**3. Analytical models**

#### **3.1 Proton exchange membrane fuel cells**

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 reduction reaction (ORR) at the cathode catalyst layer

$$\begin{array}{ccccc}\text{anode}: & H\_2 & \rightarrow & 2H^+ + 2e^-\\\text{cathode}: & \frac{1}{2}O\_2 + 2H^+ + 2e^- & \rightarrow & H\_2O\end{array} \tag{1}$$

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

$$\begin{aligned} E &= E^0 + \Delta E\_\epsilon(T) + \Delta E\_\epsilon \left( T, p\_{H\_2}, p\_{O\_2} \right) \\ &= E^0 + \frac{1}{nF} \int\_{T^0}^T \Delta \hat{s}(T) \, dT + \frac{T}{f\_\epsilon} \ln \left[ \left( \frac{c\_{H\_2}}{c\_{H\_2}^0} \right) \left( \frac{c\_{O\_2}}{c\_{O\_2}^0} \right)^{0.5} \right] \end{aligned} \tag{2}$$

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:

anodic and cathodic CLs. Δ*V*aa and Δ*V*ac increase with the rate of charge density separation *∂tρ<sup>e</sup>* (namely the proton and electron creation at the anode and recombination at the cathode—∂*<sup>t</sup>* represents the partial time derivative), which, in steadystate operation equates the current density at the TPB, *jTPB*. Δ*V*aa and Δ*V*ac are typically modeled with the Butler-Volmer equation [4]. Due to the particular porous structure of the CLs, the area *ATPB* of the TPBs where *jTPB* is produced is much larger than the active cell cross-sectional area *A* (*ATPB*/*A* can be larger than

) and, when modeling is devoted to analyzing the cell electric performance, the

By using Eq. (6), the Butler-Volmer equation allows to write the current density

*cP e*

In this equation, the total equivalent current density *jt* accounts for the effect of hydrogen crossover on the overpotentials, that can be modeled as an equivalent internal loss current to be added to the cell internal current density and is not available at current collectors, but contributes to the activation overpotentials. *j*<sup>0</sup> is the exchange current density. The accurate evaluation of its values at the anode and cathode half-reactions is important, because they strongly affect the cell performance and round-trip efficiency. To take into account the effects of the temperature on *j*0, an Arrhenius-like dependence with *T* can be considered [5]. As a consequence of the low temperatures at which PEMFCs operate and of the exponential dependence of *jt* on Δ*Va*, the activation losses are the major responsible factors for voltage drops at low current densities. In addition, Δ*Vac* is typically one order of magnitude larger than Δ*Vaa*, so that the cathodic activation losses are the

When the cell operates in load condition, the inflow of reagents and outflow of products are necessary to sustain the electrochemical reactions at the CLs. In turn, these species flows are ensured by convective mass flow in the transport channels of the bipolar plates (BPs) and by the diffusive mass transport in the diffusion layers (DLs, **Figure 1**). These flows are sustained by concentrations and pressure gradients of reagents and products, namely by values *c* and *p* at the CLs different from the bulk (and inlet) values *c* and *p* [6]. In order to model the concentration gradients *c*, Fick's first law *N* = –*D c* is often used, which invokes the medium diffusivity *D* and the gas molar flow vector *N*, related in turn to the current density vector *j* through the Faraday constant *F* and the charge carriers n as *j* = *nFN*, for both hydrogen at anode and oxygen at cathode. Since the diffusivity depends on temperature, the effect of the latter on concentration gradients can also have important role in the FC

Gas concentration and pressure gradients *c* and *p* produce the term *ΔEcl* of the Nernst equation, reducing the electromotive force (emf) with respect to the no-load

�ð Þ 1�*α fe* Δ*Va=T* (7)

<sup>0</sup>*TPB* (6)

current density is preferably reported to the cross-sectional area, as:

*j*

at the cross-sectional area of each CL as

*j <sup>t</sup>* ¼ *j* 0 *cR cR e*

*Distributed and Lumped Parameter Models for Fuel Cells*

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

dominant effect at low current densities [3].

*3.1.3 Concentration losses and limit conditions*

*3.1.2 Concentration gradients*

performance.

**25**

<sup>0</sup> <sup>¼</sup> *ATPB <sup>A</sup> <sup>j</sup>*

*<sup>α</sup> fe*Δ*Va=<sup>T</sup>* � *cP*

10<sup>3</sup>

#### **Figure 1.**

*Sketch of a PEMFC section with anode and cathode flow channels of bipolar plates (BPs), diffusion layer (DLs), catalyst layers (CLs), and proton exchange membrane (PEM). Convective (in BPs) and diffusive (in DLs) fluid flows are sketched (courtesy of Journal of Power Sources).*

$$
\Delta E\_c = \frac{T}{f\_\epsilon} \ln \left[ \left( \frac{\overline{c}\_{H\_2}}{c\_{H\_2}^0} \right) \left( \frac{\overline{c}\_{O\_2}}{c\_{O\_2}^0} \right)^{0.5} \right] + \frac{T}{f\_\epsilon} \ln \left[ \left( \frac{c\_{H\_2}}{\overline{c}\_{H\_2}} \right) \left( \frac{c\_{O\_2}}{\overline{c}\_{O\_2}} \right)^{0.5} \right] = \Delta E\_{co} + \Delta E\_{cl} \tag{3}
$$

Here *fe = nF/R,* where *R* is the gas constant, *F* is the Faraday constant, and *n* is the number of electrons transferred in the reaction. Δ*Eco* arises from the differences between the actual bulk concentrations and the standard-condition values, particularly in a no-load state. Δ*Ecl* springs from the differences between the concentrations at the triple-phase boundaries (TPBs, which are the sites of electrochemical reaction in the catalyst layers—CLs) and the bulk concentrations in the CLs, namely from the concentration gradients which arise to sustain the species molar flows needed in load conditions. In order to allow an accurate modeling over a wide temperature range, *ΔEs* can be calculated by integration. Combining the above and excluding the gradient-dependent term Δ*Ecl*, we have:

$$E\_{OC} = E^0 + \frac{1}{nF} \int\_{T^0}^{T} \Delta \hat{s}(T) \, dT + \frac{T}{f\_\epsilon} \ln \left[ \left( \frac{\overline{c}\_{H\_2}}{c\_{H\_2}^0} \right) \left( \frac{\overline{c}\_{O\_2}}{c\_{O\_2}^0} \right)^{0.5} \right] \tag{4}$$

The cell open circuit voltage (OCV) is slightly different from *EOC* because some gas crossover through the membrane occurs in every condition, including no-load, producing small concentration gradients and hence a minimal Δ*Ecl*. On the other hand, when the cell is operated at a steady-state load with an electric current density *j*, its voltage *V* differs markedly from *EOC*, due to much larger concentration gradients, and hence much larger Δ*Ecl*, and to overpotentials *ηkh* (voltage drops Δ*Vkh*, in electrical engineering terms, [3]):

$$V(j) = E\_{OC} - \eta\_{aa} - \eta\_{ac} - \eta\_{ca} - \eta\_{cc} - \eta\_{m} \tag{5}$$

The first subscript of the overpotentials indicates *a*, activation losses, or *c*, concentration losses, whereas the second subscript stands for *a*, anode, or *c*, cathode. The single subscript *m* indicates the membrane (PEM), where ohmic losses occur. All these overpotentials are strong functions of *j*.

#### *3.1.1 Exchange current density: activation losses*

The activation overpotentials or activation voltage drops, Δ*Vaa* and Δ*Vac*, are an effect of the electrochemical kinetics that appear when the species react at the

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

anodic and cathodic CLs. Δ*V*aa and Δ*V*ac increase with the rate of charge density separation *∂tρ<sup>e</sup>* (namely the proton and electron creation at the anode and recombination at the cathode—∂*<sup>t</sup>* represents the partial time derivative), which, in steadystate operation equates the current density at the TPB, *jTPB*. Δ*V*aa and Δ*V*ac are typically modeled with the Butler-Volmer equation [4]. Due to the particular porous structure of the CLs, the area *ATPB* of the TPBs where *jTPB* is produced is much larger than the active cell cross-sectional area *A* (*ATPB*/*A* can be larger than 10<sup>3</sup> ) and, when modeling is devoted to analyzing the cell electric performance, the current density is preferably reported to the cross-sectional area, as:

$$j\_0 = \frac{A\_{TPB}}{A} j\_{0TPB} \tag{6}$$

By using Eq. (6), the Butler-Volmer equation allows to write the current density at the cross-sectional area of each CL as

$$j\_t = j\_0 \left(\frac{c\_R}{\overline{c}\_R} e^{\alpha f\_s \Delta V\_a/T} - \frac{c\_P}{\overline{c}\_P} e^{-(1-a)f\_s \Delta V\_a/T} \right) \tag{7}$$

In this equation, the total equivalent current density *jt* accounts for the effect of hydrogen crossover on the overpotentials, that can be modeled as an equivalent internal loss current to be added to the cell internal current density and is not available at current collectors, but contributes to the activation overpotentials. *j*<sup>0</sup> is the exchange current density. The accurate evaluation of its values at the anode and cathode half-reactions is important, because they strongly affect the cell performance and round-trip efficiency. To take into account the effects of the temperature on *j*0, an Arrhenius-like dependence with *T* can be considered [5]. As a consequence of the low temperatures at which PEMFCs operate and of the exponential dependence of *jt* on Δ*Va*, the activation losses are the major responsible factors for voltage drops at low current densities. In addition, Δ*Vac* is typically one order of magnitude larger than Δ*Vaa*, so that the cathodic activation losses are the dominant effect at low current densities [3].

#### *3.1.2 Concentration gradients*

When the cell operates in load condition, the inflow of reagents and outflow of products are necessary to sustain the electrochemical reactions at the CLs. In turn, these species flows are ensured by convective mass flow in the transport channels of the bipolar plates (BPs) and by the diffusive mass transport in the diffusion layers (DLs, **Figure 1**). These flows are sustained by concentrations and pressure gradients of reagents and products, namely by values *c* and *p* at the CLs different from the bulk (and inlet) values *c* and *p* [6]. In order to model the concentration gradients *c*, Fick's first law *N* = –*D c* is often used, which invokes the medium diffusivity *D* and the gas molar flow vector *N*, related in turn to the current density vector *j* through the Faraday constant *F* and the charge carriers n as *j* = *nFN*, for both hydrogen at anode and oxygen at cathode. Since the diffusivity depends on temperature, the effect of the latter on concentration gradients can also have important role in the FC performance.

#### *3.1.3 Concentration losses and limit conditions*

Gas concentration and pressure gradients *c* and *p* produce the term *ΔEcl* of the Nernst equation, reducing the electromotive force (emf) with respect to the no-load

<sup>Δ</sup>*Ec* <sup>¼</sup> *<sup>T</sup> f e*

**Figure 1.**

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

*Thermodynamics and Energy Engineering*

4

!

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

*DLs) fluid flows are sketched (courtesy of Journal of Power Sources).*

excluding the gradient-dependent term Δ*Ecl*, we have:

1 *nF* ð *T*

*T*0

Δ^*s T*ð Þ*dT* þ

*T fe*

The cell open circuit voltage (OCV) is slightly different from *EOC* because some gas crossover through the membrane occurs in every condition, including no-load, producing small concentration gradients and hence a minimal Δ*Ecl*. On the other hand, when the cell is operated at a steady-state load with an electric current density *j*, its voltage *V* differs markedly from *EOC*, due to much larger concentration gradients, and hence much larger Δ*Ecl*, and to overpotentials *ηkh* (voltage drops

The first subscript of the overpotentials indicates *a*, activation losses, or *c*, concentration losses, whereas the second subscript stands for *a*, anode, or *c*, cathode. The single subscript *m* indicates the membrane (PEM), where ohmic losses

The activation overpotentials or activation voltage drops, Δ*Vaa* and Δ*Vac*, are an

effect of the electrochemical kinetics that appear when the species react at the

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

*V j*ðÞ¼ *EOC* � *ηaa* � *ηac* � *ηca* � *ηcc* � *η<sup>m</sup>* (5)

4

!

*cO*<sup>2</sup> *c*0 *O*2 3

5 (4)

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

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

Δ*Vkh*, in electrical engineering terms, [3]):

*3.1.1 Exchange current density: activation losses*

**24**

occur. All these overpotentials are strong functions of *j*.

3 5 þ *T fe*

ln *cH*<sup>2</sup> *cH*<sup>2</sup> � � *cO*<sup>2</sup>

*Sketch of a PEMFC section with anode and cathode flow channels of bipolar plates (BPs), diffusion layer (DLs), catalyst layers (CLs), and proton exchange membrane (PEM). Convective (in BPs) and diffusive (in*

Here *fe = nF/R,* where *R* is the gas constant, *F* is the Faraday constant, and *n* is the number of electrons transferred in the reaction. Δ*Eco* arises from the differences between the actual bulk concentrations and the standard-condition values, particularly in a no-load state. Δ*Ecl* springs from the differences between the concentrations at the triple-phase boundaries (TPBs, which are the sites of electrochemical reaction in the catalyst layers—CLs) and the bulk concentrations in the CLs, namely from the concentration gradients which arise to sustain the species molar flows needed in load conditions. In order to allow an accurate modeling over a wide temperature range, *ΔEs* can be calculated by integration. Combining the above and

*cO*<sup>2</sup> � �<sup>0</sup>*:*<sup>5</sup> " #

¼ Δ*Eco* þ Δ*Ecl* (3)

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

value *EOC* and constitute the concentration losses which dominate the cell's performance at high current densities. Δ*Ecl* can be split in the anodic and cathodic concentration voltage drops as:

$$-\Delta E\_{cl} = \eta\_{ca} + \eta\_{cc} = \kappa\_{ca}\frac{T}{f\_{\epsilon}}\ln\left(\frac{\overline{c}\_{H\_{2}}}{c\_{H\_{2}}}\right) + \kappa\_{cc}\frac{T}{2f\_{\epsilon}}\ln\left(\frac{\overline{c}\_{O\_{2}}}{c\_{O\_{2}}}\right) \tag{8}$$

the main cause of the difference between the open circuit emf *EOC* and the observed OCV *V*(0) [10]. It also causes a loss of stored energy that reduces round-trip efficiency. Also, oxygen crosses the PEM, but, since its diffusivity is much lower

Dissipations occurring inside the cell produce thermal gradients which affect the temperature-dependent parameters. The main loss phenomena are Peltier heating (thermodynamic heat generation), losses due to the electrochemical kinetic activity at the anode and cathode CLs, and Joule losses in the PEM, so that, inside the cell,

> *nF* <sup>þ</sup> *<sup>η</sup>a*ð Þ*<sup>j</sup>*

Heat transport inside the cell depends on conduction, diffusion, and convection and interacts with thermal capacity in dynamic conditions [3]. An accurate enough estimation of the mean temperature *T* inside the cell with respect to the room and

with *kT* a global thermal exchange coefficient. This expression can be

Butler-Volmer equation, a look-up table can be conveniently used.

*T* ¼ *Tr* þ *kt*<sup>1</sup> *j* þ *kt*<sup>2</sup> *j*

where *kt*<sup>1</sup> and *kt*<sup>2</sup> are properly fitted parameters. In the numerical implementation of such models, consistent analytical expressions can be used without introducing approximation if the electric current density *j* is chosen as the independent variable to compute all voltage terms. In order to deal with the non-invertible

DMFCs suffer from two fundamental problems: (i) the sluggish kinetics of the methanol electro-oxidation reaction and (ii) the high degree of permeation of the methanol through the membrane (crossover). Analytical and numerical models are necessary for better understanding the interactions between mass transfer and electrochemical phenomena, and for optimizing the power output and runtime to

A schematic of a typical DMFC inside a cell stack is sketched in **Figure 2**. It consists basically of an anode flow channel (AFC), an anode diffusion layer (ADL), an anode catalyst layer (ACL), a proton exchange membrane (PEM), a cathode catalyst layer (CCL), a cathode diffusion layer (CDL), and a cathode flow channel (CFC). Analytical models of DMFCs account for the following phenomena: electrochemical reactions occurring at catalyst layers, protonic conduction, and methanol crossover across the PEM, diffusion of reactants in porous media layers, fluid flow inside channels, and coupling between heat and mass transfer. Small DMFCs for portable electronics usually include a fuel reservoir with no pumps for feeding the

þ *dm σ j*

*T* ¼ *Tr* þ *kTPdiss* (12)

<sup>2</sup> (11)

<sup>2</sup> (13)

than hydrogen's [11], this effect is usually neglected.

*Distributed and Lumped Parameter Models for Fuel Cells*

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

*Pdiss* <sup>¼</sup> *<sup>j</sup>* � *<sup>T</sup>Δ*^*<sup>s</sup>*

the dissipated power per unit area is:

gas inlet temperature *Tr* can be obtained by:

**3.2 Methanol fuel cells**

interface electronics [3].

**27**

*3.2.1 DMFC analytical models*

reformulated as a function of the current density as:

where *κ* are the mass transfer coefficients. These two terms provide the anodic and cathodic current density limits *j La* and *j Lc*, namely the theoretical values of the current densities which cause the CL concentrations and pressures to vanish at the TPBs and the half-reactions stop, when the fuel cell starvation occurs. Since the smaller limit current density occurs at the cathode, due to the lower diffusivity of oxygen compared to hydrogen, *j Lc* sets the device's current density limit.

#### *3.1.4 Membrane ohmic losses*

The voltage drop in the membrane is proportional to the current and to its thickness and inversely proportional to its conductivity *σ* which depends on temperature and hydration, which is the ratio *λ* = *c*w/*cas* (with *cas =* 1970 mol m�<sup>3</sup> ) between water and sulfonic acid concentrations that varies in the range 0–22 for typical Nafion® membranes. The dependence of conductivity on hydration can be expressed as:

$$
\sigma(\lambda) = \sigma\_0(T) B \lambda \tag{9}
$$

The linear dependence on *λ* via the dimensionless coefficient *B* is the adaptation of an empirical model [7] aimed at avoiding a negative value of *σ* at lower *λ*. The temperature dependence can be expressed with the model of Vogel-Tamman-Fulcher [8]. Although *λ* varies along the PEM's thickness according to backdiffusion and electro-osmotic drag [9], the average between the PEM boundary values *λ<sup>a</sup>* and *λ<sup>c</sup>* can be used, consistently with a linear profile between *λ<sup>a</sup>* and *λc*. These values depend on the water activities *awa* and *awc* of the reacting gases at the CLs, and are computed with an empirical polynomial [7], which depends on the water vapor saturation pressure *pws* which is also a function of the temperature [3].

#### *3.1.5 Crossover*

Fuel crossover consists of hydrogen that, instead of reacting at the anode, migrates through the PEM and reacts with oxygen at the cathode, without producing electric power. This is a major side effect that affects the FC performance and efficiency and depends on two mechanisms, diffusion, and electro-osmotic drag. These two contributions of hydrogen mass flow can be modeled as equivalent current densities so that the resulting equivalent crossover current density is:

$$j\_{co} = \frac{nF D\_{mH\_2}}{d\_m} c\_{H\_2} + n\xi \dot{\chi} \tag{10}$$

where *cH*<sup>2</sup> is the hydrogen concentration at the anodic CL, *DmH*<sup>2</sup> is the hydrogen diffusivity (with an Arrhenius-like temperature dependence), *dm* is the membrane thickness and *ξ* is a dimensionless electro-osmotic drag coefficient (giving a maximum value *ν<sup>w</sup>* = *ξλ* = 2.5 at full hydration *λ* = 22, as reported). Crossover hydrogen is *Distributed and Lumped Parameter Models for Fuel Cells DOI: http://dx.doi.org/10.5772/intechopen.89048*

value *EOC* and constitute the concentration losses which dominate the cell's performance at high current densities. Δ*Ecl* can be split in the anodic and cathodic con-

> *T f e*

*La* and *j*

current densities which cause the CL concentrations and pressures to vanish at the TPBs and the half-reactions stop, when the fuel cell starvation occurs. Since the smaller limit current density occurs at the cathode, due to the lower diffusivity of

The voltage drop in the membrane is proportional to the current and to its thickness and inversely proportional to its conductivity *σ* which depends on temperature and hydration, which is the ratio *λ* = *c*w/*cas* (with *cas =* 1970 mol m�<sup>3</sup>

between water and sulfonic acid concentrations that varies in the range 0–22 for typical Nafion® membranes. The dependence of conductivity on hydration can be

The linear dependence on *λ* via the dimensionless coefficient *B* is the adaptation of an empirical model [7] aimed at avoiding a negative value of *σ* at lower *λ*. The temperature dependence can be expressed with the model of Vogel-Tamman-Fulcher [8]. Although *λ* varies along the PEM's thickness according to backdiffusion and electro-osmotic drag [9], the average between the PEM boundary values *λ<sup>a</sup>* and *λ<sup>c</sup>* can be used, consistently with a linear profile between *λ<sup>a</sup>* and *λc*. These values depend on the water activities *awa* and *awc* of the reacting gases at the CLs, and are computed with an empirical polynomial [7], which depends on the water vapor saturation pressure *pws* which is also a function of the temperature [3].

Fuel crossover consists of hydrogen that, instead of reacting at the anode, migrates through the PEM and reacts with oxygen at the cathode, without producing electric power. This is a major side effect that affects the FC performance and efficiency and depends on two mechanisms, diffusion, and electro-osmotic drag. These two contributions of hydrogen mass flow can be modeled as equivalent current densities so that the resulting equivalent crossover current density is:

*j*

*co* <sup>¼</sup> *nFDmH*<sup>2</sup> *dm*

where *cH*<sup>2</sup> is the hydrogen concentration at the anodic CL, *DmH*<sup>2</sup> is the hydrogen diffusivity (with an Arrhenius-like temperature dependence), *dm* is the membrane thickness and *ξ* is a dimensionless electro-osmotic drag coefficient (giving a maximum value *ν<sup>w</sup>* = *ξλ* = 2.5 at full hydration *λ* = 22, as reported). Crossover hydrogen is

ln *cH*<sup>2</sup> *cH*<sup>2</sup> 

where *κ* are the mass transfer coefficients. These two terms provide the anodic

þ *κcc T* 2*f e*

*Lc* sets the device's current density limit.

*σ λ*ð Þ¼ *σ*0ð Þ *T Bλ* (9)

*cH*<sup>2</sup> þ *nξλj* (10)

ln *cO*<sup>2</sup> *cO*<sup>2</sup> 

*Lc*, namely the theoretical values of the

(8)

)

centration voltage drops as:

and cathodic current density limits *j*

*Thermodynamics and Energy Engineering*

oxygen compared to hydrogen, *j*

*3.1.4 Membrane ohmic losses*

expressed as:

*3.1.5 Crossover*

**26**

�Δ*Ecl* ¼ *ηca* þ *ηcc* ¼ *κca*

the main cause of the difference between the open circuit emf *EOC* and the observed OCV *V*(0) [10]. It also causes a loss of stored energy that reduces round-trip efficiency. Also, oxygen crosses the PEM, but, since its diffusivity is much lower than hydrogen's [11], this effect is usually neglected.

Dissipations occurring inside the cell produce thermal gradients which affect the temperature-dependent parameters. The main loss phenomena are Peltier heating (thermodynamic heat generation), losses due to the electrochemical kinetic activity at the anode and cathode CLs, and Joule losses in the PEM, so that, inside the cell, the dissipated power per unit area is:

$$P\_{\rm diss} = j \left( -\frac{T\Delta\hat{s}}{nF} + \eta\_a(j) \right) + \frac{d\_m}{\sigma} j^2 \tag{11}$$

Heat transport inside the cell depends on conduction, diffusion, and convection and interacts with thermal capacity in dynamic conditions [3]. An accurate enough estimation of the mean temperature *T* inside the cell with respect to the room and gas inlet temperature *Tr* can be obtained by:

$$T = T\_r + k\_T P\_{\text{dis}} \tag{12}$$

with *kT* a global thermal exchange coefficient. This expression can be reformulated as a function of the current density as:

$$T = T\_r + k\_{t1}j + k\_{t2}j^2 \tag{13}$$

where *kt*<sup>1</sup> and *kt*<sup>2</sup> are properly fitted parameters. In the numerical implementation of such models, consistent analytical expressions can be used without introducing approximation if the electric current density *j* is chosen as the independent variable to compute all voltage terms. In order to deal with the non-invertible Butler-Volmer equation, a look-up table can be conveniently used.

#### **3.2 Methanol fuel cells**

DMFCs suffer from two fundamental problems: (i) the sluggish kinetics of the methanol electro-oxidation reaction and (ii) the high degree of permeation of the methanol through the membrane (crossover). Analytical and numerical models are necessary for better understanding the interactions between mass transfer and electrochemical phenomena, and for optimizing the power output and runtime to interface electronics [3].

#### *3.2.1 DMFC analytical models*

A schematic of a typical DMFC inside a cell stack is sketched in **Figure 2**. It consists basically of an anode flow channel (AFC), an anode diffusion layer (ADL), an anode catalyst layer (ACL), a proton exchange membrane (PEM), a cathode catalyst layer (CCL), a cathode diffusion layer (CDL), and a cathode flow channel (CFC). Analytical models of DMFCs account for the following phenomena: electrochemical reactions occurring at catalyst layers, protonic conduction, and methanol crossover across the PEM, diffusion of reactants in porous media layers, fluid flow inside channels, and coupling between heat and mass transfer. Small DMFCs for portable electronics usually include a fuel reservoir with no pumps for feeding the

overcome the activation energy of the electrochemical reaction, and causes an energy loss. A relationship similar to Eq. (15) holds for the cathode side, where the atmospheric oxygen is provided at the catalyst layer through the gas diffusion layer

*Distributed and Lumped Parameter Models for Fuel Cells*

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

crossover, and the potential and concentration distributions in electrodes.

In the one-dimensional model proposed in [12], the cell performance is assessed by describing mass transport in the porous electrode structures, including methanol

Multiphase flow is accounted for in [13], where liquid-vapor mixtures at anode and cathode compartments are considered. A two-dimensional analytical model of a DMFC, which describes electrochemical reactions on the anode and cathode and main transport phenomena in the fuel cell including methanol crossover, diffusion of reactants in porous media layers, and fluid flow in the reactants distributor, is

Previous models, mathematically and physically based, cannot be used directly in CAD software for electronic circuits, which is used to design the DMFC-circuitry interface. By using the one-dimensional assumption, a lumped circuit model for simulating the DMFC runtime is derived in [15]. The methanol consumption at the reservoir is simulated by mass conservation equation and the charge/discharge electrode dynamics on the short time scale is simulated by a fictitious equivalent capacitance. A dynamic nonlinear circuit model for passive DMFC, including water mass flow and membrane hydration, is presented in [16]. By synthesizing physics equations in circuit form, this model is able to take into account mass transport, current generation, electronic and protonic conduction, methanol adsorption, and electrochemical kinetics. Adsorption and oxidation rates, which affect the cell dynamics, are modeled by a detailed two-step reaction mechanism. The fuel cell discharge and methanol consumption are computed by combining mass transport and conservation equations. As a result, the runtime of a DMFC can be predicted

Most DMFC performance models combine differential and algebraic equations with empirically determined parameters. Simpler empirical expressions can be derived from such models, allowing designers and engineers to predict the fuel cell performance as a function of different operating conditions (such as pressure, temperature, or fuel concentration). In [17], a semi-empirical approach to account for the limiting current behavior is proposed. The general expression for the polarization curve is based on analytical formulations of the catalyst layer reaction and includes the overpotential due to transport limitation in diffusion backing layer and the effect of methanol crossover. A simple semi-empirical model for evaluating the cell voltage of a DMFC is provided in [18] by using the semi-empirical Meyers-Newman rate equation for assessing the anode overpotential and a Tafel-type kinetic model for the cathode. Ohmic losses are accounted for by considering constant and uniform membrane conductivity. Such semi-empirical models are useful to describe the steady-state fuel cell behavior. Dynamic models can resort as well to semi-empirical relationships in order to limit the model complexity in timedomain simulations, notably when implementing models in real-time controllers. In [19], a completely different approach for extracting equivalent parameters is used. Instead of fitting current-voltage equations, a nonlinear equivalent circuit is derived from impedance fuel cell models. The equivalent circuit is composed of nonlinear electrical circuit elements that include resistors, capacitors, and inductors in order to simulate the DMFC performance in a wide operative range, including steadystate and transient conditions. A model of a micro-DMFC battery is developed

and the flow channel.

presented in [14].

from current and initial methanol concentration.

*3.2.2 DMFC semi-empirical models*

**29**

#### **Figure 2.**

*DMFC schematic with reactant and species flows (a, anode; c, cathode; pem, proton exchange membrane; fc, flow channel; dl, diffusion layer; cl, catalyst layer).*

cell and make use of atmospheric oxygen. Simplifying assumptions are typically made on both model geometry and physics to obtain the current-voltage characteristic in closed form. In 2-D models, methanol/oxygen concentrations and electric potentials depend also on the direction (y-axis) orthogonal to the species flow direction (x-axis). In 1-D models, this dependency is neglected by assuming constant concentrations along the y-axis.

The following assumptions are usually made in analytical DMFC models: (i) constant physical parameters; (ii) one-phase model (vapor phase is neglected); (iii) ideal and diluted solutions; (iv) homogeneous electrochemical reactions in the electrodes; and (v) negligible overpotential variation across catalyst layers and along the y-axis.

The basic equation for analyzing the cell voltage as a function of current density is:

$$\Delta V(f) = E\_0 + \frac{\partial \mathcal{E}}{\partial \mathcal{T}} \Delta T - \eta\_\mathbf{a}(f) - \eta\_c(f) - \frac{\delta\_\mathbf{m}}{\sigma\_\mathbf{m}} \left[ f - R\_\circ f \right] \tag{14}$$

where *Eo* is the (constant) standard cell potential, *E* is the fuel cell standard potential,*T* is the temperature, *σ<sup>m</sup>* and *δ<sup>m</sup>* are the electric conductivity and the thickness of the membrane, *η* is the activation overpotential*, J* is the current density, and *Rs* is the overall contact area specific resistance (ASR). This resistance per unit cross section, typically assumed constant, accounts for all resistances between gas diffusion layers and current collectors. The current density at the anode in Eq. (14) is related to the activation voltage overpotential at the anode *ηa*. In the simplifying assumption of the Tafel equation, it can be expressed as:

$$J\_a = J\_{a, \text{ref}} \frac{\text{C}\_{\text{ac}}}{\text{C}\_{\text{ac}, \text{ref}}} \exp\left(\frac{a\_a F}{RT} \eta\_a\right) \tag{15}$$

where *Ja,ref* and *Cac,ref* are the reference current density and the reference concentration at the anode, respectively. The voltage overpotential *η<sup>a</sup>* is required to

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

overcome the activation energy of the electrochemical reaction, and causes an energy loss. A relationship similar to Eq. (15) holds for the cathode side, where the atmospheric oxygen is provided at the catalyst layer through the gas diffusion layer and the flow channel.

In the one-dimensional model proposed in [12], the cell performance is assessed by describing mass transport in the porous electrode structures, including methanol crossover, and the potential and concentration distributions in electrodes. Multiphase flow is accounted for in [13], where liquid-vapor mixtures at anode and cathode compartments are considered. A two-dimensional analytical model of a DMFC, which describes electrochemical reactions on the anode and cathode and main transport phenomena in the fuel cell including methanol crossover, diffusion of reactants in porous media layers, and fluid flow in the reactants distributor, is presented in [14].

Previous models, mathematically and physically based, cannot be used directly in CAD software for electronic circuits, which is used to design the DMFC-circuitry interface. By using the one-dimensional assumption, a lumped circuit model for simulating the DMFC runtime is derived in [15]. The methanol consumption at the reservoir is simulated by mass conservation equation and the charge/discharge electrode dynamics on the short time scale is simulated by a fictitious equivalent capacitance. A dynamic nonlinear circuit model for passive DMFC, including water mass flow and membrane hydration, is presented in [16]. By synthesizing physics equations in circuit form, this model is able to take into account mass transport, current generation, electronic and protonic conduction, methanol adsorption, and electrochemical kinetics. Adsorption and oxidation rates, which affect the cell dynamics, are modeled by a detailed two-step reaction mechanism. The fuel cell discharge and methanol consumption are computed by combining mass transport and conservation equations. As a result, the runtime of a DMFC can be predicted from current and initial methanol concentration.

## *3.2.2 DMFC semi-empirical models*

Most DMFC performance models combine differential and algebraic equations with empirically determined parameters. Simpler empirical expressions can be derived from such models, allowing designers and engineers to predict the fuel cell performance as a function of different operating conditions (such as pressure, temperature, or fuel concentration). In [17], a semi-empirical approach to account for the limiting current behavior is proposed. The general expression for the polarization curve is based on analytical formulations of the catalyst layer reaction and includes the overpotential due to transport limitation in diffusion backing layer and the effect of methanol crossover. A simple semi-empirical model for evaluating the cell voltage of a DMFC is provided in [18] by using the semi-empirical Meyers-Newman rate equation for assessing the anode overpotential and a Tafel-type kinetic model for the cathode. Ohmic losses are accounted for by considering constant and uniform membrane conductivity. Such semi-empirical models are useful to describe the steady-state fuel cell behavior. Dynamic models can resort as well to semi-empirical relationships in order to limit the model complexity in timedomain simulations, notably when implementing models in real-time controllers. In [19], a completely different approach for extracting equivalent parameters is used. Instead of fitting current-voltage equations, a nonlinear equivalent circuit is derived from impedance fuel cell models. The equivalent circuit is composed of nonlinear electrical circuit elements that include resistors, capacitors, and inductors in order to simulate the DMFC performance in a wide operative range, including steadystate and transient conditions. A model of a micro-DMFC battery is developed

cell and make use of atmospheric oxygen. Simplifying assumptions are typically made on both model geometry and physics to obtain the current-voltage characteristic in closed form. In 2-D models, methanol/oxygen concentrations and electric potentials depend also on the direction (y-axis) orthogonal to the species flow direction (x-axis). In 1-D models, this dependency is neglected by assuming con-

*DMFC schematic with reactant and species flows (a, anode; c, cathode; pem, proton exchange membrane;*

The following assumptions are usually made in analytical DMFC models: (i) constant physical parameters; (ii) one-phase model (vapor phase is neglected); (iii) ideal and diluted solutions; (iv) homogeneous electrochemical reactions in the electrodes; and (v) negligible overpotential variation across catalyst layers and

The basic equation for analyzing the cell voltage as a function of current

where *Eo* is the (constant) standard cell potential, *E* is the fuel cell standard potential,*T* is the temperature, *σ<sup>m</sup>* and *δ<sup>m</sup>* are the electric conductivity and the thickness of the membrane, *η* is the activation overpotential*, J* is the current density, and *Rs* is the overall contact area specific resistance (ASR). This resistance per unit cross section, typically assumed constant, accounts for all resistances between gas diffusion layers and current collectors. The current density at the anode in Eq. (14) is related to the activation voltage overpotential at the anode *ηa*. In the simplifying

> *Cac Cac,ref*

where *Ja,ref* and *Cac,ref* are the reference current density and the reference concentration at the anode, respectively. The voltage overpotential *η<sup>a</sup>* is required to

*<sup>Δ</sup><sup>T</sup>* � *<sup>η</sup>*að Þ� *<sup>J</sup> <sup>η</sup>c*ð Þ� *<sup>J</sup>* <sup>δ</sup><sup>m</sup>

*exp <sup>α</sup>aF RT <sup>η</sup><sup>a</sup>* 

σ<sup>m</sup>

*J* � *RsJ* (14)

(15)

∂E ∂T

stant concentrations along the y-axis.

*fc, flow channel; dl, diffusion layer; cl, catalyst layer).*

*Thermodynamics and Energy Engineering*

*V J* ð Þ¼ *E*<sup>0</sup> þ

assumption of the Tafel equation, it can be expressed as:

*Ja* ¼ *Ja,ref*

along the y-axis.

density is:

**28**

**Figure 2.**

in [20] to capture all pertinent dynamic and steady-state electrical performance parameters, including capacity and its dependence on current and temperature, open circuit voltage, methanol-crossover current, polarization curve and its dependence on concentration, internal resistance, and time-dependent response under various loading conditions. The main advantage of this model is that it is able to predict the DMFC runtime for a given usable energy capacity of the methanol reservoir. A 1-D analytical model for simulating both DMFC static and dynamic operations is interfaced to a PSO (particle swarm optimization) stochastic algorithm in order to maximize the battery duration while minimizing methanol crossover [21]. In [22], a semi-analytical model of a passive-feed DMFC with non-isothermal effects and charge conservation phenomenon is proposed. The 1-D model is suitable for predicting the current-voltage curve by resolving iteratively both (bi-phase) mass transfer, electrochemical, and heat transfer equations, starting from imposed cell current, ambient temperature, and methanol feed concentration. The phenomenon of cathodic mixed potential, which is related to methanol crossover is accounted for as well by the DMFC model. Adaptive operations for voltage stabilization are proposed in [23] by using a simplified semi-empirical model combined to an on-demand control system. A rapid response of DMFC (less than 5 s) to variations of operating parameters is experimentally observed, ensuring the applicability of the proposed adaptive control strategy.

conditions, hole concentration is about 102 times greater than the zirconia intrinsic concentration. The probabilistic dependence of the charge carrier mobility on the energy yields an exponential dependence of the conductivity on temperature:

A typical conductivity value is *σ* ffi 0.02 S/cm at *T* = 1000 K, with activation energy *Wa* = 0.5–1.2 eV. This is a rather high conductivity value but not the highest among other types of FC electrolytes and, consequently, in SOFCs, ionic conduction

FC models must strike a balance between opposite requirements. On the one hand, they should be extremely rich in order to be able to represent the complete behavior of the cell by capturing the tridimensional distribution and time evolution of the physical quantities inside the cell. Such performance is achievable with a multiphysics, three-dimensional model, described by partial differential equations (PDEs) and characterized by a large number of physical parameters. On the other hand, they should be sufficiently simple to be included in other numerical procedure, such as a stochastic optimization loop, that is, the model should be numerically computable in a very short time, and they should be characterized by a relatively small number of parameters in order to avoid the *curse of dimensionality* issue. In such cases, a zero-dimensional stationary model is preferable since it avoids PDE numerical discretization and their inherent computational burden,

The fuel-cell balance equations can be arranged so as to correspond to lumped equivalent circuits. Indeed, several of them can be identified, around the same basic concept, depending on the required level of accuracy and the behaviors to be

In **Figure 3**, *E* is the open circuit voltage (the reversible voltage, i.e., the ideal electromotive force provided by the Nernst equation that depends on temperature and concentrations), *Rion* is the electrolyte ionic resistance, *Ra* and *Rc* are the equivalent resistances of the electrode kinetics (at anode and cathode), equal to the

�*Wa=kT* (16)

*σ*ð Þ¼ *T σ<sup>o</sup> e*

losses are relatively higher.

**4.1 Circuit models**

highlighted.

**Figure 3.**

**31**

*Simple lumped equivalent circuit of a fuel cell.*

**4. Lumped parameter models**

*Distributed and Lumped Parameter Models for Fuel Cells*

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

allowing to run the algorithm on standard PCs.

#### **3.3 Solid oxide fuel cells**

A solid oxide fuel cell (SOFC) consists of an anode and a cathode with a ceramic electrolyte between them that transfers oxygen ions. Oxidant reduction occurs in the cathode catalyst layer, oxygen ions are transported through the electrolyte and oxidation of the fuel occurs at the anode catalyst layer. A SOFC typically operates between 700°C and 1000°C, that is, temperatures at which the ceramic electrolyte becomes sufficiently conductive of oxygen ions. In fact, the electrolyte ionic conductivity is a strongly increasing function of the operating temperature. Moreover, since the operating temperature accelerates electrochemical reactions, precious metal catalysts are not required to promote the reactions and cheaper materials such as nickel can be used as catalysts. In addition, the SOFC can be fed with conventional hydrocarbon fuels, reforming being performed inside the cell. SOFCs can be planar or tubular, with the former having gained increasing success because of easier manufacturing and higher performance.

Among the ceramic electrolytes used in SOFCs, the most important is yttriastabilized zirconia (YSZ), which is the most conductive. The anode is generally made of nickel/yttria-stabilized zirconia cermet and the cathode is an LSM layer chemically expressed as La1*x*Sr*x*MnO3. The open circuit potential of a SOFC is given by the Nernst equation, whereas the activation overpotentials in both electrodes are high, so that the electrochemical kinetics of the both electrodes can be approximated by the Tafel equation, with the concentration dependence of exchange current density given by [24]. Zirconia (zirconium oxide, Zr O2) has a crystalline structure and is stabilized in the allotropic cubic form YSZ with the addition (doping) of Yttria molecules (yttrium oxide, Y2O3). Moreover, Yttria significantly increases oxygen holes, namely an Y3+ ion replaces a Zr4+ inducing an oxygen hole to maintain electrical neutrality and such oxygen holes facilitate the transport of O<sup>+</sup> ions in the lattice, by means of jumps. Doping with yttrium therefore dramatically increases the concentration of charge carriers (holes) and conductivity is proportionally improved. However, if the concentration of Yttria increases too much, the holes interact with each other thus reducing their mobility. Maximum conductivity is obtained with doping concentrations of about 8%: in such conditions, hole concentration is about 102 times greater than the zirconia intrinsic concentration. The probabilistic dependence of the charge carrier mobility on the energy yields an exponential dependence of the conductivity on temperature:

$$
\sigma(T) = \sigma\_o e^{-W\_a/kT} \tag{16}
$$

A typical conductivity value is *σ* ffi 0.02 S/cm at *T* = 1000 K, with activation energy *Wa* = 0.5–1.2 eV. This is a rather high conductivity value but not the highest among other types of FC electrolytes and, consequently, in SOFCs, ionic conduction losses are relatively higher.
