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2009

Volume 70, Issue 2, pp. vii-639

* Special Issue on Fuel Cells: Modeling, Analysis, and Computation

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Special Issue on Fuel Cells: Modeling, Analysis, and Computation

A. M. Herring and P. A. Martin, Guest Editors

SIAM J. Appl. Math. 70, pp. vii-viii ( pages)

Online Publication Date: July 17, 2009

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As fossil energy resources become scarcer and as the difficult transition to renewable energy unfolds, it is critical that energy conversion devices with the highest efficiencies be chosen. Fuel cells have dramatically higher potential energy conversion efficiencies than Carnot limited heat engines at the moderate temperatures of operation typically used. This is because in a fuel cell, as in a battery, chemical energy is directly converted to electrical energy. Furthermore, because fuel cells store their fuel externally from the device, they have much higher energy densities than batteries with much higher achieved levels of safety. Fuel cells also have potentially longer lifetimes than batteries, depending on how they are operated. Unfortunately, there are still some rather severe limits on fuel cell commercialization above the 10,000 or so units currently being manufactured annually.
It is true to say that applied mathematicians and computational scientists could contribute to all the major fuel cell barriers. These primarily concern cost, operational parameters, catalysis, and durability. Low temperature devices based on polymeric proton exchange membranes (PEM) need the PEM to be fully hydrated with liquid water to achieve adequate proton conduction and hence practical power densities. As the fuel cell produces water, this leads to complex water management issues. Furthermore, the low temperature of operation requires very high loadings of platinum to catalyze the oxygen reduction reaction, which is not well understood in terms of the effect of water and of the PEM itself. High temperature fuel cells based on solid oxide conducting ceramics have complex heat management issues on start up and cool down, use rather expensive oxide materials, and face sealing issues if high power densities are to be achieved. Fuel cells based on other electrolytes or even biological systems also have their unique advantages and challenges.
It is clear that the subject of fuel cells is important, broad, and truly multidisciplinary. To develop successfully, contributions are needed from a wide range of scientists and engineers. Therefore, it was decided that a special issue of the SIAM Journal on Applied Mathematics, entitled “Fuel Cells: Modeling, Analysis, and Computation,” would be appropriate. The stated goals were to display new research on fuel cells involving applied mathematics, interpreted broadly, and to encourage more activity by mathematical scientists in the modeling, analysis, and understanding of fuel cells.
The two editors of the special issue are Paul Martin (an applied mathematician) and Andrew Herring (an applied chemist), both from the Colorado School of Mines, home of the Colorado Fuel Cell Center and the Renewable Energy Materials Research Science and Engineering Center. An open call for papers was issued and all submitted papers were subject to peer review. Authors were encouraged to try to communicate with mathematical scientists who may not be experts on fuel cells. In addition, the first two papers in the special issue give overviews, again with the intention of attracting more workers into the field.
We close this short introduction by giving some general references to the science and technology of fuel cells. The definitive reference work is the four-volume set edited by Vielstich, Lamm, and Gasteiger . For a good collection of review articles, see the special issue of Chemical Reviews edited by Whittingham, Savinell, and Zawozinski . O'Hayre et al. have written a good introductory textbook, now in its second edition. The U.S. Department of Energy has a useful website . There are also websites intended for the fuel cell industry . Finally, the Smithsonian Institution has a website dedicated to the history of fuel cells .
The editors thank all the authors and reviewers for their contributions to this special issue.

PEM Fuel Cells: A Mathematical Overview

Keith Promislow and Brian Wetton

SIAM J. Appl. Math. 70, pp. 369-409 (41 pages)

Online Publication Date: July 17, 2009

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We present an overview of the mathematical issues that arise in the modeling of polymer electrolyte membrane fuel cells. These issues range from nanoscale modeling of network structures arising in pore formation within the polymer and the formation of nanostructured agglomerates within the catalyst layer, to macroscale models of multiphase flow and water management, degradation of catalyst layers and membrane, and development of stack level codes. The dominant themes are the development and analysis of multiscale models and their reduction to simplified forms that are implementable in stack-level computations.

A Critical Overview of Computational Fluid Dynamics Multiphase Models for Proton Exchange Membrane Fuel Cells

Vladimir Gurau and J. Adin Mann,, Jr.

SIAM J. Appl. Math. 70, pp. 410-454 (45 pages) | Cited 3 times

Online Publication Date: July 17, 2009

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This paper presents an overview of the mathematical issues and the current situation in the computational fluid dynamics (CFD) modeling of multiphase transport in hydrogen-operated proton exchange membrane fuel cells (PEMFCs) at a macroscopic scale. The paper overviews multiphase models which are based on the finite volume approach and which cover water transport from anode to cathode throughout the membrane electrode assembly (MEA). We review conceptual models of water transport in the diffusion media focusing on the formulation of the balance equations and of the constitutive relations and discuss weaknesses and inconsistencies of current approaches based on experimental and theoretical evidence. A major incentive of this review is to stress the impact on water management of the widely ignored phenomena at the subgrid-scale distributed interfaces and at the macroscopic-scale interfaces between the fuel cell components. We discuss how misinterpretation of the physical meaning of various terms in the macroscopic transport equation of water in catalyst coated membranes (CCMs) might have led to faulty numerical treatments of this equation. We also recommend new evolving approaches.

Asymptotic Reduction for Numerical Modeling of Polymer Electrolyte Fuel Cells

M. Vynnycky, G. Shugai, P. Yakubenko, and N. Mellgren

SIAM J. Appl. Math. 70, pp. 455-487 (33 pages) | Cited 1 time

Online Publication Date: July 17, 2009

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Most existing three-dimensional (3D) models for the polymer electrolyte fuel cell (PEFC), as well as other types of fuel cells, are fully numerical and computationally expensive. While such models are undoubtedly useful, they cannot provide the qualitative understanding that comes from a complete prior nondimensional analysis of the problem. Here, earlier ideas for the derivation of a two-dimensional (2D) asymptotically reduced model to describe steady isothermal gas-phase flow in the cathode of a PEFC are extended to a 3D nonisothermal model for a whole PEFC with straight channel flow distributors. As well as providing characteristic current density and temperature difference scales for the whole cell, it is also possible to extract potential drops over individual cell components. The analysis indicates that, for realistic operating ranges, the PEFC is sufficiently isothermal with respect to the mass, momentum, and charge transport to enable the thermal problem to be decoupled from the rest, a simplification not noted previously. After the relevant nondimensional parameters have been identified, a reduced model is proposed and some preliminary numerical results comparing the polarization curves obtained from the reduced and 3D models are presented. Good agreement is found; most significantly, the reduced model is found to require between one and two orders of CPU time less than the full 3D model.

Using a Quasi-Potential Transformation for Modeling Diffusion Media in Polymer-Electrolyte Fuel Cells

Adam Z. Weber and John Newman

SIAM J. Appl. Math. 70, pp. 488-509 (22 pages)

Online Publication Date: July 17, 2009

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In this paper, a quasi-potential approach along with conformal mapping is used to model the diffusion media of a polymer-electrolyte fuel cell. This method provides a series solution that is grid independent and only requires integration along a single boundary to solve the problem. The approach accounts for nonisothermal phenomena, two-phase flow, correct placement of the electronic potential boundary condition, and multilayer media. The method is applied to a cathode diffusion medium to explore the interplay between water and thermal management and performance, the impact of the rib-to-channel ratio, and the existence of diffusion under the rib and flooding phenomena.

Triple Phase Boundaries in Solid-Oxide Cathodes

Joseph D. Fehribach and Ryan O'Hayre

SIAM J. Appl. Math. 70, pp. 510-530 (21 pages)

Online Publication Date: July 17, 2009

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Component potential modeling based on solid-oxide electrochemistry is used to study a single-particle configuration where a hemispherical LSM particle sits on a YSZ electrolyte half-space. The primary comparison is between two pathways: one where oxide ions travel on the particle surface; the other where these ions travel through the bulk particle interior. The systems that model each of the pathways are analyzed both mathematically and numerically, yielding insights into diffusion-reaction-conduction processes for this single-particle model. A broad range of parameter values are considered, particularly in regards to the least well-established value, the surface conductance for the surface pathway. This work includes a number of case studies that indicate which pathway dominates for a variety of parameter choices.

Efficient Parallel Algorithm for Fuel Cell Stack Simulation

A. A. Kulikovsky

SIAM J. Appl. Math. 70, pp. 531-542 (12 pages) | Cited 1 time

Online Publication Date: July 17, 2009

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A planar fuel cell stack is a layered structure consisting of repeated modules—membrane-electrode assemblies (MEAs) separated by bipolar plates (BPs). Generally, the distributions of voltage and temperature over the BP volume are described by three-dimensional Laplace equations. However, the thickness of a BP is much smaller than its in-plane size. This enables us to reduce a three-dimensional Laplace equation to a two-dimensional Poisson equation and to develop an efficient parallel algorithm for stack simulation. In the simplest variant, each individual module “MEA + BP” is solved on a separate processor. Typically, the number of cells in a stack is 10 to 100; this algorithm is thus most suitable for small- and medium-scale parallel machines. A much faster method is to cut every module into a number of “stripes” and to solve each stripe on a separate processor. Numerical tests with this method show that with eight stripes per module the solution of the electric problem is obtained roughly ten times faster than expected. Evidently, the striping algorithm provides much faster convergence of the iterative Poisson solver. The effect is presumably due to fast damping of high-frequency modes of potential in the iteration process. This algorithm may open up possibilities for fast simulation of real 100-cell stacks using massively parallel machines.

Vector-Space Methods and Kirchhoff Graphs for Reaction Networks

Joseph D. Fehribach

SIAM J. Appl. Math. 70, pp. 543-562 (20 pages)

Online Publication Date: July 17, 2009

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This article presents a vector-space formulation for constructing reaction routes (reaction pathways) and Kirchhoff graphs (reaction route graphs, fundamental graphs) for reaction networks. Specific examples from fuel-cell electrochemistry are included throughout to illustrate the more general theoretical discussion. Some of the mathematical aspects of Kirchhoff graphs, such as their representation of the fundamental theorem of linear algebra, are also discussed.

Uniqueness of Current Reconstructions for Magnetic Tomography in Multilayer Devices

Roland Potthast and Martin Wannert

SIAM J. Appl. Math. 70, pp. 563-578 (16 pages)

Online Publication Date: July 17, 2009

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Magnetic tomography is an important emerging technique for the nondestructive investigation and monitoring of electrical devices. Measurements of the magnetic field of the currents in a device are used to reconstruct the current distribution. Here we investigate the uniqueness problem for current reconstructions from multilayer devices. The general magnetic tomography problem is well known to be highly nonunique and unstable. Here, as a new result for the single-layer and multilayer device case we will study the splitting procedure and equivalence of the full nullspace to the nullspaces of operators supported on lower-dimensional subsets of the device. We will base our results on the uniqueness of wave source splitting combined with tools from potential theory and explicit estimates for particular surface integrals involving the Biot–Savart integral operator.

A Mathematical Model for the Corrosion of Metallic Bipolar Plates in PEM Fuel Cells: Numerical and Experimental Issues

Benedetto Bozzini and Ivonne Sgura

SIAM J. Appl. Math. 70, pp. 579-599 (21 pages)

Online Publication Date: July 17, 2009

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A two-dimensional mathematical model for corrosion of metallic bipolar plates in a PEM fuel cell (FC) is presented. Numerical approximations by a novel finite difference technique to deal with nonlinear and mixed boundary conditions are illustrated. Numerical computations and comparisons with experimental data obtained with a laboratory PEMFC are reported. Good qualitative agreement of the simulations has been found with the time dependence of the average current density and with the space distribution of the corrosion rate.

New Numerical Techniques for a Three-Dimensional Liquid-Feed Direct Methanol Fuel Cell Model

Pengtao Sun, Guangri Xue, Chaoyang Wang, and Jinchao Xu

SIAM J. Appl. Math. 70, pp. 600-620 (21 pages) | Cited 3 times

Online Publication Date: July 17, 2009

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In this paper, a three-dimensional two-phase transport model of a liquid-feed direct methanol fuel cell (DMFC) is studied, where flow, species, charge-transport, and energy equations are simultaneously addressed. Some new numerical techniques are specifically investigated in order to achieve fast and convergent numerical simulation. An updated source term of the water transport equation based on a combination of physical and numerical considerations, and a series of efficient numerical algorithms and discretizations by means of a combined finite element–upwind finite volume method, are designed and analyzed to assist in approaching this target. The three-dimensional numerical simulations demonstrate that the convergent physical solutions can be obtained within hundreds of steps, in contrast to the long-running and even nonconvergent nonlinear iterations operated by standard finite element/volume methods if no new numerical technique is adopted.

Optimal Control of Load Changes for Molten Carbonate Fuel Cell Systems: A Challenge in PDE Constrained Optimization

Kurt Chudej, Hans Josef Pesch, and Kati Sternberg

SIAM J. Appl. Math. 70, pp. 621-639 (19 pages)

Online Publication Date: July 17, 2009

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Molten carbonate fuel cells provide a promising technology for the operation of future stationary power plants. In order to enhance service life, a detailed understanding of the dynamical behavior of such fuel cell systems is necessary. In particular, fast load changes shall be simulated, (resp., optimized) without risking material stress due to the extreme temperature differences usually accompanying fast load changes. Fast load changes are important for daily operations in order to react on varying demands. Material stress may lead to irreparable damage of the fuel cell stack. For these contradicting goals, a family of hierarchically ordered mathematical models has been developed with the aim of simulating and optimizing the temporal and spatial dynamical behavior of the gas streams, chemical reactions, and potential fields within the fuel cells. Altogether, the most complicated system, which is investigated in the present paper, results in a Pareto-optimal control problem with constraints in form of a huge system of 28 partial differential algebraic equations and ordinary integro-differential algebraic equations and boundary conditions which are themselves partly given by an ordinary differential algebraic system of dimension 9. The PDEs are of parabolic and hyperbolic type; some are degenerate. Moreover, the variables involved in the different submodels of this fully coupled multiphysical system live on considerably different time scales. Optimal control results are presented for a compromise between sufficiently fast load changes and sufficiently small temperature differences within the cell's solid part by means of a specially tailored formulation of a chain of optimal control problems. This procedure benefits from the different time scales of the state variables and keeps the problem manageable and computable despite its tremendous complexity and scale, although standard numerical methods are employed.
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