Since water is the reactant in water electrolysis, you may be excused for thinking there will be no diffusion limitations. However, at the anode of an alkaline water electrolyzer the reactant is hydroxide (OH-), produced at the cathode. Although usually present at very high concentrations c0 of 6 or 7 M, these ions may deplete at the anode, leading to a limiting current density given by:
With a typical separator thickness L of 0.5 mm and effective diffusivity D– of 10-9 m2/s this gives about 0.5 A/cm2, 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 i0 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:
Haverkort, J. W., & Rajaei, H. (2020). Electro-osmotic flow and the limiting current in alkaline water electrolysis. Journal of Power Sources Advances, 6, 100034.0034
Haverkort, J. W. (2020). Modeling and Experiments of Binary Electrolytes in the Presence of Diffusion, Migration, and Electro-Osmotic Flow. Physical Review Applied, 14(4), 044047
In many electrochemical stacks, a large fraction of the total volume goes to flow channels that transport reactants and products into and out of the cell. In the conventional bipolar ‘sandwich’ design, each flow channel borders only a single electrode. Pim van der Stighel, then a second year BSc student, in a process engineering & thermodynamics class suggested an alternative in which each electrode faces four other electrodes – sufficiently novel and promising to file a patent application together. Another team of BSc students, in co-supervision with Hadi Rajaei, made this into a metal-3D-printed prototype producing hydrogen. Our recent paper shows that electrolysers for hydrogen production can be easily made up to 1.5-2 times less voluminous, while for fuel cells or flow batteries the volume of the flow channels can even be reduced by a factor 3-4.
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.
Patent – Paper
Exit ‘sandwich’ – enter ‘checkerboard’ stacks
Most electrodes used in commercial applications like batteries and fuel cells contain electrodes that are porous to increase the reactive surface area. Strangely, a hitherto largely unanswered but important question is how thick such electrodes should ideally be. Thick electrodes will gives a high resistance while thin electrodes less surface area. Unsurprisingly there is an optimum, graphically shown in the below graph.
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
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
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
Particle trajectories in a parabolic and constant velocity profile
Capturing particles or droplets from a flow is relevant for various applications like continuous separators, aerosol removal, and magnetic drug targeting. In a laminar flow, doubling the capturing of a small fraction of particles turns out to require a four times higher force or length of pipe or a four times lower flow velocity (Eq. 12). This is because particles from twice as far away have to be captured from a location where the flow velocity in a laminar flow is also twice as high. This simple scaling law, with an analytical correction for flow through cylindrical pipes (Eq. 35), turns out to hold well for a wide range of different force fields.
Magnetic particle motion in a Poiseuille flow
J. W. Haverkort, S. Kenjeres, and C. R. Kleijn
Annals of Biomedical Engineering, vol. 37, nr. 12, p. 2436-2448
© 2009 DOI: 10.1007/s10439-009-9786-y
By attaching drugs to magnetic nanoparticles, magnetic fields can concentrate them at the location in the body where they are needed. Pre-clinical trials have shown some potential for treatment of superficial cancer tumors. More applications could be envisioned when targets deeper in the body can be reached.
Our 2009 publication was perhaps the first three-dimensional simulation showing that it is possible to capture particles from the bloodstream of large arteries like the coronary and carotid artery. This opens up the possibility of applying the technique to combat also cardiovascular diseases.
Because of an old theorem, the drugs can only be held in a stable position deep inside the body using a dynamic magnetic field configuration or in a quasi-stable position using carefully tailored magnetic fields. Despite this fundamental complication, progress remains to be made today, particularly from the perspective of computational modeling.
Computational Simulations of Magnetic Particle Capture in Arterial Flows
J. W. Haverkort, S. Kenjeres, and C. R. Kleijn
Magnetohydrodynamics of insulating spheres
J. W. Haverkort, T. W. J. Peeters
A widespread notion in the field of ‘ideal’ magnetohydrodynamics describing highly conducting fluids there can be no electrical charge accumulation. Any nonzero charge will quickly redistribute over a short enough time-scale to be irrelevant. One can however easily show that in the presence of a magnetic field Lorentz forces can act to sustain a finite charge density, see equation 3. This charge and the associated electric field have a distinct influence on the current distribution and the resulting forces on non-conducting inclusions and bubbles in a conducting fluid as shown in the above picture and explained in the paper.
In the industrial processing of metals, magnetic fields are often used to stir or break liquid metals, calm free surfaces, influence turbulence properties and remove unwanted inclusions.
Even though bubbles and solid inclusions in liquid metals may not conduct electricity themselves, because the surrounding metal does, some interesting magnetohydrodynamic effects arise, like the electromagnetic Archimedes force, and electromagnetically induced drag.
In a continuous caster, liquid steel enters a mold and slowly turns into a solid slab. Under the influence of unwanted flow patterns small bubbles or inclusions can be trapped into the steel, forming elongated ‘blowholes’ after rolling the steel slabs. To suppress unwanted liquid steel flows enormous electromagnets fully surrounding the casting mold. Currents induced by the flow though this magnetic field generate Lorentz forces that brake the flows.
Through computer simulations we discovered a new effect on gas bubbles in the submerged entry nozzle of a continuous caster. We show that, depending on the shape of the magnetic field, the so-called electromagnetic Archimedes force can force Argon bubbles radially outwards. This effect may be used improving the anti-clogging capacity of gas.
Magnetohydrodynamic Effects on Insulating Bubbles and Inclusions in the Continuous Casting of Steel
J. W. Haverkort and T. Peeters