Search NASASearch

SEARCH · Search NASA

Results for “Surface area to volume ratio”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

1,315 records · Page 33

Dust Composition of Comet 81P/Wild 2 From JWST Spectroscopy Compared to Stardust’s Fine-Grained Materials and Gems-Rich IDPS

The Stardust Mission returned samples from the coma of Jupiter Family comet 81P/Wild 2 for detailed laboratory analyses. Here we present and discuss the best-fit thermal dust model for the coma dust of comet 81P/Wild 2 as observed by JWST. Comet 81P was observed through JWST GO 018xx, using NIRSpec (2.9–5.3µm, λ/∆λ≈1000) and MRS IFU (4.9–28.1µm, λ/∆λ≈3000) on 2023-03-20 UT and 2023-03-24 UT, respectively, at a heliocentric distance of 1.85 au and JWST distance of 1.43 au (phase angle of 32 degrees). The dust coma of 81P as revealed by JWST offers a salient compliment to laboratory studies of Stardust samples that typically are bigger than 2 µm and up to 60 µm in size.

D H Wooden

Automatic Dependent Surveillance-Broadcast (ADS-B) In-Trail Procedures (ITP)

Aircraft in oceanic and remote non-radar airspace frequently fly for extended periods of time in the same direction, at the same time, along similar flight paths as other aircraft. Since there is no radar surveillance, controllers use procedural separation to ensure that aircraft remain separated. Procedural separation distances are typically larger than radar separation distances and as a result aircraft operating in oceanic airspace are sometimes held at non-optimal flight levels due to conflicting traffic at intervening flight levels. Automatic Dependent Surveillance-Broadcast (ADS-B) In-Trail Procedures (ITP) were developed to enable flight level change maneuvers that would otherwise not be possible with current procedural separation standards. Aircraft operators choosing to equip with an ADS-B transceiver and an appropriate onboard decision support system would be able to take advantage of these procedures when operating in proximity to aircraft equipped with a suitable ADS-B transmitter (“ADS-B Out”). The ability to perform in-trail maneuvers to achieve more time at optimum altitudes could result in more efficient and predictable flight profiles thereby saving fuel and in some cases allowing operators to make beneficial operational decisions. NASA first began developing ADS-B ITP in 2003 as a result of a desire to develop methodologies, concepts, and procedures to reduce separation requirements for future air transportation systems using airborne ADS-B. The objectives were to provide insight into the details necessary to reduce separation requirements for the future and to develop applications that could provide incentives for operators to voluntarily equip with transformational technologies. From 2003 to 2008, NASA conducted research that supported the development of ITP including batch simulations, human-in-the-loop experiments and avionics and separation standards development. This research showed enough maturity and benefit that in 2008, the FAA Surveillance and Broadcast Services (SBS) program adopted ADS-B ITP as one of their three key, near-term applications to make use of ADS-B-In. The FAA developed an agreement with NASA to transition the technology and established an FAA project for the purpose of performing an operational trial of ADS-B ITP in revenue service in the summer of 2011. The objectives of the project are to a) validate the operational performance and economic benefits of ITP; and b) develop and validate ADS-B ITP Minimum Operational Performance Specifications (MOPS) material. As a part of this project, the FAA established agreements with United Airlines and Honeywell. The agreements include the work necessary for the development, certification and installation of onboard systems for twelve United Airlines 747-400s. ITP system development is nearly complete and certification activities are underway. The FAA project has also been working with Oakland Oceanic Control Center (ZOA) and the FAA’s Oceanic and Offshore Operations Office to develop controller procedures and safety analyses that are required to support the flight trial. The FAA has also been working on the development of an ITP Operational Specification that should be approved this April. The presentation will cover some of the key aspects of the development, challenges, and integration required to successfully transition ADS-B ITP from a concept in 2003 to flight trials in revenue service in 2011.

Kenneth M Jones

From Exploration Flight Test-1 to Artemis II--A NASA Langley's Orion Aerosciences Overview

The Orion Aerosciences program at NASA Langley has played a central role in advancing the aerodynamic and aeroheating prediction capabilities required for the Orion crew vehicle’s return from deep space. This presentation provides a technical overview of aerosciences contributions spanning Exploration Flight Test-1 (EFT-1), Artemis I, and the ongoing post-flight analysis of Artemis II. EFT-1 provided the first high-energy entry dataset for Orion, enabling critical validation of aerodynamic force and moment predictions, static and dynamic stability characteristics, and aeroheating environments at relevant flight Mach and Reynolds numbers. Flight-derived pressure data were used to refine the Flush Air Data System (FADS) methodology for atmospheric density reconstruction and to improve Best Estimated Trajectory (BET) solutions. The EFT-1 data also offered key insights into heat shield performance, including char layer recession, in-depth thermal response, and material retention behavior under flight conditions, informing updates to both thermal response models and uncertainty quantification practices. Building on EFT-1, Artemis I extended the database to true lunar-return conditions. Observations of heat shield performance, including localized char loss, bondline response, and recession variability, provided an unprecedented opportunity to reassess Thermal Protection System (TPS) and aeroheating modeling assumptions. Aerodynamic reconstruction efforts incorporated improved FADS calibration, enhanced atmospheric modeling, and refined force and moment databases to reduce trajectory and load uncertainties. Aeroheating comparisons between pre-flight predictions and flight data enabled targeted model updates, particularly in transitional flow environments and wake heating regions. For Artemis II, these lessons were systematically incorporated into the pre-flight prediction process. Updates included refined aerodynamic databases anchored to flight-validated corrections, improved density estimation and BET methodologies using enhanced database interpolation algorithm and FADS modeling, and revised aeroheating design environments informed by Artemis I material response observations. By the time of the workshop, Artemis II post-flight analysis will be underway, and preliminary findings will be presented where available, including early comparisons of aerodynamic reconstruction, atmospheric density estimation, and thermal protection system performance relative to updated predictions. Collectively, this body of work is a testament to the dedicated and multidisciplinary team whose sustained efforts have contributed to the program’s success and to the progressive maturation of Orion aerosciences modeling through numerical modeling, ground and flight data assimilation. The integrated advancement of aerodynamics, trajectory reconstruction, FADS-based density estimation, and aeroheating analysis has reduced predictive uncertainty and strengthened confidence for future crewed lunar and deep-space missions.

Orion

Battery Charge Equalizer with Transformer Array

High-power batteries generally consist of a series connection of many cells or cell banks. In order to maintain high performance over battery life, it is desirable to keep the state of charge of all the cell banks equal. A method provides individual charging for battery cells in a large, high-voltage battery array with a minimum number of transformers while maintaining reasonable efficiency. This is designed to augment a simple highcurrent charger that supplies the main charge energy. The innovation will form part of a larger battery charge system. It consists of a transformer array connected to the battery array through rectification and filtering circuits. The transformer array is connected to a drive circuit and a timing and control circuit that allow individual battery cells or cell banks to be charged. The timing circuit and control circuit connect to a charge controller that uses battery instrumentation to determine which battery bank to charge. It is important to note that the innovation can charge an individual cell bank at the same time that the main battery charger is charging the high-voltage battery. The fact that the battery cell banks are at a non-zero voltage, and that they are all at similar voltages, can be used to allow charging of individual cell banks. A set of transformers can be connected with secondary windings in series to make weighted sums of the voltages on the primaries.

Davies, Francis

The Canadian Galactic Emission Mapper: A New 8-10 GHz Telescope To Map the Polarization of the Northern Sky

The Canadian Galactic Emission Mapper (CGEM) radio telescope is mapping polarized Galactic foregrounds from 8-10 GHz to aid in the search for B-modes in the Cosmic Microwave Background (CMB). Here we describe the design and early on-sky characterization of the CGEM optics. CGEM employs an on-axis, hat feed optical design that is a body of revolution (BOR) and exhibits excellent intrinsic polarization purity. We describe how we have optimized the optics with a novel framework that can directly minimize simulated intensity to polarized intensity (T → P) leakage in angular power spectrum space. The optimized optics exhibit simulated T → P leakage that is orders of magnitude below anticipated CℓBB for r = 10 −3 when scaled from 8-10 GHz to the CMB observing window near 95 GHz. We go on to describe the mechanical design of the optics, including a novel secondary mirror support made from Astroquartz composite that has low loss, low dielectric constant, and preserves the BOR symmetry of the optics. We then showcase the early on-sky performance of the optics with observations of the Sun and satellites, which probe the beam to 40dB down from the peak. A beam model for the deployed CGEM optics based on electromagnetic simulations is in excellent agreement with the on-sky data.

cosmic microwave background

High Velocity Oxygen Fuel Spraying of Erosion and Wear Resistant Coatings on Jet Engine Parts

High velocity oxygen fuel (HVOF) spraying is the most recent development in the field of thermal spraying. The importance of this technique for the repair and new part manufacturing of jet engine parts is rapidly increasing. The HVOF uses a supersonic oxygen-fuel flame to heat and accelerate the powder particles that form the coating. The high particle velocity results in a high density and a low porosity, a high bond strength, and a high macro and micro hardness of the coating. The high quality of the HVOF coatings makes it possible to use these coatings on high loaded, rotating parts in jet engines. This paper will highlight the use of HVOF processes to apply erosion resistant cermet coatings to high pressure compressor blades. These blades are exposed to severe erosion. Next to the D-gun process, HVOF spraying is the only nonproprietary technique that can be used to apply these high performance coatings. Also the use of the HVOF process to apply wear resistant coatings and superalloys to jet engine parts will be discussed. The difference between HVOF coatings and plasma sprayed coatings will be highlighted. During HVOF spraying, the parts are exposed to a high heat flow. Solutions to avoid overheating, especially of titanium parts, will be presented.

A T J Verbeek

Use of Global Positioning System Measurements to Determine Geocentric Coordinates and Variations in Earth Orientation

Geocentric tracking station coordinates and short-period Earth-orientation variations can be measured with Global Positioning System (GPS) measurements. Unless calibrated, geocentric coordinate errors and changes in Earth orientation can lead to significant deep-space tracking errors. Ground-based GPS estimates of daily and subdaily changes in Earth orientation presently show centimeter-level precision. Comparison between GPS-estimated Earth-rotation variations, which are the differences between Universal Time 1 and Universal Coordinated Time (UT1-UTC), and those calculated from ocean tide models suggests that observed subdaily variations in Earth rotation are dominated by oceanic tidal effects. Preliminary GPS estimates for the geocenter location (from a 3-week experiment) agree with independent satellite laser-ranging estimates to better than 10 cm. Covariance analysis predicts that temporal resolution of GPS estimates for Earth orientation and geocenter improves significantly when data collected from low Earth-orbiting satellites as well as from ground sites are combined. The low Earth GPS tracking data enhance the accuracy and resolution for measuring high-frequency global geodynamical signals over time scales of less than 1 day.

R P Malla

Low Temperature Life-Cycle Testing of a Lithium-Ion Battery for Low-Earth-Orbiting Spacecraft

A flight-qualified, lithium-ion (Li-ion) battery developed for the Mars Surveyor Program 2001 Landeris undergoing life-testing at low temperature under a low-Earth-orbit (LEO) profile to assess its capability to provide long term energy storage for aerospace missions. NASA has embarked upon an ambitious course to return humans to the moon by 2015-2020 in preparation for robotic and human exploration of Mars and robotic exploration of the moons of outer planets. Li-ion batteries are excellent candidates to provide power and energy storage for multiple aspects of these missions due to their high specific energy, high energy density, and excellent low temperature performance. Laboratory testing of Li-ion technology is necessary in order to assess lifetime, characterize multi-cell battery-level performance under aerospace conditions, and to gauge safety aspects of the technology. Life-cycle testing provides an opportunity to examine battery-level performance and the dynamics of individual cells in the stack over the entire life of the battery. Data generated through this testing will be critical to establish confidence in the technology for its widespread use in manned and unmanned missions.

Reid, Concha

Comparison Testing and Analysis of Braided Cords Constructed from Varying Kevlar Fiber Types

Material advancements move quickly, resulting in material obsolescence issues that require parachute engineers to understand how the material properties change with a material upgrade and if there are expected impacts to overall performance of the parachute. The objective of this test was to measure the elongation of different Kevlar types at both the cord-level and at the yarn-level to determine if a change in Kevlar type impacted parachute performance. An Instron machine with a pin-to-pin connection was used to measure load and displacement time histories for various Kevlar cord and yarn samples in a cyclic load profile. Twenty-four cord samples and twenty-four yarn samples were tested; each sample was cycled five times to the relevant maximum load (either to breaking strength or to the 30% peak load expected in flight). In addition to the Instron measurements, a photogrammetric technique was also implemented to measure elongation of the samples using button targets. Modulus test data shows approximately a 10% difference in elongation between Kevlar 29AP and Kevlar 129, and another 10% difference in elongation between Kevlar 29 and Kevlar 29AP. Although this is a measurable change in modulus, the effect on parachute performance is likely small, particularly for non-primary structure elements of relatively small length. This test campaign provides an example of the type of testing and analysis that parachute engineers could do to understand how material upgrades affect performance of their systems.

Modulus Testing

Legacy of the Asteroid Redirect Robotic Mission (ARRM)

NASA’s proposed Asteroid Redirect Robotic Mission (ARRM) began with the recognition in a 2010 NASA study that emerging high-power solar electric propulsion technology could be used to rendezvous with, capture, and return an entire, very small (~10,000 kg), near Earth asteroids to the International Space Station. A 2011 workshop by the Keck Institute for Space Studies (KISS) extended the earlier NASA study to asteroid masses of order 500,000 kg by returning them to cislunar space. Subsequent detailed NASA studies in 2013-2014 confirmed the feasibility of this concept. This led to the establishment of the Asteroid Redirect Mission program that consisted of a robotic mission to return multiple tons of asteroid material to cislunar space and a crewed mission to rendezvous with the robotic vehicle, perform two extra vehicular activities (EVAs), collect samples of the asteroid material, and return this material to Earth. Implementation of ARRM got midway through Phase B before being cancelled in April 2017. Although ARRM was cancelled, it has left a near-term legacy of positive impacts to the human spaceflight community, the planetary defense community, the deep space science community, and asteroid mining interests.

John R Brophy

NASA Aerospace Flight Battery Systems Program: An Update

The major objective of the NASA Aerospace Flight Battery Systems Program is to provide NASA with the policy and posture to increase and ensure the safety, performance, and reliability of batteries for space power systems. The program was initiated in 1985 to address battery problems experienced by NASA and other space battery users over the previous ten years. The original program plan was approved in May 1986 and modified in 1990 to reflect changes in the agency's approach to battery related problems that are affecting flight programs. The NASA Battery Workshop is supported by the NASA Aerospace Flight Battery Systems Program. The main objective of the discussions is to aid in defining the direction which the agency should head with respect to aerospace battery issues. Presently, primary attention in the Battery Program is being devoted to issues revolving around the future availability of nickel-cadmium batteries as a result of the proposed OSHA standards with respect to allowable cadmium levels in the workplace. The decision of whether or not to pursue the development of an advanced nickel-cadmium cell design and the qualification of vendors to produce cells for flight programs hinges on the impact of the OSHA ruling. As part of a unified Battery Program, the evaluation of a nickel-hydrogen cell design options and primary cell issues are also being pursued to provide high performance NASA Standards and space qualified state-of-the-art cells. The resolution of issues is being addressed with the full participation of the aerospace battery community.

Manzo, Michelle A.

Comparison Testing and Analysis of Braided Cords Constructed from Varying Kevlar Fiber Types

Material advancements move quickly, resulting in material obsolescence issues that require parachute engineers to understand how the material properties change with a material upgrade and if there are expected impacts to overall performance of the parachute. The objective of this test was to measure the elongation of different Kevlar types at both the cord-level and at the yarn-level to determine if a change in Kevlar type impacted parachute performance. An Instron machine with a pin-to-pin connection was used to measure load and displacement time histories for various Kevlar cord and yarn samples in a cyclic load profile. Twenty-four cord samples and twenty-four yarn samples were tested; each sample was cycled five times to the relevant maximum load (either to breaking strength or to the 30% peak load expected in flight). In addition to the Instron measurements, a photogrammetric technique was also implemented to measure elongation of the samples using button targets. Modulus test data shows approximately a 10% difference in elongation between Kevlar 29AP and Kevlar 129, and another 10% difference in elongation between Kevlar 29 and Kevlar 29AP. Although this is a measurable change in modulus, the effect on parachute performance is likely small, particularly for non-primary structure elements of relatively small length. This test campaign provides an example of the type of testing and analysis that parachute engineers could do to understand how material upgrades affect performance of their systems.

Parachutes

Motor Configuration Selection for A New Technical Challenge to Develop A 5 MW Cryogenic Motor and Drive

Due to aviation’s appreciable and growing share of humanity’s impact on our environment and estimates that CO2emissions only account for 34% of aviation’s total effective radiative forcing [1], there is a need to reach beyond climate goals that focus only on CO2emissions, such as the US Aviation Climate Action Plan’s [3] goal to reach net-zero carbon emissions by 2050.There is motivation to develop technology that pushes toward future large transport aircraft with net zero climate impact that are highly electrified (i.e., have higher power electrical propulsion system components). This paper describes a new, 6-year technical challenge to address this need by developing a 5 MW superconducting motor and cryogenic drive. Section 1 will detail the motivation for this work. Section 2 will describe the technical challenge and the selected specifications for the motor. Section 3 will present the results of a motor configuration trade study and the down selection of one configuration to develop a detailed design for. The technical challenge focuses on the design of a5 MW superconducting motor and cryogenic drive and demonstration of it at a 2+ MW scale to achieve TRL 3. Both fully superconducting (superconducting stator and rotor) and fully cryogenic (superconducting rotor and cryogenic stator) machine configurations will be explored. An emphasis will be placed on addressing the key tall poles for high power superconducting machines. Further details will be included in the full paper. The requirements and goals of the motor will be detailed. The rated speed (2,000 to 3,000 rpm) is defined to be appropriate for directly driving multi-MW fans or propellers. A range of rated speed is permitted because the motor is not designed for a specific aircraft and to provide design flexibility if AC losses in the stator winding are found to be a significant constraint (i.e., a lower speed can be selected to reduce electrical frequency). Relatively conservative requirements for efficiency (99%) and specific power (20 kW/kg) are defined, because TRL advancement and pushing toward flight readiness is emphasized over performance optimization. However, more aggressive efficiency and specific power goals are specified (99.9% and 40 kW/kg). The 3rd section will present the results of a motor configuration trade study. The study started with a qualitative assessment of sixteen motor configurations based on geometric, mechanical, thermal, and electromagnetic criteria. This assessment has been completed with three evaluators scoring all nine criteria. A configuration down select was made by prioritizing the sixteen configurations into four tiers based on each configuration’s total score and consideration of manufacturability, complexity, and support hardware (e.g., rotary vacuum seals, bearings). Configurations in priority A and B will be further evaluated through quantitative assessments, whereas those in priority C will only be further evaluated if time permits and priority D will not be further evaluated. Eight of the sixteen configurations were down selected for quantitative assessment, which will include analytical calculations and low-to moderate-fidelity finite element analysis to produce a preliminary Pareto front of efficiency versus specific power for each configuration. This assessment emphasizes the calculation of AC losses in the stator winding and an exploration of thermal management approaches to remove that heat and maintain cryogenic temperature. The final paper will include a description of each motor configuration that was considered. The quantitative assessments are underway, and an assessment of one configuration is complete for multiple stator conductor options. The remaining assessments are scheduled to be completed by late March so that the final down select to one configuration can be included in this paper.

Net Zero

Atacama Cosmology Telescope A Measurement of Galaxy Cluster Temperatures Through Relativistic Corrections to the Thermal Sunyaev-Zeldovich Effect

The high electron temperature in galaxy clusters (> 1 keV or > 107 K) leads to corrections at the level of a few percent in their thermal Sunyaev-Zeldovich effect signatures. Both the size and frequency dependence of these corrections, which are known as relativistic temperature corrections, depend upon the temperature of the objects. In this work we exploit this effect to measure the average temperature of a stack of Compton-y selected clusters. Specifically, we apply the “spectroscopic method” and search for the temperature that best fits the clusters’ signal measured at frequencies from 30 to 545 GHz by the Atacama Cosmology Telescope and Planck satellite. We measure the average temperature of clusters detected in the Atacama Cosmology Telescope maps to be 8.5 2.4 keV, with an additional systematic error of comparable amplitude dominated by passband uncertainty. Upcoming surveys, such as the Simons Observatory and CMB-S4, have the potential to dramatically improve upon these measurements and thereby enable precision studies of cluster temperatures with millimeter observations. The key challenge for future observations will be mitigating instrumental systematic effects, which already limit this analysis.

William R Coulton

Improved Treatment of Global Positioning System Force Parameters in Precise Orbit Determination Applications

Data collected from a worldwide 1992 experiment have been processed at JPL to determine precise orbits for the satellites of the Global Positioning System (GPS). A filtering technique has been tested to improve modeling of solar-radiation pressure force parameters for GPS satellites. The new approach improves orbit quality for eclipsing satellites by a factor of two, with typical results in the 25- to 50-cm range. The resultant GPS-based estimates for geocentric coordinates of the tracking sites, which include the three DSN sites, are accurate to 2 to 8 cm, roughly equivalent to 3 to 10 nrad of angular measure.

Y Vigue

Enabling Mission Flexibility to Battery Driven Deep Space Endeavors With Generalized Battery-Health-Monitoring Using Physics-Based and Data-Driven Reduced-Order Models

The needs and requirements for an electrochemical energy storage for deep space exploration is well explored. It is often understood that different mission sites and environmental conditions require different battery chemistries or technologies. Additionally, various engineering solutions are deployed to overcome specific chemical challenges. One often overlooked need is the “health” monitoring of an electrochemical storage system. The term generalized health monitoring, as envisioned in this work, refers to the monitoring of various aspects such as electrode health, electrolyte health, reaction pathway health, cooling system health, sensor health, and BMS health [1]. Generalized health monitoring allows mission leads, engineers, and scientists to incorporate flexibility in mission designs, make on-the-fly mission changes, and extend the duration of science missions. Moreover, it enables automation and data-driven decision-making without compromising safety and performance. Recently, our group developed a hierarchy of thermal reduced-order models (TROM) by combining a physics-based modeling approach and data-driven model reduction techniques applied to flight data [2]. The resulting TROMs were found to be not only accurate but also identifiable from the flight data. Consequently, the coefficient of variance of the model parameters is small over the course of hundreds of flights, allowing for monitoring the parameter evolution trajectories as the battery ages and degrades. These parameters constitute the metrics of the generalized health of a battery. Monitoring their evolution allows such models to be used for anomaly detection and prognostics, improving early detection of abnormal behavior and thus enabling timely maintenance, longer battery life, and enhanced battery safety. For this presentation, the practicality of the thermal model will be validated on a pack of 14cells under various topology configurations such as 1S14P, 2P7S, 7S2P, and 1P14S. It is well known that manufacturing and non-uniform aging lead to variability in the performance of a cell, which is exacerbated by cell balancing during active load. Additionally, in extreme scenarios, the paramount objective is to complete the mission, regardless of the stresses on the battery. Topology-induced balancing issues further stress the battery. The goal of this study is to determine if the noise (identifiability) in the reduced-order thermal model parameters is sensitive to topology, cell spacing, cooling strategy, and manufacturing or age variability. The variability in cells is considered by assuming a multimodal distribution for microscopic parameters of a cell (such as porosity, tortuosity, reaction kinetics, volumetric thermal conductivity, and volumetric heat capacity). The compounded effect of manufacturing variability, topological selection, cooling strategies, and cell balancing ages each cell in a battery differently. The study aims to clarify whether the challenge in extracting maximum information depends on the minimum number of sensors or models used for data extraction.

Automation

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