**2. Main electrochemical phenomenon**

subjectivity appears in any modeling work, the approach of a mechanic/energy specialist, an electrochemist, or a physicochemist will differ mainly in the basic assumptions of modeling (model simplifications). However, whatever the cultural origin of the modeler, the numerical resolution of a multiphysical problem makes it

The modeling of an electrochemical system involves the mathematical expressions of the physical phenomena that take place there (a priori). Obviously, all model representations only offer a fragmentary assessment of the real systems. These various representations are distinguishable by their scales of time and space. However, the notion of adapted or appropriate modeling remains subjective. Indeed, as described in the literature about fuel cell models [6, 7], each of the approaches has limitations of description or prediction, and their main interest is to highlight one specific process. Despite this subjectivity, the model must prove

The validity of the model could be named external, i.e., related to theories, concepts, assumptions, and experimental data. Thus, the model is theoretically valid if it accepts theories or models already validated. In addition, if the model well matches to its potential of scientific explanation (the state of the art), one will qualify its heuristic validity. However, building a model cannot be done without solving it in all its intended range. Consequently, it is also necessary to define criteria of internal validity, which are criteria of evaluation of the model independent of the theories, results, and hypotheses. The algorithm (solver) must be appropriate, and the evaluation errors must remain within "valid" limits.

External validations that may be acceptable include empirical validity (the model corresponds to the available data) or pragmatic validity (the model satisfies

A fuel cell is a nonlinear and strongly coupled dynamic system. It is a multiinput multi-output system based on multiphase flow, electrochemical reactions, and heat transfer. For example, the control strategies of PEMFC can be built on a prediction of the future output of the system to compute the current control action [8]. In practice, the current control action is obtained by solving online an optimization problem. The aim of the optimization problem is to find the optimum of a cost function that minimizes the mean squared difference between predicted

To spare computation time due to computing of multiphysic fuel cell models, artificial intelligence (AI) techniques are useful as alternate approaches to conventional multiphysic modeling: e.g., artificial neural network (ANN) simulator could be employed to predict the fuel cell behavior [9–12]. The ANN could be trained with a reduced amount of data generated by a validated cell model [13]. Once this network is trained, it can predict different operational parameters of the fuel cell reducing the computation time [14]. This strategy has many possibilities [15]: spectroscopic analysis, prediction of reactions, chemical process control, and the analysis of electrostatic potentials. The ANN is trained to learn the internal relationships from data. These data may be taken from the real process even if there are noisy. The database should have a significant size and contain the maximum

possible to assure three major functions in the phase of development [5]:

• Assistance with the understanding of experimental results

• Study and optimization of design

*Thermodynamics and Energy Engineering*

• The prediction of performances

its validity.

the intended use).

outputs and target values.

**80**

An electrochemical cell is characterized by the I-V (current–voltage) behavior: the current that passes across the cell to the applied cell voltage. The I-V relation depends on various physical phenomena and is fundamental to achieve efficiently electrochemical conversion. When current is drawn using computational tools, the current density may not be uniformly distributed on the electrode surfaces. The performance and lifetime of electrochemical cells, such, is often improved by a uniform current density distribution. Therefore, it is necessary to optimize the current distribution. The electric current is a flow of electric charges: through the electrolyte between the anode and the cathode in the ions form and within the wires and current collectors/electrode materials in the electrons form. When the overall current of the cell is equal to zero, for example, on the disconnection of an electrode from the power supply or the load, the cell voltage is equal to UOCV the open circuit voltage:

$$I = 0; U\_{cell} = U\_{OCV} = E\_{a,0} - E\_{c,0} \tag{1}$$

where *Ea*,0 *and Ea*,0 are the potential of each electrodes at OCV. It is shown that the *UOCV* is related to the difference of free enthalpy differences of each reaction, involving the number n of electrons exchanged and the Faraday constant:

$$U\_{OCV} = \frac{\Delta G\_i}{nF} \tag{2}$$

where *ΔGi* is the Gibbs energy of the species i with specified cell temperature (T) and cell pressure (P). At a non-zero current, the cell voltage of an electrochemical reactor (electrolyzer) is greater than *UOCV*, and electrochemical generator (fuel cell or battery) is smaller than *UOCV* due to the various irreversibilities standing in the electrochemical conversion. The electrode potentials differ from the equilibrium values *Ea*,0 and *Ea*,0, and this difference is called overvoltage:

$$
\eta\_a = E\_a - E\_{a,0} \tag{3}
$$

$$
\eta\_c = E\_c - E\_{c,0} \tag{4}
$$

According to this description, an anode overvoltage is positive, while the cathode overvoltage is negative in all cases. The overvoltages depend on the current density at the electrode, depending on the involved electrochemical reactions, the electrode materials, and several operating conditions: concentration species, flow rate, etc. The current density is an extensive quantity that can be defined in any point of the electrochemical device, i.e., at the surface of the electrodes and through the electrolyte. In addition Ohm's law expresses the current density according to the local potential gradient, using the conductivity as follows:

$$
\overrightarrow{I} = -\sigma \overrightarrow{\nabla} \overrightarrow{E} \tag{5}
$$

(MEA), placed between two graphite bipolar plates, which feed the device with gases and cool it down. At the anode, fuel H2 is oxidized, liberating electrons and producing protons. The electrons flow to the cathode via an external circuit, where they combine with the proton and the dissolved oxidant O2 to produce water

> ψ UOCV Ucell

Proton transfer from the anode to the cathode via the membrane closes the electrical circuit. A careful water management is required to ensure that the membrane remains fully hydrated in order to improve the ionic conductivity and to avoid the electrode flooding (which occurs when an excess of liquid water restrains the active species access to the active layer). The gas diffusion electrodes (GDEs) are made up of two distinct areas: an inactive area (backing layer) and an active area (active layer) which is a place of the electrochemical and chemical reactions (**Figure 1**). Therefore, in the first approach it is possible to observe separately secondary current distribution (only in the active layer) and tertiary current

PEMFC and all solid electrolyte cells are the model devices in order to sketch the primary (in the polymeric membrane), the secondary (in active layer), and tertiary (in backing layer) current distributions. In addition, operating cell potential and each overvoltage could be compared to thermodynamic potential (Gibbs energy) to

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

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

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

can be employed to evaluate the overvoltages at both electrodes:

The efficiency of fuel cell is easily computed according to:

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

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

distribution (only in the backing layer).

access enthalpic balance.

electrode assembly.

**83**

H2 ⇆ 2H<sup>þ</sup> þ 2e� (8)

(10)

2H<sup>þ</sup> þ 2e� þ ½O2 ⇆ H2O (9)

and heat:

where *σ* is the ionic conductivity of an electrolyte. In the case of a onedimensional system where the electrolyte is confined in a finite space between two surface electrodes S and separated from each other by the distance e, this potential difference (called ohmic drop *ηohm*) can be expressed as follows:

$$
\eta\_{ohm} = -\frac{e}{\sigma \mathcal{S}} I \tag{6}
$$

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

$$\mathbf{U\_{cell}} = \mathbf{U\_{OCV}} - \eta\_{\mathbf{a}} + \eta\_{\mathbf{c}} - \eta\_{\text{ohm}} \tag{7}$$

Inside the nonaqueous electrochemical device (**Figure 1**), the primary current distribution is well controlled, because a solid electrolyte is confined in a finite space between two surface electrodes and only the misalignment of both electrodes could affect the current distribution through the electrolyte. In nonaqueous case, the optimization endeavor is devoted to secondary (electrochemical activation) and tertiary (mass transport limitation) current distributions. In this context, fuel cells are the best example to scrutinize energetic balance of electrochemical devices.

The main advantage associated with fuel cells is that they are not limited by Carnot efficiency. Besides, no moving parts are required to convert thermal energy into mechanical energy. The energy release from interatomic bonds of the reactants is converted efficiently into electrical energy. In this document, we consider protonexchange membrane fuel cells (PEMFCs): a PEMFC consists of a polymer electrolyte sandwiched between two electrodes to form a membrane electrode assembly

**Figure 1.** *Schematic description of a MEA.*

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

(MEA), placed between two graphite bipolar plates, which feed the device with gases and cool it down. At the anode, fuel H2 is oxidized, liberating electrons and producing protons. The electrons flow to the cathode via an external circuit, where they combine with the proton and the dissolved oxidant O2 to produce water and heat:

$$\text{H}\_2 \oplus 2\text{H}^+ + 2\text{e}^- \tag{8}$$

$$\text{\textbullet2H}^+ + 2\text{e}^- + \text{\textbulletO}\_2 \oplus \text{H}\_2\text{O} \tag{9}$$

The efficiency of fuel cell is easily computed according to:

$$
\Psi \frac{\mathbf{U}\_{\text{OCV}}}{\mathbf{U}\_{\text{cell}}} \tag{10}
$$

Proton transfer from the anode to the cathode via the membrane closes the electrical circuit. A careful water management is required to ensure that the membrane remains fully hydrated in order to improve the ionic conductivity and to avoid the electrode flooding (which occurs when an excess of liquid water restrains the active species access to the active layer). The gas diffusion electrodes (GDEs) are made up of two distinct areas: an inactive area (backing layer) and an active area (active layer) which is a place of the electrochemical and chemical reactions (**Figure 1**). Therefore, in the first approach it is possible to observe separately secondary current distribution (only in the active layer) and tertiary current distribution (only in the backing layer).

PEMFC and all solid electrolyte cells are the model devices in order to sketch the primary (in the polymeric membrane), the secondary (in active layer), and tertiary (in backing layer) current distributions. In addition, operating cell potential and each overvoltage could be compared to thermodynamic potential (Gibbs energy) to access enthalpic balance.
