Energy Fuels 2010, 24, 5097–5108 Published on Web 08/10/2010
: DOI:10.1021/ef100610w
)
Nature-Inspired Energy- and Material-Efficient Design of a Polymer Electrolyte Membrane Fuel Cell )
Signe Kjelstrup,*,†,‡,§ Marc-Olivier Coppens,†,‡, J. G. Pharoah,†,‡,^ and Peter Pfeifer†,‡,# §
Department of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim, Norway. Howard P. Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180. ^Fuel Cell Research Centre, Kingston, Ontario K7L 5L9, Canada. #Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211. †All authors contributed equally to this work. ‡This collaborative work was initiated at the Center for Advanced Study, The Norwegian Academy of Science and Letters, Drammensveien 78, Oslo, Norway. Received May 14, 2010. Revised Manuscript Received July 13, 2010
A design procedure is presented that improves the energy efficiency and saves catalyst material of a polymer electrolyte membrane fuel cell (PEMFC). The method is demonstrated for a single cell in a stack and uses the theorem of equipartition of entropy production to maximize energy efficiency. The gas supply and water outlet systems, designed to produce entropy uniformly, have a fractal structure inspired by the human lung. The tree-like gas distributor engraved in the bipolar plates may eliminate the need for porous transport layers. Mathematical solutions are given for the optimal height, macroporosity, and nanoporous column width of the electro-catalytic layer beneath the gas supply system. It is shown that the optimal macroporosity of the catalytic layer is equal to 1/2 for the model chosen and that the optimal height of the catalytic layer depends upon the coefficient for first-order reaction kinetics at the cathode, the diffusion constant for oxygen in the gas phase, and the oxygen concentration of the inlet flow. An upper bound can be used for the column width. The general results are illustrated using the standard E-TEK Elat/Std/DS/V2 gas diffusion electrode with 0.5 mg of Pt/cm2 membrane area and 20% Pt/C on Vulcan XC-72 as a support material. It is indicated that the amount of catalyst can be reduced by a factor of 4-8, while the energy efficiency can be increased by 10-20% at high current densities.
minimization procedure6,8-13 with a new catalyst optimization procedure.14,15 PEM and other fuel cells have, thus far, mostly been assembled without any clear, fundamental governing principle. The suggested procedure should lead to less trial and error in the production of single cells and in the assembly of fuel cell stacks. The performance of a fuel cell is evident from its polarization curve, the relation between the electric potential and the current density in the cell.16 This curve shows that a large portion of the total energy available for work is dissipated as heat in the cell at normal operating conditions. The losses vary with the current density. There are always losses connected with the electrode overpotential(s) and with the ohmic resistance loss. However, at large current densities, an additional
1. Introduction Wide-spread use of fuel cells in the transportation sector is now hampered for several reasons: the catalyst and energy efficiencies are not good enough,1-4 and the cell lifetime is too short.5 State-of-the art polymer electrolyte membrane fuel cells (PEMFCs) are using a Pt loading near 0.3 mg/cm2. This amount is too high to be commercially viable.3,4 The U.S. Department of Energy (DOE) target of 0.03 mg/cm2 seems difficult to achieve. This work introduces a systematic design procedure that will bring us closer to target. Improved fuel cell designs will facilitate commercialization, which is what motivated this study. Taking the PEMFC as an example, we shall show how to optimize the energy efficiency of a cell and, at the same time, the catalyst use, by combining an entropy production
(9) Bejan, A. Shape and Structure, from Engineering to Nature; Cambridge University Press: Cambridge, U.K., 2000. (10) Johannessen, E.; Nummedal, L.; Kjelstrup, S. Int. J. Heat Mass Transfer 2002, 45, 2649–2654. (11) Gheorghiu, S.; Kjelstrup, S.; Pfeifer, P.; Coppens, M.-O. In Fractals in Biology and Medicine; Losa, G. A., Merlini, D., Nonnenmacher, T. F., Weibel, E. R., Eds.; Birkh€auser Verlag: Basel, Switzerland, 2005; Vol 4, pp 31-42. (12) Johannessen, E.; Kjelstrup, S. Chem. Eng. Sci. 2005, 60, 3347– 3361. (13) Kjelstrup, S.; Bedeaux, D.; Johannessen, E.; Gross, J. Nonequilibrium Thermodynamics for Engineers; World Scientific: Singapore, 2010. (14) Johannessen, E.; Wang, G.; Coppens, M.-O. Ind. Eng. Chem. Res. 2007, 46, 4245–4256. (15) Wang, G.; Johannessen, E.; Kleijn, C. R.; de Leeuw, S. W.; Coppens, M.-O. Chem. Eng. Sci. 2007, 62, 5110–5116. (16) O’Hayre, R.; Cha, S.-W.; Colella, W.; Prinz, F. B. Fuel Cell Fundamentals; Wiley: New York, 2006.
*To whom correspondence should be addressed: Department of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim, Norway. Telephone: þ47-73594179. Fax: þ47-73550877. E-mail:
[email protected]. (1) Eikerling, M.; Kornyshev, A. A. J. Electroanal. Chem. 1998, 453, 89–106. (2) Okada, T.; Kaneo, M. Molecular Catalysts for Energy Conversion; Springer: Berlin, Germany, 2008; Springer Series in Material Sciences. (3) Hugl, H.; Nobis, M. Top. Organomet. Chem. 2008, 23, 1–17. (4) Mock, P. J. Power Sources 2009, 190, 133–140. (5) Wang, Z.-B.; Zuo, P.-J.; Wang, X.-P.; Lou, J.; Yang, B.-Q.; Yin, G.-P. J. Power Sources 2008, 184, 245–250. (6) Tondeur, D.; Kvaalen, E. Ind. Eng. Chem. Res. 1987, 26, 50–56. (7) Tondeur, D.; Luo, L. Chem. Eng. Sci. 2004, 59, 1799–1813. (8) Bejan, A. Entropy Generation Minimization; CRC Press: Boca Raton, FL, 1996. r 2010 American Chemical Society
5097
pubs.acs.org/EF
Energy Fuels 2010, 24, 5097–5108
: DOI:10.1021/ef100610w
Kjelstrup et al.
Figure 1. Schematic illustration of a MEA-containing membrane and nanoporous catalytic layers. The PTL and serpentine gas supply channels (c) are also indicated. The land above the PTL is given the symbol L.
loss may occur from mass-transfer limitations; e.g., see refs 1 and 17-22 and see ref 23 by Burheim et al. for an alternative explanation. The overpotential then has contributions from concentration polarization. We will show how this last contribution can be minimized, through optimization of the second-law efficiency and catalyst use,11-15 keeping the ohmic resistance constant. Diffusion limitations have their origin in the porous nature of the various layers in the cell. Pores, more or less clogged with water, will limit the access of reactants to the catalytic site and prevent product water from leaving.24 There are typically five active layers in the cell, all with different characteristic pore sizes. The cell with five layers is pictured in the upper portion of Figure 1. The central layer is the gas-tight ionexchange membrane (typically a Nafion membrane) filled with water, in which protons conduct charge and where water transport takes place by electro-osmosis and diffusion.16 The two catalytic layers on both sides of the membrane are nanoporous with dispersed platinum, carbon, and polymer. The carbon grains (20-40 nm) form agglomerates of 200300 nm size, leading to pores of 20-40 nm inside the grains and 40-200 nm pores in the void space between agglomerates; see ref 1 for a good description of geometries and trade-off situations concerning different rate-limiting steps in a nanoporous electrode. The three central layers are called the membrane electrode assembly (MEA). The MEA is covered on both sides with a porous transport layer (PTL), having micrometer-sized pores. Supplying oxygen fast enough through the PTL and the catalytic layer to the active sites can thus be a large problem. The lack of a fast supply leads to
gradients perpendicular as well as parallel to the membrane. The loss in potential because of the first mentioned gradients can easily amount to 0.2 V,25 while the last mentioned gradients give contributions of 0.1 V.21 The clogging of pores by water can give similar losses.20,24 Hydrophobic materials have therefore been added to these layers with the intention to facilitate water transport.24 The present work addresses attempts to avoid these problems. The procedure for minimization of entropy production is a rigorous methodology that was already successfully applied to process units, such as heat exchangers, chemical reactors, and distillation towers.13 An interesting property emerged from these studies.6,10,12,13,8,9 If the process can take place with sufficient degrees of freedom, the optimal process will follow a trajectory of thermodynamic states, a so-called highway in state space,12 which is characterized by constant local entropy production. Surprisingly, the human lung functions accordingly.11 The lung has a fractal structure with two distinct scaling regimes: 14-16 upper generations of branches (the bronchi), in which flow dominates transport, and 7-9 lower generations of space-filling acini lined by alveoli, with a large membrane surface/volume ratio, in which diffusion dominates transport. It was shown that the entropy production is uniformly distributed in both regimes.11 In this context, it is interesting to see that Kloess et al.26 recently have used the lung as a model in an empirical bio-inspired work on the flow system of a PEMFC. These observations are now prompting us to pose the following question related to fuel cell design: Given that constant local entropy production is beneficial to some process units, even to nature, can we take advantage of this as a guiding principle for fuel cell design? Which gas supply system do we find when the fuel cell is designed using this principle from the outset? Extrapolating from the studies mentioned, we shall use the hypothesis that optimal performance is related to uniform entropy production across the layered structure of the unit. We shall see that we can take advantage of the distributor design presented by Tondeur and Luo7 for uniform delivery of reactants and obtain a structure of a
(17) Jaouen, F.; Lindbergh, G.; Sundholm, G. J. Electrochem. Soc. 2002, 149, A437–A447. (18) Jaouen, F.; Lindbergh, G. J. Electrochem. Soc. 2003, 150, A1699– A1710. (19) Jaouen, F.; Lindbergh, G.; Wiezell, K. J. Electrochem. Soc. 2003, 150, A1711–A1717. (20) Ihonen, J.; Mikkola, M.; Lindbergh, G. J. Electrochem. Soc. 2004, 151, A1152–A1161. (21) Sun, W.; Peppley, B. A.; Karan, K. Electrochim. Acta 2005, 50, 3359–3374. (22) Harvey, D.; Pharoah, J. G.; Karan, K. J. Power Sources 2008, 179, 209–219. (23) Burheim, O.; Vie, P. J. S.; Møller-Holst, S.; Pharoah, J.; Kjelstrup, S. Electrochim. Acta 2010, 55, 935–942. (24) Weber, A.; Newman, J. J. Electrochem. Soc. 2005, 152, A677– A688.
(25) Berning, T.; Djilali, N. J. Power Sources 2003, 124, 440–452. (26) Kloess, J. P.; Wang, X.; Liu, L.; Shi, Z.; Guessous, L. J. Power Sources 2009, 188, 132–140.
5098
Energy Fuels 2010, 24, 5097–5108
: DOI:10.1021/ef100610w
Kjelstrup et al.
single cell. The potential is expressed as E tot ¼ - ΔGtot =F - RI - ηtot
single PEMFC that in good approximation has this property. The optimal pore size distribution of the porous catalysts will be determined under constraints of uniform entropy production and maximum yield.14,15 Also, to obtain numerical insight, we shall use a single PEMFC operating on hydrogen and oxygen or air at 1 bar and 80 C. The objective is thus 2-fold: to find the optimal structure of the gas flow distributor in the cell, as seen from the perspective of energy efficiency of the whole cell, and to determine the minimum amount of catalyst (in this case, Pt) in the electrocatalytic layer that accomplishes a certain production. The two aims are not conflicting but can be decoupled and subsequently combined. We shall see that they can lead to a significant reduction in catalyst need, by reducing masstransfer limitations significantly and improving the transport path for water.
ð1Þ
where ΔGtot is the Gibbs energy for the reaction 1 1 1 O2 ðgÞ þ H2 ðgÞ f H2 OðlÞ 4 2 2
ð2Þ
calculated from the inlet and outlet states of reactants and product, respectively, F is Faraday’s constant, R is the total cell resistance, I is the electric current produced, and ηtot is the total cell overpotential. The total entropy production of the cell is related to the last two terms of eq 1; see ref 27 for a more detailed expression. The origins of the potential losses are the total ohmic resistance, the reaction overpotential, in particular at the cathode,28-32 and possible mass-transfer limitations.1,18,19,21,22 The value of ηtot is frequently more than 0.4 V out of the theoretical 1.23 V, available at 298 K. On the local scale, we consider an area element ΔxΔy of the electrode area on each side of the membrane sheet. The membrane sheet is coarse-grained, such that it can be considered pseudo-homogeneous and continuous, without phase changes. In this coarse-grained representation, the electrodes are subdivided into small areas ΔxΔy around points (x, y), where Δx and Δy are substantially larger than any pore channel diameter in the catalytic layer but smaller than L and M. When the cell potential across the membrane is measured between these two points, its value is
2. System Variables Consider again the cell in Figure 1. The catalytic layers are nanoporous as described above but appear to be homogeneous on the micrometer scale. The close-up in the bottom of the figure shows the uniformly packed nanoporous material. The catalytic layers are connected to electric current collectors at the sides of the layers (not shown in the figure). The rectangular membrane sheet has the area L M. A point on the planar membrane surface has coordinates (x, y). The catalytic layers on each side of the membrane are of a height H, typically 10-20 μm. This height, measured along the z axis with the membrane-catalyst interface at position z = 0 (see the bottom of Figure 1) is one design variable in the study. The other variable is the macroporosity field ε(x, y), where the macroporosity ε is the volume fraction occupied by macropores, i.e., pores wider than 50 nm. The catalytic layers are supplied with oxygen (air) or hydrogen gas. The oxygen (air) supply system is sketched in Figure 1 as a serpentine channel. There are numerous flow-field designs for the gas supply system,49,50 and their large impact on the cell performance is well-established.21 Serpentine channels or parallel pathways are most frequent in PEMFC. The flow field is machined as channel(s) into the so-called bipolar plates. The bipolar plates are clamping the MEA. The close-up in Figure 1 illustrates the porous transport layer and the nanoporous layer. Oxygen is consumed along the serpentine channel, as indicated by the graded coloring. The oxygen concentration at the boundary of the catalytic layer is therefore a function of x, y, and z. Oxygen is distributed to the nanoporous catalytic layer by diffusion from the open bottom of the flow-field channel, first into a porous transport layer (denoted PTL in the figure) and then into the catalytic layer. The PTL functions as a gas distribution system. Gas obtains access to the area under the land, L, of the bipolar plate via the micrometer-sized pores in the PTL. The solution to the questions of higher energy efficiency with minimum amount of catalyst shall be found by redesigning the gas supply system and introducing an appropriate macroporosity field ε(x, y) and height H of the catalytic layer.
E ¼ - ΔG=F - Rjðx, yÞΔxΔy - ηc
ð3Þ
The values of E, ΔG, and the overpotential ηc refer to the average concentrations in the local area ΔxΔy. We have used superscript c to indicate from now on that we will associate the overpotential with the cathode reaction only, an assumption that can be relaxed later. The current density is j(x, y), and R is again the cell resistance. We shall assume that the electric resistance of the layer, R, has the same value everywhere. This is reasonable when the resistance to proton transport is constant, which it is when the membrane is saturated with water.16,27 The values of E, ΔG and ηc are functions of the local concentrations at the catalyst surface. Concentration gradients can develop in the z direction as well as the x and y directions in the catalytic layer, the porous transport layer, as well as the nanoporous layer.18,19,21,22 Sun et al.21 demonstrated variations in ηc(x, y) up to 0.1 V in the lateral direction from such gradients. The production of electric current can thus be diffusion-limited. The total electric current, I, is the integral of the local current density j(x, y) over the membrane area Z LZ M jðx, yÞdxdy ð4Þ I ¼ 0
0
(27) Kjelstrup, S.; Røsjorde, A. J. Phys. Chem. B 2005, 109, 9020– 9033. (28) Hoare, J. P. The Electrochemistry of Oxygen; Interscience: New York, 1968. (29) Trasatti, S. In Electrochemical Hydrogen Technologies; Wendt, H., Ed.; Elsevier: Amsterdam, The Netherlands, 1990; p 1. (30) Kinoshita, K. Electrochemical Oxygen Technology; Interscience: New York, 1992. (31) Parthasarathy, A.; Martin, C. R.; Srinivasan, S. J. Electrochem. Soc. 1992, 138, 916–921. (32) Parthasarathy, P.; Srinivasan, S.; Appleby, J. J. Electrochem. Soc. 1992, 139, 2530–2537.
3. Optimal Thermodynamic State 3.1. Global and Local Properties. The aim is to find local designs that result in the optimal global performance of the fuel cell. The global and local cell potential and current densities are therefore central. By global (or total) cell potential Etot, we mean the cell potential measured between the terminals of the current collectors from a 5099
Energy Fuels 2010, 24, 5097–5108
: DOI:10.1021/ef100610w
Kjelstrup et al.
gradient obeys
While j(x, y) can vary, we shall always compare cases for which I is the same. The total consumption of oxygen per unit membrane area in the stationary state, J0, is proportional to the production of electricity I j ¼ ð5Þ J0 ¼ 4FLM 4F
rpðx, yÞ ¼ C1
where C1 is a constant. Gheorghiu et al. found that this condition was obeyed in the bronchial tree of the human lung. In the electrocatalytic layers, uniform entropy production means that the overpotential is the same everywhere or that ð10Þ ηc ðx, yÞ ¼ C3
and the consumption per unit volume of electro-catalyst I j ¼ ð6Þ Y ¼ J 0 =H ¼ 4FLMH 4FH
Condition 10 can be restated as
where j is defined in eq.11. In an electrochemical cell, unlike a chemical reactor, the yield is set by the current density and cannot change once that is given. The optimization will therefore not increase the yield but reduce the overpotential from the one observed in the presence of mass-transfer limitations ηc to the one given by the Butler-Volmer equation. The Butler-Volmer equation relates the current density at the cathode to the cathode overpotential in the absence of mass-transfer limitations33 j ¼ j0 ½eð1 - βÞη
BV
F=Rg T
- e- βη
BV
F=Rg T
jðcx , cy Þ ¼ j ¼ C4 cðHÞ ¼ c0
ð11Þ
When the current density is the same everywhere, the concentration at z = H is constant. In the ideal case, the concentration is equal to the one at the inlet to the flow system, c0. The overpotential is by far the largest contributor to the entropy production in the cell.27 The supply system is important because it is essential to realize condition 10, but its entropy production is negligible. Condition 9 is therefore needed to realize 10, because the overpotential strongly depends upon the oxygen concentration, cf. eqs 7 and 8. This allows us to separately optimize the gas supply system and the catalytic layer.
ð7Þ
where β denotes the overall transfer coefficient and the convention is used that the cathode current and overpotential are both negative. The overpotential from the ButlerVolmer equation is given superscript BV to distinguish it from the measured ηc outside of the Butler-Volmer regime. A sign of mass-transfer limitations is that |ηc| > |ηBV|. In the absence of mass-transfer limitations, the cathode reaction has been observed to follow first-order kinetics with respect to oxygen (see refs 31 and 32 and references therein), giving Y ¼ kc
ð9Þ 11
4. Optimal Gas Supply System According to eqs 9 and 11, the gas supply system at each side of the MEA must obey the following: (1) Gas must be delivered as uniformly as possible to all positions (x, y, H) at time t. (2) Water must be removed as uniformly as possible from all positions (x, y, H) at time t. In addition, in the steady state, the rate of supply per unit membrane area, J0, must match the rate of consumption, j/4F, cf. eq 6. Required, therefore, is a gas supply system that delivers gases to and removes water from any position (x, y, H) as uniformly as possible and according to conditions 9-11. A continuous serpentine channel, as the one pictured in Figure 1, does not fulfill these criteria. Neither does a parallel flow field. To find an optimal structure starting from scratch, with all geometrical variables free, is not trivial. It is known, however,11 that the human lung as a gas distributor is characterized by the very same conditions as given by eqs 9 and 10. The human lung has two flow regimes: one for convective flow and one for diffusion.34 The structure is such that the entropy production is constant in both parts, indicating that the entropy production is minimal for the total structure.11 This is exactly the situation that we want to achieve and why we take the bronchial tree as a source of inspiration. The first part of the bronchial tree is a transport network, which has as sole purpose to deliver oxygen rapidly and uniformly to the second part, the alveolar sacs.34 There are 14-16 generations of branches in the first part. This is precisely the number of branches that are needed to slow the flow of gas to a rate compatible with the rate in the diffusional regime.35-37
ð8Þ
where k is the kinetic coefficient of the forward reaction (in s-1) and c is the concentration of oxygen gas (in mol/ m3). Equations 6-8 show that k depends upon the overpotential ηBV , as also discussed by Sun et al.21 3.2. Criteria for Minimum Entropy Production. It was shown numerically that processes with a constant demand on the production13,12 and with sufficient degrees of freedom obtain their optimum thermodynamic efficiency (i.e., the state with minimum entropy production) when the process is operated along a path with constant entropy production. In the present case, the process path consists of a series of five steps. The first step is the diffusion of oxygen and water through the PTL at the cathode. The next step is the electrochemical reaction and transport in the electrocatalytic layer. The third step is the proton and water transport between the anode and cathode. The fourth and fifth steps are the reaction at the anode and the transport through the PTL on the anode side. Each of these steps should have constant entropy production according to the highway hypothesis. There is one constraint, namely, that of constant total electricity production, I, from the area L M. The local electricity production j(x, y) is free in the outset. There are no geometric or other variable constraints. Under these conditions, an optimal state can be postulated for the fuel cell, with constant local entropy production. In the gas supply and water removal systems, this means that the pressure gradients are constant in all directions.11 The construction must be such that the pressure
(34) Weibel, E. R. Am. J. Physiol. 1991, 261, L361–L369. (35) Hou, C.; Gheorghiu, S.; Coppens, M.-O.; Huxley, V.; Pfeifer, P. In Fractals in Biology and Medicine; Losa, G. A., Merlini, D., Nonnenmacher, T. F., Weibel, E. R., Eds.; Birkh€auser Verlag: Basel, Switzerland, 2005; Vol 4, pp 17-31. (36) Murray, C. D. Proc. Natl. Acad. Sci. U.S.A. 1926, 12, 207–214. (37) Huang, W.; Zhou, B.; Sobiesiak, A. J. Electrochem. Soc. 2006, 153, A1945–A1954.
(33) Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University Press: Oxford, U.K., 1994.
5100
Energy Fuels 2010, 24, 5097–5108
: DOI:10.1021/ef100610w
Kjelstrup et al.
Figure 2. Schematic illustration of a single polymer electrode fuel cell with the suggested optimal gas distributor system and optimal catalytic layer. The layer consists of nanoporous columns of width w containing agglomerates of platinum distributed on carbon, plus pores of diameter d (in micrometers), with free access of oxygen gas. The catalytic layer has height Hopt and an oxygen supply with concentration c0.
The gas supply system for the fuel cell should be compared to the first part of the bronchial tree. According to the statements above, a tree-like channel structure, with the channels only open at the tips of all branches, can obey conditions 1 and 2. The system was investigated in detail by Tondeur and Luo.7 They designed a structure to distribute an input flow equally on a surface with 8 outlet ports/cm2 and gave the appropriate channel dimensions. While the gas supply system in the lung is three-dimensional, the gas supply system of the fuel cell can be quasi-two-dimensional when the channel diameters are small. It is furthermore practical to give the distributor channels the same height, meaning that the closed channels can be machined into planar bipolar plates. A proposal that matches these considerations is illustrated in the upper part of Figure 2, showing 16 gas outlets for the sake of illustration. Note that in each fractal distributor the distance between the inlet and outlet is identical, leading to uniform distribution. Therefore, the gas is delivered as uniformly as possible and practical over the membrane area, and products are collected likewise, to conform with conditions 9-11. For reasons that will become clear in the next section, we return to the water removal system after we have discussed the optimal catalytic layer. Four generations of branches are shown in Figure 2. This is here only an example. Two considerations are important for the number of branches. The tips should reach out to most of the membrane area. A large number of branches is favorable from this point of view. On the other hand, it should not be too difficult to manufacture the system. Space limitations and construction complications may thus limit the number. Also, the pressure drop increases with the number of generations. Right angles were chosen in the illustration in Figure 2 for reasons of simplicity. The particular form of branching of the fractal tree and, in particular, the angle of a branch with the inlet tube could give uneven pressure reductions at large gas velocities. This is probably not critical, because the entropy production in the gas distributor is negligible.27 The main purpose of the structure is not so much to minimize the entropy production in the structure itself but rather to provide boundary conditions for the MEA that produce minimum entropy in the next step of the process path.
We can now return to eq 11 and see the role of the PTL in view of the above design. If the supply and removal systems are carved into the bipolar plates, these plates can, in principle, be put directly on to the catalytic layer. Whether this is feasible depends upon the number of outlets. With the current ways to machine on the micrometer level, this may not be a limiting factor. It may then be an advantage to eliminate the porous transport layer, because its performance depends upon operating conditions.24 A PTL will under many conditions lead to masstransfer limitations and produce an extra diffusional regime. With a gas distributor system carved into the bipolar plate, on the other hand, one has pressure-driven flow all of the way up to the catalytic layer. This prevents the establishment of diffusional layers much better than the design shown in Figure 1. The single fuel cell in Figure 2 is thus drawn without the porous transport layer. The total height of the single cell is therefore smaller than that in Figure 1, making concentration gradients less likely than in Figure 1. 5. Optimal Catalytic Layer We next address the question, which local macroporosity, ε(x, y), and layer height, H, lead to the highest value of the total cell potential, Etot, as defined by eq 1, when cells are compared for constant oxygen consumption and constant j(x, y), cf. eq 11. Thus far, nanoporous catalytic layers have typically been used in the construction of membrane electrode assemblies (cf. Figure 1), and the focus in catalyst research has been on the chemical nature of the active layer.2 However, a nanoporous catalytic layer in the MEA might prevent optimal use of expensive catalyst material because of diffusion limitations. Some of the catalyst (Pt) might not be accessible in a construction like that in Figure 1 and, therefore, be used inefficiently. An additional macroporosity, ε, of this layer enhances reactant supply to the active sites in the catalyst, removing or reducing diffusion limitations. A too high macroporosity decreases the overall catalyst conversion per unit volume, because of the extra open space. The question therefore is how 5101
Energy Fuels 2010, 24, 5097–5108
: DOI:10.1021/ef100610w
Kjelstrup et al.
terms of the Thiele modulus, φ0, is R tanh φ0 V kcdV ¼ ηE ð1 - εÞkc0 V φ0
a certain oxygen consumption (current production) can be maintained over the membrane sheet as a whole, using a minimum amount of catalyst? The constant oxygen consumption per unit area, J0, was given by eq 5. The close-up in Figure 2 illustrates the situation that we want to optimize. The diameter of the macropores is denoted by d, and the width of the columns in the layer that contain the nanoporous catalyst agglomerates by w. The two parameters are related via the macroporosity by ε w ð12Þ d ¼ 1-ε
The Thiele modulus, φ0, is based on the macropores15 and is equal to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 1 - 1 ¼ Hq ð16Þ φ0 ¼ H D0 ε sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 1 with q ¼ -1 D0 ε
How can the reactants gain easier access to the active sites in the catalytic layer than is pictured in Figure 1 given that conditions 10 and 11 apply? Given a uniform concentration of gases at z = H of the catalytic layer, sufficient macroporosity should be introduced in this layer, hereby reducing the amount of catalyst needed while keeping up the same production of electricity. It was shown by Johannessen et al.14 and Wang et al.15 that a maximum yield is obtained within a macroporous catalyst layer of optimal porosity ε = εopt using a uniform distribution of macropores with a constant diameter d = dopt. It will be shown that similar formulas for εopt and dopt can be derived for the electrocatalytic layer of a PEMFC. Consider therefore a nanoporous catalytic layer penetrated by macropores, as sketched in the inset of Figure 2. The remaining part of the layer contains columns of nanoporous material with dispersed Pt. The volume of the catalyst contained in the layer is Vcat ¼ Hð1 - εÞLM
The reason for this is that, in the optimal catalyst layer, diffusion in the macropores dominates transport (diffusivity D0 and volume fraction ε), while the reaction occurs exclusively in the nanoporous catalyst matrix (volume fraction 1 - ε). These two definitions and eq 14 allow us to express the oxygen flux 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kð1 - εÞ5 ð17Þ J 0 ¼ c0 D0 kðε - ε2 Þtanh4H D0 ε In terms of system variables, the Lagrangian of the constrained optimization problem is therefore 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 1 0 0 -1 -J 5 f ¼ ð1 - εÞH þ λ4c D0 kðε - ε2 Þ tanh H D0 ε ð18Þ
ð13Þ
where λ is the Lagrange parameter. This function should be minimized with respect to ε, to minimize the amount of catalytic material in the volume V = (1 - ε)HLM, for a given current density or oxygen flux J0. The intrinsic rate constant k is constant in the optimization. The value of the constant is the one compatible with the (hypothetically valid) ButlerVolmer equation and the current density in question. We return to its determination in the next section. From the partial derivatives ∂f/∂H = ∂f/∂ε = ∂f/∂λ = 0, expressions are found for the optimal thickness of the catalyst layer, Hopt, and the optimal porosity of the catalyst layer, εopt. The equation for ∂f/∂H = 0 is solved for the Lagrange multiplier λ, and the result is substituted into ∂f/∂ε. This reduces the three equations for three unknowns to the following two equations:
rather than HLM. We assume a constant concentration of Pt within the nanoporous columns. The amount of Pt is directly proportional to Vcat. The macropores in the layer serve to distribute oxygen to the catalyst in the nanoporous columns by bulk diffusion. If the macroporosity is very low (ε f 0), the path becomes long and the observed rate of a diffusion-limited reaction becomes controlled by the effective diffusivity, De, through the nanopores. Typically, De , D0, the bulk diffusivity of oxygen at the same temperature and pressure. If the macroporosity is large, the macropores will serve as “highways” for the oxygen to reach the active Pt sites in the nanoporous material more easily. If there are too many such highways or they are too broad, the overall activity will decrease for lack of active catalytic material. The intrinsic activity of the catalyst is directly related to the geometrical and chemical structure at the (sub)nanometer scale, e.g., the nanopore size of the support, Pt/C particle sizes, and Pt/support interaction.1 These quantities are not treated as design variables in the optimization but rather as system properties. The results of the presented optimization can be adapted to other kinetics or a different local effective oxygen diffusivity, De. We show in the Appendix that the overall oxygen flux, that is, the flow per unit area of the catalytic layer, J0, can be derived in terms of the intrinsic rate constant from eq 8, the thickness, the macroporosity of the layer, and the catalyst effectiveness factor, ηE J 0 ¼ ð1 - εÞHηE kc0
ð15Þ
J 0 ¼ c0 D0 εq tanhðHqÞ 1 ε ¼ 2
ð19Þ
for the two unknowns ε and H, with φ0 = Hq. Equation 19 has a physical solution if there exists a pair (ε, H) such that the following flux condition is satisfied: 1 pffiffiffiffiffiffiffiffiffi ð20Þ J 0 e c0 D0 k ¼ J max 2 This condition states that sufficient system resources (c0, D0, and k) must be available to produce the desired flux. Clearly, not much oxygen reacts if c0, D0, and k are minuscule. The dimensionless ratio J0/Jmax can be used to measure how far the system operates from the limiting value Jmax. Equation 21 defines the limiting property R J0 J0 1 ffiffiffiffiffiffiffiffiffi e p ð21Þ ¼ R ¼ 0 2 2J max c D0 k
ð14Þ
This assumes that the fraction (1 - ε) of the layer volume is catalytically active. The definition of ηE and its expression in 5102
Energy Fuels 2010, 24, 5097–5108
: DOI:10.1021/ef100610w
Kjelstrup et al.
If eq 21 is fulfilled, then there is a unique solution εopt and Hopt given by 1 ! ! 2rffiffiffiffiffiffi rffiffiffiffiffiffi D0 2J 0 D0 J0 -1 -1 pffiffiffiffiffiffiffiffiffi ¼ ¼ tanh tanh k k J max c0 D0 k
εopt ¼ Hopt
ð22Þ Remarkably, the optimal catalyst structure for this example is universally given by the macroporosity εopt = 1/2. Furthermore, Hopt only depends upon system variables. This means that each MEA should best be designed with a tailored height Hopt and a uniform macroporosity of 1/2. This optimum value was also found by Johannessen et al.,14 even though their constraints were different. Johannessen et al.14 maximized the yield using a constant volume rather than minimizing the amount of catalyst with a constant production. As already mentioned in sections 1 and 2, it is a premise that the ohmic resistance of the cell is kept constant in the optimization. It is reasonable that the proton conductivity is constant when the membrane is fully hydrated.27,16 The electronic conductivity of the carbon supports is high and can be taken constant.27 The electric resistance of the catalytic layer is dominated by the proton transport of the monomers of this layer. This is unknown, but it is reasonable that it does not change much in the optimization, because numerous nanoporous polymer columns are kept in contact with the membrane. The relation between the width of the nanoporous catalyst columns in the layer (w) and the pore channel diameter (d) is given by eq 12, and because εopt = 1/2 wopt ¼ dopt
Figure 3. Dimensionless volume of the catalytic layer, Vcat/(J0LM/ kc0)), as a function of macroporosity ε and stress factor R.
ð23Þ
The minimum nanoporous catalyst volume hence becomes ! rffiffiffiffiffiffi 1 D0 J0 -1 Vcat, opt ¼ LM ð24Þ tanh 2 k J max According to Wang and Coppens,5 there is an upper limit for the pore wall thickness, to avoid strong diffusion limitations in the walls. This wall thickness wopt satisfies5 rffiffiffiffiffiffi De wopt e 0:2 ð25Þ k
Figure 4. Dimensionless volume of the catalytic layer, Vcat/(J0LM/ kc0), as a function of macroporosity ε, for R = 0.20 (outer curve) and 0.48 (central curve). Asymptotic values for R are shown by vertical, dotted lines.
For every porosity there is a unique catalyst volume, Vcat = Vcat(ε) that produces the given flux, J0. This catalyst volume away from the minimum is given by " # qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2J max LM J 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vcat ðεÞ ¼ ðε - ε2 Þ tanh kc0 2J max ðε - ε2 Þ pffiffiffiffiffiffiffiffiffiffiffiffi! ε - ε2 J 0 LM R -1 pffiffiffiffiffiffiffiffiffiffiffiffi tanh ¼ R kc0 ε - ε2
This equation, which serves as an upper bound for the nanoporous column width also here, must be used in combination with the result that wopt = dopt, in the production of the perforated layer. To study the sensitivity of the minimum catalyst volume to the porosity, which may be difficult to control in practice, consider the variation ð26Þ ε- e ε e ε þ
ð29Þ
where
2
ε(
3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u u 0 16 J 7 ¼ 41 ( t1 5 2 J max
From eqs 24 and 29, we see that Vcat(εopt) = Vcat(1/2) = Vcat, opt. It is also of interest to see how close Vcat(ε) is to the maximum production limit, as given by eq 21. The value of the ratio Vcat/(J0LM/kc0) is plotted as a function of ε and R in Figure 3. In the figure, ε( is given by eq 27 and R by eq 21. We see from Figure 3 that the minimum is rather flat for a range of macroporosities around 1/2. The higher the value of R, the smaller the allowed variation. Clearly, if we set a value for J0 so that R is close to 1/2, the system becomes stressed in the sense that it will tolerate less
ð27Þ
For every such ε, there is a unique catalyst layer height such that the flux condition 24 is satisfied. This height is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # D0 ε J0 -1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HðεÞ ¼ ð28Þ tanh kð1 - εÞ 2J max ðε - ε2 Þ 5103
Energy Fuels 2010, 24, 5097–5108
: DOI:10.1021/ef100610w
Kjelstrup et al. Table 1. Predicted Savings on Catalyst Material Using a Pure Oxygen Feed at 353 Ka current density (A m-2) overall transfer factor, β overpotential, ηBV (V) intrinsic rate coefficient, k0 (s-1) intrinsic rate coefficient, k (s-1) optimal layer height, Hopt (μm) Thiele modulus, φ0 effectiveness factor, ηE
500 1.0 0.192 3.8 NA 20. 0.0165 1.00
15000 1.0 0.296 NAb 117 19.6 0.089 0.997
15000 0.8 0.371 NA 415 5.5 0.048 0.999
a The exchange current density is 0.9 A/m2 membrane. b NA = not applicable.
Table 2. Predicted Savings on Catalyst Material Using an Air Feed at 353 Ka
Figure 5. Minimum dimensionless catalytic volume, Vcat/(J0LM/ kc0)), as a function of stress factor R, at an optimal value 0.5 of the macroporosity ε.
current density (A m-2) transfer factor, R overpotential, ηBV (V) intrinsic rate coefficient k0 (s-1) intrinsic rate coefficient k (s-1) optimal layer height, Hopt (μm) Thiele modulus, φ0 effectiveness factor, ηE
deviations from the calculated optimum. This is shown in more detail in Figures 4 and 5. Figure 4 presents the dimensionless volume as a function of ε. It is shown that we can tolerate a deviation in the macroporosity of (0.3 for R = 0.2 (blue curve). The tolerance is still high ((0.15) when R approaches 0.5 (purple curve). This means that the optimal solution is not so sensitive to the value of ε when the current density is low enough. Figure 5 shows that small catalyst volumes are associated with small productions of oxygen, large oxygen concentrations, or large diffusion coefficients and/or kinetic rate constants, via eq 29. The system becomes stressed and requires relatively more catalyst volume when the combination of these variables leads to R > 0.3. According to eq 23, the width of the nanoporous columns in the optimal layer is equal to the pore width. We do not have exact information for these parameters, only their upper boundary, eq 25. Figures 3-5 show that the results are not so sensitive to the macroporosity, even though it is important to keep within the limiting values of R. However, as long as the porosity is 0.5 on the average and eq 25 is obeyed, we may introduce pores of micrometer dimensions, as tightly and uniformly packed as practical. In the actual production of the catalytic layer, one may consider leaving a minor part of the nanoporous catalytic layer closest to the membrane, to protect the membrane on the anode side from drying out. For PEMFCs, the catalyst density is normally given as a loading in milligrams per area of a particular catalytic layer of height Hnano. The minimum catalyst loading is then Hopt F Hopt mopt ¼ Fð1 - εopt Þ ¼ ð30Þ 2 Hnano Hnano
500 1.0 0.235 19 NA 20 0.04 1.00
15000 1.0 0.339 NAb 577 20 0.2 0.986
15000 0.8 0.423 NA 2760 4.1 0.09 0.997
The exchange current density is 0.2 A/m2 membrane. b NA = not applicable. a
For a numerical illustration of the optimization method, consider the standard E-TEK Elat/Std/DS/V2 gas diffusion electrode.39,40 The catalyst layer in the E-TEK electrode has a loading of 0.5 mg/cm2 Pt and is homogeneous on a micrometer scale.38 A typical thickness is Hnano = 10 μm.
The example is meant to demonstrate the method and indicate that savings are plausible. Other porous electrodes and catalysts can be dealt with similarly. We consider two cases: when the system is fed by pure oxygen and when it is fed by air. Results are given in Tables 1 and 2, respectively. The procedure consists of the following steps: (1) Determine the parameters of the Butler-Volmer equation, eq 7, for the catalyst in question and, accordingly, the corresponding intrinsic reaction rate coefficient k = k0, described in section 3.1. The rate coefficient k0 represents conditions without mass-transfer limitations. (2) Consider a region of the polarization curve where mass-transfer limitations can be expected, that is a value of the current density that gives ηc > ηBV, where ηc is measured and ηBV is calculated from eq 7. Determine the intrinsic rate coefficient, k(ηBV), corresponding to ηBV( j) for the given current density or the rate of oxygen consumption J0 from eq 5. (3) Calculate the maximum oxygen consumption of the layer per unit membrane area, Jmax, given the characteristic transport data and eq 20 to assess the possibility for better catalyst use. (4) If J0 < Jmax, we can take advantage of the solution eq 22 that gives optimal height of the catalytic layer. The optimized macroporosity (here 0.5) should next be introduced in the catalytic layer. (5) Estimate the upper boundary of the pore width from eq 25 to assess whether it is feasible to implement the macropores in the layer. (6) Compare the original height Hnano of an existing nanoporous layer to the calculated optimal height Hopt of the catalytic layer with macropores, to estimate the savings from eq 30. 6.1. Parameters of the Butler-Volmer Equation, The Intrinsic Rate Coefficient. A good estimate for j0 is 0.9 A/m2 membrane in the presence of oxygen,41,42 while β = 1.0 is typical for the high-voltage regime.18,21 This corresponds to 0.8 μA/cm2 Pt, with the specific area of Pt in this electrode.41 We also used β = 0.8; see Tables 1 and 2. By scaling this value
(38) Møller-Holst, S. Denki Kagaku 1996, 64, 69–705. (39) Meland, A.-K.; Kjelstrup, S.; Bedeaux, D. J. Membr. Sci. 2006, 282, 96–108. (40) Meland, A.-K.; Kjelstrup, S. J. Electroanal. Chem. 2007, 610, 171–178.
(41) Meland, A.-K. Impedance diagrams of the electrodes in the polymer electrolyte membrane fuel cell. Ph.D. Thesis, Norwegian University of Science and Technology, Trondheim, Norway, 2007. (42) Ciureanu, M.; Wang, H. J. Electrochem. Soc. 1999, 146, 4031– 4040.
where F is the amount of Pt per area of the nanoporous catalyst layer. 6. Optimization of an E-TEK Electrocatalytic Layer
5104
Energy Fuels 2010, 24, 5097–5108
: DOI:10.1021/ef100610w
Kjelstrup et al.
variables do not differ too much from the optimum.46,47 This means, for the present case, that the macropores could form a slightly tortuous pore space with macroporosity ε ≈ 0.5 μm around particles of a size dopt. 6.5. Optimal Catalytic Layer. With these values in mind, we proceed to optimize a nanoporous electrode that shows mass-transfer limitations. We know that the new electrocatalytic layer should have a macroporosity of 0.5, given the model that we have used. The optimal layer height is calculated from eq 22. We see then from Tables 1 and 2 for j = 15 kA/m2 that the high potential value β = 1.0 does not give any savings, while a more likely value of 0.8 units leads to substantial savings; compare the last columns of both tables. The tables show that the optimal thickness of the catalytic layer becomes 5.5 μm in oxygen and 4.1 μm in air, respectively. A further reduction of β to 0.7 will cut the value to 2.0 μm. In comparison to a typical thickness of 10 μm for such membranes and with a macroporosity of 0.5, these typical values of β represent a 4-8-fold reduction in the need of Pt, with higher saving in air than in oxygen. This reduction is substantial. The Thiele modulus of the optimal layer is small in both cases, as expected; hence, the efficiency factor of the optimal catalyst layer, ηE ≈ 1. By replacing oxygen by air, we obtain a system that operates much closer to the theoretical limit of R; see Figure 5. The performance then becomes more sensitive to the choice of geometric variables. This is also expected. In this manner, one can use the Thiele modulus approach to reduce mass-transfer limitations. The optimal value of H is a consequence of enhanced access to the catalytic site and the intrinsic rate coefficient of the layer. The optimization did not increase the yield but reduced the overpotential from the one observed in the presence of mass-transfer limitations ηc to the one given by the Butler-Volmer equation ηBV. 6.6. Gas Supply System. The overall structure of the gas supply system (as well as the water removal system; see below) now follows from eqs 9 and 11. The dimensioning of the channels and the number of branches follow from the gas supply and water removal rates. A total of 16 outlets are drawn. A larger number of outlets may be needed to eliminate the need for a PTL altogether. The ease of construction is an important aspect in industrial applications. The hierarchical structure and modularity of the present design will allow the designer to make optimal use of advances in micromachinery, templating methods, and lithography. We refer to microelectromechanical systems (MEMS), microfluidics, and microreactors that are now increasingly employed in various industries, from microelectronics to chemical production and energy systems. 6.7. Water Removal System. The water transport problem in the fuel cell can also be addressed in view of the above findings. It follows from the optimal solution, eqs 9, 10, and 22, that water transport out of the cell can be enhanced, because the new situation has macropores in the catalytic layer. Water is produced in the nanoporous layer, leading to a local pressure buildup and clogging of nanopores near the
with respect to the oxygen concentration, we obtain j0 = 0.2 A/m2 membrane for the cell in the presence of air, in agreement with Eikerling and Kornyshev.1 Exchange current densities for pure Pt surfaces are larger.31,32 To establish a reference value for the intrinsic rate coefficient, k0, we used a current density well within the ButlerVolmer regime, 500 A/m2. The corresponding oxygen consumption per unit membrane area was J0 = 1.3 10-3 mol m-2 s-1, giving Y = 130 mol m-3 s-1 from eq 6, with H = 10 μm. The oxygen concentration corresponding to 1 bar pure oxygen is c0 = 34.0 mol/m3, while it is 6.8 mol/m3 in air of 1 bar. The corresponding rate coefficients become k0 = 3.9 s-1 in oxygen and k0 = 19 s-1 in air; see Tables 1 and 2. The value does not change substantially with the relevant values of β. The thickness of the optimal layer, Hopt, becomes 20 μm in these cases. All of the catalyst is used. 6.2. Intrinsic Rate Coefficient in a Regime with MassTransfer Limitations. As a representative current density in the regime where mass-transfer limitations may occur, we chose 15 kA/m2. In this regime, ηc > ηBV (absolute values are considered from now on), because the overpotential has contributions from concentration gradients. The intrinsic rate coefficient k that we would like to achieve is the one that corresponds to the (hypothetical) situation with no masstransfer limitations. It can be calculated from eq 7, assuming that the Tafel equation applies ! k βðηBV - ηBV 0 ÞF ¼ exp ð31Þ k0 Rg T is the overpotential calculated for 500 A/m2. where ηBV 0 Results for k for j = 15 kA/m2 were next found from eq 31; see Tables 1 and 2. 6.3. Maximum Oxygen Consumption of a Layer of Thickness Hnano. The diffusion coefficient for oxygen at 1 bar and 80 C is D0 = 5.6 10-6 m2/s. The maximum flux Jmax can be calculated for this E-TEK catalytic layer from eq 20 using the value of k for the high current density. The maximum flux becomes about 2 orders of magnitude larger than the yield at this current density for pure oxygen as well as for air, indicating that the catalyst material is underused and that an optimization is in place. 6.4. Upper Pore Width of the Macroporous Layer. The effective diffusivity in the nanoporous catalyst from the effective medium theory is De = 3 10-9 m2/s.22 Using this value and k from eq 31, we find an upper boundary eq 25 for the pore diameter; the highest value is 0.5 μm. The catalytic layer should accordingly be evenly intersparsed with macropores of diameter approximately 0.5 μm or smaller. This diameter is realizable in practice. Fortunately, according to Figures 4 and 5, its value is not critical for optimal performance. The macropore diameter is equal to the width of the nanoporous columns. The macropores do not need to be parallel to achieve minimum diffusion limitations.14,15 The results are more sensitive to ε than to d or w, as long as the latter (43) Kearney, M. M. Chem. Eng. Prog. 2000, 96, 61–68. (44) Coppens, M.-O. Method for operating a chemical and/or physical process by means of a hierarchical fluid injection system. U.S. Patent 6,333,019, 2001. (45) Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design, 2nd ed.; Wiley: New York, 1990.
(46) Wang, G.; Coppens, M.-O. Ind. Eng. Chem. Res. 2008, 47, 3847– 3855. (47) Coppens, M.-O.; Wang, G. In Design of Heterogeneous Catalysts; Ozkan, U., Ed.; Wiley: Weinheim, Germany, 2009; pp 25-58.
5105
Energy Fuels 2010, 24, 5097–5108
: DOI:10.1021/ef100610w
Kjelstrup et al.
catalytic site. The problem can be reduced by opening up this layer. Additions of hydrophobic material to the nanoporous catalyst exposed to the macropores may further reduce the resistance to flow. From the perspective of water removal only, a PTL, such as in Figure 1, is a disadvantage.20,24 If the PTL is not needed for gas distribution above, the layer could be removed. For the bipolar plate, it further follows that a hierarchical scheme will facilitate water management. The number of distributor/collector tips and their diameters, the coating of the materials, and the width and organization of the large pore channels are design variables that require further optimization. In this second step, experimental studies will be very important and are on the way. 6.8. Improvements in the Energy Efficiency. The improvement in the energy efficiency that follows from the optimization of the catalytic layer follows from the reduction obtained in the overpotential. The outcome of the optimization is that the same electricity is now produced with the overpotential ηBV rather than the larger value ηc. The difference in these values varies between 0.1 and 0.2 V over a range of high current densities.18,19,21,22 To avoid such a reduction means a large improvement in the energy efficiency. Recent results in the literature add support to our thesis on the gas supply system. Kloess et al.26 reported a 30% improvement in the performance of the fuel cell, by making the supply structure leaf-like. Large savings like this might well be within reach, according to the above findings.
catalytic layer, keeping the anode side of the membrane covered by a very thin layer to prevent it from drying out. An upper boundary exists for the pore width, but the performance is not so sensitive to the operating conditions (given by Figures 3-5), provided that operation takes place sufficiently far away from the maximum available current, which was also calculated. The gas supply system should deliver reactants uniformly to the whole membrane sheet, meaning that all outlets of the quasi-two-dimensional construct have the same distance from the inlet, which can be achieved by a fractal distributor. Water should be removed via a similar construct. Our result is supported by the finding of Wang et al.,46 who were able to improve the catalytic activity for removal of NOx from flue gases by a factor of 2-3 by introducing a macroporosity of 0.2-0.4. We have thus presented a method that predicts that significant catalyst savings are possible. It is reassuring to find that the optimal solution for the catalytic layer is the same as that obtained in other contexts.14,15,46 This adds credibility to the present results, despite the boundary conditions being somewhat different here. Special here is that the intrinsic kinetic constant is a function of the overpotential. To calculate the optimal height of the catalytic layer, we need to know the diffusion coefficient, the oxygen concentration, the oxygen demand, and the rate constant, cf. eq 22. The last two parameters vary, but their variations may not have a large impact on the results within a certain range, as demonstrated by Figures 4 and 5. To quantify the savings, a consistent data set is needed on these variables as well as on the ohmic resistance for the set of materials to be optimized. We have predicted a fuel cell design compatible with the hypothesis for the state of minimum entropy production. The gas distributor system, inspired by nature,11 enables us to achieve constant local entropy production, mimicking fractal distributors and injectors proposed for a variety of chemical mixing, separation,43 and multi-phase reaction44 processes. Bio-inspired flow channel designs have been proposed and tested already.26 These authors aimed for a more uniform distribution of gases over the membrane sheet, without formulating a physical reason for it, however. Their results give support to ours. Their leaf and lung designs gave a 30% improvement in energy performance of the PEMFCs. The present work not only explains why this is so but also gives a mathematical procedure for further improvements, resulting from a rational, nature-inspired design procedure rather than from an empirical, biomimetic procedure. The question of cell lifetime can be addressed against the background of the condition set, of uniform distribution of entropy production or energy dissipation. More uniform conditions across the cell represent less stress on the catalyst and the transport system. It is therefore expected that also the cell lifetime benefits from the new design. Also, a constructal theory9 has been used to successfully predict the optimal size of distributor systems (organs). According to this theory, an optimum exists, given by the trade-off between losses of useful energy by flow enhancement and losses for energy maintenance of the same structure. Such a trade-off argument is not sufficient for an analysis of the fuel cell, however. The entropy production (the loss of useful energy) in the fuel cell occurs mainly in the MEA, while the distributor system is used to minimize the losses outside the distributor and achieve uniformity. Modeling as well as experimental results of many authors show that clogging of nanosized pores is responsible for
7. Discussion and Conclusions This work has demonstrated a method to design optimal catalytic layers and gas supply systems in fuel cells, inspired by the geometry of the lung and the hypothesis for the state of minimum entropy production of irreversible thermodynamics.12,13 We were able to combine principles previously used in reactor and catalyst designs48,15,14 with thermodynamic principles12,13 and present a new, systematic procedure for energy and material cost reductions. The two optimization methods have been used before but only separately and for different systems. Chemical reactors and other process units have been optimized with the entropy production as an objective function, 12,13 and catalyst reduction methods have been studied in reactor and catalyst designs.14,15 This is the first time that both methods are applied in combination and applied to fuel cells. The presented methodology is general and applies to any type of catalyst in a nanoporous catalytic layer. A numerical example of its application to the E-TEK electrode, standard in low-temperature fuel cells, is promising. We have found that the amount of catalyst can be lowered by a factor of 4, using published values for the standard E-TEK electrode. This electrode has a loading of 0.5 mg/cm2, but more efficient electrodes are now available, making coatings of 0.2 mg/cm2 more typical. From the present optimization method, we therefore see that the target set by the U.S. DOE of 0.03 mg/cm2 comes within reach. According to the present work, one way to achieve this goal is to first introduce an optimized macroporosity into the (48) Coppens, M.-O. Ind. Eng. Chem. Res. 2005, 44, 5011–5019. (49) Li, X.; Sabir, I. Int. J. Hydrogen Energy 2005, 30, 359–371. (50) Debe, M. K.; Herdtle, T. Flow field. U.S. Patent 6,780,536 B2, Aug 24, 2004.
5106
Energy Fuels 2010, 24, 5097–5108
: DOI:10.1021/ef100610w
Kjelstrup et al.
significant performance losses, of the order of 10% of the total potential or more; e.g., see ref 20 and references therein. When macropores are introduced in the catalytic layer, one may enhance greatly the possibility for water to escape. This possibility to escape is further enhanced when we can avoid the porous transport layer altogether. On top of that, one may be able to tailor the water removal channels, which mimic the supply gas distributor of Figure 2, by making the walls hydrophobic. Heuristically, it follows that a stack is optimal in the thermodynamic sense when all single cells obey optimal conditions. The constants may differ between the cells in a stack if the oxygen condition at the inlet varies from one cell to the next. However, they may be the same if all cells are fed by the same supply. Construction of cell stacks can thus be made following the same guidelines. The design of each membrane sheet in the stack is optimized as a function of its individual oxygen consumption and the particular oxygen concentration that is fed to the sheet. The method presented is not specific to the PEMFC and may thus also prove beneficial to other fuel cells or electrochemical generators with microporous electrodes. The results remain to be validated experimentally by building a cell, proving that a better energy efficiency can indeed be realized in practice for the proposed structure. It is important to establish all cell characteristics, because the loss at low potentials may not only be due to mass-transfer limitations.23 The present results, supported by the studies of Kloess et al.,26 indicate that this may be worthwhile, because the gains seem to be significant.
enters the catalyst layer, in either large pore channels with diffusivity D0 or the nanoporous catalytically active part with effective diffusivity De, where De , D0. The production per unit interfacial (external) membrane area is Z Z D0 Dc=Dz þ ð1 - εÞ De Dc=Dz , z ¼ H J0 ¼ ε pores
If the catalyst layer thickness H , L, M and d, w , H, Johannessen et al.14 showed that the catalyst layer can be described by an effective one-dimensional continuum model, so that dc ,z ¼ H ð35Þ J 0 ¼ εD0 dz Steady-state diffusion and reaction in the optimal slab is hence described by d dc εD0 ¼ kð1 - εÞc ð36Þ dz dz In the optimized catalyst, there are no local diffusion limitations inside the particles or walls of nanoporous material (width w), so that the resistance is confined to the macropores. Therefore, the effective diffusivity is dominated by the macropores (εD0). The presence of tortuous macropores would introduce a correction (εD0/τ, with tortuosity τ ≈ 1-3). We assume straight macropores, that is, τ = 1. The solution to this equation is well-known.45 Using the definition of the Thiele modulus for this problem sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 1 -1 ð37Þ φ0 ¼ H D0 ε
Acknowledgment. The authors are grateful to the Center for Advanced Study at The Norwegian Academy of Science and Letters for extraordinary sabbatical stays. This material is also based upon work supported in part by the U.S. National Science Foundation under Grant DGE-0504361 (MOC) and the U.S. Department of Energy under Award DE-FG0207ER46411 (PP).
as well as the boundary condition for dc/dz at z = 0, integrating eq 36 gives the concentration z ð38Þ cðzÞ ¼ A cosh φ0 H and
Note Added after ASAP Publication. The affiliations were modified from the original version published on August 10, 2010. This version was reposted on September 1, 2010.
A steady-state diffusion and first-order reaction in a nanoporous, Pt-containing catalyst are described by D Dc D Dc D Dc þ þ ¼ kc ð32Þ De De De Dx Dx Dy Dy Dz Dz
A ¼
HJ 0 1 εD0 φ0 sinh φ0
and the concentration variation HJ 0 cosh φ0 z=H cðzÞ ¼ εD0 φ0 sinh φ0
Oxygen flux continuity can be imposed at the interface between the large pores and the nanoporous material. The same applies for the concentration, because the degree of adsorption of the gases on carbon is low. A set of boundary conditions in agreement with eqs 9-11 are 0, x ¼ L : Dc=Dx ¼ 0 0, y ¼ M : Dc=Dy ¼ 0 H: c ¼ c0 0: Dc=Dz ¼ 0
ð39Þ
giving us the constant A
Diffusion in large pore channels, where there is no reaction, satisfies D Dc D Dc D Dc D0 þ D0 þ D0 ¼ 0 ð33Þ Dx Dx Dy Dy Dz Dz
¼ ¼ ¼ ¼
dc φ z ¼ A 0 sinh φ0 dz H H
where A is a constant to be determined. The oxygen flux at H becomes dc φ εD0 J 0 ¼ εD0 A sinh φ0 ¼ 0 ð40Þ dz z ¼ H H
Appendix
x y z z
cat
For the boundary concentration z = H, we thus have HJ 0 1 ¼ c0 cðHÞ ¼ εD0 φ0 tanh φ0 which leads to the following expression for the flux: εD0 φ0 J 0 ¼ c0 tanh φ0 H
ð34Þ
ð41Þ
ð42Þ
ð43Þ
ð44Þ
Introducing the definition of the Thiele modulus and the value for the effectiveness factor ηE given by eq 15, we regain the
The last boundary condition at the external surface of the membrane electrode assembly (z = H) describes that oxygen 5107
Energy Fuels 2010, 24, 5097–5108
: DOI:10.1021/ef100610w
Kjelstrup et al. 2
oxygen flux that was used in eq 14 J ¼ ð1 - εÞHηE kc 0
0
j(x, y) = local current density (A/m ) j0 = equilibrium exchange current (A/m2) J0 = specific rate of oxygen consumption (mol m-2 s-1) k = rate constant for the cathode reaction (s-1) L and M = membrane sheet side lengths (m) p = pressure R = cell resistance (Ω) Rg = universal gas constant, 8.314 J K-1 mol-1 w = width of the nanoporous catalyst column (m) x, y, and z = Cartesian coordinates (m) T = temperature (K) v = gas velocity (m/s) Y = yield (mol m-3 s-1)
ð45Þ
or equivalently
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kð1 - εÞ5 J 0 ¼ c0 D0 kðε - ε2 Þ tanh4H D0 ε
ð46Þ
which is eq 17. Nomenclature 0
c = oxygen concentration, inlet (mol/m3) c = oxygen concentration (mol/m3) Ci = constants D0 = diffusivity in macropores (m2/s) De = effective diffusivity in the nanoporous catalyst (m2/s) db = width of branch, gas supply system (m) d = macropore width (m) E = cell potential (V) F = Faraday’s constant, 96 500 C/mol f = objective function G = Gibbs energy (J) H = catalyst layer thickness (m) I = electric current (A) j = average current density (A/m2)
Greek Symbols R = stress factor β = overall transfer factor ε = macroporosity of the catalyst layer ηc = cathode overpotential (V) ηBV=overpotential according to the Butler-Volmer equation (V) ηE = catalyst effectiveness factor λ = Lagrange parameter φ0 = Thiele modulus, on the basis of macropores F = bulk catalyst density (kg/m3)
5108
