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Using an ILU/Deflation Preconditioner for Simulation of a PEM Fuel Cell Cathode Catalyst Layer

Published online by Cambridge University Press:  03 June 2015

Kyle J. Lange ,
Pang-Chieh Sui  and
Ned Djilali
Kyle J. Lange*
Affiliation:
Institute of Integrated Energy Systems, University of Victoria, BC, Canada Currently at Lawrence Livermore National Laboratory, Livermore, CA, USA
Pang-Chieh Sui*
Affiliation:
Institute of Integrated Energy Systems, University of Victoria, BC, Canada
Ned Djilali*
Affiliation:
Institute of Integrated Energy Systems, University of Victoria, BC, Canada Department of Mechanical Engineering, University of Victoria, BC, Canada
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
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Abstract

Numerical aspects of a pore scale model are investigated for the simulation of catalyst layers of polymer electrolyte membrane fuel cells. Coupled heat, mass and charged species transport together with reaction kinetics are taken into account using parallelized finite volume simulations for a range of nanostructured, computationally reconstructed catalyst layer samples. The effectiveness of implementing deflation as a second stage preconditioner generally improves convergence and results in better convergence behavior than more sophisticated first stage pre-conditioners. This behavior is attributed to the fact that the two stage preconditioner updates the preconditioning matrix at every GMRES restart, reducing the stalling effects that are commonly observed in restarted GMRES when a single stage preconditioner is used. In addition, the effectiveness of the deflation preconditioner is independent of the number of processors, whereas the localized block ILU preconditioner deteriorates in quality as the number of processors is increased. The total number of GMRES search directions required for convergence varies considerably depending on the preconditioner, but also depends on the catalyst layer microstructure, with low porosity microstructures requiring a smaller number of iterations. The improved model and numerical solution strategy should allow simulations for larger computational domains and improve the reliability of the predicted transport parameters. The preconditioning strategies presented in the paper are scalable and should prove effective for massively parallel simulations of other problems involving nonlinear equations.

Keywords

65F0865F1065F1565F50DeflationPEM fuel cellcatalyst layerpore scale modelporous mediapreconditioner
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 3 , September 2013 , pp. 537 - 573
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Djilali, N. and Sui, P., Transport phenomena in fuel cells: From microscale to macroscale, Int. J. Comput. Fluid Dyn., 22 (2008), 115–133.CrossRefGoogle Scholar
[2]Schwarz, D. H. and Djilali, N., 3d modeling of catalyst layers in pem fuel cells, J. Electrochem. Soc., 154 (2007), B1167–B1178.CrossRefGoogle Scholar
[3]Springer, T. E., Zawodzinski, T. A. and Gottesfeld, S., Polymer electrolyte fuel cell model, J. Electrochem. Soc., 138 (1991), 2334–2342.Google Scholar
[4]Nguyen, T. V. and White, R. E., A water and heat management model for proton-exchangemembrane fuel cells, J. Electrochem. Soc., 140 (1993), 2178–2186.Google Scholar
[5]Nam, J. H. and Kaviany, M., Effective diffusivity and water-saturation distribution in singleand two-layer pemfc diffusion medium, Int. J. Heat Mass Trans., 46 (2003), 4595–4611.CrossRefGoogle Scholar
[6]Bernardi, D. and Verbrugge, M., Mathematical model of a gas diffusion electrode bonded to a polymer electrolyte, AICHE J., 37 (1991), 1151–1163.Google Scholar
[7]Bernardiand, D. M.Verbrugge, M.W., A mathematical model of the solid-polymer-electrolyte fuel cell, J. Electrochem. Soc., 139 (1992), 2477–2491.Google Scholar
[8]Fuller, T. F. and Newman, J., Water and thermal management in solid-polymer-electrolyte fuel cells, J. Electrochem. Soc., 140 (1993), 1218–1225.Google Scholar
[9]Kulikovsky, A. A., Divisek, J. and Kornyshev, A. A., Modeling the cathode compartment of polymer electrolyte fuel cells: Dead and active reaction zones, J. Electrochem. Soc., 146 (1999), 3981–3991.Google Scholar
[10]Bradean, R., Promislow, K. and Wetton, B., Transport phenomena in the porous cathode of a proton exchange membrane fuel cell, Numer. Heat Transfer, Part A, 42 (2002), 121–138.Google Scholar
[11]Yuan, J., Rokni, M. and Sundeén, B., A numerical investigation of gas flow and heat transfer in proton exchange membrane fuel cells, Numer. Heat Transfer, Part A, 44 (2003), 255–280.Google Scholar
[12]Berning, T. and Djilali, N., A 3D, multiphase, multicomponent model of the cathode and anode of a pem fuel cell, J. Electrochem. Soc., 150 (2003), A1589–A1598.CrossRefGoogle Scholar
[13]Iczkowski, R. P. and Cutlip, M. B., Voltage losses in fuel cell cathodes, J. Electrochem. Soc., 127 (1980), 1433–1440.Google Scholar
[14]Rho, Y. W., Srinivasan, S. and Kho, Y. T., Mass transport phenomena in proton exchange membrane fuel cells using o2/he, o2/ar, and o2/n2 mixtures, J. Electrochem. Soc., 141 (1994), 2089–2096.CrossRefGoogle Scholar
[15]Antoine, O., Bultel, Y., Durand, R. and Ozil, P., Electrocatalysis, diffusion and ohmic drop in pemfc: Particle size and spatial discrete distribution effects, Electrochim. Acta, 43 (1998), 3681–3691.Google Scholar
[16]van Bussel, H. P. L. H., Koene, F. G. H. and Mallant, R. K. A. M., Dynamic model of solid polymer fuel cell water management, J. Power Sources, 71 (1998), 218–222.Google Scholar
[17]Pisani, L., Murgia, G., Valentini, M. and D’Aguanno, B., A working model of polymer electrolyte fuel cells, J. Electrochem. Soc., 149 (2002), A898–A904.CrossRefGoogle Scholar
[18]Secanell, M., Karan, K., Suleman, A. and Djilali, N., Multi-variable optimization of pemfc cathodes using an agglomerate model, Electrochim. Acta, 52 (2007), 6318–6337.Google Scholar
[19]Das, P. K., Li, X. and Liu, Z.-S., A three-dimensional agglomerate model for the cathode catalyst layer of pem fuel cells, J. Power Sources, 179 (2008), 186–199.Google Scholar
[20]Wang, G., Mukherjee, P. P. and Wang, C.-Y., Direct numerical simulation (dns) modeling of pefc electrodes: Part ii. random microstructure, Electrochim. Acta, 51 (2006), 3151–3160.Google Scholar
[21]Kim, S. H. and Pitsch, H., Reconstruction and effective transport properties of the catalyst layer in pem fuel cells, J. Electrochem. Soc., 156 (2009), B673–B681.Google Scholar
[22]Yu, Z. and Carter, R. N., Measurement of effective oxygen diffusivity in electrodes for proton exchange membrane fuel cells, J. Power Sources, 195 (2010), 1079–1084.Google Scholar
[23]Lange, K. J., Sui, P.-C. and Djilali, N., Pore scale simulation of transport and electrochemical reactions in reconstructed pemfc catalyst layers, J. Electrochem. Soc., 157 (2010), B1434–B1442.CrossRefGoogle Scholar
[24]Shen, J., Zhou, J., Astrath, N. G., Navessin, T., Liu, Z.-S. S., Lei, C., Rohling, J. H., Bessarabov, D., Knights, S. and Ye, S., Measurement of effective gas diffusion coefficients of catalyst layers of pem fuel cells with a loschmidt diffusion cell, J. Power Sources, 196 (2011), 674–678.Google Scholar
[25]Siddique, N. and Liu, F., Process based reconstruction and simulation of a three-dimensional fuel cell catalyst layer, Electrochim. Acta, 55 (2010), 5357–5366.CrossRefGoogle Scholar
[26]Lange, K. J., Sui, P.-C. and Djilali, N., Pore scale modeling of a proton exchange membrane fuel cell catalyst layer: Effects of water vapor and temperature, J. Power Sources, 196 (2011), 3195–3203.Google Scholar
[27]Vuik, C., Segal, A. and Meijerink, J. A., An efficient preconditioned cg method for the solution of a class of layered problems with extreme contrasts in the coefficients, J. Comput. Phys, 152 (1999), 385–403.CrossRefGoogle Scholar
[28]Frank, J. and Vuik, C., On the construction of deflation-based preconditioners, SIAM J. Sci. Comput., 23 (2001), 442–462.Google Scholar
[29]Vuik, C., Segal, A., Yaakoubi, L. el and Dufour, E., A comparison of various deflation vectors applied to elliptic problems with discontinuous coefficients, Appl. Num. Math., 41 (2002), 219–233.Google Scholar
[30]Nabben, R. and Vuik, C., A comparison of deflation and coarse grid correction applied to porous media flow, SIAM J. Numer. Anal., 42 (2004), 1631–1647.Google Scholar
[31]Behie, A. and Vinsome, P. K. W., Block iterative methods for fully implicit reservoir simulation, SPE Journal, 22 (1982), 658–668.Google Scholar
[32]Dawson, C. N., Klie, H., Wheeler, M. F. and Woodward, C. S., A parallel, implicit, cell-centered method for two-phase flow with a preconditioned Newton Krylov solver, Comp. Geo-sciences, 1 (1997), 215–249.Google Scholar
[33]Scheichl, R., Masson, R. and Wendebourg, J., Decoupling and block preconditioning for sedimentary basin simulations, Comp. Geosciences, 7 (2003), 295–318.Google Scholar
[34]Aksoylu, B. and Klie, H., A family of physics-based preconditioners for solving elliptic equations on highly heterogeneous media, Appl. Num. Math., 59 (2009), 1159–1186.Google Scholar
[35]Bramble, J. H., Pasciak, J. E. and Schatz, A. H., An iterative method for elliptic problems on regions partitioned into substructures, Math. Comput., 46 (1986), 361–369.Google Scholar
[36]Aarnes, J. and Hou, T. Y., Multiscale domain decomposition methods for elliptic problems with high aspect ratios, Acta Math. Appl. Sinica, 18 (2002), 63–76.CrossRefGoogle Scholar
[37]Graham, I. G., Lechner, P. O. and Scheichl, R., Domain decomposition for multiscale pdes, Numer. Math., 106 (2007), 589–626.CrossRefGoogle Scholar
[38]Erhel, J., Burrage, K. and Pohl, B., Restarted gmres preconditioned by deflation, J. Comput. Appl. Math., 69 (1996), 303–318.Google Scholar
[39]Burrage, K. and Erhel, J., On the performance of various adaptive preconditioned gmres strategies, Numer. Linear Algebra Appl., 5 (1998), 101–121.Google Scholar
[40]Sun, P., Xue, G., Wang, C. and Xu, J., Fast numerical simulation of two-phase transport model in the cathode of a polymer electrolyte fuel cell, Communications in Computational Physics, 6 (2008), 49–71.Google Scholar
[41]Lee, S., Mukerjee, S., McBreen, J., Rho, Y., Kho, Y. and Lee, T., Effects of nafion impregnation on performances of pemfc electrodes, Electrochim. Acta, 43 (1998), 3693–3701.CrossRefGoogle Scholar
[42]Yan, Q., Toghiani, H. and Wu, J., Investigation of water transport through membrane in a pem fuel cell by water balance experiments, J. Power Sources, 158 (2006), 316–325.Google Scholar
[43]Ge, J., Higier, A. and Liu, H., Effect of gas diffusion layer compression on pem fuel cell performance, J. Power Sources, 159 (2006), 922–927.Google Scholar
[44]Probst, N. and Grivei, E., Structure and electrical properties of carbon black, Carbon, 40 (2002), 201–205.Google Scholar
[45]Sone, Y., Ekdunge, P. and Simonsson, D., Proton conductivity of nafion 117 as measured by a four-electrode ac impedance method, J. Electrochem. Soc., 143 (1996), 1254–1259.Google Scholar
[46]Neyerlin, K. C., Gu, W., Jorne, J. and Gasteiger, H. A., Determination of catalyst unique parameters for the oxygen reduction reaction in a pemfc, J. Electrochem. Soc., 153 (2006), A1955– A1963.Google Scholar
[47]Cussler, E., Diffusion: Mass Transfer in Fluid Systems, Cambridge University Press, 1997.Google Scholar
[48]Mason, E. A., Malinauskas, A. P. and III, R. B. E., Flow and diffusion of gases in porous media, J. Chem. Phys., 46 (1967), 3199–3216.Google Scholar
[49]Lee, K., Ishihara, A., Mitsushima, S., Kamiya, N. and Ota, K. ichiro, Effect of recast temperature on diffusion and dissolution of oxygen and morphological properties in recast nafion, J. Electrochem. Soc., 151 (2004), A639–A645.Google Scholar
[50]Zhao, Q., Majsztrik, P. and Benziger, J., Diffusion and interfacial transport of water in nafion, J. Phys. Chem. B, 115 (2011), 2717–2727.Google Scholar
[51]Zawodzinski, T. A., Davey, J., Valerio, J. and Gottesfeld, S., The water content dependence of electro-osmotic drag in proton-conducting polymer electrolytes, Electrochim. Acta, 40 (1995), 297–302.Google Scholar
[52]Donnet, J.-B., Bansal, R. C. and Wang, M.-J., Carbon Black: Science and Technology, Marcel Dekker, Inc., New York, NY, USA, 1993.Google Scholar
[53]Khandelwal, M. and Mench, M., Direct measurement of through-plane thermal conductivity and contact resistance in fuel cell materials, J. Power Sources, 161 (2006), 1106–1115.Google Scholar
[54]Wolfram—Alpha, , Thermal conductivity of moist air, http://www.wolframalpha.com, 2011.Google Scholar
[55]Weber, A. Z. and Newman, J., Coupled thermal and water management in polymer electrolyte fuel cells, J. Electrochem. Soc., 153 (2006), A2205–A2214.Google Scholar
[56]Lampinen, M. J. and Fomino, M., Analysis of free energy and entropy changes for half-cell reactions, J. Electrochem. Soc., 140 (1993), 3537–3546.Google Scholar
[57]Dembo, R. S., Eisenstat, S. C. and Steihaug, T., Inexact newton methods, SIAMJ. Numer. Anal., 19 (1982), 400–408.Google Scholar
[58]Saad, Y. and Schultz, M. H., Gmres: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), 856–869.Google Scholar
[59]Grama, A., Gupta, A., Karypis, G. and Kumar, V., Introduction to Parallel Computing, Addison-Wesley, 2003.Google Scholar
[60]Chen, K., Matrix Preconditioning Techniques and Applications, Cambridge University Press, Cambridge, UK, 2005.CrossRefGoogle Scholar
[61]Nakajima, K. and Okuda, H., Parallel iterative solvers with localized ilu preconditioning for unstructured grids on workstation clusters, Int. J. Comput. Fluid Dyn., 12 (1999), 315–322.Google Scholar
[62]Saad, Y., Iterative Methods for Sparse Linear Systems, SIAM, 2000.Google Scholar
[63]Lange, K. J., Misra, C., Sui, P.-C. and Djilali, N., A numerical study on preconditioning and partitioning schemes for reactive transport in a pemfc catalyst layer, Comput. Meth. Appl. Mech. Eng., 200 (2011), 905–916.Google Scholar
[64]Chapman, A. and Saad, Y., Deflated and augmented krylov subspace techniques, Numer. Linear Algebra Appl., 4 (1997), 43–66.3.0.CO;2-Z>CrossRefGoogle Scholar
[65]Tang, J. and Vuik, C., Efficient deflation methods applied to 3-D bubbly flow problems, Electron. Trans. Numer. Anal., 26 (2007), 330–349.Google Scholar