Abstract
A mathematical model for polymer electrolyte membrane (PEM) fuel cells is developed and rigorous analytical solutions of the model are obtained. The modeling domain consists of the cathode gas channel, gas diffuser, catalyst layer, and the membrane. To account for the composite structure of the gas diffuser and for its gradient in liquid water content, the gas diffuser is modeled as a series of parallel layers with different porosity and tortuosity. Starting from the oxygen transport equations and Ohm's law for proton migration, expressions for the oxygen mass fraction distribution in the gas channel, gas diffuser, and catalyst layer, and current density and membrane phase potential in the catalyst layer and membrane are derived. The solutions are presented in terms of the physical and thermodynamic parameters of the fuel cell. The polarization curve is expressed parametrically as a function of the surface overpotential. Expressions for cathode internal and overall effectiveness factors, active fraction of the catalyst layer, catalyst layer resistance, limiting current density, and the slope of the polarization curve are also presented. Due to the advantage of the closed‐form solutions this model can be easily used as a diagnostic tool for a PEM fuel cell operating on 
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