The purpose of a good physical-chemical model is to aid cell design and operation. All such models must obey balance equations as well as constitutive equations of transport. The set of transport equations must conform with the entropy balance. The difficulty is to choose the right scale of description. The pore scale may be too small, when the cell’s overall performance is of interest. But the variables given by classical thermodynamics are obviously too crude, when we want to know the location, say, of hot spots. For the material designer, knowledge of hot spots is central. The method of coarse-graining of the porous medium or the choice of the representative elementary volume (REV) becomes then central [1,2].

In this work, we present a method for integration across various porous electrodes and heterogeneous layers of a fuel cell. This entropy balance is used to check the model for consistency [1]. The REV is defined in terms of Gibbs excess properties [1,2]. The electrode surfaces appear as dynamic boundary conditions for the transport processes in the electrolyte, and connecting leads. We show how the theory of non-equilibrium thermodynamics can be used to find a consistent thermodynamic model.

The fuel cell is taken as example, and we show a simultaneous set of cell profiles for electric potential, concentration, temperature, entropy production and simultaneous heat fluxes out of the cell on both sides. The heat flux through the cell is illustrated in Fig.1. The figure shows how the heat flux varies through the membrane-electrode assembly for various current densities, 500, 2500 and 5000 A.m^-2. The electrode regions are indicated as finite slabs. The external temperature is kept constant at 340 K. Heat is leaving the cell on both sides, slightly more on the anode side that the cathode side.

One advantage of this formulation is the insight provided by the link from molecular mechanisms, via coupling coefficients for heat, mass and charge transport, to the integrated overall performance. The importance of the boundary conditions for the results will be demonstrated in the talk, in particular the effect of changing the boundary temperature. We conclude that the formulation gives a good base for cell performance optimization [3,4].

Acknowledgement The authors are grateful to the Research Council of Norway through its Center of Excellence funding scheme, PoreLab project no. 262644.

References

1. Signe Kjelstrup and Dick Bedeaux, Nonequilibrium thermodynamics of heterogeneous systems, World Scientific, Singapore, 2008, Chapter 19
2. Signe Kjelstrup, Dick Bedeaux, Alex Hansen, Bjørn Hafskjold, Olav Galteland, Non-isothermal transport of multi-phase fluids in porous media. The entropy production. Frontiers in Physics 6 (2018) 126
3. S. Kjelstrup, M-O Coppens, J. G. Pharoah and P. Pfeifer, Nature-Inspired Energy- and Material- Efficient Design of a Polymer Electrolyte Membrane Fuel Cell, Energy & Fuels, 24 (2010) 5097-5108
4. A. Zlotorowicz, Kaushik. Jayasayeed, P.I. Dahl, M. S. Thomassen and S. Kjelstrup, Tailored Porosities of the Cathode Layer for Improved Polymer Electrolyte Fuel Cells, J. Power Sources, 287 (2015) 472-477

Fig. 1. The heat flux through the polymer electrolyte fuel cell at various current densities (in A.m^-2) 500 (unbroken line), 2500(dotted curve) and 5000 (dashed curve). The membrane is sandwiched between electrode layers at positions around 0.2 and 0.3 mm. We see a discontinuity in the heat flux at the electrode interfaces due to the reaction. The temperatures at both end points are here the same, 340 K.