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Journal of Power Sources 218 (2012) 268e279

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Journal of Power Sources

journal homepage: www.elsevier .com/locate/ jpowsour

Modeling and optimization of the DMFC system: Relating materials propertiesto system size and performance

Brent Bennett*, Babar M. Koraishy, Jeremy P. Meyers 1

Department of Mechanical Engineering, The Center for Electrochemistry, The University of Texas at Austin, 1 University Station, C2200, Austin, TX 78712-0292, USA

h i g h l i g h t s

< Analytic model of oxygen concentration in the direct methanol fuel cell.< Fuel cell power output is compared to power consumption of system components.< Performance of new membrane materials is compared to that of Nafion.< Optimum flow rates are suggested for a variety of materials and system parameters.

a r t i c l e i n f o

Article history:Received 29 April 2012Received in revised form26 June 2012Accepted 28 June 2012Available online 5 July 2012

Keywords:Direct methanol fuel cell (DMFC) modelMethanol crossoverDMFC performanceDMFC efficiencyDMFC system design

* Corresponding author. Tel.: þ1 432 528 0899; faxE-mail address: brent.bennett@utexas.edu (B. Ben

1 Present address: EnerVault Corporation, 1244 ReCA 94089, USA.

0378-7753/$ e see front matter � 2012 Elsevier B.V.http://dx.doi.org/10.1016/j.jpowsour.2012.06.095

a b s t r a c t

In designing a direct methanol fuel cell and evaluating the appropriateness of new materials for inclu-sion, it is helpful to consider the impact of material properties on the performance of a complete system:to some degree, poor fuel utilization and performance losses from methanol crossover can be mitigatedby proper system design. Simple engineering models can be useful tools in facilitating this type of systemdesign. In this paper, an analytical model is developed to determine the oxygen concentration profile inthe cathode backing layer and flow channel along a one-dimensional cross-section of the fuel cell. Anexisting analytical model is then used to determine the methanol concentration profile in the anodebacking layer and membrane and the methanol crossover current along the same cross-section. Applyinga fixed cell potential and using Tafel kinetics to describe the charge-transfer reactions at the anode andthe cathode, the local current density and the rates of methanol and oxygen consumption are deter-mined. The process repeats for a fixed number of cross-sections down the entire length of the flowchannels, and the average current density and stoichiometric ratios are calculated at the end. The modelis applied to examine the effects of new low-crossover membranes and to suggest new design param-eters for those membranes. Also, an analysis is presented in which the tradeoff between stack power andthe size of system components is examined for a range of methanol and oxygen flow rates.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

The direct methanol fuel cell (DMFC) is an electrochemicalenergy conversion device that operates by converting methanoland water to carbon dioxide and protons at the anode,CH3OH þ H2O # CO2 þ 6Hþ þ 6e�, and by reducing oxygen at thecathode, O2 þ 4Hþ þ 4e� # 2H2O. The primary advantage of theDMFC is that it can be operated with a liquid anode feed andtherefore does not require a reformer or a bulky gas storage tank.

: þ1 512 471 1045.nett).amwood Avenue, Sunnyvale,

All rights reserved.

Also, because the oxidation reaction produces six electrons permethanol molecule, the DMFC delivers a high specific energy. Thesimplicity and compactness of the fuel delivery system makes theDMFC attractive for portable applications.

A paramount problem in the implementation of the DMFC ismethanol crossing over to the cathode, where it is oxidized,resulting in depolarization of the positive electrode and wastedfuel. The origin of crossover lies in the fact that methanol iscompletely soluble in water and that water is needed to swell theproton-exchange membrane and to impart protonic conductivityacross the membrane. One way to mitigate this problem is to createa low-crossover membrane to replace standard perfluorosulfonicacid membranes such as Nafion�. Increasing the efficiency of thereactions at the anode and the cathode through the design of thecatalyst layers can also minimize the harmful effects of crossover.

Fig. 1. Cross-section of the cell including a qualitative rendering of the methanol andoxygen concentration profiles.

B. Bennett et al. / Journal of Power Sources 218 (2012) 268e279 269

Finally, it is important to optimize the design of the flow channelsand operating parameters such as the methanol and oxygen feedconcentrations and the temperature.

During the past two decades, numerous approaches tomodelingthe DMFC have been taken. Most of these models focus on specificaspects of the fuel cell, such as polarization, multiphase transport,or methanol crossover. In the majority of the models, fuel utiliza-tion, system efficiency and other system-level engineering prob-lems, if considered at all, is only given brief discussion. However,there are still many DMFC models in the literature that focusexplicitly on issues such as fuel utilization [1e7], system efficiency[8,9], optimization of methanol and oxygen flow throughout thefuel cell stack [10e14], exergy and thermal analyses [14e17], andsystem size and cost [11,16,18]. What most of these models lack aresimple connections among materials properties, operatingparameters, and system constraints.

This work seeks to make those connections by incorporating asmany analytic expressions as possible. Specifically, the goal here isto examine the effect of materials properties, especially propertiesrelating to the proton-exchange membrane, on the net poweroutput and total size of the fuel cell system. System size is a criticalfactor for the cost and portability for small DMFCs. Running at highutilization rates (low flow rates) will lower the size of the systemcomponents such as the methanol tank and the air blower but willreduce performance and require a larger fuel cell stack to achievea certain power output. Conversely, running at low utilization rates(high flow rates) will increase power output and reduce the stacksize but will require a larger methanol tank and air blower. Toperform this analysis, we designed a model that incorporates asmany analytic solutions as possible for simple use and fast runtimes in a MATLAB simulation environment.

To create the desired model, we seek to derive analytical solu-tions for the methanol and oxygen concentration profiles alonga single one-dimensional cross-section of the fuel cell. The solu-tions for the methanol concentration in the anode backing layerand membrane taken from our previous publication [11]. Similarequations are derived in this work for the oxygen concentration inthe cathode backing layer and flow channel. We then apply Tafelkinetics with a fixed cell potential to determine the local currentdensity and the consumption of methanol and oxygen reactantsalong a single cross-section of the cell. Using a forward differenceapproach, the methanol and oxygen concentrations at the nextpoint in the flow channel are calculated. The model is steppeddown the flow channels in this manner, and the average currentdensity is calculated by dividing the sum of all the cross-sectionalcurrents by the number of steps.

In the first part of the analysis, we simulate polarization data fora cell operating at fixed inlet concentrations and flow rates ofmethanol and oxygen. In order to validate themodel,we compare themodel’s predictions with experimental data for the performance ofDMFCs running at lowutilization rates of both oxygen and air at 65 �Cand 90 �C. This data provides a base case for setting the standardkinetic parameters in the model, such as the exchange currentdensities and the symmetry coefficients.We then apply themodel toanticipate the effects of using new low-crossover membranes fromZhu et al. on fuel cell performance, and we offer suggestions on howto design and operate fuel cells that use the new membranes.

In the second part of the analysis, we consider the system-leveltradeoff between the power output of the cell and the size of themethanol storage tank and air blower. We set a desired currentoutput for the cell (50 mA cm�2 or 300 mA cm�2) and calculate thepower output at different oxygen and methanol flow rates andmethanol concentrations. We also consider the system-levelbenefits of using the new membranes. The size of the methanoltank and the air blower should scale linearly with the flow rates

while the marginal increase in power output will diminish as theflow rates increase. Therefore, given a set of performanceconstraints, we expect to find flow rates that minimize the total sizeof the fuel stack in combinationwith the other system components.

2. Model development

2.1. Assumptions

In Section 2.1, we develop analytic solutions for the oxygenconcentration in the cathode backing layer and across the thicknessof the cathode flow channel. For the analytic solutions of themethanol concentration profile in the anode backing layer and inthe membrane, we again refer the reader to our previous publica-tion [4]. The notation from that publication is carried into this work.Section 2.2 shows how the local current density is found using theTafel equations and assuming a constant cell voltage. In Section 2.3,we use the local current density to calculate the local methanol andoxygen consumption in the flow channels and then apply a forwarddifference method to determine the methanol and oxygenconcentration profiles down the entire length of the channels. Atthe end, the stoichiometric coefficients are calculated using theaverage current density in the fuel cell.

In the development of themodel that follows, we assume that allfluxes and current densities are defined per unit of superficial area,and transport properties in the backing layers and membrane areeffective transport properties, modified from bulk values byappropriate corrections for volume fraction and tortuosity. The term“current” will always refer to the specific current density unlessotherwise noted.We also neglect convection across the thickness ofthe cell. The methanol flux across the cell is dominated by diffusionin the anode backing layer and is a combination of diffusion andelectroosmotic drag in the membrane, while the oxygen flux is duesolely to convection in the flow channel and molecular diffusion inthe cathode backing layer. The methanol concentration is driven tozero at the boundary between the membrane and the cathodebacking layer, whereas the oxygen partial pressure maintainsa finite value at that same boundary (Fig. 1).

B. Bennett et al. / Journal of Power Sources 218 (2012) 268e279270

The most critical assumption in this model is that any effectsstemming from water transport or carbon dioxide gas removal areignored or incorporated into effective diffusion coefficients. Thisassumption allows for analytic solutions for the oxygen concen-tration to be developed, but it also restricts the validity of themodelto low-power applications. In systems using high current densityand high methanol concentration (above 2 M), or in any scenariowhere water flux in the cathode and carbon dioxide generation inthe anode and cathode are substantial, the model’s ability toprovide accurate performance predictions is limited. However, inapplications such as portable communications, where system effi-ciency and system size are paramount, the fuel cell is often oper-ated at low current densities. These systems need to run lean onfuel in order to carry the methanol oxidation reaction to comple-tion, thereby avoiding wasted fuel and the production ofintermediaries. In these low-power systems, the model shouldhave a wide range of applicability.

Another important simplification is the assumption of Tafelkinetics for both methanol oxidation and oxygen reduction. Bothreactions are multi-step, and Tafel kinetics only applies to the rate-determining step in each case. However, in the cases describedhere, the exchange current densities for both reactions, w10�5 formethanol oxidation and w10�7 for oxygen reduction, are at leasttwo orders of magnitude below the cell current density, whichranges from about 1 mA cm�2 to 1 A cm�2 in the experimental dataused to verify the model. Therefore, the reaction is forward-biasedto such an extent that we can assume the intermediate stepsproceed rapidly and apply the Tafel equation to the rate-determining step. As long as the current density is not so highthat adsorption of the reactants on the catalyst sites becomes rate-determining, Tafel kinetics should be sufficient to describe theentirety of the electrode reactions. See, for example, an analysis ofmethanol reaction kinetics under relevant conditions in Ref. [26].

The methanol solution is assumed to be an ideal incompressibleliquid in an isothermal environment such that any pressure or localtemperature effects can also be ignored. The air in the cathode isassumed to be an ideal gas, and only water in the vapor phase isconsidered explicitly at the cathode.Muchwork has been done,mostnotably by Scott et al. [19] and byWang andWang [20], to show thattwo-phase effects at the anode can be important even in liquid-feedDMFCs, but this model does not consider those effects. Instead, weassume that the behavior of the backing layers can be characterizedsimply by an effective diffusion coefficient and by their thickness.

PO2;c ¼ ��PN2þ PH2O

�þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�PO2;s þ PN2

þ PH2O�2�2ðiþ ixÞ

�RT4F

�0@ PN2

DeffO2�N2

þ PH2O

DeffO2�H2O

1ALback;c

vuuut : (4)

2.2. Oxygen pressure in the cathode

2.2.1. Cathode backing layerIn the cathode, we assume that the incoming air stream is

a pure mixture of oxygen, nitrogen (if air is used instead of pureoxygen), and water vapor (if humidified). If we assume thatcathode feed stream obeys the ideal gas law, then cO2

¼ PO2=RT ,

where PO2¼ yO2

Ptotal is the absolute oxygen pressure. We againemphasize that the effects of any liquid water present in thebacking layer or the flow channel are incorporated into theeffective diffusion coefficients. If the feed stream is humidifiedand if the saturation vapor pressure of water is solely a functionof temperature [21]

PH2O;sat ¼ exp�11:6832� 3816:44

ðT � 46:13Þ�; (1)

then the oxygen partial pressure will be 20.946% of the remainingtotal pressure for an air feed. A study by Weber and Newman [22]confirms that when the cathode backing layer in a PEM fuel cell hasa high porosity and a relatively small pressure gradient across it,convection and Knudsen diffusion can be safely ignored. Therefore,in order to obtain an analytic solution of the oxygen concentrationprofile across the backing layer, we use StefaneMaxwell diffusionto fully describe the transport of oxygen in that region

PtotalRT

vybackO2

vx¼0@� yN2

DeffO2�N2

� yH2O

DeffO2�H2O

1ANback

O2

þ0@ yO2

DeffO2�N2

1ANback

N2þ0@ yO2

DeffO2�H2O

1ANback

H2O : (2)

Again note that DeffO2�N2

and DeffO2�H2O

are effective diffusion coef-ficients for the components in the porous backing layer. The flux ofwater and nitrogen across the backing layer is negligible, allowingus to ignore the last two terms. Substituting partial pressures forthe mole fractions and Nback

O2¼ ðiþ ixÞ=4F for the oxygen flux, we

find an expression for the change in absolute oxygen pressureacross the backing layer

vPbackO2

vx¼ RT

Ptotal

0@� PN2

DeffO2�N2

� PH2O

DeffO2�H2O

1A�iþ ix

4F

�; (3)

where Ptotal ¼ PbackO2þ PN2

þ PH2O with PN2and PH2O constant. PH2O

is always set to the saturated value because there is assumed to besome liquid water in the cathode backing layer at all times, and PN2

is either 79.054% of the remaining total pressure for an air feed orzero for an oxygen feed.

Since PbackO2is included in Ptotal, it is now on both sides of this

equation, which means that the pressure profile in the backinglayer is nonlinear. The appropriate boundary conditions arePbackO2

¼ PO2 ;s at x ¼ 0 and PbackO2¼ PO2;c at x ¼ Lback;c, where PO2 ;s

is the oxygen pressure at the boundary between the backing layerand the flow channel and PO2 ;c is the oxygen pressure at theboundary between the membrane and the backing layer. Afterintegrating with these boundary conditions, we find the expression

2.2.2. Oxygen transport from cathode flow channel to backing layerA crucial difference between the treatment of the anode and the

cathode is the need to consider the movement of oxygen gas fromthe bulk of the flow channel to the surface of the cathode backinglayer. The change in oxygen partial pressure across this distancetakes the same form as Eq. (3)

vPconvO2

vx¼ RT

Ptotal

�� PN2

DO2�N2

� PH2O

DO2�H2O

��iþ ix4F

�; (5)

where PconvO2is the absolute oxygen pressure in the convective

region between the bulk flow channel and the backinglayer surface. Note that the diffusion coefficients lack the eff

B. Bennett et al. / Journal of Power Sources 218 (2012) 268e279 271

subscript because we are outside the porous region of the backinglayer.

As with PbackO2; PconvO2

is included in Ptotal and is on both sides of Eq.(5), so the result is a nonlinear pressure profile. The appropriateboundary conditions are PconvO2

¼ PO2 ;f at x ¼ 0 and PconvO2¼

PO2 ;s at x ¼ dh=ShF , i.e. the backing layer surface, where dh is thehydraulic diameter of the channel and ShF ¼ 2.71 is the Sherwoodnumber for a square channel [23]. PO2 ;f is the oxygen pressure in thebulk flow channel and is determined according to the solutionmethod given in Section 2.4.2. Integrating Eq. (5) with theseboundary conditions results in an expression for the oxygen pres-sure at the surface of the cathode baking layer that is almostidentical in form to the expression for PO2 ;c.

PO2 ;s ¼ ��PN2þ PH2O

�þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPO2;f þ PN2

þ PH2O

2�2ðiþ ixÞ�RT4F

� �PN2

DO2�N2

þ PH2O

DO2�H2O

�dhShF

s: (6)

We then substitute PO2;s directly into Eq. . to obtain thefollowing expression for PO2 ;c terms of PO2 ;f

PO2 ;c ¼ ��PN2þ PH2O

�þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPO2;f þ PN2

þ PH2O

2�2ðiþ ixÞ�RT4F

�24� PN2

DO2�N2

þ PH2O

DO2�H2O

�dhShF

þ0@ PN2

DeffO2�N2

þ PH2O

DeffO2�H2O

1ALback;c

35

vuuut : (7)

The resulting expression cannot be simplified and must besubstituted in this form when we solve for the desired current i.Aside from i, the other unknown in Eq. (7) is the oxygen pressure inthe bulk flow channel PO2;f . The solution methods for i and PO2;f aredescribed in Sections 2.2 and 2.3.2.

2.2.3. Oxygen transport in the cathode catalyst layerWhile maintaining the assumption of thin catalyst layers, we

will choose to define an effectiveness factor to account for the masstransport of oxygen in the cathode catalyst layer because theoxygen reduction reaction is first-order. The methanol oxidationreaction is still assumed to be zero-order, so there is no need todefine an effectiveness factor there. The effectiveness factor isa dimensionless quantity that relates the actual rate of reaction tothe rate if the entire interior of the catalyst layer was exposed to theconditions that exist at the catalyst layer/backing layer boundary. Ineffect, it is a measure of how far into the catalyst layer the oxygenpenetrates and is a simpler and more convenient method ofaccounting for mass transport in the catalyst layer than applyinga full model containing terms for molecular diffusion, Knudsendiffusion (particularly important in catalyst layers and micro-porous layers), and convection.

Assuming Tafel kinetics, at steady-state the governing equationfor the material balance of oxygen in the catalyst layer is [1]

d2cðxÞdx2

� i0c4FLclDeff

O2;cl

RTPO2;c

exp�� acF

RTðVcathodeÞ

�cðxÞ ¼ 0: (8)

where ic0 is the exchange current density in A cm�2, Lcl is the catalystlayer thickness, PO2 ;c is the oxygen pressure at the catalyst layersurface, ac is the cathodic transfer coefficient, and Vcathode is the

cathodic activation overpotential. Because of the complexity ofoxygen diffusion in the catalyst layer, which has relatively lowporosity and high water saturation, we choose to use an effectivediffusion coefficient of oxygen Deff

O2;clbased on the idea that the

catalyst layer is fully flooded and that the diffusion of oxygen onlyoccurs in the regions containing Nafion [24]

DeffO2;cl ¼ DO2 ;Nafion

hð1� εclÞfNafion

i1:5: (9)

whereDO2;Nafion ¼ 5:23� 10�6exp½ð23849=RÞ*ðð1=333:15Þ � ð1=TÞÞ�cm2

s�1 [25], εcl is the porosity of the catalyst layer, and fNafion is thevolume fraction of Nafion in the catalyst layer. Making the gov-

erning equation dimensionless leads us to define the Thielemodulus as follows [1]:

q2 ¼ i0c Lcl4FDeff

O2 ;cl

RTPO2;c

exp�� acF

RTðVcathodeÞ

�: (10)

Subsequently, the effectiveness factor is a function of the Thielemodulus: h ¼ (tanh q)/q. For an effectiveness factor close to unity,internal mass transport in the catalyst layer is minimal, and thereactant penetrates fully into the catalyst layer. If the effectivenessfactor is closer to zero, mass transport in the catalyst layer isa significant issue, and the catalyst layer is not being fully utilized. InDMFC cathodes the effectiveness factor is much less of an issue thanin PEMFCs due to lower operating current densities. Even with therelatively thick cathode catalyst layer in thismodel (Lcl¼ 10 mm), theeffectiveness factor is rarely below 0.9. However, because includingit does not add noticeably to the computation time, we choose toinclude it for the sake of completeness and accuracy.

2.3. Overpotentials and current solution method

2.3.1. Determination of overpotentials and cell potentialNow that the concentration profiles for methanol and oxygen

have been developed for a one-dimensional cross-section of the cellin terms of the desired current i, all that remains is to solve for thatcurrent. As stated in the introduction, our strategy is to assume thecell potential remains constant and numerically solve for the localcurrent along each cross-section. Given the high in-plane conduc-tivity of the bipolar plates, any voltage difference along the plateswill be quickly dissipated, so the assumption of a constant cellpotential appears reasonable. The cell potential is definedas follows:

Vcell ¼ Uq � iLmem

kmem� i2

�Lclakcla

þ Lclckclc

�� Vanode þ Vcathode; (11)

Fig. 2. Methanol consumption in the fuel cell.

B. Bennett et al. / Journal of Power Sources 218 (2012) 268e279272

where kmem is themembrane conductivity in S cm�1, kcla and kclc arethe anode and cathode catalyst layer conductivity in S cm�1 andUq¼ 1.21 V for the DMFC. Implicit in this equation is the assumptionthat the proton conductivity is uniform throughout the thickness ofthe catalyst layers and the membrane. While each is tracked sepa-rately, and preserves the ability to describe restricted proton accessin the catalyst layers, where the volume fraction of ionomer is lowerthan in the membrane separator, we do neglect the possibility ofvariation through the thickness of any of the components.Furthermore, the factor of ½ appears in front of the catalyst layerohmic terms because we assume that the protons travel on averagehalf the distance across each catalyst layer.While this assumption isnot true in a situation where there are substantial mass-transportlimitations in the catalyst layers, zero-order kinetics at the anodeand an effectiveness factor that is almost always above 0.9 at thecathode support the notion that the reactions are occurring some-what uniformly across the catalyst layers.

As mentioned previously, both the methanol oxidation andoxygen reduction reactions are assumed to follow Tafel kinetics,with the former being zero-order and the later being first-order.The assumption of zero-order kinetics has been shown to beaccurate to very low concentrations or operation very near thelimiting current, and is tantamount to assuming that the CO-siteson the anode electrocatalyst can quickly become saturated [26].

Vanode ¼ RTFaa

ln

ii0a

!(12)

Vcathode ¼ RTFac

ln

h

PO2;c

PO2;ref

i0ciþ ix

!(13)

Note the presence of the effectiveness factor in the cathodicoverpotential. Maintaining zero-order methanol kinetics meansthat the anode overpotential does not depend on the methanolconcentration at the anode catalyst layer. This assumption is validas long as we keep the methanol stoichiometry high enough toavoid depleting methanol at the anode catalyst layer and incurringa transition to first-order methanol kinetics [26]. Substituting Eqs.(12) and (13) into Eq. (11) results in the following expression for thecell potential:

Vcell ¼ Uq� ikmem

Lmem� i

2Lclkcl

� RTFaa

ln

ii0a

!þ RTFac

ln

hPO2;c

PO2;ref

i0ciþ ix

!:

(14)

It is clear that we must use a numerical method to solve for thecurrent in Eq. (14). However, in order to find the desired current, wemust first find an expression for the crossover current ix in terms ofthe desired current i and other known variables.

2.3.2. Determination of crossover current and total currentTo write ix in terms of i we refer to our previous work [11]

concerning the methanol concentration in the anode backinglayer and in the membrane. The crossover current is described interms of the feed concentration cfeed and the concentration at theanode catalyst layer canode as follows:

ix ¼ 6F

0@Deff

MeOH�H2O

Lback;a

1Acfeed � canode

� i: (15)

The crossover current can also be described using the dimen-sionless parameter n.

ix ¼ �ilim;a � i

� n

1þ n

: (16)

Substituting Eq. (15) into Eq. (16) and solving for canode producesa relationship between canode and cfeed

canode ¼ 1� i

ilim;a

!�1

1þ n

�cfeed: (17)

Further substitution of ilim;a ¼ 6FðDeffMeOH�H2O

=Lback;aÞcfeedresults the following expression for canode:

canode ¼ jcfeed � c0feedjð1þ nÞ : (18)

Eq. (18) shows that canode is dependent on cfeed at the correspondinglocation in the flow channel, the initial feed concentration c0feed, thedimensionless ratio n, and j ¼ i0lim;a=i, which is the limiting currentat the anode entrance normalized by the desired current.

Removing j from the denominator and substituting the aboveexpression for canode into Eq. (15) results in

ix ¼ ilim;a

cfeed

ii0lim;a

jncfeed þ c0feed

ð1þ nÞ

!� i: (19)

Noting that ilim;a=i0lim;a ¼ cfeed=c

0feed, we arrive at the desired

relation for ix in terms of i.

ix ¼ i

jvcfeed þ cofeedcofeedð1þ vÞ � 1

!: (20)

Nowwe are prepared tomake the appropriate substitutions andsolve Eq. (14) for i. The solution is performed in MATLAB using theroot-finding function fzero along each cross-section of the cell. Toobtain an average cell current the sum of the cross-sectionalcurrents is divided by the number of steps.

2.4. Consumption of methanol and oxygen in the flow channels

2.4.1. Anode flow channelHaving the ability to find the local current density as a function

of the local methanol concentration and oxygen pressure in theflow channels, we now must determine the consumption ofmethanol and oxygen in the flow channels as we move from onestep to the next (Fig. 2).

Assuming the velocity vMeOH of methanol in the flow channel isconstant, the total flow rate of methanol in the anode flow channelis given by

B. Bennett et al. / Journal of Power Sources 218 (2012) 268e279 273

NchanMeOH ¼ cfeed Achan vMeOH: (21)

where Achan ¼ wchanhchan. The change in feed concentrationbetween each step as we move down the flow channel is

vcfeedvz0

¼�NbackMeOH

�wsepLchanAchanvMeOH

�¼�

�iþix6F

��wsepLchanAchanvMeOH

�: (22)

where z0 ¼ z/Lchan is the normalized distance down the channel, andwsep is the channel separation width, that is, the width of electrodeperpendicular to the channel flow that is fed by a single channel.wsepLchan is the area of the anode that receives the methanolflowing through the channel. The effects of other parallel channelsare ignored.

Noting the presence of the crossover current term ix, wesubstitute Eq. (20) to obtain

vcfeedvz0

¼ ��

i6F

� jncfeed � c0feedc0feedð1þ nÞ

!�wsepLchanAchanv

�: (23)

We can now define the stoichiometric ratio for the methanolflow as

sMeOH ¼ c0feedi=6F

Achanv

Lchanwsep(24)

and derive an expression for the change in the feed concentration

vcfeedvz0

¼ ��

1sMeOH

� jncfeed � c0feed

ð1þ nÞ

!: (25)

The stoichiometric ratio is the rate at which methanol is deliv-ered to the inlet of the cell relative to the amount ofmethanol that isconsumed in the preferred electrochemical reaction. Hence, itrepresents the inverse of fuel utilization or fuel efficiency. In order todiscretize the flow channel and apply the one-dimensional modelfrom Sections 2.1 to 2.3, we apply a first-order forward difference,f(xþh)¼ f(x)þhf0x,whereh¼ Lchan/n is the step size andn is the totalnumber of steps. Thenwe create the following expression for cfeed atstep k þ 1 based on cfeed and the current i at step k.

cfeedðkþ1Þ ¼ cfeedðkÞ��

1siz

��1

sMeOH

� jvcfeed� cofeed

ð1þ vÞ

!: (26)

2.4.2. Cathode flow channelIn thecathodeflowchannel,where theoxygenreactant isagas,we

must consider the pressure drop of the air stream in addition to theconsumptionof oxygen.We cannot assume that the velocityof the airstream is constant as we do for the velocity of the liquid methanol/water solution in the anode flow channel. We do assume that theabsolute pressure of water vapor remains constant throughout theflow channel. If the inlet flow is dry, then thewater vapor pressure isset to zero, and if it is humidified, then the water vapor pressure isassumedtobe its saturatedvaluegiven inEq. (1). This simplification isnecessary given that changes in water content are not treated in thebacking layer, but it can lead to significant error at high current whenthe water generation is high. Because of these assumptions, thepressure drop only affects the nitrogen and oxygen pressures. Ingeneral terms, according to Darcy’s Law, the pressure drop is

vPair;fvz

¼ �mairK

vair;f ¼ �mairK

Nair;f

Achan

RTPair;f

: (27)

where K ¼ D2h=2fRe represents the cross-sectional area to flow. Re is

theReynoldsnumber, f is a friction factor, and their product is equal to14.2296 for a square channel [23]. Dh is the hydraulic diameter and isequal to four times the area of the channel divided by the perimeter.mair is the dynamic viscosity of the airmixture and is a function of themole fraction of all three components in the mixture

mair ¼XNi¼1

yimiPNj¼1 yiFi

: (28)

where N is the total number of species in the mixture, yi and yj arethe mole fractions of species i and j, and Fi is a sum of dimen-sionless ratios. For an in depth analysis of the derivation of mair, werefer the reader to pp. 153e155 in Ref. [23]. At 65 �C, mair is about1.6 � 10�5 Pa s�1, depending on the values of the mole fractions.

Substituting for k and noting that Nair;f =Pair;f ¼ NO2;f =PO2 ;f ; weobtain a final expression for the pressure drop as a function of theoxygen flow rate, oxygen pressure, and known parameters

vPair;fvz

¼ �2fRemairD2h

NO2;f

Achan

RTPO2 ;f

: (29)

Using a forward difference to discretize this equation, as we didwith Eq. (25), results in

Pair;f ðkþ1Þ¼Pair;f ðkÞ��Lchansiz

� 2fRemairðkÞ

D2h

! NO2;f ðkÞAchan

RTPO2;f ðkÞ

!:

(30)

Therefore the pressure drop at step k þ 1 is determined by theratio of the oxygen flow rate to the oxygen pressure at step k. Thechange in oxygen flow rate is simply a function of the total current,and that expression can be discretized to solve for NO2 ;f ðkÞ.

NO2;f ðkþ 1Þ ¼ NO2;f ðkÞ �iþ ix4F

�wsepLchan

siz

�: (31)

Determining the local oxygen pressure PO2 ;f ðkÞ is a bit morechallenging. The expression for the consumption of oxygen in thecathode flowchannel is derived in a similar manner as formethanolin the anode. The change in oxygen pressure due to consumption is

vPO2;f

vz¼ �Nback

O2

wsepRTAchanvO2;f

!¼ �

�iþ ix4F

� PO2;f

NO2;f

!wsep: (32)

where the substitution vO2;f ¼ vair;f ¼ ðNO2;f =AchanÞðRT=PO2;f Þ ismade. Substituting Eq. (20) for ix, we find an expression for thechange in oxygen pressure that is similar to Eq. (23)

vPO2;f

vz¼ �

�i4F

� jncfeed � c0feedc0feedð1þ nÞ

! PO2;f

NO2;f

!wsep: (33)

Discretizing Eq. (33) and adding it to Eq. (30) results in anexpression for the change in the total pressure of the air inletstream at each step

Pair;f ðkþ1Þ ¼ Pair;f ðkÞ��wsepLchan

siz

��i4F

� jncfeedðkÞ�c0feed

c0feedð1þnÞ

!

� PO2;f ðkÞNO2;f ðkÞ

!��Lchansiz

� 2fRemairðkÞ

D2h

!

� NO2;f ðkÞAchan

RTPO2;f ðkÞ

!:

(34)

Table 2Kinetic and resistive properties for Nafion membranes.

Parameter (unit) Value Reference

i0a (A cm�2) 3.5 � 10�5 Fitted to datai0c (A cm�2) 3.0 � 10�7 Fitted to dataaa (dimensionless) 0.8 Fitted to dataac (dimensionless) 0.875 Fitted to datakmem (S cm�1) 0.111 * e1268(1/303 � 1/T) O’Hayre et al. [23]kcla (S cm�1) 0.001 * e1268(1/298 � 1/T) Havranek and Wippermann [33]kclc (S cm�1) 0.0026 * e1268(1/298 � 1/T) Saab et al. [34]

Table 1Physical properties for experimental validation.

Parameter (unit) Value Reference

DO2�H2O (cm2 s�1) 6.436 � 10�6 * T1.823/p(in atm) O’Hayre et al. [23]DO2�N2

(cm2 s�1) 4.2 � 10�7 * T2.334/p(in atm) O’Hayre et al. [23]DMeOH�H2O (cm2 s�1) 10�1.416 � 999.778/T Yaws [30]Dmem (cm2 s�1) 4.9 � 10�6 * e2416(1/333 � 1/T) Kauranen and Skou [31]εa (dimensionless) 0.7 Wang and Wang [20]εc (dimensionless) 0.35 Fitted to dataεcl (dimensionless) 0.1 AssumedfNafion (dimensionless) 0.058 MeasuredLback,a ¼ Lback,c (cm) 0.02 AssumedLmem (cm) 0.0128 AssumedrH2O (g cm�3) 1.1603 � 5.371 � 10�4 * T Incropera et al. [21]xH2O (dimensionless) 2.5 Zawodzinski et al. [32]x0MeOH (dimensionless) xH2O*18=rH2O Calculatedwsep (cm) 0.2 MeasuredLchan (cm) 25 MeasuredAchan (cm2) 0.01 Measured

B. Bennett et al. / Journal of Power Sources 218 (2012) 268e279274

Because the total pressure is the sum of the partial pressures,with the water vapor pressure being constant, we can relate thenitrogen and oxygen pressures in order to make substitutionsand solve for each partial pressure individually: PO2 ;f ¼ PN2 ;fðNO2;f =NN2 ;f ÞandPN2 ;f ¼ PO2;f ðNN2;f =NO2 ;f Þ. The analysis aboveapplies to a system where the flow channels are in a coflowconfiguration, in which the methanol and oxygen flow in the samedirection through the fuel cell. If we desire, we can add the changein oxygen pressure between steps instead of subtracting it andthereby simulate a counterflow configuration, in which the meth-anol and oxygen flow in opposite directions. The only difficulty inthat situation is the need to define the oxygen pressure at thechannel outlet and the oxygen flow rate at the inlet, both of whichchange as we move down the flow channel. For the sake ofsimplicity, we choose to use a coflow configuration in the analysisbelow. The oxygen stoichiometric ratio remains very high (alwaysabove 10) in the experimental data used for validating the model,which justifies this approach. Furthermore, the flow channels havea single-serpentine design, and there is probably a large amount ofunder-rib transport. At the high flow rates we are considering, weexpect there to be little difference between coflow and counterflowin the current/potential regimes that are not mass-transfer limited.

3. Experimental validation

3.1. Nafion membranes with optimized catalyst layers

The purpose of this model is to quickly and easily determinethe effect of materials properties and operational parameters,especially the membrane permeability, conductivity, and thicknessand the methanol and oxygen flow rates, on the performance andthe required size of the DMFC system. In order to fit certainparameters and accurately predict the performance of real systems,we need to validate the model with a control set of experimentaldata. One drawback to this model is the assumption of a negligiblythin catalyst layer and simple zero-order (methanol) or first-order(oxygen) Tafel kinetics for the complex reactions taking place at theanode and cathode catalyst layers. Hence, the anodic and cathodicexchange current densities ði0a and i0c Þ and the symmetry coeffi-cients (aa and ac) must be adjusted to fit experimental data. Aftercomparison with the data, we found that the addition of a contactresistance term Rc was unnecessary. The base values that we use forall of the physical properties are listed in Table 1 and do not changein the following sections unless otherwise noted. The values of allthe kinetic and resistive properties are listed in Table 2. It isimportant to note that we assume liquid water to be almost aseffective as a solid medium at preventing gas diffusion. Therefore,we account for the effect of liquid water in the cathode by loweringthe porosity of the cathode considerably relative to the anode.

As an initial check, we used a base case scenario (Nafion 115,250 sccm oxygen flow rate, and 2mlmin�1 methanol flow rate) andvaried the temperature (65 �C or 90 �C) and the cathode flowcomposition (air or pure oxygen). The experiments were performedwith a 1 M MeOH feed at the anode and 20 psi of backpressure atthe cathode. The resulting four cases for a single cell with a 5 cm2

MEA are shown in Fig. 1 below. The MEAs were fabricated byspraying the respective catalyst layer inks onto each side of theNafion membrane and then hot-pressing the carbon cloth backinglayers onto each side. The catalyst loading was 2.5 mg cm�2 Pt/Ruon the anode and 2.5 mg cm�2 Pt on the cathode on unsupportedcarbon black. This validation was performed to determine theaccuracy of the temperature-dependent diffusion coefficients,conductivities, and exchange current densities as well as to verifyour assumption of a constant water vapor pressure in the cathode.These experiments also tested the validity of our choice to

incorporate the effects of liquid water and carbon dioxide into aneffective diffusion coefficient rather than consider those compo-nents explicitly.

The primary discrepancy between the model and the experi-ments exists in the mass-transport region, especially in the 65 �Ccasewith air. We believe this discrepancy exists mostly because ourmodel does not allow for varyingwater content in the catalyst layer,gas diffusion layer, and flow channel of the cathode. High currentdensity generally corresponds with high water crossover andgeneration rates and lower effective oxygen diffusion coefficients,which decreases performance. However, accounting for thesevariations would have eliminated our ability to derive analyticexpressions for the gas concentrations across the thickness of thecell. We decided that a modeling scheme which can match theexperimental performance over the current and potential ranges ofinterest (generally less than 300 mA cm�2) with a simple set ofassumptions and a fast run time (usually less than 1 min) is ofgreater interest to the engineering problems of system sizing andstack optimization.

Experimental results, including the results shown in Fig. 3, oftenuse deliberately high flow rates in order to maximize performance.However, in a real system, it might behoove the designer to takea small cut in performance by lowering the flow rate to say,50 sccm, such that the air blower and the compressor consume lesspower. The ability of this model to accurately predict fuel cellperformance for a variety of methanol and oxygen flow rates will beimportant when we consider optimizing the size of systemcomponents in Section 4.

3.2. Low-crossover crosslinked blend membranes

The cases above all employ Nafion membranes, but much workhas been done in recent years to develop new membranes withlower methanol crossover than Nafion and comparable protonconductivity and mechanical properties. In particular, Zhu et al.have reported novel crosslinked membranes based on a blend of

Fig. 5. Performance of the C-10 blend membrane as a function of membrane thickness.The parameters and operating conditions are identical to what is used for the low-crossover membrane in Fig. 4.

Fig. 3. Comparison of simulations to experimental data at 65 �C and 90 �C. The basecase scenario presented here uses a Nafion 115 membrane, 250 sccm oxygen flow ratewith 20 psi of backpressure, and 2 ml min�1 methanol flow rate with 1 M methanolfeed concentration.

B. Bennett et al. / Journal of Power Sources 218 (2012) 268e279 275

sulfonated poly(ether ether ketone) and carboxylated polysulfonethat showed a dramatic improvement in performance compared toNafion [27]. They kindly made that data available in tabular formalong with physical property data for our use in Figs. 4 and 5. Thissection describes a case study in which we compared our modelwith the data for their C-10 membrane. These tests were run at65 �C and 1 MMeOH using a humidified oxygen feed at a 200 sccm

Fig. 4. Comparison of experiments and simulations for the C-10 blend membrane andNafion 115 membrane at 65 �C. This scenario uses a 2.5 ml min�1 methanol flow ratewith 1 M methanol feed concentration, and a 200 sccm oxygen flow rate witha humidified pure oxygen feed and no backpressure. Experimental data is from Zhuet al. [13].

flow rate. The MEAs in this case used a different catalyst loadingthan in the previous case (2.0 mg cm�2 vs. 2.5 mg cm�2 above) andwere fabricated by painting the catalyst inks onto the carbon clothbacking layers and then hot-pressing each of the two sets ontoopposite sides of the Nafionmembrane. Therefore, we had to adjustthe exchange current densities and symmetry coefficients toaccount for the differences in the catalyst layers. The new kineticparameters, as well as the new membrane properties, are given inTable 3.

Fig. 4 above compares Nafion 115 to the new membranes withour simulation overlapping both data sets. Our model predictsa noticeable improvement when using the new membranes, butnot as a large an improvement as the experiments show. Indeed,the experimental results at high current are surprising because inthat region we would expect the higher ohmic resistance in theblend membranes to counteract performance gains from lowermethanol crossover. There are several experiments in the literature,most notably from Ge and Liu [28], running under similar condi-tions that show much better performance from Nafion membranesthan what is shown in Fig. 4. We believe the discrepancy at highercurrents is due to mass-transport problems in the experiment,specifically water management, for which our model does notaccount. Also, a different construction of the catalyst layers, as wasdone with the experiments shown in Fig. 3, might lead to betterperformance for both the Nafion and the blend membranes. Finally,the discrepancies at low current could be the result of themethanoldiffusion coefficient in the blend membranes being higher duringthe experiment than what was measured prior to the experiment.

One advantage of the blendmembranes is that they can bemadethinner without significantly increasing the methanol crossover,which can more than compensate for the decrease in specificconductivity when compared to Nafion. Fig. 5 simulates theperformance of the C-10 membrane reported by Zhu et al. [27] for

Table 3Kinetic and membrane properties for blend membranes.

Parameter (unit) Value Reference

i0a (cm�2 s�1) 3.0 � 10�5 Fitted to datai0c (cm�2 s�1) 2.0 � 10�7 Fitted to dataaa (dimensionless) 0.875 Fitted to dataac (dimensionless) 0.925 Fitted to datakmem (S cm�1) 0.107 Zhu et al.Lmem (cm) 0.01 Zhu et al.Dmem (cm2 s�1) 1.0 � 10�8 Measured

B. Bennett et al. / Journal of Power Sources 218 (2012) 268e279276

a variety of thicknesses under the conditions given in Section 3.Fig. 6 does the same with Nafion membranes. While the thinnerNafion membranes suffer from noticeable performance declines atlow current due to crossover, the blend membranes show no dropin performance. As we mentioned above, the measured methanoldiffusion coefficient might be lower than the actual value, but themodel still clearly shows that the blend membranes should bemade as thin as manufacturing tolerances allow. The reduction inohmic resistance more than compensates for the small increase inmethanol crossover.

Aside from improved performance, the lower methanol cross-over rates of the new membranes results in improved fuel effi-ciency. If we want to calculate the maximum fuel efficiency of thefuel cell we can perform an iterative reduction of the methanolstoichiometric ratio until the methanol concentration at the anodereaches the first-order transition value. Also of importance when itcomes to optimizing the DMFC system with new membranes arethe properties of the backing layers. The thickness and porosity ofthe backing layers must strike a balance between allowing reac-tants to diffuse to the catalyst layers and providing an effectivebarrier, both physically and chemically, between the reaction zonesand the flow channels. The anode backing layer is of particularimportance because it must allow enough methanol to reach theanode catalyst layer so as to promote the desired reactionwhile notallowing so much methanol so as to promote crossover. Ourprevious work [4] addresses in detail the issues of fuel efficiencyand backing layer properties, so in order to avoid repetition, thispaper will not discuss these issues at length.

4. The effect of methanol and oxygen flow rates on systemsize and performance

Because portability is of primary importance in many applica-tions where DMFCs might be used, we want to use this model tocompare the power consumption of the air blower and compressorto the stack power output for a range of methanol and air flow rates.The less methanol and oxygen that are used and the more slowlythose reactants are passed through the flow channels, the smallerwe can make the supporting system components. Ideally, therewould be no methanol crossover, and all of the reactants in thesystem would either be consumed in the desired reaction or exitthe flow channels, perhaps being recirculated through the systemto be consumed later. In this case, the size of the supporting systemcomponents would be solely dependent on the demand for current.

Fig. 6. Performance of the Nafion membranes as a function of membrane thickness.The parameters and operating conditions are identical to what is used for the Nafion115 membrane in Fig. 3.

However, the existence of crossover and the potential for wastedfuel means that the size of the tank and the blower also depends onthe fuel efficiency.

If reducing the size of the system components were our onlyconcern, then we would like to run the cell as lean as possible witha very low stoichiometric ratio. However, running a lean feed willresult in performance losses compared to running at a high stoi-chiometric ratio. We are especially concerned with avoiding thetransition from zero-order to first-order kinetics for the methanoloxidation reaction. Rather than model this complicated transition,which would require a substantial increase in overpotential at theanode and a corresponding loss in performance [26], we ensurethat the optimized system remains above any such transition byrequiring the feed concentration to remain above a minimumvaluethroughout the flow channel. In doing so, we impose a constrainton the minimum allowable stoichiometric ratio at the anode. At thecathode, our only constraint on the stoichiometric ratio is that itremains above unity.

It is important to note that the analysis in this section does notconstitute an optimization study in the true spirit of the term. Weiterate through a defined set of flow rates for a few specificscenarios instead of using an optimization function to explore theentire parameter space. A more thorough optimization analysiscould be done with this model, but we choose not to do so here forthe sake of brevity. Hence, the more accurate term to describe whatis done here is a sensitivity analysis. The goal of this section is toillustrate how the model can be applied across a wide range ofa given parameter space and how it can provide guidelines forcertain parameters.

For a single fuel cell, the size of the fuel tank Vtan k scales linearlywith the flow rate at the anode inlet Ninput

MeOH, the time of operation tand the molar volume of the methanol and water feed streamV feed : Vtan k ¼ Ninput

MeOHV feedt; where NinputMeOH ¼ c0feed Achan vMeOH as

in Eq. (21). We have already seen in Fig. 6 how the performance ofthe fuel cell is largely independent of the anode flow rate due tozero-order kinetics, so in that sense, wewant to run at as low a flowrate as possible to avoid crossover and wasted fuel. However, theremust be a point at which the flow rate is not high enough to sustainzero-order kinetics. Using the model to predict this minimum flowrate would be useful in determining parameters for fuel cell designand operation.

The scenario presented in Fig. 7 above compares the poweroutput of a fuel cell operating at 300 mA cm�2 current using Nafion115 to fuel cells using blend membranes of the same thicknesses(represented by the blue lines) as in Fig. 5. The operating conditions

Fig. 7. Cell output power as a function of methanol flow rate for a desired current of0.3 A cm�2 using Nafion 115 and C-10 blend membranes. All other parameters andoperating conditions are identical to what is used in Fig. 3.

Fig. 8. Net power output of a 100 cell stack as a function of oxygen flow rate with 0.05and 0.3 A cm�2 currents using Nafion 115 and C-10 blend membranes. All otherparameters and operating conditions are identical to what is used in Fig. 3.

B. Bennett et al. / Journal of Power Sources 218 (2012) 268e279 277

and the cell design are the same for the Nafion and blendmembranes. The blend membranes are very effective at preventingmethanol crossover, so the model tells us that we should manu-facture them as thin as possible in order to lower the ohmicresistance, especially at high current. Of course, the limits of themanufacturing process and the desire to maintain goodmechanicalproperties and durability will limit our ability to reduce themembrane thickness.

In this case, the minimum methanol flow rate is4 � 10�6 mol s�1, which corresponds to 0.25 ml min�1 and anaverage methanol stoichiometric ratio of 1.6. Since the modelcannot handle the transition to first-order kinetics, the three pointsbelow that transition are generated by lowering the desired currentuntil the methanol concentration remains above the transitionvalue all the way to the anode exit. Hence, the power output of thecell must decrease either by incurring a drop in cell voltage due tothe transition to first-order kinetics or by lowering the current untilthat transition is avoided. In designing a system under our givenconditions, we would want to keep the methanol flow rate close tothis value but safely above it to maximize performance andefficiency.

The situation is similar when we consider the oxygen flow rateunder the same set of conditions. Given the low flow rates that weare using, the power consumption of the air blower and aircompressor will scale linearly with the flow rate in the cathodeflow channel. Therefore, as with the anode flow rate, we wouldexpect at to see a point at which the extra power generated by thefuel cell does not justify increasing the flow rate. The power outputof the compressor is given by [7]

Powercomp ¼24371TNinput

O2g

ðg� 1ÞEcomp

35" P0O2;f

PO2;ref

!ð1�ð1=gÞÞ�1

#: (35)

where g is the heat capacity at constant pressure divided by that atconstant volume (g ¼ 1.4), P0O2 ;f

is the inlet oxygen pressure in thecathode flow channel, Ecomp is the air compressor efficiency(assumed to be 0.8) and the coefficient of 371 has units of J K�1 m�3.

The blower power output can be found using the specific fanpower (SFP) as follows:

Powerblower ¼ SFP*NinputO2

: (36)

The lowest SFP values for small commercial fans operatingunder normal conditions is approximately 500W s m�3 [29], so wechoose to use that value through this analysis. Note the need toconvert the oxygen flow rate from mol s�1 to m3 s�1 throughmultiplication by the molar volume. Although the compressor usesmuch more power than the blower, the performance gain achievedby raising the cathode input pressure is substantial and the needto maintain a high oxygen concentration in the presence of meth-anol crossover is imperative. Therefore, we chose to includea compressor throughout this analysis. Subtracting these parasiticlosses from the power output of the fuel cells provides us with anexpression for the net power output of the system

Powersystem ¼ VcellIcellð#cellsÞ � Powercomp � Powerblower

� Powerpump:

(37)

Because the analysis below is only concerned with the oxygenflow rate, the parasitic losses from the methanol pump Powerpumpare ignored in the calculation of the system power output.

Fig. 8 above shows the power output of a DMFC stack containing100 5 cm2 cells as a function of oxygen flow rate. As expected, the

80 mm thick blend membranes outperform the Nafion 115membranes, and the 1 M MeOH feed stream is always better forperformance than 2 M MeOH due to zero-order methanol kinetics.Similar to what was done in Fig. 7, the desired current is lowered atlow flow rates in order to keep the oxygen pressure above zeroacross the entire cathode. However, in some cases at 50 mA cm�2

the current cannot be lowered any further, and so the fuel cellcannot run at all below those cutoff flow rates. The 2 M Nafion 115case in particular shows a much higher cutoff flow rate than theblend membranes. This low current scenario is where the blendmembranes have the greatest advantage over the Nafionmembranes.

In all of these cases, the power demand of the compressor andthe blower is a substantial fraction of the power output of the fuelcell stack. At the optimum flow rate in the 300 mA cm�2 scenario,the power output of the 100 cell stack is about 40 W, and thepower consumption of the compressor and the blower is about4 W, or 10% of the total power output. That fraction only getslarger as the flow rate increases because the performance gainsbecome smaller and smaller while the power consumption of thesystem components continues to increase linearly. The oxygenreduction reaction is first-order, so there is a noticeable perfor-mance penalty to be paid for operating close to the cutoff flowrate. However, until the flow rate is within about 2 � 10�6 mol s�1

of the cutoff flow rate, that penalty is more than compensated forby the lower power consumption of the system components.Therefore, as with the methanol flow rate, it is important to keepthe oxygen flow rate as low as possible while not allowing theoxygen pressure to fall close to zero such that substantial perfor-mance losses are incurred.

B. Bennett et al. / Journal of Power Sources 218 (2012) 268e279278

5. Conclusion

In this work, a simple engineering model is developed thatdirectly relates the materials properties of the direct methanol fuelcell e including diffusion coefficients, membrane thickness, andproton conductivity e and operating parameters e most notablythemethanol and oxygen flow ratese to the stack performance andthe size of system components such as the methanol storage tankand the air blower. The model is created by deriving analyticsolutions for the methanol and oxygen concentrations across thethickness of the fuel cell and by applying Tafel kinetics to determinethe average cell current density for a fixed cell voltage. The benefitsof the model include a run time of less than a minute for a singlecurrentevoltage curve in a MATLAB simulation environment andthe ability to handle a wide range of parameter spaces.

The model was validated using experimental currentevoltagedata taken at 65 �C and 90 �C with air oxygen cathode feeds. Itwas then applied to forecast the benefits of using new low-crossover membranes and was compared to published data fromZhu et al. Finally, themodel was used in a case study to illustrate theeffect of changing the methanol and oxygen flow rates on fuel cellperformance and the net power output of the fuel cell stack. Thetradeoff that exists between stack performance, the methanolstorage tank, and the power consumption of the air blower, aircompressor, and other components dictates that there exists anoptimum set of operating conditions for any portable DMFC withsize constraints.

Acknowledgments

Financial support by the Office of Naval ResearchMURI grant No.N00014-07-1-0758 is gratefully acknowledged.

Glossary

Roman SymbolsA superficial area of the anode, cm2

Achan cross-sectional area of the flow channel, cm2

canode methanol concentration at the anode catalyst layer,mol cm�3

cfeed methanol concentration in theanode feedstream,mol cm�3

c0feed methanol feed concentration at channel inlet (z ¼ 0),mol cm�3

cH2O concentration of liquid water in the membrane, mol cm�3

Di � j diffusion coefficient of component i in component, cm2 s�1

Deffi�j effective diffusion coefficient of component i in

component j, cm2 s�1

Dmem diffusioncoefficientofmethanol in themembrane, cm2 s�1

DO2;cl Deff of oxygen in the cathode catalyst layer, cm2 s�1

dh hydraulic diameter of the flow channel, cmEcomp air compressor efficiency, dimensionlessf friction coefficient of air in the flow channel,

dimensionlessh size of each iteration step in the flow channels, cmi current through the external circuit, A cm�2

ix crossover current, A cm�2

i0a anodic reference exchange current density, A cm�2

i0c cathodic reference exchange current density, A cm�2

ilim,a limiting current in the anode backing layer, A cm�2

i0lim;a backing layer limiting current at anode channel inlet,C cm�2 s�1

k cross-sectional area to flow in cathode flow channel, cm2

Lback,a thickness of the anode backing layer, cmLback,c thickness of the cathode backing layer, cm

Lmem thickness of the membrane, cmLcl thickness of the catalyst layers, cmLchan length of the flow channels, cmn total number of iterations down the flow channel,

dimensionlessNji flow rate of component i through region j, mol s�1

Ni,f flow rate of component i in the cathode flow channel,mol s�1

NinputMeOH methanol flow rate into the anode flow channel inlet,

mol s�1

NinputO2

oxygen flow rate into the cathode flow channel inlet,mol s�1

Pi absolute pressure of component i, N m�2

Pi,f pressure of component i in the bulk flow channel, N m�2

PO2;s oxygen pressure at the cathode backing layer surface,N m�2

PO2;c oxygen pressure at the cathode catalyst layer surface,N m�2

PO2;ref reference oxygen pressure, N m�2

P0O2;finput oxygen pressure in the cathode flow channel, Nm�2

PH2O;sat saturated vapor pressure of water, N m�2

PbackO2oxygen pressure in the cathode backing layer, N m�2

PconvO2oxygen pressure in convective flow channel regions,N m�2

Ptotal total pressure in the cathode backing layer N m�2

Powercomp power consumed by the air compressor, WPowerblower power consumed by the air blower, WRe Reynolds number of cathode air flow, dimensionlesssi stoichiometric ratio of methanol or oxygen,

dimensionlessShF Sherwood number for a square channel, dimensionlessSFP specific fan power, W s m�3

t time of fuel cell operation, sUq standard cell potential, VVanode anodic activation overpotential, VVcathode cathodic activation overpotential, VVcell total cell potential, VV feed molar volume of the methanol and water feed stream,

cm3 mol�1

vMeOH velocity of methanol in the anode flow channel, cm s�1

vair,f velocity of air in the cathode flow channel, cm s�1

vO2 ;f velocity of oxygen in the cathode flow channel, cm s�1

wsep flow channel separation width, cmyi mole fraction of component i, dimensionlessz distance down the flow channel, cmz0 normalized distance down the flow channel,

dimensionless

Greek Symbolsaa anodic transfer coefficient, dimensionlessac cathodic transfer coefficient, dimensionlessεa porosity of the anode backing layer, dimensionlessεc porosity of the cathode backing layer, dimensionlessεcl porosity of the catalyst layers, dimensionlesskmem proton conductivity of the membrane, S cm�1

kcla proton conductivity of the anode catalyst layer, S cm�1

kclc proton conductivity of the cathode catalyst layer, S cm�1

mi dynamic viscosity of component i, Pa s�1

h effectiveness factor of cathode catalyst layer,dimensionless

fNafion Nafion volume fraction in the catalyst layer,dimensionless

Fi sum of dimensionless ratios (see pp.153e155 in Ref. [23]),dimensionless

B. Bennett et al. / Journal of Power Sources 218 (2012) 268e279 279

j ratio of i0lim to the current through the external circuit,dimensionless

rH2O density of liquid water, g cm�3

q Thiele modulus, dimensionlessn ratio of limiting current in backing layer and membrane,

dimensionlessxH2O electroosmotic drag coefficient of water, cm3 mol�1

x0MeOH electroosmotic drag coefficient of methanol, cm3 mol�1

xMeOH electroosmotic drag factor of methanol, dimensionlessg heat capacity ratio (gp/gv) dimensionless

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