The high mass transfer to or from gas-evolving electrodes is an attractive feature of electrochemical reactors, which can be partly attributed to the large convective flows that arise due to the buoyancy of bubbles. We derive exact analytical expressions for mass transfer coefficients for the case of constant gas flux boundary conditions. For the mass transport both Dirichlet and Neumann boundary conditions are considered. We deploy a recently derived self-similar solution of laminar two-phase flows, with density, hydrodynamic diffusivity, and viscosity dependent on the local gas fraction. Combining this with the Lévêque approximation, new mass transfer coefficients are obtained analytically. These new results are relevant for various electrochemical processes with gas evolution as well as boiling. The new formulation shows the mass transfer coefficient to scale with the vertical coordinate z proportional to z^−1/5 for short electrodes and low current densities and z^−4/15 for long ones and high current densities. The former limit also applies when buoyancy is due to temperature or concentration differences in the case that density differences are small. We provide a general overview considering all possible gas and mass boundary conditions combinations and a comparison with the Boussinesq approximation of small density differences.

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Electrochemical engineering deals with electrochemical devices like electrolysers, fuel cells, and batteries. While several excellent books exist in this long-standing and still growing field, their focus is usually on chemistry or phenomenology. In this textbook, we focus on mathematical modelling of the physical phenomena involved. Instead of resorting to numerical modelling, the aim is to derive simplified analytical models that maximise understanding. Porous electrodes, ion mass transport, and multiphase flow are central themes in this book. Examples include modelling the water saturation in a fuel cell diffusion layer, the gas fraction and current distribution in an alkaline water electrolyser, the potential distribution in a binary electrolyte inside porous battery electrode, and the concentration distribution in the flow channel of a redox flow battery. This makes for a diverse, challenging, and stimulating journey, for both students and researchers. @en

The development of a bubble plume from a vertical gas-evolving electrode is driven by buoyancy and hydrodynamic bubble dispersion. This canonical fluid mechanics problem is relevant for both thermal and electrochemical processes. We adopt a mixture model formulation for the two-phase flow, considering variable density (beyond Boussinesq), viscosity and hydrodynamic bubble dispersion. Introducing a new change of coordinates, inspired by the Lees–Dorodnitsyn transformation, we obtain a new self-similar solution for the laminar boundary layer equations. The results predict a wall gas fraction and gas plume thickness that increase with height to the power of 1/5 before asymptotically reaching unity and scaling with height to the power 2/5, respectively. The vertical velocity scales with height to the power of 3/5. Our analysis shows that self-similarity is only possible if gas conservation is entirely formulated in terms of the gas specific volume instead of the gas fraction.@en

The anodic co-production of hydrogen peroxide (H₂O₂) during alkaline water electrolysis has gained interest as a sustainable alternative for anthraquinone oxidation. However, electrochemical H₂O₂ production is often studied with idealized laboratory setups to determine the H₂O₂ formation kinetics. In this work, we perform the reaction with industrially relevant operating principles using a flow cell with separately recirculating anolyte and catholyte. We then fit the data to an analytical model that we derive based on mole balances that accounts for anodic generation, anodic oxidation, and bulk disproportionation of H₂O₂, as well as electrolyte volumes and electrode surface area. We performed experiments at 100, 200, and 300 mA cm^-2 to derive values for the reaction system. At 200 mA cm^-2, we found a generation rate of 0.037 mmol min^-1 cm^-2 (FE_H2O2 = 59%) and an anodic decomposition rate constant of 0.304 cm min^-1, with a bulk disproportionation rate constant of 1.85 × 10^-3 min^-1. We successfully applied our model to two sources in literature to derive values for their systems as well. In all cases, the contribution of anodic oxidation of H₂O₂ was found to be the larger loss mechanism in comparison to bulk disproportionation. Using the analytical model, we show that decreasing the reservoir volume is a simple way to increase the H₂O₂ concentration over time. Further refinement of the model can be achieved through the use of mass transfer relationships based on electrolyzer geometries to describe the anodic oxidation of H₂O₂ in the mole balance equations.

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In flow-by capacitive deionization (CDI) brackish water flows between two electrodes that capacitively remove salt. We assume low inlet concentrations so “salt shocks” appear in the electrodes and the process becomes diffusion-limited. For unit charge efficiency, a simplified model is derived consisting of two coupled partial differential equations. We obtain approximations, and exact solutions in terms of the Lambert W function, for the salt concentration as a function of time and space and for the equilibrium charge-voltage relation. These surprisingly simple solutions compare well with the results from comprehensive two-dimensional simulations. Useful analytical expressions are obtained for optimal geometrical and operational parameters that maximize the productivity and minimize the specific energy losses. By making cells much thinner the productivity can be increased an order of magnitude compared to typical values in the literature. The optimal electrode is found to be roughly six times thinner than the spacer. The associated pressure drop is around 0.4 bar per 1 mM of inlet salt concentration, making our recommendations practically feasible only for relatively low concentrations. The obtained model and analytical expressions provide useful guidance to strongly improve the design process.

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Poor mass transport to or from vertical gas-evolving electrodes can adversely impact energy efficiency and product purity in the production of hydrogen, chlorine, and various metals. A proper description that combines natural convection with micromixing of growing, coalescing, and departing bubbles is presently lacking. This work develops a simple, physically sound analytical model that includes the influence of bubble size, flow regime, and bubble surface coverage. By comprehensively reviewing mass transfer measurements from the water electrolysis literature, we observe that the surface coverage of oxygen bubbles increases much more strongly with increasing current density than an often-used square root scaling predicts. Strong differences are observed in the degree of micromixing of hydrogen and oxygen bubbles in alkaline and acidic electrolytes. These varied results can all be explained by a combination of electrocapillarity, and coalescence induced by either a high surface coverage or Marangoni flows.

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Microfluidic fuel cells, electrolyzers, and redox flow batteries utilize laminar flow channels to provide reactants, remove products and avoid their crossover. These devices often also employ porous flow-through electrodes as they offer a high surface area for the reaction and excellent mass transfer. The geometrical features of these electrodes and flow channels strongly influence energy efficiency. We derive explicit analytical relations for the optimal flow channel width and porous electrode volumetric surface area from the perspective of energy efficiency. These expressions are verified using a two-dimensional tertiary current distribution and porous electrode flow model in COMSOL and are shown to be able to predict optimal parameters in commonly used flow-through and interdigitated flow fields. The obtained analytical models can dramatically shorten modelling time and expedite the industrial design process. The optimal channel width and pore sizes we obtain, in the order of 100 microns and 1 micron respectively, are much smaller than those often used. This shows that there is a significant room for improvement of energy efficiency in flow cells that can sustain the resulting pressure drop.

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Understanding multiphase flow close to the electrode surface is crucial to the design of electrolyzers, such as alkaline water electrolyzers for the production of green hydrogen. Vertical electrodes develop a narrow gas plume near their surface. We apply the integral method to the mixture model. Considering both exponentially varying and step-function gas fraction profiles, we derive analytical relations for plume thickness, velocity profile, and gas fraction near the electrode as a function of height and current density. We verify these analytical relations with the numerical solutions obtained using two-dimensional mixture model simulations. We find that for low gas fractions, the plume thickness decreases with an increase in current density for an exponentially varying gas fraction profile. In contrast, the plume thickness increases with increasing current density at high gas fractions for an approximately step-function-shaped gas fraction profile, in agreement with experiments from the literature.

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The use of gas diffusion electrodes that supply gaseous CO₂ directly to the catalyst layer has greatly improved the performance of electrochemical CO₂ conversion. However, reports of high current densities and Faradaic efficiencies primarily come from small lab scale electrolysers. Such electrolysers typically have a geometric area of 5 cm², while an industrial electrolyser would require an area closer to 1 m². The difference in scales means that many limitations that manifest only for larger electrolysers are not captured in lab scale setups. We develop a 2D computational model of both a lab scale and upscaled CO₂ electrolyser to determine performance limitations at larger scales and how they compare to the performance limitations observed at the lab scale. We find that for the same current density larger electrolysers exhibit much greater reaction and local environment inhomogeneity. Increasing catalyst layer pH and widening concentration boundary layers of the KHCO₃ buffer in the electrolyte channel lead to higher activation overpotential and increased parasitic loss of reactant CO₂ to the electrolyte solution. We show that a variable catalyst loading along the direction of the flow channel may improve the economics of a large scale CO₂ electrolyser.

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This review provides a comprehensive overview of the dynamics of low-temperature water electrolyzers and their influence on coupling the three major technologies, alkaline, Proton Exchange Membrane (PEM) and, Anion Exchange Membrane (AEM) with photovoltaic (PV) systems. Hydrogen technology is experiencing considerable interest as a way to accelerate the energy transition. With no associated CO2 emissions and fast response, water electrolyzers are an attractive option for producing green hydrogen on an industrial scale. This can be seen by the ambitious goals and large-scale projects being announced for hydrogen, especially with solar energy dedicated entirely to drive the process. The electrical response of water electrolyzers is extremely fast, making the slower variables, such as temperature and pressure, the limiting factors for variable operation typically associated with PV-powered electrolysis systems. The practical solar-to-hydrogen efficiency of these systems is in the range of 10% even with a very high coupling factor exceeding 99% for directly coupled systems. The solar-to-hydrogen efficiency can be boosted with a battery, potentially sacrificing the cost. The intermittency of solar irradiance, rather than its variability is the biggest challenge for PV-hydrogen systems regarding operation and degradation.@en

Membraneless parallel-plate electrolyzers use electrolyte flow to avoid product crossover. Using a mixture model neglecting inertia, and assuming an exponential gas fraction profile, we derive approximate analytical expressions for the velocity profile and pressure drop for thin plumes. We verify these expressions using numerical solutions obtained with COMSOL and validate them using experimental data from the literature. We find that the wall gas fraction increases rapidly at small heights, but becomes fairly constant at larger heights. These expressions serve as a guiding framework for designing a membraneless parallel-plate electrolyzer by quantifying the maximum possible height. We find that buoyancy driven membraneless parallel-plate electrolyzers with a 3 mm gap can be designed with a maximum height of around 7.6 cm at 1000 A/m² for operation with 98% product purity at atmospheric pressure. For a forced flow at Re=1000, the same electrolyzer can be made around 17.6 cm tall at 1000 A/m². These limits can be further improved with smaller bubbles or higher pressure.

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The diffusion layer is a crucial part of most fuel cells and electrolyzers. We analytically solve a simplified set of visco-capillary equations for the gas and liquid saturation profiles inside such layers. Contrary to existing numerical simulations, this approach allows us to obtain general scaling relations. We derive simple explicit equations for the limiting current density associated with reactant starvation, flooding, and membrane dehydration, including the effect of fluid properties, contact angle, tortuosity, and the pore size distribution. This is the first explicit, extensive and thorough analytical modeling framework for the two-phase transport in an electrochemical cell that provides useful insights into the performance characteristics of the diffusion layer. A more even pore size distribution generally allows higher currents. Explicit expressions for the inimmum pore size and maximum layer thickness show that modern diffusion layers are typically well-designed.@en

Reducing the gap between the electrodes and diaphragm to zero is an often adopted strategy to reduce the ohmic drop in alkaline water electrolyzers for hydrogen production. We provide a thorough account of the current–voltage relationship in such a zero-gap configuration over a wide range of electrolyte concentrations and current densities. Included are voltage components that are not often experimentally quantified like those due to bubbles, hydroxide depletion, and dissolved hydrogen and oxygen. As is commonly found for zero-gap configurations, the ohmic resistance was substantially larger than that of the separator. We find that this is because the relatively flat electrode area facing the diaphragm was not active, likely due to separator pore blockage by gas, the electrode itself, and or solid deposits. Over an e-folding time-scale of ten seconds, an additional ohmic drop was found to arise, likely due to gas bubbles in the electrode holes. For electrolyte concentrations below 0.5 M, an overpotential was observed, associated with local depletion of hydroxide at the anode. Finally, a high supersaturation of hydrogen and oxygen was found to significantly increase the equilibrium potential at elevated current densities. Most of these voltage losses are shown to be easily avoidable by introducing a small 0.2 mm gap, greatly improving the performance compared to zero-gap.

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The electrochemical reduction of CO₂ on planar electrodes is limited by its prohibitively low diffusivity and solubility in water. Gas-diffusion electrodes (GDEs) can be used to reduce these limitations, and facilitate current densities orders of magnitude higher than the limiting current densities of planar electrodes. These improvements are accompanied by increased variation in the local environment within the cathode, with significant effect on Faradaic efficiency. By developing a simple and freely available analytical model of a cathodic catalyst layer configured for the production of CO, we investigate the relationships between electrode reaction kinetics, cell operation conditions, catholyte composition and cell performance. Analytical methods allow us to cover parameter ranges that are intractable for numerical and experimental studies. We validate our findings against experimental and numerical results and provide a derivation and implementation of the analytical model.

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Both daily and seasonal fluctuations of renewable power sources will require large-scale energy storage technologies. A recently developed integrated battery and electrolyzer system, called battolyser, fulfills both time-scale requirements. Here, we develop a macroscopic COMSOL Multiphysics model to quantify the energetic efficiency of the battolyser prototype that, for the first time, integrates the functionality of a nickel-iron battery and an alkaline electrolyzer. The current prototype has a rated capacity of 5 Ah, and to develop a larger, enhanced system, it is necessary to characterize the processes occurring within the battolyser and to optimize the individual components of the battolyser. Therefore, there is a need for a model that can provide a fast screening on how the properties of individual components influence the overall energy efficiency of the battolyser prototype. The model is validated using experimental results, and new configurations are compared, and the energy efficiency is optimized for the scale-up of this lab-scale device. Based on the modeling work, we find an optimum electrode thickness for the nickel electrode of 3 and 2.25 mm for the iron electrode with optimal electrode porosities in the range of void fraction of 0.15-0.35. Additionally, electrolyte conductivity and the gap thickness are found to have a small effect on the overall efficiency of the device.

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JOREK is a massively parallel fully implicit non-linear extended magneto-hydrodynamic (MHD) code for realistic tokamak X-point plasmas. It has become a widely used versatile simulation code for studying large-scale plasma instabilities and their control and is continuously developed in an international community with strong involvements in the European fusion research programme and ITER organization. This article gives a comprehensive overview of the physics models implemented, numerical methods applied for solving the equations and physics studies performed with the code. A dedicated section highlights some of the verification work done for the code. A hierarchy of different physics models is available including a free boundary and resistive wall extension and hybrid kinetic-fluid models. The code allows for flux-surface aligned iso-parametric finite element grids in single and double X-point plasmas which can be extended to the true physical walls and uses a robust fully implicit time stepping. Particular focus is laid on plasma edge and scrape-off layer (SOL) physics as well as disruption related phenomena. Among the key results obtained with JOREK regarding plasma edge and SOL, are deep insights into the dynamics of edge localized modes (ELMs), ELM cycles, and ELM control by resonant magnetic perturbations, pellet injection, as well as by vertical magnetic kicks. Also ELM free regimes, detachment physics, the generation and transport of impurities during an ELM, and electrostatic turbulence in the pedestal region are investigated. Regarding disruptions, the focus is on the dynamics of the thermal quench (TQ) and current quench triggered by massive gas injection and shattered pellet injection, runaway electron (RE) dynamics as well as the RE interaction with MHD modes, and vertical displacement events. Also the seeding and suppression of tearing modes (TMs), the dynamics of naturally occurring TQs triggered by locked modes, and radiative collapses are being studied.

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Flow-through electrolyzers, with flow parallel to the current, are used in a wide range of industrial applications. The presence of flow avoids concentration gradients but can also be used to separate evolved gases, allowing membrane-less operation. In this work, we propose a simple multiphase flow-through electrode model. We derive and experimentally validate an analytical expression for the minimum velocity required to ensure effective gas separation. From this relation, we analytically derive design parameters that show that significant energy savings can be made using flow, compared to a physical separator.

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A new compact electrode architecture with hollow pillar-shaped anodes and cathodes arranged in a ‘checkerboard’ pattern is analysed and shown to be equivalent to a particular arrangement of corrugated plate electrodes. Because all four sides of the flow channels are electrodes, this design takes up at least 1.5 to two times less volume compared to conventional ‘sandwich’-type configurations. The assumption underlying this theoretical scaling is illustrated with a 3D-printed metal prototype for alkaline water electrolysis using natural convection. For mass-transfer-limited electrolysers, fuel cells, electrowinning cells, and flow-batteries, the expected volume savings easily increase to a factor three or more.

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