On the wellposedness of a fuel cell problem
DOI:
https://doi.org/10.33044/revuma.3697Abstract
This paper investigates the existence of weak solutions to a fuel cell problem modeled by a boundary value problem (BVP) in the multiregion domain. The BVP consists of the coupled Stokes/Darcy-TEC (thermoelectrochemical) system of elliptic equations, with Beavers–Joseph–Saffman and regularized Butler–Volmer boundary conditions being prescribed on the interfaces, porous-fluid and membrane, respectively. The present model includes macrohomogeneous models for both hydrogen and methanol crossover. The novelty in the coupled Stokes/Darcy-TEC system lies in the presence of the Joule effect together with the quasilinear character given by (1) temperature dependence of the viscosities and the diffusion coefficients; (2) the concentration-temperature dependence of Dufour–Soret and Peltier–Seebeck cross-effect coefficients, and (3) the pressure dependence of the permeability. We derive quantitative estimates of the solutions to clarify smallness conditions on the data. We use fixed-point and compactness arguments based on the quantitative estimates of approximated solutions.
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J. J. Baschuk and X. Li, A general formulation for a mathematical PEM fuel cell model, J. Power Sources 142 no. 1–2 (2005), 134–153. DOI
J. J. Baschuk and X. Li, A comprehensive, consistent and systematic mathematical model of PEM fuel cells, Applied Energy 86 no. 2 (2009), 181–193. DOI
H. Beirão da Veiga, An $L^p$-theory for the $n$-dimensional, stationary, compressible Navier–Stokes equations, and the incompressible limit for compressible fluids. The equilibrium solutions, Comm. Math. Phys. 109 no. 2 (1987), 229–248. DOI MR Zbl
P. Berg and K. Ladipo, Exact solution of an electro-osmotic flow problem in a cylindrical channel of polymer electrolyte membranes, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 no. 2109 (2009), 2663–2679. DOI MR Zbl
R. G. Carbonell and S. Whitaker, Heat and mass transfer in porous media, in Fundamentals of transport phenomena in porous media, NATO ASI Series. Series E, Applied Sciences no. 82, Martinus Nijhoff, Dordrecht, 1984, pp. 121–198.
S.-J. Chern and P.-C. Huang, On the existence of a weak solution of a half-cell model for PEM fuel cells, Math. Probl. Eng. (2010), Article no. 701096. DOI MR Zbl
L. Consiglieri, Heat-conducting viscous fluids over porous media, Commun. Math. Sci. 10 no. 3 (2012), 835–857. DOI MR Zbl
L. Consiglieri, Dynamic bilateral boundary conditions on interfaces, Riv. Math. Univ. Parma (N.S.) 4 no. 1 (2013), 81–111. MR Zbl
L. Consiglieri, Quantitative estimates on boundary value problems, Lambert Academic Publishing, Saarbrücken, 2017.
L. Consiglieri, Weak solutions for multiquasilinear elliptic-parabolic systems: application to thermoelectrochemical problems, Bol. Soc. Mat. Mex. (3) 26 no. 2 (2020), 535–562. DOI MR Zbl
J. Deny and J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier (Grenoble) 5 (1954), 305–370. DOI MR Zbl
N. Djilali, Computational modelling of polymer electrolyte membrane (PEM) fuel cells: Challenges and opportunities, Energy 32 no. 4 (2007), 269–280. DOI
R. D. Donaldson and B. R. Wetton, Solving steady interface problems using residual velocities, IMA J. Appl. Math. 71 no. 6 (2006), 877–897. DOI MR Zbl
C. J. van Duijn and J. D. Fehribach, Analysis for a molten carbonate fuel cell, Electron. J. Differential Equations (1993), No. 06. MR Zbl Available at http://eudml.org/doc/118561.
E. Eggenweiler and I. Rybak, Unsuitability of the Beavers–Joseph interface condition for filtration problems, J. Fluid Mech. 892 (2020), Article no. A10. DOI MR Zbl
J. Fernández Bonder and J. D. Rossi, On the existence of extremals for the Sobolev trace embedding theorem with critical exponent, Bull. London Math. Soc. 37 no. 1 (2005), 119–125. DOI MR Zbl
Y. Fu, S. Poizeau, A. Bertei, C. Qi, A. Mohanram, J. D. Pietras, and M. Z. Bazant, Heterogeneous electrocatalysis in porous cathodes of solid oxide fuel cells, Electrochimica Acta 159 (2015), 71–80. DOI
T. F. Fuller and J. Newman, Water and thermal management in solid‐polymer‐electrolyte fuel cells, J. Electrochem. Soc. 140 no. 5 (1993), 1218–1225. DOI
K. Gröger, A $W^{1,p}$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann. 283 no. 4 (1989), 679–687. DOI MR Zbl
K. Gröger and J. Rehberg, Resolvent estimates in $W^{-1,p}$ for second order elliptic differential operators in case of mixed boundary conditions, Math. Ann. 285 no. 1 (1989), 105–113. DOI MR Zbl
S. A. Hajimolana, M. A. Hussain, W. M. A. Wan Daud, M. Soroush, and A. Shamiri, Mathematical modeling of solid oxide fuel cells: A review, Renewable Sustainable Energy Rev. 15 no. 4 (2011), 1893–1917. DOI
F. Jaeger, O. K. Matar, and E. A. Müller, Bulk viscosity of molecular fluids, J. Chem. Phys. 148 no. 17 (2018), Paper no. 174504. DOI
O. Kavian and M. Vogelius, On the existence and “blow-up” of solutions to a two-dimensional nonlinear boundary-value problem arising in corrosion modelling, Proc. Roy. Soc. Edinburgh Sect. A 133 no. 1 (2003), 119–149. DOI MR Zbl
S. Mazumder and J. V. Cole, Rigorous 3-D mathematical modeling of PEM fuel cells: II. Model predictions with liquid water transport, J. Electrochem. Soc. 150 no. 11 (2003), A1510–A1517. DOI
J. K. Nørskov, J. Rossmeisl, A. Logadottir, L. Lindqvist, J. R. Kitchin, T. Bligaard, and H. Jónsson, Origin of the overpotential for oxygen reduction at a fuel-cell cathode, J. Phys. Chem. B 108 no. 46 (2004), 17886–17892. DOI
T. E. Springer, T. A. Zawodzinski, and S. Gottesfeld, Polymer electrolyte fuel cell model, J. Electrochem. Soc. 138 no. 8 (1991), 2334–2342. DOI
K. Sundmacher, L. K. Rihko-Struckmann, and V. Galvita, Solid electrolyte membrane reactors: Status and trends, Catalysis Today 104 no. 2–4 (2005), 185–199. DOI
S. Um, C.-Y. Wang, and K. S. Chen, Computational fluid dynamics modeling of proton exchange membrane fuel cells, J. Electrochem. Soc. 147 no. 12 (2000), 4485–4493. DOI
M. Uzunoglu and M. Alam, Dynamic modeling, design, and simulation of a combined PEM fuel cell and ultracapacitor system for stand-alone residential applications, IEEE Trans. Energy Conversion 21 no. 3 (2006), 767–775. DOI
J. L. Vázquez, The porous medium equation: Mathematical theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. MR Zbl
S. A. Vilekar, I. Fishtik, and R. Datta, Kinetics of the hydrogen electrode reaction, J. Electrochem. Soc. 157 no. 7 (2010), B1040–B1050. DOI
C. Wang, M. H. Nehrir, and S. R. Shaw, Dynamic models and model validation for PEM fuel cells using electrical circuits, IEEE Trans. Energy Conversion 20 no. 2 (2005), 442–451. DOI
J. X. Wang, T. E. Springer, and R. R. Adzic, Dual-pathway kinetic equation for the hydrogen oxidation reaction on Pt electrodes, J. Electrochem. Soc. 153 no. 9 (2006), A1732–A1740. DOI
Z. H. Wang and C. Y. Wang, Mathematical modeling of liquid-feed direct methanol fuel cells, J. Electrochem. Soc. 150 no. 4 (2003), A508–A519. DOI
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