Limiting Current Operation of Proton Exchange Membrane Fuel Cells

The oxygen reduction reaction is

At high current densities, mass transport of all the reactants and products may be important. However, oxygen is the focus of this study, thus requiring separation of the different mass transport overpotentials. State of the art materials and design are characterized by negligible ohmic losses (electron mass transport). In comparison, water requires more attention. Water vapor transport losses are minimized by using saturated inlet streams (absence of gas phase transport by minimizing concentration gradient), which also increases proton transport in the membrane by both migration and diffusion.² Liquid water is more problematic, especially due to its effect on both mass transport and kinetic losses.³ Even if several methods have already been developed to measure liquid water amounts within operating PEMFCs, including neutron imaging, X-ray microtomography, proton nuclear resonance imaging microscopy, and residence time distribution,^4–9 their sophistication is not yet sufficient for quantification, or property databases have not yet been established. Neutron imaging suffers from insufficient spatial resolution (an order of magnitude increase is needed) and is unable to distinguish between liquid water within the membrane and that within the cathode and anode flow field.^{5, 7} X-ray microtomography requires development to minimize ring artifacts arising from abrupt changes in density and atomic number at the porous gas diffusion layer interface.⁶ Proton nuclear magnetic resonance microscopy has insufficient temporal resolution for transient studies and is subject to interference from magnetic fuel cell materials.⁴ Residence time distribution techniques do not provide spatial resolution.^{8, 9} Mathematical modeling cannot presently be used to predict liquid water content within gas diffusion electrodes because the properties associated with liquid water transport in porous media, saturation, capillary pressure, intrinsic and relative permeability, are largely unknown.^{10, 11} Because measurement or prediction of the liquid water content within a gas diffusion electrode is not presently possible, liquid water effects were minimized by limiting production rates. This was achieved by reducing the oxygen concentration and thus the limiting current density. This approach also has the effect of decreasing the oxygen mass transport rate, while maintaining the proton mass transport rate, thus reinforcing the oxygen transport rate limiting assumption. This approach is reasonable considering some of the recent experimental neutron imaging characterization results.⁷ The gas diffusion electrode liquid water content is estimated at 5–18% of the pore volume based on Fig. 7 data of Ref. ⁷. Because oxygen was used, these water content values are considered an upper limit and are expected to be smaller with diluted air (lower current densities and water production rates).

The main assumptions are¹²

• (1)  
  Unit cell has straight reactant channels.
• (2)  
  Cell performance is averaged over the cross-channel direction (plug flow reactor).
• (3)  
  Cell voltage is constant along the cell length.
• (4)  
  Isothermal operation.
• (5)  
  Isobaric operation.
• (6)  
  Reactants are saturated with water vapor.
• (7)  
  Reactants obey the ideal gas law.

The inlet oxygen flux in the cathode channel is first considered on an equivalent current density basis

The nitrogen flux is constant (absence of crossover to the anode compartment and leaks) and is expressed as an oxygen flux multiple

The oxygen channel concentration is

Under low voltage conditions, as a result of a reduced stoichiometry or high power demand (low external load resistance), the local current density at every position along the cell is considered to have reached the limiting value. With this observation, the local mass balance as a result of oxygen consumption by the electrochemical reaction takes the form

The limiting current is defined on the basis of a linearized first Fick's law

The parameter δ has crucial importance characterizing the transport properties of the gas diffusion electrode (design parameter). It can be interpreted as an overall mass transfer coefficient encompassing oxygen transport in both the gas (gas diffusion layer) and ionomer (catalyst layer) phases. Combination of Eq. 4, 5, 6 leads to

Equation 7 is solved in a universal way with one family of functions describing all low cell voltage operation. The cell length and oxygen flow rate are first rescaled [, ]. Replacement of these scaled variables in Eq. 7 and dropping primes leads to

The number can be described as a dimensionless ratio of the maximum current density permitted by mass transport to the equivalent current of the inlet oxidant stream. The implicit solution of Eq. 8 takes the form

Equation 9 has solutions in terms of the universal function , where solves

It is possible to write an explicit formula for in terms of the Lambert- function. Note that represents the oxygen flux fraction consumed in the cell (cell outlet is scaled to the position ) also known as the conversion often denoted by . Figure 2 illustrates the shape of as a function of the parameter . The maximum current density occurs at the cell inlet (Eq. 4, 6 combined with Eq. 8 inlet condition)

The more general derivation of Eq. 11 leads to (combination of Eq. 4, 6)

The universal scaled low voltage current density that can be derived from the oxygen flux curves is

Figure 3 illustrates the shape of the universal scaled current density as a function of the parameter . The scaled average current density produced in the cell is (using definition from Eq. 8 and Eq. 11)

Equation 14 also defines the minimum stoichiometry to obtain the average current density

Figure 4 illustrates the shape of the minimum stoichiometry as a function of the scaled average current density for different values of the parameter . In practice, the main equation describing diffusional limitations (Eq. 8) can be simplified for large values (low oxygen concentration in the reactant stream)

This is also convenient from a cell design point of view because, for air, large limiting current densities are expected, requiring design changes to accommodate modifications in electrical and thermal requirements. The solution to Eq. 16 has the form

This exponential decay is consistent with Kulikovsky's derivation.¹³ The universal scaled low voltage current density distribution that can be derived from the oxygen flux curves (simplified Eq. 11, 12, 13 for large values combined with Eq. 17) is also exponential

The average current density is

For and after rearrangement, Eq. 19 is rewritten as

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The local polarization curve, as previously described for the -dimensional unit cell model,¹ has the following form

with the limiting current density defined as (Eq. 6). Equation 22 was modified by grouping several constants into a single parameter

This last expression was used in its exponential form (numerically more stable) to fit the polarization data illustrated in Fig. 5 (, tests A) using SIGMAPLOT 9.0 (Marquardt-Levenberg algorithm). The three fitted parameters, respectively, characterize kinetic , ohmic , and mass transport limitations. The fit is good especially at the low oxygen concentration values, where liquid water interactions are expected to be negligible (lower limiting current density and water production values). The value of the mass transfer coefficient is close but somewhat different than the value found in previous work .¹ The difference is mostly ascribed to operating conditions changes (Table I) and absence of limiting current density data in the older work. The limiting current density values are plotted in Fig. 6 and also correspond well to the theoretical expression based on the fitted mass transfer coefficient value. Therefore, the data support the assumption that a sharp limiting current density exists at least locally.

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However, the good agreement between the zero-dimensional model and the experimental data is deceptive because the actual oxidant stoichiometry at the limiting current density was as low as 1.4 (Table I). This is ascribed to the implication of operation at the limiting current density. Under these limiting conditions, oxygen is locally consumed at a significant rate. Uniform conditions can only be attained if the local oxygen consumption rate is compensated by large reactant flow rates. For low oxygen concentrations, this results in increase pressure drop and nonuniform conditions further compounded by potential convection in the gas diffusion electrode. For high oxygen concentrations, the limiting current density is increased, leading to even more acute nonuniformities. These considerations are independent of both flow field and cell active area size and justify the development of a one-dimensional (channel length) low cell voltage model.

For cases where the reactant concentration can be considered constant and the current density is varied, Eq. 23 is rewritten with Eq. 6 as the equivalent form

Equation 24 represents a simple empirical expression for the cell performance with a physical explanation for the mass transport loss (existence of a limiting current). From this standpoint, Eq. 24 represents an alternative to several other similar empirical expressions, which do not necessarily have a physical meaning attached to the mass transport loss term.^15–17 More recently, an attempt has been made to derive an empirical expression from fundamental principles.¹⁸ The resulting empirical expression contains four parameters for the mass transport loss term (includes the effect of liquid water) rather than one for Eq. 24.

Experimental current distribution data obtained for full size cells and different current values are illustrated in Fig. 7a and 7b (tests B). A shift in the current density distribution towards the cell inlet is observed for larger current values. At the limiting current, the current density distribution no longer evolves. Agreement with the one-dimensional low voltage model (Eq. 13, 14) is poor at the limiting current (Fig. 7b) with an observed shift towards the cell outlet. The model curve was generated by first determining ν and from experimental and values (Eq. 15). Then, Eq. 10 is solved (Mathcad 2000 professional, secant, and Mueller methods) for using , , and the experimental value (assumption that the one-dimensional unit cell low voltage model applies to the data). Equation 8 is then used to compute δ using the operating conditions and values. The δ values from the three current densities of Fig. 7b are averaged to compute an average value. Equations 13, 14 are then used to compute the ratio of the local limiting current density to the average limiting current density. This last step is performed using the average value, , from Eq. 10 (experimental value, ) and from Eq. 10 (experimental value, θ varying from 0 to the average value).

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The corresponding local cell voltage values are illustrated in Fig. 8a and 8b, showing a gradual progression toward lower values at higher currents. Additionally, a larger cell voltage gradient along the cell length is developing as the current value is increased. At the limiting current, this evolution is no longer operative. The significant cell voltage gradient of at the limiting current favors in-plane current redistribution from the cell inlet to the cell outlet. This current redistribution, as previously described,^{19, 20} is responsible for the significant discrepancy existing between the model predictions and experimental current density data. An in-plane current of is computed using Ohm's law, the observed voltage gradient of , the bipolar plate resistivity, and thickness, and Mk902 active area length and width.¹⁹ This in-plane current value favorably compares to the values determined from the inlet and outlet current density over average current density ratio deviations from the model curve (respectively, and , Fig. 7b). Because the average current to each segregated element of the current distribution plate is ( elements), the observed change in current for the elements at the plate extremities are 2.6 and ( and ).

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Recent air data obtained by Kulikovsky et al. under potentiostatic conditions avoid the concern of current redistribution.²¹ For this case, the model predictions (Eq. 12) are in good agreement with the experimental data (Fig. 9), resulting in a mass transfer coefficient of ( standard deviation). This value is of the same order as values reported in the previous section. Differences can be explained in terms of operating conditions and membrane-electrode assembly construction. The model curves were generated using the same procedure as for Fig. 7b except that individual values and Eq. 12 are used for the different currents.

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Limiting current density data were also obtained under potentiostatic conditions using low oxidant concentrations (Fig. 10, tests C). Each curve follows the expected dependency on reactant concentration. However, as the reactant flow rate increases, a concurrent increase in the limiting current density is also observed. Also, the rate of current density increase decreases for larger reactant flow rates. These observations are consistent with nonuniform reactant distribution along the channel length. For very large reactant flow rates, the limiting current density is no longer expected to change, as reactant consumption would be insignificant in comparison to the reactant supply provided to the cell. Figure 10 data were replotted according to Eq. 20 (Fig. 11) using experimental , ν, and values. A plot of vs has a slope equivalent to using Eq. 16, 20. Again, a good correlation was obtained from linear regression , with a single value for the mass transfer coefficient lower than the value derived from Fig. 5 (tests A). The main reason for the discrepancy relates to the higher concentration used to compute the limiting current density because the channel effect decrease was not taken into account in Eq. 23. For a stoichiometry of 1.4, the average dimensionless reactant concentration along the channel length is . If the tests A mass transfer coefficient is corrected by this average dimensionless reactant concentration, a value of is found, resulting in a closer agreement with the tests C result. The remaining difference may relate to the higher temperature used for tests A (Table I) indicating the predominant effect of increased oxygen diffusivity on the mass transfer coefficient.

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In the previous sections, several references were made to properly select operating conditions to achieve limiting current conditions which can be easily interpreted and which are free of interactions from other sources. In this regard, cathode kinetics should be controlling. This is achieved by minimizing anode kinetics by the use of high fuel stoichiometries and minimizing membrane losses by the use of saturated reactant gases (Table I). Cell design is also important to ensure uniform conditions (other than the oxygen concentration) along the channel length. Reactant pressure drop is minimized using adequate flow-field designs. Temperature rise is controlled using either small active areas or high coolant flow rates. As for the effect of liquid water, its impact is reduced by using low oxidant concentrations leading to relatively smaller water production rates.

Some of these considerations are well illustrated considering some of the data of Williams et al. (their Table 2).²² The mass transfer coefficients for the dual-layer in-house gas diffusion layer are illustrated in Fig. 12. These were generated using the reported reactant utilization to compute ν and (Eq. 15). Then, Eq. 10 is solved for using , , and the experimental value. Equation 8 is then used to compute δ using the operating conditions (temperature, pressure, average relative humidity) and values. The most important conclusion relates to the mass transfer coefficient, which is not constant and contrasts with the present model assumption. The effect of the different operating conditions (reactant concentration, flow rate and relative humidity, cell temperature) provides the necessary clues to formulate a reasonable assumption for this behavior. Nonuniformities were introduced with the use of a subsaturated reactant gas, affecting oxygen mass transport in the ionomer. Oxygen permeability through Nafion and recast Nafion increases with temperature and relative humidity.²³ The oxygen mass transfer coefficient decreases with reactant flow rate (Fig. 12) due to the drying effect of the subsaturated reactant (average relative humidity within the cell decreases) negatively affecting ionomer permeability. This also explains the smaller mass transfer coefficient values for the 4% oxygen reactant (in comparison to air). Because the flow rate was the same for both reactants and the limiting current was lower for the 4% oxygen case (less water production within the cell), the ionomer is relatively drier. For the combined effect of increased temperature and decreased relative humidity, the trend is not so clear and indicates the conflicting effects of ionomer dryness and increased permeability. These trends were also observed for the other gas diffusion layers. Figure 13 illustrates the comparison between the experimental data for the dual-layer in-house gas diffusion layer and model predictions. The experimental data were generated using the reported limiting current density and reactant utilization (to compute ν). The model curves were generated using the computed δ values (as indicated at the beginning of this paragraph). Three δ values were selected (average, those corresponding to the minimum and maximum reactant flow rate). The corresponding equivalent current density range was then computed from the measured limiting current densities and reactant utilizations. This range in turn defines a range using Eq. 8 and appropriate operating conditions corresponding to the selected δ values. Equation 10 is then solved using experimental values to compute , , and . Figure 13 shows that the model does not fit the experimental data as already discussed due to the nonuniform operating conditions leading to variations in δ values. Operating conditions which would lead to the absence of liquid water cannot be used to further analyze this situation because they may lead to a transient or pseudo-steady-state behavior and cell failure (if both outlet reactant streams are less than saturated).²⁴

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Oxygen mass transfer limitations in the gas diffusion layer under the flow-field channel landings (serpentine channel design) were modeled, leading to analytical solutions for the local oxygen concentration.²⁵ The results were extended to the entire channel length by assuming a linear oxygen concentration profile. In view of Eq. 13, 18, valid for straight channel designs, this assumption may be viewed as inappropriate, thus defining a need to measure the limiting current density distribution for serpentine channel designs. Additionally, polarization curves do not include data in the limiting current density regime, precluding further validation of the present model.

The final requirement to ensure reliable limiting current values is a sufficiently low cell voltage. The analysis is simplified by using the inlet cell voltage, which represents the largest observed value in either galvanostatic (Fig. 8a and 8b) or potentiostatic experiments. This precaution ensures that the requirement will be met at all locations along the cell. Three methods were investigated to quantify the required potential. The first method uses the zero-dimensional cell voltage model (Eq. 23) without mass transport losses evaluated at the limiting current

The second method also uses the zero-dimensional cell voltage model but evaluated at a predetermined fraction of the limiting current

The third method relies on the Damköhler number definition (surface reaction rate/mass transport rate)

The cathode overpotential η corresponds to (Eq. 22) and with the definition (Eq. 23), Eq. 27 reduces to Eq. 21. Additionally, η corresponds to the last term on the right of Eq. 22. If this term is replaced in Eq. 27, the resulting expression is

Figure 14 illustrates the tests B data fitted to the zero-dimensional cell voltage model. The same procedure was used as for Fig. 5 except that was assumed to achieve a better fit because cell voltage data were not available at low current densities. Figure 7a and 8a show that, for inlet cell voltages between and , the current distribution changes. However, Fig. 7b and 8b show that the current distribution no longer changes below a cell voltage of . Figure 14 illustrates that method 1, which leads to a critical voltage of , is not appropriate. Method 2, which leads to cell voltages of 0.515 and for respective values of equal to 0.999 and 0.9999, is appropriate to define the critical voltage.

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The polarization data of Fig. 5 were replotted in Fig. 15 using the Damköhler number to further emphasize the limiting current behavior and its adequate representation by a different theoretical framework. A good correlation is obtained between the experimental data and Eq. 28, demonstrating the potential of using values to define the critical voltage. If the previously identified values of are substituted for in Eq. 28, values between 999 and 9999 are required to achieve the critical voltage. As expected, based on the physical meaning of the number, the 999 to 9999 range is larger than the usually accepted value of required for mass transport control. The critical cell voltage value determined here, –, is still relatively large in comparison to minimum practical cell operating voltages of –. This is fortunate because in practice load banks do not operate below a certain voltage (typically ).

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