CCL, polarization curve

Appendix B Accurate Explicit Approximations for General CCL Polarization Curve (6.17)... 

To derive an explicit dependence of Id on jo, the CCL polarization curve is needed. This curve is obtained using the following arguments. Setting jc = 1 and c = ci in Equation 4.74 gives... 

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). 

This equation determines the CCL polarization curve at large currents, as discussed... 

Figure 4.24 shows the CCL polarization curves for the range of parameter between zero (no crossover) and 0.5 (large crossover). Note that Figure 4.24 shows the electrode potential calculated according to Ecath = E oi — t]ox,o. The crossover dramatically (by 300-600 mV) lowers the electrode potential at the OCP (Figure 4.24). Note that for jS between 0 and 0.2, there is a range of currents (50 to 150 mA cm ), where the polarization curve is nearly flat (Figure 4.24). This is a feature of DMFC an increase in the useful current jo lowers the crossover current jcross,o so that the sum jo + jcross.o and hence the cell potential do not change significantly.

The equations in the section CCL Polarization Curves of Chapter 4 contain the oxygen concentration ci at the CCL/GDL interface. This concentration is, itself, related... 

The value of Ct appears in the expression for the rate of ORR (6.3). We, therefore, can immediately obtain the general polarization curve of the cathode side by simply replacing Ct in the general voltage current curve of the CCL (6.17) with (6.28). Substituting (6.28) into (6.17) and omitting the subscript 0 we get [12]... 

Thus, when e is small and jo < 1 the polarization curve of the CCL is linear. Equation (2.46) in dimension variables takes the form... 

The high-current polarization curve of the CCL follows from Ek[. (2.54). Setting X = 0 in this equation we get... 

With (2.70) this equation is the general polarization curve for the CCL with ideal proton transport. Of particular interest are the two limiting cases. 

Consider first the low-current regime of CCL operation. The low-current polarization curve of a CCL is given by (2.44). To simplify calculations we will assume that parameter e (2.13) is large, so that coth(l/e) e (this situation is typical of PEFCs). Equation (2.44) then reduces to... 

Polarization curves for different j3 and A are shown in Figure 4.26 (the other parameters are listed in Table 4.3). An increase in the rate of crossover (3 reduces the cell open-circuit voltage V),c (Figure 4.26(a)). Qualitatively, this is what we could expect since methanol crossover reduces the amount of oxygen in the CCL. 

Another consequence of membrane contamination by cationic impurity can be a decrease in the limiting current on polarization curve measured at high contamination levels. Due to proton deficiency at the cathode, the ORR current may become limited by diffusion of protons, but not oxygen diffusion through the CCL. This effect observed in experimental systems (Uribe et al, 2002 Halseid et al, 2006b) was qualitatively described using model assumptions proposed by Kienitz et al. (2009). 

This equation is the simplest polarization curve of the CCL, demonstrating a typical exponential dependence of the converted current density on the cathode overpotential. Solving Equation 1.75 for t]q results in... 

This chapter is devoted entirely to performance models of conventional catalyst layers (type I electrodes), which rely on reactant supply by gas diffusion. It introduces the general modeling framework and employs it to discuss the basic principles of catalyst layer operation. Structure-based models of CCL rationalize distinct regimes of performance, which are discernible in polarization curves. If provided with basic input data on structure and properties, catalyst layer models reproduce PEFC polarization curves. Consistency between model predictions and experimental data will be evaluated. Beyond polarization curves, performance models provide detailed maps or shapes of reaction rate distributions. In this way, the model-based analysis allows vital conclusions about an optimal design of catalyst layers with maximal catalyst utilization and minimal transport losses to be drawn. 

The reaction penetration depths. Id or la, are highly insightful parameters to evaluate catalyst layer designs in view of transport limitations, uniformity of reaction rate distributions, and the corresponding effectiveness factor of Pt utilization, as discussed in the sections Catalyst Layer Designs in Chapter 1 and Nonuniform Reaction Rate Distributions Effectiveness Factor in Chapter 3. Albeit, these parameters are not measurable. The differential resistances, Rd or Ra, can be determined experimentally either as the slope of the polarization curve or from electrochemical impedance spectra (Nyquist plots) as the low-frequency intercept of the CCL semicircle with the real axis. The expressions in Equation 4.33 thus relate the reaction penetration depths to parameters that can be measured. 

FIGURE 4.6 (a) Effect of the Nafion weight fraction, 7 /, in CCL with uniform composition on the fuel cell voltage, Eceih evaluated at different values of the current density, yo. Experimental data taken from Passalacqua et al. (2001) (crosses) are shown for comparison. (b) Comparison of polarization curves, calculated in the model of composition-dependent performance, with experimental data of Uchida et al. (1995a,b). 

FIGURE 4.9 Polarization curve calculated with a CCL model that accounts for the full coupling between porous structure, liquid water accumulation, and performance. The dashed lines represent Hmiting scenarios corresponding to the ideally wetted state and the fuUy saturated state. Bistability occurs in the transition region. (Reprinted from Electrochim. Acta, 53(13), Liu, J., and Eikerling, M. Model of cathode catalyst layers for polymer electrolyte fuel cells The role of porous structure and water accumulation. 4435 1446. Copyright (2008), Elsevier. With permission.)... 

Figure 4.12b shows the variation of the total effectiveness factor of the CCL with current density for the polarization curves in Figure 4.12a. Fstat and F p are functions of composition and microstructure of the CCL neglecting degradation effects, these parameters should remain constant. However, the agglomerate effectiveness factor decreases with current density. The dependence of the effectiveness factor on current density is stronger at low and high current densities, jo < 0.4 A cm and jo > I A cm . Effectiveness factor values are similar over a wide range of jo for the studies of Suzuki et al. (2011) and Soboleva et al. (2011). However, the higher propensity for flooding of the GDL results in a sharper drop of the effectiveness factor at jo > I A cm in Suzuki et al. (2011).

This equation is the polarization curve for the CCL with ideal proton transport (Figure 4.15, dashed curve). Of particular interest are the limiting cases of small and large fo, corresponding to low and high cell current, respectively. In these cases, relations (4.77) and (4.78) can be simplified to express rjo and Id through the cell current jo. 

This is the general polarization curve of the CCL with ideal oxygen transport for the case of large Equation 4.128 describes normal Tafel kinetics at small currents, double Tafel kinetics at large currents and the transition region between these two regimes. Moreover, this equation reduces to a correct limit at jo -> 0. 

The exact numerical and the approximate (Equation 4.128) polarization curves of the CCL with y from Equation 4.127 are depicted in Figure 4.18. As can be seen. Equation 4.128 with y from Equation 4.127 provides excellent accuracy in the whole range of cell currents. 

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 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... 

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. 

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 model below shows that this approach is correct unless the cell current is not large (Kulikovsky, 2012c). At small currents, the MOR runs close to the membrane, while the ORR is shifted toward the GDL. In this regime, the DMFC cathode represents a complete short-circuited fuel cell. However, at large currents, the MOR and ORR share the same domain of the CCL, leading to a rapidly decreasing, resistive-like polarization curve. 

The model allows deriving the approximate polarization curve of the CCL in the supercritical (high-current) regime. In this regime, Coxfi — 0 and hence... 

FIGURE 4.27 (a) The points the DMFC polarization curves for the indicated working temperatures (data from Argyropoulos et al. (2(X)2)). The dotted lines the linear fit according to Equation 4.215. The slopes of the straight line (the CCL resistivities) are collected in Table 4.3. (b) The calculated electrode polarization curve versus the sum of the useful and crossover current densities jo + jcrossfl for tbe indicated parameter jS. ... 

In other words, the onset of the high-current linear regime is determined by the characteristic current of oxygen transport through the CCL, while the polarization curve in this regime is determined by the transport of methanol and protons. 

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. 

What is the effect of finite A, at high currents In this section, the case will be considered for poor oxygen diffusivity and ideal proton conductivity of the CCL. In this case, the local polarization curve of the CCL is given by Equation 4.87. Substituting Equation 5.41 into Equation 4.87, one obtains the local polarization curve with the term describing oxygen transport in the GDL ... 

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