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1D + 1D model of a DMFC: localized solutions and mixedpotential
Electrochemistry Communications 6 (2004) 1259–1265 www.elsevier.com/locate/elecom
1D + 1D model of a DMFC: localized solutions and mixedpotential A.A. Kulikovsky
*
Institute for Materials and Processes in Energy Systems (IWV-3), Research Center ‘‘Ju¨lich’’, D-52428 Ju¨lich, Germany Received 9 September 2004; received in revised form 28 September 2004; accepted 4 October 2004 Available online 19 October 2004
Abstract Our 1D + 1D model of DMFC reveals a new effect. At infinitely small total current in the cell, near the channel inlet forms a ‘‘bridge’’, a narrow region with finite local current density. The bridge short-circuits the electrodes, thus reducing cell open-circuit voltage. In our previous work the effect is described for the case of equal methanol ka and oxygen kc stoichiometries. In this Letter, we analyze the general case of arbitrary ka and kc. In the case of ka > kc current may occupy finite domain of the cell surface. Asymptotic solution for the case of ka kc shows, that the size of this domain is proportional to oxygen stoichiometry. In the opposite limit of ka kc local current exponentially decreases with the distance along the channel. Asymptotic solutions suggest that the bridge forms regardless of the relationship between ka and kc. In all cases local current density in the bridge increases with the rate of methanol crossover and decreases with the growth of the ‘‘rate-determining’’ stoichiometry. The expression for voltage loss at open-circuit is derived. 2004 Elsevier B.V. All rights reserved. Keywords: DMFC; Analytical model; Mixedpotential; Open-circuit voltage
1. Introduction Methanol crossover through the membrane dramatically decreases performance of a direct methanol fuel cell (DMFC). During the past decade a huge work has been done worldwide to reduce crossover [1]. Though the negative effect of crossover is well recognized, the physics of lowering of cell voltage due to crossover is still poorly understood. Crossover dramatically reduces DMFC voltage at small currents. This effect, called mixedpotential is well known in DMFC studies. Several mechanisms can contribute to this reduction. Methanol, permeated through the membrane is oxidized on the cathode side. ‘‘Burning’’ of permeated methanol requires additional flux of oxygen through the cathode gas diffusion (backing)
*
On leave from Moscow State University, Research Computing Center, 119992 Moscow, Russia. Fax: +49 2461 61 6695. E-mail address: [email protected]. 1388-2481/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.elecom.2004.10.002
layer, which is equivalent to additional resistance [2]. One may speculate about poisoning of the cathode catalyst layer by the intermediates of methanol oxidation [1]. Another candidate to explain mixedpotential is flooding of the cathode side by water produced in the methanol–oxygen reaction [3,4]. However, due to the lack of kinetic and transport data it is difficult to estimate the respective voltage loss and the nature of mixedpotential is still controversial. Our 1D + 1D model of DMFC suggests the new explanation of mixedpotential. Analytical solution to model equations reveals the new effect: formation of a narrow ‘‘bridge’’ of local current near the inlet of the feed channels [5]. Local current density in the bridge remains finite at vanishingly small total current in the cell. The bridge ‘‘short-circuits’’ the electrodes, thus reducing cell open-circuit voltage. Analysis [5] is performed for the case of equal methanol and oxygen stoichiometries ka and kc. In that case local current density exponentially decreases with the distance along the channel.
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A.A. Kulikovsky / Electrochemistry Communications 6 (2004) 1259–1265
Nomenclature b c cref Db E F i* J J~ j jcross jlim h lb lm lt L Ncross p q Rn
Tafel slope (V) total molar concentration in the channel (mol cm3) reference molar concentration (mol cm3) diffusion coefficient of feed molecules in the backing layer (cm2 s1) total voltage loss (V) Faraday constant exchange current density per unit volume (A cm3) mean current density in a cell (A cm2) dimensionless mean current density and total current in a cell local current density (A cm2) equivalent crossover current density (A cm2) limiting current density (A cm2) channel height (cm) thickness of the backing layer (cm) thickness of the membrane (cm) thickness of the catalyst layer (cm) channel length (cm) molar flux of methanol through the membrane (mol cm2 s1) ratio of Tafel slopes (p = bc/ba) dimensionless parameter contact resistance (X cm2)
In this Letter, we analyze the case of arbitrary ka and kc. In the case of ka > kc the problem has a new type of solutions: due to the lack of oxygen current density appears to be localized in a finite domain. The bridge then has finite spread over the cell surface. Asymptotic solutions to a problem suggest that the bridge forms regardless of the relationship between ka and kc. In all cases local current density in the bridge appears to be proportional to the rate of methanol crossover and inversely proportional to the ratedetermining stoichiometry.
2. Model 2.1. Continuity equations In the next two sections we briefly repeat the derivation of the governing equations [5]. Molar fractions of methanol w and oxygen n in the respective channel are determined by continuity equations: ha v a c a
ow j ¼ N cross ; oz 6F
ð1Þ
cell voltage (V) cell open circuit voltage (V) flow velocity in the channel (cm s1) coordinate along the channel (cm)
Vcell Voc v z
Superscripts 0 at the inlet (at z = 0) a anode side c cathode side Subscripts 0 at zero total current b backing layer lim limiting t catalyst layer Greek symbols b dimensionless parameter of crossover b1 b1 = b/(1 + b) c dimensionless parameter g polarization voltage (V) j dimensionless local current density at the inlet k stoichiometry of the feed flow n oxygen molar fraction in the channel w methanol molar fraction in the channel
on j 3 ¼ N cross : ð2Þ oz 4F 2 The superscripts ‘‘a’’ and ‘‘c’’ refer to the anode and the cathode side, respectively, z is the distance along the channel, h is the channel height, v is the flow velocity, c is the total molar concentration of the mixture in the channel, j(z) is the local current density. Eqs. (1) and (2) are written assuming plug flow conditions in both channels. The flux of methanol through the membrane Ncross is given by [6] ja b j N cross ¼ lim 1 a ; ð3Þ jlim 6F 1 þ b
h c vc c c
where jalim ¼ 6FDab ca w=lab is local limiting current density due to methanol transport through the backing layer and b is a ratio of methanol mass transfer coefficients in the membrane and in the anode backing layer b¼
Dm lab : Dab lm
ð4Þ
Here Dm and Dab are diffusion coefficients of methanol in the membrane and in the anode backing layer, respectively, lm and lab are the thicknesses of the membrane
A.A. Kulikovsky / Electrochemistry Communications 6 (2004) 1259–1265
and of the backing layer, c is the total molar concentration of the mixture in the channel. The dominating mechanism of crossover is diffusion due to the concentration gradient (see [6] for detailed discussion). The flux Ncross (3) linearly decreases with the growth of current density j, which agrees with the experiments [7–9]. To analyze the governing Eqs. (1) and (2) we introduce dimensionless variables ~j ¼ j ; ja0 lim
z ~z ¼ ; L
~¼ w; w w0
n ~ n¼ 0; n
ð5Þ
where L is channel length, the superscript ‘‘0’’ indicates the values at the channel inlet (at z = 0) and ja0 lim
Da ca w0 ¼ 6F b a lb
ð6Þ
is methanol-limiting current density at the channel inlet. Introducing stoichiometries ka and kc of methanol and oxygen flow, respectively ka ¼
6Fha va ca w0 ; LJ
kc ¼
4Fhc vc cc n0 ; LJ
Eqs. (1) and (2) transform to ka
~ ~ ~jÞ ~j þ b1 ðw ow ¼ ; ~ o~z J
ð7Þ
kc
~ ~jÞ ~j þ b1 ðw o~ n ¼ : ~ o~z J
ð8Þ
Here J~ ¼ J =ja0 lim is dimensionless mean current density in a cell and b1 ¼
b : 1þb
ð9Þ
The mean current density J and the total current in the cell I differ only by a constant factor, the area of cell surface. In dimensionless variables these values coincide; for that reason we will treat J~ also as the total current in the cell. Eqs. (7) and (8) show that the rates of methanol and oxygen consumption are proportional to the sum ~j þ ~jcross , where equivalent crossover current ~jcross is ~ ~ ~jcross N cross =ð6Fja0 lim Þ ¼ b1 ðw jÞ:
ð10Þ
2.2. Polarization voltages The expressions for polarization voltages ga, gc of the anode and the cathode side, respectively, are derived in [6] within the scope of 1D model of DMFC. With the dimensionless variables (5) these expressions take a form
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~j ~j ga þ lnð1 þ bÞ; ¼ ln ln 1 ~ ~ ba w qw ! ~ ~jÞ ~j ~j þ b1 ðw gc ¼ ln ; ln 1 bc c~n aq~n
ð11Þ
ð12Þ
where la ia la lc i c c a q ¼ t a ba ; a ¼ at a ref 6FDb cref lt i ccref 2Dcb lab cc n0 c¼ 3Dab lcb ca w0
c c n0 ; ca w 0
ð13Þ
are dimensionless parameters. Here lt is the thickness of the catalyst layer, i* is the exchange current density per unit volume and cref is the reference molar concentration of the feed molecules. Physically, the first terms on the right sides of (11) and (12) describe reaction activation and concentration overpotentials. The second terms there describe voltage loss due to the transport of methanol and oxygen, respectively, through the backing layer. The second logarithm in (12) takes also into account transport of oxygen required to burn permeated methanol. The term ln(1 + b) in (11) represents voltage loss due to the lowering of methanol concentration in the anode catalyst layer induced by crossover. ~ and ~n in (11) and (12) In 1D model the values ~j, w represent the mean current density in the cell and the average molar fractions of methanol and oxygen in the respective channel. Within the scope of 1D + 1D ~ approach (11) and (12) contain local values ~j, w a c ~ and n. Therefore, generally g and g are functions of z. DMFC electrodes are equipotential; the sum of polarization voltages ga(z) + gc(z) = E is thus constant along ~z: ~j ~j þ lnð1 þ bÞ ln ln 1 ~ ~ w qw " !# ~ ~jÞ ~j ~j þ b1 ðw ~ þ p ln ¼ E: ð14Þ ln 1 c~n aq~n Here p¼
bc ba
ð15Þ
~ ¼ E=ba is the total is the ratio of Tafel slopes and E dimensionless voltage loss. Eqs. (7), (8) and (14) form a system of 3 equations for ~ ~n and ~j. This is a 1D + 1D model of 3 unknowns: w, DMFC: the local rates of methanol and oxygen consumption in the respective channel (the right sides of (7) and (8)) are determined by equipotentiality
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condition (14), which takes into account local transport of reactants across the cell.
3. Solutions 3.1. Numerical solution
2.3. Equation for local current density For further analysis we will convert (14) into differential equation. Differentiating (14) with respect to ~z and using (7) and (8) we come to h o~j ~ ckc ~ ~ ¼ ~jð~j þ bwÞ n þ ðpcka b1 ð1 þ pÞkc Þw o~z i 1 þ ðpb 1Þð1 þ bÞ kc pcka ~j ka kc J~ ~ ~j þ ð1 þ pÞ pcð1 þ bÞ~ n þ ð1 pbÞw i ~ : ~ ~ bw cð1 þ bÞn w
h
ð16Þ
Initial condition for this equation is ~jð0Þ ¼ j. Equa~ ¼ ~n ¼ 1 tion for j is obtained if we put ~j ¼ j, and w in (14) j ln ln ð1 jÞ þ lnð1 þ bÞ q j j þ b1 ð1 jÞ ~ þ p ln ln 1 ¼ E: ð17Þ aq c Note, that (16) does not contain a and q. These parameters thus enter into the solution only through the initial condition (17). It means that a and q simply re-scale the curve ~jð~zÞ as a whole, not affecting its shape. ~ in the Mean current density J~ and voltage loss E system (7), (8), (16) and (17) determine the point on polarization curve. Parameter j is excluded using the following condition. Total current in the system is J~; solution to (16) must, therefore, obey the relation Z ~zmin ~j d~z ¼ J~; ð18Þ
Numerical solution to the full system of Eqs. (7), (8) and (16) can be obtained by the standard Runge–Kutta method. Fig. 1 shows dimensionless polarization curves ~ as a function of mean of the cell (the total voltage loss E ~ current density J ) for 3 values of parameter b. In all ~ forms the plateau at small currents (Fig. 1). This cases E plateau indicates formation of a ‘‘bridge’’ (see below). The length of the plateau increases with b (Fig. 1). The profiles ~jð~zÞ at several points of the polarization curve for b = 1 (Fig. 1) are depicted in Fig. 2. At small J~ current is localized in a finite domain of the cell (Fig. 2). As J~ decreases, the point ~z0 , where ~j vanishes moves to zero, whereas ~jð0Þ ¼ j remains constant (Fig. 2). We see that near the inlet forms a bridge, a narrow region, which supports finite local current density at infinitely small current J~ in the load. The bridge is formed due to crossover. Qualitatively, at small J~ the equivalent current density of crossover ~jcross (10) is maximal. The domain of current generation then is localized close to the oxygen channel inlet, where oxygen concentration is maximal. In the rest part of the cell all available oxygen is consumed in the reaction with permeated methanol. ~ (14) depends on the local current Since voltage loss E density rather than on the total current, formation of the bridge immediately decreases cell voltage. The bridge,
0
where ~zmin ¼ minf~z0 ; 1g (see below). Iterations are re~ which provides fulfillment of quired to determine E, (18). Cell voltage is then calculated from V cell ¼ V oc E Rn j;
ð19Þ
where Rn is a sum of all contact resistances. For further analysis we will need the asymptotic form of (16) for small ~j. Expanding the right side of (16) over ~j and retaining only linear term we get h i c~ a c ~ ~ o~j b ck n þ ðcpk b1 ð1 þ pÞk Þw j h i : ¼ o~z ~ cð1 þ bÞ~ n ka kc J~ð1 þ pÞ bw
ð20Þ
Fig. 1. Voltage–current curve of DMFC in dimensionless coordinates. ~ ¼ ðga þ gc Þ=ba vs mean current density Shown is total voltage loss E J~ ¼ J =ja0 for indicated values of crossover parameter lim ~ axis b ¼ Dm lab =ðDab lm Þ (small b means small crossover). Note, that E is directed downward. Methanol and oxygen stoichiometries are ka = 4 and kc = 2, respectively. The other parameters a = 20.04, c = 2.803, q = 1.727 · 105 represent typical operating conditions of DMFC [5]. The profiles of local current density in the black points are shown in Fig. 2. Diamonds: the points, where ~z0 ¼ 1 (see Fig. 2 and the text).
A.A. Kulikovsky / Electrochemistry Communications 6 (2004) 1259–1265
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Eq. (23) is valid provided that ~j b1 . Physically, current should be much less, than the maximal crossover current; oxygen consumption is then determined by crossover. Solution to (23) gives linear decrease of oxygen fraction with the distance ~n ¼ 1 ~z ; ~zox
ð24Þ
where ~zox ¼
Fig. 2. The profiles of local current density ~j ¼ j=ja0 lim along the channel for indicated values of dimensionless mean current density J~ in a cell. L is the channel length. Crossover parameter b = 1, the other parameters are the same as in Fig. 1. ~z0 is the point, where local current density vanishes. The respective polarization curve is shown in Fig. 1 (the curve for b = 1). Note that for J~ 6 0:3 the local current density j at the channel inlet does not change.
therefore, manifests itself as a decrease in cell ‘‘open-circuit’’ voltage. Analytical solution for the case of ka = kc gives the bridge with exponential decrease of current along ~z [5]. In the case of ka > kc shown in Figs. 1 and 2 the bridge has a finite spread over the cell surface. To rationalize the effect of a localized current on cell performance, in the next section we derive asymptotic solution to a system (7), (8) and (16) in the limiting case of large methanol stoichiometry.
kc J~ b1
ð25Þ
is the point where ~n vanishes. Physically, at ~zox all available oxygen is consumed in the reaction with permeated methanol. Substituting (24) into (22) and solving the resulting equation we find p 1þp ~j ¼ j 1 ~z ; ð26Þ ~z0 where
1 1 ~z0 ¼ k J b1 c c~
ð27Þ
is the point, where ~j vanishes. Local current density (26) for several values of J~ is shown in Fig. 3 along with the solutions of the full system of Eqs. (7), (8) and (16). If ~z0 < 1 current is localized in the domain 0 6 z 6 ~z0 (Fig. 3). CurrentR density at the inlet j is obtained from the ~z condition 0 0 ~j d~z ¼ J~. Integrating (26) we find
3.2. Large methanol stoichiometry, small current Large methanol stoichiometry (ka kc) means that methanol fraction is almost constant along ~z, that is ~ ’ 1. Putting in (20) w ~ ¼ 1 we find w h i c~ a c ~ o~j b ck n þ cpk b1 ð1 þ pÞk j h i : ¼ ð21Þ o~z n ka kc J~ð1 þ pÞ b cð1 þ bÞ~ Estimate shows that for typical conditions c . 1, p . 1. Since ~ n 6 1 and ka kc, we can omit terms with c k in the numerator of the expression on the right side of (21). This leads to o~j bcp~j h i; ¼ c o~z k J~ð1 þ pÞ b cð1 þ bÞ~ n
~jð0Þ ¼ j:
ð22Þ
Equation for ~ n in this limit is obtained from (8) if we ~ ¼ 1 there and neglect ~j on the right side: put w kc
o~ n b ¼ 1; o~z J~
~ nð0Þ ¼ 1:
ð23Þ
Fig. 3. The profiles of local current density along the channel for indicated values of the total current J~ in the cell. Solid lines – asymptotic solution (26), dashed lines – numerical solution of the full system of Eqs. (7), (8) and (16). Parameters: b = 1, ka = 100, kc = 8, p = 1, c = 2. The other parameters for numerical solution: a = 20.04, q = 1.727 · 105.
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j¼
A.A. Kulikovsky / Electrochemistry Communications 6 (2004) 1259–1265
cb1 ð1 þ 2pÞ : kc ðc b1 Þð1 þ pÞ
ð28Þ
Thus, current density at the inlet j does not depend on J~, whereas ~z0 (27) is proportional to J~ (Fig. 3). At small J~ this leads to formation of the bridge. Indeed, meanR current density in the current – carrying ~z domain ~jb ¼ ~z10 0 0 ~j d~z. Using here (26) and calculating integral we find 1 ~jb ¼ ð1 þ pÞj ¼ kc 1 1 : ð29Þ 1 þ 2p b1 c We see that ~jb does not depend on J~. As J~ ! 0, the current – carrying domain transforms into a narrow bridge, which supports constant local current density ~jb . Fig. 3 helps to clarify the physics of bridge formation. Oxygen concentration vanishes at ~zox > ~z0 . In the region ~z0 < z 6 ~zox all available oxygen is consumed in the reaction with permeated methanol. Current appears to be localized in a finite domain ½0; ~z0 , since to the right of ~z0 there is no oxygen to generate current. Note that the length D~z ¼ ~zox ~z0 ¼ kc J~=c increases with the current in the cell J~. Suppose that we are moving along the polarization curve from large to small currents J~ (Fig. 1). At large J~ we have ~z0 > 1, that is local current ‘‘covers’’ the whole surface of the cell. When ~z0 becomes equal to 1, the mean current density ~jb in the current – carrying domain ½0; ~z0 ceases to change. Further decrease in J~ occurs due to shrinking of this domain i.e., due to the shift of ~z0 to zero (Fig. 3). The value of J~, at which ~z0 ¼ 1 is clearly seen on the polarization curves: it is the point, where the curve goes into plateau (diamonds in Fig. 1). As in the case of ka = kc [5], the bridge supports finite current density ~jb regardless of the value of J~. The bridge, therefore, ‘‘short-circuits’’ DMFC electrodes as J~ ! 0. The respective loss in cell open-circuit voltage ~ is given by (17) with j (28). Indeed, since the sum of E polarization voltages is constant along ~z, we may calcu~ ¼~ ~ at ~z ¼ 0, where w late E n ¼ 1 and ~j ¼ j. Note, that ~ on kc (17) with j (28) gives also the dependencies of E and b. The shapes of the bridge for different kc are depicted in Fig. 4. The increase in kc decreases j and ‘‘smears out’’ the bridge over the larger domain. In view of (17) lower j means higher open-circuit voltage.
Fig. 4. The profiles of local current density along the channel at small total current in the cell (the shape of the bridge) for indicated values of oxygen stoichiometry kc. Parameters: b = 1, J~ ¼ 0:1, ka ! 1; the other parameters are the same as in Fig. 1.
~ in this limit is obtained from (7) if we Equation for w ~ neglect j on the right side of this equation: ka
~ ~ ow bw ¼ 1 : o~z J~
ð31Þ
Quite analogous to (23), this equation is valid provided that ~j b1 . Physically, small ~j means that methanol consumption is determined mainly by crossover. Solution to (31) is ~ ¼ exp ~z ; w ð32Þ ~zM where the characteristic scale of methanol consumption is ~zM ¼
ka J~ : b1
ð33Þ
Substituting (32) into (30) and solving the resulting equation we find ~j ¼ j exp ~z ; ð34Þ ~zj where ~zj ¼
~zM ka J~ ¼ 1 þ p b1 ð1 þ pÞ
ð35Þ
and 3.3. Large oxygen stoichiometry, small current ~ ¼ 1 and neglect the In that case in (20) we can put n term with ka in the numerator. This yields h i ~ ~j b c b1 ð1 þ pÞw o~j h i: ¼ ð30Þ o~z ka J~ð1 þ pÞ bw ~ cð1 þ bÞ
1 1 : j ¼ b1 ka ð1 þ pÞ 1 exp ~zj
ð36Þ
We see that local current decreases exponentially along ~z with the characteristic scale (35), which is (1 + p) times smaller, than the scale of methanol variation (33). The ‘‘exponential’’ bridge is quite analogous to those obtained in [5] for the case of ka = kc.
A.A. Kulikovsky / Electrochemistry Communications 6 (2004) 1259–1265
Pre-exponential factor j in (34) is local current density at the inlet. As before, this parameter determines mean current density of the bridge ~jb Z 1 ~z b ~jb ¼ 1 : ð37Þ j exp d~z ¼ a 1 ~zj 0 ~zj k ð1 þ pÞ Comparing this to (36) we see that ~jb ¼ j0 ¼ lim j. J~!0
4. Discussion At small current J~ in the load resistance, current density in the bridge does not depend on J~. In the case of ka = kc it was shown in [5]; in the limiting cases of ka kc and ka kc this is shown above. Furthermore, comparing (29) and (37) we conclude that in both limiting cases of ka kc and ka kc current density in the bridge ~jb increases with b and decreases with the growth of stoichiometry of the ‘‘rate-determining’’ flow. The same behavior exhibits ~jb in the case of ka = kc [5]. We, therefore, may expect that the bridge retains all these properties for arbitrary ka and kc. What physically means small total current J~? All currents are normalized to the methanol-limiting current density at the inlet (6). If methanol concentration is large enough, total current in the cell is limited by oxygen. Small J~ then in dimension units may be quite large, on the order of the oxygen-limiting current density. In that case the solution, shown in Fig. 3 gives a picture of DMFC operation in a quite large range of operating current densities. In our laboratory we get evidences that part of DMFC may turn into electrolysis regime [10]. Analysis of the case of ka kc suggests that the mixed regime of cell operation is realized if ~z0 < 1 (Fig. 3). To the right of ~z0 , on the cathode side there is not enough oxygen for complete oxidation of permeated methanol and methanol may be split into protons and electrons. The protons then move to the ‘‘anode’’ side, where they recombine with electrons to yield hydrogen. Therefore, to the left of ~z0 the cell generates power, whereas to the right of this point it may consume power to produce hydrogen. Condition ~z0 P 1 thus provides the regime of power generation over the whole surface area of the cell. With (27) this condition reads 1 1 1 P c þ ; b1 k J~ c
ð38Þ
Therefore, small b1 (low crossover) in all cases helps to maintain the cell in power generation regime. If crossover is large, it is beneficial to increase the product kc J~ or/and c in order to keep the cell in power generation regime. Parameter c is proportional to oxygen molar con-
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centration at the inlet ccn0 (13). Thus, under large crossover one has to increase oxygen concentration or/ and stoichiometry of oxygen flow.
5. Conclusions Our 1D + 1D model of a direct methanol fuel cell [5] is further analyzed. The model couples transport of oxygen and methanol across the cell (including methanol crossover) to the transport of these reactants in the respective channel. Numerical solution to model equations for arbitrary methanol and oxygen stoichiometries exhibits the effect of ‘‘bridge’’ formation, described in [5] for the case of ka = kc. A narrow region close to the channel inlet (a bridge) supports finite local current density at vanishingly small total current in the cell. The bridge ‘‘short-circuits’’ DMFC electrodes, thus reducing cell open-circuit voltage. In case of ka > kc the system has a new type of solutions: current density appears to be localized in a finite domain of the cell. Asymptotic solutions give the properties of the bridge in the cases of ka kc and ka kc. In the case of ka kc the bridge has exponential shape, quite analogous to those described in [5]. Taking into account analytical result for ka = kc [5] and asymptotic solution for ka kc we may expect, that for any ka 6 kc local current ‘‘covers’’ the whole surface of the cell. If ka > kc and total current is small, local current flows only in the bridge. In all cases the bridge is a domain of non-vanishing local current density as J~ ! 0. In both cases of ka kc and ka kc local current density in the bridge increases with the rate of crossover and decreases with the growth of stoichiometry of the rate-determining flow. The formula for voltage loss at open circuit is derived.
References [1] S. Srinivasan, A.S. Arico`, V. Antonucci, Fuel Cells 1 (2001) 133. [2] W. Vielstich, V.A. Paganin, F.H.B. Lima, E.A. Ticianelli, J. Electrochem. Soc. 148 (2001) A502. [3] A. Ku¨ver, W. Vielstich, J. Power Sources 74 (1998) 211. [4] S.C. Thomas, X. Ren, S. Gottesfeld, P. Zelenay, Electrochim. Acta 47 (2002) 3741. [5] A.A. Kulikovsky, On the nature of mixed potential in a DMFC, J. Electrochem. Soc. (2004, submitted). [6] A.A. Kulikovsky, Electrochem. Commun. 4 (2002) 939. [7] R. Jiang, D. Chu, J. Electrochem. Soc. 151 (2004) A69. [8] X. Ren, P. Zelenay, S. Thomas, J. Davey, S. Gottesfeld, J. Power Sources 86 (2000) 111. [9] V. Gogel, T. Frey, Z. Yongsheng, K.A. Friedrich, L. Jo¨rissen, J. Garche, J. Power Sources 127 (2004) 172. [10] M.Mu¨ller, H. Dohle, Private communication, 2003.
1D + 1D model of a DMFC: localized solutions and mixedpotential A.A. Kulikovsky
*
Institute for Materials and Processes in Energy Systems (IWV-3), Research Center ‘‘Ju¨lich’’, D-52428 Ju¨lich, Germany Received 9 September 2004; received in revised form 28 September 2004; accepted 4 October 2004 Available online 19 October 2004
Abstract Our 1D + 1D model of DMFC reveals a new effect. At infinitely small total current in the cell, near the channel inlet forms a ‘‘bridge’’, a narrow region with finite local current density. The bridge short-circuits the electrodes, thus reducing cell open-circuit voltage. In our previous work the effect is described for the case of equal methanol ka and oxygen kc stoichiometries. In this Letter, we analyze the general case of arbitrary ka and kc. In the case of ka > kc current may occupy finite domain of the cell surface. Asymptotic solution for the case of ka kc shows, that the size of this domain is proportional to oxygen stoichiometry. In the opposite limit of ka kc local current exponentially decreases with the distance along the channel. Asymptotic solutions suggest that the bridge forms regardless of the relationship between ka and kc. In all cases local current density in the bridge increases with the rate of methanol crossover and decreases with the growth of the ‘‘rate-determining’’ stoichiometry. The expression for voltage loss at open-circuit is derived. 2004 Elsevier B.V. All rights reserved. Keywords: DMFC; Analytical model; Mixedpotential; Open-circuit voltage
1. Introduction Methanol crossover through the membrane dramatically decreases performance of a direct methanol fuel cell (DMFC). During the past decade a huge work has been done worldwide to reduce crossover [1]. Though the negative effect of crossover is well recognized, the physics of lowering of cell voltage due to crossover is still poorly understood. Crossover dramatically reduces DMFC voltage at small currents. This effect, called mixedpotential is well known in DMFC studies. Several mechanisms can contribute to this reduction. Methanol, permeated through the membrane is oxidized on the cathode side. ‘‘Burning’’ of permeated methanol requires additional flux of oxygen through the cathode gas diffusion (backing)
*
On leave from Moscow State University, Research Computing Center, 119992 Moscow, Russia. Fax: +49 2461 61 6695. E-mail address: [email protected]. 1388-2481/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.elecom.2004.10.002
layer, which is equivalent to additional resistance [2]. One may speculate about poisoning of the cathode catalyst layer by the intermediates of methanol oxidation [1]. Another candidate to explain mixedpotential is flooding of the cathode side by water produced in the methanol–oxygen reaction [3,4]. However, due to the lack of kinetic and transport data it is difficult to estimate the respective voltage loss and the nature of mixedpotential is still controversial. Our 1D + 1D model of DMFC suggests the new explanation of mixedpotential. Analytical solution to model equations reveals the new effect: formation of a narrow ‘‘bridge’’ of local current near the inlet of the feed channels [5]. Local current density in the bridge remains finite at vanishingly small total current in the cell. The bridge ‘‘short-circuits’’ the electrodes, thus reducing cell open-circuit voltage. Analysis [5] is performed for the case of equal methanol and oxygen stoichiometries ka and kc. In that case local current density exponentially decreases with the distance along the channel.
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A.A. Kulikovsky / Electrochemistry Communications 6 (2004) 1259–1265
Nomenclature b c cref Db E F i* J J~ j jcross jlim h lb lm lt L Ncross p q Rn
Tafel slope (V) total molar concentration in the channel (mol cm3) reference molar concentration (mol cm3) diffusion coefficient of feed molecules in the backing layer (cm2 s1) total voltage loss (V) Faraday constant exchange current density per unit volume (A cm3) mean current density in a cell (A cm2) dimensionless mean current density and total current in a cell local current density (A cm2) equivalent crossover current density (A cm2) limiting current density (A cm2) channel height (cm) thickness of the backing layer (cm) thickness of the membrane (cm) thickness of the catalyst layer (cm) channel length (cm) molar flux of methanol through the membrane (mol cm2 s1) ratio of Tafel slopes (p = bc/ba) dimensionless parameter contact resistance (X cm2)
In this Letter, we analyze the case of arbitrary ka and kc. In the case of ka > kc the problem has a new type of solutions: due to the lack of oxygen current density appears to be localized in a finite domain. The bridge then has finite spread over the cell surface. Asymptotic solutions to a problem suggest that the bridge forms regardless of the relationship between ka and kc. In all cases local current density in the bridge appears to be proportional to the rate of methanol crossover and inversely proportional to the ratedetermining stoichiometry.
2. Model 2.1. Continuity equations In the next two sections we briefly repeat the derivation of the governing equations [5]. Molar fractions of methanol w and oxygen n in the respective channel are determined by continuity equations: ha v a c a
ow j ¼ N cross ; oz 6F
ð1Þ
cell voltage (V) cell open circuit voltage (V) flow velocity in the channel (cm s1) coordinate along the channel (cm)
Vcell Voc v z
Superscripts 0 at the inlet (at z = 0) a anode side c cathode side Subscripts 0 at zero total current b backing layer lim limiting t catalyst layer Greek symbols b dimensionless parameter of crossover b1 b1 = b/(1 + b) c dimensionless parameter g polarization voltage (V) j dimensionless local current density at the inlet k stoichiometry of the feed flow n oxygen molar fraction in the channel w methanol molar fraction in the channel
on j 3 ¼ N cross : ð2Þ oz 4F 2 The superscripts ‘‘a’’ and ‘‘c’’ refer to the anode and the cathode side, respectively, z is the distance along the channel, h is the channel height, v is the flow velocity, c is the total molar concentration of the mixture in the channel, j(z) is the local current density. Eqs. (1) and (2) are written assuming plug flow conditions in both channels. The flux of methanol through the membrane Ncross is given by [6] ja b j N cross ¼ lim 1 a ; ð3Þ jlim 6F 1 þ b
h c vc c c
where jalim ¼ 6FDab ca w=lab is local limiting current density due to methanol transport through the backing layer and b is a ratio of methanol mass transfer coefficients in the membrane and in the anode backing layer b¼
Dm lab : Dab lm
ð4Þ
Here Dm and Dab are diffusion coefficients of methanol in the membrane and in the anode backing layer, respectively, lm and lab are the thicknesses of the membrane
A.A. Kulikovsky / Electrochemistry Communications 6 (2004) 1259–1265
and of the backing layer, c is the total molar concentration of the mixture in the channel. The dominating mechanism of crossover is diffusion due to the concentration gradient (see [6] for detailed discussion). The flux Ncross (3) linearly decreases with the growth of current density j, which agrees with the experiments [7–9]. To analyze the governing Eqs. (1) and (2) we introduce dimensionless variables ~j ¼ j ; ja0 lim
z ~z ¼ ; L
~¼ w; w w0
n ~ n¼ 0; n
ð5Þ
where L is channel length, the superscript ‘‘0’’ indicates the values at the channel inlet (at z = 0) and ja0 lim
Da ca w0 ¼ 6F b a lb
ð6Þ
is methanol-limiting current density at the channel inlet. Introducing stoichiometries ka and kc of methanol and oxygen flow, respectively ka ¼
6Fha va ca w0 ; LJ
kc ¼
4Fhc vc cc n0 ; LJ
Eqs. (1) and (2) transform to ka
~ ~ ~jÞ ~j þ b1 ðw ow ¼ ; ~ o~z J
ð7Þ
kc
~ ~jÞ ~j þ b1 ðw o~ n ¼ : ~ o~z J
ð8Þ
Here J~ ¼ J =ja0 lim is dimensionless mean current density in a cell and b1 ¼
b : 1þb
ð9Þ
The mean current density J and the total current in the cell I differ only by a constant factor, the area of cell surface. In dimensionless variables these values coincide; for that reason we will treat J~ also as the total current in the cell. Eqs. (7) and (8) show that the rates of methanol and oxygen consumption are proportional to the sum ~j þ ~jcross , where equivalent crossover current ~jcross is ~ ~ ~jcross N cross =ð6Fja0 lim Þ ¼ b1 ðw jÞ:
ð10Þ
2.2. Polarization voltages The expressions for polarization voltages ga, gc of the anode and the cathode side, respectively, are derived in [6] within the scope of 1D model of DMFC. With the dimensionless variables (5) these expressions take a form
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~j ~j ga þ lnð1 þ bÞ; ¼ ln ln 1 ~ ~ ba w qw ! ~ ~jÞ ~j ~j þ b1 ðw gc ¼ ln ; ln 1 bc c~n aq~n
ð11Þ
ð12Þ
where la ia la lc i c c a q ¼ t a ba ; a ¼ at a ref 6FDb cref lt i ccref 2Dcb lab cc n0 c¼ 3Dab lcb ca w0
c c n0 ; ca w 0
ð13Þ
are dimensionless parameters. Here lt is the thickness of the catalyst layer, i* is the exchange current density per unit volume and cref is the reference molar concentration of the feed molecules. Physically, the first terms on the right sides of (11) and (12) describe reaction activation and concentration overpotentials. The second terms there describe voltage loss due to the transport of methanol and oxygen, respectively, through the backing layer. The second logarithm in (12) takes also into account transport of oxygen required to burn permeated methanol. The term ln(1 + b) in (11) represents voltage loss due to the lowering of methanol concentration in the anode catalyst layer induced by crossover. ~ and ~n in (11) and (12) In 1D model the values ~j, w represent the mean current density in the cell and the average molar fractions of methanol and oxygen in the respective channel. Within the scope of 1D + 1D ~ approach (11) and (12) contain local values ~j, w a c ~ and n. Therefore, generally g and g are functions of z. DMFC electrodes are equipotential; the sum of polarization voltages ga(z) + gc(z) = E is thus constant along ~z: ~j ~j þ lnð1 þ bÞ ln ln 1 ~ ~ w qw " !# ~ ~jÞ ~j ~j þ b1 ðw ~ þ p ln ¼ E: ð14Þ ln 1 c~n aq~n Here p¼
bc ba
ð15Þ
~ ¼ E=ba is the total is the ratio of Tafel slopes and E dimensionless voltage loss. Eqs. (7), (8) and (14) form a system of 3 equations for ~ ~n and ~j. This is a 1D + 1D model of 3 unknowns: w, DMFC: the local rates of methanol and oxygen consumption in the respective channel (the right sides of (7) and (8)) are determined by equipotentiality
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A.A. Kulikovsky / Electrochemistry Communications 6 (2004) 1259–1265
condition (14), which takes into account local transport of reactants across the cell.
3. Solutions 3.1. Numerical solution
2.3. Equation for local current density For further analysis we will convert (14) into differential equation. Differentiating (14) with respect to ~z and using (7) and (8) we come to h o~j ~ ckc ~ ~ ¼ ~jð~j þ bwÞ n þ ðpcka b1 ð1 þ pÞkc Þw o~z i 1 þ ðpb 1Þð1 þ bÞ kc pcka ~j ka kc J~ ~ ~j þ ð1 þ pÞ pcð1 þ bÞ~ n þ ð1 pbÞw i ~ : ~ ~ bw cð1 þ bÞn w
h
ð16Þ
Initial condition for this equation is ~jð0Þ ¼ j. Equa~ ¼ ~n ¼ 1 tion for j is obtained if we put ~j ¼ j, and w in (14) j ln ln ð1 jÞ þ lnð1 þ bÞ q j j þ b1 ð1 jÞ ~ þ p ln ln 1 ¼ E: ð17Þ aq c Note, that (16) does not contain a and q. These parameters thus enter into the solution only through the initial condition (17). It means that a and q simply re-scale the curve ~jð~zÞ as a whole, not affecting its shape. ~ in the Mean current density J~ and voltage loss E system (7), (8), (16) and (17) determine the point on polarization curve. Parameter j is excluded using the following condition. Total current in the system is J~; solution to (16) must, therefore, obey the relation Z ~zmin ~j d~z ¼ J~; ð18Þ
Numerical solution to the full system of Eqs. (7), (8) and (16) can be obtained by the standard Runge–Kutta method. Fig. 1 shows dimensionless polarization curves ~ as a function of mean of the cell (the total voltage loss E ~ current density J ) for 3 values of parameter b. In all ~ forms the plateau at small currents (Fig. 1). This cases E plateau indicates formation of a ‘‘bridge’’ (see below). The length of the plateau increases with b (Fig. 1). The profiles ~jð~zÞ at several points of the polarization curve for b = 1 (Fig. 1) are depicted in Fig. 2. At small J~ current is localized in a finite domain of the cell (Fig. 2). As J~ decreases, the point ~z0 , where ~j vanishes moves to zero, whereas ~jð0Þ ¼ j remains constant (Fig. 2). We see that near the inlet forms a bridge, a narrow region, which supports finite local current density at infinitely small current J~ in the load. The bridge is formed due to crossover. Qualitatively, at small J~ the equivalent current density of crossover ~jcross (10) is maximal. The domain of current generation then is localized close to the oxygen channel inlet, where oxygen concentration is maximal. In the rest part of the cell all available oxygen is consumed in the reaction with permeated methanol. ~ (14) depends on the local current Since voltage loss E density rather than on the total current, formation of the bridge immediately decreases cell voltage. The bridge,
0
where ~zmin ¼ minf~z0 ; 1g (see below). Iterations are re~ which provides fulfillment of quired to determine E, (18). Cell voltage is then calculated from V cell ¼ V oc E Rn j;
ð19Þ
where Rn is a sum of all contact resistances. For further analysis we will need the asymptotic form of (16) for small ~j. Expanding the right side of (16) over ~j and retaining only linear term we get h i c~ a c ~ ~ o~j b ck n þ ðcpk b1 ð1 þ pÞk Þw j h i : ¼ o~z ~ cð1 þ bÞ~ n ka kc J~ð1 þ pÞ bw
ð20Þ
Fig. 1. Voltage–current curve of DMFC in dimensionless coordinates. ~ ¼ ðga þ gc Þ=ba vs mean current density Shown is total voltage loss E J~ ¼ J =ja0 for indicated values of crossover parameter lim ~ axis b ¼ Dm lab =ðDab lm Þ (small b means small crossover). Note, that E is directed downward. Methanol and oxygen stoichiometries are ka = 4 and kc = 2, respectively. The other parameters a = 20.04, c = 2.803, q = 1.727 · 105 represent typical operating conditions of DMFC [5]. The profiles of local current density in the black points are shown in Fig. 2. Diamonds: the points, where ~z0 ¼ 1 (see Fig. 2 and the text).
A.A. Kulikovsky / Electrochemistry Communications 6 (2004) 1259–1265
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Eq. (23) is valid provided that ~j b1 . Physically, current should be much less, than the maximal crossover current; oxygen consumption is then determined by crossover. Solution to (23) gives linear decrease of oxygen fraction with the distance ~n ¼ 1 ~z ; ~zox
ð24Þ
where ~zox ¼
Fig. 2. The profiles of local current density ~j ¼ j=ja0 lim along the channel for indicated values of dimensionless mean current density J~ in a cell. L is the channel length. Crossover parameter b = 1, the other parameters are the same as in Fig. 1. ~z0 is the point, where local current density vanishes. The respective polarization curve is shown in Fig. 1 (the curve for b = 1). Note that for J~ 6 0:3 the local current density j at the channel inlet does not change.
therefore, manifests itself as a decrease in cell ‘‘open-circuit’’ voltage. Analytical solution for the case of ka = kc gives the bridge with exponential decrease of current along ~z [5]. In the case of ka > kc shown in Figs. 1 and 2 the bridge has a finite spread over the cell surface. To rationalize the effect of a localized current on cell performance, in the next section we derive asymptotic solution to a system (7), (8) and (16) in the limiting case of large methanol stoichiometry.
kc J~ b1
ð25Þ
is the point where ~n vanishes. Physically, at ~zox all available oxygen is consumed in the reaction with permeated methanol. Substituting (24) into (22) and solving the resulting equation we find p 1þp ~j ¼ j 1 ~z ; ð26Þ ~z0 where
1 1 ~z0 ¼ k J b1 c c~
ð27Þ
is the point, where ~j vanishes. Local current density (26) for several values of J~ is shown in Fig. 3 along with the solutions of the full system of Eqs. (7), (8) and (16). If ~z0 < 1 current is localized in the domain 0 6 z 6 ~z0 (Fig. 3). CurrentR density at the inlet j is obtained from the ~z condition 0 0 ~j d~z ¼ J~. Integrating (26) we find
3.2. Large methanol stoichiometry, small current Large methanol stoichiometry (ka kc) means that methanol fraction is almost constant along ~z, that is ~ ’ 1. Putting in (20) w ~ ¼ 1 we find w h i c~ a c ~ o~j b ck n þ cpk b1 ð1 þ pÞk j h i : ¼ ð21Þ o~z n ka kc J~ð1 þ pÞ b cð1 þ bÞ~ Estimate shows that for typical conditions c . 1, p . 1. Since ~ n 6 1 and ka kc, we can omit terms with c k in the numerator of the expression on the right side of (21). This leads to o~j bcp~j h i; ¼ c o~z k J~ð1 þ pÞ b cð1 þ bÞ~ n
~jð0Þ ¼ j:
ð22Þ
Equation for ~ n in this limit is obtained from (8) if we ~ ¼ 1 there and neglect ~j on the right side: put w kc
o~ n b ¼ 1; o~z J~
~ nð0Þ ¼ 1:
ð23Þ
Fig. 3. The profiles of local current density along the channel for indicated values of the total current J~ in the cell. Solid lines – asymptotic solution (26), dashed lines – numerical solution of the full system of Eqs. (7), (8) and (16). Parameters: b = 1, ka = 100, kc = 8, p = 1, c = 2. The other parameters for numerical solution: a = 20.04, q = 1.727 · 105.
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j¼
A.A. Kulikovsky / Electrochemistry Communications 6 (2004) 1259–1265
cb1 ð1 þ 2pÞ : kc ðc b1 Þð1 þ pÞ
ð28Þ
Thus, current density at the inlet j does not depend on J~, whereas ~z0 (27) is proportional to J~ (Fig. 3). At small J~ this leads to formation of the bridge. Indeed, meanR current density in the current – carrying ~z domain ~jb ¼ ~z10 0 0 ~j d~z. Using here (26) and calculating integral we find 1 ~jb ¼ ð1 þ pÞj ¼ kc 1 1 : ð29Þ 1 þ 2p b1 c We see that ~jb does not depend on J~. As J~ ! 0, the current – carrying domain transforms into a narrow bridge, which supports constant local current density ~jb . Fig. 3 helps to clarify the physics of bridge formation. Oxygen concentration vanishes at ~zox > ~z0 . In the region ~z0 < z 6 ~zox all available oxygen is consumed in the reaction with permeated methanol. Current appears to be localized in a finite domain ½0; ~z0 , since to the right of ~z0 there is no oxygen to generate current. Note that the length D~z ¼ ~zox ~z0 ¼ kc J~=c increases with the current in the cell J~. Suppose that we are moving along the polarization curve from large to small currents J~ (Fig. 1). At large J~ we have ~z0 > 1, that is local current ‘‘covers’’ the whole surface of the cell. When ~z0 becomes equal to 1, the mean current density ~jb in the current – carrying domain ½0; ~z0 ceases to change. Further decrease in J~ occurs due to shrinking of this domain i.e., due to the shift of ~z0 to zero (Fig. 3). The value of J~, at which ~z0 ¼ 1 is clearly seen on the polarization curves: it is the point, where the curve goes into plateau (diamonds in Fig. 1). As in the case of ka = kc [5], the bridge supports finite current density ~jb regardless of the value of J~. The bridge, therefore, ‘‘short-circuits’’ DMFC electrodes as J~ ! 0. The respective loss in cell open-circuit voltage ~ is given by (17) with j (28). Indeed, since the sum of E polarization voltages is constant along ~z, we may calcu~ ¼~ ~ at ~z ¼ 0, where w late E n ¼ 1 and ~j ¼ j. Note, that ~ on kc (17) with j (28) gives also the dependencies of E and b. The shapes of the bridge for different kc are depicted in Fig. 4. The increase in kc decreases j and ‘‘smears out’’ the bridge over the larger domain. In view of (17) lower j means higher open-circuit voltage.
Fig. 4. The profiles of local current density along the channel at small total current in the cell (the shape of the bridge) for indicated values of oxygen stoichiometry kc. Parameters: b = 1, J~ ¼ 0:1, ka ! 1; the other parameters are the same as in Fig. 1.
~ in this limit is obtained from (7) if we Equation for w ~ neglect j on the right side of this equation: ka
~ ~ ow bw ¼ 1 : o~z J~
ð31Þ
Quite analogous to (23), this equation is valid provided that ~j b1 . Physically, small ~j means that methanol consumption is determined mainly by crossover. Solution to (31) is ~ ¼ exp ~z ; w ð32Þ ~zM where the characteristic scale of methanol consumption is ~zM ¼
ka J~ : b1
ð33Þ
Substituting (32) into (30) and solving the resulting equation we find ~j ¼ j exp ~z ; ð34Þ ~zj where ~zj ¼
~zM ka J~ ¼ 1 þ p b1 ð1 þ pÞ
ð35Þ
and 3.3. Large oxygen stoichiometry, small current ~ ¼ 1 and neglect the In that case in (20) we can put n term with ka in the numerator. This yields h i ~ ~j b c b1 ð1 þ pÞw o~j h i: ¼ ð30Þ o~z ka J~ð1 þ pÞ bw ~ cð1 þ bÞ
1 1 : j ¼ b1 ka ð1 þ pÞ 1 exp ~zj
ð36Þ
We see that local current decreases exponentially along ~z with the characteristic scale (35), which is (1 + p) times smaller, than the scale of methanol variation (33). The ‘‘exponential’’ bridge is quite analogous to those obtained in [5] for the case of ka = kc.
A.A. Kulikovsky / Electrochemistry Communications 6 (2004) 1259–1265
Pre-exponential factor j in (34) is local current density at the inlet. As before, this parameter determines mean current density of the bridge ~jb Z 1 ~z b ~jb ¼ 1 : ð37Þ j exp d~z ¼ a 1 ~zj 0 ~zj k ð1 þ pÞ Comparing this to (36) we see that ~jb ¼ j0 ¼ lim j. J~!0
4. Discussion At small current J~ in the load resistance, current density in the bridge does not depend on J~. In the case of ka = kc it was shown in [5]; in the limiting cases of ka kc and ka kc this is shown above. Furthermore, comparing (29) and (37) we conclude that in both limiting cases of ka kc and ka kc current density in the bridge ~jb increases with b and decreases with the growth of stoichiometry of the ‘‘rate-determining’’ flow. The same behavior exhibits ~jb in the case of ka = kc [5]. We, therefore, may expect that the bridge retains all these properties for arbitrary ka and kc. What physically means small total current J~? All currents are normalized to the methanol-limiting current density at the inlet (6). If methanol concentration is large enough, total current in the cell is limited by oxygen. Small J~ then in dimension units may be quite large, on the order of the oxygen-limiting current density. In that case the solution, shown in Fig. 3 gives a picture of DMFC operation in a quite large range of operating current densities. In our laboratory we get evidences that part of DMFC may turn into electrolysis regime [10]. Analysis of the case of ka kc suggests that the mixed regime of cell operation is realized if ~z0 < 1 (Fig. 3). To the right of ~z0 , on the cathode side there is not enough oxygen for complete oxidation of permeated methanol and methanol may be split into protons and electrons. The protons then move to the ‘‘anode’’ side, where they recombine with electrons to yield hydrogen. Therefore, to the left of ~z0 the cell generates power, whereas to the right of this point it may consume power to produce hydrogen. Condition ~z0 P 1 thus provides the regime of power generation over the whole surface area of the cell. With (27) this condition reads 1 1 1 P c þ ; b1 k J~ c
ð38Þ
Therefore, small b1 (low crossover) in all cases helps to maintain the cell in power generation regime. If crossover is large, it is beneficial to increase the product kc J~ or/and c in order to keep the cell in power generation regime. Parameter c is proportional to oxygen molar con-
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centration at the inlet ccn0 (13). Thus, under large crossover one has to increase oxygen concentration or/ and stoichiometry of oxygen flow.
5. Conclusions Our 1D + 1D model of a direct methanol fuel cell [5] is further analyzed. The model couples transport of oxygen and methanol across the cell (including methanol crossover) to the transport of these reactants in the respective channel. Numerical solution to model equations for arbitrary methanol and oxygen stoichiometries exhibits the effect of ‘‘bridge’’ formation, described in [5] for the case of ka = kc. A narrow region close to the channel inlet (a bridge) supports finite local current density at vanishingly small total current in the cell. The bridge ‘‘short-circuits’’ DMFC electrodes, thus reducing cell open-circuit voltage. In case of ka > kc the system has a new type of solutions: current density appears to be localized in a finite domain of the cell. Asymptotic solutions give the properties of the bridge in the cases of ka kc and ka kc. In the case of ka kc the bridge has exponential shape, quite analogous to those described in [5]. Taking into account analytical result for ka = kc [5] and asymptotic solution for ka kc we may expect, that for any ka 6 kc local current ‘‘covers’’ the whole surface of the cell. If ka > kc and total current is small, local current flows only in the bridge. In all cases the bridge is a domain of non-vanishing local current density as J~ ! 0. In both cases of ka kc and ka kc local current density in the bridge increases with the rate of crossover and decreases with the growth of stoichiometry of the rate-determining flow. The formula for voltage loss at open circuit is derived.
References [1] S. Srinivasan, A.S. Arico`, V. Antonucci, Fuel Cells 1 (2001) 133. [2] W. Vielstich, V.A. Paganin, F.H.B. Lima, E.A. Ticianelli, J. Electrochem. Soc. 148 (2001) A502. [3] A. Ku¨ver, W. Vielstich, J. Power Sources 74 (1998) 211. [4] S.C. Thomas, X. Ren, S. Gottesfeld, P. Zelenay, Electrochim. Acta 47 (2002) 3741. [5] A.A. Kulikovsky, On the nature of mixed potential in a DMFC, J. Electrochem. Soc. (2004, submitted). [6] A.A. Kulikovsky, Electrochem. Commun. 4 (2002) 939. [7] R. Jiang, D. Chu, J. Electrochem. Soc. 151 (2004) A69. [8] X. Ren, P. Zelenay, S. Thomas, J. Davey, S. Gottesfeld, J. Power Sources 86 (2000) 111. [9] V. Gogel, T. Frey, Z. Yongsheng, K.A. Friedrich, L. Jo¨rissen, J. Garche, J. Power Sources 127 (2004) 172. [10] M.Mu¨ller, H. Dohle, Private communication, 2003.