In stacks of electrolysers, redox flow batteries, or fuel cells with liquid electrolytes, ions can leave their half-cell through the fluid inlet or outlet ports and enter an entirely different half-cell. These pathways have a much higher resistance than the intended route through the membrane. However, in bipolar stacks, the driving force, the potential difference between half-cells that are far apart, is much larger. These ‘shunt currents’ can deteriorate the efficiency of electrolysers or drain a flow battery through internal self-discharge.

Therefore, many scientific works have measured shunt currents and successfully modelled them by numerically solving an equivalent circuit. It turns out that a very simple and intuitive approximation is also possible. The average shunt current in the manifold can be approximated by the average potential difference between cells divided by the series resistance of the manifold R_mn and the inlet/outlet port resistance R_io. However, the latter should be multiplied by 12/N(N+1), where N is the total number of cells in the stack. This can be shown both from a limiting-case solution, possibly dating back to a 1930s Russian book, and from correcting an overlooked German-language exact solution.

J.W. Haverkort, An analytical model for shunt currents in electrolyser, fuel cell, and flow battery stacks, Journal of Power Sources, Volume 670, 2026, 239447.

Hydrogen and oxygen are generated at the electrodes of an electrolyser in dissolved form. When the solubility is exceeded, the electrolyte supersaturates with dissolved gas, and bubbles nucleate at the surface. They grow by taking up dissolved gas until they are large enough that buoyancy or interactions with the flow or other bubbles cause them to leave.

After departure, modellers usually assume that the bubbles maintain their departure size. However, by analysing videos from the open literature, we found that they continue to grow as they rise.

This means that some distance from the electrode, the electrolyte is still supersaturated. This is important to know and model accurately because it can cause increased transport of hydrogen towards the oxygen side or oxygen towards the hydrogen side, potentially leading to explosive mixture conditions.

Venkatesh, N. K., Krikke, I. R. C., Valle, N., & Haverkort, J. W. (2025). Bubble growth after release from an electrode: A validated CFD model. Electrochimica Acta, 147827.

In many electrolysers, including water electrolysers for the production of green hydrogen, gas bubbles are created that can be used to drive natural electrolyte circulation. These bubbles rise due to buoyancy, setting the liquid electrolyte into motion. Normally, pumps are used to induce this electrolyte flow, which moves heat out of the cells. However, by designing an electrolyser well, pumps are no longer needed.

We developed a model that accurately predicts the liquid flow rate (Q_l) as a function of the gas flow rate (Q_g), matching our in-house experiments very well:

It includes the most important physical effects, including:

• viscous losses in the downcomer (Q_v)
• inertial losses due to bends and turbulence (Q_i)
• bubble slip (Q_s)
  
  From this equation we derived expressions for the optimal electrode-wall distance that, for example, maximizes the electrolyte flow rate. In this way our model will allow for the future design of more efficient and pump-free electrolysers.

Deiters, G. B., de Vries, J. R., & Haverkort, J. W. (2025). An experimentally validated analytical model for natural electrolyte recirculation in an alkaline water electrolyser. International Journal of Hydrogen Energy, 188, 152053.

Electrodes in modern water electrolysers for the production of green hydrogen contain holes to let bubbles through, but what size should these holes be? Despite more than half a century since its first usage, there is no clear guidance on this question available in the open literature.

Therefore, we performed an extensive experimental campaign under industrially relevant conditions, testing 17 electrodes with varying hole sizes and shapes. The most important feature, by far, determining the electrode performance turned out to be the hole size, not its shape, the electrode thickness, or other features like the presence of pillars.

We found that for most relevant current densities and atmospheric pressure a hole size of several millimeters is optimal. Much larger holes lead to larger cell voltages as the current becomes more inhomogeneously distributed, increasing the resistance. But particularly, much smaller holes lead to excessive overpotentials.

To study why this is, we looked at the behavior of gas bubbles, as they leave the holes from the back, but also from the front, through a transparent membrane. Holes small enough to be filled by a bubble equal to the hole size many times per second, tend to become clogged. As a result, a large gas film forms between the electrode and the diaphragm. This activates large parts of the electrode surface, forcing the reaction to the backside, increasing the resistance.

Millimeter-sized holes are largely immune to this clogging and perform much better, which explains why commercial expanded metal electrodes have holes in this size range.

J.W. Haverkort, A.S. Aghdam, E.J.B. Craye. The optimal electrode hole size in zero gap alkaline water electrolysis: A combined electrochemical, theoretical, and bubble imaging approach. Int. J. Hydrogen Energy (2025), 171, 150919.

Of course, the volume fraction of hydrogen gas bubbles cannot exceed 100%. Consider spherical gas bubbles of equal size, the maximum theoretical packing fraction is much lower at about 74 %, or even roughly 64 % for a random packing. The space between bubbles can be taken up by smaller bubbles and the space between those smaller bubbles again by smaller bubbles, so that theoretically there is no strict limit below 100%. Also, gas bubbles could coalesce to form a big bubble filling the entire electrolyser space so that the gas fraction could approach 100%.

However, it turns out that bubbles in strong electrolytes do not coalesce so easily and prefer to stay at a small distance from each other. This makes that these hydrogen and oxygen bubbles behave somewhat like solid particles. Therefore, we introduced into our models what in granular matter modelling is called solid pressure. As the gas fraction increases, a repulsive pressure avoids a further increase.

Figure: Simulations of the gas fraction near a zero-gap electrode. Red is about 60%.

We find that a maximum gas fraction around 65 % corresponds well with experimental results on the bubble-induced resistance. In this way, the extra ingredient of a solid pressure helps simulations correspond more to reality and as a result converge more easily as extremely high gas fractions are avoided.

W.L. van der Does, N. Valle, J.W. Haverkort, Multiphase alkaline water electrolysis simulations: The need for a solid pressure model to explain experimental bubble overpotentials, Int. J. Hydrogen Energy (2025), 102, p 295-303

CO₂ and water can be electrochemically converted to the more valuable and industrially useful mixture of CO and H₂ (syngas). However, as more CO₂ is converted relatively more H₂ is produced. Therofore, a tempting strategy is to use less catalyst downstream and produce relatively more of the desireable CO at lower catalyst costs.

We first made a simple model that we optimised for our self-defined “effectiveness” giving

On the left is the average CO₂ reaction rate coefficient <k> times the residence time L/U. Here q is the ratio between the inlet CO₂ and H₂ reaction rate constants.

By using less catalyst downstream, <k> decreases and a smaller residence time is better. However...the same effectiveness is obtained!

A similar conclusion is obtained for a more elaborate model including more physical and economical complexity. For a given residence time the catalyst distribution can be optimised using our freely available Matlab script. But in this case the improvements in effectiveness can be larger than those obtained by simply increasing the residence time, while at the same time using less catalyst!

Blake, J. W., Haverkort, J. W., & Padding, J. T. (2024). Less is more: Optimisation of variable catalyst loading in CO2 electroreduction. Electrochimica Acta, 145177.

Mass transfer at electrodes with gaseous products is excellent, because bubbles that coalesce or leave the electrode mix the fluid very well. Additionally, rising bubbles set the electrolyte into motion causing additional flow transport. Traditionally, these two contributions, k_μ and k_f, have been simply added. To improve upon this oversimplification, a new simple ‘addition rule’ was devised

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.

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A promising way to remove low salt concentrations from brackish water is capacitive deionization (CDI). Instead of removing the water from the salt, as for example in reverse osmosis, in CDI the salt ions are removed from the water by applying an electric field. The ions are removed from the main flow and stored in the electric double layers of the porous electrodes flanking the main flow channel. Despite the fact that this is an inherently two-dimensional and transient process, usually modelled with several partial differential equations, we managed to simplify the problem to two coupled ordinary differential equations and obtained an explicit analytical solution. Here is a comparison of the salt concentration from our analytical model (left) with that from a comprehensive computational COMSOL model (right):

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.

Various electrochemical cells including microfluidic fuel cells, membraneless redox flow batteries, and microfluidic (CO₂) electrolysers include a channel between their two electrodes. The wider this channel, the larger the ohmic drop. The thinner this channel, the larger the pressure drop. We obtained a simple analytical formula for the optimal channel width that minimizes the combined associated energy dissipation, which we verified using numerical simulations.

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).

Bhadra, A., & Haverkort, J. W. (2023). The optimal electrode pore size and channel width in electrochemical flow cells. Journal of Power Sources, 579, 233240.