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At least 325 records · Page 18

Modeling the Effect of Hydrogen on Tungsten Grain Boundary Properties

The objective of this study is to assess the effect of hydrogen isotopes on mechanical properties of tungsten that are relevant to first-wall applications in a tokamak. We focus on the effects of hydrogen segregation to grain boundaries and its impact on the grain boundary structure and properties. Through molecular dynamics and other atomistic techniques, we are able to make predictions of grain boundary structure and properties related to it, including grain boundary mobility and impurity diffusivity along the grain boundary. The grain boundary structure can change significantly and abruptly as conditions change, including due to heating and radiation damage. These grain boundary phase transformations can have a pronounced effect on mobility and impurity diffusivity, even when no hydrogen is present. How this behavior changes in the presence of hydrogen is not known. The phase may affect the level of hydrogen segregation and hydrogen storage, and the hydrogen may induce significant grain boundary structure changes. Our objective is to predict whether they do. The results inform thermomechanical modeling of tungsten at tokamak conditions.

36 MATERIALS SCIENCE↗

Monte Carlo Technique Used to Model the Degradation of Internal Spacecraft Surfaces by Atomic Oxygen

Atomic oxygen is one of the predominant constituents of Earth's upper atmosphere. It is created by the photodissociation of molecular oxygen (O2) into single O atoms by ultraviolet radiation. It is chemically very reactive because a single O atom readily combines with another O atom or with other atoms or molecules that can form a stable oxide. The effects of atomic oxygen on the external surfaces of spacecraft in low Earth orbit can have dire consequences for spacecraft life, and this is a well-known and much studied problem. Much less information is known about the effects of atomic oxygen on the internal surfaces of spacecraft. This degradation can occur when openings in components of the spacecraft exterior exist that allow the entry of atomic oxygen into regions that may not have direct atomic oxygen attack but rather scattered attack. Openings can exist because of spacecraft venting, microwave cavities, and apertures for Earth viewing, Sun sensors, or star trackers. The effects of atomic oxygen erosion of polymers interior to an aperture on a spacecraft were simulated at the NASA Glenn Research Center by using Monte Carlo computational techniques. A two-dimensional model was used to provide quantitative indications of the attenuation of atomic oxygen flux as a function of the distance into a parallel-walled cavity. The model allows the atomic oxygen arrival direction, the Maxwell Boltzman temperature, and the ram energy to be varied along with the interaction parameters of the degree of recombination upon impact with polymer or nonreactive surfaces, the initial reaction probability, the reaction probability dependence upon energy and angle of attack, degree of specularity of scattering of reactive and nonreactive surfaces, and the degree of thermal accommodation upon impact with reactive and non-reactive surfaces to be varied to allow the model to produce atomic oxygen erosion geometries that replicate actual experimental results from space. The degree of erosion of various interior locations was compared with the erosion that would occur external to the spacecraft. Results of one cavity model indicate that, at depths into a two-dimensional cavity that are equal to 10 cavity widths, the erosion on the walls of the cavity is less than that on the top surface by over 2 orders of magnitude. Wall erosion near the surface of a cavity depends on which wall is receiving direct atomic oxygen attack. However, deep in the cavity little difference is present. Testing of various cavity models such as these gives spacecraft designers an indication of the level of threat to sensitive interior surfaces for different geometries. Even though the Monte Carlo model is two-dimensional, it can be used to provide qualitative information about spacecraft openings that are three-dimensional by offering reasonable insight as to the nature of the attenuation of damage that occurs within a spacecraft in low Earth orbit. As shown, there is more erosion on the side seeing direct atomic oxygen attack until a depth of approximately 5 times the width of the opening, where the erosion is the same on both sides.

Banks, Bruce A.↗

Damage progression and failure of SiC/SiC composite tubes under hard-contact radial expansion

The response of silicon carbide (SiC) fiber-reinforced SiC matrix (SiC/SiC) composite cladding to mechanical interaction with fissile fuel is a knowledge gap that must be overcome to design and assess SiC-based cladding systems for advanced nuclear applications. This study developed the relevant mechanical testing capability and identified the failure behavior and the critical microstructural features and processing defects. Sections of SiC composite tube were subjected to a modified expansion-due-to-compression (EDC) test in an X-ray computed tomography microscope: a polyurethane plug pressed surrogate Al 2 O 3 into the inner walls of the SiC/SiC composite tubes to achieve hard contact. A pure EDC test with just a polyurethane plug was also performed as a reference. Through the use of displacement fields, digital volume correlation revealed inhomogeneous deformation fields in the tubes, even for pure EDC, which was related to the inherent defects in the structure. Deep learning–aided segmentation and systematic data analysis revealed that the presence of inhomogeneous deformation applied by the hard contact was exaggerated by the presence of inner surface imperfections left behind from the matrix densification process. In conclusion, the findings provide insights into the applications, highlighting the necessity for improvements in inner surface roughness and the incorporation of localized contacts in pellet–cladding mechanical interaction computational models.

Composites↗

Mesh and model requirements for capturing deep-stall aerodynamics in low-Mach-number flows

Here, this work presents a comprehensive computational fluid dynamics investigation of the effects of grid resolution and turbulence-model choice for capturing the unsteady three-dimensional aerodynamic performance of NACA 0012 and 0021 airfoils, with specific focus on the deep-stall regime. At high angles of attack (α), wind turbine blades routinely experience vortex-induced vibrations, which can cause significant structural damages. Accurate predictions of post-stall aerodynamics can identify the frequencies at which such vibrations maybe triggered. In this context, the NACA 0012 airfoil simulations are conducted at a chord-based Reynolds number, Re c =2×10 6 , with the k-ω Shear-Stress Transport Reynolds-Averaged Navier-Stokes (RANS) and Improved Delayed Detached Eddy Simulation (IDDES) hybrid RANS-Large Eddy Simulation turbulence models. The effect of mesh resolution both in the wall-normal and spanwise directions is investigated. Only the IDDES model with a minimum spanwise resolution of 24 cells per chord length correctly predicts the aerodynamic forces. Spectral analysis shows the peak primary shedding frequency at α=30°, which signifies the end of the stall region. In the post-stall regime, both lift and drag frequencies drop asymptotically with increasing α. The Strouhal number, based on normalised chord length, remains nearly constant in this region. Based on this study, NACA 0021 airfoil runs are performed with IDDES for Re c =2.7×10 5 and 2.0×10 6 on the finest wall-normal mesh and three spanwise grids. Simulations conducted on the finer spanwise grids demonstrate grid independence and show good agreement with experiments. The effect of varying Rec on the airfoil frequency statistics is investigated. Additionally, comparison studies are presented to investigate the impact of airfoil thickness on the frequency content at Re c =2.0×10 6 . The results from the study provide guidance on the choice of mesh resolution with the IDDES model to accurately capture aerodynamic quantities for complex industrial applications.

17 WIND ENERGY↗

Results of the AFRSI rewaterproofing systems screening test in the NASA/Ames Research Center (ARC) 2 x 2-foot transonic wind tunnel

An experimental investigation was conducted in the NASA/Ames Research Center 2x2-foot Transonic Wind Tunnel to evaluate two AFRSI rewaterproofing systems and to investigate films as a means of reducing blanket joint distortion. The wind tunnel wall slot configuration influenced on the flow field over the test panel was investigated; primarily using oil flow data, and resulted in a closed slot configuration to provide a satisfactory screening environment flow field for the test. Sixteen AFRSI test panels, configured to represent the test system or film, were subjected to this screening environment (a flow field of separated and reattached flow at a freestream Mach numnber of 0.65 and q = 650 or 900 psf). Each condition was held until damage to the test article was observed or 55 minutes if no damage was incurred. All objectives related to AFRSI rewaterproofing and to the use of films to stiffen the blanket fibers were achieved.

Marroquin, J.↗

Scanning structural mapping at the Life Science X-ray Scattering Beamline

This work describes the instrumentation and software for microbeam scattering and structural mapping at the Life Science X-ray Scattering (LiX) beamline at NSLS-II. Using a two-stage focusing scheme, an adjustable beam size between a few micrometres and a fraction of a millimetre is produced at the sample position. Scattering data at small and wide angles are collected simultaneously on multiple Pilatus detectors. A recent addition of an in-vacuum Pilatus 900k detector, with the detector modules arranged in a C-shaped configuration, has improved the azimuthal angle coverage in the wide-angle data. As an option, fluorescence data can be collected simultaneously. Fly scans have been implemented to minimize the time interval between scattering patterns and to avoid unnecessary radiation damage to the sample. For weakly scattering samples, an in-vacuum sample environment has been developed here to minimize background scattering. Data processing for these measurements is highly sample-specific. To establish a generalized data process workflow, first the data are reduced to reciprocal coordinates at the time of data collection. The users can then quantify features of their choosing from these intermediate data and construct structural maps. As examples, results from in-vacuum mapping of onion epidermal cell walls and 2D tomographic sectioning of an intact poplar stem are presented.

36 MATERIALS SCIENCE↗

Design, Manufacture and Test of Cryotank Components

On the composite cryotank technology development (CCTD) project, the Boeing Company built two cryotanks as a means of advancing technology and manufacturing readiness levels (TRL and MRL) and lowering the risk of fabricating full-scale fuel containment vessels.1 CCTD focused on upper stage extended duration applications where long term storage of propellants is required. The project involved the design, analysis, fabrication, and test of manufacturing demonstration units (MDU), a 2.4 m (precursor) and a 5.5 m composite cryotank. Key design features included one-piece wall construction to minimize overall weight (eliminating the need for a bellyband joint), 3-dimensionally (3D) reinforced y-joint material to alleviate stress concentrations at the tank to skirt interface and a purge-able fluted core skirt to carry high axial launch loads. The tanks were made with OoA curing pre-impregnated (prepreg) carbon/epoxy (C/E) slit-tape tow (STT) that contained thin micro-crack resistant plies in the tank wall center to impede permeation. The tanks were fabricated at Boeing's Seattle-based Advanced Development Center (ADC) using RAFP and multipiece break-down tooling. The tooling was designed and built by Janicki Industries (JI) at Sedro Woolley, Washington. Tank assemblage consisted of co-bonded dome covers, one-piece fluted core skirts and mechanical fastened cover/sump. Ultrasonic inspection was performed after every cure or bond and a structural health monitoring system (SHMS) was installed to identify potential impact damage events (in-process and/or during transportation). The tanks were low temperature tested at NASA's George C. Marshall Space Flight Center (MSFC) in Huntsville, Alabama. The testing, which consisted of a sequence of fill/drain pressure and thermal cycles using LH2, was successfully concluded in 2012 on the 2.4 m tank and in 2014 on the 5.5 m tank. Structural, thermal, and permeation performance data was obtained. 2 Critical design features and manufacturing advancements, which helped to validate 25% weight and 30% cost reduction projections, were matured. These advancements will help to guide future composite tank integration activities on next generation long duration aircraft and space launch vehicles. Because CCTD addressed innovative design features, heavy lift size scale-up, multipiece captured tooling, new generation automated material placement (AMP) equipment and OoA materials, this chapter should be of interest to educators, students and manufacturers of composite hardware and flight vehicles.

McCarville, Douglas A.↗

Modeling of thin, back-wall silicon solar cells

The performance of silicon solar cells with p-n junctions on the nonilluminated surface (i.e., upside-down or back-wall cells) was calculated. These structures consisted of a uniformly shaped p-type substrate layer, a p(+)-type field layer on the front (illuminated) surface, and a shallow, n-type junction on the back (nonilluminated) surface. A four-layer solar cell model was used to calculate efficiency, open-circuit voltage, and short-circuit current. The effect on performance of p-layer thickness and resistivity was determined. The diffusion length was varied to simulate the effect of radiation damage. The results show that peak initial efficiencies greater than 15 percent are possible for cell thicknesses or 100 micrometers or less. After 10 years of radiation damage in geosynchronous orbit, thin (25 to 50 micrometers thick) cells made from 10 to 100 ohm cm material show the smallest decrease (approximately 10 percent) in performance.

Baraona, C. R.↗

Cleanroom Contamination Identification Method Development

During fabrication, assembly, and testing of spacecraft and flight hardware it is vital to avoid contaminants that can cause degradation and could result in significant failure. Yet, there is no existing contamination monitoring method that provides the identity of airborne particles in a cleanroom facility. Knowing the particle identities, would allow scientists and engineers to determine the source of the contaminants and prevent setbacks before they occur or cause damage. Current cleanliness monitoring methods include airborne particle counters (APCs), fallout filters, and visual inspections. Particle counts from APCs are the primary metric used to define a cleanroom class and hence its level of cleanliness, but do not provide identification nor can they differentiate between large and small sizes of particles. In addition, using fallout filters is not a proactive, timely, or representative approach to cleanroom contamination monitoring because these samples are only retrieved after 30 days and are placed away from spacecraft processing to avoid interference with operations. In contrast, the forced air sampling method can collect a sample within an hour at any location required and provide results in less than a day. This system uses a cassette and filter sample medium to capture airborne particles which are then taken to a scanning electron microscope with energy dispersive spectroscopy (SEM/EDS) to identify and size the captured particles. Development of forced air sampling into an established laboratory capability will allow for fast sampling and routine identification of unknown contamination sources within the cleanroom. The test method development required market research for an air sampling cassette that increases sample collection efficiency and a filter with low enough background contamination to allow differentiation between a blank (control) and the collected sample. It was determined that a conductive black cassette and a polycarbonate filter were the best options. Conductive black cassettes, in comparison to the standard styrene, are manufactured using polypropylene filled with carbon. This makes the cassette conductive and minimizes the tendency of particles to stick to the wall of the cassette due to electrostatic force. In previous trials a mixed cellulose ester (MCE) filter was used to capture the contaminants, however the rougher surface of the filter contributed to entrapment of the particles within the filter structure and made it harder to identify the particles. In comparison, track etched polycarbonate filters have random cylindrical pores and a smooth surface which contributes to uniform sample distribution on the surface of the filter. Future work includes: testing the system using control samples to determine the efficiency and suitability of the medium, performing sample collection in various environments to establish ideal operating parameters and analyzing contaminant particles using SEM/EDS and assistant characterization techniques. Once fully developed, employing the forced air sampling method will help to prevent damage to spacecraft, avoid schedule delays, and allow for mission success.

Hernandez Melendez, Jailyn M.↗

Falling particle impact erosion testing for particle based CSP systems

CSP systems are currently being examined as an effective method for using solar energy to generate electricity. While much of this research has been conducted using molten salt as the heat transfer medium, in recent years, solid particles have been posited as a cheaper and more energy efficient heat transfer medium for use in CSP systems. Many researchers have looked into the effectiveness of solid particles as a heat transfer medium. However, a concern with the use of particles is that the falling particles will impact various components within the CSP system, such as the hopper, the heat exchanger, and the insulating material, resulting in surface erosion and damage. With repeated impacts from falling particles, material from the surfaces under impact is removed. The degree of erosion depends on whether the substrate is made of brittle or ductile material. Furthermore, factors such as particle shape, size, hardness, concentration, impact angle, particle velocity, and substrate hardness will contribute to the erosion process and determine how much and how quickly erosion will takes place in a system. Three types of erosion will be observed simultaneously: impact erosion from particle impact on receiver, particle storage, and heat exchanger walls, abrasion erosion from particle sliding motion along walls, and attrition erosion as the particles breakdown from particle-to-particle contact and particle-to-wall interactions. It is imperative to understand each of these erosions individually for a more comprehensive understanding and prediction of CSP system durability. In this work, we developed an experimental test setup capable of measuring impact erosion at conditions relevant to 1MW CSP plant. Subsequently, Impact erosion measurements were performed for three different candidate substrate materials at low particle velocities using HSP 40/70 ceramic particles.

14 SOLAR ENERGY↗

Micro-Vibration Measurements on Thermally Loaded Multi-Layer Insulation Samples in Vacuum

Some scientific missions require to an extreme extent the absence of any on-board microvibration. Recent projects dedicated to measuring the Earth's gravity field and modeling the geoid with extremely high accuracy are examples. Their missions demand for extremely low micro-vibration environment on orbit for: (1) Not disturbing the measurement of earth gravity effects with the installed gradiometer or (2) Even not damaging the very high sensitive instruments. Based on evidence from ongoing missions multi-layer insulation (MLI) type thermal control blankets have been identified as a structural element of spacecrafts which might deform under temperature variations being caused by varying solar irradiation in orbit. Any such deformation exerts tiny forces which may cause small reactions resulting in micro-vibrations, in particular by exciting the spacecraft eigenmodes. The principle of the test set-up for the micro-vibration test was as follows. A real side wall panel of the spacecraft (size about 0.25 m2) was low-frequency suspended in a thermal vacuum chamber. On the one side of this panel, the MLI samples were fixed by using the standard methods. In front of the MLI, an IR-rig was installed which provided actively controlled IR-radiation power of about 6 kW/m2 in order to heat the MLI surface. The cooling was passive using the shroud temperature at a chamber pressure <1E-5mbar. The resulting micro-vibrations due to MLI motion in the heating and the cooling phase were measured via seismic accelerometers which were rigidly mounted to the panel. Video recording was used to correlate micro-vibration events to any visual MLI motion. Different MLI sample types were subjected to various thermal cycles in a temperature range between -60 C to +80 C. In this paper, the experience on these micro-vibration measurements will be presented and the conclusions for future applications will be discussed

Deutsch, Georg↗

Shields for Enhanced Protection Against High-Speed Debris

A report describes improvements over the conventional Whipple shield (two thin, spaced aluminum walls) for protecting spacecraft against high-speed impacts of orbiting debris. The debris in question arise mainly from breakup of older spacecraft. The improved shields include exterior bumper layers composed of hybrid fabrics woven from combinations of ceramic fibers and high-density metallic wires or, alternatively, completely metallic outer layers composed of high-strength steel or copper wires. These shields are designed to be light in weight, yet capable of protecting against orbital debris with mass densities up to about 9 g/cm3, without generating damaging secondary debris particles. As yet another design option, improved shields can include sparsely distributed wires made of shape-memory metals that can be thermally activated from compact storage containers to form shields of predetermined shape upon arrival in orbit. The improved shields could also be used to augment shields installed previously.

Christiansen, Eric L.↗

Shields for Enhanced Protection Against High-Speed Debris

A report describes improvements over the conventional Whipple shield (two thin, spaced aluminum walls) for protecting spacecraft against high-speed impacts of orbiting debris. The debris in question arises mainly from breakup of older spacecraft. The improved shields include exterior "bumper" layers composed of hybrid fabrics woven from combinations of ceramic fibers and high-density metallic wires or, alternatively, completely metallic outer layers composed of high-strength steel or copper wires. These shields are designed to be light in weight, yet capable of protecting against orbital debris with mass densities up to about 9 g/cubic cm, without generating damaging secondary debris particles. As yet another design option, improved shields can include sparsely distributed wires made of shape memory metals that can be thermally activated from compact storage containers to form shields of predetermined shape upon arrival in orbit. The improved shields could also be used to augment shields installed previously.

Christiansen, Eric L.↗

Concorde noise-induced building vibrations: International Airport Dulles

A series of studies were conducted to assess the noise-induced building vibrations associated with Concorde operations. The vibration levels of windows, walls, and floors were measured along with the associated noise levels of Concorde, subsonic aircraft and some nonaircraft events. Test sites included Sully Plantation which is adjacent to Dulles International Airport and three residential homes located in Montgomery County, Maryland. The measured vibration response levels due to Concorde operations were found to be: (1) higher than the levels due to other aircraft, (2) less than the levels due to certain household events which involve direct impulsive loading such as door and window closing, (3) less than criteria levels for building damage, and (4) comparable to levels which are perceptible to people.

Mayes, W. H.↗

Radiation Protection Using Carbon Nanotube Derivatives

BHA and BHT are well-known food preservatives that are excellent radical scavengers. These compounds, attached to single-walled carbon nanotubes (SWNTs), could serve as excellent radical traps. The amino-BHT groups can be associated with SWNTs that have carbolyxic acid groups via acid-base association or via covalent association. The material can be used as a means of radiation protection or cellular stress mitigation via a sequence of quenching radical species using nano-engineered scaffolds of SWNTs and their derivatives. It works by reducing the number of free radicals within or nearby a cell, tissue, organ, or living organism. This reduces the risk of damage to DNA and other cellular components that can lead to chronic and/or acute pathologies, including (but not limited to) cancer, cardiovascular disease, immuno-suppression, and disorders of the central nervous system. These derivatives can show an unusually high scavenging ability, which could prove efficacious in protecting living systems from radical-induced decay. This technique could be used to protect healthy cells in a living biological system from the effects of radiation therapy. It could also be used as a prophylactic or antidote for radiation exposure due to accidental, terrorist, or wartime use of radiation- containing weapons; high-altitude or space travel (where radiation exposure is generally higher than desired); or in any scenario where exposure to radiation is expected or anticipated. This invention s ultimate use will be dependent on the utility in an overall biological system where many levels of toxicity have to be evaluated. This can only be assessed at a later stage. In vitro toxicity will first be assessed, followed by in vivo non-mammalian screening in zebra fish for toxicity and therapeutic efficacy.

Conyers, Jodie L., Jr.↗

Design Considerations for Lightweight Space Radiators Based on Fabrication and Test Experience With a Carbon-Carbon Composite Prototype Heat Pipe

This report discusses the design implications for spacecraft radiators made possible by the successful fabrication and proof-of-concept testing of a graphite-fiber-carbon-matrix composite (i.e., carbon-carbon (C-C)) heat pipe. The prototype heat pipe, or space radiator element, consists of a C-C composite shell with integrally woven fins. It has a thin-walled furnace-brazed metallic (Nb-1%Zr) liner with end caps for containment of the potassium working fluid. A short extension of this liner, at increased wall thickness beyond the C-C shell, forms the heat pipe evaporator section which is in thermal contact with the radiator fluid that needs to be cooled. From geometric and thermal transport properties of the C-C composite heat pipe tested, a specific radiator mass of 1.45 kg/sq m can be derived. This is less than one-fourth the specific mass of present day satellite radiators. The report also discusses the advantage of segmented space radiator designs utilizing heat pipe elements, or segments, in their survivability to micrometeoroid damage. This survivability is further raised by the use of condenser sections with attached fins, which also improve the radiation heat transfer rate. Since the problem of heat radiation from a fin does not lend itself to a closed analytical solution, a derivation of the governing differential equation and boundary conditions is given in appendix A, along with solutions for rectangular and parabolic fin profile geometries obtained by use of a finite difference computer code written by the author.

Juhasz, Albert J.↗

Acoustic fatigue and sound transmission characteristics of a ram composite panel design

An experimental study to determine the acoustic fatigue characteristics of a flat multi-layered structural panel is described. The test panel represented a proposed design for the outer skin of a research application module to be housed within the space shuttle orbiter vehicle. The test specimen was mounted in one wall of the Wyle 100,000 cu ft reverberation room and exposed to a broadband acoustic environment having an overall level of 145 db. The test panel was exposed to nine separate applications of the acoustic environment, each application consisting of 250 seconds duration. Upon completion of the ninth test run, the specimen was exposed to a simulated micrometeoroid impact near the panel center. One additional test run of 250 seconds duration was then performed to complete the overall simulation of 50 flight missions. The experimental results show that no significant fatigue damage occurred until the test specimen was exposed to a simulated micrometeoroid impact. The intermediate foam layer forming the core of the test specimen suffered considerable damage due to this impact, causing a marked variation in the dynamic characteristics of the overall test panel. During the final application of the acoustic environment, the strain and acceleration response spectra showed considerable variation from those spectra obtained prior to impact of the panel. Fatigue damage from acoustic loading however, was limited to partial de-bonding around the edges of the composite panel.

Cockburn, J. A.↗

Design Considerations for Lightweight Space Radiators Based on Fabrication and Test Experience with a Carbon-Carbon Composite Prototype Heat Pipe

This report discusses the design implications for spacecraft radiators made possible by the successful fabrication and Proof-of-concept testing of a graphite-fiber-carbon-matrix composite (i.e., carbon-carbon (C-C)) heat pipe. The proto-type heat pipe, or space radiator element, consists of a C-C composite shell with integrally woven fins. It has a thin-walled furnace-brazed metallic (Nb-1%Zr) liner with end caps for containment of the potassium working fluid. A short extension of this liner, at increased wall thickness beyond the C-C shell, forms the heat pipe evaporator section which is in thermal contact with the radiator fluid that needs to be cooled. From geometric and thermal transport properties of the C-C composite heat pipe tested, a specific radiator mass of 1.45 kg/m2 can be derived. This is less than one-fourth the specific mass of present day satellite radiators. The report also discusses the advantage of segmented space radiator designs utilizing heat pipe elements, or segments, in their survivability to micro-meteoroid damage. This survivability is further raised by the use of condenser sections with attached fins, which also improve the radiation heat transfer rate. Since the problem of heat radiation from a fin does not lend itself to a closed analytical solution, a derivation of the governing differential equation and boundary conditions is given in appendix A, along with solutions for rectangular and parabolic fin profile geometries obtained by use of a finite difference computer code written by the author.

Juhasz, Albert J.↗