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Results for “Theoretical Kinetics”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 325 records · Page 18
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

System efficiency and power assessment of the all-aqueous copper thermally regenerative ammonia battery

Thermally regenerative ammonia batteries (TRABs) use low-temperature (T < 100 °C) heat to provide stationary energy and power with higher power densities and efficiency relative to other waste heat devices. TRABs are an active area of research in waste heat devices, but currently there is little consensus on what aspect of the system is limiting TRAB performance and what maximum efficiencies are possible. Experiments and numerical models were used here to examine the sensitivity of the battery and distillation column in a TRAB system to key operating variables, thereby establishing practical limits and identifying focus areas for improving performance. Battery power was eight times more sensitive to ohmic losses than kinetic and mass transfer losses, regardless of the operating temperature, and the peak power density was simulated to be 18.8 mW cm –2 at 75 °C. Theoretical energy efficiency limits were defined for a series of ammonia concentrations and operating pressures, ranging from 5 – 12%, which is 2–3 times higher than previous experimental estimations. Atmospheric pressure column operation used a larger amount of waste heat compared to sub-atmospheric pressure. It is estimated that the volume of the battery would take up 9.2m 3 for every 1% of the power output of a natural gas turbine, but with realistic improvements to cell conductivity, the size would reduce to 2.5 m 3 . Finally, the results presented in this work will help streamline future development by focusing on minimizing ohmic losses and provide specific data for full system evaluation of future TRABs.

25 ENERGY STORAGE↗
DOE OSTI2024

Thermal and dynamic partition of dumbbell interstitials in complex concentrated alloys

Complex concentrated alloys (CCAs) are promising candidates for applications in extreme conditions, such as irradiation where interstitial mediated diffusion is important. In CCAs with N principal elements, $\frac{N(N+1)}{2}$ types of dumbbell interstitials exist. Currently, there is no way to predict the thermal partition (fractional concentration at equilibrium) and the dynamic partition (fractional time an interstitial spends during diffusion) of each type of dumbbell interstitial. To mitigate this issue, this work proposes a theoretical model for computing the equilibrium concentrations and thermal partition of dumbbell interstitials in CCAs and validates the model using grand canonical Monte Carlo simulations. Lattice kinetic Monte Carlo simulations show that the thermal partition is equivalent to the dynamic partition, and both are governed by composition and formation energies of dumbbells. As a result, the model proposed provides a foundation for understanding radiation enhanced diffusion and induced segregation in CCAs under irradiation.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Electronic Spectroscopy and Dissociation Dynamics of Vinyl-Substituted Criegee Intermediates: 2-Butenal Oxide and Comparison with Methyl Vinyl Ketone Oxide and Methacrolein Oxide Isomers

The 2-butenal oxide Criegee intermediate [(CH 3 CH=CH)CHOO], an isomer of the four-carbon unsaturated Criegee intermediates derived from isoprene ozonolysis, is characterized on its first π * ←π electronic transition and by the resultant dissociation dynamics to O ( 1 D) + 2-butenal [(CH 3 CH=CH)CHO] products. The electronic spectrum of 2-butenal oxide under jet-cooled conditions is observed to be broad and unstructured with peak absorption at 373 nm, spanning to half maxima at 320 and 420 nm, and in good accord with the computed vertical excitation energies and absorption spectra obtained for its lowest energy conformers. The distribution of total kinetic energy released to products is ascertained through velocity map imaging of the O ( 1 D) products. About half of the available energy, deduced from the theoretically computed asymptotic energy, is accommodated as internal excitation of the 2-butenal fragment. A reduced impulsive model is introduced to interpret the photodissociation dynamics, which accounts for the geometric changes between 2-butenal oxide and the 2-butenal fragment, and vibrational activation of associated modes in the 2-butenal product. Application of the reduced impulsive model to the photodissociation of isomeric methyl vinyl ketone oxide reveals greater internal activation of the methyl vinyl ketone product arising from methyl internal rotation and rock, which is distinctly different than the dissociation dynamics of 2-butenal oxide or methacrolein oxide.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Striations in moderate pressure dc driven nitrogen glow discharge

Plasma stratification has been studied for more than a century. Despite the many experimental studies reported on this topic, theoretical analyses and numerical modeling of this phenomenon have been mostly limited to rare gases. In this work, a one-dimensional fluid model with detailed kinetics of electrons and vibrationally excited molecules is employed to simulate moderate-pressure (i.e. a few Torrs) dc discharge in nitrogen in a 15.5 cm long tube of radius 0.55 cm. The model also considers ambipolar diffusion to account for the radial loss of ions and electrons to the wall. The proposed model predicts self-excited standing striations in nitrogen for a range of discharge currents. The impact of electron transport parameters and reaction rates obtained from a solution of local two-term and a multi-term Boltzmann equation on the predictions are assessed. In-depth kinetic analysis indicates that the striations result from the undulations in electron temperature caused due to the interaction between ionization and vibrational reactions. Furthermore, the vibrationally excited molecules associated with the lower energy levels are found to influence nitrogen plasma stratification and the striation pattern strongly. A balance between ionization processes and electron energy transport allows the formation of the observed standing striations. Simulations were conducted for a range of discharge current densities from ~0.018 to 0.080 mA cm –2 , for an operating pressure of 0.7 Torr. Parametric studies show that the striation length decreases with increasing discharge current. Furthermore, the predictions from the model are compared against experimental measurements and are found to agree favorably.

42 ENGINEERING↗
DOE OSTI2022

Determination of the characteristic magnetic pre-sheath length at divertor surfaces using micro-engineered targets on DiMES at DIII-D

The magnetic pre-sheath (MPS) width, L MPS , is a critical parameter to define the sheath potential, which controls the ion trajectory of low-Z species (D, T, He, and C), as well as the prompt re-deposition of high-Z species. In this work, to determine L MPS , we fabricated micro-trenches (30×30×4 µm) via focused ion beam (FIB) milling on a silicon surface and exposed them to L-mode deuterium plasmas in DIII-D via the Divertor Material Evaluation System (DiMES) removable sample exposure probe. The areal distribution of impurity depositions, mainly consisting of carbon, was measured by energy-dispersive X-ray spectroscopy (EDS) to reveal the deuterium ion shadowing effect on the trench floors. The carbon deposition profiles showed that the erosion was maximized for the azimuthal direction of φ = -40° (referenced to the toroidal magnetic field direction) as well as the polar angle of θ = 80°. A Monte Carlo equation-of-motion model, based on a collisionless MPS, was used to calculate the azimuthal and polar deuterium ion angle distributions (IADs) for a range of L MPS = k × ρ i , where ρ i is the ion gyro radius and k = 0.5-4. Then, gross erosion profiles were calculated by a Monte Carlo micro-patterning and roughness (MPR) code for ion sputtering using as input the calculated azimuthal and polar IADs for each value of k . Good agreement with the experimental C deposition profiles was obtained for the case k = 2.5-3.5. This result is consistent with a previous kinetic modeling prediction of k ~ 3, as well as previous analytical investigations that predicted the L MPS to be several ion gyro radii. A validation of theoretical sheath models supports its applicability to ITER and pilot plant divertors to successfully predict plasma-materials interactions.

Erosion↗
DOE OSTI2020

Characterization of light-induced potentials in the strong-field dissociation of O 2 +

In this work, we investigate theoretically the imprints of light-induced potentials (LIPs) on the dissociation dynamics of O 2 + molecular ions, as observable in angle-resolved fragment kinetic-energy-release (KER) spectra. Following the vibrational and rotational dynamics of the initial pump-laser-excited cationic nuclear wave packet, while accounting for the dipole coupling between the O 2 + ( a 4 Π u ) and O 2 + ( f 4 Π g ) electronic states in 800-nm 40-fs probe-laser pulses with peak intensities between 10 13 and 10 14 W / cm 2 , we calculate angle-resolved KER spectra which reveal characteristic energy- and angle-dependent fringe structures. These fringes shift downward in energy as the molecular alignment angle θ relative to the probe-pulse polarization direction increases from 0 to π / 2 . The angle-dependent shifts in the KER fringes increase for larger probe-pulse peak intensities and follow the angle and light-wave-intensity dependence of the vibrational spectrum in the associated Floquet bond-hardening well, which is a manifestation of transient O 2 + nuclear-probability trapping in the LIP during dissociation. By examining the rovibrational dynamics of the dissociating molecular cation near the light-induced conical intersection (LICI) in the cationic LIP surface at θ = π / 2 , we identify related angle-dependent structures in the KER spectra, suggesting a means for assessing the significance of LICIs in molecular dissociation pathways.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2026

Elucidating the Microscale Behavior and Phase Separation Kinetics of Thermally Responsive Ionic Liquid–Water Mixtures

Thermally responsive ionic liquids (ILs) exhibit liquid-liquid phase separation into a water-rich (WR) and ionic-liquid-rich (ILR) phase when heated above a lower critical solution temperature (LCST). This phase behavior has been leveraged for applications ranging from forward osmosis (FO) desalination, where the IL acts as a draw solute, to refrigeration and dehumidification cycles, where the IL acts as a liquid desiccant. While significant effort has been devoted to characterizing the thermodynamic and thermophysical properties of LCST ILs, their phase separation kinetics have not been investigated. In this work, we describe the macroscale phase separation kinetics (phase separation time) by gleaning insight into the microscale colloidal behavior of aqueous mixtures of four different materials, P 4444 TFA (tetrabutylphosphonium-2,4-trifluoroacetate), P 4444 DMBS (tetrabutylphosphonium-2,4-dimethyl-benzenesulfonate), N 4444 Sal (tetrabutylammonium salicylate), and P 4444 Sal (tetrabutylphosphonium salicylate) as a function of IL concentration at a separation temperature of 70 °C. We report the discontinuous microscale size distributions for each material and correlate their theoretical settling velocities to experimental phase separation times. The results indicate that a simple Stokes' law model can predict the phase separation time within reasonable accuracy. Overall, this work lays the foundation for understanding the micro- to macroscale phase separation behavior and kinetics of LCST ILs for various water-energy applications.

LCST↗
DOE OSTI2025

Cosmic millicharge background and reheating probes

We demonstrate that the searches for dark sector particles can provide probes of reheating scenarios, focusing on the cosmic millicharge background produced in the early universe. We discuss two types of millicharge particles (mCPs): either with, or without, an accompanying dark photon. These two types of mCPs have distinct theoretical motivations and cosmological signatures. We discuss constraints from the overproduction and mCP-baryon interactions of the mCP without an accompanying dark photon, with different reheating temperatures. We also consider the ∆n eff constraints on the mCPs from kinetic mixing, varying the reheating temperature. The regions of interest in which the accelerator and other experiments can probe the reheating scenarios are identified in this paper for both scenarios. These probes can potentially allow us to set an upper bound on the reheating temperature down to ~ 10 MeV, much lower than the previously considered upper bound from inflationary cosmology at around ~ 10 16 GeV. In addition, we derive a new “distinguishability condition”, in which the two mCP scenarios may be differentiated by combining cosmological and theoretical considerations. Finally, we discuss the implications of dedicated mCP searches, future CMB-S4 observations, and the target for experiments when considering the minimally allowed reheating temperature.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Harnessing Photoelectrochemistry for Wastewater Nitrate Treatment Coupled with Resource Recovery

Wastewater is a misplaced resource well suited to recover nutrients, value-added chemicals, energy, and clean water. A photoelectrochemical device is proposed to transform wastewater nitrates to ammonia and nitrous oxide, coupled with water oxidation. Numerical models were developed to quantify the dependence of process efficiencies and nitrogen-removal rates on light absorber band gaps, electrocatalytic kinetic parameters, competing oxygen reduction and hydrogen evolution reactions, and the reacting nitrate species concentrations that affect the mass-transfer limited current densities. With a single light-absorber and state-of-the-art catalysts, optimal solar-to-chemical efficiencies of 7% and 10% and nitrogen-removal rates of 260 and 395 gN m -2 day -1 are predicted for nitrate reduction to ammonia and nitrous oxide, respectively. The influence of competing reactions on the performance depends on the nitrate concentration and band gap of the light absorber modeled. Oxygen reduction is more dominant than hydrogen evolution to compete with the nitrate reduction reaction, but it is mass-transfer limited. Even with kinetic parameters that enhanced the driving forces for the competing reactions, the performance is only minimally affected by these reactions for optimally selected band gaps and nitrate concentrations larger than 100 mM. Here, theoretically predicted peak nitrogen removal rates and specific energy intensities are competitive with reported estimates for electrochemical and Sharon-Anammox processes for ammonia recovery and nitrogen removal, respectively. This result, together with the added benefit of harnessing sunlight to produce value-added products, indicates promise in the photoelectrochemical approach as a tertiary pathway to recover nutrients and energy from wastewater nitrates.

25 ENERGY STORAGE↗
DOE OSTI2025

Complexions at the iron-magnetite interface

Synthesizing distinct phases and controlling crystalline defects are key concepts in materials design. These approaches are often decoupled, with the former grounded in equilibrium thermodynamics and the latter in nonequilibrium kinetics. By unifying them through defect phase diagrams, we can apply phase equilibrium models to thermodynamically evaluate defects—including dislocations, grain boundaries, and phase boundaries—establishing a theoretical framework linking material imperfections to properties. Using scanning transmission electron microscopy (STEM) with differential phase contrast (DPC) imaging, we achieve the simultaneous imaging of heavy Fe and light O atoms, precisely mapping the atomic structure and chemical composition at the iron-magnetite (Fe/Fe 3 O 4 ) interface. We identify a well-ordered two-layer interface-stabilized phase state (referred to as complexion) at the Fe[001]/Fe 3 O 4 [001] interface. Using density-functional theory (DFT), we explain the observed complexion and map out various interface-stabilized phases as a function of the O chemical potential. The formation of complexions increases interface adhesion by 20% and alters charge transfer between adjacent materials, impacting transport properties. Our findings highlight the potential of tunable defect-stabilized phase states as a degree of freedom in materials design, enabling optimized corrosion protection, catalysis, and redox-driven phase transitions, with applications in materials sustainability, efficient energy conversion, and green steel production.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Prehydrodynamic evolution and its impact on quark-gluon plasma signatures

State-of-the-art hydrodynamic models of heavy-ion collisions have considerable theoretical model uncertainties in the description of the very early prehydrodynamic stage. Here we add a new computational module, K T Iso, that describes the prehydrodynamic evolution kinetically, based on the relativistic Boltzmann equation with collisions treated in the isotropization time approximation. As a novelty, K T Iso allows for the inclusion and evolution of initial-state momentum anisotropies. To maintain computational efficiency K T Iso assumes strict longitudinal boost invariance and allows collisions to isotropize only the transverse momenta. We use it to explore the sensitivity of hadronic observables measured in relativistic heavy-ion collisions to initial-state momentum anisotropies and microscopic scattering during the prehydrodynamic stage.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2020

Reactive Molecular Dynamics Simulations and Quantum Chemistry Calculations To Investigate Soot-Relevant Reaction Pathways for Hexylamine Isomers

Sooting tendencies of a series of nitrogen-containing hydrocarbons (NHCs) have been recently characterized experimentally using the yield sooting index (YSI) methodology. This work aims to identify soot-relevant reaction pathways for three selected C6H15N amines, namely, dipropylamine (DPA), diisopropylamine (DIPA), and 3,3-dimethylbutylamine (DMBA) using ReaxFF molecular dynamics (MD) simulations and quantum mechanical (QM) calculations and to interpret the experimentally observed trends. ReaxFF MD simulations are performed to determine the important intermediate species and radicals involved in the fuel decomposition and soot formation processes. QM calculations are employed to extensively search for chemical reactions involving these species and radicals based on the ReaxFF MD results and also to quantitatively characterize the potential energy surfaces. Specifically, ReaxFF simulations are carried out in the NVT ensemble at 1400, 1600, and 1800 K, where soot has been identified to form in the YSI experiment. These simulations account for the interactions among test fuel molecules and pre-existing radicals and intermediate species generated from rich methane combustion, using a recently proposed simulation framework. ReaxFF simulations predict that the reactivity of the amines decrease in the order DIPA > DPA > DMBA, independent of temperature. Both QM calculations and ReaxFF simulations predict that C2H4, C3H6, and C4H8 are the main nonaromatic soot precursors formed during the decomposition of DPA, DIPA, and DMBA, respectively, and the associated reaction pathways are identified for each amine. Both theoretical methods predict that sooting tendency increases in the order DPA, DIPA, and DMBA, consistent with the experimentally measured trend in YSI. This work demonstrates that sooting tendencies and soot-relevant reaction pathways of fuels with unknown chemical kinetics can be identified efficiently through combined ReaxFF and QM simulations. Overall, predictions from ReaxFF simulations and QM calculations are consistent, in terms of fuel reactivity, major intermediates, and major nonaromatic soot precursors.

fuel decomposition↗
DOE OSTI2020

Understanding how chemical structure affects ignition-delay-time $\phi$-sensitivity

$\phi$-sensitivity is the change in ignition delay time (IDT) with respect to the fuel-to-air equivalence ratio ($\phi$). High $\phi$-sensitivity is a desirable fuel property for applications in advanced compression ignition and multi-mode engine designs. Understanding how $\phi$-sensitivity depends on chemical structure is essential for selecting promising biofuels from the ever-growing list of proposed candidates. Here, we investigate the effect of chemical structure on $\phi$-sensitivity with experiment, simulation, and theory. Experimental Advanced Fuel Ignition Delay Analyzer (AFIDA) measurements for 2,4-dimethylpentane and diisopropyl ether provide evidence that branching and functional groups strongly impact $\phi$-sensitivity. Further insights into this dependence are obtained with 0-D kinetic simulations with existing mechanisms for n-pentane, diethyl ether, 3-pentanone, n-heptane, 2-methylhexane, 2,4-dimethylpentane, and 2,2,3-trimethylbutane. Quantum mechanical (QM) G4 calculations of low-temperature reactions help explain the observed experimental and simulation trends. Specifically, these QM calculations provide theoretical estimates of the ketohydroperoxide (KHP) dissociation rates, the HO 2 formation rates from peroxy radical (ROO), and the “cross-over” temperatures, i.e., the temperature at which ROO dissociation is favored compared to hydroperoxyl radical (QOOH) formation. Each of these reaction rates is compared to the n-alkane reference point to determine the impact of branching and different functional groups. Although kinetic mechanisms typically assume that KHP dissociation rates are invariant of chemical environment, our QM results suggest that this rate can span a range of roughly two orders of magnitude. We also discuss the importance of including the peroxy-hydroperoxy (OO-OOH) hydrogen transfer reaction for branched ethers. Finally, the insights gained assist in proposing a highly $\phi$-sensitive compound, namely, isopropyl propyl ether.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Understanding the Role of Borohydride Doping in Electrochemical Stability of Argyrodite Li 6 PS 5 Cl Solid‐State Electrolyte

This work elucidates the mechanism by which lithium borohydride (LiBH 4 ) doping into argyrodite-type Li 6 PS 5 Cl (LBH-LPSCl) solid-state electrolyte (SSE) enhances electrochemical stability. State-of-the-art electrochemical performance is achieved with 5 wt% borohydride. Symmetric cells achieve critical current density (CCD) of 7.3 mA cm −2 , versus 2.6 mA cm −2 for baseline-LPSCl. All solid-state batteries (ASSBs) employing lithium metal and NMC811 cathode are stable over 400 cycles at 0.5C, with capacity retention of 83%. An anode-free ASSB (AF-ASSB) is stable over 600 cycles, with capacity loss of 0.04% per cycle. 5LBH-LPSCl allows for enhanced low temperature operation, down to −14 °C. Yet the difference in electrolytes’ bulk microstructures and hardnesses are minimal, while ionic conductivity is incrementally improved (≈50%). Theoretical modeling indicates limited effect of substitution on thermodynamic stability of PS 4 3- units, which decompose when contacting Li. Instead, enhanced electrochemical stability is site-specific kinetic effect: In situ electrodeposition experiments using X-ray photoelectron spectroscopy (XPS) and time-of-flight secondary ion mass spectrometry (TOF-SIMS) reveal tri-layer SEI based predominately on Li 3 P/LiBH 4 /Li 2 S that blocks electrons while facilitating ion transport. This SEI manifests reduced interface resistance and accelerated nucleation and growth of metallic Li. With baseline-LPSCl the SEI based on Li 3 P/Li 2 S is substantially thicker, generating localized stresses that promote interfacial cracking while cycling.

argyrodite↗
DOE OSTI2020

Conformational Panorama and Chirality Controlled Structure–Energy Relationship in a Chiral Carboxylic Acid Dimer

Chirality recognition in dimers of tetrahydro-2-furoic acid (THFA) was studied in a conformer-specific manner using rotational spectroscopy and theoretical approaches. THFA shows a strong preference for the trans- over the cis-COOH configuration. Two drastically different scenarios are possible for the detectable (THFA) 2 : a kinetically preferred dimer bound by feeble interactions between two trans-THFAs or a thermodynamically favored dimer with a double hydrogen-bonded ring structure between two cis subunits. To identify the conformers responsible for the extremely dense rotational spectra observed, it was essential not only to locate several hundred homo/heterochiral (THFA) 2 minima in ab initio calculations but also to evaluate the energetic connectivities among the minima. The study further reveals an interesting chirality dependent structure-energy ordering relationship. A method for enantiomeric excess (ee) determination of THFA is presented using a recently proposed chiral self-tag approach.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Probing High-Temperature Amine Chemistry: Is the Reaction NH 3 + NH 2 ⇌ N 2 H 3 + H 2 Important?

The reaction NH 3 + NH 2 ⇌ N 2 H 3 + H 2 (R1) has been identified as a key step to explain experimental results for pyrolysis and oxidation of ammonia. However, no direct experimental or theoretical evidence for the reaction has been reported. In the present work, the reaction was studied by ab initio theory and by reinterpretation of experimental data. We could not locate a transition state for R1 occurring as a direct process, but alternative mechanisms yielded an upper bound to k 1 of 1.5 × 10 13 exp(–58.9 kcal mol –1 /RT) cm 3 mol –1 s –1 over 1000–2500 K, several orders of magnitude lower than values applied in modeling. Consistent with the theoretical work, re-evaluation of NH 3 pyrolysis data supported a very low value of k 1 . However, this finding opens up a novel unresolved issue. Current kinetic models cannot capture the NH 3 oxidation behavior in a number of laminar flow reactor and jet-stirred reactor experiments without adopting an improbably high value for k 1 . Furthermore, important oxidation steps might be underestimated or missing from mechanisms.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Nature of the bonded-to-atomic transition in liquid silica to TPa pressures

First-principles calculations and analysis of the thermodynamic, structural, and electronic properties of liquid SiO 2 characterize the bonded-to-atomic transition at 0.1–1.6 TPa and 10 4 –10 5 K (1–9 eV), the high-energy-density regime relevant to understanding planetary interiors. We find strong ionic bonds that become short-lived due to high kinetics during the transition, with sensitivity of the transition temperature to pressure, and our calculated Hugoniots agree with past experimental data. Furthermore, these results reconcile previous experimental and theoretical findings by clarifying the nature of the bond dissociation process in early Earth and “rocky” (oxide) constituents of large planets.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2021

Analytical modeling for redox flow battery design

Deeper market penetration of redox flow batteries requires optimization of the cell performance. Though important for optimization, detailed analytical solutions have not been developed for electrolyte flow, mass and charge transport, and reaction kinetics within redox flow batteries. To this end, here we present analytical solutions to active species concentration and over-potential based on advection-diffusion transport for ions and Bulter-Volmer model for interface reaction kinetics. The solutions were validated with results from a finite element model. These solutions were then applied to investigate the relationship between over-potential and state of charge, current density, reaction rate constant, flow velocity, diffusivity, total active species concentration, and electrode structure. Explicit formulas were identified for minimum activation over-potential and limiting current density as well as their dependence on electrolyte properties, operation conditions, and electrode structure. With our new mathematical formulas, this work provides a theoretical framework for flow battery design.

25 ENERGY STORAGE↗