reactant mass balance between production and consumption, and pressure-dependent storage requirements. To enable reliable convergence, the optimization problem was reformulated to reduce dimensionality and improve numerical stability, with subsystem models organized for efficient evaluation. Problem dimensionality was reduced by consolidating lower-level design variables into higher-level representative quantities, and subsystem behavior was evaluated within the optimization loop. A multi-start initialization strategy was employed to mitigate sensitivity to local minima and improve solution quality, while nonlinear relationships were solved using robust numerical methods. The results showed that convergence was achieved across a range of required output power values. Specific energy reached a maximum at a critical mission power level, where the electrolyzer power matched the available solar input and operated near its voltage and current density limits. Beyond this point, further increases in required power resulted in less mass-efficient operation, increasing total system mass and reducing overall performance. The developed model represents an advancement in RFC system-level optimization by enabling analysis of a broader and more tightly coupled design space than previous considerations. While convergence behavior and computational cost remain challenges, the methods introduced improve solvability and allow inclusion of additional design variables with minimal loss of physical fidelity. However, the numerical results should not be interpreted as definitive design recommendations, as the model includes simplifying assumptions and omits several higher-order effects. Future work should extend this framework by incorporating additional subsystems and loss mechanisms, such as thermal management, parasitic power consumption, and reactant losses, to improve fidelity and ensure more representative design conclusions.

Electrochemistry↗
NASA NTRSDate not supplied

Solvent-Free Preparation of High Energy, Binder-Free Electrodes Enabled by Dry Compressible Holey Graphene

Graphene is an atomically thick sheet consisting of a graphitic carbon network with excellent mechanical strength, electrical and thermal conductivity, and chemical stability. Holey graphene, a structural derivative of graphene, has an array of through-the-thickness holes across the lateral surface of the nanosheet. The presence of these holes has minimal detrimental effect on the graphene properties and leads to enhanced performance in applications such as electronics, sensors, and energy storage. For example, these holes allow more facile cross-plane ion and gas transport than intact graphene, making holey graphene an ideal electrode material for electrochemical energy storage. This presentation will focus on the ability of holey graphene to be compression molded into robust articles or architectures under solvent-free conditions without the need for potentially parasitic binders. The unique dry compressibility of holey graphene has enabled facile fabrication of high mass loading electrodes with both high density and high porosity, which have found use in supercapacitors and various high-energy battery systems such as lithium-oxygen, lithium-sulfur, and lithium-selenium batteries.

Yi Lin↗
NASA NTRS1995

Single-Drop Reactive Extraction/Extractive Reaction with Forced Convective Diffusion and Interphase Mass Transfer

An algorithm has been developed for the forced convective diffusion-reaction problem for convection inside and outside a droplet by a recirculating flow field hydrodynamically coupled at the droplet interface with an external flow field that at infinity becomes a uniform streaming flow. The concentration field inside the droplet is likewise coupled with that outside by boundary conditions at the interface. A chemical reaction can take place either inside or outside the droplet or reactions can take place in both phases. The algorithm has been implemented and results are shown here for the case of no reaction and for the case of an external first order reaction, both for unsteady behavior. For pure interphase mass transfer, concentration isocontours, local and average Sherwood numbers, and average droplet concentrations have been obtained as a function of the physical properties and external flow field. For mass transfer enhanced by an external reaction, in addition to the above forms of results, we present the enhancement factor, with the results now also depending upon the (dimensionless) rate of reaction.

Leonid S Kleinman↗
NASA NTRS2023

Satellite characterization of global stratospheric sulfate aerosols released by Tonga volcano

Large volcanic eruptions create an enhanced layer of sulfate aerosols in the stratosphere. These sulfuric acid droplets persist for many months, altering the climate and stratospheric chemistry. Sulfate aerosols scatter sunlight back to space, cooling the surface of the Earth and absorb outgoing thermal radiation, heating the stratosphere. The calculation of the climate impact of sulfate aerosols depends on their physical properties such as droplet size and chemical composition. These properties are not well known, and this uncertainty contributes to the errors in climate model predictions. Here we derive the first empirical formula that predicts the composition of stratospheric sulfate aerosols from volcanic eruptions from the air temperature and water vapor pressure. Measurements of atmospheric infrared transmittance of the Hunga Tonga-Hunga Ha'apai sulfate aerosol plume by the Atmospheric Chemistry Experiment (ACE) satellite were analyzed to determine composition (weight percent of sulfuric acid) and median particle radius. These data are supplemented by measurements of the Raikoke and Nabro eruptions. Our analysis allows the properties of volcanic aerosols in the stratosphere to be predicted reliably in atmospheric models.

P Bernath↗
NASA NTRS1995

Single-Drop Reactive Extraction/Extractive Reaction With Forced Convective Diffusion and Interphase Mass Transfer

An algorithm has been developed for time-dependent forced convective diffusion-reaction having convection by a recirculating flow field within the drop that is hydrodynamically coupled at the interface with a convective external flow field that at infinity becomes a uniform free-streaming flow. The concentration field inside the droplet is likewise coupled with that outside by boundary conditions at the interface. A chemical reaction can take place either inside or outside the droplet, or reactions can take place in both phases. The algorithm has been implemented, and for comparison results are shown here for the case of no reaction in either phase and for the case of an external first order reaction, both for unsteady behavior. For pure interphase mass transfer, concentration isocontours, local and average Sherwood numbers, and average droplet concentrations have been obtained as a function of the physical properties and external flow field. For mass transfer enhanced by an external reaction, in addition to the above forms of results, we present the enhancement factor, with the results now also depending upon the (dimensionless) rate of reaction.

Leonid S Kleinman↗
NASA NTRS2026

Evaluation of Superalloy 718 Additively Manufactured Lattice Blocks

Lattice block structures made up of high temperature superalloys could offer advantages in density, efficiency, and tailored properties. Three candidate lattice designs were screened through additive manufacturing of superalloy 718. Laser powder bed fusion was applied with 718 powder to produce lattice blocks for three-dimensional octahedral unit cell designs of three heights. As-designed versus additively manufactured 718 lattices were compared, including structural dimensions, densities, defects, grain structures, crystallographic texture, and surface residual stresses. Manufactured lattice I diverged the most from its design, with both corner and center nodes merged into continuous columns. Manufactured lattice III came closest to its design, though the actual nodes and struts were still larger in cross section than in the design. Only modest grain coarsening and negligible chemical segregation were observed for all three designs, suggesting uniform thermo-physical and mechanical properties are possible. Significant tensile residual stresses were present at the surface of the manufactured blocks, with highest stresses in the building direction. Current findings indicate this concept could produce useful superalloy structures for high-temperature-capable static components requiring tailored density, and thermo-physical/mechanical properties.

super alloys↗
NASA NTRS1995

Extensional Flow Convecting a Reactant Undergoing a First Order Homogeneous Reaction and Diffusional Mass Transfer From a Sphere at Low to Intermediate Peclet and Damkohler Numbers

Forced convective diffusion-reaction is considered for viscous axisymmetric extensional convecting velocity in the neighborhood of a sphere. For Peclet numbers in the range 0.1 ≤ Pe ≤ 500 and for Damkohler numbers increasing with increasing Pe but in the overall range 0.02 ≤ Da ≤ 10, average and local Sherwood numbers have been computed. By introducing the eigenfunction expansion c(r,Θ) = Σ c n (r)P n (cosΘ) into the forced convective diffusion equation for the concentration of a chemical species undergoing a first order homogeneous reaction and by using properties of the Legendre functions P n (cosΘ), the variable coefficient PDE can be reduced to a system of N+1 second order ODEs for the radial functions C n (r), n=0,1,2, ... ,N. The adaptive grid algorithm of Pereyra and Lentini can be used to solve the corresponding 2(N+ 1) first order differential equations as a two-point boundary value problem on 1 ≤ r ≤ r •• . Convergence of the expansion for a specific value of N can thus be established and provides "spectral" behavior as well as the full concentration field c(r,Θ).

N Y Shah↗