**4.1 Circuit models**

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 highlighted.

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

**Figure 3.** *Simple lumped equivalent circuit of a fuel cell.*

activation overpotentials divided by the electrode-generated current. Depending on the analysis purpose, both the incremental and differential values can be used:

$$R = \frac{\eta\_c}{i'} \text{ or } R = \frac{d\eta\_c}{di'} \tag{17}$$

The latter resistance characterizes transient responses to small variation of the cell's working point. Expanding it in the neighborhood of a working point for which the Tafel approximation holds provides:

$$R = \frac{d}{di} \left( -\frac{RT}{anF} \ln j\_0 + \frac{RT}{anF} \ln i \right) = \frac{RT}{anF} \frac{}{i} = \frac{RT}{anF \ i\_0 e^{anF\eta/RT}} \tag{18}$$

that is an adynamic circuit element, strongly dependent on the current or voltage. *Cdl* is the double layer capacity, related to TPB interfacial charge behavior, *Jl* are leakage current densities, due to various phenomena and particularly to the reactant crossover through the electrolyte, which make the output electric current smaller than the electrode-generated current (they are controlled source elements since such currents depend on the cell useful currents and on concentrations), *Zw* is an element accounting for the losses due to concentration gradients. This is a dynamic element because it synthesizes the diffusion of reactant concentration at the electrodes, which are governed by the diffusion dynamics equations. In case of pulsating variations at an angular frequency ω around a steady-state working point, this is a Warburg nonlinear impedance that can be expressed as:

$$Z\_w = \frac{RT}{A\left(nF\right)^2 \overline{c}\sqrt{D}\sqrt{\alpha}}\tanh\left(\delta\sqrt{\frac{j\alpha}{D}}\right)e^{j\frac{\pi}{4}}\tag{19}$$

where *δ* is the electrode thickness. By taking the limit *ω* ! 0 it reduces to:

$$Z\_w(\mathbf{0}) = \frac{RT}{A \, (nF)^2 \overline{c}} \frac{\delta}{D} \tag{20}$$

**Figure 4.**

**33**

*(courtesy of IEEE Trans. on Industrial Electronics).*

*Distributed and Lumped Parameter Models for Fuel Cells*

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

*Multiphysics equivalent circuits capable of simulating electrical, chemical, and mass transport interactions*

The dynamic elements *C*dl and *Z*<sup>w</sup> limit the response dynamics of the FC. In other words, an FC is not able to face arbitrary power variations and when these are too rapid it is necessary to support the FC with a storage device capable of faster response. This solution is adopted in automotive FCs, which are coupled with a lithium-ion battery or a supercapacitor, sized to meet the fastest transients and capable of reversible operations, enabling regenerative braking, but unable to accumulate as much energy as the H2 tank of the FC.

A more sophisticated equivalent circuit of a DMFC is presented in **Figure 4** in the case of a DMFC. Here, the dependences of the leakage currents on the reagent gradients and on the electrolyte currents have been separated and the concentration losses have been represented as voltage sources controlled by the methanol and oxygen gradients. Three separated equivalent circuits have been added, to represent the water, methanol, and oxygen behavior. Each of these includes capacitances, which account for the volumes of species accumulating at electrolyte, catalyst, and diffusion layers, whereas the controlled current sources describe drain and source at the catalyst layers and current-driven crossover.

In any case, equivalent circuits provide an approximated description of the complex reactions and transport events occurring inside the FC. They have the

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

activation overpotentials divided by the electrode-generated current. Depending on the analysis purpose, both the incremental and differential values can be used:

<sup>0</sup> or *<sup>R</sup>* <sup>¼</sup> *<sup>d</sup>η<sup>c</sup>*

<sup>¼</sup> *RT*

that is an adynamic circuit element, strongly dependent on the current or voltage. *Cdl* is the double layer capacity, related to TPB interfacial charge behavior, *Jl* are leakage current densities, due to various phenomena and particularly to the reactant crossover through the electrolyte, which make the output electric current smaller than the electrode-generated current (they are controlled source elements since such currents depend on the cell useful currents and on concentrations), *Zw* is an element accounting for the losses due to concentration gradients. This is a dynamic element because it synthesizes the diffusion of reactant concentration at the electrodes, which are governed by the diffusion dynamics equations. In case of pulsating variations at an angular frequency ω around a steady-state working point, this is

*<sup>α</sup>nF i* <sup>¼</sup> *RT*

The latter resistance characterizes transient responses to small variation of the cell's working point. Expanding it in the neighborhood of a working point for which

*di*<sup>0</sup> (17)

*<sup>α</sup>nF i*0*e<sup>α</sup> n F <sup>η</sup>=RT* (18)

*<sup>R</sup>* <sup>¼</sup> *<sup>η</sup><sup>c</sup> i*

the Tafel approximation holds provides:

*<sup>α</sup>n F* ln *<sup>j</sup>*

0 þ

a Warburg nonlinear impedance that can be expressed as:

*Zw* <sup>¼</sup> *RT A nF* ð Þ<sup>2</sup>

mulate as much energy as the H2 tank of the FC.

the catalyst layers and current-driven crossover.

**32**

*c* ffiffiffiffi *D* p ffiffiffi *ω* p

*Zw*ð Þ¼ 0

where *δ* is the electrode thickness. By taking the limit *ω* ! 0 it reduces to:

The dynamic elements *C*dl and *Z*<sup>w</sup> limit the response dynamics of the FC. In other words, an FC is not able to face arbitrary power variations and when these are too rapid it is necessary to support the FC with a storage device capable of faster response. This solution is adopted in automotive FCs, which are coupled with a lithium-ion battery or a supercapacitor, sized to meet the fastest transients and capable of reversible operations, enabling regenerative braking, but unable to accu-

A more sophisticated equivalent circuit of a DMFC is presented in **Figure 4** in the case of a DMFC. Here, the dependences of the leakage currents on the reagent gradients and on the electrolyte currents have been separated and the concentration losses have been represented as voltage sources controlled by the methanol and oxygen gradients. Three separated equivalent circuits have been added, to represent the water, methanol, and oxygen behavior. Each of these includes capacitances, which account for the volumes of species accumulating at electrolyte, catalyst, and diffusion layers, whereas the controlled current sources describe drain and source at

In any case, equivalent circuits provide an approximated description of the complex reactions and transport events occurring inside the FC. They have the

*RT A nF* ð Þ<sup>2</sup> *c δ*

tanh *δ*

ffiffiffiffiffi *jω D* ! r

*ej π*

<sup>4</sup> (19)

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

� �

*RT <sup>α</sup>n F* ln *<sup>i</sup>*

*di* � *RT*

*Thermodynamics and Energy Engineering*

*<sup>R</sup>* <sup>¼</sup> *<sup>d</sup>*

#### **Figure 4.**

*Multiphysics equivalent circuits capable of simulating electrical, chemical, and mass transport interactions (courtesy of IEEE Trans. on Industrial Electronics).*

merit to highlight separately single events (such as the ohmic losses in the electrolyte), thus allowing to easily study the effects of the variation of single quantities/parameters on the FC behavior. The main drawback of circuit models is that complex interactions and nonlinearity are not simulated in detail. More importantly, they can be implemented into circuit simulation software, to study the electrical interface of the FC with the power management electronics and system supervisor that is devoted to provide FC control, in order to study the overall system dynamics [16].

complexes which once freed from sulfonic acid groups can move through the membrane. In these conditions, that is, proper hydration, proton conduction strongly depends on the water content and the temperature of the membrane, and

As briefly mentioned above, protonic conduction in Nafion® strongly depends on the temperature, since the mechanism is based on charged particles jumping from site to site, with a rate described by the diffusivity *D*. This statistical parameter depends on the activation barrier energy which exhibits an exponential dependence

where *Do* is a diffusivity reference value, *Wai* is activation barrier energy, and *k*

*<sup>μ</sup>* <sup>¼</sup> j j *<sup>z</sup> F D*

with |*z*| the ion charge number, so that the proton conductivity *σ* = *ρ<sup>c</sup> μ* can be

Apart from the temperature, the proton conductivity also depends on the water content in the membrane. A common modeling approach to represent such dependence is based on the hydration *λ*, that is, the ratio between the number of water molecules and the number of charge sites available for proton conduction. In the specific case of Nafion®, such ratio can be rewritten in a modified form using the water concentration *cw* and the sulfonic acid concentration *cas*, that is, *λ* = *cw*/*cas*. In absence of more sophisticated models, a linear dependence of conductivity on

where *λ* is derived from a correlation empirically derived for Nafion® [7]. Com-

with *Wai*/*k* = 1268 K for ions hopping. This is a more sophisticated model than the one expressed by Eq. (9). The conductivity *σ* influences the scalar potential *φ*

*Wai k* 1

�*Wai=kT* <sup>¼</sup> *<sup>σ</sup><sup>o</sup> <sup>e</sup>*

*σ* ¼ *α λ* ¼ 0*:*5139 *λ S=m* (24)

∇ � *σ λ*ð Þ *, T* ∇*φ* ¼ 0 (26)

<sup>303</sup>�<sup>1</sup> ð Þ*<sup>T</sup>* (25)

�*Wai=k T* (21)

*RT* (22)

�*Wai=kT* (23)

*D* ¼ *Do e*

is Boltzmann's constant. The charged particle mobility *μ* is proportional to *D*

*<sup>σ</sup>* <sup>¼</sup> ð Þ *z F* <sup>2</sup> *cDo*

*RT <sup>e</sup>*

bining the above, the factorized expression of *σ*(*λ*,*T*) is obtained:

*σ λ*ð Þ¼ *, T α λ e*

according to the charge conservation equation in quasi-static conditions:

which shows that the distribution of *φ* depends on *λ* and *T*.

being *ρ<sup>c</sup>* = |*z*|, *Fc* the charge density, and *c* the molar concentration.

can reach values as high as 20 S m�<sup>1</sup> at 100°C.

*Distributed and Lumped Parameter Models for Fuel Cells*

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

on the temperature *T* according to the law:

*5.1.1 Electrical conductivity model*

according to the Einstein relation:

hydration can be assumed:

written as:

**35**
