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.

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 (CO_{2}) 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).

Hydrogen bubbles generated at an electrode of a water electrolyzer rise due to their buoyancy and set the surrounding liquid into motion. Alternatively, the bubbles make the mixture lighter causing it to rise – similar to the hot fluid near a heated plate. An important difference with this well-studied ‘thermal natural convection’ case is that the gas fraction has a certain maximum, below one, which influences the plume in a previously unknown way. This understanding is important, for example, to describe heat and mass transport. Knowing the bubble layer thickness is also useful to properly choose the dimensions of an electrolyzer.

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.

CO_{2}-electrolysis is a promising way of making carbon-based fuels and chemicals from CO_{2}, instead of from fossil fuels. Affordable electrolyzers that can efficiently convert CO_{2} and water into for example syngas (CO and H_{2}) hold great promise for transforming the way carbon-based products can be industrially made in the future. There has been enormous research progress over the last few years, typically in lab set-ups with dimensions of a few centimeter. In a recent publication we tried to answer the question what would happen in a much taller electrolyzer. We find that much more of the CO_{2} entering the electrolyser is converted and also the fraction of the desirable CO in the outlet stream is increased. Unfortunately, our simulations show that this seemingly positive effect arises because more CO_{2} undergoes undesirable side-reactions with the electrolyte. As a partial remediation, we show that by varying the amount of catalyst along the electrolyzer this problem can be substantially reduced.

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

Water electrolyzers without a membrane have the potential to make green hydrogen more energy efficiently and cheaper. Perhaps the simplest type of membraneless electrolyzer consists of two vertical parallel plate electrodes with upwards electrolyte flow separating hydrogen and oxygen bubbles, avoiding the formation of an explosive mixture. The faster the flow, the thinner the bubble plumes and the closer the electrodes can be placed together, resulting in a higher energy efficiency. Natural convection can only provide modest velocities. Forced flow can provide higher velocities, but to avoid turbulent mixing of the bubble plumes, these are limited to similarly modest laminar flow velocities.

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.

The electrochemical reduction of CO_{2} from flue gas, or even directly from the atmosphere, to useful products is an alluring prospect. Liquid fuels like gasoline can be made our of just air, water, and renewable electricity. One of the most promising routes is to reduce CO_{2} and H_{2}O to CO and H_{2}, or syngas. This can be done in a gas-diffusion electrode using silver catalyst particles.

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.

In alkaline water electrolysers for hydrogen production, mesh electrodes are separated by a porous separator to avoid mixing of hydrogen and oxygen bubbles. Since the ohmic resistance increases with the distance over which the current travels, you would expect that minimizing the distance between the electrodes leads to the highest energy efficiency. Therefore, the so-called zero-gap configuration, in which the electrodes are placed directly adjacent to the separator, has become the standard in industrial electrolysers.

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.

When making hydrogen in an electrolyser, at the same time also oxygen is produced. To avoid an explosive mixture, a sub-microporous separator is placed in between the electrodes. However, this separator leads to large energy losses and allows dissolved gas to pass (see also https://jwhaverkort.weblog.tudelft.nl/?p=140).

Electrolyte flow can be used to separate all hydrogen and oxygen and make for a membraneless electrolyser.

The question is whether flow can ensure separation more energy-efficiently than a physical separator. Using a combination of modelling and experiments we found the answer to be: yes!

Placing the electrodes approximately half a millimetre apart the ohmic losses can be made much smaller than with a separator, while the pumping power adds only a small additional loss.

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