Fuel Cell Equivalent Circuit Modeling

In the equivalent circuit analog, resistors represent conductive pathways for ion and electron transfer. As such, they represent the bulk resistance of a material to charge transport such as the resistance of the electrolyte to ion transport or the resistance of a conductor to electron transport. Resistors are also used to represent the resistance to the charge-transfer process at the electrode surface. Capacitors and inductors are associated with space-charge polarization regions, such as the electrochemical double layer, and adsorption/ desorption processes at an electrode, respectively. 

The PEMFC physical picture, equivalent circuit, and the Nyquist plot using the impedance model. 

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It should also be pointed out that an equivalent circuit is not unique. In describing the same AC impedance spectrum, several circuits may exhibit the same result. For example, a model that includes elements without any chemical basis and practical meaning can demonstrate a perfect fit. Various equivalent circuit models used in PEM fuel cells will be discussed in detail in Chapter 4. 

When we begin to investigate an electrochemical system, we normally know little about the processes or mechanisms within the system. Electrochemical impedance spectroscopy (EIS) can be a powerful approach to help us establish a hypothesis using equivalent circuit models. A data-fitted equivalent circuit model will suggest valuable chemical processes or mechanisms for the electrochemical system being studied. From Chapter 1, we know that a fuel cell is actually an electrochemical system involving electrode/electrolyte interfaces, electrode reactions, as well as mass transfer processes. Therefore, EIS can also be a powerful tool to diagnose fuel cell properties and performance. 

Ahn et al. have developed fibre-based composite electrode structures suitable for oxygen reduction in fuel cell cathodes (containing high electrochemically active surface areas and high void volumes) [22], The impedance data obtained at -450 mV (vs. SCE), in the linear region of the polarization curves, are shown in Figure 6.22. Ohmic, kinetic, and mass transfer resistances were determined by fitting the impedance spectra with an appropriate equivalent circuit model. 

FIGURE 6.20 Example of a generalized equivalent circuit model for a single-cell fuel cell. Models regarding real fuel cells often contain constant phase elements (CPE). (Adapted from Scribner Associates, 2010. http //www. scribner.com/technical-papers.html)... 

Figure 3.7 Simple equivalent circuit model of a fuel cell.

EIS data can be analyzed by modeling or fitting the impedance spectrum with an equivalent circuit to extract the physically meaningful properties of the studied system. However, the design of the equivalent circuit is very important, and sometimes, the complexity of the PEM fuel cell system makes this process difficult. Depending on the shape of the EIS spectrum, the equivalent circuit model is usually composed of resistors (R), conductors (L), and capacitors (C), which are connected in series or in parallel, as shown in Fig. 3.11 in this equivalent circuit, R, R, and Rmt represent the membrane resistance, charge transfer resistance, and mass transfer resistance, respectively, and CPEi and CPE2 represent the Rt and Rmt associated capacitances, respectively. 

The common equivalent circuit models used to interpret simple EIS data are shown in Table 8.4 along with fuel cell representative components and Nyquist plots. 

There are several fuel cell system dynamic models in the literature that are specifically developed for vehicle propulsion. The simplest dynamic model of a fuel cell stack is the representation of the stack with an equivalent circuit whose operating parameters are based on the polarization curve obtained from the manufacturer data sheet at nominal conditions of temperature and pressure. An equivalent single-cell model with semiempirical equation for the fuel cell polarization curve can also be used for this purpose. 

The next set of models examined in this section is impedance models. Impedance is often used to determine parameters and understand how the fuel cell is operating. By applying only a small perturbation during operation, the system can be studied in situ. There are many types of impedance models. They range from very simple analyses to taking a complete fuel-cell model and shifting it to the frequency domain. The very simple models use a simple equivalent circuit just to understand some general aspects (for examples, see refs 302—304). 

While a good equivalent-circuit representation of the transport processes in a fuel cell can lead to an increased understanding, it is not as good as taking a 1-D sandwich model and taking it into the frequency domain. These models typically analyze the cathode side of the fuel cell. °2.3i3 3i4 pj g j ost comprehensive is probably that of Springer et al. °2 The use of impedance models allows for the calculation of parameters, like gas-phase tortuosity, which cannot be determined easily by other means, and can also allow for the separation of diffusion and migra-... 

In summary, simple combinations of elements and basic equivalent circuits for electrochemical systems have been introduced in this section. Although these models are relatively simple, they are commonly employed in the investigation of electrochemical systems, including fuel cells. A real electrochemical system may be much more complicated. However, complicated electrochemical systems can still be constructed from these basic equivalent circuits. 

Figure 5.30. Schematic of the catalyst layer geometry and its composition, exhibiting the different functional parts, a A sketch of the layer, used to construct a continuous model, b A one-dimensional transmission-line equivalent circuit where the elementary unit with protonic resistivity Rp, the charge transfer resistivity Rch and the double-layer capacitance Cj are highlighted [34], (Reprinted from Journal of Electroanalytical Chemistry, 475, Eikerling M, Komyshev AA. Electrochemical impedance of the cathode catalyst layer in polymer electrolyte fuel cells, 107-23, 1999, with permission from Elsevier.)...

Based upon the results of detailed models of these two types, overall models for the performance of fuel cells may be formulated in terms of simple equivalent electric circuit models that parametrise the loss terms and allow calculations of overall efficiencies as a function of such parameters. 

Figure 3.3B. Simplified equivalent circuit for the fuel cell reactor model. The battery voltage, Vb, is the result of the chemical potential difference for hydrogen between the anode and cathode. The internal resistance, R ib, is primarily the resistance of the membrane. Rl is the external load impedance. V is the voltage across the external load and I is the current through the load.

On the basis of this model and the equivalent circuit shown in Figure 4.5.67, the changes and differences, depending on the used anode in the fuel cell (Pt/C or PtRu/C) in the impedance spectra during the experiment, are dominated by the changes of the charge transfer resistance of the anode (Raj), the surface relaxation impedance (Rg, tg) and the finite diffusion impedance (Z ). 

A number of models describing supercapacitor resistor and capacitor behaviors used to mimic their performances in power systems have been reported and include classical equivalent, ladder circuit, and lumped or distributed parameter electrical and Debye polarization cell models [6]. An established design of a dynamic model of the often-used polymer electrolyte membrane fuel cell (PEMFC) is included in MATLAB and Simulink software to simulate performance under varying conditions specific to applications. 

In addition to the equivalent circuit method, the impedance results can also be analyzed using mathematical models based on physicochemical theories. Guo and White developed a steady-state impedance model for the ORR at the PEM fuel cell cathode [15]. They assumed that the electrode consists of flooded ionomer-coated spherical agglomerates surrounded by gas pores. Stefan-Maxwell equations were used to describe the multiphase transport occurring in both the GDL and the catalyst layer. The model predicted a high-frequency loop due to the charge transfer process and a low-frequency loop due to the combined effect of the gas-phase transport resistance and the charge transfer resistance when the cathode is at high current densities. 

CHR 08] Chrenko D., Pera M.C., Hissel D., et a/., Macroscopic modeling of a PEFC system based on equivalent circuits of fuel and oxidant supplying , ASME Journal of Fuel Cell Science and Technology, vol. 5, no. 1, pp. 011015-1/011015-8,2008. 

FIGURE 3.11 An equivalent circuit for EIS spectrum modeling of PEM fuel cells [43],... 

EIS data are commonly analyzed by fitting them to an equivalent electrical circuit model corresponding to a fuel cell component or components. Most of the circuit elements in the model are common electrical elements such as resistors, capacitors, and inductors. As an example, the electrolyte ohmic resistance can be represented with a resistor. Very few electrochemical cells... 

Modern EIS test systems are equipped with an equivalent circuit analysis software capability, greatly simplifying data analysis. The EIS analysis can be conducted on an individual electrode or on a full fuel cell for qualitative comparison of the various losses between different materials, electrodes, fuel cell design, or operating conditions. Note that fitting the EIS data to a given electrical circuit does not guarantee the model is correct, only that the data fit the assumed model. 

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