which takes into account the surface coverage *θ* of bubbles and a single empirical micromixing parameter *a*.

Next, this formula was compared with all available literature data for water electrolysis, revealing large differences. For hydrogen in alkaline electrolytes, bubbles hardly mix and the mass transfer is determined by natural convection. For hydrogen bubbles in sulphuric acid, on the other hand, natural convection can be usually neglected and for other cases, both mechanisms can be important.

The reason why there are these large differences in the parameter *a* is hypothesized to be due to solutal Marangoni convection, which repels hydrogen bubbles away from each other and the electrode in alkaline electrolytes while it attracts hydrogen bubbles towards each other and the electrodes in sulphuric acid.

Haverkort, J. W. (2024). A general mass transfer equation for gas-evolving electrodes *Int. J. Hydrogen Energy*, *74*, 283-296.

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.

]]>The power of such an analytical solution lies in making optimization much easier. Instead of seven individual physical and geometrical parameters, time, and a spatial coordinate, our solution primarily depends on a single dimensionless number that is a combination of these parameters.

We find that the optimal porous electrodes are roughly six times thinner than the spacer. By minimizing the energy losses and maximizing the amount of water processed, we find that an optimal design can increase the latter metric by an order of magnitude compared to typical values in the literature.

J.W. Haverkort, B. Sanderse, J.T. Padding, and J.W. Blake (2024). An analytical

ow-by capacitive deionization mode. Desalination 582, 117408.

To increase surface area while avoiding diffusion limitations, sometimes porous flow-through electrodes are used. Here, a similar optimization can be performed. Smaller pore sizes give more reactive surface area, but also a larger pressure drop. Also here an analytical formula was obtained, which works well even for the popular interdigitated flow configuration.

We compared our formulas with values from various papers in the literature and found that typically an order of magnitude too large channels and pores are used. Therefore, significant improvements in energy efficiency can be obtained by further miniaturization.

The flow velocity (arrows) and reactant concentration (color) inside a repeating unit of an interdigitated flow field (left). The combined activation and pumping dissipation from simulations (diamonds) and the obtained analytical expression (solid line) along with the optimal (dashed vertical line) volumetric surface area a (roughly the inverse of pore size).

]]>Therefore, we made an analytical model to describe the plume thickness and associated velocity profile. To validate this model, we performed computational simulations. A comparison between the analytical and computational models is shown in the below figure. The resulting formulas will be very useful for scaling up electrolyzers and understanding the effect of height and current density on gas hold-up and heat transport.

Rajora, A., & Haverkort, J. W. (2023). An analytical model for the velocity and gas fraction profiles near gas-evolving electrodes. *International Journal of Hydrogen Energy*.

Joseph W. Blake, Vojtecȟ Konderla, Lorenz M. Baumgartner, David A. Vermaas, Johan T. Padding, and J. W. Haverkort, Inhomogeneities in the Catholyte Channel Limit the Upscaling of CO_{2} Flow Electrolysers, *ACS Sustainable Chem. Eng.* 2023, 11, 7, 2840–2852

To determine how tall a membraneless electrolyzer can be made while still avoiding overlap between the oxygen and hydrogen gas plumes, we developed an analytical model that we verified with a more complete computational model and validated with experimental data from the literature. Based on our model we show that natural convection can allow safe and efficient atmospheric membraneless electrolysers up to about 5-10 cm height, while forced flow adds another 10 cm. At higher pressure, or by inducing smaller bubbles, taller or more energy efficient electrolysers of this type can be made.

See this previous post for an alternative type of membraneless electrolyzer.

]]>In a previous post we reported on a fully analytical multiphase model of the gas-diffusion layer (GDL). Recently, we succeeded in making an analytical model of the catalyst layer (CL), including transport, electrochemical reactions and homogeneous reactions involving CO_{2}, OH^{–}, HCO_{3}^{–}, and CO_{3}^{2-}.

Our publication comes with a spreadsheet you can download that implements our analytical model.

]]>However, it has been found, time and again, that the ohmic resistance of zero-gap configurations is substantially higher than expected from the separator resistance. This can be partially attributed to the current lines not being straight in a zero-gap configuration, see the below figure. With detailed measurements we find that over the course of approximately 10 seconds the ohmic drop further increases. This can be attributed to the formation of gas bubbles, see the below figure. With this, we can now finally explain the anomalously high ohmic drops found in zero-gap water electrolysers.

Besides these bubble-induced losses, our detailed measurements further quantify the often-overlooked effect of dissolved gases on the equilibrium potential (*E _{eq}* in the below figure) as well as the concentration overpotential due to hydroxide depletion, only relevant at low electrolyte concentrations.

We also show that with a very small gap the ohmic drop can be significantly reduced, see the above figure. This initially counter-intuitive result can now be understood by the gap being large enough for bubbles to escape, allowing the current lines to become straighter. This simple and cheap way of increasing the efficiency of hydrogen production likely also leads to improved durability of the electrodes and separators as well as reduced gas cross-over. This can increase the turndown ratio, important for flexible operation of electrolysers with renewable energy.

]]>Many types of fuel cells and electrolyzers contain a diffusion layer, in between the reactant flow channels and the membrane. This porous layer protects the catalyst layer and provides an electronic connection to the current collector.

To allow reactants to reach the catalyst layer by diffusion, this layer should not be too thick. Liquid products can block the pores. We needed a somewhat lengthy and analysis to show that the resulting maximum current density reads

The first part describes the well-known diffusion-limited current density. The second term describes the reduction of the limiting current due to products blocking the pores. Here *j _{n}* is a characteristic current density depending on e.g. reactant viscosity and surface tension, and λ is a parameter describingthe the pore size-distribution .

The effect of the second term in this equation is usually not dominant, decreasing the limiting current by less than half. Therefore, perhaps somewhat underwhelming, the result of our mathematical exercise is that modern diffusion layers are quite well-designed. At least we now know why and under what conditions.

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