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PEM Fuel Cells, Energy Conversion, and Mathematics

Cross-section of a unit cell. Water (liquid or vapor) is formed at the cathode catalyst. Liquid water hydrates the membrane and increases its protonic conductivity, but may also flood the catalyst layer pores, preventing oxygen from reaching the catalyst layers. Promislow and Wetton, 2009. Click to enlarge.

An open-access paper published on 17 July in the Society for Industrial and Applied Mathematics (SIAM) Journal on Applied Mathematics: Special Issue on Fuel Cells examines the mathematical issues that arise when modeling PEM fuel cells. The paper PEM Fuel Cells: A Mathematical Overview is co-authored by Keith Promislow of the Michigan State University and Brian Wetton of the University of Vancouver.

PEM fuel cells are good examples of energy conversion systems that have several levels of interacting functional structures. The interactions range from proton exchange at the nanoscale level to interactions at the macroscale level among the layered media of which the cells are made. Accurately simulating the resulting multiscale interactions requires carefully constructed mathematical models that faithfully represent the physics at the various scales.

Modeling and analysis of PEM fuel cell structures, their construction, performance, and degradation also requires the development of new mathematical solutions and highly structured and highly adaptive numerical techniques. Mathematical analysis and scientific computation will play a large role in the resolution of these important issues and as a result will affect the progress of PEM fuel cell research and development.

Polymer electrolyte membrane (PEM) fuel cells employ a hydrophobic polymer, functionalized by acidic sidechains, as the electrode separator. They enjoy rapid startup and shut-down transients and efficient operation when hydrogen gas is used as the fuel. Moreover, they possess a very rich, multiscale structure that requires many levels of models to describe. Indeed, an emerging theme in the realm of energy conversion is the need for nanostructured composites composed of interpercolating networks with selective transport properties feeding reactants to catalyst sites with large surface area densities.

We focus our attention on the mathematical description of their operation with attention towards the optimization of performance. This includes the development of structured nanocomposites, such as the catalyst layers and electrode separator; the understanding of the macroscopic properties that influence the fuel cell performance under different operating conditions, such as the multiphase flow associated with water management; the modeling of various parasitic reactions that can degrade the fuel cell’s performance over time, particularly corrosion of carbon in the catalyst layer; and finally the integration of subscale models into stack-level codes that describe the operation of hundreds of fuel cells in series.

—Promislow and Wetton


  • Keith Promislow and Brian Wetton (2009) PEM Fuel Cells: A Mathematical Overview. SIAM J. Appl. Math. Volume 70, Issue 2, pp. 369-409 doi: 10.1137/080720802



Ballard made huge improvements to the GE PEM design in the 1990s. They just continued to work on the design until it was viable. I think we need to continue with that. The Honda FCX Clarity is an example of improving on fuel cells for cars day by day until the are about the best that the state of the art can provide.

Henry Gibson

The mathematics about fuel cells can go on to deal with the efficiencies of conversions of any form of energy to hydrogen and the efficiencies of storing and transporting hydrogen along with the costs of such systems. The costs of fuel cell systems must be compared to the costs of any other converter system. It would probably be shown that diesel plug-in-hybrids will be more energy efficient at far lower cost. Hydrogen can be fed simply with the intake air to replace a very high percentage of the diesel.

There are now people in the world who have enough money to buy a car powered with the Plutonium isotope Pu238. This plutonium cannot be used to make a bomb in any amount or combination, nor can it be used by itself as the fuel metal in any fission reactor. It is safe enough to have been used in long life pacemakers. It can provide enough heat to power a stirling engine or even a steam turbine. It is far less dangerous in crashes than gasoline or even diesel. And certainly it is less dangerous than some or even most lithium batteries.

In France you can buy a car that is powered by nuclear power for the lowest CO2 release per mile possible. Even hydroelectric facilities release Methane and CO2 from the anaerobic decomposition of the organic materials from the river tributaries. ..HG..


The PNGV cars were all diesel hybrids that got 70-80 mpg, that is really good. The car makers said no one would buy them at the price they would have to charge. If gasoline becomes $5 per gallon in the U.S. a car that gets over 60 mpg would look pretty good even with a $30,000 price tag.

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