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At least 541 records · Page 30

Regioisomeric Engineering for Multicharge and Spin Stabilization in Two-Electron Organic Catholytes

Developing multicharge and spin stabilization strategies is fundamental to enhancing the lifetime of functional organic materials, particularly for long-term energy storage in multiredox organic redox flow batteries. Current approaches are limited to the incorporation of electronic substituents to increase or decrease the overall electron density or bulky substituents to sterically shield reactive sites. With the aim to further expand the molecular toolbox for charge and spin stabilization, we introduce regioisomerism as a scaffold-diversifying design element that considers the collective and cumulative electronic and steric contributions from all of the substituents based on their relative regioisomeric arrangements. Through a systematic study of regioisomers of near-planar aromatic cyclic triindoles and nonplanar nonaromatic cyclic tetraindoles, we demonstrate that this regioisomeric engineering strategy significantly enhances the H-cell cycling stability in the above two new classes of 2e – catholytes, even when current strategies failed to stabilize the multicharged species. Density functional theory calculations reveal that the strategy operates by redistributing the charge and spin densities while highlighting the role of aromaticity in charge stabilization. The most stable 2e – catholyte candidate was paired with a viologen derivative anolyte to achieve a proof-of-concept all-organic flow battery with 1.26–1.49 V, 98% capacity retention, and only 0.0117% fade/h and 0.00563% fade/cycle over 400 cycles (192 h), which is the highest capacity retention ever reported over 400 cycles in a multielectron all-organic flow battery setup. We anticipate regioisomeric engineering to be a promising strategy complementary to conventional electronic and steric approaches for multicharge and spin stabilization in other functional organic materials.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Geochemical Impact of Acid Spearhead and Slickwater Stimulation on Wolfcamp Shale from the Hydraulic Fracturing Test Site

The Hydraulic Fracturing Test Site 1 (HFTS-1) was a field study performed in the Wolfcamp Formation in the West Texas Permian (Midland) Basin, USA, with a focus on improving the efficiency of hydraulic fracturing. Investigating site-specific rock-fluid geochemical interactions during hydraulic fracturing is an important step to understanding the impact on formation shale porosity, permeability, and long-term shale gas production. During field operations in this region, hydraulic fracturing fluid (HFF) injection usually starts with a concentrated acid spearhead for rapid rock dissolution, followed by the injection of near-neutral pH slickwater containing chemicals and proppants. A multistep sequential injection approach was used to investigate different stages of rock-fluid interactions. The carbonate content in the host rock is important when acid spearhead is considered, as carbonate mineral dissolution is rapid and can result in porosity and permeability changes in the shale matrix. Here, in this study, we designed flow-through experiments using fractured carbonate-rich and clay-rich Wolfcamp shale cores with (1) a short-time acid soaking step and (2) a long-term slickwater flow-through step to simulate the injection method used at HFTS-1. The fluid chemistry was analyzed. A thorough mineralogical progression [e.g., Calcium (Ca) dissolution and iron (Fe) redox progression] in the cores during HFF injection was also characterized and imaged by synchrotron microprobe. Reactive transport modeling was performed based on the experimental setup. The results showed that the acid spearhead is a crucial step in creating a reaction front by mineral dissolution, especially in carbonate-rich shales. A slight layer of ferrihydrite precipitated during the slickwater flow-through period. This study provides insights into potential geochemical impact due to hydraulic fracturing operations in the Permian Basin.

clastic rock↗

Modulation of Uptake and Reactivity of Nitrogen Dioxide in Metal-Organic Framework Materials

We report the modulation of reactivity of nitrogen dioxide (NO 2 ) in a charged metal–organic framework (MOF) material, MFM-305-CH 3 in which unbound N-centres are methylated and the cationic charge counter-balanced by Cl - ions in the pores. Uptake of NO 2 into MFM-305-CH 3 leads to reaction between NO 2 and Cl - to give nitrosyl chloride (NOCl) and NO 3 - anions. A high dynamic uptake of 6.58 mmol g -1 at 298 K is observed for MFM-305-CH 3 as measured using a flow of 500 ppm NO 2 in He. In contrast, the analogous neutral material, MFM-305, shows a much lower uptake of 2.38 mmol g -1 . The binding domains and reactivity of adsorbed NO 2 molecules within MFM-305-CH 3 and MFM-305 have been probed using in situ synchrotron X-ray diffraction, inelastic neutron scattering and by electron paramagnetic resonance, high-field solid-state nuclear magnetic resonance and UV/Vis spectroscopies. The design of charged porous sorbents provides a new platform to control the reactivity of corrosive air pollutants.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Automated Control for Nuclear Thermal Propulsion Start-Up using MOOSE-based Applications

This report presents a Griffin/Bison/RELAP-7 numerical model of a prototypical NTP system that features fuel assemblies arranged in rings, and which was designed to simulate rapid startup transients. The physics modeled include full-core neutronics, assembly-wise heat conduction, and conjugate heat transfer, with the balance of plant mainly imposed through boundary conditions. In addition, various forms of automated reactivity control were deployed by using the MOOSE to autonomously drive the model and simulate the reactor transitioning from assumed initial conditions to nominal power in a fraction of a minute. To generate the cross-sections of the neutronics model, and in an effort to simultaneously account for the tremendous axial temperature gradients in the reactor and to limit the number of state points required for cross-section generation, the average component temperatures and hydrogen densities in the cooling channels were correlated to the average fuel and moderator temperatures, and fixed axial profiles were derived for nominal conditions and then used during the transient. With this approximation, a tractable cross-section library tabulated with fuel/moderator temperatures and CD angles was generated using Serpent. The full-core SPH correction procedure and the CD decusping technology in Griffin, respectively, ensure preservation of the multiplication factor and reaction rates at state points, along with a reasonably accurate reactivity worth between tabulated CD angles, despite using a coarse mesh. Feedback from other physics was calculated by modeling one representative fuel assembly per ring, along with the corresponding fuel and moderator cooling channels. To limit power overshoots during startup, another layer of multiphysics coupling was added to the model in order to automatically control the drums. Two different technologies presented herein showed outstanding performance in this regard: (1) a novel hybrid PID controller based on both power and reactivity signals, and (2) a PGC that relies on kinetics parameters and reactivity coefficients to predict future behavior and adjust the desired signal accordingly. A challenging benchmark was devised, featuring a power demand curve that exponentially increases by a factor of 500 within 30 seconds, then levels out after that. Both control approaches create a simulated power curve that closely follows the power demand curve and limits power overshoots to 1% or less. While the former approach requires more tuning of the internal parameters, the latter requires additional knowledge of the reactivity feedback coefficients and rates of change of the corresponding variables, including fuel and moderator temperature, which could be difficult to dynamically measure for a real NTP system. Fortunately, some inaccuracy in these quantities will not drastically degrade the PGC performance. Subsequently, a more realistic startup sequence was considered, in which the mass flow rate and outlet pressures are ramped up to model bootstrap and thrust build-up phases prior to reaching steady-state conditions, demonstrating the ability of the hybrid PID and PGCs to handle such transients, with both types of controllers exhibiting very similar behavior. Nevertheless, a significant chamber temperature overshoot was observed, caused by the demanded power signal and assumed mass flow rate. This issue could be mitigated by deploying a reactor controller that follows the chamber temperature signal and actuates both the control valves and drums (rather than using a power signal based solely on the drums to control reactivity). Enhancement of the hydrogen fluid properties available in MOOSE, as well as a better understanding of prototypical initial conditions, are also needed to further enhance this startup model. Finally, a study was performed to model decay heat post-shutdown, and to prepare for extending this model to predict shutdown behavior and post-shutdown pulsed cooling requirements.

33 ADVANCED PROPULSION SYSTEMS↗

Computer Programs For Analysis Of Thermally Reactive Tracer Tests In Geothermal Reservoirs

The Tracer Analysis Toolbox (TAT) is a stand-alone executable software package that focuses on reservoir monitoring, using changes in concentration of thermally degrading tracers between and injection well and a production well to reflect the temperature-time histories along flow paths through the reservoir. By conducting such tests periodically, the thermal evolution of the flow paths connecting injection wells to production wells can be monitored. Because this approach is sensitive to temperature along the whole path, it can detect changes well before they occur at the distal end of the path (i.e., the production well). Thus, the thermally degrading tracer method provides early detection of thermal decline before it impacts that well. The goal of TAT is to simplify planning of a reactive tracer monitoring campaign and interpretation of results by providing: (1) tracer test planning and analysis tools to define the necessary kinetic parameters, injection concentrations, and other tracer parameters that are suitable for the expected reservoir residence time and temperature, (2) identification of tracers with suitable characteristics (defined by (1)) and (3) a suite of analysis programs to analyze tracer test results.

Plummer, MitchellAaron [Idaho National Laboratory ↗

Mode Specificity in the OH + HO 2 → H 2 O + O 2 Reaction: Enhancement of Reactivity by Exciting a Spectator Mode

A reaction usually involves a few active modes while the other modes are largely preserved throughout the reaction as spectators. Excitation of an active mode is expected to promote the reaction, but depositing energy in a spectator mode typically has no effect, because of the differing ability for energy flow to the reaction coordinate. Here, we report a surprising case of mode specificity in a key radical–radical reaction OH + HO 2 → H 2 O + O 2 , where such canonical expectations fail to hold. Despite its spectator nature, the vibrational excitation of the OH reactant is shown at low collision energies to enhance the reactivity significantly. This unique effect can be attributed to the increased attraction with HO 2 due to the larger dipole of the stretched OH. At low collision energies, the stronger attraction increases the chance of capturing the reactants to form a hydrogen-bonded complex, thus of passing through the submerged barrier. The novel mechanism differs from the conventional vibrational enhancement via coupling to the reaction coordinate at the transition state, enriching our understanding of mode specificity in chemistry.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

3D Quantification of Pore Networks and Anthropogenic Carbon Mineralization in Stacked Basalt Reservoirs

Basalt formations are promising candidates for the geologic storage of anthropogenic CO 2 due to their storage capacity, porosity, permeability, and reactive geochemical trapping ability. The Wallula Basalt Carbon Storage Pilot Project demonstrated that supercritical CO 2 injected into >800 m deep Columbia River Basalt Group stacked reservoir flow tops mineralizes to ankerite-siderite-aragonite on month-year timescales, with 60% of the 977 metric tonnes of CO 2 converted within two years. The potential impacts of mineral precipitation and consequent changes on rock porosity, pore structure, pore size, and pore size distributions have likely been underestimated hitherto. Herein, we address these knowledge gaps using X-ray Micro Computed Tomography (XMT) to evaluate the pore network architecture of sidewall cores recovered two years after CO 2 injection. In the present study, we performed a detailed quantitative analysis of the CO 2 -reacted basalt cores by XMT imaging. Reconstructed 3D images were analyzed to study the distribution and volumetric details of porosity and carbonate nodules in the cores along with the various other phases, providing insight into paragenesis and carbonate growth mechanisms, including mineralogic/chemical zonation. Finally, these findings are being used to parametrize multiphase reactive transport models to predict the fate and transport of subsurface CO 2 , enabling scale-up to commercial-scale geologic carbon storage in basalts and other reactive mafic-ultramafic formations.

54 ENVIRONMENTAL SCIENCES↗

Nuclear modules for space electric propulsion

Analysis of interplanetary cargo and piloted missions requires calculations of the performances and masses of subsystems to be integrated in a final design. In a preliminary and scoping stage the designer needs to evaluate options iteratively by using fast computer simulations. The Oak Ridge National Laboratory (ORNL) has been involved in the development of models and calculational procedures for the analysis (neutronic and thermal hydraulic) of power sources for nuclear electric propulsion. The nuclear modules will be integrated into the whole simulation of the nuclear electric propulsion system. The vehicles use either a Brayton direct-conversion cycle, using the heated helium from a NERVA-type reactor, or a potassium Rankine cycle, with the working fluid heated on the secondary side of a heat exchanger and lithium on the primary side coming from a fast reactor. Given a set of input conditions, the codes calculate composition. dimensions, volumes, and masses of the core, reflector, control system, pressure vessel, neutron and gamma shields, as well as the thermal hydraulic conditions of the coolant, clad and fuel. Input conditions are power, core life, pressure and temperature of the coolant at the inlet of the core, either the temperature of the coolant at the outlet of the core or the coolant mass flow and the fluences and integrated doses at the cargo area. Using state-of-the-art neutron cross sections and transport codes, a database was created for the neutronic performance of both reactor designs. The free parameters of the models are the moderator/fuel mass ratio for the NERVA reactor and the enrichment and the pitch of the lattice for the fast reactor. Reactivity and energy balance equations are simultaneously solved to find the reactor design. Thermalhydraulic conditions are calculated by solving the one-dimensional versions of the equations of conservation of mass, energy, and momentum with compressible flow.

Difilippo, F. C.↗

Control-oriented core-SOL-divertor model to address integrated burn and divertor control challenges in ITER

The real-time regulation of a burning plasma’s temperature and density, or burn control, will be necessary to produce high fusion power in future tokamaks like ITER. This is made more challenging due to the plasma’s nonlinear characteristics and the interdependence between the core-plasma and edge-plasma regions. For example, a raising plasma temperature leads to increasing reactivity and therefore to more alpha-particle heating, which further increases temperature. Furthermore, a raise of the fusion power increases the heat flow through the scrape-off-layer (SOL), which can compromise the integrity of the divertor without proper safeguards. For control design, a model-based approach is attractive because it can directly incorporate the nonlinear, coupled, burning-plasma dynamics into the design. To facilitate this design approach, a control-oriented core-SOL-divertor (CSD) model is presented in this work. In this CSD model, a core-plasma model captures the nonlinear dynamics of the core’s density and temperature, and a SOL-divertor model defines the plasma conditions at the separatrix and divertor including the heat load on the target plates. The core-plasma and SOL-divertor models are coupled through the exchange of various variables. In particular, the SOL-divertor model yields the separatrix temperature and the influx of recycled particles into the core-plasma. Further, these variables influence the power and particle balances captured by the core-plasma model. In return, the core-plasma model determines the intensity of the heat and particles fluxes across the separatrix, and this outflow strongly impacts the SOL-divertor model. Therefore, the power and density of the core-plasma, which can be readily modulated through external heating systems and pellet injection, can be viewed as control knobs for the SOL-divertor region in addition to the gas puffing. In simulations of the CSD model, it is demonstrated how external actuation can be utilized to meet burn control and divertor control objectives simultaneously.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Effect of CO 2 -brine-rock reactions on pore architecture and permeability in dolostone: Implications for CO 2 storage and EOR

Geologic carbon sequestration (GCS) is considered a feasible technology for storing substantive volumes of greenhouse gases in subsurface geological formations. In the reservoir, far from carbon dioxide (CO 2 ) injection wells or in post-injection scenarios, diffusion dominates over advection. This condition conjoins with spatially distributed geochemical reactions to induce heterogeneous changes in pore architecture, i.e. pore body and throat sizes or surface roughness. These changes can affect CO 2 transport properties and storage capacity. In this work, we investigated mineral dissolution and precipitation in dolomite samples saturated with a CO 2 -saturated brine at 93 °C and 34.5 MPa, aged without flow. Two rock types samples, i.e. intergranular- and vuggy-dominant, were selected to investigate changes in pore size, porosity and permeability under reactive conditions. Mineral dissolution and precipitation were characterized using scanning electron microscopy. Changes in pore size were quantified via time-domain nuclear magnetic resonance (TD-NMR) transverse relaxation time (T 2 ) and diffusion coefficient (D) distributions. We show that mineral dissolution likely occurs in highly permeable pathways. These observations are confirmed through analysis of (T 2 ) and diffusion coefficient (D) distributions. In contrast to results during CO 2 -enriched brine continuous injection, mineral precipitation was observed in micropores. The leftward shift of the T 2 peaks, corresponding to micropores, also evidenced mineral precipitation in lowpermeability zones. However, microscale alterations resulted only in a subtle increase in porosity and permeability. Results in this study shed light on effects of geochemical reactions on alteration of rock properties in diffusion-dominated regions during CO 2 storage.

58 GEOSCIENCES↗

Bias-pulsed atomic layer etching of 4H-silicon carbide producing subangstrom surface roughness

A new approach to atomic layer etching (ALE) has been demonstrated, and its application to 4H-SiC is reported here. By pulsing only the DC bias for an Ar/Cl2 inductively coupled plasma-reactive ion etching system, the etch cycle duration is reduced by more than an order of magnitude relative to conventional ALE processes. Gas flows are not changed throughout the ALE process. With this process protocol, we achieved an etch rate of 2.48±0.09 Å/cycle with 6 s cycles, an RMS surface roughness (Rq) of 0.83±0.08 Å, and an ALE synergy value of S = 99%. The parameters explored within this ALE process demonstrate effective subangstrom smoothening of 4H-SiC surfaces and is well-suited for a variety of classical and quantum device nanofabrication.

42 ENGINEERING↗

CompLaB v1.0: a scalable pore-scale model for flow, biogeochemistry, microbial metabolism, and biofilm dynamics

Abstract. Microbial activity and chemical reactions in porous media depend on the local conditions at the pore scale and can involve complex feedback with fluid flow and mass transport. We present a modeling framework that quantitatively accounts for the interactions between the bio(geo)chemical and physical processes and that can integrate genome-scale microbial metabolic information into a dynamically changing, spatially explicit representation of environmental conditions. The model couples a lattice Boltzmann implementation of Navier–Stokes (flow) and advection–diffusion-reaction (mass conservation) equations. Reaction formulations can include both kinetic rate expressions and flux balance analysis, thereby integrating reactive transport modeling and systems biology. We also show that the use of surrogate models such as neural network representations of in silico cell models can speed up computations significantly, facilitating applications to complex environmental systems. Parallelization enables simulations that resolve heterogeneity at multiple scales, and a cellular automaton module provides additional capabilities to simulate biofilm dynamics. The code thus constitutes a platform suitable for a range of environmental, engineering and – potentially – medical applications, in particular ones that involve the simulation of microbial dynamics.

58 GEOSCIENCES↗

Soft-seal valve holds hazardous fluids safely

Valve assembly allows transfer of hazardous or reactive fluids such as liquid fluorine without corrosion of valve face and seat material. The assembly consists of a plug to block bulk flow and a soft-seal outer seat to effect zero-leak stoppage.

Source record↗

Pump-and-Treat Impacts on Aquifer Conditions and the Development of System-Wide Performance Strategies - 20377

The pump-and-treat (P and T) system on the Hanford Site's Central Plateau currently treats multiple contaminants within the 200-ZP-1 operable unit, including nitrate through its biological treatment processes. Although it was initially designed to support the 200-ZP-1 Remedial Action Objectives, the P and T system has evolved in use to address contaminants from other regions of the aquifer as well. The P and T influents are biogeochemically distinct, treating groundwaters with elevated concentrations of cyanide, uranium, technetium-99, and nitrate. Given the varying geochemical signatures from current and potential future sources, operational conditions, and amendments considered for treatment processes, there is a need to identify the impacts that the P and T effluent has on the aquifer, and how the aquifer affects P and T performance. To this end, laboratory efforts have focused on characterizing P and T effluent and its biogeochemical effects on the sediments through a series repacked column experiments. The P and T effluent is rich in particulate matter. A systematic inspection of this particulate matter in the P and T effluent showed significant levels of manganese/iron oxides, calcite crystals, relatively large organic matter molecules, and biomass. Therefore, the column experiments were designed and conducted for four different treatment systems: artificial groundwater, filtered P and T effluent, non-filtered P and T effluent, and chlorinated P and T effluent. Different levels of microbiologically- and geochemically-induced changes in the columns occurred and modified the flow systems. For example, physical filtration occurred in the sediments where the particulate material had accumulated in the top portion of the column. Reactive transport modeling of the laboratory experiments is being carried out so that the reaction network can be extrapolated to the field scale. These simulations will provide predictive evaluations of system performance based on P and T operational changes and aquifer mass loading and capacity estimates. This information can be used to address combined subsurface and operational challenges, such as the need for frequent well redevelopment to maintain injection capacity. This information also provides the technical basis for decisions related to the P and T system operations that support remedy optimization efforts and short- and long-term remedy decisions related to multiple operable units on the Hanford Site Central Plateau. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

An experimental study of reactive turbulent mixing

An experimental study of the turbulent mixing of two coaxial gas streams, which react very rapidly (pseudo-hypergolic chemical kinetics), was performed to investigate the mixing characteristics of turbulent flow fields. The center stream consisted of a CO-N2 mixture and the outer annular stream consisted of air vitiated by H2 combustion. The streams were at equal velocity (50 m/sec) and temperature (1280 K). Turbulence measurements were obtained using hot film anemometry. A sampling probe was used to obtain time averaged gas compositions. Six different turbulence generators were placed in the annular passage to alter the flow field mixing characteristics. The turbulence generators affected the bulk mixing of the streams and the extent of CO conversion to different degrees. The effects can be related to the average eddy size and the bulk mixing. Higher extents of conversion of CO to CO2 were found by increasing the bulk mixing and decreasing the average eddy size.

Cooper, L. P.↗

Multiphysics Analysis of the MSRE Experiment Using Griffin-SAM Coupled Code System

Molten salt reactors (MSRs) with flowing fuel have the unique feature of utilizing the fuel salt for heat generation and extraction at the same time since the fuel salt is circulating through the whole primary loop of the reactor. This movement of the fuel salt results in a partial decay of the delayed neutron precursors (DNPs) outside the core and corresponding redistribution in the active core region. To capture this phenomenon accurately, neutronics and thermal hydraulics computational tools need to be modified to handle the movement of the DNPs and their decay. Idaho and Argonne National Laboratories are actively working on developing neutronics and thermal hydraulics tools to model MSRs with flowing fuel. The neutronics code Griffin and the thermal hydraulics code SAM were extended to handle flowing fuel with the drift of the DNPs under the MOOSE framework. In this work, a simplified Multiphysics model of the Molten Salt Reactor Experiment (MSRE) is developed and utilized to perform steady-state and transient analyses. The MSRE experiments were designed to show the impact of the DNPs losses on core reactivity. The following section provides a description of the MSRE experiment with the core main parameters used to develop the model. Then, the developed Multiphysics model of the MSRE is presented. Finally, steady-state verification tests and unprotected loss of flow transient tests are discussed followed by a summary and conclusions.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Thermochemical reduction modeling in a high-temperature moving-bed reactor for energy storage: 1D model

The design of robust and efficient high-temperature thermochemical reactors and determination of operating conditions are critical steps toward enabling high-efficiency long-duration solar energy storage. This work presents a computational model for the thermal reduction of a metal oxide material (Mg-Mn-O) up to 1450 °C and the coupled complex transport phenomena in a novel tubular thermal reactor design that features the capability for a high extent-of-reduction (high energy storage density) and inherent heat recuperation. A one-dimensional model coupling counter-current gas–solid flow, two-phase heat transfer, thermochemical redox reactions, and species transport in a moving-bed reactor is developed. Simplified versions of the model are validated with published results in the literature for packed beds with both inert and reactive particles; the fully coupled model is also validated with experimental measurements of a moving-bed reactor in terms of local temperatures and oxygen release at the exit. Detailed comparisons on the effects of different boundary conditions in the reaction zone (prescribed wall temperature vs. heat flux conditions) and formulations based on a simple uniform flow assumption vs. plug flow using Ergun equation for gas flow are investigated. The results are compared with experimental measurements, and for all cases, the energy flow components in the reactor system and the thermal to chemical conversion efficiency and overall system efficiency are computed. Finally, the predicted high thermal-to-chemical efficiency ~95% and system efficiency ~30% agree with experimental measurements.

25 ENERGY 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.

Kleinman, Leonid S.↗