Mathematical Modeling of Fuel Cells

Electrochemistry Laboratory, Paul Scherrer Institut, Villigen, Switzerland 

During the past two decades, modeling has become an important tool for fuel cell technology development. The mathematical simulation work has two main goals  

Understanding. As electrochemical reactors with gradients of species, temperature, and potential in all dimensions and over a broad range of scales from nanometers to meters, fuel cells are complex devices. Changing a single parameter (e.g., the gas humidity in a PEMFC) can result in effects on the scale of reaction kinetics and other parameters, all the way to temperature distribution in a fuel ceU stack. This multitude of effects, as well as their consequences as felt in important parameters such as efficiency or power density, is difficult or impossible to comprehend without modeling approaches. 

Fuel Cells Problems and Solutions, Second Edition. Vladimir S. Bagotsky. 

A wide range of modeling approaches—from the nanoscale of components to the large scale of stack and system levels—have been developed in fuel cell research, and the different procedures can be classified in various ways. 

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The differential PEM fuel cell reactor is motivated by considering a small element in the serpentine flow channel fuel cell as shown in Figure 3.1 [12]. Mathematical models of fuel cells use differential mass, momentum and energy balances around the differential element as the defining equations for modeling larger and more complex flow fields [12]. In the differential element the only compositional variations are transverse to the membrane. The key element of a differential fuel cell is that the compositions in the gas phases in the flow channels at the anode and cathode are uniform. 

Darling RM, Meyers JP. 2005. Mathematical model of platinum movement in PEM fuel cells. J Electrochem Soc 152 A242-A247. 

In PEM fuel cells, catalyst activity and catalyst efficiency are still significant issues. Russell and Rose summarize fundamental work involving X-ray absorption spectroscopy on catalysts in low temperature fuel cell systems. These types of studies are very useful for developing a detailed understanding of the mechanisms of reactions at catalyst surfaces and could lead to the development of new improved efficient catalysts. Important in the development of fuel cell technology are mathematical models of engineering aspects of a fuel cell system. Wang writes about studies related to this topic. 

In order to study cathode flooding in small fuel cells for portable applications operated at ambient conditions, Tuber et al.81 designed a transparent cell that was only operated at low current densities and at room temperature. The experimental data was then used to confirm a mathematical model of a similar cell. Fig. 4 describes the schematic top and side view of this transparent fuel cell. The setup was placed between a base and a transparent cover plate. While the anodic base plate was fabricated of stainless steel, the cover plate was made up of plexiglass. A rib of stainless steel was inserted into a slot in the cover plate to obtain the necessary electrical connection. It was observed that clogging of flow channels by liquid water was a major cause for low cell performance. When the fuel cell operated at room temperature during startup and outdoor operation, a hydrophilic carbon paper turned out to be more effective compared with a hydrophobic one.81... 

Mathematical modeling of physical processes in fuel cells inevitably involves some assumptions that may or may not be valid under all circumstances. Furthermore approximations have to be introduced to make the computational models robust and tractable. These approximations in the mathematical models lead to the so called modeling errors . That is if the equations posed are solved exactly, the difference between this exact solution and the corresponding true but usually unknown physical reality is known as the modeling error. However, it is rarely the situation that the solution to the mathematical models is exact due to the inherent numerical errors such as round off errors, iteration convergence and discretization errors, among oth-... 

Bemardi DM, Verbrugge MW (1992) A mathematical model of the solid-polymer-electrolyte fuel cell. J Electrochem Soc 139 2477-91... 

Chapters I to III introduce the reader to the general problems of fuel cells. The nature and role of the electrode material which acts as a solid electrocatalyst for a specific reaction is considered in chapters IV to VI. Mechanisms of the anodic oxidation of different fuels and of the reduction of molecular oxygen are discussed in chapters VII to XII for the low-temperature fuel cells and the strong influence of chemisorhed species or oxide layers on the electrode reaction is outlined. Processes in molten carbonate fuel cells and solid electrolyte fuel cells are covered in chapters XIII and XIV. The important properties of porous electrodes and structures and models used in the mathematical analysis of the operation of these electrodes are discussed in chapters XV and XVI. 

DjUah, N. and Lu, D., Mathematical modelling of the transport phenomena and the chemical/electrochemical reactions on solid oxide fuel cells, Int. J. Therm. Sci. 41 (2002) 29 0. 

Figure 1.2. Equilibrium concentration of mobile platinum species in solution versus platinum electrode relative to standard hydrogen electrode R. M. Darling and J. P. Meyers Mathematical Model of Platinum Movement in PEM Fuel Cells, J. Electrochem. Soc., 152, A242 (2005).. From Ref. [31]. Reproduced with permission of the Electrochemical Society.

D. M. Bernardi and M. W. Verbrugge, A Mathematical Model of the Solid-Polymer Electrolyte Fuel Cell, J. Electrochem. Soc., 139, TAll (1992). 

K. Promislow and B. Wetton, A Simple, Mathematical Model of Thermal Coupling in Fuel Cell Stacks, accepted in the J. Power Sources, 150,129-135. February, (2005). 

Sousa R, Gonzalez ER (2005) Mathematical modeling of polymer electrolyte fuel cells. J Power Sources 147 32-45... 

Wang ZH, Wang CY (2003) Mathematical modeling of liquid-feed direct methanol fuel cells. J Electrochem Soc 150 A508-A519... 

Garcia BL, Sethuraman VA, Weidner JW, White RE (2004) Mathematical model of a direct methanol fuel cell. J Fuel Cell Sci Technol 1 43-48... 

H. Zhu and R. J. Kee. A general mathematical model for analyzing the performance of fuel-cell membrane-electrode assemblies. J. Power Sources 117, (2003) 61-74. 

In the following, a mathematical model of a single MCFC is presented and discussed. In order to keep the complexity of the model at a reasonable level, the focus is only on the fuel cell itself here. Furthermore, no model of the electrode processes is given here instead, an interface between the fuelcell model and any arbitrary electrode model is defined, and different electrode models are discussed afterwards (Section 28.3). 

Promislow, K. and Wetton, B. (2005) A simple, mathematical model of thermal coupling in fuel cell stacks. J. Power Sources, 150, 129-135. 

Stroma Z, Watanabe S, Yasuda K, Fukuta K, Yanagi H (2011) Mathematical modeling of the concentration profile of carbonate ions in an anion exchange membrane fuel cell. J Electrochem Soc 158 B682-B689. doi 10.1149/1.3576120... 

M. M. Hussain, X. Li, and I. Dincer. Multi-component mathematical model of solid oxide fuel cell anode. Int. J. Energy Res., 29 1083-1101, 2005. 

A. Rowe and X. Li. Mathematical modeling of proton exchange membrane fuel cells. J. 

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