Analytical polarization curve

In this section, we construct several analytical polarization curves of PEMFCs and high-temperature polymer electrolyte membrane fuel cells (HT-PEMFCs). In these types of cell, owing to the excellent kinetics of the hydrogen oxidation reaction, the polarization voltage of the anode is negHgible. The voltage loss in a PEMFC is determined by the oxygen transport, ORR kinetics, and the cell resistivity. 

Figure 4.20 compares the exact numerical and analytical polarization curves (the latter is calculated with Equations 4.170 and 4.127 and/c = 1). The numerical curve is a solution of the system of Equations 4.156 and 4.55, valid at arbitrary oxygen transport limitations in the CCL. As can be seen. Equation 4.170 as is describes the exact polarization curve up to e 0.2 well. However, a simple correction factor of the form... 

Equation 4.172 with yi from Equation 4.173 and yo from Equation 4.127 solve the problem. With this. Equation 4.170 is an explicit analytical polarization curve. 

Nordlund and Lindbergh (2004) suggested a simple analytical polarization curve. In the notations of this section, a kinetic part of their polarization equation (not accounting for the methanol transport in the ABL) has the form... 

Kulikovsky, A. A. 2013a. Analytical polarization curve of DMFC anode. 

Indicator electrodes are used both for analytical purposes (in determining the concentrations of different substances from values of the open-circuit potential or from characteristic features of the polarization curves) and for the detection and quantitative characterization of various phenomena and processes (as electrochemical sensors or signal transducers). One variety of indicator electrode are the reference electrodes, which have stable and reproducible values of potential and thus can be used to measure the potentials of other electrodes. 

The first photoelectric fhiorimeter was described by Jette and West in 1928. The instrument, which used two photoemissive cells, was employed for studying the quantitative effects of electrolytes upon the fluorescence of a series of substances, including quinine sulfate [5], In 1935, Cohen provides a review of the first photoelectric fluorimeters developed until then and describes his own apparatus using a very simple scheme. With the latter he obtained a typical analytical calibration curve, thus confirming the findings of Desha [33], The sensitivity of these photoelectric instruments was limited, and as a result utilization of the photomultiplier tube, invented by Zworykin and Rajchman in 1939 [34], was an important step forward in the development of suitable and more sensitive fluorometers. The pulse fhiorimeter, which can be used for direct measurements of fluorescence decay times and polarization, was developed around 1950, and was initiated by the commercialization of an adequate photomultiplier [35]. 

A perfect prototype of an ideally cation-permselective interface is a cathode upon which the cations of a dissolved salt are reduced. Experimental polarization curves measured on metal electrodes fit the theory very closely. Since in dimensional units the limiting current is proportional to the bulk concentration, the polarization measurements on electrodes may serve for determining the former. This is the essence of the electrochemical analytical method named polarography. (For the theory of polarographical methods see [28]—[30].)... 

Fig. 8. Potentiodynamic polarization curve for a rotating Cu disc in 0.1 M NaOH with two analytical Pt half rings detecting soluble Cu+ (i jji) and Cu2+ ion formation (ijq) [62],...

It is necessary to note that (44) is an approximation, because the value of y is lower than unity. This approximation is widely used in qualitative discussions, because it permits the simple mathematical treatment of electrochemical processes with relatively small errors and with clear physical meaning. If y 1 is included in the derivation of the general polarization curve equation, simple analytical solutions are not available and numerical solutions are required. 

Dependence of the total overpotential of the electrode reaction on current can be found by the mutual solution of Equations (1.54) and (1.58). However, this solution cannot be found by analytical calculation. The curves turn of this dependence is defined by the 7°gm/ °me ratio. The calculated polarization curves are presented in Figure 1.20. The cathode current was restricted by the value of /°gm> and the polarization curves become similar to the Taffel dependencies with the increase of Tj. 

Figure 23.2 The general polarization curve of the catalyst layer with ideal feed transport. Solid lines, analytical solutions for the low- and high-current regimes points, the exact numerical solution to the system [Eqs.(23.6) and (23.7)] dashed line, the approximate analytical curve, Eq. (23.11),...

Based on an analytical model for PFFCs, Kulikovsky et al. presented a two-step procedure to evaluate the parameters Tafel slope, exchange current density, and cell resistance from two sets of polarization curves for an HT-PFFC [36]. The method was validated with experimental data. Shamardina et al. described an analytical model taht accounts for the crossover of gases through the membrane [37]. The model is pseudo-two dimensional and describes mainly the effects across the MFA. Temperature and pressure variations in the cell were not considered. From these analytical studies, it follows that the crossover effect has a considerable influence only at a low stoichiometry of oxygen. 

Figure 2.4 The general polarization curve of the catalyst layer with ideal transport of the reactant (oxygen). Solid lines analytical solutions, points— the exact numerical curve. Parameter = 0.1 (left panel) and = 10 (right panel). Linear domain is described by fjo = sjo, Tafel region is given by fjo = arcsinh ( jo coth (l/ )) and double-Tafel law is fjo = 21n( jo)-...

Figure 2.10 Upper solid analytical high-current polarization curve for uniform loading. Crosses the exact numerical polarization curve for uniform loading. Lower solid polarization curve of the active layer with optimal shape of catalyst loading. Short-dashed curve nonuniform loading third and fourth order derivatives in Eq. (2.92) are taken into accoimt.

The performance of a PEMFC can be expressed through the analytical formulation of the polarization curve. Simultaneous estimation of the parameters through analysis of the polarization curve can be helpful in diagnosis of PEMFC degradation and identification of degradation mechanism. Some models and testing methods have been developed to characterize PEMFC performance through polarization curve analysis [25,26]. 

When the diagrams of sin x and 1/2 sin x are plotted in polar coordinates, they represent a good example of the orientation dependence of tensor components. In this case, particular components are plotted in every direction along the polar axis, which results is a family of analytical lemniscate curves with two petals for p and four petals for t. In the case of a uniaxial extension, the shear components are twice shorter than the extension components, b. Uniaxial compression (e.g., of a pillar). In this case, one also has F/S = P, but the value of P is negative. The p and t stress rosettes in this case have the same appearance, with the only exception that the p petals have an inverse sign ... 

There is a great variety of approaches to fuel cell performance modeling. The simplest approach used in system simulations deals with the semiempirical polarization curves of the cell or stack under investigation. Such curves are obtained by fitting a simple analytical model equation to measured data. This philosophy is very useful in the optimization of FC systems with numerous peripheral components (blowers. 

FIGURE 4.18 The exact numerical (open circles) and the analytical (sohd hue, Equation 4.128) polarization curves of the CCL with the ideal oxygen transport. Dashed lines the low- and high-current curves. Equations 4.129 and 4.133, respectively. The exact numerical points are calculated with y heing the exact solution to Equation 4.122, while the analytical curve (4.128) is calculated with the approximate y given hy Equation 4.127. The current density is normalized to j = Opb/lci- Parameter ci = 1 and e = 100 (PEEC cathode. Table 5.7). Note the transition from normal to double Tafel slope at the current densities around io = 2. 

FIGURE 4.20 Polarization curves of the CCL for the dimensionless diffusion coefficients of 2.59. The points the exact numerical solution. The dashed hne the analytical Equation 4.170 with fc = 1 and y calculated from Equation 4.127. The dotted line Eiquation 4.170 with fc given hy Equation 4.171. The parameter s = 866. 

The analytical CCL polarization curve Equation 4.189 is compared to the exact numerical solution of the system (4.53) and (4.54) in Eigure 4.21. A reference value ofDrt = 1.37 10 cm s is taken from measurements (Shen et al., 2011). The curves in Eigure 4.21 correspond to the indicated ratios D/Dre/- Clearly, as this ratio tends to infinity, the analytical and numerical results tend to the diffusion-free polarization curve Equation 4.141. Note that as b decreases, the overpotential due to oxygen transport increases, and the accuracy of the model drops. Nonetheless, in the region of currents Jo < 1, the model works well for D/Dref as small as 0.1 (Figure 4.21). 

FI G U RE 4.21 Exact numerical (points) and analytical (Equation 4.189) (solid lines) polarization curves of the cathode catalyst layer with the finite rate of oxygen transport. The indicated parameter for the curves is the ratio D/Dref, where Dref = 1.37 10 cm s is the CCL oxygen diffusivity measured in Shen et al. (2011). The bottom solid line is the curve for infinitely fast oxygen transport in the CCL (Equation 4.141). 

The quality of Equation 4.248 can be checked by comparing this equation to the exact numerical polarization curve, following from the direct solution of the system (4.232) and (4.233). For the sake of comparison, zero transport loss is assumed in the ABL and zero crossover, therefore in Equation 4.248, Cmt = 1 is set. Figure 4.30 shows that the analytical curve works well up to the limiting current density. Further numerical tests show that Equation 4.248 only fails in a small vicinity of the adsorption-limiting current (Figure 4.30). 

FIGURE 4.30 Exact numerical (points) and analytical (lines, Equation 4.248) polarization curves of the anode catalyst layer for the indicated values of parameter f. In both the cases, parameter e = 1. 

FIGURE 4.31 Analytical (lines) and experimental polarization curves of DMFC anode measured by Nordlund and Lindbergh (2004) for the indicated methanol concentrations and cell temperature of (a) 50°C and (b) 70°C. The fitting parameters are listed in Table 4.4. 

In many applications (for example, in automatic control systems), an analytical equation for the cell polarization curve is highly desirable. Such an equation is also useful in cell performance and aging studies fitting an analytical equation to the measured curve may help to understand the contribution of kinetic and transport processes to the total potential loss in the cell. 

A number of attempts have been made to suggest semiempirical analytical equations for the cell polarization curve (Boyer et al., 2000 Kim et al., 1995 Squadrito et al., 1999). However, the empirical equations usually do not follow from the conservation laws, which makes them unreliable. The predicting capability of these equations is limited. 

In this section, conservation laws are used to derive analytical solutions for the polarization curve of the cathode side at finite oxygen stoichiometry, when either oxygen or proton transport in the CCL is poor. These equations help in understanding the type of transport loss in the CCL by fitting the cell polarization curve. Furthermore, the results of this chapter could be used as MEA submodels in CFD models of cells and stacks. Last but not least, the solutions below are simple enough to be used in real-time control systems. 

To model an electrochemical reaction and determine its mass and charge transfer parameters quantitatively, an electrochemical data fitting tool has been devdojjed in our research group. From an analytical approach, it is designed to extract a quantitative reaction mechanism from polarization curves. 

For the analytical modeling of the reaction, also 11 identical experimental curves are performed, at 500 and 1000 rpm. In the reduction of Ru NH3), an xmexpected variation of the current values in the region of the limiting current is observed. This behavior differs from the characteristic curve of an ET mechanism and might be due to additional reactions in the supporting electrolyte. This contribution of the supporting electrolyte is measured and subtracted from the experimental polarization curves. 

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