Modeling polymer electrolyte fuel cells: an innovative approach

Journal of Power Sources 101 (2001) 275±286 Modeling polymer electrolyte fuel cells: an innovative approach G. Maggio*, V. Recupero, L. Pino CNR-TAE Institute, Via Salita S., Lucia sopra Contesse 5, 98126 Santa Lucia, Messina, Italy Received 18 December 2000; received in revised form 2 March 2001; accepted 5 March 2001 Abstract In this paper, a mathematical simulation model is proposed to describe the water transport in proton conductive membranes, used in polymer electrolyte fuel cells (PEFCs). The model, which includes the calculation of electrochemical parameters of a PEFC, represents a quite innovative approach. In fact, it is based on the use of original mathematical relationships taking into account diffusional and ohmic overpotentials for electrode ¯ooding and membrane dehydration problems. The calculated performance of polymer fuel cells using a Na®on 117 membrane clearly demonstrates the model validation (3% variation with respect to experimental data). Besides, analysis of model results allows a useful comparison of two different membranes (Na®on 117, Dow) in order to de®ne the best membrane/electrode assembly. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Electrode ¯ooding; Membrane dehydration; Modeling; Polymer electrolyte fuel cells; Water transport 1. Introduction The polymer electrolyte fuel cell (PEFC) operates at a low temperature (70±908C), and promises to be one of the most serious candidates for both stationary and automotive applications as a substitute of traditional systems (thermoelectric power plants, internal combustion engine, etc.), due to wellknown advantages (high ef®ciency, no pollutant emission, etc.). The stringent requirements in terms of compactness, high energy density, performance stability and low cost, move the R&D in the direction of optimizing the different aspects of the PEFC systems. One approach is the development of theoretical analysis in order to identify and to solve problems related with the above-mentioned items. In this respect, several mathematical models have been proposed in literature for PEFCs [1±25]; all these models aimed at determining the potential or cell voltage as a function of the current density and, in general, of the operating conditions. One of the key objectives of the current research is an adequate water content in the electrolyte to achieve a good ionic conductivity. In fact, membrane dehydration causes an increase in resistance to the proton ¯ow (the ionic charges move through the sulfonic groups of the polymeric * Corresponding author. Tel.: ‡39-90-624227; fax: ‡39-90-624247. E-mail address: [email protected] (G. Maggio). membrane surrounded by water molecules) and possible deadhesion of the membrane from the electrode. On the other hand, for excess water content, the cathode is ¯ooded and the diffusion of the reactant gas through its pores is hindered. Therefore, an accurate evaluation of the water transport in PEFCs, often made by modeling approaches [1± 5], is desirable. Nevertheless, a formulation that directly relates the performance of a polymer fuel cell with the cathode ¯ooding and/or the dehydration problems at the membrane/electrode interfaces has never been proposed. As evidenced in literature [11], the cathode ¯ooding, typical of high current density operation, involves a decrease in the porosity available for gas diffusion in the electrode. Likely, it can be inferred that the membrane dehydration phenomenon, that especially occurs on the anode side at high current densities, involves a decrease in ionic conductivity [7±9,11,26]. Starting from these considerations, a mathematical model has been developed. It has been validated and applied to polymer fuel cells based on Na®on 117 and Dow membranes. 2. Model development In this paragraph, a description of the mathematical model developed for the solid polymer electrolyte fuel cell is presented. The main assumptions of the model are: (i) one-dimensional treatment; (ii) isothermal and steady state 0378-7753/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 7 7 5 3 ( 0 1 ) 0 0 7 5 8 - 3 276 G. Maggio et al. / Journal of Power Sources 101 (2001) 275±286 Nomenclature A cH‡ d DH‡ DO2 N2 EW F I Ilim Io Io,Pt kp kF ld‡ lm M M1,max M2,max OCV p psat w R Rcell Rmemb SPt T UPt v Vcell Vm W WPt x superficial electrode area (cm2) fixed-charge concentration (mol cm 3) membrane average pore diameter (cm) protonic diffusion coefficient (cm2 s 1) oxygen/nitrogen binary diffusion coefficient at standard conditions (cm2 s 1) membrane equivalent weight (g mol 1) Faraday constant (96,487 C per eq.) current density (A cm 2) limiting current density (A cm 2) exchange current density (A cm 2) exchange current density per Pt surface (A cm 2 Pt) membrane hydraulic permeability (cm2) membrane electrokinetic permeability (cm2) cathode gas-diffusion layer thickness (cm) wet membrane thickness (cm) water amount to be added to the cell (mol s 1) water amount corresponding to complete flooding (mol s 1) water amount corresponding to complete dehydration (mol s 1) open circuit voltage (V) gas pressure (atm) saturated vapor pressure (atm) universal gas constant (8.3143 J mol 1 K 1) cell resistance (O cm2) membrane resistance (O cm2) catalyst surface area (cm2 mg 1) cell temperature (K) catalyst utilization water velocity in membrane pores (cm s 1) cell voltage (V) standard molar volume (22,414 cm3 mol 1) water flow rate (mol cm 2 s 1) catalyst loading (mg cm 2) gas-phase mole fraction Greek symbols d pore-water density (mol cm 3) ddry membrane dry density (g cm 3) d eg gas porosity in gas-diffusion layer em membrane water porosity w Z cell overpotential (V) k membrane ionic conductivity (O 1 cm 1) m pore-water viscosity (kg m 1 s 1) membrane resistance/cell resistance ratio tR o empirical constant for diffusional overpotential (O cm2 K 1) x membrane water-transport ratio z stoichiometric ratio Subscripts and superscripts act activation conv dif el in o ohm out sat tr um w ‡ convective diffusional electrochemical reaction cell inlet initial or inlet to gas chamber ohmic cell outlet saturated condition water transport humidification condition water cathode side anode side conditions; (iii) ideal and uniformly distributed gases; (iv) electrode pores for gas ¯ow are separated from pores for liquid water; (v) the inlet gas temperatures are equal to the cell temperature. A set of input data is used for model calculations that, for convenience, have been divided into two distinct sections: water balance calculations and electrochemical calculations. It must be remarked that the gas porosity in the cathode diffusional layer and the ionic conductivity of the membrane, which affect diffusional and ohmic overpotentials, strictly depend on the water balance conditions calculated at any current density value. The core of the calculations is preceded by the determination of the membrane properties. 2.1. Membrane properties calculations To calculate the membrane properties, the following equations have been used: cH ‡ ˆ kF ˆ ddry 1 em w† ; EW (1) 2 em wd ; 80 (2) m ˆ 1:002  10 1:3272 293 T† 0:001053 293 T†2 †= T 168†† 3 ; (3) d ˆ 3:014  10 2 ‡ 4:179  10 DH‡ ˆ DH‡ ko ˆ 22 C† 4 ‡ 2:216  10 T 10 3 7 2 5:842  10 T T ; (4) T m 22 C† ; m 295 (5) F2 D ‡c ‡; RT H H (6) where m(228C† ˆ 0:9548  10 3 kg m 1 s 1 is the water viscosity at 228C. In particular, the equations for fixedcharge concentration (Eq. (1)), electrokinetic permeability (Eq. (2)), and initial ionic conductivity (Eq. (6)) have already been reported in literature [2,27]. The equations for porewater viscosity (Eq. (3)) and density (Eq. (4)) have been obtained by extrapolation of data reported in [28], in the 277 G. Maggio et al. / Journal of Power Sources 101 (2001) 275±286 proper ranges of temperature (20±100 and 40±1008C, respectively). Finally, the protonic diffusion coefficient (Eq. (5)) has been derived on the basis of the relationship Dm=T ˆ constant [2]. At the ®rst calculation loop, corresponding to OCV conditions, it will be k ˆ ko : (7) 2.2. Water balance equations The model makes a set of calculations in order to verify on the basis of ®xed experimental conditions, the water transport characteristics of the membrane and to suggest some solutions in order to achieve an optimal water transport in the fuel cell. As already mentioned, a proper water balance is critical for good operation of a polymer fuel cell, also during transient response. It is necessary to identify operative conditions able to avoid electrode ¯ooding and membrane dehydration problems which can drastically reduce the cell performance. In particular, at low current density, dehydration of the cathode side of the membrane can occur: the water produced within the cathode and that dragged through the membrane from the anode to the cathode due to electro-osmosis (with H‡ protons) could be insuf®cient. On the contrary, at high current density, the water production at the cathode can cause a ¯ooding of the electrode. Moreover, the electroosmotic ¯ow of water increases with current, and the anode side of the membrane is thus subjected to dehydration, which is only partially counterbalanced by back diffusion (from cathode to anode). Then, with the exception of the intermediate current density region, at low and high current densities, a proper water balance can be obtained only by using appropriate expedients, e.g. by humidi®cation of the gases entering the fuel cell or by supplying (removing) liquid water. Fig. 1 outlines the water ¯ows inside the cell. According to [29], the amount of water produced by chemical reaction due to cross-over of gases can be neglected. The following equations have been used to determine the water ¯ow rates related to the different phenomena: Water transport : Wtr ˆ Electrochemical reaction : Anode humidification : I x: F (8) Wel ˆ W um ˆ I : 2F I z 2F 1 (9) xum w xum w : (10) W‡um ˆ I z 4F ‡ 1 1‡ xoN2 xoO2 ! Anode outlet : : (11) W out ˆ I z 2F Cathode outlet : " ! xoN2 I out z 1‡ o W‡ ˆ 4F ‡ xO 2 1 1† # xsat w : 1 xsat w xsat‡ w : 1 xsat‡ w (12) (13) According to [1,3,4], the membrane water-transport ratio can be calculated as x ˆ em wF dv ; I (14) where v is the water velocity in the membrane pores, determined by the following equation:  1   F 2 c2H‡ kF kF cH‡ FI kp dp vˆ ; (15) 1‡ mk mk m dz which establishes that the water transport in the membrane depends on two contributions: the electro-osmotic transport associated with the flow of the H‡ protons, and the diffusive transport due to a pressure gradient in the membrane (dp/dz). This last term can be obtained by known relationships [1,3,4]. The gas-phase mole fractions of water corresponding to saturated condition have been calculated as xsat ˆ w psat w ; p (16) xsat‡ ˆ w psat w ; p‡ (17) with  psat ˆ exp 13:669 w Cathode humidification : xum‡ w xum‡ w Fig. 1. Schematic of water balance model Ð g.-d.l.: gas-diffusion layer; c.l.: catalyst layer; W um , W‡um : water entering the cell due to fuel and oxidant humidification, respectively; W in , W‡in : water directly added (e.g. by wicking) to the anode and cathode compartment, respectively; Wel: water produced in the cell (on the cathode side) from the electrochemical reaction; Wtr: water transported through the membrane from anode to cathode due to drag (electro-osmosis) and diffusion (or back diffusion) phenomena; W out , W‡out : water outgoing from anode and cathode compartment, respectively.  5096:23 : T (18) Analogously, the mole fractions of water corresponding to um‡ humidification condition (xum w , xw ) can be determined by 278 G. Maggio et al. / Journal of Power Sources 101 (2001) 275±286 considering the humidification temperatures, instead of the cell temperature in Eq. (18). The water balance is expressed by the following relationships: At the anode : W out ˆ W in ‡ W um W‡out At the cathode : ˆ W‡in ‡ W‡um Wtr ; (19) ‡ Wel ‡ Wtr : (20) The amounts of water to be supplied at the anode and at the cathode compartment to achieve the water balance are in M ˆ W A; (21) M‡ ˆ W‡in A: (22) To calculate the different overpotentials, the following relationships have been considered:   RT I ln : (25) Activation overpotential : Zact ˆ F Io ohmic overpotential : Zohm ˆ IRcell : (26)   Ilim : ˆ oTI ln Ilim I (27) Diffusional overpotential : Zdif Convective overpotential : Zconv ˆ FcH‡ v lm : k (28) As well known [30,31], under operative conditions, the voltage of a fuel cell decreases, compared with the ideal or reversible theoretical value, because of the irreversible losses associated with activation, ohmic, and diffusional polarizations. Once estimated accurately such effects, it is possible to obtain a suf®ciently correct description of the cell performance. Cell voltage can be determined by subtracting the overpotentials related to the cell irreversibilities from the open circuit voltage (OCV), In Eq. (26), the cell resistance is considered to vary as a function of the current density, depending on the water balance conditions. Eq. (27) for diffusional overpotential is different from that commonly reported in most handbooks on fuel cells [30,31]. Here, the pre-logarithmic term is assumed to be dependent on current density, according to [16,34], whose purpose was to simulate the cell behavior in the entire current density region through a rational description of mass-transport problems. Moreover, unlike the classical approaches, the limiting current density (Ilim) varies as a function of the cell current density, and decreases in consequence of the ¯ooding of the cathode gas-diffusion layer (see Eq. (31)). Eq. (28) provides the overpotential associated with proton convection, which depends on the protonic concentration and water velocity in the membrane pores [1±4]. Exchange current density per geometrical electrode surface, cell resistance, and limiting current density have been determined by the following equations: Vcell ˆ OCV Io ˆ Io;Pt SPt WPt UPt ; It is evident that when these last equations give negative values, a removal, instead of addition, of the corresponding water amounts is required to obtain the balance. 2.3. Electrochemical calculation equations Zact Zohm Zdif Zconv ; (23) where Zact is the activation overpotential, Zohm the ohmic overpotential, and Zdif the diffusional overpotential; the last term Zconv has been introduced on the basis of the following considerations. The ¯ux of a dissolved species in the membrane pores is due to three effects: migration, diffusion, and convection [2,32,33]. The ®rst one corresponds to the ohmic term related to the membrane (already included in Zohm), the second is associated with the so-called diffusion potential, and the third to the effect of the potential gradient on the velocity of the charged ¯uid. In the case of a PEFC, the only mobile ions in the membrane pores are hydrogen ions, whose concentration in the membrane is constant; thus, there is no proton transport due to diffusion [2,33]. Therefore, only the proton transport for convection, to which the overpotential Zconv is associated, has to be considered. The OCV has been calculated by using the following equation [1±4]: OCV ˆ 1:23 0:9  10 3 T 298† ‡ RT ln p2H2 pO2 †: 4F (24) Rcell ˆ Ilim ˆ (29) Rmemb lm ˆ ; tR ktR 2FDO2 N2 edg †1:5 T=273†0:823 ln 1 Vm ld‡ (30) xO2 †; (31) where Io,Pt is the exchange current density referred to Pt surface, SPt the catalyst surface area, WPt the catalyst loading in the electrode, UPt the catalyst utilization coefficient, tR  1 the membrane resistance/cell resistance ratio, assumed to be constant (Eq. (31) from [11]). Eq. (31) gives the dependency of the limiting current on the gas porosity in the cathode diffusional layer, in agreement with the results obtained in [1±4] and with the considerations explained in [11]: gas transport limitations in the cathode diffusional layer determine the limiting current of the cell. Finally, the gas porosity in the cathode diffusional layer and the membrane ionic conductivity have been determined from the following expressions:   jM‡ j d d;o eg ˆ eg 1 (32) ; if M‡ < 0; M1;max G. Maggio et al. / Journal of Power Sources 101 (2001) 275±286 8  > > > ko 1 > < kˆ  > > > o > :k 1  M‡; ; M2;max  M‡ 1 M2;max if M‡; > 0 and M ;‡  0;  M ; if M‡ > 0 and M > 0; M2;max being egd;o the initial porosity in the gas-diffusion layer, ko the initial membrane ionic conductivity calculated by Eq. (6), M‡ and M the amounts of water required for water balance on the cathode and on the anode side, respectively, derived from previous calculations (Eqs. (21) and (22)), M1,max the water amount corresponding to a complete flooding, and M2,max the water deficit corresponding to complete dehydration. As a first approximation, it has been assumed M2;max ˆ M1;max ; (34) that is, the water deficit that results in complete dehydration at the membrane/electrode interfaces is equal to the excess water amount that induces the complete flooding of the cathode gas-diffusion layer. 3. Innovative aspects of the proposed model All the mathematical models for polymer fuel cells so far proposed assume the gas porosity in the electrode diffusional layer to be constant (for example, 40% in [1±4]). Nevertheless, Springer et al. [11] have previously evidenced that this assumption does not explain the limiting current experimentally obtained and their dependence on gas composition and pressure. As was observed, the average effective porosity of the gas-diffusion layer should be signi®cantly lower (from 25 to 18.6% between 0 and 1.5 A cm 2 in [11]) than the ``natural'' porosity (ca. 40%) of the carbon cloth electrode. The reason is a noticeable ``partial ¯ooding'' of the cathode by the liquid water produced that slows down the oxygen transport. To express such an effect, we assumed that the gas porosity in the diffusional layer decreases, in case of cathode ¯ooding (M‡ < 0), according to Eq. (32). In other words, it is assumed that the gas porosity in the diffusional layer depends on current density; it decreases when the latter increases, in agreement with [11]. In the mathematical model presented in [11], the effective porosity (liquid water-free pores) could be ``adjusted'' and was considered either independent from the current density or decreased in proportion to the rate of water production at the cathode. To obtain reliable values of the effective porosities corresponding to a given fuel cell, the authors were forced to realize a simultaneous ®tting of several polarization curves of the cell [11] recorded under different operative conditions. Our approach, differently, allows direct calculation of the gas porosity through Eq. (32). 279 (33) As the cathode ¯ooding involves a decrease in the porosity available for the gas that diffuses through the electrode, it can be argued that the membrane dehydration involves a decrease in ionic conductivity. As a consequence, the overall cell resistance, to which the membrane strongly contributes, will correspondingly increase. Even though such an effect is neglected in mathematical models for fuel cells (except [9]), where a constant resistance is considered, it is evidenced in some experimental works [7,8,11,26]: e.g. a resistance increase in the order of 10% going from 0 to 0.8 A cm 2 for a cell based on Na®on 117 is observed in [26]. Therefore, in analogy with the method used for the gas porosity, we assumed that the membrane ionic conductivity decreases, in case of cathode (M‡ > 0) and/or anode dehydration (M > 0), according to Eq. (33). Eq. (33) is equivalent to the assumption that the ionic conductivity of the membrane depends on current density, by decreasing when the latter increases, in agreement with [26]. A decrease in the membrane ionic conductivity and thus an increase in the cell resistance at low current densities due to the incidental dehydration at membrane/cathode interface (M‡ > 0) is also expected from Eq. (33). The innovative character of the proposed model is evident. In particular, the more original aspect consists in the correlation introduced between the performance of the polymer fuel cell and the water balance conditions. In fact, although it is evident that possible problems of electrode ¯ooding and/or membrane/electrodes dehydration cannot be unin¯uential on the operation of this type of fuel cell, no literature model explicitly expresses this. Such a result has been obtained here through the use of two mathematical relationships (Eqs. (32) and (33)) which state that the gas porosity in the cathode diffusional layer and the membrane conductivity vary with the current density, in agreement with some literature considerations [11,26]. As a consequence, the diffusional (Eq. (27)) and ohmic overpotentials (Eq. (26)) increase in case of ¯ooding and dehydration problems, respectively. This effect is shown, from a qualitative point of view, in Figs. 2 and 3, where the overpotentials are reported in arbitrary unities (A.U.). In particular, in Fig. 3 the membrane/cathode dehydration problems have been emphasized. Appropriate humidi®cation conditions would let us exclude the occurrence of this phenomenon. Nevertheless, as already mentioned, if the humidi®cation conditions are not adequate, at low current densities, the water produced at the cathode and that dragged from anode to cathode due to 280 G. Maggio et al. / Journal of Power Sources 101 (2001) 275±286 Fig. 2. Qualitative effect of cathode flooding on diffusional overpotential, based on Eqs. (26), (31) and (32). electro-osmosis could not be suf®cient to avoid dehydration on the cathode side of the membrane. Another innovative element of the proposed model is represented by Eq. (27) used to calculate the diffusional overpotential, which is partly based on some modeling approaches of semi-empirical type [16,34]. In fact, the relationship commonly reported in literature [30,31] in this respect is very general, and does not seem to provide a good description of the diffusional problems at high current densities: the diffusional overpotential of the PEFC results underestimated. Some models proposed for PEFCs are not able to provide an adequate description of the cell behavior in the whole range of current densities [12,13]. In fact, they often result approximate in the high current density region, just where the diffusional overpotentials (mass-transport problems) prevail, especially when air is used as reactant gas. On the contrary, the innovative elements introduced in our model allowed to obtain an excellent agreement with the experimental results of a polymer fuel cell. 4. Model input data The proposed model has been validated through application to polymer fuel cells with a Na®on 117 membrane, Table 1 Input data relative to the cell and the electrodes Datum Value 2 Current densities (A cm ) Cell temperature (8C) Electrode surface (cm2) Electrode gas-diffusion layer thickness (mm) Electrode hydraulic permeability (cm2) Cathode feed gas Anode gas pressure (atm) Cathode gas pressure (atm) Anode inlet gas-flow rate (Nl min 1) Cathode inlet gas-flow rate (Nl min 1) Anode humidifying temperature (8C) Cathode humidifying temperature (8C) a 0.5  10 3  0.850a 70a (base-case) 50a 360a 4.73  10 15 [1] Aira 2.5a (base-case) 3.0a (base-case) 0.53a 3.0a 85a (base-case) 75a (base-case) Experimental data CNR-ITAE. whose experimental results coming from our laboratory are available. Afterwards, on the basis of literature data, the model has also been applied to a fuel cell based on a Dow membrane in order to make a comparison between the two membranes. The input data are summarized in Tables 1±3. As regards the parameters required for electrochemical calculations listed in Table 2, the initial porosity of the gas-diffusion Fig. 3. Qualitative effect of membrane dehydration on ohmic overpotential, based on Eqs. (25), (30) and (33). 281 G. Maggio et al. / Journal of Power Sources 101 (2001) 275±286 Table 2 Input data required for electrochemical calculations Datum Value Initial gas porosity in gas-diffusion layer (%) Catalyst surface area (cm2 mg 1) Catalyst loading (mg cm 2) Catalyst utilization Exchange current density per Pt surface (A cm 2) Membrane resistance/cell resistance ratio Empirical constant for diffusional overpotential (O cm2 K 1) Water amount for complete flooding (mol s 1) 30a 1100a 0.15a 0.5a 6.0  10 0.77d 3.4  10 3.0  10 9 (Nafion 117)b; 6.6  10 9 (Dow)c 4e 3f a Experimental data CNR-ITAE. This value comes from elaboration of experimental cell potential by a program developed with the software package MathematicaTM [35]. c From data reported in [36]: the ratio between the exchange current densities Dow/Nafion 117 is about 1.1 at 708C and cathode pressure 3 atm. d From experimental measurements carried out at CNR-ITAE, a membrane resistance (Nafion 117) of 0:16  0:02 O cm2 results, and the global cell resistance ˆ 0:20  0:02 O cm2 in the intermediate current densities region. e From extrapolation of data reported in [34]. f This value is fixed so as to have about a 10% increase in resistance from 0 to 0.8 A cm 2, in agreement with data reported in [26]. b Table 3 Membrane properties required as input data Property Nafion 117, 7 mil Dow, 7 mil Wet thickness (mm) Dry density (g cm 3) Equivalent weight (g mol 1) Hydraulic permeability (cm2) Ê) Average pore diameter (A Water porosity (%) H‡ diffusion coefficient at 228C (cm2 s 1) 230 [1,37] 1.84 [38] 1100 [36,38] 1.8  10 14 [1] 55 [1,37] 28 [1,37] 1.4  10 5 [27] 230 [37] 2.143a 800 [36] 1.8  10 44 [1,37] 44 [1,37] 2.0  10 a This value is derived from Eq. (1), based on a fixed-charge concentration of 1.5  10 reported in the table. layer and the data relative to the catalyst refer to electrodes manufactured in house; the other parameters (Io,Pt, tR, o, and M1,max) have been determined according to the speci®cations accompanying the table. In general, when the experimental data of cell potential are available, these model parameters are ``®tting parameters'', which allow to adjust the behavior of the theoretical curve. The exchange current density per Pt surface (Io,Pt) in¯uences the low current density region controlled by activation. The ratio between membrane and cell resistance (tR) in¯uences the intermediate current density region, where the ohmic overpotentials prevail. The empirical constant o and the water amount corresponding to a complete ¯ooding (M1,max) in¯uence the high current density region, dominated by diffusional problems; even if, on the basis of assumption (34), the parameter M1,max partly conditions the ohmic losses. It must be observed that, apart from the value of exchange current density per Pt surface, the values required for electrochemical calculations (Table 2) have been assumed to be equal for the two membranes, as a ®rst approximation. Nevertheless, such a hypothesis should be better veri®ed in further investigations. For the membrane properties (Table 3), we made reference to literature data [1,27,36±38] relative to 7 mil (175 mm) thick membranes in the dry, H‡ state. 3 mol cm 3 14 5 [1] [27] [1], and on the porosity and equivalent weight values 5. Results and discussion The properties calculated by the model for the two membranes considered are reported in Table 4. The comparison shows that some properties (saturated vapor pressure, water viscosity and density), depending only on the temperature of membrane operation (708C in this case), have the same value for both membranes. The electrokinetic permeabilities of the two membranes, calculated by Eq. (2), are also equal; but in such a case, this result comes from a balance between the volume of the membrane pores and their size. In fact, the Dow membrane Table 4 Membrane properties calculated at 708C Property Nafion 117, 7 mil Dow, 7 mil Saturated vapor pressure (atm) Pore-water viscosity (kg m 1 s 1) Pore-water density (mol cm 3) H‡ diffusion coefficient (cm2 s 1) Initial ionic conductivity (O 1 cm 1) Fixed-charge concentration (mol cm 3) Electrokinetic permeability (cm2) 0.3046 4.04  10 4 0.0543 3.8  10 5 0.151 1.2  10 3 1.06  10 15 0.3046 4.04  10 4 0.0543 5.5  10 5 0.269 1.5  10 3 1.06  10 15 282 G. Maggio et al. / Journal of Power Sources 101 (2001) 275±286 is characterized by a larger porosity (44 versus 28%), but it Ê ). Other membrane has a lower pore diameter (44 versus 55 A characteristics calculated by the model show better values for Dow, in terms of ®xed-charge concentration, protonic diffusion, and ionic (initial) conductivity (78% higher than Na®on 117), in agreement with literature [1]. Besides, calculations [39] based on a concentrated solution theory [40] showed that in spite of a higher frictional coef®cient between water and polymer, the Dow membrane takes advantage of a reduced friction of the hydrogen ions inside the membrane. This result agrees with some studies [37] that attribute to the Dow membrane, characterized by a high protonic diffusion coef®cient (see Table 4), reduced water ¯ow rates, and a better proton transport (compared with Na®on 117). A substantial difference between the two membranes is represented by the water-transport number (Eq. (14)), whose calculated values for Dow membrane are lower than Na®on 117 (the average difference is 0.31 H2O/H‡, and lies from a minimum of 0.26 H2O/H‡ at 0.250 A cm 2 to a maximum of 8.04 H2O/H‡ at OCV), in agreement with literature [41,42]. This difference leads to different water requirements (M and M‡) to achieve water balance conditions. So that, in any case, the ¯ooding and/or dehydration problems will be more signi®cant for the fuel cell based on Na®on 117 membrane. In Figs. 4±9, the model results corresponding to the electrochemical calculations are presented. In particular, Fig. 4 shows the behavior of diffusional and ohmic overpotential calculated for the fuel cell based on Na®on 117 Fig. 4. Influence of flooding and dehydration problems on diffusional and ohmic overpotentials calculated by the model in the case of a Nafion 117 membrane. Fig. 5. Overpotential contributions calculated by the model in the case of a Nafion 117 membrane. G. Maggio et al. / Journal of Power Sources 101 (2001) 275±286 283 Fig. 6. Percentage of overpotential contributions calculated for Nafion 117 membrane at four current density values: (a) 0.2 A cm 2; (b) 0.4 A cm 2; (c) 0.6 A cm 2; (d) 0.8 A cm 2. membrane (base-case), according to two different hypotheses: in the absence of ¯ooding (edg constant: 30%) and dehydration phenomena (Rcell constant: 0.198 O cm2), and in the presence of such problems (edg and Rcell vary with current density). In compliance with the expectations anticipated in Figs. 2 and 3, it can be observed that the diffusional overpotential is enhanced by the ¯ooding problems of the cathode gas-diffusion layer, going from 0.094 V (in the absence of ¯ooding) to 0.263 V at 0.850 A cm 2. Analogously, the ohmic overpotential suffers from the dehydration of the anode side of the membrane; in fact, it increases from 0.168 to 0.189 V at 0.850 A cm 2. Vice versa, there is no evidence of any dehydration of the cathode side of the membrane (possible, sometimes, at low current densities); but this is justi®ed by the humidi®cation conditions chosen. As already mentioned, the behaviors presented in Fig. 4 are due to the two negative effects: a decrease, caused by ¯ooding of the gas porosity in the diffusional layer; an increase, caused by dehydration of the overall cell resistance. In fact, the gas porosity passes from the initial 30% (at OCV) to 22.7% (24% less!) at 0.850 A cm 2; while, the cell resistance goes from 0.198 O cm2 (at OCV) to 0.222 O cm2 at 0.850 A cm 2, with a rise of about 11%. As Fig. 4 reveals, both these effects become evident (>5 mV) only for current densities larger than about 0.5 A cm 2. It must be observed that the value of the (initial) ionic conductivity calculated by Eq. (6) allows to obtain (Eq. (30)) Fig. 7. Polarization curves: comparison between model and experimental results for the base-case of Nafion 117 membrane. 284 G. Maggio et al. / Journal of Power Sources 101 (2001) 275±286 Fig. 8. Comparison between experimental (bullets) and theoretical (dashed lines) polarization curves under different operative conditions (Nafion 117 membrane). an overall cell resistance (0.204 O cm2 at 0.4 A cm 2) in excellent agreement with the value measured in the intermediate current density region, equal to 0:20  0:02 O cm2. Besides, as it will be shown later, it has not been necessary to artfully change the value of the membrane ionic conductivity to achieve a good ®tting of the experimental results. Instead, a conductivity equal to 0.07 O 1 cm 1 at 808C has been considered in [2] to reach a satisfactory ®tting of the polarization experimental curves, even though the value calculated by Eq. (6) was signi®cantly higher (i.e. 0.17 O 1 cm 1). From Fig. 5, it is possible to distinguish the different overpotential contributions calculated by the model for the fuel cell based on Na®on 117 membrane. At low current densities (<0.1 A cm 2), the activation overpotential is almost entirely responsible for the cell voltage losses. For current densities > 0:3 A cm 2, the ohmic losses due to the membrane and the electrodes become more important, and the activation overpotential reaches a relatively constant value. The results also show the effect due to the oxygen mass-transport limitations, which involve an enhanced diffusional overpotential for current densities > 0:6 A cm 2. The overpotential due to proton convection, which in Fig. 5 is coupled to the diffusional overpotential, is not noticeable: it reaches a maximum of 11.2 mV at 0.850 A cm 2. A fundamental difference, compared with analogous results presented in literature [1,3,4], is an appreciable limitation due to mass-transport phenomena. Actually, in [1,3,4], the only parameter which in¯uences the high current density region is the gas porosity in the diffusional layer edg, but this is a datum of the model and does not vary with current density. Two extreme cases were presented in [1,3,4]: when the porosity edg ˆ 50%, no transport limitations were observed; for edg ˆ 11% instead, the diffusional Fig. 9. Polarization and power density curves calculated by the model: comparison between results relative to Nafion 117 (base-case) and Dow membranes. 285 G. Maggio et al. / Journal of Power Sources 101 (2001) 275±286 problems become evident, and the oxygen-limiting current density reaches 0.750 A cm 2. However, the potential drop related to the mass-transport problems appears to be too sharp (the cell voltage is still very high, i.e. about 0.660 V at 0.650 A cm 2) compared with our predictions (Figs. 5±7). Fig. 6 allows to discern the relative share of the calculated overpotential contributions corresponding to four current density values. The activation overpotential is 89.5% of the total irreversibilities at 0.2 A cm 2, but decreases at 54.3% for 0.8 A cm 2; the percentages of the other contributions, instead, increases with current density. Thus, at 0.8 A cm 2, ohmic and diffusional overpotential both represent more than 21% of the total losses, while convective overpotential does not exceed 1.3%. Is it signi®cant that for constant gas porosity and membrane ionic conductivity, the percentages at 0.8 A cm 2 become: 63.2% activation, 23.8% ohmic, 11.8% diffusional, and 1.2% convective overpotential. Fig. 7 shows the comparison between the model and the experimental results of the fuel cell based on a Na®on 117 membrane. The agreement obtained between the polarization curve predicted by the simulation model and the experimental curve is very satisfactory. In fact, the mean error is in the order of 5.7 mV, with two relative maxima of ‡26.6 mV at 0:5  10 3 A cm 2 (corresponding to ‡2.8%) and of 15.6 mV at 0.7 A cm 2 (equal to 2.9%). Thus, the maximum error for the entire curve is less than 3%. Moreover, leaving out the ®rst point (0:5  10 3 A cm 2) for which there are unquestionable measurement dif®culties (recorded values for ®ve experimental curves range from 0.968 to 0.981 V!), the mean error between the two curves reduces to 4.8 mV; i.e. in percentage terms, 0.7%. From a more general point of view, apart from few points (0:5  10 3 , 2:5  10 3 , 0.400, 0.800, and 0.850 A cm 2), it can be noticed that the calculated cell voltage is always slightly lower than the experimental one. However, peculiar behaviors of the error corresponding to the three current density regions controlled by activation, ohmic, and diffusional overpotentials are not observed. Then, as a matter of fact, these three contributions seem to be described with a good accuracy by the mathematical model here proposed, whose validation has been consequently attained. In fact, further simulations carried out at different operative conditions con®rmed the good agreement between the model and the experimental results (Fig. 8). Fig. 9 shows a comparison of the polarization and power density curves calculated by the model for a fuel cell having the same characteristics of the base-case, with the two different types of membranes. The results, obtained by considering the data reported in Tables 1±3, show a superior performance for the Dow-based fuel cell. Such a performance is justi®ed by several factors that can be easily understood by comparing the overpotential values calculated at 0.850 A cm 2 for the two cases (Table 5). In fact, the higher exchange current density associated with the Dow-based fuel cell, owing to its higher protonic concentration, implies a slightly lower activation Table 5 Comparison between calculated overpotentials at 0.850 A cm 2 Overpotential Cell with Nafion 117 membrane (mV) Cell with Dow membrane (mV) Difference (mV) Activation ohmic Diffusional Convective 424.33 189.21 263.36 11.15 421.52 101.31 190.92 3.26 2.81 87.90 72.44 7.89 overpotential (3 mV less in all the current density range). Moreover, the higher protonic concentration of the Dow membrane results in a reduced overpotential for protonic convection: about 8 mV less at 0.850 A cm 2. But, what makes the big difference between the two fuel cells are the ohmic and the diffusional overpotentials, respectively, 88 and 72 mV less at 0.850 A cm 2 for the Dow-based fuel cell. Thus, altogether the cell voltage calculated at 0.850 A cm 2 results a good 0.171 V higher for the case of the fuel cell with Dow membrane: 0.480 V against 0.309 V for the Na®on-based cell. This is clearly due to a lower cell (initial) resistance (0.111 O cm2 against 0.198 O cm2 at OCV), i.e. a higher ionic conductivity of the Dow membrane, but also, in particular, to a less-pronounced effect of the dehydration and ¯ooding problems. In fact, the resistance undergoes an increase equal to 6.7%, and the gas porosity in the diffusional layer suffers a decrease equal to 19.8%, when we pass from OCV to 0.850 A cm 2, versus 11.1 and 24.2%, respectively, for the Na®on-based fuel cell. This is also related to the values of the water-transport number (Eq. (14)), which in the case of the Dow membrane are lower than those of the Na®on 117 membrane (e.g. 0.67 H2O/H‡ versus 0.97 H2O/H‡ at 0.850 A cm 2). Thus, the water transport for electro-osmosis through the Dow membrane is slower, and it contributes to simultaneously reduce the membrane dehydration (at the anode interface) and the cathode ¯ooding. Obviously, the power density of the fuel cell is also higher for the cell using the Dow membrane, with a maximum of about 0.438 W cm 2 at 0.750 A cm 2; whereas, for the cell based on the Na®on 117 membrane, the maximum power density is approximately equal to 0.370 W cm 2 at 0.700 A cm 2. The results obtained by the model are, from a qualitative point of view, in agreement with literature [1,3,4,14,21,26], where experimental and/or theoretical polarization curves of PEFCs based on Na®on 117 and Dow membranes are presented, with a clear superior performance for the latter. 6. Conclusions In this paper, a mathematical model for polymer fuel cells is described; it represents an innovative approach compared 286 G. Maggio et al. / Journal of Power Sources 101 (2001) 275±286 with others reported in literature. In fact, it is based on the introduction of some original relationships that envisage a decrease in the cell resistance and in the porosity available for the gas in the electrode, respectively, due to dehydration on the anode side of the membrane and to cathode ¯ooding. Once the operative conditions are ®xed, such relationships allow to determine the ohmic and diffusional overpotentials as a function of the water balance conditions of the fuel cell. The cell voltage is accordingly calculated. The model results here presented, for the case of a polymer fuel cell using a Na®on 117 membrane, demonstrate an excellent agreement with the experimental voltage values of a cell realized at the CNR-TAE Institute: the maximum error is less than 3%. The simulations carried out at different operative conditions con®rm this agreement. Moreover, a superior performance (0.480 V against 0.309 V at 0.850 A cm 2) was predicted in the case of a fuel cell, with the same characteristics (electrodes, operative, and humidi®cation conditions) based on a Dow membrane, whose relative ¯ooding and dehydration problems are less critical. In conclusion, the developed model represents a suitable theoretical tool to provide a valid support for an advanced design of the membrane/electrode assembly of PEFCs for the realization of polymeric membranes at low cost and for the optimization of cell-operating conditions. [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] References [1] D.M. Bernardi, M.W. Verbrugge, General Motors Report GMR-7360, 15 May 1991. [2] D.M. Bernardi, M.W. Verbrugge, AIChE J. 37 (1991) 1151±1163. [3] D.M. Bernardi, M.W. Verbrugge, in: R.E. White, M.W. Verbrugge, J.F. 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