Effect of unsaturated hydrocarbonsInitial testing was performed using research grade (> 99.99%) propane, however, it was found that no power could be produced; the PEMFC simply did not “start” and the open circuit voltage (OCV) was only 0.05 V even at cell temperatures up to 90 °C. In contrast, the direct propane PEMFC study reported by Varela and Savagado18 employed CP-grade propane (> 99% purity) whose impurities likely included some unsaturated hydrocarbons (UHs). With this motivation, we systematically studied the effect of UHs on PEMFC performance. It was found that the addition of small quantities of UHs to the research-grade propane fuel stream did in fact initiate the PEMFC operation with OCV ≈ 0.85 V. Figure 1a shows the efficacy of different UHs on power output; of the 4 UHs tested, ethylene is clearly the most effective on a mole basis and even more so on a mass basis. At a current density of 36 mA cm−2 and additive concentration 2540 ppm, with ethylene the cell operation continues for > 1000 s with slowly decreasing power output whereas for the least effective additive (isobutylene) the power abruptly ceases after ≈ 200 s, a phenomenon we will term “extinguishment” by analogy with flames. It was subsequently found that fuel cell power generation could be initiated with a short (≈ 5 s) period of trace UH flow and the cell would continue to operate for much longer periods of time after the trace UH flow ceased. Figure 1b shows that at a current density of 36 mA cm−2, with no continuing flow of UH, power continues for ≈ 70 s and furthermore that the time before extinguishment increases with increasing UH in the fuel stream up to about 2000 ppm where operation is essentially continuous. In what follows, unless otherwise noted 2000 ppm of ethylene was added to the fuel stream for 5 s to initiate the PEMFC then shut off before starting data collection. Additionally, it was determined that with the addition of small concentrations of UHs, the PEMFC could operate even at ambient room temperature, albeit with a reduced OCV of ≈ 0.6 V and a maximum power density of ≈ 0.5 mW cm−2.Figure 1(a) Effect of addition of 2540 ppm (mole basis) of 4 distinct UHs into the fuel stream on power generation at a constant current density of 36 mA cm−2 (in the absence of UHs, power generation is negligible); (b) effect of ethylene concentration on power generation at 36 mA cm−2 (for 0 ppm, the cell was initially activated with 5 s exposure of 2000 ppm of ethylene in the fuel stream). Note the logarithmic time scales on both plots.The power generated by the cell when operated continuously with propane and a UH additive cannot be attributed solely to the additive. The residence time with the cell (ratio of cell gas volume to volume flow rate) is only 0.19 s yet the effect of the initial trace of ethylene used to start the cell can last hundreds of times longer, even with no steady-state flow of ethylene. Moreover, as shown later, when the fuel cell is operated with 1.2 L min−1 of pure ethylene as fuel, the power density obtained is actually lower than when operated with a combination of 1.2 L min−1 of propane and 2540 ppm of ethylene. These observations indicate that the propane must be reacting and generating power in the fuel cell to obtain the results shown in Fig. 1.Effect of current scan rate and current densityAs could be anticipated based on the time-dependent behavior of the direct propane PEMFC shown in Fig. 1, it was found that current scan rates have a significant effect on performance. While direct propane PEMFCs have been studied previously16,17,18 (see Introduction), these authors did not report such time-dependence and no information about the scan rates for their polarization curves was reported in those studies. In the current-scan process employed in this work, each current level was held for 3 s before increasing to the next level required to obtain the specified current scan rate. For example, at a scan rate of 0.66 mA cm−2 s−1, after each 3 s hold period the current was increased by 2 mA cm−2. The slower scan rates significantly degrade the performance and eventually extinguish the fuel cell, especially at relatively high currents, as shown in Fig. 2a and b. For example, at a slow scan rate (0.66 mA cm−2 s−1), the maximum power density is 13 mW cm−2, increasing to 21 mW cm−2 at 6.66 mA cm−2 s−1. By comparison, no such time-dependent behavior was observed in the same PEMFC using hydrogen (maximum power density 380 mW cm−2) or dimethyl ether (maximum power density 46 mW cm−2) (see Supplemental Data, Fig. 1). Consequently, the marked time-dependence of the power generation appears to be a phenomenon unique to hydrocarbon fuels.Figure 2Dynamics of the fuel cell behavior at different scan rates. (a) Voltage and power density polarization curves for direct propane PEMFCs at varying current scan rates. (b) dynamics of direct propane PEMFCs for different values of (constant) applied current (note logarithmic time scales). All cases: flow rates 1.2 L min−1 and 0.8 L min−1 for propane and oxygen, respectively, no added UHs after initiation, cell temperature 80 °C.These results show that traditional polarization curves are inappropriate for characterizing hydrocarbon-fueled PEMFCs. We propose that a more appropriate characterization method is to employ a galvanostatic mode (constant current) and observe the transient voltage (thus power) behavior and repeat at different current levels. An example of this approach is shown in Fig. 2b. At current densities of about 25 mA cm−2 or lower, the power reaches a nearly steady value after an initial transient period (note the logarithmic time scale) whereas at higher currents the power slowly decreases over time then rapidly drops to zero. The time until extinguishment decreases as current increases. It might be expected that the cell extinguishment results from some material coating the anode sites and that a cleaning process would be required to restore cell operation, however, this was not the case; in fact, once the current was removed for more than a few seconds the cell “reset” as if such an extinguishment had never occurred. A proposed mechanism for these fuel cell dynamics is discussed in Sect. “Fuel cell dynamic model”. Moreover, membranes of varying thicknesses, Nafion N-212 (2 mm) and Nafion N-1110 (10 mm), showed polarization curves nearly identical to that of Nafion N-117 (7.2 mm) at 1.33 mA cm−2 (see Supplemental data, Fig. 2), demonstrating that membrane characteristics such as proton resistivity do not affect the results reported in this work. Additionally, we explored the impact of reduced catalyst loading, with 4 mg cm−2 platinum on the anode and 2 mg cm−2 platinum on the cathode. Under these conditions, the attained power density was approximately 4 mW cm−2 at 1.33 mA cm−2 (see Supplemental data, Fig. 3). This demonstrates that over the range that we examined, the maximum power density obtained from our PEMFC scales approximately linearly with catalyst loading.Figure 3Polarization and power density curves of direct propane PEMFCs at different operating temperatures and Arrhenius plot of maximum power vs. temperature. (a) Polarization curves with current scan rate 0.133 mA cm−2 s−1. (b) Arrhenius plot from which apparent activation energy is estimated.Effect of cell temperaturesFigure 3a shows the effect of temperatures on fuel cell performance. As expected, the maximum power and current density increases with temperature. Above 114 °C, the fuel cell abruptly stopped producing power, which may be due to a glass transition of the Nafion® membrane, first Tg ≈ 130 °C21,22,23,24. From the slope of an Arrhenius plot (Fig. 3b) of the natural logarithm of maximum power vs. 1/T, an effective activation energy for electro-oxidation of propane on Pt catalyst can be inferred to be ≈ 5.2 kcal mol−1 (\({\text{slope}} = – \frac{{E_{a} }}{R};\) Ea = effective activation energy & R = ideal gas constant). Ahn et al.25 reported a similar value (6.4 kcal mol−1) for thermal oxidation of propane on NH3-treated Pt catalyst over a similar temperature range. For hydrogen-fueled PEMFCs over this temperature range, the corresponding activation energy is about 2.9 kcal/mole26, which again points to the differences between hydrogen- and hydrocarbon-fueled PEMFCs.Load interrupt modeSince stopping current for short periods of time reset the cell performance, a series of tests were performed to determine if repeated cycling the current off and back on again at regular intervals (a technique we term “load-interrupt mode”) could lead to higher average power than if the same average current were applied in a steady mode. Figure 4 shows the power densities at different current densities. At low current densities (e.g., 12 mA cm−2, Fig. 4a), where extinguishment does not occur, there is no significant difference between load-interrupt and constant-current modes (6.2 vs. 6.1 mW cm−2). At intermediate current densities (e.g., 24 mA cm−2, Fig. 4b), the load-interrupt mode achieves a higher average power density than the constant-current mode (9.4 vs. 7.6 mW cm−2). At higher current densities (e.g., 37 mA cm−2, Fig. 4c), constant-current operation is not possible, but the load-interrupt mode achieves an average power density of 11.6 mW cm−2. From these data, a pseudo polarization curve was generated by calculating the average power density and voltage over 1000 s using the load-interrupt mode (Fig. 4d). Comparison to a traditional quasi-steady polarization curve indicates that the load-interrupt mode can yield a higher average power density and thus may be a potentially useful operation mode for direct hydrocarbon PEM fuel cells.Figure 4Direct propane PEMFC performance in galvanostatic and load-interrupt modes. (a) Low current density experiment. (b) Medium current density experiment. (c) High current density experiment. (d) Pseudo-polarization curve in load-interrupt mode averaged over 1000 s compared to galvanostatic mode. For the load-interrupt mode, the fuel cell was operated at the indicated current density for 20 s and then the current was removed for 5 s.CO poisoning and effect of fuel typeIt is well known that small amounts of CO in the anode stream can have a significant detrimental effect on PEMFC performance27. For CO mitigation, Gottesfeld and Pafford28 showed that blending 4.5% O2 into a hydrogen fuel stream containing 100 ppm CO significantly increases PEMFC power. With this motivation, and to assess whether CO is a likely contributor to the aforementioned extinguishing behavior we observed, we tested the effect of blending small concentrations of O2 into the propane feed stream. It was found that this immediately rendered the PEMFCs inoperative; the amount of O2 required to suppress power output was roughly equal to the electron-receiving capacity of the oxygen flow based on 100% coulombic efficiency of the anode for oxygen, meaning that, unsurprisingly, the anode has far more affinity for oxygen (if present) than hydrocarbon fuel. Extinguishment times were nearly identical whether the same molar flow rate of O2 was supplied to the cathode in the form of air or pure O2; if O2 cross-over to the anode were a factor, extinguishment would occur on a significantly shorter time scale with O2 supplied to the anode. Moreover, the PEMFC exhaust is at ambient pressure and all components upstream of the exhaust are at pressures slightly above ambient, thus ambient air leakage into the anode supply stream is unlikely. Additionally, a platinum/ruthenium anode catalyst, frequently used to suppress CO poisoning29, was tested but no improvement in PEMFC performance was observed. In contrast, if a significant amount (14 mol percent) of carbon monoxide (CO) was intentionally introduced into the fuel stream, it was observed that the presence of CO actually prevented extinguishment of the cell (Fig. 5b) in a manner similar to that of UHs (Fig. 1), though CO is far less effective in this role than UHs. In fact, the fuel cell could operate on pure CO albeit with considerably lower power densities (≈ 4 mW cm−2; not shown) compared to hydrocarbons. These results strongly suggest that neither CO nor O2 accumulation on the anode is a significant contributor to the observed extinguishing behavior of hydrocarbon fueled PEMFCs that we have observed.Figure 5Comparison of PEMFC performance for propane, isobutane, n-butane, and 86% propane + 14% CO. (a) Polarization curves for current scan rate 0.133 mA cm−2 s−1. (b) power output vs. time in galvanostatic mode at 36 mA cm−2 (note logrithmic time scale.) No UH was added for any fuel except for initiation.For portable power generation devices, it may be beneficial to employ butane rather than propane as fuel because the saturation pressure of butane at room temperature is ≈ 2.5 atm compared to ≈ 10 atm for propane and thus butane is more readily stored in lightweight fuel tanks; it is for this reason that butane is used in many portable applications. With this motivation, the performance of n-butane and isobutane was explored. Figure 5 shows that n-butane exhibits a similar performance to that of propane whereas isobutane shows somewhat lower performance, perhaps due to the larger number of stronger primary C–H bonds in isobutane. All three hydrocarbons show the sudden extinguishment behavior at high currents (Fig. 5b), indicating that the dynamics we have observed are common in PEMFCs using saturated hydrocarbons.The feasibility of using a pure UH as fuel was also explored. Ethylene was employed since it is the most effective fuel additive for saturated hydrocarbons (Fig. 1a). Results are shown in Fig. 6. It was found that the maximum current was ≈ 90 mA cm−2 compared to ≈ 40 mA cm−2 for propane, however, the maximum power density (≈ 8 mW cm−2) was somewhat less than that of propane due to much lower resulting voltages. Furthermore, no extinguishing behavior was observed even at high current densities. These results indicate that propane or butane with addition of > 2000 ppm of ethylene is a preferred fuel mixture because it results in higher power densities than pure ethylene and does so without extinguishment present at high currents without added UH.Figure 6Performance of a direct ethylene PEMFC. (a) Polarization curve generated using values of voltage and power density at each current density averaged over 1000 s. (b) power vs. time for various constant current densities.Effluent analysisChemical analyses were conducted to determine the products of the anode reactions and in particular to search for intermediate species such as CO and hydrocarbons that were not present in the reactants. Theoretically, the half-cell and overall reactions for propane are$${\text{C}}_{{3}} {\text{H}}_{{8}} + {\text{6H}}_{{2}} {\text{O}} \to {\text{3CO}}_{{2}} + {2}0{\text{H}}^{ + } + {2}0{\text{e}}^{ – } \left( {{\text{Anode}}} \right)$$$${\text{5O}}_{{2}} + {2}0{\text{H}}^{ + } + {2}0{\text{e}}^{ – } \to {1}0{\text{H}}_{{2}} {\text{O}}\left( {{\text{Cathode}}} \right)$$$${\text{C}}_{{3}} {\text{H}}_{{8}} + {\text{5O}}_{{2}} + {\text{6H}}_{{2}} {\text{O}} \to {\text{3CO}}_{{2}} + {1}0{\text{H}}_{{2}} {\text{O}}\left( {{\text{Overall}}} \right)$$To maximize the possibility of detecting such intermediate species, the residence time of the reactants was maximized by operating the fuel cell at a steady condition then closing valves at both the inlet and outlet of the anode and operating the cell until extinguishment. A sample of anode chamber gas was then extracted and analyzed on a gas chromatograph using flame ionization and thermal conductivity detectors. This analysis showed that CO2 and unreacted C3H8 were the dominant species. Trace amounts of CH4 were detected (> 75 × smaller than CO2 concentrations) along with even smaller amounts of C2H6. No other hydrocarbons, CO, or H2 were detected. The liquid effluent was also analyzed on a liquid chromatograph—mass spectrometer system but no organic compounds were detected. Consequently, it can be stated that fuel molecules are primarily either fully oxidized to CO2 and H2O or not reacted at all. Similar behavior has been observed in the thermal oxidation of propane on Pt catalyst at low temperatures25, which in that case was due to the fact that the activation energy for desorption of intermediates is far greater than that of CO2 and H2O and thus once adsorption and decomposition of the hydrocarbon on the catalyst surface commences, only the final products have significant probability of desorbing 30.From the tests described in the previous paragraph the “Carbon Conversion Ratio” (CCR), i.e., the ratio of C atoms in the form of CO2 to the total C atoms in the extracted samples (which are essentially all in the form of CO2 or C3H8 as mentioned above) could be calculated. Figure 7a shows the CCR as a function of the current density. The average over several tests is shown for each current density; the error bars represent one standard deviation above and below this average. The CCR decreases as current density increases, again indicating that extinguishment is accelerated at higher currents. Figure 7a also shows that there is a very close correspondence between the CCR and the total charge created (fuel cell current integrated over time up to extinguishment). This indicates that the number of electrons produced per C3H8 molecule consumed is nearly independent of current density. From the known volume of the anode chamber (3.8 cm3) and the number density of C3H8 in the chamber before reaction and after extinguishment (as determined by the GC analysis), the number of moles of C3H8 consumed and thus moles of electrons produced per C3H8 molecule consumed can be calculated. Figure 7b shows that the electrons produced per C3H8 molecule is indeed nearly constant and close to the maximum possible (20) for complete oxidation of propane. The inferred values may slightly exceed 20, which is probably due to incomplete mixing of the gas within the closed anode chamber causing a slight underestimation of the fuel consumed. Nevertheless, the data indicate that the fuel molecules are nearly completely consumed electrochemically without residual intermediates, which is consistent with the conclusion based on the chemical analysis described in the previous paragraph. Furthermore, from the total electrical energy produced (power integrated over time) from the time the valves to the anode chamber were closed until the cell extinguished and the total free energy change of the reacted C3H8 molecules, the overall energy efficiency of the cell is readily determined. Figure 7b shows that this efficiency varies from a maximum of 73% at low current densities to 25% at high densities. At the low current densities obtained in this work, ohmic losses across the membrane are expected to be minimal, as are mass transport losses. Thus, the decreased efficiencies at higher current densities are likely due to anode activation polarization losses, however, it is not practical to employ a Butler-Volmer type of analysis to estimate these losses due to the transient nature of these experiments.Figure 7Effect of current density on anode product composition. (a) Carbon Conversion Ratio (see text) and total charge production. (b) Number of electrons reacted per propane molecule and efficiency of conversion of reacted fuel to electrical energy.Fuel cell dynamic modelThe experimental results indicate that (1) the fuel cell will extinguish if operated for sufficiently long times at higher current densities but not at lower current densities, (2) removing the current for a few seconds will effectively “reset” the anode history and (3) the addition of small concentrations of UHs can prevent cell extinguishment. These results suggest that there is a competition between the conversion of active (power-producing) anode sites to inactive sites and the reverse of this process. With this motivation, a simple physically-based model of the dynamics of fuel cell performance is proposed using as few assumptions and as few empirical constants as possible to reproduce the key features of the experimental results.First a “binary” model is assumed, i.e., there are only two types of anode sites, namely active sites (that produce power in proportion to their number) and inactive sites; their mole fractions of the total available anode sites are denoted as \({X}_{a}\) and \({X}_{i}\), respectively. The sum of these two types of sites is constant, i.e.,$$X_{a} + X_{i} = 1$$
(1)
Next it is assumed that the rate at which the inactive sites recover to form active sites (the source of active sites) follows the Law of Mass Action, i.e., the rate is proportional to the fraction of inactive sites:$$\frac{{dX_{i} }}{dt} = – K_{1} X_{i} \;{\text{or}}\;\frac{{dX_{a} }}{dt} = K\left( {1 – X_{a} } \right)$$
(2)
where K1 is a constant (units s−1). It is assumed that the rate of change of states depends only on the current state (Xa) and not any higher-order temporal derivatives, i.e., the system has no “momentum.” Furthermore, it is assumed that the rate at which active sites are converted to inactive sites (the sink of active sites) is proportional to the fraction of active sites available to be converted into inactive sites with an additional term related to the fraction of already existing inactive sites:$$\frac{{dX_{a} }}{dt} = K_{2} X_{a} – K_{3} X_{i}^{b} = – K_{2} X_{a} – K_{3} \left( {1 – X_{a} } \right)^{b}$$
(3)
where K2, K3 (units s−1) and b (dimensionless) are constants. While the − K2Xa term is straightforward and analogous to the − K1Xi term in Eq. (2), the − K3Xib term is unusual (it does not follow the Law of Mass Action) but necessary; without such a term the long-time behavior of the system is simply an equilibrium value of Xa given by Xa,eq = K1/(K1 + K2) and no extinguishment (defined as Xa → 0 in finite time) as seen in the experiments can occur. The − K3Xib term represents a self-accelerating process where the rate of conversion of active to inactive sites increases with the increasing fraction of existing inactive sites. This could occur, for example, if once an inactive site is formed there is a higher probability of formation of inactive sites adjacent to existing inactive sites than at active sites not adjacent to inactive sites. A potential scenario for this would be the formation of a polymer-like chain of inactive sites starting from a “seed” of a single fuel molecule adsorbing on the anode and forming said seed instead of decomposing to initiate the electrochemical process resulting in power production and desorption of products as in nominal anode operation. (The chemical analysis of the gaseous and liquid effluent from the anode did not indicate any polymers; if they do form on the anode, they revert to monomers once desorbed.) While extinguishment in finite time can occur even for 0 < b ≤ 1, the decrease in Xa over time is more abrupt (and thus closer to the experimental observation) if b > 1. Physically this might correspond to a multi-site reaction, e.g., with b = 2 then two adjacent inactive sites are needed to initiate or propagate the chain of inactive sites.Combining sources and sinks of active sites (Eqs. 2 and 3) yields an expression for the total rate of change of active sites:$$\frac{{dX_{a} }}{dt} = K_{1} – \left( {K_{1} + K_{2} } \right)X_{a} – K_{3} \left( {1 – X_{a} } \right)^{b}$$
(4)
If we select b = 2 in order to obtain a relatively simple equation with an exact solution that mimics rapid extinction as seen in experiments (values of b larger than 2 were examined via numerical integration but did not exhibit substantively different trends), the result is a type of Riccati equation:$$\frac{{dX_{a} }}{dt} = AX_{a}^{2} + BX_{a} + C;\quad A = – K_{3} ;\,B = 2K_{3} – K_{1} – K_{2} ;\;C = K_{1} – K_{3}$$
(5)
With the initial condition Xa = Xa,o at t = 0, the physically meaningful solutions of Eq. (5) for t ≥ 0, 0 ≤ Xa ≤ 1 are$$\begin{gathered} {\text{Case}}\;{\text{I}}:\;4AC – B^{2} > 0:\;X_{a} (t) = \frac{1}{2A}\left\{ {R\tan \left[ {\frac{R}{2}\left( {t + \tau_{1} } \right)} \right] – B} \right\};\;\;\tau_{1} \equiv \frac{2}{R}\tan^{ – 1} \left( {\frac{{2AX_{a,0} + B}}{R}} \right) \hfill \\ {\text{Case}}\;{\text{II:}}\;4AC – B^{2} < 0,\;\left| {2AX_{a,0} + B} \right| < R:X_{a} (T) = \frac{1}{2A}\left\{ {R\tanh \left[ {\frac{R}{2}\left( {\tau_{2} – t} \right)} \right] – B} \right\};\;\tau_{2} \equiv \frac{2}{R}\tanh^{ – 1} \left( {\frac{{2AX_{a,0} + B}}{R}} \right) \hfill \\ {\text{Case}}\;{\text{III:}}\;4AC – B^{2} < 0,\;\left| {2AX_{a,0} + B} \right| > R:X_{a} (T) = \frac{1}{2A}\left\{ {R{\text{coth}}\left[ {\frac{R}{2}\left( {\tau_{3} – t} \right)} \right] – B} \right\};\;\tau_{3} \equiv \frac{2}{R}\coth^{ – 1} \left( {\frac{{2AX_{a,0} + B}}{R}} \right) \hfill \\ {\text{where}}\;R \equiv \sqrt {\left| {4AC – B^{2} } \right|} = \sqrt {\left| {4K_{2} K_{3} – \left( {K_{1} + K_{2} } \right)^{2} } \right|} \hfill \\ \end{gathered}$$
(6)
For Case I (unconditional extinction), extinction will always occur in finite time for any Xa,o; the solution is valid until the extinction time text, i.e., the smallest positive value of t for which Xa = 0, which corresponds to$$t_{ext} = \frac{2}{R}\tan^{ – 1} \left( \frac{B}{R} \right) – \tau_{1} \quad \left( {{\text{Case}}\;{\text{I}}} \right)$$
(7)
For Case II (unconditional equilibrium), Xa always approaches an equilibrium value (Xa,eq) as t → ∞ given by$$X_{a,eq} = – \frac{R + B}{{2A}}$$
(8)
which is a generalization of the aforementioned result for K3 = 0, namely Xa,eq = K1/(K1 + K2). Xa,o will eventually reach this equilibrium state even if Xa,o = 0, i.e., the cell will recover even from a fully extinguished condition. For Case III (conditional equilibrium), equilibrium will be achieved if$$\frac{B}{R} > 1\;{\text{and}}\;\frac{{B + 2AX_{a,o} }}{R} < – 1$$
(9)
with Xa,eq as in Eq. (8), otherwise extinction occurs in finite time at$$t_{ext} = \tau_{3} – \frac{2}{R}\coth^{ – 1} \left( \frac{B}{R} \right)\quad \left( {{\text{Case}}\;{\text{III}}} \right)$$
(10)
To obtain quantitative comparisons of the predictions of this simple model to experimental observations, estimated values of K1, K2, and K3 are needed. K1 should be independent of the current density (I) and UH concentration since it represents the rate at which active sites are restored from inactive sites when no current is flowing. Figure 4 shows that when I = 0, power and thus active sites are restored with a time constant (63% restoration) of approximately 2 s thus we set K1 = 0.5 s−1, independent of I (see Table 1). With K1 set, the combination of K2 and K3 will determine the power level at the “knee” of the curve just before the onset of rapid approach to extinction, which is typically 40% of the initial power level (Fig. 1b; Fig. 3b). It was found that the baseline combination K2 = 0.25 s−1 and K3 = 0.57 s−1 produced this behavior with fuel cell dynamic response and extinguishment time (65 s) for I = 36 mA cm−2 and no UH additives close to that found experimentally (Fig. 8a).
Table 1 Parameters for sigmoid-fit (Eq. (11)) to rate constants for PEMFC dynamic model (Eq. (6)).Figure 8Comparison of direct propane PEMFC dynamic model to experimental data. (a)–(d) current density I = 36 mA cm−2, varying UH (ethylene) concentration; (a, e, f) UH = 0, varying I. For values of K1, K2 and K3 employed in the model predictions refer to Eq. (11) and Eq. (12) and Table 1.It is expected that the rate constant for conversion of active to inactive sites (K2) should be mostly independent of UH concentration because power is mostly independent of UH concentration (Fig. 1b) until very near the extinction time text. The rate constant for the term responsible for abrupt extinction (K3) will decrease with increasing UH concentration since this decreases the propensity for extinction to occur. We have not identified a first-principles model of the expected functional relationship between K3 and UH concentration but for I = 36 mA cm−2 an empirical relation of the form$$K_{3} = \frac{{K_{{3,X_{UH} = 0}} }}{{1 + 5.05X_{UH} }}$$
(11)
where \({K}_{3,{X}_{UH}=0}\)= 0.57 s−1 and XUH is the mole fraction of ethylene in the fuel stream (Fig. 8a–d). This functional form is sensible because it provides a simple expression for the effect of XUH using only one additional constant that yields the correct behavior for both XUH = 0 and for XUH ≠ 0 over the entire range of XUH examined in this work.Furthermore, the experimental data show that K2 and K3 will depend on the current density (I). K2 and K3 will decrease with decreasing I because decreasing current increases the time to extinguishment (text) (Fig. 3b). Again, we have not identified a first-principles model of the expected functional relationship between K2, K3 and I, but a sigmoid function relationship$$K_{i} = a_{i} + \frac{{b_{i} }}{{1 + c_{i} e^{ – dI} }}$$
(12)
results in model predictions that are consistent with the experimental observations for a range of I (Fig. 8a,e,f). The best-fit values of ai, bi, ci, and d for K2 and K3 are shown in Table 1; as might have been expected, it was found that a single constant d = 0.23 cm2/mA accounted for the effect of I on both K2 and K3. In the limit I = 0, the values of K2 and K3 are 0.13 s−1 and 0.37 s−1, respectively, which in turn yield A = − 0.37 s−1, B = 0.12 s−1, C = 0.13 s−1 and thus 4AC- B2 = − 0.21 s−2 and R = 0.45 s−1, which corresponds to Case II, namely equilbrium, which means that an extinguished cell (Xa,o = 0) always recovers when I is set to zero, a result that is consistent with experimental observations. This is significant because if the values of K2 and K3 at I = 0 obtained from these correlations corresponded to Cases I or III (extinction), an extinguished cell could never recover, which is contrary to the experimental observations.With the rate constants given in Table 1, the model predictions (Eq. 6) closely reproduce the dynamic behavior of hydrocarbon-fueled PEMFCs (Figs 8a–f) over the entire range of current densities and UH concentrations studied experimentally. In particular, the sudden extinction at high current densities, steady-state behavior at lower current densities, and the shape of the power vs. time histories are close to the measured values. The impact of the K3 term describing the effect of the concentration of inactive anode sites on the growth rate of additional inactive sites is apparent, since no extinction in finite time is possible without this term.
Dynamics of direct hydrocarbon PEM fuel cells
