0
Research Paper

# An Effective Combined Finite Element-Upwind Finite Volume Method for a Transient Multiphysics Two-Phase Transport Model of a Proton Exchange Membrane Fuel Cell

[+] Author and Article Information
Pengtao Sun

Department of Mathematical Sciences,
Las Vegas 4505 Maryland Parkway,
Las Vegas, NV 89154
e-mail: pengtao.sun@unlv.edu

Su Zhou

College of Automotive Studies/New Energy Automotive Engineering Center,
Tongji University,
Shanghai 201804, China
e-mail: suzhou@tongji.edu.cn

Qiya Hu

Institute of Computational Mathematics and Scientific Engineering Computing,
Beijing 100080, China
e-mail: hqy@lsec.cc.ac.cn

Contributed by the Advanced Energy Systems Division of ASME for publication in the JOURNAL OF FUEL CELL SCIENCE AND TECHNOLOGY. Manuscript received August 26, 2012; final manuscript received January 13, 2013; published online May 14, 2013. Editor: Nigel M. Sammes.

J. Fuel Cell Sci. Technol 10(3), 031004 (May 14, 2013) (11 pages) Paper No: FC-12-1079; doi: 10.1115/1.4023837 History: Received August 26, 2012; Revised January 13, 2013

## Abstract

In this paper, an effective combined finite element-upwind finite volume method is studied for a three-dimensional transient multiphysics transport model of a proton exchange membrane fuel cell (PEMFC), in which Navier–Stokes–Darcy coupling flow, species transports, heat transfer, electrochemical processes, and charge transports are fully considered. Multiphase mixture (M2) formulation is employed to define the involved two-phase model. Kirchhoff transformation is introduced to overcome the discontinuous and degenerate water diffusivity that is induced by the M2 model. By means of an adaptive time-stepping fourth-order multistep backward differencing formula (BDF), we design an effective temporal integration scheme to deal with the stiff phenomena arising from different time scales. In addition, all the governing equations are discretized by a combined finite element-upwind finite volume method to conquer the dominant convection effect in gas channels, while the diffusion and reaction effects are still dealt with by finite element method. Numerical simulations demonstrate that the presented techniques are effective to obtain a fast and convergent nonlinear iteration within a maximum 36 steps at each time step; in contrast to the oscillatory and nonconvergent iteration conducted by commercial CFD solvers and standard finite element/finite volume methods.

<>

## References

Wang, Y., and Wang, C. Y., 2007, “Two-Phase Transients of Polymer Electrolyte Fuel Cells,” J. Electrochem. Soc, 154, pp. B636–B643.
Wang, Y., and Wang, C. Y., 2005, “Transient Analysis of Polymer Electrolyte Fuel Cells,” Electrochim. Acta, 50, pp. 1307–1315.
Wang, Y., and Wang, C. Y., 2006, “Dynamics of Polymer Electrolyte Fuel Cells Undergoing Load Changes,” Electrochim. Acta, 51, pp. 3924–3933.
Wang, C. Y., and Cheng, P., 1997, “Multiphase Flow and Heat Transfer in Porous Media,” Adv. Heat Transfer, 30, pp. 93–196.
Wang, C. Y., 2004, “Fundamental Models for Fuel Cell Engineering,” Chem. Rev., 104, pp. 4727–4766. [PubMed]
Pasaogullari, U., and Wang, C. Y., 2004, “Liquid Water Transport in Gas Diffusion Layer of Polymer Electrolyte Fuel Cells,” J. Electrochem. Soc., 151, pp. A399–A406.
Wang, Z. H., Wang, C. Y., and Chen, K. S., 2001, “Two-Phase Flow and Transport in the Air Cathode of Proton Exchange Membrane Fuel Cells,” J. Power Sources, 94, pp. 40–50.
Pasaogullari, U., and Wang, C. Y., 2005, “Two-Phase Modeling and Flooding Prediction of Polymer Electrolyte Fuel Cells,” J. Electrochem. Soc., 152, A380–A390.
Liu, F. Q., and Wang, C. Y., 2007, “Mixed Potential in a Direct Methanol Fuel Cell—Modeling and Experiments,” J. Electrochem. Soc., 154, pp. B514–B522.
Liu, W., and Wang, C. Y., 2007, “Three-Dimensional Simulations of Liquid Feed Direct Methanol Fuel Cells,” J. Electrochem. Soc., 154, pp. B352–B361.
Liu, W., and Wang, C. Y., 2007, “Modeling Water Transport in Liquid Feed Direct Methanol Fuel Cells,” J. Power Sources, 164, pp. 189–195.
Sun, P., Xue, G., Wang, C.-Y., and Xu, J., 2009, “Fast Numerical Simulation of Two-Phase Transport Model in the Cathode of a Polymer Electrolyte Fuel Cell,” Commun. Comput. Phys., 6, pp. 49–71.
Sun, P., Xue, G., Wang, C.-Y., and Xu, J., 2009, “A Domain Decomposition Method for Two-Phase Transport Model in the Cathode of a Polymer Electrolyte Fuel Cell,” J. Comput. Phys., 228, pp. 6016–6036.
Sun, P., and Wang, Y., 2010, “A New Formulation and an Efficient Numerical Technique for a Nonisothermal, Anisotropic, Two-Phase Transport Model of PEMFC,” Proceedings of the Eighth International Fuel Cell Science, Engineering and Technology Conference, Brooklyn, NY, June 14–16.
Sun, P., Zhou, S., Hu, Q., and Liang, G., 2012, “Numerical Study of a 3D Two-Phase PEM Fuel Cell Model Via a Novel Automated Finite Element/Finite Volume Program,” Commun. Comput. Phys., 11, pp. 65–98.
Mattheij, R. M., Reinstra, S. W., and ten Thije Boonkkamp, J. H. M., 2005, Partial Differential Equations: Modeling, Analysis, Computation, SIAM, Philadelphia.
Shampine, L. F., and Gear, C. W., 1979, “A User's View of Solving Stiff Ordinary Differential Equations,” SIAM Rev., 21, pp. 1–17.
Li, M., and Tang, T., 2001, “A Compact Fourth-Order Finite Difference Scheme for Unsteady Viscous Incompressible Flows,” J. Sci. Comput., 16, pp. 29–45.
Strikwerda, J. C., 1997, “High-Order-Accurate Schemes for Incompressible Viscous Flow,” Int. J. Numer. Meth. Fluids, 24, pp. 715–734.
Wang, Y., 2008, “Modeling of Two-Phase Transport in the Diffusion Media of Polymer Electrolyte Fuel Cells,” J. Power Sources, 185, pp. 261–271.
Wang, Y., and Wang, C. Y., 2006, “A Nonisothermal Two-Phase Model for Polymer Electrolyte Fuel Cells,” J. Electrochem. Soc, 153, pp. A1193–A1200.
Pasaogullari, U., Mukherjee, P., Wang, C. Y., and Chen, K., 2007, “Anisotropic Heat and Water Transport in a PEFC Cathode Gas Diffusion Layer,” J. Electrochem. Soc., 154, pp. B823–B834.
Wang, C. Y., and Cheng, P., 1996, “A Multiphase Mixture Model for Multiphase, Multicomponent Transport in Capillary Porous Media—I. Model Development,” Int. J. Heat Mass Transfer, 39, pp. 3607–3618.
Sun, P., 2011, “Modeling Studies and Efficient Numerical Methods for Proton Exchange Membrane Fuel Cell,” Comput. Methods Appl. Mech. Eng., 200, pp. 3324–3340.
Kulikovsky, A. A., 2003, “Quasi-3D Modeling of Water Transport in Polymer Electrolyte Fuel Cells,” J. Electrochem. Soc., 150, pp. A1432–A1439.
Crank, J., 1984, Free and Moving Boundary Problems, Clarendon, New York.
Eyres, N., Hartree, D., Ingham, J., Jackson, R., Sarjant, R., and Wagstaff, S., 1946, “The Calculation of Variable Heat Flow in Solids,” Philos. Trans. R. Soc. London Sect. A, 240, pp. 1–57.
Alt, H., and Luckhaus, S., 1983, “Quasilinear Elliptic-Parabolic Differential Equations,” Math. Z., 183, pp. 311–341.
Arbogast, T., Wheeler, M., and Zhang, N., 1996, “A Nonlinear Mixed Finite Element Method for a Degenerate Parabolic Equation Arising in Flow in Porous Media,” SIAM J. Numer. Anal., 33, pp. 1669–1687.
Rose, M., 1983, “Numerical Methods for Flows Through Porous Media. {I},” Math. Comp., 40, pp. 435–467.
Sun, P., Xue, G., Wang, C.-Y., and Xu, J., 2009, “New Numerical Techniques for a Liquid Feed 3D Full Direct Methanol Fuel Cell Model,” SIAM Appl. Math., 70, pp. 600–620.
Sun, P., Wang, C., and Xu, J., 2010, “A Combined Finite Element-Upwind Finite Volume Method for Liquid-Feed Direct Methanol Fuel Cell Simulations,” J. Fuel Cell Sci. Tech., 7, p. 041010.
Feistauer, M., Felcman, J., Lukáčová-Medvid'Ová, M., and Warnecke, G., 1999, “Error Estimates for a Combined Finite Volume-Finite Element Method for Nonlinear Convection-Diffusion Problems,” SIAM J. Numer. Anal., 36, pp. 1528–1548.
Sun, P., and Zhou, S., 2011, “Numerical Studies of Thermal Transport and Mechanical Effects Due to Thermal-Inertia Loading in PEMFC Stack in Sub-Freezing Environment,” J. Fuel Cell Sci. Tech., 8, p. 011010.
Tezduyar, T. E., 1992, “Stabilized Finite Element Formulations for Incompressible Flow Computations,” Adv. Appl. Mech., 28, pp. 1–44.
Brezzi, F., and Pitkaranta, J., 1984, “On the Stabilization of Finite Element Approximations of the Stokes Problem,” Efficient Solutions of Elliptic Systems, Notes on Numerical Fluid Mechanics, Vol. 10, W. Hackbusch, ed., Viewig, Wiesbaden, Germany, pp. 11–19.
Hughes, T. J. R., Franca, L. P., and Balestra, M., 1986, “A New Finite Element Formulation for Computational Fluid Dynamics: V. Circumventing the Babuška-Brezzi Condition: A Stable Petrov-Galerkin Formulation of the Stokes Problem Accommodating Equal-Order Interpolations,” Comput. Meth. Appl. Mech. Eng., 59(1), pp. 85–99.

## Figures

Fig. 1

Water diffusivity DH2O(CH2O) (left) and DmH2O,eff(CH2O) (right) at 80 °C

Fig. 2

Computational domain and mesh of a single-channel PEFC

Fig. 3

Convergence history of our numerical method (left) versus that of standard FEM/FVM (right) at the 449th time step (13.79 s), where the iteration error tolerance is 10-6

Fig. 6

Liquid saturation s at 1 s (left), 3 s (middle), and 9 s (right)

Fig. 7

Liquid saturation in GDLs at the anode (top part) and the cathode (bottom part) along flow direction at 9 s (top), 59 s (middle), and 100 s (bottom)

Fig. 4

Hydrogen concentration CH2 at 1 s (left), 3 s (middle), and 9 s (right)

Fig. 5

Oxygen concentration CO2 at 1 s (left), 3 s (middle), and 9 s (right)

## Errata

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections