**2. Modeling challenges**

The performance of a fuel cell depends on the materials used, on the sizes of the cell components, on their geometry and arrangement, and on the combination and interactions of all these factors, involving interdisciplinary effects of, notably, interface, mass charge and heat transport, electrochemistry, catalysis, and materials science. Consequently, identifying an optimized stack configuration is a very demanding task that can require long and expensive experimental programs. In such a framework, models are very helpful in exploring possible system behavior and addressing the search for optimal solutions. However, due to the diversity and complexity of such phenomena, which occur at multidimensional level and on a wide range of length and time scales, model analyses have to necessarily be carried out by computer simulations. The numerical modeling of fuel cells, often dubbed CFCD (computational fuel cell dynamics), deals with multidimensional mass transport phenomena, electrochemical kinetics, and transport of charge (electrons and ions), in complex temperature-dependent relationships that are strongly coupled to each other. Namely, they are strongly coupled nonlinear problems and their solutions require advanced iterative algorithms capable of efficiently ensuring converged and accurate solutions. The analyses can be conveniently formulated in terms of electric potentials of the electronic and electrolyte phases, whose equations are strongly nonlinear and coupled to each other at electrochemical kinetics level. Techniques have been developed for efficiently solving these potential equations. Fuel cells present a stratified structure, made of thin layers of different materials where interactions occur. These domains with very different dimensions along coordinate axes give rise to additional computational challenges. Electrochemical activity sites generate current densities that can reach values in the order of 1 A/cm<sup>2</sup> (referred to the cross-sectional area) so that a current of 500 A requires membrane cross sections of 20 20 cm or more, whereas typical thicknesses of gas diffusion layers (GDLs) are in the order of 2–<sup>3</sup> <sup>10</sup><sup>2</sup> cm. These GDLs are separated by proton-exchange membranes (PEMs) which are thin electrolytes, with widths in the order of 1–<sup>5</sup> <sup>10</sup><sup>3</sup> cm.

and heat transport in the diffusion layers; and mass, heat, and charge transport in the bipolar plates and their flow channels. In fact, these parameters are needed in the equations (Nernst equation, Butler-Volmer equation, Darcy's equation, Fourier's law, Ohm's law, etc.) constituting FC models [3]. Since these equations are strongly nonlinear and coupled, the models present a strong vulnerability to parameter uncertainties. A number of diagnostic methods, such as cyclic voltammetry, thinfilm rotating ring-disc electrode (CV-TF-RRDE), electrochemical impedance spectroscopy (EIS), and broadband electrical spectroscopy (BES), can provide accurate ex situ measurements. Nevertheless, the use of the measurement data thus obtained to model fuel cells in working conditions presents several setbacks. On the other hand, parameter values fully suitable for operative conditions can be achieved by means of *in situ* measurements, but few are available and they are often difficult to carry out, allowing to determine a limited number of parameters. Examples are EIS, neutron radiography, and voltammetric and chronoamperometric approaches in the "driven-cell" mode. To cope with material characterization and parameters extractions, a number of sophisticated numerical techniques that resort to optimization methods for extracting more parameters at once from a large number of experimental data have been developed in recent years. Stochastic methods have proven to be particularly successful to meet such target. Efficient modeling suitable to provide a comprehensive understanding of fuel cell dynamics thus involves: physicochemical model identification, advanced numerical algorithms, materials parameters extraction, and model validation against detailed distribution data. This last aspect arises since data of global nature, such as I-V curves, can result inadequate to

PEMFCs have been subject to a vast body of studies aimed at modeling and some of the most important issues are described hereafter. In PEMFCs*,* the hydrogen oxidation reaction (HOR) occurring at the anode catalyst layer (CL) and the oxygen

anode : *H*<sup>2</sup> ! 2*H*<sup>þ</sup> þ 2*e*

are segregated by the proton exchange membrane (**Figure 1**). According to the Nernst equation, the cell's reversible voltage *E* varies with temperature *T* and gas

> *T f e*

where *E*<sup>0</sup> = 1.229 V is the value in standard temperature and pressure conditions, Δ*Es* is the entropic variation due to Δ^*s*, and Δ*Ec* is the term related to the variation of gas pressures and hence of gas concentrations. By introducing the "bulk" (undisturbed) concentrations *cH*<sup>2</sup> *,cO*<sup>2</sup> , Δ*Ec* can be split into two terms:

*; pO*<sup>2</sup> � �

> ln *cH*<sup>2</sup> *c*0 *H*<sup>2</sup>

4

! *cO*<sup>2</sup>

!<sup>0</sup>*:*<sup>5</sup> 2

*c*0 *O*<sup>2</sup>

*, pO*<sup>2</sup> (or equivalently, with concentrations *cH*<sup>2</sup> *, cO*<sup>2</sup> ) [4]:

Δ^*s T*ð Þ*dT* þ

*O*<sup>2</sup> þ 2*H*<sup>þ</sup> þ 2*e*

�

� ! *<sup>H</sup>*2*<sup>O</sup>* (1)

3

5 (2)

capture the material parameters with their correlations.

*Distributed and Lumped Parameter Models for Fuel Cells*

*DOI: http://dx.doi.org/10.5772/intechopen.89048*

reduction reaction (ORR) at the cathode catalyst layer

2

*<sup>E</sup>* <sup>¼</sup> *<sup>E</sup>*<sup>0</sup> <sup>þ</sup> <sup>Δ</sup>*Es*ð Þþ *<sup>T</sup>* <sup>Δ</sup>*Ec <sup>T</sup>; pH*<sup>2</sup>

*T*0

1 *nF* ð *T*

cathode : <sup>1</sup>

<sup>¼</sup> *<sup>E</sup>*<sup>0</sup> <sup>þ</sup>

**3.1 Proton exchange membrane fuel cells**

**3. Analytical models**

pressures *pH*<sup>2</sup>

**23**

PEMs allow ion transport while preventing the passage of electrons, a key feature of electrochemical devices in order to force electrons to flow in the external electric circuits. Ion-conductive membranes allow the flow of protons when sufficiently hydrated. In optimal conditions, the proton conductivity can reach values as high as 0.2 S/cm at 100°C, which is a fairly good value for ions, but a poor one if compared with the electronic conductivity of metals, so that in order to reduce the inherent voltage and power losses such electrolytes must be very thin. At the same time, the relative fluctuations of the thickness due to manufacturing must be small, to ensure the membrane performance to be very uniform. The numerical simulation of such a domain, with aspect ratio exceeding 10<sup>3</sup> , involves severe size problems: a regular hexahedral tessellation with 10 elements in the thickness direction implies 10<sup>9</sup> nodes, namely a very demanding computational dimension [1]. Commercial CFD codes commonly use un-structured meshes, but this fact does not alter the dimensional complexity of the problem. As a further concern, depending on the transient time scale, a large number of time steps may be needed in order to accurately compute time dynamics. These features raise very challenging problems that can only be faced with parallel computing and multiprocessor computers. Analyzing a fuel cell behavior requires the full characterization of the materials used, that is, the determination of their chemical, physical, thermal, and electrical parameters [2], which are involved in: mass, heat, and charge transport in the electrolyte; thermodynamics and electrokinetics in the catalyst layers; mass, charge,

### *Distributed and Lumped Parameter Models for Fuel Cells DOI: http://dx.doi.org/10.5772/intechopen.89048*

**2. Modeling challenges**

*Thermodynamics and Energy Engineering*

the order of 1–<sup>5</sup> <sup>10</sup><sup>3</sup> cm.

**22**

of such a domain, with aspect ratio exceeding 10<sup>3</sup>

The performance of a fuel cell depends on the materials used, on the sizes of the cell components, on their geometry and arrangement, and on the combination and interactions of all these factors, involving interdisciplinary effects of, notably, interface, mass charge and heat transport, electrochemistry, catalysis, and materials science. Consequently, identifying an optimized stack configuration is a very demanding task that can require long and expensive experimental programs. In such a framework, models are very helpful in exploring possible system behavior and addressing the search for optimal solutions. However, due to the diversity and complexity of such phenomena, which occur at multidimensional level and on a wide range of length and time scales, model analyses have to necessarily be carried out by computer simulations. The numerical modeling of fuel cells, often dubbed CFCD (computational fuel cell dynamics), deals with multidimensional mass transport phenomena, electrochemical kinetics, and transport of charge (electrons and ions), in complex temperature-dependent relationships that are strongly coupled to each other. Namely, they are strongly coupled nonlinear problems and their solutions require advanced iterative algorithms capable of efficiently ensuring converged and accurate solutions. The analyses can be conveniently formulated in terms of electric potentials of the electronic and electrolyte phases, whose equations are strongly nonlinear and coupled to each other at electrochemical kinetics level. Techniques have been developed for efficiently solving these potential equations. Fuel cells present a stratified structure, made of thin layers of different materials where interactions occur. These domains with very different dimensions along coordinate axes give rise to additional computational challenges. Electrochemical activity sites generate current densities that can reach values in the order of 1 A/cm<sup>2</sup> (referred to the cross-sectional area) so that a current of 500 A requires membrane cross sections of 20 20 cm or more, whereas typical thicknesses of gas diffusion layers (GDLs) are in the order of 2–<sup>3</sup> <sup>10</sup><sup>2</sup> cm. These GDLs are separated by proton-exchange membranes (PEMs) which are thin electrolytes, with widths in

PEMs allow ion transport while preventing the passage of electrons, a key feature of electrochemical devices in order to force electrons to flow in the external electric circuits. Ion-conductive membranes allow the flow of protons when sufficiently hydrated. In optimal conditions, the proton conductivity can reach values as high as 0.2 S/cm at 100°C, which is a fairly good value for ions, but a poor one if compared with the electronic conductivity of metals, so that in order to reduce the inherent voltage and power losses such electrolytes must be very thin. At the same time, the relative fluctuations of the thickness due to manufacturing must be small, to ensure the membrane performance to be very uniform. The numerical simulation

regular hexahedral tessellation with 10 elements in the thickness direction implies 10<sup>9</sup> nodes, namely a very demanding computational dimension [1]. Commercial CFD codes commonly use un-structured meshes, but this fact does not alter the dimensional complexity of the problem. As a further concern, depending on the transient time scale, a large number of time steps may be needed in order to accurately compute time dynamics. These features raise very challenging problems that can only be faced with parallel computing and multiprocessor computers. Analyzing a fuel cell behavior requires the full characterization of the materials used, that is, the determination of their chemical, physical, thermal, and electrical parameters [2], which are involved in: mass, heat, and charge transport in the electrolyte; thermodynamics and electrokinetics in the catalyst layers; mass, charge,

, involves severe size problems: a

and heat transport in the diffusion layers; and mass, heat, and charge transport in the bipolar plates and their flow channels. In fact, these parameters are needed in the equations (Nernst equation, Butler-Volmer equation, Darcy's equation, Fourier's law, Ohm's law, etc.) constituting FC models [3]. Since these equations are strongly nonlinear and coupled, the models present a strong vulnerability to parameter uncertainties. A number of diagnostic methods, such as cyclic voltammetry, thinfilm rotating ring-disc electrode (CV-TF-RRDE), electrochemical impedance spectroscopy (EIS), and broadband electrical spectroscopy (BES), can provide accurate ex situ measurements. Nevertheless, the use of the measurement data thus obtained to model fuel cells in working conditions presents several setbacks. On the other hand, parameter values fully suitable for operative conditions can be achieved by means of *in situ* measurements, but few are available and they are often difficult to carry out, allowing to determine a limited number of parameters. Examples are EIS, neutron radiography, and voltammetric and chronoamperometric approaches in the "driven-cell" mode. To cope with material characterization and parameters extractions, a number of sophisticated numerical techniques that resort to optimization methods for extracting more parameters at once from a large number of experimental data have been developed in recent years. Stochastic methods have proven to be particularly successful to meet such target. Efficient modeling suitable to provide a comprehensive understanding of fuel cell dynamics thus involves: physicochemical model identification, advanced numerical algorithms, materials parameters extraction, and model validation against detailed distribution data. This last aspect arises since data of global nature, such as I-V curves, can result inadequate to capture the material parameters with their correlations.
