On the smallness conditions for a PEMFC single cell problem - Consiglieri, Luisa (Journal of engineering mathematics, 2025)
The aim of the present paper is to determine the conditions necessary for existence of a solution to a problem involving fuel cells. In the first part, we present the model for a proton exchange membrane fuel cell (PEMFC) single cell and we clarify the interactions of the different components namely, velocity, pressure, density, temperature, and potential. The final mathematical model consists of the Stokes–Darcy system in the fluid-porous domain altogether with the heat, charge, and diffusion equations, which involve the Fourier, Ohm and Fick fluxes, the Dufour–Soret (or thermodiffusion effect), and Peltier–Seebeck (or thermoelectrical effect) cross effects, the Nernst–Einstein relation, and the Joule heating effect. We complete the quasilinear elliptic system by considering the Beavers–Joseph–Saffman condition on the fluid-porous interface, and a Butler–Volmer-type condition, under the presence of a known limiting current, on the membrane interface. The proof of existence of weak solutions relies on the Tychonoff fixed point theorem, by providing some regularity and some smallness conditions. We divide the actual system into two systems of equations and study them separately. The novelty of the present work is to establish quantitative estimates for improving the technical hypotheses and, in particular, the smallness conditions in the two-dimensional case. Indeed, the smallness conditions only can be explicit if quantitative estimates are established. To this aim, we also establish quantitative estimates for the Poincaré and Sobolev inequalities and for some trilinear terms.