**3. One-dimensional modeling of electrochemical membrane cell**

A single cell can be described schematically as an assembly of several layers constituting four distinct areas. A fuel cell is a composite structure of anode, cathode, and electrolyte. Good electrochemical performance of the cell requires effective electrocatalysts. On both sides of the cell, the interconnect plates cumulate three functions: current collector, gas feeding via the gas channels, and thermal control thanks to cooling water channels. Flow field is used to supply and distribute the fuel and the oxidant to the anode and cathode electrocatalysts, respectively. The distribution of flow over the electrodes should ideally be uniform to try to ensure a uniform performance of each electrode across its surface. Thus, it is possible to develop a single-cell 1D model. Although most of the models used are one-dimensional, they correctly predict the electrochemical behavior of membrane electrode assembly.

In order to build a 1D model, a particular attention is required on water flux. The water managements is the key issue in single-cell modeling. In order to predict cell performance, the single-cell model must take into account gas diffusion in the porous electrodes, water diffusion, and electroosmotic transport through the polymeric membrane. Ramousse et al. [16] have developed one-dimensional coupled charge and mass transfer model in the electrodes. The three main types of model can be employed to evaluate the overvoltages at both electrodes:

*I* !

difference (called ohmic drop *ηohm*) can be expressed as follows:

*Thermodynamics and Energy Engineering*

**Figure 1.**

**82**

*Schematic description of a MEA.*

where *σ* is the ionic conductivity of an electrolyte. In the case of a onedimensional system where the electrolyte is confined in a finite space between two surface electrodes S and separated from each other by the distance e, this potential

*<sup>η</sup>ohm* ¼ � *<sup>e</sup>*

where J is the total current through electrochemical device: J is the average current density I multiplied by the electrode surface S. For other geometries, it will be necessary to calculate *ηohm* from the relation (Eq. 5) by integration. The ohmic drop in the electrolytic solution of an electrochemical cell is one of the parameters to be evaluated to optimize the cell efficiency. This limitation is also called primary current distribution. Thus, the cell potential is calculated from following expression:

Inside the nonaqueous electrochemical device (**Figure 1**), the primary current distribution is well controlled, because a solid electrolyte is confined in a finite space between two surface electrodes and only the misalignment of both electrodes could affect the current distribution through the electrolyte. In nonaqueous case, the optimization endeavor is devoted to secondary (electrochemical activation) and tertiary (mass transport limitation) current distributions. In this context, fuel cells are the best example to scrutinize energetic balance of electrochemical devices. The main advantage associated with fuel cells is that they are not limited by Carnot efficiency. Besides, no moving parts are required to convert thermal energy into mechanical energy. The energy release from interatomic bonds of the reactants is converted efficiently into electrical energy. In this document, we consider protonexchange membrane fuel cells (PEMFCs): a PEMFC consists of a polymer electrolyte sandwiched between two electrodes to form a membrane electrode assembly

*σS*

Ucell ¼ UOCV � η<sup>a</sup> þ η<sup>c</sup> � ηohm (7)

¼ �*σ*∇*<sup>E</sup>* ! (5)

*I* (6)

• The Tafel law results from experimental observations: this simple and widely used description is based on the phenomenological Eq. (12).

*I* ¼ ðð *S*

*How to Build Simple Models of PEM Fuel Cells for Fast Computation*

*DOI: http://dx.doi.org/10.5772/intechopen.89958*

(Eq. 12) to provide corrected Tafel mode l as follows:

from thermodynamic equilibrium.

**Figure 2.**

**85**

*catalyst layer model.*

**Figure 2** exhibits the evolutions of the cathodic and anodic overpotentials, calculated in the same operating conditions, as functions of the current density. There are great differences between the overpotentials predicted by the three models, which emphasize the important influence of the active layer on fuel cell performances. The porous catalyst layer model (agglomerate model) and nonporous catalyst layer models require only intrinsic parameters: active layer thickness, platinum load (roughness factor), and kinetic coefficients (exchange current density). The polarization curves obtained with the porous and nonporous models exhibit Tafel behavior. Due to the slow diffusion of active species in the homogeneous solid phase, the nonporous model tends to overestimate the overpotentials. Particular attention should be paid to the use of the Tafel law. The effective platinum area being higher than the MEA section and the exchange current density i0 should be multiplied by a geometric parameter such as the roughness factor γ

<sup>η</sup>a*=*<sup>c</sup> <sup>¼</sup> ba*<sup>=</sup>*<sup>c</sup> log <sup>i</sup>

However, Singh et al. [19] suggested that a multidimensional model could improve the description, since gas composition and temperature vary along the feeding channels. In addition, these phenomena become critical for large cell areas used to obtain great current intensities. The flow field allows gas to flow along the length of the electrode while permitting mass transport to the electrocatalyst normal to its surface. One of the simplest flow field designs consists of a series of narrow parallel rectangular channels where fuel or oxidant is fed at one end and removed from the opposite side. **Figure 3** shows a schematic illustration of the PEMFC that consists of a membrane sandwiched between two gas diffusion electrodes, this assembly being pressed between two current collectors. Within a PEMFC, reactant depletion and cathodic water production occur along the length of

*Influence of electrode description on overvoltages à T = 60°C. (a) Anodic overvoltage; c, cathodic overvoltage; 1, Tafel law corrected with the roughness factor; 2, porous catalyst layer model (agglomerate model); 3, nonporous*

In **Figure 2**, the anodic prediction is incoherent because the anodic overvoltage is negative. Therefore, the Tafel law is only adapted to high current density, far

γj 0*:*a*=*c

*i dS* (11)

(12)


In these three models of active layer, the authors have computed the same values of kinetic coefficients *bi* and *io a*ð Þ ,*<sup>c</sup>* (**Table 1**).

The simulations presented below describe, in steady state, one-dimensional mass transfer in the whole cell and charge and mass transfers in the electrodes. Then, only a numerical solution can be accessed. Thanks to numerical computation the cell overpotential *ηa/c* and the ohmic drop through the membrane *ηohm* can be simulated as a function of current density.

The total current is then calculated by integrating the local current density all over the MEA surface:


#### **Table 1.**

*Common parameter values of PEMFC cells.*

*How to Build Simple Models of PEM Fuel Cells for Fast Computation DOI: http://dx.doi.org/10.5772/intechopen.89958*

• The Tafel law results from experimental observations: this simple and widely

• The agglomerate model takes into account the existence of gas pore and the ohmic drop in the agglomerate. This model results to considering a cylindrical

In these three models of active layer, the authors have computed the same values

The simulations presented below describe, in steady state, one-dimensional mass transfer in the whole cell and charge and mass transfers in the electrodes. Then, only a numerical solution can be accessed. Thanks to numerical computation the cell overpotential *ηa/c* and the ohmic drop through the membrane *ηohm* can be

The total current is then calculated by integrating the local current density all

**Parameter Values** *γ*: roughness factor (�) 100 *<sup>δ</sup>*: gas backing layer thickness (m) <sup>230</sup> � <sup>10</sup>�<sup>6</sup> *LCA*: catalyst Layer thickness (m) <sup>10</sup> � <sup>10</sup>�<sup>6</sup> *<sup>δ</sup>2*: size of the agglomerate (m) <sup>3</sup> � <sup>10</sup>�<sup>6</sup> *Lm: membrane* thickness (m) <sup>125</sup> � <sup>10</sup>�<sup>6</sup>

*GDL*: gas backing layer porosity 0.8

*CL*: active layer porosity 0.5

*bc*: Tafel slope (mV/dec) 120 *ba*: Tafel slope (mV/dec) 30 νe: total number of exchanged electrons 4

*EW*: equivalent weight (kg/mole) 1.1

*τ0*: osmotic coefficient 2.5/22

) 4 � <sup>10</sup>�<sup>7</sup>

) 1 � <sup>10</sup>�<sup>2</sup>

/s) 1.63 � <sup>10</sup>�<sup>4</sup>

/s) 3.20 � <sup>10</sup>�<sup>5</sup>

/s) 2.41 � <sup>10</sup>�<sup>5</sup>

/s) 3.35 � <sup>10</sup>�<sup>5</sup>

/s) <sup>3</sup> � <sup>10</sup>�<sup>10</sup>

CA) 1.5

/s) <sup>10</sup>�<sup>9</sup> � (<sup>ε</sup>

) 2020

used description is based on the phenomenological Eq. (12).

model of pores inside the electrode [17, 18].

of kinetic coefficients *bi* and *io a*ð Þ ,*<sup>c</sup>* (**Table 1**).

simulated as a function of current density.

*j0c*: exchange current density (A/cm2

*j0a*: exchange current density (A/cm<sup>2</sup>

*<sup>H</sup>*2*=H*2*<sup>O</sup>*: effective diffusion coefficients (m<sup>2</sup>

*<sup>O</sup>*2*=H*2*<sup>O</sup>*: effective diffusion coefficient (m2

*DCL*: diffusion coefficient in the CA (m2

*ρdry*: dry *Nafion* density, (kg/m3

*Common parameter values of PEMFC cells.*

: effective diffusion coefficients (m2

*:* effective diffusion coefficients (m2

*Dm*: effective diffusion coefficients of water in the membrane (m2

(Appendix 1).

*Thermodynamics and Energy Engineering*

over the MEA surface:

*ε*

*ε*

*Deff*

*Deff*

*Deff O*2*=N*<sup>2</sup>

*Deff H*2*O=N*<sup>2</sup>

**Table 1.**

**84**

• The classical catalyst layer model exhibited in the previous section

$$I = \iint\_{S} i \, d\mathcal{S} \tag{11}$$

**Figure 2** exhibits the evolutions of the cathodic and anodic overpotentials, calculated in the same operating conditions, as functions of the current density. There are great differences between the overpotentials predicted by the three models, which emphasize the important influence of the active layer on fuel cell performances. The porous catalyst layer model (agglomerate model) and nonporous catalyst layer models require only intrinsic parameters: active layer thickness, platinum load (roughness factor), and kinetic coefficients (exchange current density). The polarization curves obtained with the porous and nonporous models exhibit Tafel behavior. Due to the slow diffusion of active species in the homogeneous solid phase, the nonporous model tends to overestimate the overpotentials. Particular attention should be paid to the use of the Tafel law. The effective platinum area being higher than the MEA section and the exchange current density i0 should be multiplied by a geometric parameter such as the roughness factor γ (Eq. 12) to provide corrected Tafel mode l as follows:

$$\mathfrak{h}\_{\mathbf{a}/\mathbf{c}} = \mathbf{b}\_{\mathbf{a}/\mathbf{c}} \log \frac{\dot{\mathbf{i}}}{\eta \dot{\mathbf{j}}\_{0.\mathbf{a}/\mathbf{c}}} \tag{12}$$

In **Figure 2**, the anodic prediction is incoherent because the anodic overvoltage is negative. Therefore, the Tafel law is only adapted to high current density, far from thermodynamic equilibrium.

However, Singh et al. [19] suggested that a multidimensional model could improve the description, since gas composition and temperature vary along the feeding channels. In addition, these phenomena become critical for large cell areas used to obtain great current intensities. The flow field allows gas to flow along the length of the electrode while permitting mass transport to the electrocatalyst normal to its surface. One of the simplest flow field designs consists of a series of narrow parallel rectangular channels where fuel or oxidant is fed at one end and removed from the opposite side. **Figure 3** shows a schematic illustration of the PEMFC that consists of a membrane sandwiched between two gas diffusion electrodes, this assembly being pressed between two current collectors. Within a PEMFC, reactant depletion and cathodic water production occur along the length of

#### **Figure 2.**

*Influence of electrode description on overvoltages à T = 60°C. (a) Anodic overvoltage; c, cathodic overvoltage; 1, Tafel law corrected with the roughness factor; 2, porous catalyst layer model (agglomerate model); 3, nonporous catalyst layer model.*

cross-section and ignore the influence of viscosity. Dohle et al. [24] have proposed a pseudo two-dimensional model of the cathode compartment of a PEMFC. The model is based on the continuity equations for the gases and assumes constant pressure and plug flow conditions. The influence of hydrodynamics was not considered. Regardless of the latter point, the model shows some interesting effects for a simple, parallel flow field channel configuration. The mole fraction change along

through the gas diffusion layer (GDL) allowing to the electrochemical processes in the active layer of the GDL and the corresponding current density changes. Note that this normal flux also depends on z. The mass balance for a single-phase species i

dz <sup>¼</sup> NGDL

where Fi is respectively the molar flow rate for the component Xi. Both oxygen and hydrogen fluxes are proportional to current density, while the nitrogen flux is

dFi

zero because nitrogen is neither consumed nor produced in the fuel cell:

NGDL H2O <sup>þ</sup> <sup>N</sup><sup>m</sup>

water diffusion and water electroosmotic fluxes [25].

**4.2 Pseudo 2D modeling: closed-form expression**

NGDL <sup>i</sup> <sup>¼</sup> <sup>ν</sup>ii

For the specific case of PEMFC, water produced at the cathode can flow through the gas diffusion layer to reach the gas channel and transport through the

H2O <sup>¼</sup> <sup>i</sup>

Even if water transport processes into the membrane are not well understood, the phenomenological model of water transport in the membrane takes into account

Dohle et al. [24] have proposed an analytical solution of the oxygen profile along the channel. This model assumed that oxygen reduction is the determining rate step

*Lζ*

*L*1

*exp* � <sup>2</sup>*:*<sup>3</sup> *<sup>η</sup><sup>c</sup> bc*

(16)

(17)

(18)

and neglects water transport. The oxygen concentration profile is given by:

where L<sup>ζ</sup> is the characteristic length of oxygen consumption.

*<sup>L</sup><sup>ζ</sup>* <sup>¼</sup> <sup>4</sup>*<sup>F</sup> h v*<sup>0</sup> *XO*<sup>2</sup> ½ � <sup>∗</sup> *γ j* 0,*c*

*XO*<sup>2</sup> ½ �¼ *XO*<sup>2</sup> ½ �<sup>0</sup> <sup>1</sup> � ð Þ <sup>1</sup> � *<sup>ζ</sup> <sup>z</sup>*

*XO*<sup>2</sup> ½ �<sup>0</sup> *XO*<sup>2</sup> ½ � <sup>∗</sup> <sup>1</sup>�*<sup>ζ</sup>*

where *h* is the channel height and *v0* is the inlet flow velocity, assuming a kinetic Tafel law for oxygen reduction (Eq. (12)). Note that for *γ* = 1, the oxygen profile

*XO*<sup>2</sup> ½ �¼ *XO*<sup>2</sup> ½ �<sup>0</sup> *exp* � *<sup>z</sup>*

GDL in the y-direction

<sup>i</sup> (13)

<sup>ν</sup>eF (14)

2F (15)

the gas channel length results from the normal molar flux Ni

*How to Build Simple Models of PEM Fuel Cells for Fast Computation*

*DOI: http://dx.doi.org/10.5772/intechopen.89958*

(= O2, H2, N2) may be written as:

membrane:

distribution becomes:

**87**

#### **Figure 3.**

*Scheme of a proton exchange membrane fuel cell and the equivalent electrical circuit for the whole PEMFC for DC and AC solutions.*

the fuel cell due to the electrochemical processes. Gas supply to the catalytic active site takes place in the porous gas diffusion electrode that contains a GDL consisting of hydrophobic gaseous pores and a reaction region (CL). Namely, slow diffusion in the GDL and CL can induce oxygen depletion in the case of air cathode. The membrane, commonly Nafion®, acts as both a separator and an electrolyte. Water transport across the membrane results from the electroosmotic water dragging with proton migration from anode to cathode and water "back-diffusion" that reduces the concentration gradient. During the operation, these effects can be responsible of either membrane dehydration or flooding on the porous electrode.

A finite volume method using a computational grid [20] can be used to solve mass, charge, energy, momentum balances including transport through porous media, and chemical and electrochemical reactions within the porous electrodes in a gas diffusion electrode model. Freshly, Zhang et al. [21] have developed a twophase multidimensional model to properly handle mass transport. The results of these computations reveal that liquid water transport inside and across the polymeric membrane plays an important role in PEM fuel cell water distributions. In addition, it was shown that increasing the contact angle at GDL/channel interface is found to be able to improve the water removal process in channels. However, the resolution of fuel cell models requires important calculation times to obtain the detailed variations of flow field, species concentrations, temperature, liquid saturation, and electronic and ionic phase potentials.

### **4. Simple multidimensional modeling**

#### **4.1 Pseudo 2D modeling**

To cut down these difficulties, the pseudo 2D as 1D + 1D approach implies that (i) meander-like channel is replaced with the straight one and (ii) water and oxygen concentrations in the channel provide "boundary conditions" for local transport of these species across the cell. PEMFC cell model, which couples 1D transport across the cell with 1D description of the flow in the feed channel, has been proposed [22, 23]. An approach is to model the flow as an ideal gas in a straight channel of

*How to Build Simple Models of PEM Fuel Cells for Fast Computation DOI: http://dx.doi.org/10.5772/intechopen.89958*

cross-section and ignore the influence of viscosity. Dohle et al. [24] have proposed a pseudo two-dimensional model of the cathode compartment of a PEMFC. The model is based on the continuity equations for the gases and assumes constant pressure and plug flow conditions. The influence of hydrodynamics was not considered. Regardless of the latter point, the model shows some interesting effects for a simple, parallel flow field channel configuration. The mole fraction change along the gas channel length results from the normal molar flux Ni GDL in the y-direction through the gas diffusion layer (GDL) allowing to the electrochemical processes in the active layer of the GDL and the corresponding current density changes. Note that this normal flux also depends on z. The mass balance for a single-phase species i (= O2, H2, N2) may be written as:

$$\frac{\text{dF}\_{\text{i}}}{\text{dz}} = \text{N}\_{\text{i}}^{\text{GDL}} \tag{13}$$

where Fi is respectively the molar flow rate for the component Xi. Both oxygen and hydrogen fluxes are proportional to current density, while the nitrogen flux is zero because nitrogen is neither consumed nor produced in the fuel cell:

$$\mathbf{N}\_{\mathrm{i}}^{\mathrm{GDL}} = \frac{\nu\_{\mathrm{i}}\mathbf{i}}{\nu\_{\mathrm{e}}\mathbf{F}} \tag{14}$$

For the specific case of PEMFC, water produced at the cathode can flow through the gas diffusion layer to reach the gas channel and transport through the membrane:

$$\mathbf{N}\_{\mathrm{H}\_2\mathrm{O}}^{\mathrm{GDL}} + \mathbf{N}\_{\mathrm{H}\_2\mathrm{O}}^{\mathrm{m}} = \frac{\mathrm{i}}{2\mathrm{F}} \tag{15}$$

Even if water transport processes into the membrane are not well understood, the phenomenological model of water transport in the membrane takes into account water diffusion and water electroosmotic fluxes [25].

#### **4.2 Pseudo 2D modeling: closed-form expression**

Dohle et al. [24] have proposed an analytical solution of the oxygen profile along the channel. This model assumed that oxygen reduction is the determining rate step and neglects water transport. The oxygen concentration profile is given by:

$$[X\_{O\_2}] = [X\_{O\_2}]\_0 \left(\mathbf{1} - (\mathbf{1} - \boldsymbol{\zeta})\frac{z}{L\_{\boldsymbol{\zeta}}}\right) \tag{16}$$

where L<sup>ζ</sup> is the characteristic length of oxygen consumption.

$$L\_{\zeta} = 4F \frac{h \, v\_0 [X\_{O\_2}]\_\*}{\gamma \, j\_{0, \varepsilon}} \left( \frac{[X\_{O\_2}]\_0}{[X\_{O\_2}]\_\*} \right)^{1 - \zeta} \exp \left( -\frac{2.3 \, \eta\_{\varepsilon}}{b\_{\varepsilon}} \right) \tag{17}$$

where *h* is the channel height and *v0* is the inlet flow velocity, assuming a kinetic Tafel law for oxygen reduction (Eq. (12)). Note that for *γ* = 1, the oxygen profile distribution becomes:

$$\mathbb{E}\left[\mathbf{X}\_{O\_2}\right] = \left[\mathbf{X}\_{O\_2}\right]\_0 \exp\left(-\frac{z}{L\_1}\right) \tag{18}$$

the fuel cell due to the electrochemical processes. Gas supply to the catalytic active site takes place in the porous gas diffusion electrode that contains a GDL consisting of hydrophobic gaseous pores and a reaction region (CL). Namely, slow diffusion in the GDL and CL can induce oxygen depletion in the case of air cathode. The membrane, commonly Nafion®, acts as both a separator and an electrolyte. Water transport across the membrane results from the electroosmotic water dragging with proton migration from anode to cathode and water "back-diffusion" that reduces the concentration gradient. During the operation, these effects can be responsible of

*Scheme of a proton exchange membrane fuel cell and the equivalent electrical circuit for the whole PEMFC for*

A finite volume method using a computational grid [20] can be used to solve mass, charge, energy, momentum balances including transport through porous media, and chemical and electrochemical reactions within the porous electrodes in a gas diffusion electrode model. Freshly, Zhang et al. [21] have developed a twophase multidimensional model to properly handle mass transport. The results of these computations reveal that liquid water transport inside and across the polymeric membrane plays an important role in PEM fuel cell water distributions. In addition, it was shown that increasing the contact angle at GDL/channel interface is found to be able to improve the water removal process in channels. However, the resolution of fuel cell models requires important calculation times to obtain the detailed variations of flow field, species concentrations, temperature, liquid

To cut down these difficulties, the pseudo 2D as 1D + 1D approach implies that (i) meander-like channel is replaced with the straight one and (ii) water and oxygen concentrations in the channel provide "boundary conditions" for local transport of these species across the cell. PEMFC cell model, which couples 1D transport across the cell with 1D description of the flow in the feed channel, has been proposed [22, 23]. An approach is to model the flow as an ideal gas in a straight channel of

either membrane dehydration or flooding on the porous electrode.

saturation, and electronic and ionic phase potentials.

**4. Simple multidimensional modeling**

**4.1 Pseudo 2D modeling**

**86**

**Figure 3.**

*DC and AC solutions.*

*Thermodynamics and Energy Engineering*

The mean current density in the cell is defined as an average over the channel length L:

$$i = i\_{lim} \left[ \mathbf{1} - \left( \mathbf{1} - (\mathbf{1} - \zeta) \frac{x}{L\_{\zeta}} \right)^{\frac{1}{1-\zeta}} \right] \tag{19}$$

current density distribution Iz must take in to account the membrane conductivity σm,z, which depends on membrane water content. Therefore, accurate numerical approaches should be proposed in order to calculate the electrical characteristics (current density, ohmic resistance, overpotential) as well as gas concentration

A first rigorous 3D modeling of PEM single cell using CFD programs based on commercial finite volume technic solver has been developed by He et al. [26], to solve the fully coupled governing equations. The model assumes that liquid film is formed on the electrode surface during liquid water condensation and computes the diffusion flux, electroosmotic drag force, and water back-diffusion in order to assess to water management. According to *Zhang and Jiao* [27], despite the multiplicity of 3D CFD models available in the literature, a satisfactory 3D multiphase CFD model which is able to simulate the detailed gas and liquid two-phase flow in channels and reflect its effect on PEM fuel cell performance precisely still was not real, because it is difficult to solve coupling physics and computation amount is real barrier. Nandjou et al. [28] have proposed a pseudo 3D model to reduce the computational cost (i.e., the transport equations are formulated using a pseudo 3D approximation and coupled to an analytical electrochemical model at the catalyst layers/membrane interface). Nevertheless, the experimental measurement cannot easily emphasize or "validate" the computing results. Krewer et al. [29] have showed that 3D CFD results could be compared to experimental results via assuming that the experimental RTD of the inlet and outlet pipes can be used as input signal for the CFD simulations. Diep et al. [30], thanks to RTD experiments, have shown that a reduction in the unit cell model dimensionality to 1 + 1 or 1 + 0 D based on scaling arguments and contrasts with higher dimensional 3D models is accurate. **Figure 5** exhibits a comparison between the model and the outlet RTD. Good agreement between laminar flow 1+ 0 D-based model and experimental data was

*Comparison between tracer step outlet RTD model fuel cell measurements and one-dimensional numerical*

distributions along the gas channel.

*DOI: http://dx.doi.org/10.5772/intechopen.89958*

found.

**Figure 5.**

**89**

*model computations (cathode) [30].*

**4.3 Continuous stirred tank reactor approach to fuel cell**

*How to Build Simple Models of PEM Fuel Cells for Fast Computation*

The limiting current density (ilim) represents the limiting current density, which is attained when all the available oxygen is consumed:

$$i\_{lim} = 4F \frac{h \,\, \nu\_0 [X\_{O\_2}]\_0}{L} \tag{20}$$

The cathode overpotential is then given by:

$$\eta\_c = \frac{b\_C}{2.3} \ln \left[ \frac{K\_\zeta}{1 - \zeta} \left( 1 - \left( 1 - \frac{\overline{i}}{i\_{lim}} \right) \right)^{1 - \zeta} \right] \tag{21}$$

where:

$$K\_{\zeta} = 4F \frac{h \, v\_0 [X\_{O\_2}]\_\*}{\chi \dot{j}\_{0, \varepsilon} \, L} \left( \frac{[X\_{O\_2}]\_0}{[X\_{O\_2}]\_\*} \right)^{1 - \zeta} \tag{22}$$

Based on the previous physical models, the experimental polarization curves can be approximated. The fitting parameters are generally the exchange current density, the Tafel slope, and the limiting current density. **Figure 4** displays the effect of the channel length. The decrease in channel length L increases the limiting current density (Ilim). The decrease in L not only increases Ilim but improves cell performance in the whole range of current densities. Physically, for lower L, oxygen concentration in the channel is more uniform, and this improves the overall performance per unit cross-section area.

This single-cell model exhibits that the reactive gas diffusion becomes a limiting step. However, the water management and electrochemical kinetics play a critical role. For a given cell voltage Ucell assumed to be constant along the gas channel, the

**Figure 4.** *Voltage–current curves for the three indicated values of channel length L (cm) [24].*

## *How to Build Simple Models of PEM Fuel Cells for Fast Computation DOI: http://dx.doi.org/10.5772/intechopen.89958*

The mean current density in the cell is defined as an average over the channel

*Lζ*

1�*ζ*

*<sup>L</sup>* (20)

(19)

(21)

(22)

� � <sup>1</sup>

<sup>1</sup> � <sup>1</sup> � *<sup>i</sup>*

Based on the previous physical models, the experimental polarization curves can be approximated. The fitting parameters are generally the exchange current density, the Tafel slope, and the limiting current density. **Figure 4** displays the effect of the channel length. The decrease in channel length L increases the limiting current density (Ilim). The decrease in L not only increases Ilim but improves cell performance in the whole range of current densities. Physically, for lower L, oxygen concentration in the channel is more uniform, and this improves the overall per-

This single-cell model exhibits that the reactive gas diffusion becomes a limiting step. However, the water management and electrochemical kinetics play a critical role. For a given cell voltage Ucell assumed to be constant along the gas channel, the

� � � � <sup>1</sup>�*<sup>ζ</sup>* " #

*ilim*

*XO*<sup>2</sup> ½ �<sup>0</sup> *XO*<sup>2</sup> ½ � <sup>∗</sup> � �<sup>1</sup>�*<sup>ζ</sup>*

" #

The limiting current density (ilim) represents the limiting current density, which

*ilim* <sup>¼</sup> <sup>4</sup>*<sup>F</sup> h v*<sup>0</sup> *XO*<sup>2</sup> ½ �<sup>0</sup>

*<sup>i</sup>* <sup>¼</sup> *ilim* <sup>1</sup> � <sup>1</sup> � ð Þ <sup>1</sup> � *<sup>ζ</sup> <sup>z</sup>*

is attained when all the available oxygen is consumed:

*Thermodynamics and Energy Engineering*

The cathode overpotential is then given by:

*<sup>η</sup><sup>c</sup>* <sup>¼</sup> *bC* 2*:*3

formance per unit cross-section area.

*ln <sup>K</sup><sup>ζ</sup>* 1 � *ζ*

*<sup>K</sup><sup>ζ</sup>* <sup>¼</sup> <sup>4</sup>*<sup>F</sup> h v*<sup>0</sup> *XO*<sup>2</sup> ½ � <sup>∗</sup> *γ j* 0,*<sup>c</sup> L*

*Voltage–current curves for the three indicated values of channel length L (cm) [24].*

length L:

where:

**Figure 4.**

**88**

current density distribution Iz must take in to account the membrane conductivity σm,z, which depends on membrane water content. Therefore, accurate numerical approaches should be proposed in order to calculate the electrical characteristics (current density, ohmic resistance, overpotential) as well as gas concentration distributions along the gas channel.
