Energy Fuels 2010, 24, 1882–1888 Published on Web 02/19/2010
: DOI:10.1021/ef901519f
Simulation of the Proton-Exchange Membrane (PEM) Fuel Cell Life-Cycle Performance with Data-Driven Parameter Estimation T.-W. Lee,*,† A. A. Tseng,† K.-S. Bae,† and Y. H. Do‡ † Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, Arizona 85287-6106, and ‡Department of Mathematics, Kyungbuk National University, Daegu 702-701, Korea
Received December 11, 2009. Revised Manuscript Received January 25, 2010
Degradation parameters are included in a computational model to simulate the fuel cell output voltage as a function of time, with these degradation parameters optimized using experimental data. The contributions of the loss of catalytic activity and membrane/electrode conductivity to the fuel cell output voltage are represented through the degradation parameters. This is achieved without having to include detailed electrochemical mechanisms or their rate data but, instead, using the data on global fuel cell voltage as a function of time. Using parameter estimation based on output data, complex degradation effects are summed into a small number of parametric equations, so that the overall behavior can be reproduced. The simulation results are in excellent agreement with the data and also point to the main mechanisms of degradation. Also, it is shown that at least two parameters are needed, one for the loss of catalytic activity and another one for the ohmic losses, to faithfully simulate the loss of fuel cell performance across different current densities over time. This approach can be extended to predict the lifetime performance of fuel cells in general, if there is a minimum amount of data to apply the parameter estimation algorithm for the determination of a small number of degradation parameters.
From the operational standpoints, it is of interest to be able to determine the degradation characteristics, so that the fuel cell performance and reliability can be optimized or at least predicted, so that overhaul can be scheduled and requisitioned in a systematic manner. For this reason, a large amount of work exists in modeling both the steady state and degradation of fuel cell processes in proton-exchange membrane (PEM)1-6 and solid oxide fuel cells (SOFCs).6-9 Detailed mechanical and chemical modeling of the degradation mechanisms may be the most accurate and rigorous way, but this approach is often quite involved and time-consuming to execute. In addition, there will likely be a need for extensive data to validate the detailed mechanisms for the large variety of the fuel cell types and geometries that exist. For example, some very detailed models of SOFC individual cells, stacks, or the entire system can be found in refs 7-9. In work by Gazzari and Kessler,7 the degradation in the electrode contact and corresponding increase in the cell resistance are modeled to produce the impedance spectrum. One of the aims of such studies is to establish a diagnostic method to monitor the cell degradation and not so much to predict the lifetime performance of the fuel cell. In the work by Virkar,8 the effect of the cell voltage drop
Introduction Fuel cell processes can be used to generate electrical energy in an efficient and clean manner and, thus, are a strong candidate to augment the conventional power generation methods. However, one of the limiting factors is the performance and reliability over the fuel cell life cycle, which involves the degradation of the fuel cell components. There are many issues regarding the deterioration of the overall fuel cell system performance that include thermal damage, deactivation of the catalyst, and material and contamination issues. Here, we focus on the degradation of the fuel cell core involving the loss of the catalytic activity and increase in the membrane resistance. The above factors are manifest through a decrease in the cell voltage as a function of time, typically measured over thousands of hours over some portion of the expected lifetime of the fuel cell. There are several suspected mechanisms that lead to the above degradations. For the loss of the catalytic activity, for example, carbon support for the platinum catalyst particles can undergo oxidation under the presence of air, which results in platinum particle aggregation and, therefore, the loss in reactive surface area. The platinum itself can be dissolved in the medium, again causing a reduction in the catalytic reactivity. This dissolution process contributes to the “Oswald ripening”, where the dissolved substance migrates and agglomerates on the larger crystals, accelerated by the cycling of the loads on the fuel cell. A loss in the fuel cell voltage also arises from the increase in the ohmic loss or the overall resistivity of the fuel cell. Even small levels of contamination or corrosion of the electrodes and membranes can lead to an appreciable increase in the ohmic loss.
(1) Amphlett, J. C.; Baumert, R. M.; Mann, R. F.; Peppley, B. A.; Roberge, P. R.; Rodriques, A. J. Power Sources 1994, 49, 349–356. (2) Fowler, M. W.; Mann, R. F.; Amphlett, J. C.; Peppley, B. A.; Roberge, P. R. J. Power Sources 2002, 106, 274–283. (3) Mann, R. F.; Amphlett, J. C.; Hooper, M. A. I.; Jensen, H. M.; Peppley, B. A.; Roberge, P. R. J. Power Sources 2000, 86, 173–180. (4) Suwanwarangkul, R.; Croiset, E.; Pritzker, M. D.; Fowler, M. W.; Douglas, P. L.; Entchev, E. J. Power Sources 2006, 154, 74–85. (5) Shao, Y.; Yin., G.; Gao, Y. J. Power Sources 2007, 171, 558–566. (6) Yu, X.; Ye, S. J. Power Sources 2007, 172, 145–154. (7) Gazzarri, J. I.; Kesler, O. J. Power Sources 2007, 167, 100–110. (8) Virkar, A. V. J. Power Sources 2007, 172, 713–724. (9) Gemmen, R. S.; Johnson, C. D. J. Power Sources 2006, 159, 646–655.
*To whom correspondence should be addressed. Telephone: 480-9657989. Fax: 480-965-1384. E-mail:
[email protected]. r 2010 American Chemical Society
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Energy Fuels 2010, 24, 1882–1888
: DOI:10.1021/ef901519f
Lee et al.
on the stack performance is considered through an electrical circuit analysis. System-level considerations of the SOFC stack are made by Gemmen and Johnson.9 For PEM fuel cells, the work by Mann and co-workers1-4 involves the thermodynamic phenomenological model of the fuel cell operation and degradation is embedded in the model. More detailed aspects of the degradation mechanisms in PEM fuel cells are being examined (for examples, see refs 5 and 6), and such details can be incorporated in degradation models using the data as they become available. Full considerations of the detailed mechanisms and kinetics of the degradation can obviously become very complex.5,6 For example, full three-dimensional models of PEM fuel cells exist,10 and in principle, detailed spatial and temporal variations in the degradation kinetics can be embedded in such models, again at the expense of complexity and data requirements for validations. On the other hand, oversimplification in the modeling of physical processes included fuel cell degradation can limit the applicability of the results and also hinder the understanding of the underlying physical mechanisms. It is a matter of trade-off between practicality and complexity, often dictated by the availability of the data to rigorously test the model approach. In this work, we take a global modeling approach, where the degradation mechanisms are incorporated through a small number of estimatable parameters. The estimation of the model parameters in turn is based on long-term experimental data.11 A baseline fuel cell model used by Mann and co-workers1-4 is first setup to simulate the PEM fuel cell performance, and within this model some of the operating parameters are allowed to vary over time to match the observed fuel cell voltage. The degradation parameters are systematically optimized using a generic parameter estimation algorithm driven by the experimental data. The model framework of Mann and co-workers1-4 is quite useful in this regard, because it incorporates many of the relevant thermal and electrochemical parameters and timedependent degradation parameters can be easily embedded and optimized within the model. This approach can also be used to predict the lifetime performance of other types of fuel cells. In such applications, the baseline model will have to be constructed to reflect the electrochemical and physical processes for a given type of fuel cell and also a minimum amount of data are needed to apply the parameter estimation algorithm for the determination of the simulation parameters. The current approach of parameter estimation using global data circumvents the requirements for detailed electrochemical steps and corresponding degradation rate parameters and yet is able to compute the time-dependent degradation of the fuel cell output voltage quite correctly. Further data can then be used to add and refine the degradation mechanisms to be applicable over a wider range of conditions and materials, as needed.
electrochemical parameters can be embedded and optimized against experimental data. There are many different methods for estimating model parameters, starting from simple regression fits to a maximum entropy method. As noted in the Introduction, because of the complexity of most energy systems, it is just quite difficult to fully simulate the entire details of the processes because that means a large number of model parameters need to determined or estimated. Because the baseline model of the fuel cell as formulated by Mann and co-workers1-4 is essentially algebraic, a Gauss-Newton method for parameter estimation can be applied. We use the relatively recent experimental data on the degradation of output voltage in PEM fuel cells.11 Thus, some of the “internal” parameters are estimated on the basis of the comparison of the model output for an “external” variable, fuel cell voltage, with experimentally observed data. The utilities of this kind of systematic parameter estimation are 2-fold: first, it allows for the optimization of the parameter values based on the experimental data, so that the model is able to reproduce output variables that closely match the real system, and second, perhaps more importantly, it illustrates how the model parameters vary as a function of time, so that some insights into the origin of the degradation processes may be gained. The details of the basic fuel cell model are best described in the aforementioned references by Mann and co-workers,1-4 and we herein give only the outline of the model to illustrate our approach of incorporating and systematically estimating the degradation process. The baseline model2 calculates the fuel cell output voltage based on the Nernst voltage (VNernst) and overpotential terms (ηact,a, ηact,c, ηohmic, and ηconc). VC ¼ VNernst þ ηact, a þ ηact, c þ ηohmic þ ηconc
ð1Þ
The Nernst equation for VNernst is written for the generic hydrogen-oxygen reactions using the following entropy change parameters: VNernst ¼ 1:229 -ð8:5 10-4 ÞðT -298:15Þ 1 þ ð4:308 10-5 Þ ln pH2 þ pO2 2
ð2Þ
* and p* are the partial pressures of hydrogen and Here, pH O2 2 oxygen in atmospheres at the anode and cathode interface, respectively, and T is the temperature in kelvin. The overpotential terms in eq 1 are the activation loss terms at the anode and cathode and the ohmic loss term that represents the resistance effect of the proton conductivity in the electrolyte and other internal components. The activation loss terms at the anode and cathode are combined, for simplicity, and the following model equation is used:
ηact ¼ ξ0 þ k1 ξ1 T þ k2 ξ2 T lnðcO2 Þ þ k3 ξ3 T lnðiÞ
ð3Þ
Here, the terms c*O2 and i are the oxygen concentration at the cathode in mol/cm3 and the current density in A/m2, respectively. The initial values of the parameters in the above expression are as follows1,2,5 by setting k1 = k2 = k3 = 1.0:
Computational Methods
ξ0 ¼ -0:9514
The baseline model of the PEM fuel cell is based on the work by Mann and co-workers,1-4 while a parameter estimation algorithm developed in this laboratory is used to determine the degradation parameters to calculate the fuel cell output voltage over a long period of time. As noted above, this model framework provides a sound basis, where many of the relevant thermal and
k1 ξ1 ¼ 0:00312 k2 ξ2 ¼ 7:4 10-5 k3 ξ3 ¼ -0:000187
ð4Þ
These initial values are somewhat arbitrary in the sense that they are, strictly speaking, applicable to a different type of PEM fuel cell. However, the parameter estimation algorithm will start from these initial values but will zero in on the final, optimized values in any event through the multipliers, k1, k2, and k3, based on the
(10) Hu, G.; Fan, J. Energy Fuels 2006, 20 (2), 738–747. (11) Cleghorn, S. J. C.; Mayfield, D. K.; Moore, M. A.; Moore, J. C.; Rusch, G.; Sherman, T. W.; Sisofo, N. T.; Beuscher, U. J. Power Sources 2006, 158, 446–454.
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any loss in proton conductivity across the membrane: " 2 2:5 # i T i þ 0:062 181:6 1 þ 0:03 A 303 A FM ¼ " # " # i T -303 k4 -0:634 -3 exp 4:18 A T
experimental data, as described below. Physically, the first parameter, ξ0, describes the effects of the free energy of activation at the anode and cathode. The second parameter, k1ξ1, represents rate constants for the anode and cathode reactions. k2ξ2 and k3ξ3 are constants that describe the change in activation loss as a function of the oxygen concentration and current, respectively. The form of eq 3 and the above set of parameters arise from analyses of the overvoltages at the anode and cathode,12 where thermodynamic considerations of the respective chemical reaction rates lead to terms that are constant, linearly dependent upon the temperature, linearly dependent upon the temperature and oxygen concentration, and linearly dependent upon the temperature and logarithm of the current. Again, although the thermodynamic parameters vary from one fuel cell type to another, the parameter values in eq 4 are used only as initial guesses in the current parameter optimization scheme and optimal values are found by matching the results with experimental data. Any time-dependent degradation effects will affect the fuel cell output voltage, and this will be reflected in the model as changes in the above parameters and also in the parameter for the ohmic loss term below. Thus, time-dependent modification factors (k1, k2, and k3) need to be embedded for the above model parameters and optimized using experimental data. For example, the number of active sites in the carbon-support catalyst directly affects the conversion rate of hydrogen ions and oxygen to water molecule. Amorphous carbon in the support material can be lost because of oxidation by abundant oxygen molecules, in which case the Pt particles will tend to aggregate, resulting in surface area loss and, therefore, reduced active sites. This effect is incorporated into the parameter, ξ1, by writing, k1(t)ξ1, where k1(t) is the time-dependent factor to describe the decrease in the cathode reaction rate. Other loss terms are similarly written as k2(t)ξ2 and k3(t)ξ3. However, as will be seen later, the cathode reaction rate change as a function of time is much more significant than other loss terms. The concentration overvoltage is a function of the current density and written using the standard Stefan-Maxwell equation.13 i ð5Þ ηconc ¼ BT log 1 imax
ð7Þ
The empirical expression models the membrane resistivity at zero current and 30 °C, with the temperature and current effects added as multiplicative factors as shown in eq 7. Thus, the degradation parameter, k4, attempts to model the change in the resistivity at zero current and 30 °C, while assuming that the current and temperature modifiers will remain the same in time. In the above algorithm, all of the model equations are algebraic and we have designated four time-dependent model parameters, k1-k4. We can use a vector notation of these model parameters. 2 3 k1 6 k2 7 7 k ð8Þ B ¼6 4k 5 3 k4 It is now assumed that the experimental data for the output parameter, fuel cell voltage, are available also to be put in a vector form. y BE ¼ ½yE1 ; yE2 ; :::; yEm ¼ experimental data for the fuel cell voltage
ð9aÞ y BC ¼ f ðt; kBÞ ¼ fuel cell voltage computed from the model ð9bÞ
Then, we can set up an objective function to minimize the difference between the computed and experimental fuel cell voltage. T Sðk BÞ ¼ ½y BE -f ðt, k BÞ Q B½y BE -f ðt, kBÞ
ð10Þ
Q is a unit matrix. Now, the optimization can be performed in a number of ways, including the Gauss-Newton method, where the gradient vectors are written as follows: !T ∂f T G , i ¼ 1, 2, 3, and 4 ð11Þ B ¼ ∂ki
The parameter B is initially taken to be 0.000 053, and imax=2.5.13 Because the degradation losses occur mostly because of reduced reaction rates at the anode and cathode and increased resistance (ohmic loss), the concentration overvoltage is assumed to be constant over time. Finally, the ohmic loss terms are written in terms of the electronic and proton resistance terms, where the resistance to the electron transfer at the carbon collector contributes to the former and the resistance to proton transfer in the polymer membrane contributes to the latter F L ηohmic ≈ - iRproton ¼ -i M ð6Þ A
Then, the increment in the parameter vector, ΔkB, can be found from a matrix equation A BΔ k B ¼b B
ð12Þ
, T
T
where A BG B and b B ¼G B Q B½y BE -f ðt, kBÞ B ¼G B Q
ð13Þ
The parameter vector is updated until a sufficiently close agreement between the data and computed fuel cell voltage is achieved, using
,where FM is the membrane-specific resistivity for the hydrated proton transport in ohms centimeters, L is the thickness of the membrane in centimeters, and A is the membrane surface area in centimeters squared. The degradation in other components in the fuel cell, such as the bipolar plates or the gas diffusion layer, is not separately considered in eq 6. The above component performance can also affect the fuel cell performance over time and, in principle, be embedded in the current parameter estimation algorithm. The membrane-specific resistivity in turn is written as follows, where k4 is again kept as a time-varying parameter to account for
ð jþ1Þ
k B
ð jÞ
¼ kB
þ μΔk B
ð14Þ
where μ is a stepping parameter between 0 and 1. Once the parameters have thus converged, the model can be run with the set parameter values under various operation conditions for simulation of the fuel cell output voltage as a function of time, as discussed below. This approach can, in principle, be generalized to linear, nonlinear, and rate-dependent systems and generates optimized model parameters in a systematic manner. Although we only narrow the parameters down to two because of redundancies in the model parameters, a larger set can be handled in most cases other than excessive computing time expended for convergence. One unique advantage of this method is that instead of relying on determination of the parameter values from the first principles, which may involve many assumptions
(12) Berger, C. Handbook of Fuel Cell Technology; Prentice Hall: New York, 1968. (13) Amphlett, J. C.; Baumert, R. M.; Harris, T. J.; Mann, R. F.; Peppley, B. A.; Roberge, P. R. J. Electrochem. Soc. 1995, 142, 1.
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Figure 1. Effect of the degradation parameter k1.
Figure 3. Effect of the degradation parameter k3.
Figure 2. Effect of the degradation parameter k2.
Figure 4. Effect of the degradation parameter k4.
leading to inaccuracies and estimates, the parameters are optimized systematically using available experimental data. Additional parameters or operating conditions can then be added, as long as data are available to optimize the new parameters or processes.
into parameter k4. Indeed, using only two parameters k1 and k4, the shape and magnitude of the polarization curves can be adjusted to match the experimental data quite closely. There may be competing effects among the four parameters initially selected. However, because we optimize the parameters based on only the observed polarization curve, the data concerning the internal processes are not available and we can only discern the effects of the above two parameters k1 and k4. Thus, within the current scope of the parameter estimation, it is true that the effects of k1 and k2 or k3 and k4, respectively, are indiscernible, again because the data for the specific processes governed by these parameters are absent and very hard to measure in any event. For example, Figures 3 and 4 show that the slopes in the polarization curves are affected by these two parameters. Thus, the effects of the membrane resistance (k4) and the current-dependent activation loss (k3), in principle, cannot be separated when only the polarization curve data are available. To “drive” the parameter estimation algorithm, we need experimental data for the fuel cell voltage as a function of time and as a function of as many control variables as possible, e.g., current density, temperature, etc. The data by Cleghorn et al.11 is used for this purpose. A 25 cm2 single-cell polymer electrolyte hardware (Fuel Cell Technologies, Albuquerque, NM) was tested over a time duration of some 26 334 h, and the fuel cell voltage as a function of the current density in mA/cm2
Results and Discussion Figures 1-4 show the changes in the polarization curve (fuel cell voltage versus current density) as the model parameters k1-k4 are varied. The parameters k1 and k2 are the linear multiplicative terms to the activation overpotential terms in eq 3, and thus, a decrease in these parameters should reduce the fuel cell voltage, as shown in Figures 1 and 2. Both parameters result in a uniform decrease in the fuel cell voltage over the entire range of the current density. Because of the fact that only the current density is varied in the experimental data, it is not possible to discern the effects of parameters k1 and k2 and we select k1 as the parameter that reflect the decrease in the catalytic reaction rates as well as the effects of the oxygen concentration. To identify the exact contribution of the latter effect, we would need a data set that measures, for example, the fuel cell voltage as a function of the temperature and oxygen concentration. For a similar reason, the effects of k3 and k4 are not easily discernible because both parameters result in a decrease of fuel cell voltage that is biased at higher current densities. Thus, we elect to absorb the current effects 1885
Energy Fuels 2010, 24, 1882–1888
: DOI:10.1021/ef901519f
Lee et al.
Figure 6. Time dependence of the parameter k1.
Figure 5. Comparison of the simulated polarization curves over time, with experimental data.
is reported at various time intervals. The anode and cathode supply flow was of serpentine, co-flow geometry, while the electrolyte consisted of the Gore PRIMEA Series 5621 MEA, with an operational pressure of 150 psi. The fuel used for the time up to 23 000 h was pure hydrogen (99.999%), while the air flow was filtered, compressed, and dried. The fuel cell was run continuously, without cycling except for brief interruptions at 500 h for electrochemical diagnostics, at a constant current density of 800 mA/cm2. Again, full details of the experimental setup are found in ref 11. A comparison of the experimental data with model results is shown in Figure 5, where the time-dependent degradation parameters are estimated through eqs 8-14. It can be seen that an excellent agreement with the measured polarization curves can be achieved through the use of the parameter estimation algorithm that optimizes the two parameters k1 and k4 based on the experimental data. Polarization curves are faithfully reproduced at all times, starting from 0 to 25 334 h, with the fuel cell voltage decreasing sharply after longer periods of operation. Normally, the fuel cell voltage as a function of time at a given current density is of interest, which is presented later based on the simulation results shown in Figure 5. The changes in these parameters that lead to accurate simulations of the fuel cell voltage in Figure 5 are plotted in Figures 6 and 7, where both k1 and k4 decrease in time. The first parameter, k1, is a linearly decreasing function of time with some scatter, while the second parameter, k4, follows an exponential decay. As noted in the previous section, k1 describes the changes in the cathode reaction rate. The fact that this parameter linearly decreases indicates that the preexponential factor (and not the activation energy inside the exponential function) in the reaction rate equation must be decreasing, attributable to the loss of reaction sites as the cathode or the catalyst platinum particles degrade over time. In particular, the loss of the carbon support through oxidation is believed to cause the platinum particles to agglomerate, which reduces the available surface area for catalytic reactions. The second parameter, k4, tracks the internal resistance loss, which can arise from a number of mechanisms, including proton-transport retardation, interfacing degradation, and carbon oxidation. The significance of the above two trends in the parameters is that, because we now know the functional
Figure 7. Time dependence of the parameter k4.
form for the key parameters as a function of time, these parameter variations can be estimated using short-time data, as opposed to the 26 000 plus hours of test data in Cleghorn et al.,11 to simulate the fuel cell performance over long periods of time. The converged parameter values tend to oscillate depending upon the initial estimates and the number of iterations, which is intrinsic in multivariate parameter optimization. However, these oscillations were relatively small, less than (10% of the mean values for k1 and (8% for k4. The fuel cell voltage at any current density level can be calculated as a function of time using the current simulation approach with optimized parameters, as shown in Figure 8. It can be seen that the fuel cell operation without significant degradation in the output voltage is achieved only at low current densities, with the decrease in the voltage becoming steeper at higher current densities by large margins. The errors associated with the parameter value uncertainties (arising from the oscillations noted in the previous paragraph) were (12% of the mean values with 95% confidence shown in Figure 8. The effects of other variables can also be examined using the current model. Figure 9, for example, shows the effects of the operating temperature on the fuel cell output voltage as a function of time. The overall voltage is greater at higher temperatures because of the increased kinetic rates. The rate of degradation is higher at low temperatures. The pressure effect over time is more uniform, as exhibited in Figure 10. The output voltage is again higher at increased pressures, 1886
Energy Fuels 2010, 24, 1882–1888
: DOI:10.1021/ef901519f
Lee et al.
embedded, either directly or indirectly, in the model equations. Therefore, if the electrode reaction rates follow the normal thermodynamic dependence upon these two parameters, for example, then the model should be reasonably accurate for a range of pressures and temperatures near the tested operating conditions. To be sure, however, the extrapolation outside the tested operating conditions needs to be validated again with experimental data. Parameter optimization, from an operational standpoint, is also receiving much interest for fuel cell and hybrid systems.14-16 The need for optimizing the fuel cell performance by adjusting operational parameters is evident to maximize the system efficiency. The full operational model that includes degradation predictions can thus be quite useful in optimizing the operation based on the temporal projections of the fuel cell performance. Figure 8. Degradation of voltage as a function of time at various current densities.
Conclusions
Figure 10. Effect of the pressure on the fuel cell output voltage as a function of time.
Degradation parameters are embedded in a computational model from refs 1-4 and optimized using the experimental data in ref 11 to determine the fuel cell output voltage. The contributions of the loss of catalytic activity and membrane/ electrode conductivity to the fuel cell output voltage are reproduced in the degradation model. This is achieved without detailed electrochemical mechanisms but, instead, using the data on global fuel cell voltage as a function of time. As noted at the outset, the degradation of the catalytic activity can occur through agglomeration of the platinum particles or through the Oswald ripening, both of which are difficult to quantify. The current model of course circumvents such detailed degradation kinetics for either the catalytic activity or the ohmic losses, and degradation parameters are systematically estimated on the basis of experimental data using a small number of parametric equations. Construction of more detailed models will inevitably generate more parameters to be estimated, which in turn will require more detailed measured data. The simulation results are in excellent agreement with the data and also point to the main mechanisms of degradation. Also, it is shown that at least two parameters are needed, one for the loss of catalytic activity and another for the ohmic losses, to faithfully simulate the loss of fuel cell performance across different current densities over time. This work shows that computational models, such as the one used by Mann and co-workers1-4 and herein, are quite useful in that degradation parameters can easily be embedded and optimized, so that lifetime performance of fuel cells can be predicted in general, if there is a minimum amount of data to apply the parameter estimation algorithm for the determination of a small number of degradation parameters. The current approach of parameter estimation using global data circumvents the requirements for such detailed electrochemical steps and rate parameters yet is able to compute the time-dependent degradation of the fuel cell output voltage quite correctly. Further data can then be used to refine the degradation mechanisms to be applicable over a wider range of operating conditions and materials, as needed. Some operating parameters, such as pressure and temperature, are embedded, either
but the degradation is more or less uniform at all pressure levels. Strictly speaking, the parameters are optimized only against the test conditions of Cleghorn et al.11 However, some operating parameters, such as pressure and temperature, are
(14) Seo, S. H.; Lee, C. S. Energy Fuels 2008, 22, 1212–1219. (15) Chen, X; Lin, B.; Chen, J. Energy Fuels 2009, 23, 6079–6084. (16) Norheim, A.; Warnhus, I.; Brostom, M.; Hustad, J. E.; Vik, A. Energy Fuels 2007, 21, 1098–1101.
Figure 9. Effect of the temperature on the fuel cell output voltage as a function of time.
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operating conditions needs to be validated with experimental data.
directly or indirectly, in the model equations. Therefore, if the electrode reaction rates follow the normal thermodynamic dependence upon these two parameters, for example, then the model should be reasonably accurate for a range of pressures and temperatures near the tested operating conditions. Strictly speaking, the parameters in the current degradation model are optimized only against the test conditions of Cleghorn et al.,11 and extrapolation outside the tested
Acknowledgment. One of the authors (Y. H. Do) was supported from the World Class University (WCU) Project (No. R32-2009-00-20021-0) from the MEST/KOSEF, Korea, through Kyungpook University. The authors would like to thank Dr. Baek, Hunki of the Kyungpook National University for useful discussions on mathematical optimization.
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