We previously demonstrated (here) that, counterintuitively, moving your electrodes further apart can sometimes decrease the ohmic resistance. A small 0.2 mm gap between a perforated electrode and the diaphragm was found to increase energy efficiency compared to a zero-gap configuration. The reason is that a small gap allows electrolyte flow and gas bubbles to escape more easily.

A natural next question is, what is the optimal value of this gap? We found that a 0.06 mm gap still gives slightly better results than a 0.2 mm gap. Smaller values are unlikely to be significantly better, since bubbles are of this size and are likely to get trapped.

For longer distances, we found that multiphase computational fluid dynamics simulations were in good agreement with our experiments. We even developed an approximate analytical model of the gas fraction and resistance as functions of the gap distance and height. This surprisingly simple model allows for estimating how much the current density will decrease as a function of height.

van der Does, W. L., Valle, N., & Haverkort, J. W. (2026). Bubble resistance in near-zero-gap alkaline water electrolysis. Electrochimica Acta, 148509.

An often mentioned simple way to improve water electrolysis is to modulate the current, for example, by rapidly switching it on and off periodically. Many papers report improvements without a fair comparison with the non-modulated (DC) case. In such a comparison, the same amount of hydrogen is produced per unit time, so that the average current over time is equal.

Unfortunately, doing exactly this, we found no positive effect of modulating the current in the 0.01-1000 kHz range.

This is easy to understand, because energy losses usually increase more than linearly with current density: to obtain the same average current density, the lower energy losses at lower-than-average current are more than offset by the higher energy losses during periods of higher-than-average current.

There are a few noteworthy exceptions: for example, at low currents, the stack losses can decrease with increasing current density due to shunt currents. Also, by modulating at the time scale of gas build-up, a small advantage may be obtained. Finally, there may be a regime at extremely high frequencies at which the catalysis will be influenced. However, in general, claims of improvement should be considered with healthy scepticism.

For a cell voltage V₀+ARj, the larger the difference between the high current (j_high) and low current (j_high), and the larger the fraction of the time D or 1-D that the current is high or low, the higher the specific energy consumption (SEC) compared to DC (j_av) :

This prediction was nicely validated by in-house experiments with an AEM cell.

Anamika Ghosh, J.W. Haverkort, Modulating the current in water electrolysis does not increase energy efficiency compared to a constant equal average current, International Journal of Hydrogen Energy, Volume 221, 2026, 154195

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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