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A comparison of Fick and Maxwell–Stefan diffusion formulations in PEMFC gas diffusion layers


This paper explores the mathematical formulations of Fick and Maxwell–Stefan diffusion in the context of polymer electrolyte membrane fuel cell cathode gas diffusion layers. The simple Fick law with a diagonal diffusion matrix is an approximation of Maxwell–Stefan. Formulations of diffusion combined with mass-averaged Darcy flow are considered for three component gases. For this application, the formulations can be compared computationally in a simple, one dimensional setting. Despite the models’ seemingly different structure, it is observed that the predictions of the formulations are very similar on the cathode when air is used as oxidant. The two formulations give quite different results when the Nitrogen in the air oxidant is replaced by helium (this is often done as a diagnostic for fuel cells designs). The two formulations also give quite different results for the anode with a dilute Hydrogen stream. These results give direction to when Maxwell–Stefan diffusion, which is more complicated to implement computationally in many codes, should be used in fuel cell simulations.

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  1. Wagner W, Kretzschmar HJ (2008) International steam tables: properties of water and steam based on the industrial formulation IAPWS-IF97, 2nd edn. Springer-Verlag, Berlin, Heidelberg

  2. Lide DR (ed) (2009) CRC handbook of chemistry and physics, 90th edn. CRC press (Taylor and Francis Group), Boca Raton, FL

  3. Bear J, Bachmat Y (1990) Introduction to modelling of transport phenomena in porous media. Kluwer, Dordrecht

    Book  MATH  Google Scholar 

  4. Borup R, Meyers J, Pivovar B, Kim YS, Mukundan R, Garland N, Myers D, Wilson M, Garzon Fernando, Wood D, Zelenay P, More K, Stroh K, Zawodzinski T, Boncella James, McGrath James E, Inaba M, Miyatake K, Hori M, Ota K, Ogumi Z, Miyata S, Nishikata A, Siroma Z, Uchimoto Y, Yasuda K, Kimijima K, Iwashita N (2007) Scientific aspects of polymer electrolyte fuel cell durability and degradation. Chem Rev 107(10):3904–3951

    Article  Google Scholar 

  5. Burheim OS, Ellila G, Fairweather JD, Labouriau A, Kjelstrup S, Pharoah JG (2013) Ageing and thermal conductivity of porous transport layers used for PEM fuel cells. J Power Sources 221:356–365

    Article  Google Scholar 

  6. Cayan Fatma N, Pakalapati Suryanarayana R, Elizalde-Blancas F, Celik I (2009) On modeling multi-component diffusion inside the porous anode of solid oxide fuel cells using fick’s model. J Power Sources 192(2):467–474

    Article  Google Scholar 

  7. Chang P, Kim GS, Promislow K, Wetton B (2007) Reduced dimensional computational models of polymer electrolyte membrane fuel cell stacks. J Comput Phys 223(2):797–821

    Article  MathSciNet  MATH  Google Scholar 

  8. Jeon DH, Greenway S, Shimpalee S, Van Zee JW (2008) The effect of serpentine flow-field designs on PEM fuel cell performance. Int J Hydrog Energy 33(3):1052–1066

    Article  Google Scholar 

  9. Martinez Michael J, Shimpalee S, Van Zee JW (2008) Comparing predictions of PEM fuel cell behavior using Maxwell–Stefan and CFD approximation equations. Comput Chem Eng 32(12):2958–2965

    Article  Google Scholar 

  10. Promislow K, Chang P, Haas H, Wetton B (2008) Two-phase unit cell model for slow transients in polymer electrolyte membrane fuel cells. J Electrochem Soc 155(7):A494–A504

    Article  Google Scholar 

  11. Promislow K, St-Pierre J, Wetton B (2011) A simple, analytic model of polymer electrolyte membrane fuel cell anode recirculation at operating power including nitrogen crossover. J Power Sources 196(23):10050–10056

    Article  Google Scholar 

  12. Promislow K, Stockie J, Wetton B (2006) A sharp interface reduction for multiphase transport in a porous fuel cell electrode. Proc R Soc A Math Phys Eng Sci 462(2067):789–816

    Article  MathSciNet  MATH  Google Scholar 

  13. Reshetenko T, St Pierre J (2014) Separation method for oxygen mass transport coefficient in gas and Ionomer phases in PEMFC GDE, J Electrochem Soc 161:F1089–F1100

    Article  Google Scholar 

  14. St-Pierre J (2011) Hydrogen mass transport in fuel cell gas diffusion electrodes. Fuel Cells 11(2):263–273

    Article  Google Scholar 

  15. Stockie J, Promislow K, Wetton B (2003) A finite volume method for multicomponent gas transport in a porous fuel cell electrode. Int J Numer Methods Fluids 462:186–789

    MATH  Google Scholar 

  16. Suwanwarangkul R, Croiset E, Fowler MW, Douglas PL, Entchev E, Douglas MA (2003) Performance comparison of ficks, dusty-gas and Stefan–Maxwell models to predict the concentration overpotential of a SOFC anode. J Power Sources 122(1):9–18

    Article  Google Scholar 

  17. Taylor M, Krishna R (1993) Multicomponent mass transfer. Wiley, Hoboken

    Google Scholar 

  18. Larminie J, Dicks A (2003) Fuel cell systems explained, 2nd edn. Hoboken, Wiley

    Book  Google Scholar 

  19. Wilkinson DP, Zhang J, Hui R, Fergus J, Li X (2009) Proton exchange membrane fuel cells: materials properties and performance. CRC Press, Boca Raton

    Google Scholar 

  20. Yang WJ, Wang HY, Kim YB (2013) Effects of the humidity and the land ratio of channel and rib in the serpentine three-dimensional pemfc model. Int J Energy Res 37(11):1339–1348

    Article  Google Scholar 

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The first author thanks NSERC for a graduate scholarship and the Automotive Fuel Cell Corporation (AFCC) and the MITACS Accelerate Internship programme for funding for this work. The second author acknowledges research funding support from an NSERC Canada grant. We would both like to thank the referee that suggested extending the air cathode results to the helox cathode and dilute hydrogen anode.

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Correspondence to Brian Wetton.

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Lindstrom, M., Wetton, B. A comparison of Fick and Maxwell–Stefan diffusion formulations in PEMFC gas diffusion layers. Heat Mass Transfer 53, 205–212 (2017).

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