Two-dimensional, transient, yet analytical model for capacitive deionization

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 with a comprehensive computational COMSOL model:

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.

What gap and pore size maximize the energy efficiency of electrochemical flow cells?

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 Sources579, 233240.

How thick is the bubble layer near an electrode ?

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.

Reasonable agreement is obtained between the velocity profiles of our new analytical model (solid lines) and those of a computational mixture model (arrows).

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.

What would be the performance of a 1 m tall CO2 electrolyzer?

CO2-electrolysis is a promising way of making carbon-based fuels and chemicals from CO2, instead of from fossil fuels. Affordable electrolyzers that can efficiently convert CO2 and water into for example syngas (CO and H2) 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 CO2 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 CO2 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 CO2 Flow Electrolysers, ACS Sustainable Chem. Eng. 2023, XXXX, XXX, XXX-XXX

What is the maximum height of a membraneless electrolyser?

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.

A comparison of the electrolyte velocity obtained numerically (arrows) with our analytical model (solid line)

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.

Rajora, A., & Haverkort, J. W. (2022). An Analytical Multiphase Flow Model for Parallel Plate Electrolyzers. Chemical Engineering Science, 117823.

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

Analytical CO2R Model

The electrochemical reduction of CO2 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 CO2 and H2O to CO and H2, 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 CO2, OH, HCO3, and CO32-.

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

Blake, J. W., Padding, J. T., & Haverkort, J. W. (2021). Analytical modelling of CO2 reduction in gas-diffusion electrode catalyst layers. Electrochimica Acta, 138987.

Does zero-gap minimize ohmic drop?

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.

Tentative current line distributions without (left) and with (right) gas (light gray) – near the separator (blue) and electrodes (dark gray).

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

Using a small gap, the cell voltage as a function of current density j can be significantly reduced.

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.

J.W. Haverkort, H. Rajaei (2021), Voltage losses in zero-gap alkaline water electrolysis, Journal of Power Sources

Modern Diffusion Layers are Fine

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 jn 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.

Rajora, A., & Haverkort, J. W. (2021). An Analytical Model for Liquid and Gas Diffusion Layers in Electrolyzers and Fuel Cells. Journal of The Electrochemical Society.

Can flow efficiently separate hydrogen from oxygen in an electrolyser?

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

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.

Rajaei, H., Rajora, A., & Haverkort, J. W. (2021). Design of membraneless gas-evolving flow-through porous electrodes. Journal of Power Sources491, 229364.

Why does the cathode level rise?

In a simple electrolysis experiment we found the cathode electrolyte level to decrease by several centimetre over the course of an hour, while that of the anode increased by a similar amount.

Since water is consumed at the cathode and produced at the anode and hydroxide ions drag still more water to the anode, this is an unexpected result. Our explanation is that the electric field, acting on positive charges near the separator pore walls, gives rise to an electro-osmotic flow from anode to cathode.

This flow impacts the limiting current as well as the crossover of dissolved hydrogen and oxygen and can therefore be useful in increasing the hydrogen purity and extending the operational range.

For more information, see:

Haverkort, J. W., & Rajaei, H. (2020). Electro-osmotic flow and the limiting current in alkaline water electrolysis. Journal of Power Sources Advances6, 100034.0034


Haverkort, J. W. (2020). Modeling and Experiments of Binary Electrolytes in the Presence of Diffusion, Migration, and Electro-Osmotic Flow. Physical Review Applied14(4), 044047