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At least 415 records · Page 23

Microsample Characterization of Coatings for Grcop-84 for High Heat Flux Applications

A multidisciplinary Johns Hopkins University-NASA Glenn team is undertaking a collaborative research program to elucidate and model the thermal stability and mechanical integrity of candidate coatings for GRCop-84. GRCop-84 is a high conductivity, high strength copper alloy that was recently developed at NASA Glenn for use in high temperature, high heat flux applications. With potential applications in rocket motor combustion chamber liners, nozzle ramps and other actively cooled structures, this new material offers great potential for decreasing weight and increasing reliability of third generation reusable launch vehicles. Current emphasis has turned toward the development of environmentally resistant and thermal barrier coatings for this alloy. Metallic coatings such as NiCrAlY and Cu-8-30%Cr have shown promise in: prohibiting blanching, reducing dog-house failures, increasing operating temperatures and decreasing cooling requirements. The focus of this research program is to develop a fundamental understanding of the substrate-coating interactions that occur during thermal cycling (inter-diffusion, viscoplasticity, morphological evolution, crack formation, etc.) and to derive a science-based protocol for future coating selection, optimization and reliability assurance. The microsample tensile testing approach adopted for this study allows us to characterize small-scale and highly scale-specific coatings and properties in a way not possible by conventional means. In addition to providing much needed design data, the integration of microsample testing with detailed microstructural observations provides a mechanistic foundation for coating optimization and life prediction modeling.

Hemker, Kevin↗

Analysis of Launch Vehicle Liftoff Debris: Historical Perspective from Space Shuttle and Application to Artemis I

Human exploration-class launch vehicles are inherently prone to debris due to the extreme environments generated during pre-launch operations, liftoff, and flight. The use of cryogenic propellants often requires thermal protection system (TPS) coatings, typically foam, to maintain the propellant conditions in the tank and prevent an accumulation ice on the external surface of the vehicle. Some ice growth is to be expected at umbilical interfaces, vents, flanges, or brackets where it is difficult to apply TPS. This ice may come loose at any time due to wind on the launch pad, structural vibration and acoustics after rocket ignition, or aerodynamic forces during flight. This phenomena is especially apparent on vehicles with no TPS, such as the Saturn V rockets used in the Apollo Program, see Figure 1. During propellant tanking, the thermal contraction of the underlying substrate may generate cracks in the TPS (Figure 1). Chunks of TPS can release due to the expansion of ingested gas from cryopumping or from aerodynamic forces if the crack creates an offset surface. Most foams will also have a certain amount of “popcorning” where small pieces of foam will pop off during flight because of the differential between the static surface pressure and the pressure of the gas trapped in the foam cell structure. There are a number of other coating or closeout materials that may be shed from the vehicle and become debris. During pre-launch operations and liftoff, the vehicle may also be exposed to debris originating from the launch pad or ground support equipment. This debris is separate from foreign object debris, or FOD, which is not intended to be present and is strictly controlled through operations and maintenance procedures. In this case, debris is generated from hardware and materials that are necessary for launch and are subject to the intense vibration, acoustics, and direct plume impingement of the launch environment. Examples include ice from umbilicals, tape and tie wraps that protect cables, and rust or corrosion from the launch platform. While NASA has historically been aware of debris as a potential issue that could cause a failure resulting in loss of mission, loss of vehicle, or loss of crew, the likelihood and severity of that risk was not always well understood or given sufficient weight in program and flight decisions. After the Space Shuttle Columbia accident (STS-107), the investigation found that foam TPS debris shed from the external tank was the proximate cause of the damage to the orbiter wing. Six previous observations of debris released from the foam ramp that covered the bipod connecting the forward end of the orbiter to the external tank resulted in minor changes or were determined to be accepted flight risks. Two occurrences of bipod ramp foam loss were not identified until the STS-107 investigation. Despite the damage inflicted by these debris strikes, the Shuttle Program Requirements Control Board deemed the vehicle safe to fly. During the Return to Flight effort following the Columbia disaster, NASA Engineering developed a process for the assessment of debris transport, impact, and damage tolerance to support independent assessments of risk by NASA Safety and Mission Assurance (S&MA). Under this system, each element (vehicle or ground system) defines a catalog of all expected debris based on launch history, component testing, or analysis. Debris transport analysis (DTA) is conducted using the debris catalog characteristics and potential flow transport mechanisms (e.g., vehicle aerodynamics, gravity, wind, plume-driven). The predicted debris impact locations and velocities are provided to the hardware owners, who use available test data and analysis to determine whether each component can withstand the impacts. In cases where the element hardware may be severely damaged or fail, the options are to mitigate the debris source through some change in design or operation, or to work with S&MA to try to characterize the probability of the impact and damage for program risk acceptance. Because of the differences in debris characteristics and transport, the DTA has been divided between the Liftoff and Ascent regimes. The development and application of Liftoff DTA methodology from the Shuttle Program to the current Artemis Program is the subject of this paper. Liftoff DTA covers the time from the start of pre-launch operations at the launch pad, up until the vehicle clears the launch tower and there is no longer any interaction with ground systems. Debris transport during this period is broadly classified as either gravity, wind, and plume-entrained (GWPE) or plume driven (PD). GWPE debris is generally lower speed, travelling in a forward-to-aft direction. PD transport includes flow features from the rocket ignition transient, as well as plume impingement and recirculation that occur as the vehicle lifts off the launch platform. In these cases, the debris typically moves in an aft-to-forward direction at higher speeds. The applicable transport mechanisms must be considered for each piece of debris depending on the material, and release location and time. For example, rust or metallic debris from the tower could fall (GWPE) and impact the vehicle before landing on the launch platform deck where it could be also be transported by plume impingement (PD). However, falling ice (GWPE) from an umbilical is unlikely to survive impact with the vehicle or launch platform and be available for PD transport. Modeling of debris transport is accomplished using a set of DTA tools which simulate debris trajectories subject to a reference frame acceleration (i.e., gravity) and aerodynamic drag. Where the trajectory encounters a solid surface, the debris is allowed to rebound with a specified coefficient of restitution. The drag is calculated by interpolating the fluid state at each point in the debris trajectory from high-fidelity computational fluid dynamics (CFD) simulations of the launch vehicle and pad. The CFD data may either be static (steady state or time averaged), typically for GWPE transport, or dynamic (time-accurate) for PD flow features like the ignition transient. Examples of the CFD flow field solutions for the Space Launch System (SLS) rocket and launch pad are shown in Figure 2. Typical SLS debris trajectory predictions from DTA are illustrated in Figure 3. The final version of this paper will include a more detailed examination of the Liftoff DTA process developed during the Shuttle Program, and how it has been augmented and applied to the SLS rocket under the Artemis Program. Comparisons with debris observations from the Artemis I launch will demonstrate validation of the tools and methodology.

Debris↗

Power Electronics Materials and Bonded Interfaces - Reliability and Lifetime

Wide-bandgap devices have pushed the operational limit of semiconductor devices in automotive power electronics packages to higher temperatures (>200 C). While a higher efficiency can be achieved through the use of wide-bandgap devices, the entire package must be re-designed with components that can withstand the higher temperature limits. Sintered silver and transient liquid phase bonds are potential candidates as bonded materials for use in higher temperatures however, the underlying mechanics of deformation under thermal loads and the resulting failure mechanisms in these materials are not well understood. Accelerated experiments conducted at NREL reveal that high-lead solder joints have better reliability than sintered silver under extreme thermal cycling conditions. Furthermore, efforts to develop a crack propagation model for the high-temperature materials are described.

ADVANCED PROPULSION SYSTEMS↗

Oxidation of Carbon/Carbon through Coating Cracks

Reinforced carbon/carbon (RCC) is used to protect the wing leading edge and nose cap of the Space Shuttle Orbiter on re-entry. It is composed of a lay-up of carbon/carbon fabric protected by a SiC conversion coating. Due to the thermal expansion mismatch of the carbon/carbon and the SiC, the SiC cracks on cool-down from the processing temperature. The cracks act as pathways for oxidation of the carbon/carbon. A model for the diffusion controlled oxidation of carbon/carbon through machined slots and cracks is developed and compared to laboratory experiments. A symmetric cylindrical oxidation cavity develops under the slots, confirming diffusion control. Comparison of cross sectional dimensions as a function of oxidation time shows good agreement with the model. A second set of oxidation experiments was done with samples with only the natural craze cracks, using weight loss as an index of oxidation. The agreement of these rates with the model is quite reasonab

Jacobson, N. S.↗

Analytical and experimental investigation of rubbing interaction in labyrinth seals for a liquid hydrogen fuel pump

Cracking of the titanium knife edges on the labyrinth seals of the liquid hydrogen fuel pump in the Space Shuttle main engine is considered. Finite element analysis of the thermal response of the knife edge in sliding contact with the wear ring surface shows that interfacial temperatures can be quite high and they are significantly influenced by the thermal conductivity of the surfaces in rubbing contact. Thermal shock experiments on a test specimen similar to the knife edge geometry demonstrate that cracking of the titanium alloy is possible in a situation involving repeated thermal cycles over a wide temperature range, as might be realized during a rub in the liquid hydrogen fuel pump. High-speed rub interaction tests were conducted using a representative knife edge and seal geometry over a broad range of interaction rates and alternate materials were experimentally evaluated. Plasma-sprayed aluminum-graphite was found to be significantly better than presently used aluminum alloy seals from the standpoint of rub performance. Ion nitriding the titanium alloy knife-edges also improved rub performance compared to the untreated baseline.

Dolan, F. X.↗

Microstructural analysis of the 2195 aluminum-lithium alloy welds

The principal objective of this research was to explain a tendency of 2195 Al-Li alloy to crack at elevated temperature during welding. Therefore, a study was made on the effect of welding and thermal treatment on the microstructure of Al-Li Alloy 2195. The critical roles of precipitates, boundaries, phases, and other features of the microstructure were inferred from the crack propagation paths and the morphology of fracture surface of the alloy with different microstructures. Particular emphasis was placed on the microstructures generated by the welding process and the mechanisms of crack propagation in such structures. Variation of the welding parameters and thermal treatments were used to alter the micro/macro structures, and they were characterized by optical and scanning electron microscopy. A theoretical model is proposed to explain changes in the microstructure of welded material. This model proposes a chemical reaction in which gases from the air (i.e., nitrogen) release hydrogen inside the alloy. Such a reaction could generate large internal stresses capable to induce porosity and crack-like delamination in the material.

Talia, George E.↗

Thermographic detection of cracks in thin sheets

A thermographic inspection technique for crack detection based on a 2D filter convolved with the thermal temperature images is presented. The filter is designed to approximate operating on the temperature images with a Laplacian operator. This operation yields an image which approximates changes in the heat flux in a thin plate. This filtering method results in an enhanced contrast due to the presence of cracks. Measurements have been performed on samples with fabricated electrical discharge machining (EDM) notches (both through-the-thickness and surface notches) and closed fatigue cracks around rivets. It is shown that the technique is effective for the detection of various crack lengths down to the resolution limits of the imager used.

Cramer, K. E.↗

A methodology to predict damage initiation, damage growth and residual strength in titanium matrix composites

In this research, a methodology to predict damage initiation, damage growth, fatigue life, and residual strength in titanium matrix composites (TMC) is outlined. Emphasis was placed on micromechanics-based engineering approaches. Damage initiation was predicted using a local effective strain approach. A finite element analysis verified the prevailing assumptions made in the formulation of this model. Damage growth, namely, fiber-bridged matrix crack growth, was evaluated using a fiber bridging (FB) model which accounts for thermal residual stresses. This model combines continuum fracture mechanics and micromechanics analyses yielding stress-intensity factor solutions for fiber-bridged matrix cracks. It is assumed in the FB model that fibers in the wake of the matrix crack are idealized as a closure pressure, and an unknown constant frictional shear stress is assumed to act along the debond length of the bridging fibers. This frictional shear stress was used as a curve fitting parameter to the available experimental data. Fatigue life and post-fatigue residual strength were predicted based on the axial stress in the first intact 0 degree fiber calculated using the FB model and a three-dimensional finite element analysis.

Bakuckas, J. G., Jr.↗

Use of Several Thermal Analysis Techniques on a Hypalon Paint Coating for the Solid Rocket Booster (SRB) of the Space Shuttle

White Hypalon paint is brush-applied as a moisture barrier coating over cork surfaces on each of the two Space Shuttle SRBS. Fine cracks have been observed in the Hypalon coating three times historically on laboratory witness panels, but never on flight hardware. Recent samples of the cracked and standard ("good") Hypalon were removed from cork surfaces and were tested by Thermal Gravimetric Analysis (TGA), Thermomechanical (TMA) and Differential Scanning Calorimetry (DSC) thermal analysis techniques. The TGA data showed that at 700 C, where only paint pigment solids remain, the cracked material had about 9 weight percent more material remaining than the standard material, probably indicating incomplete mixing of the paint before it was brush-applied to produce the cracked material. Use of the TMA film tension method showed that the average static modulus vs. temperature was about 3 to 6 times higher for the cracked material than for the standard material, indicating a much higher stiffness for the cracked Hypalon. The TMA data also showed than an increased coating thickness for the cracked Hypalon was not a factor in the anomaly.

Wingard, Charles D.↗

Strain isolated ceramic coatings

Plasma sprayed ceramic coatings are used in gas turbine engines to improve component temperature capability and cooling air efficiency. A compliant metal fiber strain isolator between a plasma sprayed ceramic coating and a metal substrate improves ceramic durability while allowing thicker coatings for better insulation. Development of strain isolated coatings has concentrated on design and fabrication of coatings and coating evaluation via thermal shock testing. In thermal shock testing, five types of failure are possible: buckling failure im compression on heat up, bimetal type failure, isothermal expansion mismatch failure, mudflat cracking during cool down, and long term fatigue. A primary failure mode for thermally cycled coatings is designated bimetal type failure. Bimetal failure is tensile failure in the ceramic near the ceramic-metal interface. One of the significant benefits of the strain isolator is an insulating layer protecting the metal substrate from heat deformation and thereby preventing bimetal type failure.

Tolokan, R. P.↗

Synthesis of a 316L stainless steel-copper composite by laser melting

This paper demonstrates a laser melting method to produce a metal matrix composite with a higher thermal conductivity compared to 316 L stainless steel using a Jettable Cu ink. A novel canister was used to control the oxygen level during laser processing to emulate a hybrid inkjet-powder bed fusion technique for doping the stainless steel powder bed with Cu precursor ink. Here, the thermal conductivity of the composite was found to be 187% higher than that of 316 L stainless steel, while the microhardness decreased by 39%. Microstructural results show opportunities to further enhance the thermal conductivity and mechanical properties of the composite by eliminating interfacial cracking.

36 MATERIALS SCIENCE↗

A New Perspective of Post-Weld Baking Effect on Al-Steel Resistance Spot Weld Properties through Machine Learning and Finite Element Modeling

The root cause of post-weld baking on the mechanical performance of Al-steel dissimilar resistance spot welds (RSWs) has been determined by machine learning (ML) and finite element modeling (FEM) in this study. A deep neural network (DNN) model was constructed to associate the spot weld performance with the joint attributes, stacking materials, and other conditions, using a comprehensive experimental dataset. The DNN model positively identified that the post-weld baking reduces the joint performance, and the extent of degradation depends on the thickness of stacking materials. A three-dimensional finite element (FE) model was then used to investigate the root cause and the mechanism of the baking effect. It revealed that the formation of high thermal stresses during baking, from the mismatch of thermal expansion between steel and Al alloy, causes damage and cracking of the brittle intermetallic compound (IMC) formed at the interface of the weld nugget during welding. This in turn reduces the joint performance by promoting undesirable interfacial fracture when the welds were subjected to externally applied loads. The FEM model further revealed that increase in structural stiffness, because of increase in steel sheet thickness, reduces the thermal stresses at the interface caused by the thermal expansion mismatch and consequently lessens the detrimental effect of post-weld baking on the joint performance.

36 MATERIALS SCIENCE↗

Below-Ambient and Cryogenic Thermal Testing

Thermal insulation systems operating in below-ambient temperature conditions are inherently susceptible to moisture intrusion and vapor drive toward the cold side. The subsequent effects may include condensation, icing, cracking, corrosion, and other problems. Methods and apparatus for real-world thermal performance testing of below-ambient systems have been developed based on cryogenic boiloff calorimetry. New ASTM International standards on cryogenic testing and their extension to future standards for below-ambient testing of pipe insulation are reviewed.

below-ambient↗

Kinetic and equilibrium reactions on natural and laboratory generation of thermogenic gases from Type II marine shale

The phenomenon that laboratory pyrolysis experiments produce much wetter gases than those in natural reservoirs is a long-recognized and debated problem in the investigation of natural gases in sedimentary basins. In this study, we explore the discrepancy by pyrolyzing a type II kerogen from the Woodford Shale in Oklahoma, compared with the previous results on the produced natural gases from the Arkoma Basin generated from the same source rock (Liu et al., 2019) with the discussion of gas and isotopic compositions at bulk and position-specific (PS) levels. An improved GC-pyrolysis-GC IRMS method is applied for the determination of PS δ 13 C of propane produced in the pyrolysis of the Woodford Shale at Easy %R o from 0.76 to 3.27. Kinetic and thermodynamic considerations of the chemical and isotopic compositions of the natural and laboratory pyrolysis gases suggest that the generation of light hydrocarbons involves uni-directional cracking reactions, exchange reactions with water, and likely reversible reactions among light hydrocarbons and other H-containing volatiles. After the gas generation in the unconventional Woodford Shale reservoirs, the C 1 -C 4 gases might have approached close to chemical equilibrium of C 1 -C 3 and isotope equilibrium of C 2 -C 1 and C 3 -C 1 pairs at their peak temperatures. The capping H for the generation of C 1 -C 4 in the Woodford Shale gases appears to have experienced at least partial exchange with the water, while that in the pyrolysis gases is only originated from organic-bound compounds with large kinetic isotope effects (KIE). Our findings indicate that elevated compound-specific and PS δ 13 C values of propane in the wet-gas cracking stage are significantly influenced by the breakdown of the thermally stable compounds (e.g., remaining kerogen, residues). A first synthesis of PS δ 13 C and δ 2 H isotopic compositions of propane from this study and the literature data suggests relatively similar isotopic structures of propane precursors in kerogens. Finally, this study demonstrates that PS isotope analysis of propane can contribute to identifying various geological (e.g., maturation, wet-gas cracking, H exchange, diffusion) and biodegradation processes.

58 GEOSCIENCES↗

Unveiling damage mechanisms of chromium-coated zirconium-based fuel claddings at LWR operating temperature by in-situ digital image correlation

Here, we investigate the coupled thermomechanical fracture mechanisms of coated nuclear fuel claddings at Light Water Reactor (LWR) operating temperatures. These coated claddings are a highly attractive, near-term solution, which addresses the demands for accident-tolerant fuel systems and provide greater oxidation resistance. However, the fracture mechanisms of these coatings, which may create channels for oxidation ingression, must be fully understood prior to implementation. Thus, high-temperature expanding plug experiments were conducted on coated cladding specimens at a temperature of 315 °C, consistent with the operating environment of LWRs. Furthermore, in-situ thermomechanical deformation was measured with stereo digital image correlation during heating and mechanical testing to separately resolve contributions of thermal and mechanical strain. Digital image correlation, supported by acoustic emissions (AE) detection, was also leveraged to track cracking activity during loading. Coating fracture was found to initiate at total hoop strains of 0.34%. The thermal deformation of the coated claddings was investigated via finite element simulations, revealing a bidirectional stress-state within the coating with axial and circumferential strains of 0.026 and 0.031%. This bidirectional stress-state was attributed with the generation of off-axis fracture pattern within the coating as identified via post-experiment scanning electron microscopy. Thus, this study unveiled critical, coupled thermomechanical mechanisms governing the coating fracture of coated claddings at LWR temperatures.

36 MATERIALS SCIENCE↗

Fabrication of surrogate oxide spent fuel with various cracking patterns and design of an axial gas transport apparatus

In this article, understanding gas transport behavior in nuclear fuel rods is important for the design, performance, and safety of nuclear fuels. Surrogate materials help enable efficient research by reducing both the costs and the amount of time required. A parametric study using Darcy’s law is completed that demonstrates the feasibility of observing pressure decay over short 13-to-15-cm specimens to enable full characterization of the fabricated specimens using x-ray computed tomography. This paper demonstrates that thermally shocked and mechanically compressed alumina pellets produce surrogate samples whose various cracking patterns are representative of the severity of cracking observed as a function of burnup in irradiated nuclear fuels. Furthermore, image analyses of the cracking patterns—in conjunction with gas transport testing using surrogate samples—affords a valuable accelerated basis for developing gas transport simulations.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

A Radiation Hard Multi-Channel Digitizer ASIC for Operation in the Harsh Jovian Environment

In 1995, the Galileo spacecraft arrived at Jupiter to conduct follow-up experiments on pathfinder Pioneer and key Voyager discoveries especially at Io, Europa, Ganymede and Callisto. These new observations helped expand our scientific knowledge of the prominent Galilean satellites; studies revealed diversity with respect to their geology, internal structure, evolution and degree of past and present activity. Jupiter's diverse Galilean satellites, of which three are believed to harbor internal oceans, are central to understanding the habitability of icy worlds. Galileo provided for the first time compelling evidence of a near-surface global ocean on Europa. Furthermore, by understanding the Jupiter system and unraveling the history of its evolution from initial formation to the emergence of possible habitats and life, gives insight into how giant planets and their satellite systems form and evolve. Most important, new light is shed on the potential for the emergence and existence of life in icy satellite oceans. In 2009, NASA released a detailed Jupiter Europa Mission Study (EJSM) that proposed an ambitious Flagship Mission to understand more fully the satellites Europa and Ganymede within the context of the Jovian system. Key to EJSM is the NASA led Jupiter Europa Orbiter (JEO) and the ESA led Jupiter Ganymede Orbiter (JGO). JEO and JGO would execute a choreographed exploration of the Jovian system before settling into orbit around Europa and Ganymede, respectively. The National Academies Planetary Decadal Survey, 2011 has listed the NASA-led JEO as the second highest priority mission for the decade 2013-2022, and if chosen it would be launched in 2020 with arrival at Jupiter in 2025. If the JEO mission is not chosen it is anticipated that there will be opportunities in future decadal cycles. Jupiter Orbit Insertion (JOI) begins a 30-month Jovian system tour followed by nine months of science mapping after Europa Orbit Insertion (EOI) in July 2028. The orbiter will ultimately impact the surface of Europa after the mission is completed. The current JEO mission concept includes a range of instruments on the payload, to monitor dynamic phenomena (such as Io's volcanoes and Jupiters atmosphere), map the Jovian magnetosphere and its interactions with the Galilean satellites, and characterize water oceans beneath the ice shells of Europa and Ganymede. The payload includes a low mass (3.7 Kg) and low power (< 5 W) Thermal Instrument (TI) concept for measuring possible warm thermal anomalies on Europa s cold surface caused by recent (< 10,000 years) eruptive activity. Regions of anomalously high heat flow will be identified by thermal mapping using a nadir pointing, push-broom filter radiometer that provides far-IR imagery in two broad band spectral wavelength regions, 8-20 m and 20-100 m, for surface temperature measurements with better than a 2 K accuracy and a spatial resolution of 250 m/pixel obtained from a 100 Km orbit. The temperature accuracy permits a search for elevated temperatures when combined with albedo information. The spatial resolution is sufficient to resolve Europa's larger cracks and ridge axial valleys. In order to accomplish the thermal mapping, the TI uses sensitive thermopile arrays that are readout by a custom designed low-noise Multi-Channel Digitizer (MCD) ASIC that resides very close to the thermopile linear array outputs. Both the thermopile array and the MCD ASIC will need to show full functionality within the harsh Jovian radiation environment, operating at cryogenic temperatures, typically 150 K to 170 K. In the following, a radiation mitigation strategy together with a low risk Radiation-Hardened-By-Design (RHBD) methodology using commercial foundry processes is given for the design and manufacture of a MCD ASIC that will meet this challenge.

Aslam, Shahid↗

The effects of Venus' thermal structure on buoyant magma ascent

The recent Magellan images have revealed a broad spatial distribution of surface volcanism on Venus. Previous work in modeling the ascent of magma on both Venus and Earth has indicated that the planetary thermal structure significantly influences the magmatic cooling rates and thus the amount of magma that can be transported to the surface before solidification. In order to understand which aspects of the thermal structure have the greatest influence on the cooling of ascending magma, we have constructed magma cooling curves for both plutonic and crack buoyant ascent mechanisms, and evaluated the curves for variations in the planetary mantle temperature, thermal gradient curvature with depth, surface temperature gradient, and surface temperature. The planetary thermal structure is modeled as T/T(sub 0) = 1-tau(1-Z/Z(sub 0)(exp n), where T is the temperature, T(sub 0) is the source depth temperature, tau = 1-(T(sub s)/T(sub 0)) where T(sub s) is the planetary surface temperature, Z is the depth, Z(sub 0) is the source depth, and n is a constant that controls thermal gradient curvature with depth. The equation is used both for mathematical convenience and flexibility, as well as its fit to the thermal gradients predicted by the cooling half-space models. We assume a constant velocity buoyant ascent, body-averaged magma temperatures and properties, an initially crystal-free magma, and the same liquidus and solidus for both Venus and Earth.

Sakimoto, S. E. H.↗