The effect of cathodic water on performance of a polymer electrolyte fuel cell

Electrochimica Acta 49 (2004) 5187–5196

The effect of cathodic water on performance of a polymer electrolyte fuel cell A.A. Kulikovsky∗,1 Institute for Materials and Processes in Energy Systems (IWV-3), Research Center “J¨ulich”, D-52428 J¨ulich, Germany Received 28 January 2004; received in revised form 24 March 2004; accepted 14 June 2004 Available online 7 August 2004

Abstract A simple analytical model of water transport in the polymer electrolyte fuel cell is developed. Nonlinear membrane resistance and voltage loss due to incomplete membrane humidification are calculated. Both values depend on parameter r, the ratio of mass transport coefficients of water in the membrane and in the backing layer. Simple equation for cell performance curve, which incorporates the effect of cathodic water is constructed. Depending of the value of r, the cell may operate in one of the two regimes. When r ≥ 1, incomplete membrane humidification simply reduces cell voltage; the limiting current density is determined by oxygen transport in the backing layer (oxygen-limiting regime). If r < 1, limiting current density is determined by membrane drying (water-limiting regime). In that case there exists optimal current density, which provides minimal membrane resistance. It is shown that membrane drying may lead to parasitic “in-plane” proton current. © 2004 Elsevier Ltd. All rights reserved Keywords: PEFC; Water management; Analytical modelling; Limiting current density

1. Introduction One of the key components of a polymer electrolyte fuel cell (PEFC) is polymer electrolyte membrane, which prevents mixing of feed gases and provides transport of protons from the anode to the cathode. Nafion® membrane has noticeable proton conductivity only in a wet state. Furthermore, in a wide range of operating conditions membrane conductivity is a linear function of water content. Water produced on the cathode side of PEFC usually falls short of the proper membrane humidification and additional water is supplied through the feed channels. Several mechanisms provide transport of water in a cell. If cell temperature is high enough, liquid water generated in oxygen reduction reaction (ORR) rapidly evaporates. In porous (gas diffusion and catalyst) layers water vapour is transported due to diffusion. In membrane the diffusion ∗

Tel.: +49 2461 61 5396; fax: +49 2461 61 6695. E-mail address: [email protected] (A.A. Kulikovsky). 1 On leave from Moscow State University, Research Computing Center, 119992 Moscow, Russia. 0013-4686/$ – see front matter © 2004 Elsevier Ltd. All rights reserved doi:10.1016/j.electacta.2004.06.034

flux of liquid water counterbalances electroosmotic flux caused by proton current. The distribution of water (and hence of conductivity) in membrane depends on current density and can be very non-uniform. Intelligent water management is thus essential issue for optimal PEFC performance. A model of water transport is a key component of the state-of-the-art numerical models of a PEFC [1–9]. Much less is done in analytical modelling of this transport. Okada et al. [10,11] developed 1D diffusion-type model, assuming constant diffusion coefficient of water in membrane. Nonstationary and steady-state water profiles are obtained and the expression for membrane resistance is derived. Using similar approach, Sena et al. [12] derived the formula for voltage loss in membrane, inserted it into equation for cell polarization voltage and fitted experimental voltage current curves. Solutions [10–12], however, do not take into account transport of water in the cathode backing layer. Here we show that this transport may dramatically change membrane resistance. Attempt to account for water transport in the backing layer was made in our previous work [13]. The model [13], however, ignores the discontinuity of water concentration at the catalyst

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layer/membrane interface and hence tends to underestimate back diffusion flux of water in the membrane. In this work we develop 1D “local” model of water transport across the membrane-electrode assembly (MEA) of a PEFC. The effects due to the along-the-channel variation of water concentration are discussed in the last section. In many respects this work follows the approach of Sena et al. [12]. We derive and analyse the expression for membrane resistance, which results from solution of the problem of conjugate water transport in the cathode compartment and in the membrane. A simple equation for cell polarization curve, which accounts for water management is constructed. Finally, we show that along-the-channel non-uniformity of water concentration in membrane leads to a parasitic “inplane” proton current. 2. Model 2.1. Qualitative description and basic assumptions The following picture of water transport in a cell stands behind model equations. If cell temperature is high enough, liquid water generated in ORR rapidly evaporates and fills voids of the gas diffusion (backing) layer. In the bulk membrane water is trapped in a nanoscale clusters in a bind (liquid) form [14,15]. The flux of vapour in the backing layers is caused by diffusion due to concentration gradient. The distribution of liquid water in membrane is determined by proton current (electroosmotic effect) and by diffusion. Water concentration across the cathode catalyst layer is assumed to be uniform (Fig. 1). The concentration of vapour in the cathode catalyst layer determines water content of membrane surface and thus establishes “boundary condition” for liquid water transport in the membrane.

All diffusion coefficients are assumed to be constant. The profiles of water and oxygen concentration are then straight lines (Fig. 1). Since the rate of water production/oxygen consumption is proportional to current density j, the slope of all lines in Fig. 1 depends on j. The slope of liquid water concentration in the membrane determines membrane resistance and membrane voltage loss. The slope of oxygen concentration (Fig. 1) determines voltage loss due to oxygen transport in the backing layer. The model is thus based on the following assumptions: 1. Cell temperature is high enough, so that the flux of liquid water in the backing layer can be neglected. 2. Transport of species only across the cell is considered; local 2D effects are neglected. The along-the-channel effects are discussed in Section 3.6. 3. The concentration of liquid water on the cathode side of the membrane and the concentration of water vapour in the cathode catalyst layer are related by membrane sorption isotherm. 4. Oxygen and water vapour diffusion coefficients in the cathode backing layer coincide. 5. Diffusion coefficient of liquid water in the membrane is constant. 6. Total flux of water in the membrane is zero. The first assumption is supported by recent numerical model [16], which shows that under temperature 60 ◦ C and pressure 1 bar more than 90% of water molecules is transported through the cathode backing layer in gas phase. Water saturation pressure rapidly increases with the temperature, hence at higher temperatures the role of liquid water flux is even smaller. Liquid water partially fills voids of the backing layer thus reducing oxygen diffusivity. The latter effect is taken into account by the effective diffusion coefficient of oxygen transport (see below). The third assumption physically means that membrane surface is in equilibrium with available in the catalyst layer water vapour. For typical operating conditions oxygen and water vapour diffusion coefficients in the backing layer differ only by 15% [17]. To simplify the analysis we take them to be equal. Diffusion coefficient of liquid water in membrane Dl is roughly constant at λ > 5 and decreases with water content when λ < 5 [18]. The effects due to this nonlinearity are considered in detail in [7]. Here for simplicity we assume constant Dl . The last assumption is justified for anode operating pressure 1–2 bar and temperature 60–80 ◦ C. The respective estimate is done in Section 2.3. 2.2. Water transport in membrane

Fig. 1. Sketch of oxygen, water vapour and liquid water profiles across the cell. CL and BL stand for the catalyst and backing layer respectively. The slopes of all lines depend on current density.

In this section x = 0 is located at the anode side of the membrane (ASM) and x-axis is directed towards the cathode. According to the assumptions, back diffusion and

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electroosmotic fluxes and in membrane are equal: Dl

∂cl j = nd ∂x F

Note that the variation of equivalent weight of Nafion changes the value of Kλ . (1)

where cl is the concentration of liquid water. At a constant Dl , Eq. (1) gives linear profile of cl across the membrane
  j x cl (x) = cl,t 1 + −1 (2) jl lm where cl,t is the liquid water concentration on the cathode side of the membrane, lm is the membrane thickness and jl =

FDl cl,t nd lm

(3)

Eq. (2) shows that when j = jl we have cl (0) = 0, i.e., the ASM is dry. Zero water concentration means zero proton conductivity (see below); jl is thus the limiting current density, which the membrane is able to support. Water concentration experiences discontinuity on both sides of the membrane. In membrane water is in liquid form, whereas the pores of the catalyst layer are filled with water vapour. Consider the cathode side. At the membrane/catalyst layer interface cl,t and the concentration of vapour cw,t are related by water sorption isotherm   cl,t cw,t (4) =Λ sat c H+ cw sat is the concentration of saturated vapour and c + where cw H is the molar concentration of protons/sulfonic groups in the membrane. The function Λ for Nafion, measured in [19] is shown in Fig. 2. To simplify the calculations we will approximate it by linear dependence (Fig. 2):

cl,t cw,t  Kλ sat c H+ cw

(5)

With (5), Eq. (3) takes a form jl =

FDl Kλ cH+ cw,t sat nd lm cw

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(6)

Fig. 2. Membrane water content vs. water vapour activity. Dots: experiment [19], straight line: linear fit used in our calculations.

2.3. Limits of validity Relation (6) allows us to establish the limits of validity of assumption of zero total water flux in membrane. The flux of water vapour in the anode backing layer is Nwa = −Db (∂cw /∂x), where Db is the diffusion coefficient. Hydrogen is usually well humidified; in that case Nwa is directed towards the membrane (the case of dry anode feed is considered in Appendix A). Nwa is maximal when waa / la . ter concentration at the ASM is zero: Nwa max = Db cw,h b a Here cw,h is the water molar concentration in the anode channel, lba is the backing layer thickness. The flux Nwa max is attained when the membrane carries limiting current density jl . Clearly, anodic water does not affect water balance in membrane if Nwa max is much smaller, than the electroosmotic flux a / la  n j /F . Here n is the of water in membrane: Db cw,h d l d b drag coefficient, a number of water molecules transported per one proton due to electroosmosis. Under this condition humidification of the ASM is provided mainly by back diffusion of water produced on the cathode side. For water concentraa  n j la /FD . tion in the anode channel we then have cw,h d l b b With Eq. (6), this gives a cw,h 

Dl lba cw,t K c + sat λ H Db lm cw

(7)

For the estimate we take Db = 10−2 cm2 s−1 (corrected for porosity–tortuosity), Dl = 10−5 cm2 s−1 , lba / lm ≈ sat ≈ 1, K ≈ 10 (Fig. 2) and c + = 1.2 × 10, cw,t /cw λ H −3 a  10−4 mol cm−3 . 10 mol cm−3 . Eq. (7) then gives cw,h For typical operating conditions (p ≈ 1–2 bar, T ≈ 60–80 ◦ C) this condition is fulfilled. We conclude that in a quite wide range of operating conditions the ASM is humidified mainly by water produced on the cathode side. This conclusion is supported by literature data on water transfer coefficient α, the number of water molecules transported per proton from the anode to the cathode side of the cell, taking into account back diffusion. Okada et al. [10] reported α < 0.1 for current densities up to 2 A cm−2 . Detailed measurements of Janssen and Overvelde [20] gave α < 0.2 in a wide range of operating conditions and temperatures. Fitting the along-the-channel model to experiment, Berg et al. [21] obtained α < 0.1 (in our notations) everywhere along the cathode channel, excluding small region close to the outlet. According to [21] total water flux in membrane close to the cathode channel outlet changes sign. Close to the outlet back diffusion in membrane exceeds electroosmotic flux and local transfer coefficient α ≈ −1 [21]. However, the length of this region is small, membrane there is well humidified and the contribution of this region to the overall current production is also small; we thus may neglect this region and take for the estimate α ≈ 0.1. Since water drag coefficient in Nafion nd ≈ 1 [22,23] we have nd /α  1, which means that on

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average the membrane water content is determined mainly by the balance of two large fluxes: elecroosmotic and back diffusion. 2.4. Oxygen transport in the backing layer Now x-axis with the origin at the cathode channel/backing layer interface (Fig. 1) is directed towards the anode. Diffusion flux of oxygen in the backing layer is determined by oxygen consumption in the electrochemical reaction: ∂cox j (8) = , cox |x=0 = cox,h ∂x 4F where cox,h is the oxygen concentration in the cathode channel, Db is the oxygen diffusion coefficient in the backing layer.2 Current density in a cell j is given by Tafel equation:   cox,t η (9) exp j = lt i∗ cref bT −Db

where lt is the catalyst layer thickness, i∗ is the exchange current density, cox,t is the oxygen concentration in the catalyst layer, cref is the reference oxygen concentration, bT is the Tafel slope and η is the polarization voltage of the cathode side. Eq. (9) is valid provided that j  j∗ , where j∗ = 2σt bT / lt is the characteristic current density [24]. Here σ t is the proton conductivity of the cathode catalyst layer. In this (low-current) regime overpotential η weakly depends on x and to a good approximation can be taken constant [25,24]. Integrating (8), substituting x = lb into the solution and using Eq. (9) we get cox,t = cox,h − jlb /4FDb = cox,h − (q exp ηˆ )cox,t . Solving for cox,t we find cox,h (10) cox,t = 1 + q exp ηˆ Here q=

lt i∗ lb , 4FDb cref

ηˆ =

η bT

q exp ηˆ 1 + q exp ηˆ

jD1

(12)

ε1.5 0 Db

(13)

The value of Db can be estimated using the following expression: =

 ξi i

∂cw j =2 , ∂x 4F

Di

where ε0 is the porosity of dry backing layer, ξ i is a molar fraction of ith component in the mixture and Di is the binary diffusion coefficient of oxygen and ith component [17].

cw |x=0 = cw,h

(14)

(Db for oxygen and water vapour are the same). Solving Eq. (14) and substituting x = lb into the solution we get cw,t = cw,h + (2q exp ηˆ )cox,t = cw,h +

2q exp ηˆ cox,h 1 + q exp ηˆ

(15)

where Eq. (10) is used. Total molar concentration in the channel is c: cox,h + cw,h + cb,h = c

(16)

where cb,h is the molar concentration of bulk gas (nitrogen). With (16), Eq. (15) takes a form 2q exp ηˆ cox,h 1 + q exp ηˆ   1 − q exp ηˆ = c − cb,h − ξh c 1 + q exp ηˆ

cw,t = c − cb,h − cox,h +

(17)

2.6. Polarization voltage of the cathode side For further calculations we need to express polarization voltage η through the current and oxygen fraction in the channel ξ h . From (12) we immediately obtain 

j ξ h jT





j − ln 1 − ξh jD1

 (18)

where jT = lt i∗

c is the total molar concentration of the mixture in the channel and we put cox,h = ξ h c, ξ h is the oxygen molar fraction in the channel. 2

Db

ηˆ = ln

where Db c = 4F lb

Since total flux of water in the membrane is zero, the diffusion flux of water in the cathode backing layer is determined only by water produced in ORR:

(11)

Using (10), Eq. (9) can be written as j = ξh jD1

2.5. Water transport in the backing layer

c cref

(19)

The first term on the right side of (18) takes into account activation and concentration overpotentials, the second term accounts for voltage loss due to oxygen transport in the backing layer. Eq. (18) shows that current density in the cell is limited by lim jox = ξh jD1

(20)

(oxygen-limiting current density). This equation expresses the well known fact that current in a cell with ideally humidified membrane is limited by imperfect oxygen transport through the backing layer.

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3. Results 3.1. Voltage loss and membrane resistance Voltage loss in the membrane is lm Vm = jRm = j 0

dx σm (x)

(21)

where x = 0 is located at the ASM and σ m is the membrane proton conductivity. The latter is proportional to membrane water content cl (x) σm (x) = σm0 (22) c H+ where σ m0 is the conductivity at unit water content (one water molecule per sulfonic group). Using (22) and (2) to calculate integral in Eq. (21), we get   j Vm = −bm ln 1 − (23) jl where bm =

FDl cH+ σm0 nd

(24)

and jl is given by (6). Voltage Vm tends to infinity as j → jl due to the drying of the ASM. The value jl itself depends on current density j. Indeed, with (17), Eq. (6) can be written as   1 ζh 1 − q exp ηˆ 0 j l = jl ξ h − − (25) ξh ξh 1 + q exp ηˆ where ζ h = cb,h /c is the molar fraction of bulk gas in the channel and FDl cKλ cH+ jl0 = (26) sat nd lm cw Physically jl depends on the rate of water production and on the flux of water through the backing layer; both these values in turn depend on j. Expressing q exp ηˆ in terms of j with Eq. (12), substituting the result into Eq. (25) and using he resulting expression in (23), after simple manipulations we come to   j Vm = −bm ln 1 − (27) r(ψh jD1 /2 + j) Here ψh = 1 − ζh − ξh

(28)

is the water fraction in the channel and we have used the identity q = jT /jD1 . Membrane resistance Rm = Vm /j then is   bm j Rm = − ln 1 − (29) j r(ψh jD1 /2 + j) Parameter r in (27) and (29) is 2j 0 D l l b K λ cH+ r= l = . sat jD1 2Db nd lm cw

(30)

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Physically, r is proportional to the ratio of mass transfer coefficient of liquid water in membrane to that of water vapour in the backing layer. Large r thus means high rate of back diffusion of water in membrane and/or low rate of water removal through the backing layer. Low r means poor back diffusion in membrane and/or high water flux to the channel. Clearly, the latter situation facilitates drying of the ASM. Relations (27) and (29) can be written in dimensionless form:   j˜ (31) V˜ m = − ln 1 − ˜ r(ψh /2 + j)   1 j˜ ˜ (32) Rm = − ln 1 − ˜ j˜ r(ψh /2 + j) where V , V˜ m = bm

j j˜ = , jD1

˜ = RjD1 R bm

(33)

3.2. Cell voltage Cell voltage is Vcell = Voc − η − Vm − Rc j, where Voc is the open-circuit voltage and Rc is the contact resistance. With (18) and (27) we get     j j Vcell = Voc − bT ln + bT ln 1 − ξh j T ξh jD1   j +bm ln 1 − (34) − Rc j r(ψh jD1 /2 + j) This equation does not take into account the effect of finite oxygen stoichiometric ratio λox on cell voltage [26] and hence is valid in the limit λox → ∞. The third and the fourth terms on the right side of Eq. (34) contain logarithm of the form ln(1 − j/j lim ), which tends to infinity as j → jlim . Physically, current in the cell can be limited either by oxygen transport through the backing layer (the third term) or by membrane drying (the fourth term). The relation between the respective limiting current densities determines the regime of cell operation, as discussed below. 3.3. Dry cathode mixture For dry feed mixture we put ψh = 0 in Eqs. (31) and (32). This yields   1 ˜ (35) Vm = − ln 1 − r   1 1 ˜ Rm = − ln 1 − (36) r j˜ Voltage loss in membrane no longer depends on current and is maximal for given r (Fig. 3a). Indeed, all curves with ψh = 0 (Fig. 3a) tend to V˜ m (35) as j˜ increases. In other words, V˜ m

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concentration in the channel reduces the flux of water in the backing layer and Rm increases with the decrease in j slower (Fig. 3b). When r  1 (thick backing layer − thin membrane), logarithm in (35) can be expanded to yield Vm 

sat bm 2FDb lm cw = r Kλ σm0 lb

3.4. Wet cathodic mixture Hereinafter we will assume ψh = 0. 3.4.1. Arbitrary current When the cathode mixture is wet, parameter r determines limiting current density in the cell. Indeed, V˜ m is infinite if the expression under logarithm in (31) is zero. Equating it to zero we get the limiting current density due to membrane drying (water-limiting current density) j˜ llim =

Fig. 3. (a) Dimensionless voltage loss in membrane and (b) dimensionless membrane resistance vs. dimensionless current density for indicated values of water molar fraction. For all curves parameter r = 2.

(35) coincides with the limiting voltage loss for the case of wet cathode mixture. Physically, under large current membrane state is determined by the balance of electroosmotic flux and water production regardless of water concentration in the cathode channel. Eq. (35) shows that Vm is infinite when r = 1. Physically, if r ≤ 1 the membrane cannot support any current due to fast water removal from the catalyst layer to the channel. This is fundamental reason for cell design with the backing layer on the cathode side: without this layer water produced in ORR rapidly leaks out to the cathode channel, which facilitates membrane drying. The cell with dry cathode mixture can thus operate only when r > 1. Since V˜ m (35) is constant, membrane resistance (36) decreases with the growth of j˜ (Fig. 3b). Physically, when ψh = 0 all available water is produced in electrochemical reaction. Since r > 1, low diffusivity of the backing layer retards water removal to the channel. With the growth of j˜ water accumulates on the cathode side of the membrane, membrane water content increases and membrane resistivity decreases. On the other hand, the rate of water removal through the backing layer for dry mixture exceeds this rate for wet mixture due to zero boundary condition (ψh = 0) in the channel. For dry mixture the growth of Rm as j → 0 is then the fastest, Rm ∼ 1/j (dashed line in Fig. 3b). When ψh = 0, nonzero water

rψh 2(1 − r)

(37)

This equation shows that if r > 1 or r = 1, drying of the ASM does not limit the current (j˜ llim is negative or infinite, respectively; in both cases the expression under logarithm in (31) is positive). When r ≥ 1, water flux from the catalyst layer to the channel is small and amount of water produced is always lim in the sufficient to carry oxygen-limiting current density jox membrane. Incomplete membrane humidification then simply increases the slope of performance curve (see below) not lim . affecting the limiting current density, which is jox For r < 1, however, we get finite positive j˜ llim (37). Physically, in that case the rate of water removal to the channel is high and now membrane drying may limit the current. The lim . Dimenworst-case scenario is when j˜ llim is less than j˜ ox lim = ξ (Eq. (20)); equating j˜ lim = j˜ lim we obtain sionless j˜ ox h ox l re , which provides equality of water- and oxygen-limiting currents: re =

1 1 + ψh /2ξh

(38)

Clearly, re < 1 and we may have the following cases. If r < re lim (watercurrent is limited by membrane drying: j˜ llim < j˜ ox lim lim , i.e., limiting regime). When re ≤ r < 1 we have j˜ l ≥ j˜ ox though water-limiting current is finite, it exceeds oxygenlimiting current. For r = 1 and r > 1 we have infinite and negative j˜ llim , respectively, i.e., no limitation due to membrane drying. Water, therefore, limits the current when r < re . If r > re a lack of water in membrane only reduces cell potential, not lim . Since r < 1, condiaffecting limiting current density j˜ ox e tion r = 1 gives minimal thickness of the backing layer, which maintains membrane in a wet state regardless of feed composition. Equating r = 1 we get the minimal ratio of backing

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layer and membrane thicknesses   sat 2Db nd cw lb = lm min D l K λ c H+

5193

(39)

which supports membrane in a wet sate. 3.4.2. Small current When current is small logarithm in (27) can be expanded to yield Vm 

2bm j = Rm j rψh jD1

(40)

where3 Rm =

sat 2bm lm cw lm = = rψh jD1 σm0 Kλ ψh c σm0 λ

(41)

is the low-current membrane resistance. Here λ = sat is the water content of the cathode side of the Kλ ψh c/cw membrane under small current. In this limit Rm does not depend on current density. Physically, at small current water concentration in the backing layer is uniform and is equal to the concentration in the channel ψh c. Membrane water content also does not depend on sat . Membrane conductivity then is constant x: λ = Kλ ψh c/cw l σ m = σ m0 λ and membrane resistance is Rm = 0m dx/σm = lm /σm0 λ, which coincides with (41).

Fig. 4. (a) Membrane resistance and (b) cell voltage for indicated values of parameter r. The other parameters are: ψh = ξ h = 0.2, jT = 0.05 A cm−2 , ξ h jD1 = 1 A cm−2 , bT = bm = 0.05 V. For ψh = ξ h , re = 2/3.

3.5. Parameter r and cell performance

3.6. Along-the-channel proton current

Membrane resistance and cell voltage for various r are shown in Fig. 4. As shown above, for r ≥ re membrane resistance affects only the slope of performance curve; current lim . If, however, r < r , limiting current denis limited by jox e sity is determined by membrane drying and cell performance dramatically degrades (Fig. 4b). When current is not large, membrane resistance decreases with the growth of j (Fig. 4a). Physically, at small current drying of the ASM is small and the growth of water production with j increases membrane conductivity. For larger j the electroosmotic flux dries out the ASM and Rm increases with j. Therefore, if r < 1 there exists optimal current density jopt , when voltage loss in membrane is minimal (Fig. 4a). jopt is a solution of equation dRm /dj = 0. Numerical solution to this equation is shown in Fig. 5.

The model above ignores the along-the-channel variation of oxygen and water fractions. In this section we show that this variation induces parasitic “in-plane” proton current in a fuel cell. Let z is coordinate along the channel with the origin at the channel inlet. Qualitatively, water accumulates in the cathode flow, i.e., ψh increases with z. Suppose that membrane voltage loss Vm at the inlet is large (dry cathode feed). At a distance L∗ the membrane is well humidified and Vm becomes small

3 Relation (40) is valid provided that one of the following sets of conditions is fulfilled:

r≥1

and

ψh j˜  or 2

r<1

and

j˜  min

ψ

h

2

,

rψh 2(1 − r)



. Fig. 5. Dimensionless optimal current density vs. parameter r.

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4. Discussion

Fig. 6. Sketch of the feed channel.

or even negligible. Equipotentiality of cell electrodes means that the sum η + Vm does not depend on z. Variation of Vm then induces along-the-channel gradient of η. This gradient generates along-the-channel proton current jz in the membrane. To estimate this “in-plane” proton current we write jz = −σm (∂η/∂z). Since η + Vm = const., we have ∂η/∂z 0 (V 0 /L ), = −(∂Vm /∂z) and hence jz = σm (∂Vm /∂z)  σm ∗ m 0 0 where σm and Vm are the membrane conductivity and voltage loss in membrane at the inlet. For x-component of proton 0 V 0 / l . Combining these current at z = 0 we have jx0 = σm m m relations we find jz 

lm 0 j L∗ x

For straight channels the ratio lm /L∗ is small and jz is negligible. Usually, however, the channel has a form of meander. Adjacent turns of meander are separated by distance on the order of 2a (Fig. 6), where a is the channel width. If the number of meander turns covering L∗ is n∗ , the distance between the inlet and z = L∗ along y-axis (Fig. 6) is of the order of 2an∗ . We then get y-component of proton current jy 

lm 0 j 2an∗ x

For upper estimate we take n∗ = 1, i.e. full hydration of membrane occurs at the first turn of the meander. With a = 0.1 cm, lm = 0.01 cm and jx0  j¯ we ¯ Under j¯  1 A cm−2 y-component of proget jy  0.05j. ton current is about 50 mA cm−2 . For smaller a (e.g. in micro-fuel cells) this parasitic proton current can be even higher.

If diffusion coefficients of oxygen and water vapour in the backing layer are different, the value Db in (30) should be replaced with water diffusion coefficient Db,w , whereas lim (20) the expression for oxygen-limiting current density jox should contain oxygen diffusion coefficient Db,ox . Varying Db,w one then can vary parameter r, leaving limiting current lim intact. This situation is presumably realized in the density jox gas diffusion media with the microlayer [20]. The microlayer retards water removal from the membrane, not affecting oxygen transport. In other words, microlayer increases parameter lim unchanged. r, leaving jox Diffusion coefficient of liquid water in the membrane Dl was taken to be constant. In Nafion membranes Dl decreases with λ for low water contents [18]. The role of this nonlinearity was considered in detail in [7]. Qualitatively, account of this dependence would lead to lower water-limiting current density. The vast majority of works on numerical modelling of PEFCs utilize the assumption η(z) = const. It can be shown that the constancy of η leads to the exponential decrease of local current density along the channel [27,26]. Physically, when η is constant most part of the current is produced close to the channel inlet, where oxygen concentration is the largest. This situation, however, is realized if membrane is well hydrated. If the membrane falls short of water, it cannot carry the current produced in electrochemical reaction and the exponential profile j(z) is distorted. This is the case of waterlimiting regime, considered above. Local η decreases in the regions of dry membrane, so that the reaction there generates exactly the current, which can be supported by membrane plus along-the-channel current σ m ∂η/∂z. Our quasi-3D simulation of PEFC, which resolve catalyst layers confirms this picture (to be published elsewhere). Constant η is thus a simplifying assumption, which is not valid in the case of strong local drying of the membrane. Physically, since electrodes are equipotential it is the sum η + Vm , which remains constant. The expression for Vm (27) does not contain exchange current density i∗ and Tafel slope bT . Voltage loss due to membrane drying is hence determined only by the balance of water fluxes in the membrane and in the backing layer, regardless of ORR kinetics. The thicknesses of the backing layer and membrane can hence be optimised for proper membrane humidification irrespective of kinetic properties of the catalyst layer. 5. Conclusions • Simple model of water transport in polymer electrolyte fuel cell is developed. • The expressions for membrane voltage loss and membrane resistance are obtained. These expressions take into account the balance of water fluxes in the membrane and in the backing layer.

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• The equation for cell polarization voltage is constructed. It shows that the regime of cell operation (oxygen- or waterlimiting) is controlled by parameter r, which is proportional to the ratio of mass transport coefficients of water in the membrane and in the backing layer. • When r < 1 there exists optimal current density, which provides minimal membrane resistance. • Voltage loss in the membrane and the value of parameter r do not depend on kinetics of ORR. • Membrane drying causes parasitic “in-plane” proton current.

Appendix A. Outgoing flux of water on the anode side In this section we estimate outgoing (i.e., directed from membrane towards the anode channel) flux of water in the anode backing layer. Putting x = 0 in Eq. (2). we get the concentration of liquid water on the anode side of the membrane   j a c cl,t 1− (A.1) = cl,t jl where the subscripts “a” and “c” refer to the anode and the cathode side, respectively. Taking into account Eq. (5), we obtain the relation between water vapour concentrations on both sides of the membrane   j a c cw,t = cw,t 1 − (A.2) jl Solving Eq. (14) we find c c cw,t = cw,h +

jlb 2FDbc

(A.3)

The flux of water vapour in the anode backing layer is a − ca )/ l ). Expressing ca from this relaNwa = Dba ((cw,t b w,t w,h a and cc (Eq. (A.3)) into Eq. (A.2) tion and substituting cw,t w,t we find
    Dba j jlb a c a Nw = cw,h + 1− − cw,h (A.4) lb 2FDbc jl Diffusion coefficient of water vapour in hydrogen is three to four times higher, than diffusivity of vapour in oxygen/nitrogen mixture [17]. For the estimate we put Dba = 4Dbc ; Eq. (A.4) then takes a form
     4Dbc c j 2j j a a Nw = cw,h 1 − − cw,h + (A.5) 1− lb F jl jl This shows, that outgoing flux of water is maximal when a cw,h = 0 (dry hydrogen). For that case we get  Nwa =

c 4Db cw,h

lb

+

2j F

 1−

j jl

 (A.6)

5195

Limiting current density jl is itself a function of local current density j. Comparing Eqs. (27) and (23) we see, that jl = r(ψh jD1 /2 + j). To determine the characteristic current density, for which the assumption of zero total flux in membrane is violated we substitute this jl to Eq. (A.6) and equate resulted Nwa to electroosmotic flux nd j/F. This gives   j˜ ˜ (ξh + 2j) 1 − (A.7) = nd j˜ ˜ r(ψh /2 + j) As before j˜ = j/jD1 . For the estimate we put nd = 1 and consider the two cases: (i) dry cathode feed and (ii) well-humidified cathode feed. In the first case we put ψh = 0 in Eq. (A.7). Solving the resulting equation we find ξh (r − 1) j˜ dry = 2−r

(A.8)

In a wide range of operating conditions parameter r does not exceed 1. For upper estimate we take r = 1, which physically means low rate of water removal through the cathode backing layer. With this r Eq. (A.8) becomes j˜ dry = 0 (for r < 1 j˜ dry is even negative). We see that in case of dry cathode feed the outgoing flux of water is always negligible. Physically, this is quite reasonable, since the cell operates with zero external supply of water. For the case of well-humidified cathode feed we put ψh = ξ h in Eq. (A.7). Solving the resulting equation we find ξh r j˜ wet = 2−r

(A.9)

With r = 1 we obtain j˜ wet = ξh . Dimensionless oxygenlimiting current density is ξ h ; we thus conclude, that the outgoing flux of water in the anode backing layer exceeds electroosmotic flux in membrane in the whole range of working current densities. This, however, is the “worst-case” combination of parameters, which physically corresponds to dry anode feed plus well-humidified cathode feed plus good “isolation” of water in the cathode catalyst layer (r = 1). In this particular case the model above is inapplicable. In this case, however, the membrane is well humidified and the whole problem is of minor practical interest, since voltage loss in membrane is small.

Appendix B. Nomenclature

bm bT c cH+ cl cox

characteristic voltage loss in membrane (V) Tafel slope (V) total molar concentration of the mixture in the cathode channel (mol cm−3 ) proton molar concentration in membrane (mol cm−3 ) molar concentration of liquid water in membrane (mol cm−3 ) oxygen molar concentration (mol cm−3 )

5196

cref cw sat cw Db Dl F i∗ j jD1 j∗ jllim lim jox

Kλ lb lm lt nd N r re Rc Rm Vcell Vm Voc x z

A.A. Kulikovsky / Electrochimica Acta 49 (2004) 5187–5196

reference oxygen molar concentration (mol cm−3 ) molar concentration of water vapour (mol cm−3 ) molar concentration of saturated water vapour (mol cm−3 ) oxygen and water vapour diffusion coefficient in the backing layer (cm2 s−1 ) diffusion coefficient of liquid water in membrane (cm2 s−1 ) Faraday constant (9.6495 × 104 C mol−1 ) exchange current density per unit volume (A cm−3 ) mean current density in a cell (A cm−2 ) characteristic limiting current density (A cm−2 ) characteristic current density (A cm−2 ) limiting current density due to membrane drying (A cm−2 ) limiting current density due to imperfect oxygen transport in the backing layer (A cm−2 ) dimensionless slope of membrane water sorption isotherm thickness of the backing layer (cm) thickness of the membrane (cm) thickness of the catalyst layer (cm) electroosmotic drag coefficient molar flux (mol cm−2 s−1 ) dimensionless parameter value of r, which provides equality of oxygen- and water-limiting current densities contact resistance (# cm2 ) membrane resistance (# cm2 ) cell voltage (V) voltage loss in membrane (V) cell open circuit voltage (V) coordinate across the cell (cm) coordinate along the channel (cm)

Greek symbols ζh molar fraction of bulk gas (nitrogen) in the channel η polarization voltage of the cathode side (V) λ membrane water content (a number of water molecules per sulfonic group) λox stoichiometry of oxygen flow Λ membrane water sorption isotherm ξh oxygen molar fraction in the channel σm proton conductivity of bulk membrane (#−1 cm−1 ) σ m0 proton conductivity of bulk membrane at unit water content (#−1 cm−1 ) σt proton conductivity of membrane phase in the catalyst layer (#−1 cm−1 ) molar fraction of water vapour in the channel ψh

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