**A.1 The classical catalyst layer model**

Under steady-state conditions, the dimensionless equations relating the concentrations of species (Eq. (17)) are obtained by setting *<sup>∂</sup>=∂<sup>t</sup>* <sup>¼</sup> 0:

$$\frac{\partial^2 \mathbf{X}\_i}{\partial \mathbf{Y}^2} + a\_\epsilon \frac{\nu\_i j\_{0, a/c} \mathbf{L}^2}{\left| \nu\_\epsilon \right| F D\_i^{\rm eff} \left[ \mathbf{X}\_i \right]\_\ast} \exp \left( \pm 2.3 \frac{\eta\_{a/c}}{b} \right) \mathbf{X}\_i = \mathbf{0} \tag{A1}$$

*i* ¼

*Simulated i - |η| curve for a catalyst layer.*

**Figure A1.**

**99**

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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

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

*ν<sup>i</sup>* j j*L*

*<sup>i</sup> ci*,0 *γ j*

vuut *exp* �2*:*<sup>3</sup>

0,*a=c*

A catalyst layer controlled by both kinetics and diffusion is analogous to the classical chemical engineering problem of porous catalyst pellet with a first-order reaction controlled by mass transport limitation of reactant characterized by Thiele's modulus. This explanation also holds for fuel cell model where the active layer may be considered to be a single pellet or agglomerate. When both kinetic and diffusion in the active layer are controlling the electrochemical process, it is wellknown that limiting current behavior does not occur, but rather a doubling of the effective Tafel slope on a steady-state polarization curves and a first-order dependence on the concentration [38]. The shapes of the concentration profiles depend strongly on parameter *mL* [39]. The model allows us to understand easily how the diffusion limitation within the catalyst layer acts on the DC responses. Under kinetic control (*mL < <1*), the diffusion is fast enough to supply the reacting species up to the catalyst surface in the GDE, and the concentration profile remains close to a constant value in the catalyst layer. Conversely, with increasing diffusion limitation (i.e., increasing parameter *mL*), an increasing concentration gradient is predicted. **Figure A1** shows the influence of the diffusion limitation within the active layer on the polarization curve. Under kinetic and diffusion control in the active layer, a doubling of the apparent Tafel slope is observed in the DC response. Simulation was performed by considering parameters summarized in **Table 1**.

> b ¼ 2 � 120 mV*=*dec*:* b ¼ 120 mV*=*dec*:*

*η<sup>a</sup>=<sup>c</sup>* 2 *b* � � *Xi* ½ �<sup>0</sup>

*Xi* ½ � <sup>∗</sup>

¼ 0 (A5)

*<sup>ν</sup><sup>e</sup>* j j*FDeff*

By introducing the dimensionless parameter *Xi* ¼ *Xi* ½ �*= Xi* ½ � <sup>∗</sup> and *Y* ¼ *y=LCL*, the Thiele' modulus (mL) appears as follows:

$$\left(mL\right)^{2} = a\_{\epsilon} \frac{\left|\nu\_{i}\right| j\_{0,a/\epsilon} L^{2}}{\left|\nu\_{\epsilon}\right| F D\_{i}^{\mathcal{G}\overline{\!\!\!}}\left[X\_{i}\right]\_{\ast}} \exp\left(\pm 2.3 \frac{\eta\_{a/\epsilon}}{b}\right) = 0 \tag{A2}$$

Note that this dimensionless parameter *mL* enables us to compare diffusion resistances with regard to kinetic resistance on the active layer performances. The stationary problem has an analytical solution in the regime in which the effect of charge transport can be neglected. The steady-state formulation of the mass balance equation within the CL (Eq. (33)) leads to an expression of the geometrical current density *i* in the CL versus the overpotential:

$$\dot{\alpha} = \gamma \, j\_{0, a/c} \exp\left(\pm 2.3 \frac{\eta\_{a/c}}{b}\right) \frac{[X\_i]\_0}{[X\_i]\_\*} \frac{\tanh\left(mL\right)}{mL} = \mathbf{0} \tag{A3}$$

where γ (=*ae* � *LCL*) is the roughness factor. This model predicts two distinct regions arising from control by kinetics and the other due to the control of both kinetic and diffusion. Under kinetics control, for low current density corresponding to *mL* < <1, concentration depletion in the active layer does not occur, and the current density remains equal to the kinetic one:

$$i = \gamma \, j\_{0, a/c} \exp\left(\pm 2.3 \frac{\eta\_{a/c}}{b}\right) \frac{[\mathbf{X}\_i]\_0}{[\mathbf{X}\_i]\_\*} = \mathbf{0} \tag{A4}$$

Conversely, under kinetics and diffusion control corresponding to *mL*>> 1, since *tanh(mL)* converges to unity, the current density can be approximated by [38]:

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

#### **Figure A1.** *Simulated i - |η| curve for a catalyst layer.*

Φ electrostatic potential

*Thermodynamics and Energy Engineering*

*γ* roughness factor *η* overpotential (V)

*ρ* density (kg m�<sup>3</sup>

*τ* tortuosity (�)

**A. Appendix**

*Γ<sup>i</sup>* surface concentration of a solute species *i* (mol m�<sup>2</sup>

trations of species (Eq. (17)) are obtained by setting *<sup>∂</sup>=∂<sup>t</sup>* <sup>¼</sup> 0:

*νij* 0,*a=cL*<sup>2</sup>

*<sup>ν</sup><sup>e</sup>* j j*FDeff*

*ν<sup>i</sup>* j j*j*

*<sup>ν</sup><sup>e</sup>* j j*FDeff*

0,*a=<sup>c</sup> exp* �2*:*3

*<sup>i</sup> Xi* ½ � <sup>∗</sup>

0,*a=cL*<sup>2</sup>

*<sup>i</sup> Xi* ½ � <sup>∗</sup>

Note that this dimensionless parameter *mL* enables us to compare diffusion resistances with regard to kinetic resistance on the active layer performances. The stationary problem has an analytical solution in the regime in which the effect of charge transport can be neglected. The steady-state formulation of the mass balance equation within the CL (Eq. (33)) leads to an expression of the geometrical current

> *η<sup>a</sup>=<sup>c</sup> b Xi* ½ �<sup>0</sup>

where γ (=*ae* � *LCL*) is the roughness factor. This model predicts two distinct regions arising from control by kinetics and the other due to the control of both kinetic and diffusion. Under kinetics control, for low current density corresponding to *mL* < <1, concentration depletion in the active layer does not occur, and the

0,*a=<sup>c</sup> exp* �2*:*3

*tanh(mL)* converges to unity, the current density can be approximated by [38]:

*Xi* ½ � <sup>∗</sup>

*η<sup>a</sup>=<sup>c</sup> b Xi* ½ �<sup>0</sup>

Conversely, under kinetics and diffusion control corresponding to *mL*>> 1, since

*Xi* ½ � <sup>∗</sup>

)

Under steady-state conditions, the dimensionless equations relating the concen-

By introducing the dimensionless parameter *Xi* ¼ *Xi* ½ �*= Xi* ½ � <sup>∗</sup> and *Y* ¼ *y=LCL*, the

*exp* �2*:*3

*exp* �2*:*3

*η<sup>a</sup>=<sup>c</sup> b* 

> *η<sup>a</sup>=<sup>c</sup> b*

> > *tanh mL* ð Þ

*Xi* ¼ 0 (A1)

¼ 0 (A2)

*mL* <sup>¼</sup> <sup>0</sup> (A3)

¼ 0 (A4)

*λ* water content of the membrane (�) *ν<sup>e</sup>* exchange number of electron (�)

*τ<sup>0</sup>* electroosmotic drag coefficient (�)

**A.1 The classical catalyst layer model**

*∂*2 *Xi <sup>∂</sup>Y*<sup>2</sup> <sup>þ</sup> *ae*

Thiele' modulus (mL) appears as follows:

density *i* in the CL versus the overpotential:

*i* ¼ *γ j*

current density remains equal to the kinetic one:

**98**

*i* ¼ *γ j*

ð Þ *mL* <sup>2</sup> <sup>¼</sup> *ae*

*ν<sup>i</sup>* stoichiometric coefficient of species *i* (�)

) *σ* conductivity of the solution (S m�<sup>1</sup> )

$$i = \sqrt{\frac{|\nu\_e| \overline{F D\_i^{\text{off}} c\_{i,0} \, \text{y}} \, j\_{0,a/c}}{|\nu\_i| L}} \exp\left( \pm 2.3 \frac{\eta\_{a/c}}{2 \, b} \right) \frac{[\mathbf{X}\_i]\_0}{[\mathbf{X}\_i]\_\*} = \mathbf{0} \tag{A5}$$

A catalyst layer controlled by both kinetics and diffusion is analogous to the classical chemical engineering problem of porous catalyst pellet with a first-order reaction controlled by mass transport limitation of reactant characterized by Thiele's modulus. This explanation also holds for fuel cell model where the active layer may be considered to be a single pellet or agglomerate. When both kinetic and diffusion in the active layer are controlling the electrochemical process, it is wellknown that limiting current behavior does not occur, but rather a doubling of the effective Tafel slope on a steady-state polarization curves and a first-order dependence on the concentration [38]. The shapes of the concentration profiles depend strongly on parameter *mL* [39]. The model allows us to understand easily how the diffusion limitation within the catalyst layer acts on the DC responses. Under kinetic control (*mL < <1*), the diffusion is fast enough to supply the reacting species up to the catalyst surface in the GDE, and the concentration profile remains close to a constant value in the catalyst layer. Conversely, with increasing diffusion limitation (i.e., increasing parameter *mL*), an increasing concentration gradient is predicted. **Figure A1** shows the influence of the diffusion limitation within the active layer on the polarization curve. Under kinetic and diffusion control in the active layer, a doubling of the apparent Tafel slope is observed in the DC response. Simulation was performed by considering parameters summarized in **Table 1**.

$$\mathbf{b} = 2 \times 120 \text{ mV/dec.}$$

$$\mathbf{b} = 120 \text{ mV/dec.}$$

*Thermodynamics and Energy Engineering*
