Journal of The Electrochemical Society, 150 共4兲 A430-A438 共2003兲

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Lateral Ionic Conduction in Planar Array Fuel Cells Ryan O’Hayre,a,*,z Tibor Fabian,b Sang-Joon Lee,b,** and Fritz B. Prinza,b a

Department of Materials Science and Engineering, bDepartment of Mechanical Engineering, Stanford University, Stanford, California 94305-303, USA

A performance degradation phenomenon is observed in planar array fuel cells. This effect occurs when multiple cells sharing the same electrolyte membrane are connected in series to build voltage. The open circuit voltage 共OCV兲 and low current behavior of such a series connected planar stack is lower than should be expected. The flow of ionic cross currents between cells in the array, dubbed membrane cross-conduction, is proposed as the likely cause for this loss phenomenon. This hypothesis is confirmed by experimental observations. An equivalent circuit model for a planar double cell is developed which takes into account membrane cross conduction. This model is shown to predict the observed current-voltage behavior of an experimental planar double cell while a simple series model does not. The validated model is used to investigate the impact of various fuel cell parameters on the membrane cross-conduction effect. Design rules are extracted to minimize membrane cross-conduction losses for a linear fuel cell array. It is concluded that the membrane cross-conduction phenomenon primarily affects the OCV and low current density behavior of planar fuel cell arrays. Losses due to membrane cross conduction are minimal for conservative cell spacing, but can be significant for densely packed fuel cell arrays. © 2003 The Electrochemical Society. 关DOI: 10.1149/1.1554912兴 All rights reserved. Manuscript submitted July 8, 2002; revised manuscript received October 10, 2002. Available electronically February 28, 2003.

The bulk of fuel cell research has focused on large-scale 共⬎1 kW兲 systems; however, proton exchange membrane 共PEM兲 fuel cells are interesting candidates for portable power applications 共1 – 100 W兲 as well. This is due to their good energy conversion efficiency and the high energy density of their fuel sources.1,2 In considering fuel cells for portable application, a unique set of design criteria must be applied. This is because the requirements for largescale fuel cell systems and small-scale fuel cell systems are fundamentally different. Large-scale fuel cell systems typically employ vertical-series stack configurations in combination with peripheral devices to regulate gas flows, pressures, stack temperature, membrane hydration, etc. However, for portable fuel cell devices, operation at ambient conditions with a minimum of peripheral components is desirable.3 Like large-scale fuel cells, practical application of fuel cells in portable devices necessitates interconnection of multiple cells to meet application specific voltage requirements. However, portability concerns require that the packaging of small-scale fuel cell devices be space efficient and cost effective. The unique requirements of portable fuel cell systems have led to the formulation of several design alternatives to the conventional vertical-series fuel cell stack. These design alternatives include the banded membrane configuration,4 the flip-flop configuration,5 and the single-cell fuel cell with dc-dc converter. These alternative designs are compared along with the conventional vertical-series stack in Fig. 1. A common trait for these design alternatives is that they are planar, employing a single, continuous membrane. While the single-cell dc-dc conversion fuel cell relies on electronic transformers to build device voltage to the required level, the other two designs use multiple fuel cells arrayed on a single membrane to build voltage. For small power applications, these planar designs can have advantages in terms of power density, manufacturability, and packaging flexibility.6 However, the series combination of multiple cells on a single membrane can also lead to complications. Some of these disadvantages have been discussed in the literature before, including the potentially higher Ohmic 共IR兲 losses of planar interconnected cells, and the increased difficulty of ensuring equal reactant distribution to multiple cells in a plane.7 Recently, we have observed an additional complication in planar array fuel cells. This complication arises when multiple series connected cells sharing the same electrolytic membrane are packed in a sufficiently dense array. Because the membrane is an ion conductor and the series connected cells are at different potentials, it is possible for the cells to communicate by

* Electrochemical Society Student Member. ** Electrochemical Society Active Member. z

E-mail: [email protected]

the flow of ion crosscurrents. These parasitic currents adversely affect overall fuel cell system performance, reducing the output voltage of the series array below its expected value. For the purposes of this paper, this loss phenomenon is hereby termed membrane cross conduction. This paper focuses on the experimental evidence for the membrane cross-conduction phenomenon and then develops a simple model to describe the membrane cross-conduction phenomenon. From this model, design parameters to minimize the effects of membrane cross conduction will be extracted for planar fuel cell array geometries. Experimental 16-cell planar fuel cell array description.—Using printed circuit board 共PCB兲 technology, a 16 cell banded configuration planar fuel cell array was constructed 共see Fig. 2兲. PCB technology was employed to provide both flow routing and electrical interconnection for the fuel cell array. Gas inlets and macro flow routing were implemented on the back side of the PCB. The micro-flow-channeled electrode areas were made on the front side of the PCB by selectively etching copper, which was subsequently protected by gold plating. Series electrical interconnection between cells in the planar array was accomplished via pin interconnects at the edge of the PCB board. This electrical interconnection methodology allowed the use of a single, uninterrupted membrane shared by all 16 fuel cells. Furthermore, this scheme provided individual electrical access to every pole 共cathode and anode兲 of the 16 cells in the array. Thus, the electrical relationship between the 16 cells in the array could be changed easily and arbitrarily. This allowed quick switching between series connected and unconnected states during an experiment. The behavior of any individual cell or arbitrary group of cells could be monitored during an experiment. External voltage bias could even be applied between cells in order to arbitrarily fix their potential relative to their neighbors. All tests described in this paper were conducted using the 16 cell PCB fuel cell array described above. The active dimensions of each of the cells in the array were 5 by 20 mm, giving per cell active areas of 1 cm2 共see Fig. 2兲. The 16 cells were arrayed in two columns of eight cells. The eight cells within a column were separated from each other with 4 mm spacing. The two columns were separated from each other by 13 mm. This led to two types of nearest neighbor cell spacing: 4 mm, 共termed close neighbors兲 and 13 mm 共termed far-neighbors兲. Cell measurements.—All fuel cell measurements were conducted at room temperature using dry hydrogen and dry oxygen gas at 1atm pressure. These modest conditions were used to mimic the likely environment of a portable fuel cell device. (O2 gas was used instead

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Figure 2. Dimensions and numbering scheme for the 16 cell planar fuel cell array. Active area per cell ⫽ 0.5 cm ⫻ 2.0 cm ⫽ 1 cm2. Note that the cell active area dimensions are slightly smaller than the PCB electrode dimensions. This array geometry gives rise to two types of cell spacing. Cells 12 and 13 are examples of close neighbors separated by 4 mm; cells 7 and 15 are examples of far neighbors separated by 13 mm.

of air to avoid the additional interpretation complications imposed by oxygen mass transport limitations on the present study.兲 Gas flow rates were regulated at 10 mL/min, and measurements were acquired via a Gamry PC4/750 potentiostat system linked to a PC. Electrochemical impedance spectroscopy 共EIS兲 measurements were conducted under the same cell conditions. A 10 mV amplitude sinusoidal excitation signal was used to investigate a range of frequencies from 100 kHz to 10 mHz at various cell potentials. The anode was used as the reference electrode. Evidence for Membrane Cross Conduction The membrane cross-conduction phenomenon was first observed due to its impact on the open circuit voltage 共OCV兲 of a planar, series connected fuel cell array. As is elucidated later, the crossconduction phenomenon has its most significant impact on the open circuit and low current behavior of planar fuel cell stacks. The open cell behavior of the 16 cell planar array is summarized in Fig. 3. This figure contrasts the typical OCV values for the 16 cells when they are unconnected vs. their OCV values when con-

Figure 1. Stack designs for membrane fuel cells: 共a兲 Conventional vertical 共bipolar兲 stack, 共b兲 Single cell with dc/dc conversion, 共c兲 Planar flip-flop configuration, 共d兲 Planar banded configuration.

Figure 3. 16 cell planar array OCV behavior. When in the series connected state, the cells in the planar array show lower OCV values than when in the values electrically isolated state.

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nected in series. Without exception, the cells show lower OCV in the series connected state than when they are in the unconnected state. This voltage drop phenomenon is consistent and instantaneous. If a cell initially connected with others in series is removed from the series connection, its OCV immediately rebounds. If it is later reconnected in the series configuration, its OCV immediately declines. At open circuit conditions, no external current can flow through the cell array. The fact that the OCV values of the cells drop when they are connected in series indicates the flow of some type of internal short currents. While the individual voltage drops per cell are small, 共100 mV兲 their cumulative effects cause the OCV of the 16 cell array to be more than 2 V lower than the value predicted by the sum of the individual isolated OCV. To further understand the origin of these open cell potential drops, it was necessary to consider a system with reduced array complexity. Thus, a series of planar double cell configurations were designated for study. As mentioned earlier, the 16 cell array already had two different cell spacing geometries built into it. Therefore, double-cell configurations incorporating these two spacing geometries were isolated from the 16 cell array and studied in greater detail. Test schematics of the two types of double-cell configuration are shown for reference in Fig. 4a and b. As was mentioned earlier, the ‘‘near-neighbor’’ double cell configuration had a cell spacing of 4 mm while the ‘‘far-neighbor’’ double cell configuration featured cells separated by 13 mm. It was intended that the different cell separation distances of the two double-cell configurations would give insight into the impact of cell spacing on the cross-conduction phenomenon. In a similar fashion to the full 16 cell OCV study, the OCV behaviors of double cells were investigated thoroughly. Figure 5 highlights the typical behavior of planar double cells in the unconnected and connected states. Again, it can be seen that the total OCV of a series connected pair is lower than that of an unconnected pair. Furthermore, the OCV reduction is more severe for a closely spaced cell pair 共4 mm兲 compared to a more broadly spaced cell pair 共13 mm兲. When the two cells of a double cell pair are unconnected, the cells are at approximately the same potential. In other words, both anodes are at about the same potential, and both cathodes are also at approximately the same potential, about 1 V higher than the anodes. Thus, there are no lateral potential gradients to drive the flow of an ionic crosscurrent. 共See Fig. 6a兲 However, when the two cells are linked in series, the cathode of cell A is now electrically connected to the anode of cell B. Essentially, the cathode of cell A and the anode of cell B now form a shorted, leakage fuel cell. 共See Fig. 6b兲 A current flows through this leakage cell, driving the loss phenomenon. So long as the cathode of cell A is electrically connected to the anode of cell B, this leakage current will flow, even under external open cell conditions. 共In other words, even when no external current is flowing through the double cell device, an internal current will be flowing through the A共⫹兲/B共-兲 leakage cell.兲 This means that even in the absence of an external net current, significant net 共nonequilibrium兲 reactions are occurring at both the anode of cell B and the cathode of cell A. Almost no overpotential loss occurs at the cell B anode, due to the reversibility of the hydrogen reaction, but at the cell A cathode, a significant overpotential loss is incurred. An intriguing result of this situation is that cell A should manifest an appreciable OCV reduction whereas the OCV of cell B should be virtually unaffected. Extending the logic further, all the cells in a multiple 共greater than two兲 series array of planar fuel cells will exhibit OCV reductions except for the final cell of the series, whose OCV should be unaffected. A close inspection of Fig. 3 reveals that this conclusion is confirmed experimentally. The existence of a leakage current flowing through the A共⫹兲/B共-兲 leakage cell necessitates that an ionic cross current must be flowing diagonally 共laterally兲 across the membrane between the two cells. This ionic current through the membrane between the two cells flows in concert with the electrical current through the wire between the two cells in order to complete the circuit and maintain electroneutrality. A series of experiments were therefore developed to experimentally establish the existence of this ionic cross current.

To explore the cross-current hypothesis, a fixed potential difference was imposed on cell B 共the second cell in the double-cell configuration兲 relative to cell A 共the first cell兲. The voltage offset was accomplished by adding a series voltage boost from a power supply between the cathode of cell A and the anode of cell B. This voltage boost was used to ‘‘magnify’’ the potential difference between the neighboring cells and thus attempt to drive a larger ionic cross current. As this potential difference was ramped from 0 V to 10 V, the OCV of the cell A was monitored 共see Fig. 7兲. As the potential on cell B increased, the OCV of cell A dropped, clearly verifying the existence of some type of communication between cells. As shown by the lower two curves in Fig. 7, this effect was largely linear, indicating a resistive, IR-loss-type mechanism. Furthermore, the effect was consistently larger for the closely spaced cells compared to the more widely spaced cells. From this evidence alone however, it was not possible to determine with certainty that the communication between the cells was occurring via ionic crosscurrents in the membrane. A resistive shunt between electrodes in the PCB, or an ionically conductive water film between neighboring electrodes could also account for this seeming drop in OCV. In order to identify the membrane as the definitive medium of cell-to-cell communication, the voltage-offset experiment was repeated with a discontinuous membrane. Using the same experimental setup, the membrane between cells A and B was now cut, so that cells A and B were isolated electrolytically from one another. The voltage-offset experiment was then duplicated. The results of this experiment are shown in Fig. 7 and compared to the previous trials. As the top curve in this figure shows, the OCV behavior of cell A is no longer affected by the potential of its neighbor. In other words, severing the membrane between the two cells also severs the communication between them. Equivalent Circuit Model for Membrane Cross-conduction Model description.—The above experiments served to pinpoint lateral membrane communication as the cause of the OCV performance degradation. As noted previously, the proposed mechanism for this performance degradation is the flow of lateral ionic currents between adjacent cells in the array, forming parasitic leakage cells. The magnitude of these lateral currents will be limited by the resistance of the membrane and should scale with the lateral spacing between cells. Assuming Nafion 115 with thickness t ⫽ 125 ␮m, conductivity ␴⫽ 0.1 ⍀ ⫺1 cm-1 共a typical value for Nafion兲8 and spacing between cells ⫽ 4.0 mm, the lateral membrane resistance between cells should be on the order of 300 ⍀. For a 1 V potential difference between two series connected cells, this resistance would result in an approximate 3 mA parasitic cross current. Such a cross current is sufficiently large compared to the exchange current density, I 0 , for the oxygen reaction, to reduce OCV values by up to several tenths of a volt. An equivalent circuit model 共ECM兲 has been constructed to simulate the membrane cross-conduction phenomenon. A schematic of the model is shown in Fig. 8. This model is developed for the simple case of a series connected two-cell array 共i.e., a simple planar double cell configuration兲. The model takes into account both the steady-state dc 共the current-voltage, I-V, polarization behavior兲 and the transient 共high frequency or time-dependant impedance behavior兲 of the fuel cells. 2 Model parameter estimation.—Well-developed parameter extraction methods were used to fit the fuel cell model.9,10 The important parameters of the equivalent circuit model were fit to experimental observations using standard cell polarization and EIS measurements of the fuel cell array. The kinetics of the oxygen and hydrogen reactions were modeled by the parallel combination of a double layer capacitance (C dl) and a voltage-dependent current source 关 I(␩ act)兴. To first approximation, C dl is assumed to be constant and independent of cell potential. Springer et al.,11 show an approximately twofold variation in C dl of with cell potential, but this variation has no impact on the steady-state fuel cell behavior and

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Figure 5. Open circuit behavior of planar double cells in the unconnected and series connected states. As was the case for the full 16 cell planar array, the double cells show a lower OCV when in the series connected state compared to the electrically isolated state. Note that the OCV reduction is more severe for the closer cell spacing.

only marginal impact on the predicted transient behavior. The voltage dependent current source, I(␩ act), models the activation overpotential due to the charge-transfer kinetics of the electrochemical reaction. 共In many ECMs, this activation overvoltage is simulated by a nonlinear resistor, however in P-SPICE it makes sense to model the kinetics with a voltage-dependent current source.兲 The behavior of I(␩ act 兲 is given by the Butler-Volmer equation for charge transfer kinetics 关Eq. 共1兲兴, and is fit by two parameters, ␣ and I 0 .

共

I ⫽ I0 e

␣nF␩ RT

⫺ e⫺

(1 ⫺ ␣)nF␩ RT

兲,

关1兴

All three parameters, ␣, I 0 , and C dl , are evaluated from experimental EIS and I-V polarization measurements of the planar array fuel cells using a complex nonlinear least-squares fitting algorithm. In standard fuel cell operation mode, the sluggish kinetics of the oxygen reaction dominate the impedance response. Thus, only the kinetic parameters for the oxygen reaction are extracted, under the condition that the anode is reversible and acts as a psuedo-reference electrode. Because the hydrogen reaction is much faster than the oxygen reaction, the hydrogen kinetic parameters do not influence the polarization behavior of the cell. Therefore the hydrogen kinetic parameters are arbitrarily fixed in the ECM using values typical of a

Figure 4. Depiction of the double-cell configurations. 共a兲 Close neighbor double-cell configuration with 4 mm cell spacing. 共b兲 Far neighbor doublecell configuration with 13 mm cell spacing. The remaining electrodes on the PCB array were covered with Kapton tape to ensure full isolation of the double-cell under study.

Figure 6. A schematic of double cell intercommunication. 共a兲 Unconnected double cell: There are no lateral potential gradients to drive the flow of an ionic cross-current. 共b兲 Series connected double cell: Electrical connection of cell A cathode with cell B anode sets up an internally shorted ‘‘leakage cell’’.

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Journal of The Electrochemical Society, 150 共4兲 A430-A438 共2003兲 Table I. A comparison of typical parameter values for the planar array fuel cell ECM vs. previously reported literature values. The typical planar array fuel cell values were obtained by complex nonlinear least squares fitting of experimental EIS and I-V curve data from individual planar array cells. Fitting parameters vary by 15–30% from cell to cell within an array, which can be attributed to differences in true cell active area sizes, experimental condition fluctuations, etc. The experimentally extracted model parameters for the planar array PEM cells are compared against a range of previously reported values for PEM fuel cells operating at 60–80°C.

Figure 7. Voltage offset experiments for the planar double cells configurations. Results for a discontinuous 共broken兲 membrane between the double cells are compared against results for continuous membrane double cells: 共䊏兲 Broken membrane, 4 and 13 mm cell spacing, 共䊉兲 continuous membrane, 13 mm cell spacing, 共䉱兲 continuous membrane, 4 mm cell spacing. Severing the membrane connection between the cells severs the electrical intercommunication between the cells, indicating that the membrane is responsible for the intercommunication effect.

polymer electrolyte fuel cell12; I 0,hydrogen ⫽ 200 mA/cm2 , ␣ hydrogen ⫽ 0.5. An EIS measurement made in hydrogen pump mode while hydrogen gas was supplied to both the anode and cathode verified that the I 0 value chosen for the hydrogen reaction was appropriate. In Table I, typical fitting values for the planar array fuel cells at room temperature and pressure with dry O2 and H2 gas are compared to typical literature values for a standard PEM fuel cell under fully hydrated conditions at 80°C.13–17 The oxygen I 0 values for the planar array fuel cells are an order of magnitude smaller than those reported for typical PEM cells, but are consistent with the change in temperature from 80 to 25°C. Also shown in the chart are values for I leak . This represents the constant current loss across a fuel cell due to fuel crossover from anode to cathode. This fuel crossover term is responsible for the OCV reduction of an isolated fuel cell below its E 0 value predicted from thermodynamics. Fuel crossover is modeled in the equivalent circuit as a constant current loss across the voltage source E 0 共not shown in Fig. 8兲.

Figure 8. Equivalent circuit model for a two-cell planar array fuel cell incorporating membrane cross conduction.

I 0 Hydrogen ␣ Hydrogen C dl Hydrogen I 0 Oxygen ␣ Oxygen C dl Oxygen I leak R anode ⫽ R cathode R lateral

Typical planar array cell

Previous literature values

200 mA/cm2 0.5 1 mF/cm2 .002 mA/cm2 0.37 8 mF/cm2 0.5 mA/cm2 1.2 ⍀ cm2 300 ⍀ cm2 共4 mm spacing兲

200 mA/cm2 0.5 0.1–100 mF/cm2 0.01–0.1 mA/cm2 0.2–0.5 0.1–100 mF/cm2 0–2 mA/cm2 0.1–0.5 ⍀ cm2 N/A

The IR resistances of each cell are lumped into two symmetric resistances, called R cathode and R anode . From a measurement standpoint, it is often difficult to separate the origin of the various IR cell resistances, or to determine which to ascribe to the cathode and which to ascribe to the anode. Thus, all IR resistances were lumped together and divided symmetrically between the anode and cathode. Because these are simple IR resistors, their position in the circuitmodel is somewhat unimportant. They have both been placed inside the membrane portion of the fuel cell, but as ‘‘lumped’’ resistances, they truly represent the IR resistance contributions from several discrete parts of the system, including the membrane resistance, the cathode and anode electrical resistances, and the IR resistance of the cell interconnects. The lateral IR resistance across the membrane, R lateral , provides a short-circuit path between the cells. This resistor represents the membrane cross conduction. Like the other model parameters, the values for these IR resistances were obtained from a complex nonlinear least-squares fit of the experimental EIS measurements of the planar array fuel cells. The internal IR cell resistance was simply determined from the high frequency real axis intercept of the EIS spectrum. Values for lateral resistance were determined by EIS measurements between neighboring cells. The lateral membrane resistance measurements were taken between neighboring cathodes, between neighboring anodes, and also from the cathode of one cell to the anode of another cell. All produced roughly equivalent lateral resistance values. The IR resistance values obtained for the planar array fuel cells are compared to typical literature values for a standard PEM fuel cell under fully hydrated conditions at 80°C in Table I. It is noteworthy that the cell resistance values for the planar array fuel cells are considerably larger than those reported for typical PEM cells. This is partly explained by the higher membrane resistance at 25°C under nonhumidified conditions, but is mostly attributed to the high contact resistances between the cells and the PCB electrode contacts. Model validation.—To validate the model, the close neighbor double-cell from previous experiments was revisited. Using the fitting procedure described in the previous section, model parameters were extracted for both cells of the close neighbor double cell. This was accomplished by taking EIS measurements of the two cells separately, each as single cells in the unconnected state. The lateral membrane resistance between the two cells was also measured. These parameters, summarized in Table II, were input to the circuit model, which was evaluated in P-Spice. The circuit model was used

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Table II. Parameter values used for model validation on an experimental close-neighbor double cell. Parameter values were obtained for both cell A and cell B of the double cell by complex nonlinear least-squares fitting of experimental EIS and I-V curve data from the two cells. These parameter values were input into the equivalent circuit model described in Fig. 8 to generate the simulation results shown in Fig. 9. These simulation results closely mirror the experimentally observed behavior of the double cell. Cell A I 0 Hydrogen ␣ Hydrogen C dl Hydrogen I 0 Oxygen ␣ Oxygen C dl Oxygen I leak R anode ⫽ R cathode R lateral

Cell B 2

200 mA/cm 200 mA/cm2 0.5 0.5 0.6024 mF/cm2 1.45 mF/cm2 0.002 mA/cm2 0.0022 mA/cm2 0.38 0.37 8.1 mF/cm2 7.3 mF/cm2 0.4 mA/cm2 0.4 mA/cm2 2 2.1 ⍀ cm 1.3 ⍀ cm2 340 ⍀ cm2

to predict the behavior of the two single cells in isolation and then the behavior of the series connected double cell, taking into account membrane cross-conduction. The true IV behavior of cells A and B, as well as the series connected double cell were then measured experimentally and compared to the model’s prediction. The results, shown in Fig. 9a and b, reveal good agreement between the experiment and the model. As can be seen, the model accurately predicts both the isolated single-cell behavior of the two cells and the performance degradation due to membrane cross conduction when the cells are connected in series. The membrane cross conduction model is compared against a simple series summation model that does not take into account membrane cross conduction. The membrane crossconduction model correctly predicts the observed behavior, while the simple series connection model does not. Model Results and Discussion Scenario modeling.—We now investigate a series of scenarios that explore the impact of various parameters on membrane cross conduction losses, particularly the effect on membrane crossconduction by changes in 共i兲 Lateral membrane resistance 共e.g., via cell spacing changes兲, 共ii兲 Internal cell resistance, and 共iii兲 Activation kinetics. All scenarios begin with the same idealized ‘‘base-case’’ double cell array. The model parameters used for this base case system are summarized in Table III. In order to make these scenarios as generally applicable as possible without incorporating the peculiarities, asymmetries, or variations of a particular experimental fuel cell array, the parameters chosen in Table III are based on averaged literature values for a PEMFC.17 Furthermore, the characteristics of both cells in the double cell are assumed identical. In each scenario, generally one of the model parameters will be varied while the others are held constant in order to elucidate its role in membrane cross-conduction. The effect of lateral membrane resistance.—The lateral membrane resistance can vary greatly depending on engineering parameters such as the thickness of the electrolyte membrane and the spacing between cells. Lateral resistance can also vary with operating conditions such as cell temperature and membrane hydration. In Fig. 10, all the base-case model parameters from Table III are fixed except for R lateral , which is varied between 5 ⍀ and 100 M⍀. (R lateral ⫽ 100 M⍀ effectively corresponds to the case where there is no cross conduction.兲 As can be seen from the figure, the impact of membrane cross conduction on double-cell polarization behavior is affected dramatically by the magnitude of the lateral resistance between the cells.

Figure 9. Comparison of the experimental and model results for a close neighbor double cell. 共a兲 Single-cell experiment vs. model comparison for cells A and B in the unconnected state: 共䉱兲 Cell A, 共䊏兲 Cell B. 共b兲 Experiment vs. model comparison for the series connected double cell. The low current density region is magnified for clarity. 共⽧兲 Experiment, 共䉭兲 Crossconduction model, 共*兲 Simple summation model. The membrane crossconduction model correctly predicts the observed behavior while the simple series connection model does not.

The effect of internal cell resistance.—As can be seen in Fig. 11, for a constant lateral membrane resistance, an improvement in the internal IR resistance of the cells slightly lessens the impact of membrane cross conduction. At the same time, of course, lowering cell resistance produces additional benefit by improving fuel cell performance over the entire range of cell potentials. Thus this strategy pays double rewards. In Fig. 11, all the base-case model parameters from Table III are fixed except for R cathode/anode , which is varied between .05 and 5 ⍀.

Table III. Idealized parameter values for a PEM planar array double cell. These parameter values represent average or ‘‘normal’’ values for a state of the art PEM fuel cell. These values were input into the ECM and used in the scenario modeling investigations shown in Fig. 10–12.

I 0 Hydrogen ␣ Hydrogen C dl Hydrogen I 0 Oxygen ␣ Oxygen C dl Oxygen I leak R anode ⫽ R cathode R lateral

Cell A

Cell B

200 mA/cm2 0.5 1 mF/cm2 0.1 mA/cm2 0.25 1 mF/cm2 2 mA/cm2 0.15 ⍀ cm2

200 mA/cm2 0.5 1 mF/cm2 0.1 mA/cm2 0.25 1 mF/cm2 2 mA/cm2 0.15 ⍀ cm2

300 ⍀ cm2

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Figure 10. Effect of lateral membrane resistance. Model simulation showing the effect of lateral membrane resistance on double-cell performance. The base cases given by the values in Table III 共with and without cross conduction兲 are shown by the dark curves. All other curves were obtained by holding all the base conditions constant except for R lateral , which was varied between 5 ⍀ and 100 M⍀.

The effect of cell activation kinetics.—In Fig. 12, all the basecase model parameters from Table III are fixed except for I 0 for oxygen, which is varied between 0.001 mA/cm2 and 1 mA/cm2 . As can be seen, improvements to the oxygen kinetics improve overall cell performance. However, membrane cross conduction is seen to affect all cells approximately the same, regardless of the value of I 0,02 . Variations in ␣, not displayed, show a similar trend. Design criteria for a linear fuel cell array.—As the above scenarios illustrate, the lateral membrane resistance between cells is the single greatest factor determining the magnitude of loss suffered due to membrane cross conduction. Fortunately, from an engineering sense, it is also the easiest parameter to control. Lateral membrane resistance depends strongly on cell spacing and membrane thickness. Lateral resistance will be high for thin membranes and for widely spaced cells. Thus, cells should be spaced as far apart from one another as possible. Due to power density concerns, however, it is desirable to achieve as dense an array of fuel cells on a membrane as possible. Therefore, a minimum cell spacing criteria can be speci-

Figure 12. Effect of oxygen kinetics. Model simulation showing the effect of oxygen kinetics on double-cell performance. The base cases given by the values in Table III 共with and without cross conduction兲 are shown by the dark curves. All other curves, 共solid line兲 with cross conduction, 共dashed line兲 without cross-conduction were obtained by holding the base conditions constant except for I 0,02 which was varied between 0.001 mA/cm2 and 1 mA/ cm2.

fied which reduces the effects of membrane cross conduction to an acceptable level while still allowing reasonable power densities. It is instructive to consider the pseudo-two-dimensional case of a linear fuel cell array. The important dimensions in such a system are schematized in Fig. 13. By a consideration of these dimensions, design criteria for cell spacing in this pseudo two-dimensional case can be extracted. It is assumed that the lateral resistance between cells in this linear array scales with the conductive path length between the cells. For a first approximation, this is given by Eq. 共2兲 ␳S tL 2

关2兴

␳S R lateral

关3兴

R lateral ⫽ From which tL 2 ⫽

Referring back to Fig. 10, for the case of a typical PEM fuel cell, membrane cross conduction causes approximately a 5% decrease in OCV for R lateral ⫽ 80 ⍀. If this is taken as a maximum tolerable OCV degradation limit, a linear design criterion can then be established between the cell spacing, S, and the thickness-edge length product, tL 2 . This relationship is charted in Fig. 14 for the design limit of 5% OCV degradation, using ␳ Nafion ⫽ 10 ⍀ cm. The results for several other choices of design limit criteria are also shown. This

Figure 11. Effect of internal cell resistance. Model simulation showing the effect of internal cell resistance on double-cell performance. The base cases given by the values in Table III 共with and without cross conduction兲 are shown by the dark curves. All other curves were obtained by holding base conditions constant except for R cathode and R anode , which were varied between 0.05 ⍀ and 5 ⍀.

Figure 13. Schematic of the linear planar fuel cell array geometry used to extract cell-spacing design rules. The black rectangles represent the individual cell active areas, which are arranged in a linear array of arbitrary number. The important dimensions relevant to this geometry are labeled on the diagram. 共S兲 cell spacing, (L1) cell edge length 1, (L2) cell edge length 2, 共t兲 membrane thickness.

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Figure 14. Calculated design rules for the linear fuel cell array geometry described in Fig. 13. There is a linear relationship between the minimum allowable cell spacing, S, and the thickness/edge length product of the cells in the array, tL 2 . This limits practically achievable power densities for planar array fuel cells. Several curves are shown for different choices of the allowable performance degradation due to membrane cross conduction.

analysis assumes internal cell resistance to be independent of membrane thickness, but in reality, the internal cell resistance 共on a perarea basis兲 will scale with thickness as R ohmic ⫽ C ⫹ ␳ * t, where C is a constant resistance due to contact resistances, cell interconnects, and electrodes. Thus, the above analysis is conservative. As was shown previously, the reduction in the internal cell resistance with decreasing membrane thickness will actually further lessen the impact of membrane cross-conduction. The analysis becomes considerably more complicated when fully two-dimensional arrays are considered. In the case of a twodimensional array, cells may have several neighbors among which crosscurrents can flow. Furthermore, these nearest neighbors may be at vastly different potentials 共greater than 1 V兲, further exacerbating the voltage losses. Conclusions Ionic cross-conduction in planar array PEMFC stacks has been experimentally demonstrated and a simple equivalent circuit model has been developed to simulate its effect on planar fuel cell arrays. It was shown both experimentally and through the model that membrane cross-conduction causes degradation in fuel cell array performance. Model scenarios showed that changing certain fuel cell properties can significantly change the magnitude of membrane crossconduction loss. In general, anything that reduces the size of the ionic cross currents flowing between cells will reduce membrane cross-conduction losses. Thus, changing the kinetics of the fuel cell reaction has little impact on membrane cross-conduction losses. 共Because the kinetics do not significantly impact the magnitude of the ionic cross currents between cells.兲 However, increasing the spacing between cells, increasing the membrane resistance between cells, and minimizing the potential 共voltage兲 steps between cells all significantly reduce membrane cross-conduction losses. Since loss minimization is desirable, the model findings suggest certain design directions for planar fuel cell arrays. Linear-strip geometries ensure that there are no cell-to-cell potential steps greater than 1V. Thus they may be a good option, although they may elicit power density concerns due to high perimeter to area ratios. Reducing membrane thickness is greatly desired, as it reduces intracell resistance while simultaneously increasing intercell resistance, producing a double benefit. It is also beneficial to consider ways of increasing the membrane resistance between cells to eliminate crossconduction losses. As was shown in the voltage-offset experiment, one option is to simply cut the membrane between each cell in the

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array. However, gas sealing now becomes complex and the benefits of a planar array stack are lost. Another possibility is to locally modify the membrane to destroy the lateral conductivity between cells. Each cell is then electrically isolated, just as if the membrane had been cut, but the single planar membrane continuity is preserved for gas sealing purposes. Design criteria were extracted for linear fuel cell arrays to keep membrane cross-conduction losses below acceptable limits. In general, overly ambitious cell array densities 共cell-to-cell spacing ⬍2 mm兲 combined with highly resistive cells (R cathode/anode ⬎ 1 ⍀兲 can lead to large membrane cross-conduction losses. 共⬎10% OCV reduction, ⬎5% power losses at 200 mA/cm2 兲. In contrast, a conservative cell spacing 共⬎5 mm兲 combined with low internal cell resistance (R cathode/anode ⫽ 0.1 ⍀兲 will reduce the effect of membrane cross conduction to a minimum with virtually no performance impact above 50 mA/cm2 . 共⬍2% OCV reduction, ⬍1% power loss at 50 mA/cm2 兲 While it is relatively straightforward to manage membrane crossconduction losses in arrays with a small number of cells, designers should be highly aware of the membrane cross-conduction issue when developing larger fuel cell arrays. Of particular importance is a consideration of the complications imposed by fully twodimensional planar fuel cell arrays. The present study only considered the psuedo-two-dimensional cases of planar double cells and linear fuel cell arrays. Future modeling efforts should focus on fully two-dimensional array situations. In such arrays, the possibility exists for neighboring cells to have far larger than 1 V potential differences between them. In such cases, the magnitude of the ionic cross currents flowing between cells is expected to increase. The possibilities for larger potential difference between cells and the possibilities for multiple nearest neighbors make the modeling more complicated. However, such modeling is warranted because the effect of membrane cross-conduction in fully two-dimensional arrays is expected to be considerably greater than in linear arrays. A circuit modeling package was used for all equivalent circuit modeling simulations, and was found to offer many benefits for fuel cell modeling. These benefits included quick turnaround times, multiple parametric study options, and full simulation capabilities in the steady state, time, and frequency domains. Equivalent circuit modeling should also be amenable to the simulation of multiple-cell fuel cell arrays where membrane intercommunication between cells can be represented by fully developed networks of lateral resistors. Such multicell simulations may also be approached with finite element analysis modeling. Such modeling efforts offer compelling avenues for further study. Acknowledgments The authors would like to acknowledge financial support of this work by Honda Research and Development Co., Ltd. This material is based upon work supported under a Stanford Graduate Fellowship. Stanford University assisted in meeting the publication costs of this article. References 1. S. Srinivasan, J. Electrochem. Soc. 136, 41C 共1989兲. 2. C. K. Dyer, J. Power Sources 106, 31 共2002兲. 3. A. Heinzel, C. Hebling, M. Muller, M. Zedda, and C. Muller, J. Power Sources 105, 250 共2002兲. 4. A. Heinzel, R. Nolte, K. Ledjeff-Hey, and M. Zedda, Electrochim. Acta 43, 3817 共1998兲. 5. S. J. Lee, S. W. Cha, Y. C. Liu, R. O’Hayre, and F. B. Prinz, in Micro Power Sources, K. Zaghib and S. Surampudi, Editors, PV 2000-3, The Electrochemical Society Proceeding Series, Pennington, NJ 共2000兲. 6. R. Nolte, A Kolbe, and K. Ledjeff-Hey, Proceedings of the 37th International Power Sources Conference, Cherry Hill, NJ, June 17–20, 1996, IEEE, p. 77 共1996兲. 7. A. Heinzel, C. Hebling, M. Muller, M. Zedda, and C. Muller, J. Power Sources 105, 250 共2002兲. 8. J. H. Hirschenhofer, D. B. Stauffer, R. R. Engleman, Fuel Cells, A Handbook, U.S. Department of Energy, Gilbert/Commonwealth, Inc. 共1994兲.

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14. A. Parthasarathy, S. Srinivasan, and A. J. Appleby, J. Electroanal. Chem. 339, 101 共1992兲. 15. T. E. Springer, T. A. Zawodzinski, M. S. Wilson, and S. Gottesfeld, J. Electrochem. Soc. 143, 587 共1996兲. 16. E. Antolini, L. Giorgi, A. Pozio, and E. Passalacqua, J. Power Sources 77, 136 共1999兲. 17. J. Larminie and A. Dicks, Fuel Cell Systems Explained, John Wiley and Sons, England, p. 53 共2000兲.