Characteristic length of fuel and oxygen consumption in feed channels of polymer electrolyte fuel cells

Electrochimica Acta 46 (2001) 4389– 4395 www.elsevier.com/locate/electacta

Characteristic length of fuel and oxygen consumption in feed channels of polymer electrolyte fuel cells A.A. Kornyshev, A.A. Kulikovsky * Institute for Materials and Processes in Energy Systems (IWV-3), Research Center ‘‘Ju¨lich’’, D-52425 Ju¨lich, Germany Received 17 April 2001; received in revised form 11 June 2001

Abstract How much of feed channels is actually used in a fuel cell? This problem is considered in this paper where we study gas or liquid feed consumption in the feed supplying channels of a polymer electrolyte fuel cell. Simple analytical expressions which define profiles of feed concentration along the channel and expressions for characteristic length of feed consumption are obtained. For typical conditions this length could be as low as several tens of centimeters. The dependence of consumption length on reaction rate parameters and system parameters is discussed. Analytical results are in good agreement with data of simulation, performed specially to verify the validity of the theory. © 2001 Elsevier Science Ltd. All rights reserved. Keywords: PEMFC; DMFC; Channel; Feed consumption

1. Introduction During the past decade polymer electrolyte membrane (PEM) fuel cells with hydrogen or methanol as a fuel were intensively investigated in view of promising applications for electrical vehicles and portable power sources [1]. A typical design of PEM fuel cell is shown in Fig. 1. Two metal plates have channels, which supply the pressurized feed to the reaction layers. The feed could be hydrogen, gaseous or liquid methanol at the anode and oxygen or air at the cathode. The feed channels may have a meander or spiral form and can be several meters long. The feed, supplied at the channel inlet, moves through the channel. On the way it leaks in a perpendicular direction through the porous backing

layer and reaches the catalyst layer, where the electrochemical reaction occurs. The feed concentration varies along the channel due to the consumption in the electrochemical reaction. Recently a number of numerical models of PEM fuel cells, which include the gas transport along the channel were developed [2 – 6]. The simulations show that gas consumption can be high near the channel inlet, so that there can be a lack of feed in the rest of the channel. In this paper we formulate the laws of the feed consumption and compare them with numerical simulations.

2. The law and characteristic length of gas consumption

2.1. Mass balance * Corresponding author. On leave from Moscow State University, Research Computing Center (NIVC), 119899 Moscow, Russia. Tel.: +49-2461-615396; fax: +49-2461-616695. E-mail address: [email protected], a.kulikovsky@fz. juelich.de (A.A. Kulikovsky).

The channel will be considered as straight, i.e. we will neglect the influence of curved parts on mass balance. The continuity equation for molar concentration c of feed molecules has a form

0013-4686/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 4 6 8 6 ( 0 1 ) 0 0 6 6 2 - 4

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d(6c) S j(z) =− . dz nF h

(1)

Here z is a distance along the channel, 6 the flow velocity, j, local current density at a given point z; h, the height of the channel, n, the number of electrons participating in the reaction; S the stoichiometric coefficient (the number of molecules which give or consume n electrons in electrochemical reaction), and F the Faraday constant1. Eq. (1) follows from the mass balance in an elementary volume of the channel. When the feed gas constitutes a small portion of a total gas flow (e.g. oxygen in air-breathing PEFC or DMFC), one may assume that 6(z) is approximately constant, equal to the flow velocity at the inlet 60. Here we will use this assumption.

2.2. Generation of the current Generally, the mechanism of current production in a cell involves transport of feed molecules to the reaction sites, fuel ionization in the anode catalyst layer, proton migration through the membrane and charge recombi-

nation in the cathode catalyst layer. We will consider one electrode (anode or cathode) and will neglect diffusion limitations on the transport of feed molecules to the reaction sites. The latter is a simplifying assumption. However, below we compare the resulting formulae with computer simulations that do not employ this assumption. The comparison shows that for quite realistic cases this assumption works well. If the catalyst layer is thick, the reaction takes place in a stripe of the catalyst layer, adjacent to the membrane. The characteristic width of the stripe [7] is u=

'

|RT , 2Fhi(p, c)

(2)

where | is a proton conductivity of the layer and i(p, c) is a rate of electrochemical reaction (a charge produced/consumed in unit volume per second), p the overpotential at the catalyst layer/membrane interface and h the transfer coefficient. The generated current density in that case is (3)

j = i(p, c)u

The inequality l  u, where l is the thickness of the layer specifies the case of a thick layer. In the opposite case, when l u (thin catalyst layer) the reaction takes place in the whole volume of the layer and (4)

j= i(p, c)l

Fig. 1. The sketch of the gas channel and cell cross-section. 1

For hydrogen – oxygen fuel cell n=4, S = 2 for the hydrogen electrode (anode) and S= 1 for oxygen electrode (cathode). For direct methanol fuel cell (DMFC), n= 6, S = 1 for the anode and S=3/2 for the cathode.

In expressions (3) and (4) we assumed that overpotential p does not vary significantly at the distance u or l, so that the integrals u0i(p(y)) dy and l0i(p(y)) dy may be replaced by products i(p(0))u and i(p(0))l, respectively. Here y is a distance across the catalyst layer, measured from the membrane. The dependence of p on distance along the channel z will also be neglected. The numerical simulations justify this assumption: for typical conditions the variation of p at the catalyst layer/membrane interface on both sides of the cell does not exceed several percent on a length of about 1 m (see below). The intermediate case, u # l, is described by more complicated expressions [7] and will be considered elsewhere. Here we consider the limiting cases of thick and thin layers. A thick layer produces/consumes the maximum possible current density per unit length of the channel and, therefore, gives a lower limit of feed consumption length. For the function i we will use Tafel law [8], which is a standard approximation both for the anode and the cathode reactions in fuel cells, far from equilibrium [9–11]: i(p, c) =i 

    c cref

k

exp

hF p , RT

k \0.

(5)

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Here i  is the product of exchange current density and the volume density of the reaction surface, cref is a reference molar concentration of feed [11], h the effective transfer coefficient and k is the effective reaction order.

2.3. Thin catalyst layer Substituting Eqs. (5) and (4) into Eq. (1) under the assumption 6(z)#60, we come to

 

 

1 c d c =− L1 cref dz cref

L1 =

(6)

nF h60cref hF p exp − S li  RT



(7)

 



c(z) =c0 1−(1−k) Á ÃkB1, Í Ãk\1, Ä where L k = L1

2.4. Thick catalyst layer

z Lk

n

Lk 1−k 05 zB

 

 

Substituting Eqs. (5) and (3) into Eq. (1) we get d c c 1 =− dz cref L0 1 cref

At a constant p, Eq. (6) is the ordinary differential equation with constant coefficients whose integration is trivial. Introducing the notation c0 =c(z=0), which is a molar concentration of feed at the channel inlet, we obtain: z c(z)=c0 exp − , k=1 (8) L1 and

The faster fuel consumption for a smaller reaction order is understandable. If there were no dependence of the reaction current (Eq. (5)) on c, the fuel would be consumed at all distances at the same rate and the variation of the concentration with z would be linear (Eq. (9), k= 0). For k\ 0 the rate of reaction decreases with the decrease of amount of available gas and this slows down the consumption.

k



where

4391

'

where L0 1 =

k/2

(11)



2Fh nF hF h60cref exp − p |RTi  S 2RT



(12)

We, thus, have the same type of equation as Eq. (6) and may use the results of Section 2.3 with the replacements: L1 “ L0 1 and k “ k/2. This gives

1/(1 − k)

05 z5

  c0 cref

(9)

1−k

(10)

is a characteristic length of feed consumption in the channel. Thus for k B1 there exists a distance zk =(Lk )/ (1− k), where the feed concentration vanishes, c(zk )= 0, together with the derivative, (dc/dz)z = zk =0. For k5 1 the concentration monotonically decays to zero as z tends to infinity. This rather tricky change of dependence is illustrated in Fig. 2 for the cases of k=0.5 and k= 1.5. Note that in the case of k"1 the characteristic length Lk depends also on the inlet molar concentration of the feed c0 (Eq. (10)). For typical parameters of the cathode of a gas-feed DMFC, h =0.1 cm, 60 =15 cm s − 1, T =110 °C, i  = 10 − 5 A cm − 3, h=2, k=0.5, n=4, S =1, p=0.3 V and c0 =cref the u#7.2×10 − 4 cm (7.2 mm). For the case of thin catalyst layer (l=4 mm) the characteristic length is Lk #37 cm and the ultimate consumption length 2Lk #74 cm. Here, there is no reason to make the air channel longer than 74 cm, since the rest of the channel will not contribute to the current generation.

Fig. 2. The profiles of feed concentration along the channel for different values of c0/cref and k. The example of k =0.5: c= c0(1 −(z/(2Lk )))2, 0 5 z 52Lk shows the finite length of total feed consumption. For k =1.5: c =c0(1 +(z/(2Lk ))) − 2, 0 5z 5 . For k =1 the concentration decreases exponentially. The profile of feed concentration depends on the inlet concentration c0: the larger c0 the faster the decrease in c for k \1 and the slower it is for k B 1.

     n

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z , k=2 L0 1 k z 1/(1 − k/2) c(z) =c0 1− 1− 2 L0 k c(z) =c0 exp −

L0 k 1−k/2 05 zB

Á ÃkB2, Í Ãk\2, Ä where L0 k = L0 1

0 5z5

  c0 cref

(13)

(14)

1 − (k/2)

(15)

For k B2 the ultimate consumption length is z˜ k =

L0 1 1−

k 2

,

kB 2

(16)

For the same set of parameters as in Section 2.3, L0 k # 20.4 cm. The thick catalyst layer consumes feed molecules faster than the thin one, since it offers a larger reaction volume. Therefore, consumption length is smaller. This is, of course, valid only to the accuracy of insignificance of gas diffusion limitations in the electrode.

3. Comparison with the numerical simulations To verify the theory predictions we have performed numerical simulations of gas feed DMFC using the quasi-3D numerical model [6]. The model takes into account diffusion gas transport to the catalyst layers and uses Butler –Volmer kinetics [8] for electrochemical reactions. The flow in gas channels on both sides of the cell is described by mass conservation and Euler’s equations.

3.1. Numerical model The numerical model is based on the following idea. The 2D and 3D ‘‘along-the-channel’’ models [2,3] clearly show that feed gas concentration and local current density vary but slowly along z. On a distance of the order of thickness of the cell these variations are negligible. This allows us to neglect the fluxes of gases and currents in porous media (backing and catalyst layers) in the z-direction as compared to these fluxes in the plane, perpendicular to the channel. The fully 3D model of fuel cell can then be split into two models. The first one is a 2D model for gas concentrations and currents in the plane, perpendicular to the feed channels (the ‘‘inner problem’’). The second one is a problem of gas flow in the channels on both sides of the cell, where the feed gas consumption is determined by the local current density along the channel (the ‘‘channel problem’’).

Under the given concentrations in all the channel cross-sections (Fig. 1), the inner problem can be solved. It gives local current density in nine points along each channel. These profiles then are used to solve the channel problem, which gives new values of gas concentration in every channel cross section (Fig. 1). This procedure is repeated until convergence is reached. The inner problem can further be split into nine identical problems for single cell elements (one element is shown in Fig. 1 by dashed lines). The 2D equations, which govern the distributions of gas concentration and current densities in a single element are described in Ref. [12]. Each element is simulated on two processors. On each iteration every element exchanges ‘‘boundary conditions’’ (the concentrations and potentials along the dashed lines, Fig. 1) with the neighbor elements. The advantage of that approach is that it allows simulation of any number of elements and hence channels of any length. The number of elements is limited only by a number of available processors.

3.2. The map of fuel cell Cell operation conditions and parameters are presented in Table B1. The simulation was performed for the cell with the 80 cm long channels. Each channel has a meander form. Fig. 3 shows the color maps of the methanol and oxygen concentrations, electrochemical reaction rates and membrane phase potential. Since conductivity of carbon phase is large, the overpotential keeps track of the variation of membrane phase potential. The distribution of membrane phase potential justifies the assumption that p practically does not depend on z: the variation of p along e.g. cathode catalyst layer/membrane interface does not exceed 2% (Fig. 3).

3.3. Calculation of feed consumption length The extraction of feed consumption length from the simulation data needs explanation. (The same procedure can be applied to experimental data). It needs first the evaluation of p at a given j for the specified length of the channel, L and then calculation of consumption length itself. 1. For a given mean current density j we put c= c0 (inlet feed gas concentration) in relations (2) and (5) and adjusted p in order the relation i(p, c0)u =j be satisfied. This gives an estimate of u. 2. (a) If l  u (thin catalyst layer), then the mean current density in a cell may be equated to j: (l/L) L0 i(p, c(z)) dz= j. Here c(z) is defined by Eq. (9). This gives p and then Eqs. (7) and (10) are used to calculate Lk. (b) If l  u (thick catalyst layer), p is obtained from u/L L0 i(p, c(z)) dz=j, where c(z) is given by Eq. (14). Eqs. (12) and (15) are then used to calculate L0 k.

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Fig. 3. The color maps of the parameters in a cross-section of gas feed DMFC (the sketch of the cell is shown in Fig. 1). Displayed are the maps of methanol and oxygen concentration (10 − 6 mol cm − 3), reaction rates at the anode and cathode catalyst layers, ia and ic, respectively (A cm − 3) and membrane phase potential € (mV).

3.4. Verification of analytical results For the data shown in Table B1, 300 mA cm − 2 current density is provided by overpotential p=294 mV (see Eqs. (3) and (5)). For such p the reaction stripe has a width u#9 mm. The catalyst layer has a thickness 10 mm (Table B1) and we approximately treat this as a case of a thin catalyst layer, described by Eqs. (7), (10) and (9); for lower current densities, 100 and 200 mA cm − 3, the inequality l u is warranted. Theoretical and simulated profiles of oxygen concentration along the channel are shown in Fig. 4. It is

seen that the theory fits the simulated points well. The simulation gives p # 315 mV at the cathode catalyst layer/membrane interface (Fig. 3). The small difference with the analytically calculated value (294 mV) is due to the fact that the theory neglects the diffusion loss of oxygen in the backing layer. This loss leads to a lower concentration of oxygen in the catalyst layer than in the theory and forces p to be several percent higher than the theoretical one. It is essential that the shape of p is fairly flat along the cathode catalyst layer (Fig. 3). This fully validates our theoretical predictions.

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Fig. 4. Comparison of simulated (dots) and theoretical (solid lines) profiles of oxygen concentration (10 − 6 mol cm − 3) along the channel for the three indicated values of mean current density in the cell (mA cm − 2). Here Lk exceeds 1 m and we see the ‘linear’ sections of the feed concentration decay (c.f. Fig. 2).

4. Discussion We have considered the laws of the fuel and oxygen consumption in the anode or in the cathode separately. However, in a cell the performance of the cathode and the anode is coordinated, since the same number of protons that was produced at the anode must be consumed at the cathode. Simple analysis (to be published elsewhere) shows that the consumption length is determined by the smallest consumption length of the anode and the cathode. When the feed on one side of the cell will be consumed, the remaining part of the feed on the other side will leave the channel unused. Throughout the paper we considered two cases, ‘thin’ and ‘thick’ catalyst layers. They give upper and lower estimates, respectively, for the consumption length, as far as there are no diffusion losses of the feed transport across the electrodes. Under high current densities the kinetics of oxygen reduction at the cathode of the PEM fuel cell usually obeys the Tafel law (Eq. (5)) with k=0.5 [10,11]. The oxygen then is completely consumed on a finite length lying between 3Lk /4 and 2Lk (or 3L0 k /4 and 2L0 k in the case of a thick catalyst layer). This length can be as low as several tens of centimeters. If the length of the oxygen channel exceeds this value, the remaining part of the cell does not produce electricity at all. If k " 1, then the profile of feed concentration along the channel depends on the inlet concentration (Fig. 2). The higher the c0 the slower c decreases with z for kB1 and the faster it decreases for k\1. This behavior is transparent: if k\1, the growth of c0 essentially accelerates the reaction rate (Eq. (5)) near the channel inlet and the feed concentration is rapidly exhausted. In the case of kB1, the reaction rate depends on c0 weaker, and the increase in c0 gives an inverse effect: the length of total feed consumption increases due to a higher inlet flux.

The diffusion losses hinder the transport of feed to the reaction sites. One may expect that account for such losses would increase the consumption length. The integration of Eqs. (4) and (3) along the channel with the proper function c(z) gives the current –voltage curve [13]. The cell with a long feed channel exhibits limiting current density, caused by the total feed gas consumption in the channel. Moreover, the marked voltage loss, associated with the non uniform distribution of feed gas along the channel, appears in the whole range of practically interesting current densities [13]. For the case of k B1 the dependence of mean current density on channel length has a maximum. Indeed, the feed is consumed at the finite length (Eq. (10)) and further increase L is useless: the mean current density will decrease. Note that the presence of such a maximum on a measured current –length curve for a given voltage could give information about the reaction order k. The simplest way to increase the characteristic length of feed consumption is to increase the inlet velocity 60 or the channel height h. All in all the simple formulas derived can be used for an estimate of the characteristic length of feed consumption and a rational design of the gas supplying compartment.

Acknowledgements We are grateful to H. Dohle, J. Mergel and D. Stolten for useful discussions. The simulations were performed on Cray T3E computer of the von Neumann Institute of Applied Mathematics, Research Center ‘‘Ju¨ lich’’, Germany.

Appendix A. Nomenclature c c0 cref F h i i j l L Lk L1

feed gas concentration (mol cm−3) feed gas concentration at the channel inlet (mol cm−3) reference concentration of feed gas (mol cm−3) Faraday constant (9.6495×104 Coulomb gmol−1) channel height (cm) rate of electrochemical reaction (A cm−3) exchange current density per unit volume (A cm−3) local current density in a cell (A cm−2) catalyst layer thickness (cm) length of the channel (cm) characteristic length of feed consumption (cm) characteristic length of feed consumption for k= 1 (cm)

A.A. Kornyshe6, A.A. Kuliko6sky / Electrochimica Acta 46 (2001) 4389–4395

6 60 R S T z zk h

flow velocity in the channel (cm s−1) flow velocity at the channel inlet (cm s−1) gas constant (8.314 J K−1 g mol−1) stoichiometric coefficient cell temperature (K) coordinate along the channel (cm) ultimate consumption length (cm) Butler–Volmer transfer coefficient

Greek k p | |n

symbols reaction order overpotential (V) membrane conductivity (V−1 cm−1) carbon phase conductivity (V−1 cm−1)

Appendix B Table B1 Conditions and parameters of numerical simulations

Cell temperature T (°C) Inlet gas pressure p (atm) Inlet gas velocity 60 (cm s−1) Oxygen molar fraction at the inlet Water vapor molar fraction at the inlet Nitrogen molar fraction at the inlet Methanol molar fraction at the inlet CO2 molar fraction at the inlet cO2 ref (mol cm−3) i  (A cm−3) Transfer coefficient h Reaction order k Mean pore radius in the backing layer Žr (cm) Mean pore radius in the catalyst layer Žr (cm) Membrane phase conductivity | (V−1 cm−1) Carbon phase conductivity |n (V−1 cm−1) Catalyst layer thickness (cm) Backing layer thickness (cm)

Anode side

Cathode side

110 1.5 100

110 2.0 100 0.2

0.79

0.01 0.79

0.20 0.01

0.01 0.5 1.0 10−5

3.18×10−5 1.0×10−5 2.0 0.5 10−5

10−6

10−6

0.0315

0.0315

40

40

0.001 0.01

0.001 0.01

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Table B1 (Continued)

Membrane thickness (cm) Channel width (cm) Channel height (cm) Channel length (cm)

Anode side

Cathode side

0.002 0.1 0.1 80

0.002 0.1 0.1 80

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