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 part includes the effect of the diffusion layer thickness *L*, porosity є, maximum pore size *r*_{max}, a parameter λ -with higher values denoting a more even pore size distribition-, a material-specific constant *a*, and a characteristic current density *J _{n}*, depending on e.g. reactant viscosity and surface tension.

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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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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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:

and

]]>With a typical separator thickness L of 0.5 mm and effective diffusivity D_{–} of 10^{-9} m^{2}/s this gives about 0.5 A/cm^{2}, in the operating range of modern electrolyzers.

In the following graph, the measured voltage over the separator can be seen to diverge when a current larger than i_{0} is applied. The dashed lines show the behavior expected from a simple model.

For more information on the these measurements, the model, and their relevance for hydrogen production see:

and

]]>*Worried about the practicality of making this on a large scale?* A second patent was filed for a design with the same advantages, but conveniently manufactured from corrugated plate electrodes.

]]>Exit ‘sandwich’ – enter ‘checkerboard’ stacks

**The dimensionless electrode overpotential versus electrode thickness. For the notation used see: https://doi.org/10.1016/j.electacta.2018.10.065**

The seminal 1962 paper of Newman and Tobias provided exact but *implicit* analytical solutions. Introducing a generalization of the *effectiveness factor* concept*, *I obtained approximate *explicit* current-potential relations that are insightful and easy to use. Using these, analytical expressions could be derived for both the optimal electrode thickness and porosity of catalyst layers as well as battery electrodes.

“*A theoretical analysis of the optimal electrode thickness and porosity*” can be freely accessed through: https://doi.org/10.1016/j.electacta.2018.10.065

A poster summarizing the paper presented at Modval 2019 : poster

A short presentation adapted from a talk at the ISE 2018 conference in Bologna or at the ECCM Conference in The Hague in 2019.

An excellent first introduction to the modeling porous electrodes can be found at:http://www.joshuagallaway.com/?p=215

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The extremely hot plasma inside a *tokamak *nuclear fusion reactor ‘floats’ in a magnetic field to avoid contact with the walls. Plasma rotation has several different effects on the plasma stability, succinctly summarized in a comprehensive analytical stability criterion derived in the following publication:

**Stability of localized modes in rotating tokamak plasmas**

J. W. Haverkort and H. J. de Blank. *Plasma Physics and Controlled Fusion*, vol. 53, nr. 4, p. 045008. © 2011 DOI: 10.1088/0741-3335/53/4/045008

Besides well-known terms arising from the Kelvin-Helmholtz, magneto-rotational, and Rayleigh-Taylor instabilities, the criterion also contains two stabilizing terms due to rotation. One was described in an earlier post. Another less known effect was discovered to be due the Coriolis force also responsible for the circulating weather patterns in the earth atmosphere.

For certain conditions (See Eq. 3) the stabilizing influence of flow shear outweighs other destabilizing effects, allowing differential rotation to have a positive effect on tokamak stability.

**Flow shear stabilization of rotating plasmas due to the Coriolis effect**

J. W. Haverkort and H. J. de Blank. *Physical Review E*, vol. 86, 016411. © 2012 The American Physical Society DOI: 10.1103/PhysRevE.86.016411

**Tokamak simulation**

If you move an air parcel up in a quiet atmosphere its density will be higher than that of the air surrounding it so that it will fall back. The frequency of the resulting oscillation is called the Brunt–Väisälä frequency. The hot ionized *plasma*, inside the donut-shaped nuclear fusion *tokamak*, typically rotates around its central axis. The resulting centrifugal forces, similar to the effect of gravity, cause similar oscillations as in the atmosphere.

Through this effect rotation stabilizes and helps to keep the hot plasma confined. See also another post for a second positive effect of rotation.

**The Brunt-Väisälä Frequency of Rotating Tokamak Plasmas**

J. W. Haverkort, H. J. de Blank, and B. Koren

*Journal of Computational Physics*,

© 2012 DOI: 10.1016/j.jcp.2011.03.016