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ARCTRON: A Rapid Experimental Proving Ground for TPS Experiments and Arcjet Technology Development

Innovation in high-enthalpy facilities is fundamentally limited by the cost and risk of experimentation. New concepts for plasma control, diagnostics, facility components, and plasma-material interaction often require repeated iterations that are impractical to perform in production arcjets. As a result, promising ideas may remain unexplored or reach operational facilities only after significant development effort. ARCTRON is being developed as a rapid experimental proving ground where new ideas in plasma science, arcjet engineering, diagnostics, and material response can be conceived, tested, and quantitatively evaluated before transition to large-scale facilities. The platform combines radio-frequency (RF) and DC arc plasma generation, externally applied magnetic fields, configurable gas composition, reduced-pressure operation, laser heating, electrical biasing, and modular diagnostic access. These capabilities permit the plasma source, applied forcing, test article, and measurement configuration to be modified independently, allowing individual physical mechanisms to be isolated more readily than in a traditional test environment. One class of investigations addresses fundamental plasma-surface interaction physics. Conventional material tests often expose a specimen simultaneously to convective heating, reactive species, pressure, shear, radiation, and surface-current effects. The resulting material response may be measured accurately, while the contribution of each mechanism remains difficult to identify. ARCTRON is designed to vary these effects selectively. Plasma chemistry can be changed independently through configurable gas mixtures; magnetic fields and electrical biasing can modify charged-particle transport; laser heating can provide a non-plasma thermal input; and pressure, flow, and discharge mode can be varied over a broad operating space. This enables controlled tests of hypotheses involving surface catalycity, reactive-species transport, plasma-assisted oxidation, electromagnetic effects, shear, and the relative contributions of thermal and chemical loading. A second class of investigations enabled by this approach concerns the engineering of high-enthalpy facilities themselves. Arc-heated facilities are limited by electrode erosion, unstable arc attachment, localized heating, and damage to nozzles and other plasma-facing components. ARCTRON provides a lower-cost environment for testing concepts intended to mitigate these limitations. Candidate investigations include the use of applied magnetic fields to alter current paths and reduce plasma interaction with nozzle walls, ExB forcing to introduce controlled plasma rotation, magnetic or geometric approaches for distributing arc attachment, and alternative electrode or discharge configurations intended to reduce erosion and improve stability. Because the platform is reconfigurable, these concepts can be evaluated through repeated design--build--test cycles before they are considered for implementation in operational facilities. The platform also supports the development and validation of diagnostics that may be difficult to introduce initially into a large arcjet. Current and planned measurements include spatially resolved optical emission spectroscopy, electrostatic probes, fast imaging, pyrometry, calorimetry, laser-induced fluorescence, and absorption spectroscopy. These diagnostics are intended not merely to document a nominal operating condition, but to constrain the local plasma state and its relationship to component or material response. The modular facility geometry allows diagnostic concepts to be tested, calibrated, and compared under repeatable conditions before deployment in more demanding environments. ARCTRON is also supported by an integrated software suite. Automated control and data acquisition allow discharge parameters, gas composition, magnetic fields, diagnostic timing, and test configuration to be recorded as part of each experiment (STARDAC - Software for Testing, Analysis, Research Data, and Control). The Backend for Experiment Analysis, Storage, and Traceability (BEAST) is a database that provides the infrastructure needed to associate heterogeneous measurements with facility configuration, specimen identity, calibration state, geometry, and analysis provenance. This backend is particularly important for exploratory campaigns, in which many related configurations may be tested, and the value of an individual experiment depends on its connection to earlier and subsequent iterations. Complementary analysis capabilities, including computer-vision-based transient response measurements (arcjetCV), three-dimensional surface reconstruction (STARSCAN), and model-based Bayesian inference (SHIELD), and tomography data analysis (TOMATO, PuMA) can be incorporated when required by a specific hypothesis without becoming the focus of every campaign. The central objective of ARCTRON is therefore not to maximize heat flux or reproduce a complete flight environment. Its purpose is to reduce the cost and time required to ask consequential questions about plasma behavior, plasma-facing materials, diagnostics, and arcjet technology. By providing a controlled environment for rapid reconfiguration, mechanism isolation, quantitative measurement, and iterative engineering, ARCTRON can help mature concepts that would otherwise remain too speculative or too risky for evaluation in production facilities. The resulting knowledge can then guide the design of material models, focus test objectives in larger arcjets, reduce facility-development risk, and improve the physical basis of high-enthalpy ground testing. This work will present the ARCTRON architecture, operating modes, diagnostic suite, and digital experimental workflow. Initial experimental results from the first integrated operation of the facility will be presented, including flow characterization, power limitations, and deployment of the initial diagnostic suite. Ongoing development efforts aimed at catalycity characterization, magnetic plasma control, and advanced optical diagnostics will also be discussed, illustrating how the platform supports rapid iteration from concept to experiment.

experimental diagnostics

ARCTRON: A Rapid Experimental Proving Ground for TPS Experiments and Arcjet Technology Development

Innovation in high-enthalpy facilities is fundamentally limited by the cost and risk of experimentation. New concepts for plasma control, diagnostics, facility components, and plasma-material interaction often require repeated iterations that are impractical to perform in production arcjets. As a result, promising ideas may remain unexplored or reach operational facilities only after significant development effort. ARCTRON is being developed as a rapid experimental proving ground where new ideas in plasma science, arcjet engineering, diagnostics, and material response can be conceived, tested, and quantitatively evaluated before transition to large-scale facilities. The platform combines radio-frequency (RF) and DC arc plasma generation, externally applied magnetic fields, configurable gas composition, reduced-pressure operation, laser heating, electrical biasing, and modular diagnostic access. These capabilities permit the plasma source, applied forcing, test article, and measurement configuration to be modified independently, allowing individual physical mechanisms to be isolated more readily than in a traditional test environment. One class of investigations addresses fundamental plasma-surface interaction physics. Conventional material tests often expose a specimen simultaneously to convective heating, reactive species, pressure, shear, radiation, and surface-current effects. The resulting material response may be measured accurately, while the contribution of each mechanism remains difficult to identify. ARCTRON is designed to vary these effects selectively. Plasma chemistry can be changed independently through configurable gas mixtures; magnetic fields and electrical biasing can modify charged-particle transport; laser heating can provide a non-plasma thermal input; and pressure, flow, and discharge mode can be varied over a broad operating space. This enables controlled tests of hypotheses involving surface catalycity, reactive-species transport, plasma-assisted oxidation, electromagnetic effects, shear, and the relative contributions of thermal and chemical loading. A second class of investigations enabled by this approach concerns the engineering of high-enthalpy facilities themselves. Arc-heated facilities are limited by electrode erosion, unstable arc attachment, localized heating, and damage to nozzles and other plasma-facing components. ARCTRON provides a lower-cost environment for testing concepts intended to mitigate these limitations. Candidate investigations include the use of applied magnetic fields to alter current paths and reduce plasma interaction with nozzle walls, ExB forcing to introduce controlled plasma rotation, magnetic or geometric approaches for distributing arc attachment, and alternative electrode or discharge configurations intended to reduce erosion and improve stability. Because the platform is reconfigurable, these concepts can be evaluated through repeated design--build--test cycles before they are considered for implementation in operational facilities. The platform also supports the development and validation of diagnostics that may be difficult to introduce initially into a large arcjet. Current and planned measurements include spatially resolved optical emission spectroscopy, electrostatic probes, fast imaging, pyrometry, calorimetry, laser-induced fluorescence, and absorption spectroscopy. These diagnostics are intended not merely to document a nominal operating condition, but to constrain the local plasma state and its relationship to component or material response. The modular facility geometry allows diagnostic concepts to be tested, calibrated, and compared under repeatable conditions before deployment in more demanding environments. ARCTRON is also supported by an integrated software suite. Automated control and data acquisition allow discharge parameters, gas composition, magnetic fields, diagnostic timing, and test configuration to be recorded as part of each experiment (STARDAC - Software for Testing, Analysis, Research Data, and Control). The Backend for Experiment Analysis, Storage, and Traceability (BEAST) is a database that provides the infrastructure needed to associate heterogeneous measurements with facility configuration, specimen identity, calibration state, geometry, and analysis provenance. This backend is particularly important for exploratory campaigns, in which many related configurations may be tested, and the value of an individual experiment depends on its connection to earlier and subsequent iterations. Complementary analysis capabilities, including computer-vision-based transient response measurements (arcjetCV), three-dimensional surface reconstruction (STARSCAN), and model-based Bayesian inference (SHIELD), and tomography data analysis (TOMATO, PuMA) can be incorporated when required by a specific hypothesis without becoming the focus of every campaign. The central objective of ARCTRON is therefore not to maximize heat flux or reproduce a complete flight environment. Its purpose is to reduce the cost and time required to ask consequential questions about plasma behavior, plasma-facing materials, diagnostics, and arcjet technology. By providing a controlled environment for rapid reconfiguration, mechanism isolation, quantitative measurement, and iterative engineering, ARCTRON can help mature concepts that would otherwise remain too speculative or too risky for evaluation in production facilities. The resulting knowledge can then guide the design of material models, focus test objectives in larger arcjets, reduce facility-development risk, and improve the physical basis of high-enthalpy ground testing. This work will present the ARCTRON architecture, operating modes, diagnostic suite, and digital experimental workflow. Initial experimental results from the first integrated operation of the facility will be presented, including flow characterization, power limitations, and deployment of the initial diagnostic suite. Ongoing development efforts aimed at catalycity characterization, magnetic plasma control, and advanced optical diagnostics will also be discussed, illustrating how the platform supports rapid iteration from concept to experiment.

experimental diagnostics

Recommended Practices in Laser-Induced Fluorescence (LIF) Diagnostics for Electric Propulsion

Laser-induced fluorescence (LIF) spectroscopy uses Doppler-shifted laser photons to excite a bound electron transition in an atom or ion, with subsequent fluorescence emission detected in order to measure the local velocity distribution function (VDF) and/or particle density. Due to its non-invasive nature, high spatial resolution, and acceptable difficulty of implementation, LIF has become a favored diagnostic for electric propulsion (EP) plasmas, particularly in the study of Hall thrusters. This paper presents recommended best practices for LIF measurements in electric thrusters, drawn from over three decades of implementation heritage in the EP community. Focusing on both single-photon LIF (typically used to measure VDFs) and two-photon LIF (TALIF, typically used to determine densities), the paper covers selection of atomic transitions to target, experimental setups, and interpretation of data. A number of special topics and advanced applications are discussed, including uncertainty analysis, techniques for high-speed measurements, and applications to a variety of propellants.

plasma plume

Laser Beam Welding Benchmark Experiments Performed in Reduced Gravity and Vacuum

Laser beam welding (LBW) is affected by the extreme temperatures, reduced pressure, and reduced gravity present in space environments. Gravity and pressure especially influence its melt pool and solidification dynamics. A compact, modular vacuum chamber adaptable to flight platforms from parabolic to orbital currently hosts an experiment to investigate the combined influence of reduced gravity and pressure on LBW. A swappable cartridge contains a rotating platen on which customizable workpieces can be welded under vacuum, greatly increasing experimental throughput. Instrumentation includes weld and thermal cameras observing the process, thermocouples placed on workpieces, accelerometers, and vacuum sensors. Experimental data gathered during the welding process will be combined with post-flight nondestructive evaluation, metallography, and mechanical testing to provide validation datasets for computational modeling. Phase I of this effort involves a parabolic flight campaign in low gravity while an anticipated Phase II would proceed to in-space demonstration to access extended duration microgravity.

in-space welding

Laser Welding Dissimilar Reflective Alloys

This project, jointly sponsored by Rocketdyne and CSTAR, involves the development of laser joining of materials which have heretofore been impractical to bond. Of particular interest are joints between stainless steel and copper and also aluminum 6061 to aluminum 2219. CSTAR has a unique opportunity in this area since both the process and development and diagnostics are of interest to industry. Initial results using the pulse tailored laser welding technique developed in CLA for joining crack sensitive materials have proven promising for the aluminum joints based upon metallurgical and electronic microprobe analysis. A declaration of success requires additional mechanical testing. A CW technique has been applied to the stainless-copper joining with some preliminary success. These joints are of significant interest for aeronautics and rocket propulsion applications and the project is expected to continue.

M H Mccay

Gateway Element and Payload Materials Outgassing Analyses: HALO, HERMES, and ERSA

Gateway was intended to be humanity’s first space station around the Moon, but its development has been paused as the National Aeronautics and Space Administration (NASA) shifts focus to achieving the United States’ National Space Policy goals. Instead of an orbiting lunar outpost, NASA will now pursue the development of a lunar surface base to support a sustained human presence on the Moon. Before the program’s pause, Gateway’s Induced Environments team worked to ensure payloads and elements (i.e., modules) complied with induced environment requirements. Methods developed and insights gained from this work will have applicability to NASA’s Moon Base and the potential repurposing of Gateway elements and payloads, as well as to induced environments modeling for future space stations. The Gateway program’s induced environment included molecular contamination, electric thruster plume sputter and redeposition, and lunar dust transfer from the Human Landing System (HLS). Primary sources of external molecular contamination included materials outgassing, chemical thruster plume contamination, and vacuum venting. The focus of this paper will be on element- and payload-level materials outgassing analyses performed for Gateway Configuration 1, extending the previously-developed framework for Gateway system-level external molecular contamination modeling. Gateway Configuration 1 consisted of the Power and Propulsion Element (PPE) and the Habitation and Logistics Outpost (HALO). It also included payloads like the European Radiation Sensor Array (ERSA) attached to PPE and the Heliophysics Environmental and Radiation Measurement Experiment Suite (HERMES) attached to HALO. The element- and payload-level analyses to be introduced in this paper for HALO, HERMES, and ERSA enabled high-fidelity descriptions of Gateway’s external molecular contamination environment. Approaches to geometric modeling, meshing, outgassing rate assignment, molecular transport modeling, and analysis methodology will be presented. Element and payload contaminant deposition onto sensitive Gateway receiver surfaces will be summarized and results compared to induced environment requirements. While these results incorporate refinements made over the course of the program, they were not intended to be final. Therefore, modeling assumptions and inputs, potential improvements, and lessons-learned will be documented to inform future work on Moon Base, repurposed elements and payloads, and other space stations.

Gateway

Design Criteria for Integrated Flight/Propulsion Control Systems for STOVL Fighter Aircraft

As part of NASA's program to develop technology for short takeoff and vertical landing (STOVL) fighter aircraft, control system designs have been developed for a conceptual STOVL aircraft. This aircraft is representative of the class of mixed-flow remote-lift concepts that was identified as the preferred design approach by the US/UK STOVL Joint Assessment and Ranking Team. The control system designs have been evaluated throughout the powered-lift flight envelope on Ames Research Center's Vertical Motion Simulator. Items assessed in the control system evaluation were: maximum control power used in transition and vertical flight, control system dynamic response associated with thrust transfer for attitude control, thrust margin in the presence of ground effect and hot gas ingestion, and dynamic thrust response for the engine core. Effects of wind, turbulence, and ship airwake disturbances are incorporated in the evaluation. Results provide the basis for a reassessment of existing flying qualities design criteria applied to STOVL aircraft.

James A Franklin

Atmospheric, Non-Combusting Droplet Sizing and Spray Imaging Results From a Non-Proprietary Aero Gas Turbine Engine Pressure Swirl Atomizer using Shadowgraphy and Planar Laser Scatter

Members from industry and federal agencies of the United States have developed a collaborating group focused on improving the understanding of the atomization processes common to gas turbine engines. The group is aimed on providing high quality data using x-ray diagnostics in the internal geometry and the near-field dense spray as well as optical diagnostics in the visible spectrum, downstream of the nozzle. This paper reports results from one of the two non-proprietary injectors designs from the group: the primary-secondary injector. In particular, we characterize the spray from only the primary, pressure-swirl atomizing circuit using water through the nozzle. We observed Sauter mean diameter decreased with increased water pressure drop across the nozzle. SMD was also collected at three axial distances and showed decreased size with distance from the nozzle. Planar laser sheet droplet scattering provided qualitative assessment of the spray. Higher water pressure drops provided better atomization than lower pressure drops. The addition of swirling air flow through the injector alleviates and improves the atomization quality, particularly at the lower water pressure drops.

optical diagnostics

Colors of Jupiter's large anticyclones and the interaction of a Tropical Red Oval with the Great Red Spot in 2008

The nature and mechanisms producing the chromophore agents that provide color to the upper clouds and hazes of the atmospheres of the giant planets are largely unknown. In recent times, the changes in red coloration that have occurred in large- and medium-scale Jovian anticyclones have been particularly interesting. In late June and early July 2008, a particularly color-intense red tropical oval interacted with the Great Red Spot (GRS) leading to the destruction of the red tropical oval and cloud dispersion. We present a detailed study of the tropical vortices, usually white but sometimes red, and a characterization of their color spectral signatures and dynamics. From the spectral reflectivity in methane bands we study their vertical cloud structure compared to that of the GRS and BA. Using two spectral indices we found a near-correlation between anticyclones cloud top altitudes and red color. We present detailed observations of the interaction of the red oval with the GRS and model simulations of the phenomena that allow us to constrain the relative vertical extent of the vortices. We conclude that the vertical cloud structure, vertical extent and dynamics of Jovian anticyclones are not the causes of their coloration. We propose that the red chromophore forms when background material (a compound or particles) is entrained by the vortex, transforming into red once inside the vortex due to internal conditions, exposure to ultraviolet radiation or to the mixing of two chemical compounds that react inside the vortex, confined by a potential vorticity ring barrier.

red color

Survey of Supersonic Inlets for High Mach Number Applications

The various inlet design philosophies are assessed on the basis of recent experimental results obtained at Mach numbers up to 5. The basic compression systems (external, internal, and combined external-plus-internal) are compared for both Mach 4.0 turbojet and hypersonic-cruise ramjet engines. The inlet that appears best suited for the Mach 4.0 turbojet application is that with combined external-plus-internal compression. Because of the severe cooling and weight problems associated with variable geometry at hypersonic flight speeds, the fixed-geometry self-starting characteristics of external-compression systems may be the overriding factor in the selection of an inlet for the hypersonic-cruise ramjet.

James F Connors

Lead-acid battery

A light weight lead-acid battery (30) having a positive terminal (36) and a negative terminal (34) and including one or more cells or grid stacks having a plurality of vertically stacked conductive monoplates (10, 20) with positive active material and negative active material deposited on alternating plates in the cell or grid stack. Electrolyte layers (26, 28) positioned between each monoplate are included to provide a battery cell having four sides which is capable of being electrically charged and discharged. Two vertical positive bus bars (42, 43) are provided on opposite sides of the battery cell for connecting the monoplates (10) with positive active material together in parallel current conducting relation. In addition, two negative bus bars (38, 39) on opposite sides of the battery cell each being adjacent the positive bus bars are provided for connecting the monoplates (20) with negative active material together in parallel current conducting relation. The positive (42, 43) and negative (38, 39) bus bars not only provide a low resistance method for connecting the plurality of conductive monoplates of their respective battery terminals (36, 34) but also provides support and structural strength to the battery cell structure. In addition, horizontal orientation of monoplates (10, 20) is provided in a vertical stacking arrangement to reduce electrolyte stratification and short circuiting due to flaking of positive and negative active materials from the monoplates.

Rowlette, John J.

Effects of Motion Fidelity on Pilot Performance Using Mixed-Reality Visuals

This paper investigates the effects of simulator motion fidelity on pilot performance using mixed-reality visuals. In NASA’s Vertical Motion Simulator, eleven pilots flew a lift-plus-cruise electric vertical takeoff and landing vehicle in a vertiport approach and landing task under four different motion conditions: no motion, small hexapod motion, large hexapod motion, and full motion. The simulator motion condition statistically significantly affected: 1) the severity of cybersickness, 2) motion ratings, 3) final approach and take-off boundary crossing groundspeeds, 4) longitudinal touchdown point, and 5) battery time remaining after landing. These results suggest that the use of motion systems in simulators with virtual- or mixed-reality visuals is warranted.

simulation

A Navier-Stokes Boundary Element Solver

Using global interpolation functions (GIF's) boundary element solutions are obtained for two-dimensional laminar flows. Two schemes are proposed for handling the convective terms. The first treats convection as a forcing function, and converts the flow equations to pseudo-Poisson equations. In the second scheme, some convective effect is incorporated into the fundamental solution used in constructing the pertinent integral equations. The lid-driven cavity flow is selected as the benchmark problem.

O Lafe

Observational Requirements for the Regional Scale

Observational requirements are provided for the 'regional scale' (10 2 to 10 3 km space scale; 3 to 24 h time scale). Given this range, the regional scale represents a spatial and temporal domain in which important scale-interactive processes occur that act to concentrate large vertical wind shears, significant horizontal thermal gradients, and vertical motion patterns into narrow regions. A short review of the mass and momentum adjustments associated with jet streak-induced circulations is discussed. Evidence for the need to specify the wind field in the upper troposphere to accurately simulate forcing for the transverse circulations is also presented. The importance of specifying temperature tendency to resolve the lower tropospheric portion of the transverse circulations is highlighted. The observational requirements are then discussed, along with possible approaches for meeting the requirements on the regional scale.

Louis W Uccellini

Effects of Motion Fidelity on Pilot Performance Using Mixed-Reality Visuals

This paper investigates the effects of simulator motion fidelity on pilot performance using mixed-reality visuals. In NASA’s Vertical Motion Simulator, eleven pilots flew a lift-plus-cruise electric vertical takeoff and landing vehicle in a vertiport approach and landing task with four different motion conditions: no motion, small hexapod motion, large hexapod motion, and full motion. The simulator motion condition statistically significantly affected: 1) the severity of cybersickness, 2) motion ratings, 3) final approach and take-off boundary crossing groundspeed, 4) longitudinal touchdown point, and 5) battery time remaining after landing. The findings suggest that motion systems are essential in simulators with virtual- or mixed-reality visuals when it is required to have pilot flying performance similar to that in real flight.

mixed reality

Thermal Cycle and Capacitance Testing of Mezzo Industries Phase Change Material Heat Capacitor

A phase change material heat capacitor prototype, designed and built under a NASA Small Business Innovation Research grant by Mezzo Technologies, was tested through coolant thermal cycling within the expected flow rates and temperature range of the Orion spacecraft’s propylene glycol-water mixture coolant system. Testing was performed with the phase change material contained under sealed conditions after a degassing procedure and exposed to atmosphere without prior degassing. The useful heat storage capacitance was measured and compared to Orion spacecraft requirements and theoretical N-Pentadecane storage capacity.

Propylene Glycol Water

Performance of Solar Reflectors When Applied to Storage of Cryogenic Propellants in Space

An analysis was performed to determine the effectiveness of solar reflectors in reducing heat transfer through the insulation of cryogenic fuel tanks located in deep space. A solar reflector, as used herein, is defined as a surface having a ratio of absorptance with respect to solar radiation to hemispherical emittance (α/ϵ) less than 1.0. This analysis considers the effect of surface coatings with α/ϵ values varying from 0.1 to 1.0 when applied to the external surfaces of bulk insulation or multifoil insulation. Results were obtained in terms of heat leak to two cryogenic propellants, liquid hydrogen and liquid oxygen, when stored at Mars, Earth, and Mercury orbital distances from the sun. The results show that, with sufficiently good insulation and low heat leaks to the propellant, α/ϵ is a reasonable criterion of merit for these coatings, but that, at larger heat leaks, consideration of ϵ (or α) is necessary. It was observed that selective finishes of α/ϵ = 0.1 on a given amount of bulk insulation material can reduce the heat leak to 40 to 55 percent of the heat leak obtained when gray finishes are considered. Or, for equivalent heat leaks, a reduction in bulk insulation material thickness of about one-half is possible with high-quality selective finishes (α/ϵ = 0.1) compared with flat reflectors. Larger benefits may be achieved from selective finishes painted on foils; the analysis shows that heat leak may be reduced tenfold for surfaces with an α/ϵ of 0.1 compared with gray surfaces. Again, consideration of equivalent heat leaks indicates that the use of approximately 10 times fewer foils of emissivity 0.05 is possible when spectrally selective surfaces are employed. The present state of the art in achieving such surfaces is also briefly reviewed.

Propellant Storage

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

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

Leonid S Kleinman