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

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 found.

#### **Figure 5.**

*Comparison between tracer step outlet RTD model fuel cell measurements and one-dimensional numerical model computations (cathode) [30].*

On the other hand, to simplify the hydrodynamic description (modeling) or experimental observations, it is possible to employ the ideal reactor models. The continuous stirred tank reactor (CSTR) model does not require a long computing time. Benziger et al. [31] have employed this method to explore the auto-humidification of the PEMFC. In **Figure 6**, the anode and cathode chambers in the PEM fuel cell are modeled as two stirred tank reactors (STR). The previous models can only be considered as approximate, and more refined approach shall be preferred, in particularly for larger fuel cells.

Boillot et al. [32] have demonstrated that the bipolar plate is similar to a series of continuous stirred tank reactors. These authors have studied various models with or without exchange zones. They have observed that best fitting was obtained by the simple model of CSTR in series and the optimal number of CSTR (J) varied from four to six in the range investigated (**Figure 7**).

Deseure [33] proposed series of CSTR to solve the mass balance equation inside the gas channel (**Figure 8**). The electrode is divided into *k* elementary units. With such a hypothesis, gas transport in the gas channel is assumed to be infinitely fast, leading to homogeneous gas composition for each CSTR. In particular, the partial pressure of a component in the elementary unit (*k*) is equal to that of the outlet and

to that of the elementary unit (*k* + 1) inlet. The resolutions of mass balances are numerically obtained thanks to the continuity of the flux between the GDL, the CL,

The electrode is divided into k elementary units, where the gas transport in the gas channel is assumed to be infinitely fast leading to homogeneous gas composition for each CSTR. Partial pressure of a component in the elementary unit (k) is equal to the outlet one and equal to the elementary unit (k + 1) inlet one. Like the 1D + 1D approach, all elementary units have the same voltage U due to the high electrical conductivity of bipolar plate and backing layer. Therefore, partial pressures of each component in each elementary unit vary and make necessary to solve mass balances and current density in each elementary unit at the same time. The current–voltage expression used is the one proposed previously in Section 2.1. The molar balance equation of each CSTR with electrode geometric area S<sup>k</sup> for each component in gas

and the membrane.

**Figure 8.**

The term ξ

**91**

phase in steady state is written as:

*Schematic description of the PEMFC half cell (cathode) [33].*

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

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

balance for each CSTR is given by:

*F<sup>k</sup>*,*in*

that is estimated using a numerical optimization (1 ≤ ξ

*F<sup>k</sup>*,*in <sup>O</sup>*<sup>2</sup> � *<sup>F</sup><sup>k</sup>*,*out*

*<sup>H</sup>*2*<sup>O</sup>* � *<sup>F</sup><sup>k</sup>*,*out*

*F<sup>k</sup>*,*in <sup>N</sup>*<sup>2</sup> � *<sup>F</sup><sup>k</sup>*,*out*

*<sup>H</sup>*2*<sup>O</sup>* <sup>¼</sup> *<sup>ξ</sup><sup>k</sup>*

*<sup>O</sup>*<sup>2</sup> <sup>¼</sup> *<sup>S</sup><sup>k</sup>*

*<sup>S</sup><sup>T</sup>* <sup>¼</sup> <sup>X</sup> *k*

*Sk NGDL*,*<sup>k</sup>*

<sup>k</sup> represents the ratio of gas molar flux divided by liquid molar flux

*NGDL*,*<sup>k</sup>*

*<sup>H</sup>*2*<sup>O</sup>* (23)

*<sup>O</sup>*<sup>2</sup> (24)

<sup>k</sup> ≥ 0). The global mass

*<sup>N</sup>*<sup>2</sup> ¼ 0 (25)

*S<sup>k</sup>* (26)

**Figure 6.** *STR PEM fuel cell model [31].*

**Figure 7.** *RTD measurements and modeling of the column flow pattern (signals in arbitrary units) [32].*

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

On the other hand, to simplify the hydrodynamic description (modeling) or experimental observations, it is possible to employ the ideal reactor models. The continuous stirred tank reactor (CSTR) model does not require a long computing time. Benziger et al. [31] have employed this method to explore the auto-humidification of the PEMFC. In **Figure 6**, the anode and cathode chambers in the PEM fuel cell are modeled as two stirred tank reactors (STR). The previous models can only be considered as approximate, and more refined approach shall be

Boillot et al. [32] have demonstrated that the bipolar plate is similar to a series of continuous stirred tank reactors. These authors have studied various models with or without exchange zones. They have observed that best fitting was obtained by the simple model of CSTR in series and the optimal number of CSTR (J) varied from

Deseure [33] proposed series of CSTR to solve the mass balance equation inside the gas channel (**Figure 8**). The electrode is divided into *k* elementary units. With such a hypothesis, gas transport in the gas channel is assumed to be infinitely fast, leading to homogeneous gas composition for each CSTR. In particular, the partial pressure of a component in the elementary unit (*k*) is equal to that of the outlet and

*RTD measurements and modeling of the column flow pattern (signals in arbitrary units) [32].*

preferred, in particularly for larger fuel cells.

*Thermodynamics and Energy Engineering*

four to six in the range investigated (**Figure 7**).

**Figure 6.**

**Figure 7.**

**90**

*STR PEM fuel cell model [31].*

**Figure 8.** *Schematic description of the PEMFC half cell (cathode) [33].*

to that of the elementary unit (*k* + 1) inlet. The resolutions of mass balances are numerically obtained thanks to the continuity of the flux between the GDL, the CL, and the membrane.

The electrode is divided into k elementary units, where the gas transport in the gas channel is assumed to be infinitely fast leading to homogeneous gas composition for each CSTR. Partial pressure of a component in the elementary unit (k) is equal to the outlet one and equal to the elementary unit (k + 1) inlet one. Like the 1D + 1D approach, all elementary units have the same voltage U due to the high electrical conductivity of bipolar plate and backing layer. Therefore, partial pressures of each component in each elementary unit vary and make necessary to solve mass balances and current density in each elementary unit at the same time. The current–voltage expression used is the one proposed previously in Section 2.1. The molar balance equation of each CSTR with electrode geometric area S<sup>k</sup> for each component in gas phase in steady state is written as:

$$F\_{H\_2O}^{k,in} - F\_{H\_2O}^{k,out} = \xi^k \mathcal{S}^k N\_{H\_2O}^{GDL,k} \tag{23}$$

$$F\_{O\_2}^{k,in} - F\_{O\_2}^{k,out} = \mathbf{S}^k N\_{O\_2}^{GDL,k} \tag{24}$$

$$F\_{N\_2}^{k,in} - F\_{N\_2}^{k,out} = \mathbf{0} \tag{25}$$

$$\mathcal{S}^T = \sum\_{k} \mathcal{S}^k \tag{26}$$

The term ξ <sup>k</sup> represents the ratio of gas molar flux divided by liquid molar flux that is estimated using a numerical optimization (1 ≤ ξ <sup>k</sup> ≥ 0). The global mass balance for each CSTR is given by:

$$F\_T^{k,in} - F\_T^{k,out} = \mathbf{S}^k \left( \mathbf{N}\_{O\_2}^{GDL,k} + \mathbf{N}\_{H\_2O}^{GDL,k} \right) \tag{27}$$

and the gas flux of consumption and production are expressed as:

$$N\_{H\_2O}^{GDL,k} = \frac{\dot{t}^k}{2F} - N\_{H\_2O}^{m,k} \tag{28}$$

*Fin*,0

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

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

*rO*<sup>2</sup> ð Þ¼ *T* 0*:*21

proposed at high level such as presented in **Figure 9**.

**4.4 Results of continuous stirred tank reactor approach to fuel cell**

First, the effect of water and ion transport within the membrane has been

The aim of these simulations is to emphasize the dependence of the locale electrochemical behavior on overall electrical performance. Three kinds of humidification mode are investigated: the auto-humidification, the humidification of inlet gas, and the humidification via the membrane. The steady-state equations of each elementary unit are solved thanks to the "simplex" numerical optimization method.

*The minimal stoichiometry oxygen coefficient rO2* ð Þ *to obtain a single-phase condition as a function of INLET*

cathode:

**Figure 9.**

**93**

*HRC with t = 60°C without membrane water exchange.*

*<sup>O</sup>*<sup>2</sup> <sup>¼</sup> *rO*2*S<sup>T</sup> <sup>I</sup>*

with I as the average current density of ik given by Eq. (17). In order to initialize the computing, the initial inlet molar flux density is obtained considering only one STR according to Benziger et al. [31] investigations. This model does not take into account the liquid water apparition; according to this assumption the values of *rO*<sup>2</sup> are calculated in order to obtain a gaseous condition. A simple mass balance could estimate the minimal air flux or *rO*<sup>2</sup> value for non-apparition of liquid water on the

> <sup>2</sup> <sup>þ</sup> *<sup>P</sup> P*�*Psat*ð Þ *T*

" #

*<sup>P</sup>*�*Psat*ð Þ *<sup>T</sup>* � *HRCP*

*P*�*HRCPsat*ð Þ *T*

*P*

This mass balance equation considers that the produced water by oxygen reduction is fully exhausted by the gas flux. However, this relation does not take into account the water flux provided from anode side: for this reason, the stoichiometry oxygen coefficient is proposed at high value. Indeed, **Figure 4** shows the limit of liquid water apparition as a function of water activity (HRC) of inlet gas. In any simulated cases, the selected values of *rO*<sup>2</sup> allow to avoid the flooding. To insure no saturation by water in all studied cases, the feeding flux in simulations could be

<sup>4</sup>*<sup>F</sup>* (36)

(37)

$$N\_{O\_2}^{GDL,k} = -\frac{i^k}{4F} \tag{29}$$

To satisfy saturated vapor pressure of water, the following expression is required:

$$P\_{sat} = \mathcal{e}^{\left(13.66 - \frac{5096.23}{(T + 273.15)}\right)}\tag{30}$$

The resolution of mass balance is numerically obtained thanks to the continuity of flux between the backing layer, the active layer, and the membrane. The molar flux density of oxygen and nitrogen are nil at membrane/electrode layer interface, and the molar flux density of water are given by the solution of the equation (Eq. (15)). Thus, the sum of diffusion and electroosmotic fluxes yields in steady state a first-order differential equation:

$$N\_{H\_2O}^{m,k} = \frac{\tau\_0 i^k}{F} \left[ \lambda\_a^k + \frac{\lambda\_c^k - \lambda\_a^k}{1 - e^{k\_m^k L\_m}} \right] \tag{31}$$

$$k\_m^k = \frac{EW\tau\_0 i^k}{\rho\_{dry} D\_m F} \tag{32}$$

*λk <sup>c</sup>* and *λ<sup>k</sup> <sup>a</sup>* stand for polymer water content at the membrane/active layer interface respectively at cathode and anode. The thermodynamic equilibrium between water vapor in the backing layers and liquid water in the polymer has been assumed using Eq. (30).

The ionic conductivity of polymeric membrane could be accessed thanks to the Neubrand [34] equation as follows:

$$
\sigma\_{H^{+}} = e^{\left(-E\_{4}\left(\frac{1}{7} - \frac{1}{35}\right)\right)} \left(0.0013\lambda^{3} + 0.0298\lambda^{2} + 0.2658\lambda\right) \tag{33}
$$

$$E\_A = 2\mathbf{\tilde{640e}}^{(-0.6\lambda)} + \mathbf{1183} \tag{34}$$

*λk <sup>c</sup>* and *λ<sup>k</sup> <sup>a</sup>* are determined using sorption equilibrium of [35]:

$$\lambda = 0.3 + 10.8 \left( \frac{\text{Py}\_{H\_20}}{\text{Psat}} \right) - 16 \left( \frac{\text{Py}\_{H\_20}}{\text{Psat}} \right)^2 + 14.1 \left( \frac{\text{Py}\_{H\_20}}{\text{Psat}} \right)^3 \tag{35}$$

The set of these equations could be solved by iterative methods and have needed only these inlet conditions *XO*<sup>2</sup> ½ � <sup>∗</sup> , *XN*<sup>2</sup> ½ � <sup>∗</sup> , *XN*<sup>2</sup> ½ � <sup>∗</sup> and *<sup>F</sup>in*,0 *<sup>O</sup>*<sup>2</sup> , *<sup>F</sup>in*,0 *<sup>N</sup>*<sup>2</sup> , *<sup>F</sup>in*,0 *<sup>H</sup>*2*<sup>O</sup>*, where *<sup>F</sup>in*,0 *<sup>O</sup>*<sup>2</sup> þ *Fin*,0 *<sup>N</sup>*<sup>2</sup> <sup>þ</sup> *<sup>F</sup>in*,0 *<sup>H</sup>*2*<sup>O</sup>* <sup>¼</sup> *<sup>F</sup>in*,0 *<sup>T</sup>* . The conventional way to investigate the influences of the mass flux on electrochemical behavior uses the inlet molar flux, which are proportional to the current density and the stoichiometry at the cathode. This flux is currently expressed using the stoichiometry oxygen coefficient *rO*<sup>2</sup> , and this expression is given by:

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

$$F\_{O\_2}^{in,0} = r\_{O\_2} \mathbf{S}^T \frac{I}{4F} \tag{36}$$

with I as the average current density of ik given by Eq. (17). In order to initialize the computing, the initial inlet molar flux density is obtained considering only one STR according to Benziger et al. [31] investigations. This model does not take into account the liquid water apparition; according to this assumption the values of *rO*<sup>2</sup> are calculated in order to obtain a gaseous condition. A simple mass balance could estimate the minimal air flux or *rO*<sup>2</sup> value for non-apparition of liquid water on the cathode:

$$r\_{O\_2}(T) = \textbf{0.21} \left[ \frac{\textbf{2} + \frac{P}{P - P\_{\text{sat}}(T)}}{\frac{P}{P - P\_{\text{sat}}(T)} - \frac{HR\_{\text{C}}P}{P - HR\_{\text{C}}P\_{\text{sat}}(T)}} \right] \tag{37}$$

This mass balance equation considers that the produced water by oxygen reduction is fully exhausted by the gas flux. However, this relation does not take into account the water flux provided from anode side: for this reason, the stoichiometry oxygen coefficient is proposed at high value. Indeed, **Figure 4** shows the limit of liquid water apparition as a function of water activity (HRC) of inlet gas. In any simulated cases, the selected values of *rO*<sup>2</sup> allow to avoid the flooding. To insure no saturation by water in all studied cases, the feeding flux in simulations could be proposed at high level such as presented in **Figure 9**.

#### **4.4 Results of continuous stirred tank reactor approach to fuel cell**

The aim of these simulations is to emphasize the dependence of the locale electrochemical behavior on overall electrical performance. Three kinds of humidification mode are investigated: the auto-humidification, the humidification of inlet gas, and the humidification via the membrane. The steady-state equations of each elementary unit are solved thanks to the "simplex" numerical optimization method. First, the effect of water and ion transport within the membrane has been

#### **Figure 9.**

*The minimal stoichiometry oxygen coefficient rO2* ð Þ *to obtain a single-phase condition as a function of INLET HRC with t = 60°C without membrane water exchange.*

*Fk*,*in <sup>T</sup>* � *<sup>F</sup>k*,*out*

*Thermodynamics and Energy Engineering*

state a first-order differential equation:

Neubrand [34] equation as follows:

*<sup>H</sup>*2*<sup>O</sup>* <sup>¼</sup> *<sup>F</sup>in*,0

*<sup>σ</sup><sup>H</sup>*<sup>þ</sup> <sup>¼</sup> *<sup>e</sup>* �*EA* <sup>1</sup>

*<sup>λ</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>3</sup> <sup>þ</sup> <sup>10</sup>*:*<sup>8</sup> *PyH*<sup>20</sup>

only these inlet conditions *XO*<sup>2</sup> ½ � <sup>∗</sup> , *XN*<sup>2</sup> ½ � <sup>∗</sup> , *XN*<sup>2</sup> ½ � <sup>∗</sup> and *<sup>F</sup>in*,0

required:

*λk <sup>c</sup>* and *λ<sup>k</sup>*

*λk <sup>c</sup>* and *λ<sup>k</sup>*

*Fin*,0 *<sup>N</sup>*<sup>2</sup> <sup>þ</sup> *<sup>F</sup>in*,0

**92**

given by:

using Eq. (30).

*<sup>T</sup>* <sup>¼</sup> *<sup>S</sup><sup>k</sup> <sup>N</sup>GDL*,*<sup>k</sup>*

*k* <sup>2</sup>*<sup>F</sup>* � *<sup>N</sup>m*,*<sup>k</sup>*

and the gas flux of consumption and production are expressed as:

*NGDL*,*<sup>k</sup> <sup>H</sup>*2*<sup>O</sup>* <sup>¼</sup> *<sup>i</sup>*

*Psat* ¼ *e*

*N<sup>m</sup>*,*<sup>k</sup> <sup>H</sup>*2*<sup>O</sup>* <sup>¼</sup> *<sup>τ</sup>*0*<sup>i</sup>*

*NGDL*,*<sup>k</sup> <sup>O</sup>*<sup>2</sup> ¼ � *<sup>i</sup>*

To satisfy saturated vapor pressure of water, the following expression is

*k <sup>F</sup> <sup>λ</sup><sup>k</sup> <sup>a</sup>* þ

*<sup>m</sup>* <sup>¼</sup> *EWτ*0*<sup>i</sup>*

face respectively at cathode and anode. The thermodynamic equilibrium between water vapor in the backing layers and liquid water in the polymer has been assumed

The ionic conductivity of polymeric membrane could be accessed thanks to the

� <sup>16</sup> *PyH*<sup>20</sup> *Psat* <sup>2</sup>

The set of these equations could be solved by iterative methods and have needed

flux on electrochemical behavior uses the inlet molar flux, which are proportional to the current density and the stoichiometry at the cathode. This flux is currently expressed using the stoichiometry oxygen coefficient *rO*<sup>2</sup> , and this expression is

*<sup>T</sup>* . The conventional way to investigate the influences of the mass

*kk*

*EA* ¼ 2640*e*

*Psat* 

*<sup>a</sup>* are determined using sorption equilibrium of [35]:

*<sup>O</sup>*<sup>2</sup> <sup>þ</sup> *<sup>N</sup>GDL*,*<sup>k</sup> H*2*O*

*<sup>H</sup>*2*<sup>O</sup>* (28)

<sup>4</sup>*<sup>F</sup>* (29)

(27)

(30)

(31)

(35)

*<sup>O</sup>*<sup>2</sup> þ

*<sup>H</sup>*2*<sup>O</sup>*, where *<sup>F</sup>in*,0

*k*

<sup>13</sup>*:*66� <sup>5096</sup>*:*<sup>23</sup> ð Þ *T*þ273*:*15 

> *λk <sup>c</sup>* � *<sup>λ</sup><sup>k</sup> a*

*k*

*<sup>a</sup>* stand for polymer water content at the membrane/active layer inter-

*<sup>T</sup>*� <sup>1</sup> ð Þ ð Þ <sup>353</sup> <sup>0</sup>*:*0013*λ*<sup>3</sup> <sup>þ</sup> <sup>0</sup>*:*0298*λ*<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*2658*<sup>λ</sup>* (33)

<sup>1</sup> � *ek<sup>k</sup> <sup>m</sup>Lm*

*<sup>ρ</sup>dryDmF* (32)

ð Þ �0*:*6*<sup>λ</sup>* <sup>þ</sup> <sup>1183</sup> (34)

<sup>þ</sup> <sup>14</sup>*:*<sup>1</sup> *PyH*<sup>20</sup> *Psat* <sup>3</sup>

*<sup>O</sup>*<sup>2</sup> , *<sup>F</sup>in*,0

*<sup>N</sup>*<sup>2</sup> , *<sup>F</sup>in*,0

The resolution of mass balance is numerically obtained thanks to the continuity of flux between the backing layer, the active layer, and the membrane. The molar flux density of oxygen and nitrogen are nil at membrane/electrode layer interface, and the molar flux density of water are given by the solution of the equation (Eq. (15)). Thus, the sum of diffusion and electroosmotic fluxes yields in steady

#### **Figure 10.**

*Membrane ratio conductivity as a function of hydration process and current density considering a series of two CSTR, rO2 = 2.2 and T = 60°C.*

investigated; **Figure 9** shows the minimal stoichiometry oxygen coefficient rO2 to obtain a single-phase condition without membrane water exchange. Therefore it is possible to scrutinize the average ohmic drop within the membrane versus humidification process. The ohmic behavior appears as linear contribution of overall polarization curves. **Figure 10** shows thee ratio of ideal conductivity of fully hydrated Nafion® membrane (Eq. (35)) divided per average ionic conductivity trough the membrane according to the set parameter of gathered in Table. Considering a series of two CSTR with rO2 = 2.2, it was observed that the autohumidification or the humidification of inlet gas involves similar ion conductivity performance. Of course, a humidification of inlet gas at 45°C improves the effective membrane conductivity, but the impact of water production inside catalyst layer is more significant.

In the cases of dry inlet condition which corresponds to a PEM fuel cell operating with pure hydrogen and oxygen, the stoichiometric effects have been simulated by Han and Chung [36]. The results of these computations have shown decreasing membrane ratio conductivity with increasing stoichiometry oxygen coefficient *rO*<sup>2</sup> and the main effect of water production. Therefore as highlighted by Nguyen and White [37], the membrane ratio conductivity is expected to depend on gas distribution. **Figure 11** shows a decreasing effective conductivity with increasing number of CSTR series: it is worth mentioning that piston flow distribution (20 CSTR) increases the ohmic drop and the static mixing (1 or 2 CSTR) improves membrane conductivity.

In **Figure 12** the opposite effect is observed for rO2 = 2.2; gas access is limited when static mixing is developed (two CSTR). Then, cathodic overvoltage is higher considering only 2 CSTR than using a series of 20 CSTR. Nevertheless, ionic access to the catalyst areas is modeled here, and for a large amount of gas flow (rO2 = .11), piston flow distribution increases the cathodic overvoltage.

with decreasing number of CSTR because the amount of water transported back to the anode by back-diffusion is lower than convective water flux. For higher water content in the cathode side, diffusion limitations are increased. Consequently the local current density decreases, and the net water flux across the membrane also decreases, resulting in a lower depletion rate of water from the anode gas stream, a lower production rate of water in the cathode, and a lower depletion rate of hydro-

*Simulation of cathodic overpotential with varying CSTR approach and stoichiometry oxygen coefficient,*

*considering rO2 = 2.2,T = 60°C with dry inlet conditions.*

*Membrane ratio conductivity as function of CSTR series approaches and current density, considering rO2 = 2.2,*

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

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

However, near the inlet of the cell, as the current density is high and cathodic water concentration is low, the electroosmotic drag coefficient is higher, and ionic access limits the reaction rate. This observation is possible in case of piston flow (20

gen and oxygen.

**95**

**Figure 12.**

**Figure 11.**

*T = 60°C with dry inlet conditions.*

At this stage, it is important to study the distribution of current density. In **Figure 13**, the simulations show the effects of oxygen gas access. The gas distribution is the main control on current densities along the gas channel. At the cathode inlet, dry gas feeds the fuel cell; the water content in the cathode stream increases

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

**Figure 11.**

investigated; **Figure 9** shows the minimal stoichiometry oxygen coefficient rO2 to obtain a single-phase condition without membrane water exchange. Therefore it is possible to scrutinize the average ohmic drop within the membrane versus humidification process. The ohmic behavior appears as linear contribution of overall polarization curves. **Figure 10** shows thee ratio of ideal conductivity of fully hydrated Nafion® membrane (Eq. (35)) divided per average ionic conductivity trough the membrane according to the set parameter of gathered in Table. Consid-

*Membrane ratio conductivity as a function of hydration process and current density considering a series of two*

humidification or the humidification of inlet gas involves similar ion conductivity performance. Of course, a humidification of inlet gas at 45°C improves the effective membrane conductivity, but the impact of water production inside catalyst layer is

In the cases of dry inlet condition which corresponds to a PEM fuel cell operating with pure hydrogen and oxygen, the stoichiometric effects have been simulated by Han and Chung [36]. The results of these computations have shown decreasing membrane ratio conductivity with increasing stoichiometry oxygen coefficient *rO*<sup>2</sup> and the main effect of water production. Therefore as highlighted by Nguyen and White [37], the membrane ratio conductivity is expected to depend on gas distribution. **Figure 11** shows a decreasing effective conductivity with increasing number of CSTR series: it is worth mentioning that piston flow distribution (20 CSTR) increases the ohmic drop and the static mixing (1 or 2 CSTR) improves membrane conductivity. In **Figure 12** the opposite effect is observed for rO2 = 2.2; gas access is limited when static mixing is developed (two CSTR). Then, cathodic overvoltage is higher considering only 2 CSTR than using a series of 20 CSTR. Nevertheless, ionic access to the catalyst areas is modeled here, and for a large amount of gas flow (rO2 = .11),

At this stage, it is important to study the distribution of current density. In **Figure 13**, the simulations show the effects of oxygen gas access. The gas distribution is the main control on current densities along the gas channel. At the cathode inlet, dry gas feeds the fuel cell; the water content in the cathode stream increases

ering a series of two CSTR with rO2 = 2.2, it was observed that the auto-

piston flow distribution increases the cathodic overvoltage.

more significant.

**94**

**Figure 10.**

*CSTR, rO2 = 2.2 and T = 60°C.*

*Thermodynamics and Energy Engineering*

*Membrane ratio conductivity as function of CSTR series approaches and current density, considering rO2 = 2.2, T = 60°C with dry inlet conditions.*

**Figure 12.**

*Simulation of cathodic overpotential with varying CSTR approach and stoichiometry oxygen coefficient, considering rO2 = 2.2,T = 60°C with dry inlet conditions.*

with decreasing number of CSTR because the amount of water transported back to the anode by back-diffusion is lower than convective water flux. For higher water content in the cathode side, diffusion limitations are increased. Consequently the local current density decreases, and the net water flux across the membrane also decreases, resulting in a lower depletion rate of water from the anode gas stream, a lower production rate of water in the cathode, and a lower depletion rate of hydrogen and oxygen.

However, near the inlet of the cell, as the current density is high and cathodic water concentration is low, the electroosmotic drag coefficient is higher, and ionic access limits the reaction rate. This observation is possible in case of piston flow (20

**5. Conclusion: single-cell modeling results and limits**

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

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

challenge of "Smart Grid".

*b* Tafel slope (V dec<sup>1</sup>

*d* pore diameter (m) *Di* diffusion coefficient (m<sup>2</sup> s

*h* enthalpy (J mol<sup>1</sup>

*LCL* CL thickness (m)

*T* temperature (K)

*Z* impedance (Ω) *zi* charge ()

Δ*S* entropy (J mol<sup>1</sup> K<sup>1</sup>

*δ* GDL thickness (m) *ε* porosity ()

*t* time (s) *V* potential (V) **v** bulk velocity (m s<sup>1</sup>

**97**

**i** current density (A m<sup>2</sup>

*I* current (A)

*E0* equilibrium potential (V)

*F* Faraday's constant (C mol<sup>1</sup>

*j* faradaic current density (A m<sup>2</sup>

*Mi* molecular weight of *i* (kg mol<sup>1</sup>

*xi* mole fraction of the species *Xi*

*Ri* homogeneous reaction rate (mol m<sup>3</sup> s

*ri* heterogeneous reaction rate (mol m<sup>2</sup> s

*ai* the specific surface area (m<sup>1</sup>

*Cdl* double-layer capacitance (F m<sup>2</sup>

*cT* total gas concentration (mol m<sup>3</sup>

**List of symbols**

We have shown, through a simplified but physically reasonable model of PEM fuel cell steady-state multiplicity, caused by reactant access (oxygen) or water management (a product of the reaction) and the reaction rate. Of course, catalyst loading is critical but 0-D models do not have enough accuracy to perform realistic predictions. Closed-form model (pseudo 2D) can provide relevant simulations but does not take into account water management. The succinct analogy with CSTR can provide fruitful analysis of water management through the fuel cell. This approach was settled with a view to determining the impact of gas distribution in the gas channel using a fluid dynamics observation (i.e., RTD) coupled with the usual electrochemical model. Water production and removal are analogous to heat production and removal. Therefore this analogy (energy balance) can be added to the present electrochemical pseudo 2D approach of mass balance (CSTR-PEM).

Operating conditions that can threaten the life of the PEM cell are not easy to detect; using simple models for fast computation with A.I. shall avoid the problematic lack of large experimental databases. Predictive models and A.I. can take up the

)

1 )

EW dry membrane weight per mole of sulfonate group (kg mol<sup>1</sup>

)

)

)

)

)

)

1 )

1 ) )

)

1 )

)

)

*Fi* molar flow rate for the component *Xi* (mol s<sup>1</sup>

**N***<sup>i</sup>* flux density of each dissolved species (mol m<sup>2</sup> s

) *[Xi]* total gas concentration of the species (mol m<sup>3</sup>

)

)

#### **Figure 13.**

*Simulation of local current density distribution along the channel (rO2 = 2.2 for dry conditions at T = 60°C with |ηc| = 0.55 V).*

#### **Figure 14.**

*Simulation of local oxygen (*■*) and water () molar fraction distribution along the channel (rO2 = 2.2 for dry conditions at T = 60°C with |ηc| = 0.55 V).*

CSTR). In **Figure 14**, at low stoichiometric value, a depletion of reactants can be observed close to the outlet, thus generating a cathodic overpotential increase and a dramatically current density decrease in this part. It is interesting to note that water molar fraction increases at the cathode side in connection to the electroosmotic drag of water, from anode to cathode, and the insufficient "back-diffusion" from the cathode to anode. Water transport and production at the cathode result in increasing water content at the cathode side along the gas channel while it decreases at the anode side.

This study has drawn attention to the relative advantages of using a CSTR description. Moreover liquid water phase only exists at the cathode side, thus leading to the polymer drying out at the anode and consequently increasing ohmic resistance. In the future, it will be important to include the flooding or partial flooding effects that were not taken into account in this study.
