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Journal of Power Sources 246 (2014) 239e252

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Journal of Power Sources

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

Modeling of the anode of a liquid-feed DMFC: Inhomogeneouscompression effects and two-phase transport phenomena

Pablo A. García-Salaberri, Marcos Vera*, Immaculada IglesiasDept. de Ingeniería Térmica y de Fluidos, Universidad Carlos III de Madrid, Avda. de la Universidad, 30, 28911 Leganés, Spain

h i g h l i g h t s

� An isothermal two-phase across-the-channel model for liquid-feed DMFCs is presented.� The model considers inhomogeneous compression of the GDL (non-uniform porosity).� Effective anisotropic properties of the GDL are evaluated from empirical data.� The hydrophobic Leverett J-function approach provides physically inconsistent results.� Empirical pc � s data reflecting the mixed-wettability of GDLs give much better predictions.

a r t i c l e i n f o

Article history:Received 21 February 2013Received in revised form19 June 2013Accepted 25 June 2013Available online 18 July 2013

Keywords:DMFC modelingAnodePorous layerEffective propertiesAssembly compressionCapillary transport

* Corresponding author. Tel.: þ34 91 624 9987; faxE-mail addresses: marcos.vera@uc3m.es, m_vera_cURL: http://fluidos.uc3m.es/people/mvcoello

0378-7753/$ e see front matter � 2013 Elsevier B.V.http://dx.doi.org/10.1016/j.jpowsour.2013.06.166

a b s t r a c t

An isothermal two-phase 2D/1D across-the-channel model for the anode of a liquid-feed Direct Meth-anol Fuel Cell (DMFC) is presented. The model takes into account the effects of the inhomogeneousassembly compression of the Gas Diffusion Layer (GDL), including the spatial variations of porosity,diffusivity, permeability, capillary pressure, and electrical conductivity. The effective anisotropic prop-erties of the GDL are evaluated from empirical data reported in the literature corresponding to Toraycarbon paper TGP-H series. Multiphase transport is modeled according to the classical theory of porousmedia (two-fluid model), considering the effect of non-equilibrium evaporation and condensation ofmethanol and water. The numerical results evidence that the hydrophobic Leverett J-function approachis physically inconsistent to describe capillary transport in the anode of a DMFC when assemblycompression effects are considered. In contrast, more realistic results are obtained when GDL-specificcapillary pressure curves reflecting the mixed-wettability characteristics of GDLs are taken into ac-count. The gas coverage factor at the GDL/channel interface also exhibits a strong influence on the gas-void fraction distribution in the GDL, which in turn depends on the relative importance between thecapillary resistance induced by the inhomogeneous compression, Rc ðfvpc=vεÞ, and the capillary diffu-sivity, Dc ðfvpc=vsÞ.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

The development of liquid-feed Direct Methanol Fuel Cells(DMFCs), a variant of hydrogen PEMFC technology inwhich the fuelis an aqueous methanol solution, has accelerated over the lastyears. Although the power densities that can be reached withPEMFCs are higher, liquid-feed DMFCs present two major advan-tages compared with PEMFCs: the easier delivery and storageof liquid methanol solutions, and the higher volumetric energy

: þ34 91 624 9430.@hotmail.com (M. Vera).

All rights reserved.

density of liquid methanol [1e3]. These characteristics makeDMFCs a promising candidate as a power source for portableelectronic applications, including cell phones, laptop computers,military equipment, etc. [4,5].

However, widespread commercialization of DMFCs is still hin-dered by several technological problems, such as the slow kineticsof the Methanol Oxidation Reaction (MOR) at the anode CatalystLayer (CL) and the cathode mixed potential associated with theoxidation of methanol that crosses over the polymer membranefrom anode to cathode [6]. Furthermore, the interplay betweenmass/charge/heat transport, electrochemical kinetics, inhomoge-neous compression effects, interfacial contact resistances, and two-phase transport phenomena makes it difficult to achieve optimumdesign and operating conditions [7]. Thorough study of these

P.A. García-Salaberri et al. / Journal of Power Sources 246 (2014) 239e252240

complex and interrelated phenomena is therefore necessary fromboth the experimental and the modeling points of view [8].

One key element affecting fuel cell performance is the GasDiffusion Layer (GDL). It provides several functions: a pathway forreactant access and excess product removal to/from the CLs, highelectrical and thermal conductivity, and adequate mechanical sup-port. In order to fulfill these requirements, GDLs are typically madeof porous carbon paper or woven cloth [9]. Both media are charac-terized by exhibiting significant structural anisotropy due to thepreferential orientation of carbon fibers, which typically results indifferent transport properties along the in-plane and through-planedirections [10]. In addition, the high porosity of these materialsprovides to the GDL a characteristic soft and flexible structure,susceptible of large deformations when subjected to compression[11]. This introduces important variations in the effective transportproperties between the regions under the channel and under the ribwhen the fuel cell is assembled [8,12,e15].

Even though the fundamental role of the GDL has long beenrecognized by the fuel cell community [16e18], most models pre-sented in the literature to date assume homogenous and isotropicGDL properties. However, this situation has changed during the lastfew years, and a larger effort is now devoted to the development ofmore comprehensive models reflecting the real characteristics andoperating conditions of the GDLs [19]. Thus, different aspects suchas the inhomogeneities caused by the assembly compression andmanufacturing process, as well as the inherent anisotropy of theGDL, have been recently incorporated into numerical models. Inaddition, numerous experimental and numerical works exploringthe effective transport properties of GDLs are now available in theliterature [20]. This trend should be reinforced in the near future.

The influence of GDL inhomogeneities and anisotropic proper-ties on PEMFC operation has been explored in several modelingworks [21e25]. However, these phenomena have been traditionallyignored in DMFC modeling studies (see, e.g., [5,26,27] and refer-ences therein), and only a few works can be found in the openliterature.Möst et al. [28] presented an analysis of the diffusivemasstransport in the anode GDL of a DMFC. In this work, the anisotropicdrydiffusivities of differentGDLs at various compression ratiosweremeasured, and subsequently employed in Monte Carlo simulationstaking into account inhomogeneous compression effects. Thelimiting current densities predicted by this single-phase modelwere in qualitative agreement with those obtained experimentally,but a systematic overestimation was observed. Miao et al. [29]developed a two-phase 2D across-the-channel model of a DMFCto investigate the effects of GDL anisotropic properties, inhomoge-neous compression, and electrical and thermal contact resistances.These phenomena were taken into account by considering theempirical data reported by Himanen et al. [8,13e15] as a function ofthe in-plane coordinate x. They concluded that the anisotropy of theGDL has a significant effect on the distribution of species concen-tration, overpotential, current density, and temperature; eventhough the isotropic and anisotropic models lead to very similarpolarization curves. They also observed that the electrical contactresistance at the CL interface and the GDL compression both play animportant role in determining cell performance. In a subsequentwork, Miao et al. [30] presented an upgraded version of this model,in which anisotropic heat transfer coefficients were used to capturea more realistic heat transport mechanism. This study was exclu-sively focused on the influence of GDL anisotropic properties,inhomogeneous compression and contact resistances on heat gen-eration and transport processes. They found that these phenomenahave a strong impact on heat transfer processes in DMFCs. Using amodel similar to that presented in Ref. [29], He et al. [31] investi-gated the behavior of water transport through the MEA. The nu-merical results showed that both the channel-rib pattern and the

deformation of the GDL can cause an uneven distribution of thewater-crossover flux along the in-plane direction. In addition, theyconcluded that both the contact angle and the permeability of thecathode GDL can significantly influence the water-crossover flux.

The effect of assembly compression on the performance of bothactive and passive DMFCs has been also stressed in a few experi-mental works [32,33]. Even though the number of studies related toDMFCs is notably lower compared to those available for PEMFCs[32,34e37], these results are far enough to visualize the importanceof assembly pressure on DMFC operation. Nevertheless, it would beinteresting to explore a wider range of operating conditions andMEA configurations (particularly different type and thicknesses ofthe GDL) in future experimental work.

The above literature review shows the large impact that GDLinhomogeneous compression and anisotropic properties have onmass/charge/heat transport in DMFCs, thereby affecting fuel cellperformance and lifetime. These phenomena should be investi-gated by physically sound mathematical models including detaileddescriptions of the transport processes that occur in the GDL [27].

The present work presents an isothermal two-phase 2D/1Dacross-the-channel model for the anode of a liquid-feed DMFC. Themodel takes into account the spatial variations induced by theinhomogeneous assembly compression of the GDL, as well as theinherent anisotropic properties of this key element of the cell. Theassembly process is simulated by a novel Finite Element Method(FEM) model, which fully incorporates the nonlinear orthotropicmechanical properties of the GDL (Toray� carbon paper TGP-H se-ries) [11]. The resulting porosity distribution is then used to eval-uate the effective transport properties of the GDL, i.e., massdiffusivity, permeability, capillary pressure and electrical conduc-tivity, through empirical data reported in the literature. Conceivedas an extension of our previous research activity on DMFCs [6], thepresent model brings new light on the potential influence of GDLinhomogeneous compression on capillary transport processes andgas saturation distribution in the anode of liquid-feed DMFCs. In itspresent form, the 2D/1D across-the-channel model constitutes afirst step towards the development of a full 3D-model for DMFCsthat properly takes into account the influence of inhomogeneouscompression effects on multiphase transport phenomena.

2. Numerical model

The 2D/1D DMFC model presented in this work is based on the3D/1D single-phase model previously proposed by Vera [6]. The 3Dgeometry of the anode has been reduced to a 2D across-the-channel GDL section, while the single-phase 1D model,comprising the catalyst layers, the membrane, and the cathodeGDL, presents only minor changes with respect to that presented inRef. [6]. As major improvements, the upgraded 2D anode modeltakes into account multiphase transport phenomena according tothe classical theory of porous media (two-fluid model), retainingthe effect of non-equilibrium phase change of methanol/water. Italso includes the effects of the inhomogeneous compression of theGDL, incorporated through the non-uniform porosity distributionobtained from a previous analysis of the GDL compression process[11]. Specifically, the model makes extensive use of empirical cor-relations of anisotropic carbon paper-based GDLs to evaluate theresulting non-uniform diffusivity, permeability, capillary pressureand electrical conductivity fields induced by the assembly pressure.Finally, the GDL electronic potential is now solved, with the pos-sibility of including the effect of non-uniform contact resistances atthe GDL/Rib and GDL/CL interfaces as boundary conditions. Themain contribution of this work is to explore the potential impact ofGDL inhomogeneous assembly compression on two-phase capillarytransport in the anode of a DMFC. The mathematical formulation of

Fig. 1. Schematic of the modeling domains covered by the 2D and 1D models showingthe coordinate system, and the notation used for the anode rib-half width, wrib, thedistance between the anode rib and channel symmetry planes, wagdl, the fillet radius ofthe anode rib, rrib, and the thickness of the different layers of the MEA.

P.A. García-Salaberri et al. / Journal of Power Sources 246 (2014) 239e252 241

the model can be found in the Appendix, along with all physico-chemical properties and parameters (see Table A.1). The main as-sumptions, geometry, constitutive relations (effective propertiesand phase change source terms), and the main steps followed forthe numerical implementation of the model are presented below.

2.1. Assumptions

The present model is based on several simplifying hypothesesthat define its limits of validity. The most relevant assumptionsspecific to the model presented herein are as follows: i) masstransport limitations and flooding phenomena at the cathode aresmall; ii) the temperature is constant throughout the cell; iii)contact resistances are negligible for the GDL compression ratioconsidered in this work (CR ¼ 20%); iv) the porosity of the un-compressed GDL (and CL) is homogenous (i.e., the through-planevariations of the porosity of the GDL inherent to themanufacturing process and the PTFE treatment [39e42] are nottaken into account); v) the assembly compression does not alter thephysicochemical properties of the catalyst layer and the polymermembrane; vi) the gas saturation in the anode catalyst layer doesnot reduce the catalyst active surface area; vii) carbon dioxide isonly present in the gas phase; viii) the saturation profile at the GDL/CL interface is continuous. The remaining assumptions made in thiswork are common to other DMFC modeling studies and can befound elsewhere [3,6].

Notice that assumption viii) is physically inconsistent accordingto the theory of porous media [43]. It is capillary pressure ratherthan saturation the physical variable that must be continuous at theGDL/CL interface. However, the high uncertainty found in theexperimentally reported capillary pressure curves of the CL,particularly the high sensitivity to the presence of cracks [44], madeus choose a more conservative approach. To date, all the modelingpublications that assume a continuous pressure profile at the GDL/CL interface make use of the Leverett J-function approach. Theavailability of analytical expressions for the pc�s curves of the GDLand CL makes then easier to impose the physically sound conditionof continuous capillary pressure at the GDL/CL interface. However,as will be discussed in Section 2.6, the applicability of the Leverettapproach has been challenged by numerous experimental studies,and should also be reconsidered in future fuel cell modeling efforts.

2.2. Geometry

We shall assume a parallel channel configuration for the anodeflow-field. Accordingly, the domain under study is restricted to theregion between the mid-plane of one channel and that of theneighboring rib to reduce computational cost. Symmetry boundaryconditions are considered at both boundaries. As depicted in Fig. 1,the computational domain includes the five layers composing theMEA, i.e., the anode gas diffusion layer (agdl), the anode catalystlayer (acl), the polymer electrolyte membrane (mem), the cathodecatalyst layer (ccl), and the cathode gas diffusion layer (cgdl). Thegeometry of the anode 2D model is defined by four geometricalparameters, namely the rib half-width,wrib, the uncompressed GDLthickness, dagdl, the distance between the rib and channel sym-metry planes, wagdl, and the fillet radius of the rib, rrib. The geom-etry of the 1D model (catalyst layers, membrane and cathode GDL)is simply defined by the thickness of the different layers; dacl anddccl for the anode and cathode catalyst layers, respectively, dmem forthe polymer membrane, and dcgdl for the cathode GDL. The value ofthe geometric dimensions has been kept constant throughout thework; wagdl ¼ 1 mm, wrib ¼ 0.5 mm, rrib ¼ 40 mm,dagdl ¼ dcgdl ¼ 190 mm (Toray� carbon paper TGP-H-060),dacl ¼ dccl ¼ 30 mm, dmem ¼ 178 mm (Nafion� 117).

2.3. Effective diffusivity

Following the approach of Nam and Kaviany [45], the effectivediffusivity of species i in the GDL under multiphase flow conditions,

Deff ;weti;j , where j ¼ l/g denotes liquid/gas phase, respectively, is

correlated with porosity (dry diffusivity) and saturation (relativediffusivity) through the normalized functions f(ε) and gj(s), definedas

Deff ;weti;j

Dbulki;j

¼Deff ;dryi;j

Dbulki;j|fflfflfflffl{zfflfflfflffl}f ðεÞ

Deff ;weti;j

Deff ;dryi;j|fflfflfflffl{zfflfflfflffl}gjðεÞ

¼ f ðεÞgjðsÞ (1)

where Dbulki;j is the bulk diffusivity of species i in phase j.

To account for the influence of compression on the dry, orsingle-phase, diffusivity, Deff ;dry

i;j , we have considered the experi-mental data reported by Flückiger et al. [46,47] and Möst et al. [28].Both works show good agreement between them and with otherexperimental [48,49], numerical [50e52], and analytical [53]studies presented in the literature. A good fit to the reported in-plane (ip) and through-plane (tp) experimental values for carbonpaper TGP-H-060 with different PTFE contents (0e40 wt.%) overthe porosity range 0.3<ε < 0.8 is provided by the following expo-nential functions [11]:

f ipðεÞ ¼Deff ;dry;ipi;j

Dbulki;j

¼ 2:92� 10�2expð3:80εÞ (2a)

f tpðεÞ ¼Deff ;dry;tpi;j

Dbulki;j

¼ 6:51� 10�3expð5:02εÞ (2b)

Notice that due to the in-plane arrangement of the fibers thepores are preferentially oriented in this direction, resulting in a

higher effective in-plane dry diffusivity, Deff ;dry;ipi;j > Deff ;dry;tp

i;j .

Indeed, the degree of anisotropy is about 2, and increases as theGDL is compressed due to the realignment of fibers [11,47].

The influence of liquid/gas saturation on the effective diffusivity,given by the relative diffusivity, has been modeled according to the

P.A. García-Salaberri et al. / Journal of Power Sources 246 (2014) 239e252242

isotropic Bruggeman’s effective medium theory [54], as typicallyemployed in fuel cell modeling studies:

gip=tpl ðsÞ ¼Deff ;wet;ip=tpi;j

Deff ;dry;ip=tpi;j

¼ s1:5 (3a)

gip=tpg ðsÞ ¼Deff ;wet;ip=tpi;j

Deff ;dry;ip=tpi;j

¼ ð1� sÞ1:5 (3b)

Although there are several experimental [55,56] and numerical[45,52,57,58] works related to the impact of water saturation on theeffective diffusivity of the GDL in PEMFCs, there is a lack ofknowledge on the effects of the two-phase flow conditions thatprevail in the anode of a DMFC. In general, a large research effort isstill needed to incorporate realistic descriptions reflecting theparticular multiphase characteristics of GDLs. Particularly, the in-fluence of GDL anisotropy and compression on multiphase prop-erties should be further investigated in order to improve thepredictive capabilities of numerical models. For the time being, theBruggeman correction has been considered a good starting point inthis work.

2.4. Absolute and relative permeability

The impact of compression on the absolute permeability istaken into account according to the experimental data reported byFlückiger [47] corresponding to carbon paper TGP-H-060 with 0e20 wt.% PTFE over the porosity range 0.6<ε < 0.8. The values re-ported in Ref. [47] are in the range of other empirical data presentedin the literature [59e61] and the analytical model recently pro-posed by Tamayol et al. [62,63]. The analytical expressions obtainedby fitting the reported in-plane and through-plane experimentaldata to exponential functions are as follows:

K ipðεÞ ¼ 2:52� 10�16 expð14:65εÞ �m2� (4a)

KtpðεÞ ¼ 2:30� 10�16 expð12:51εÞ �m2� (4b)

It is worth noting that the degree of anisotropy of the absolutepermeability is in general higher than that observed for the drydiffusivity. Thus, the ratio between the values along the in-planeand through-plane direction is between 2 and 5 in the porosityrange measured by Flückiger [47]. Interestingly, the degree ofanisotropy of the absolute permeability decreases with increasingcompression as opposite to the dry diffusivity.

To account for the influence of liquid/gas saturation, we haveconsidered the third-order empirical correlation for the relativepermeability originally developed for well-sorted unconsolidatedsand [64]:

kip=tprg ðsÞ ¼ ð1� sÞ3 (5a)

kip=tprl ðsÞ ¼ s3 (5b)

Although this correlation is usually adopted in fuel cell modelingstudies, it should be noted that its applicability to fibrous GDLs hasbeen debated many times. Nowadays, several GDL-specific experi-mental data can be found in the literature [61,65e67]. However, themeasurements are subject to high uncertainty and the liquidsaturation levels explored in them are quite small so as to reach afirm conclusion for the conditions typically present in the anode ofa DMFC. As noted above for the relative diffusivity, experimental

and numerical work is still necessary to incorporate more realisticcharacterizations into continuum-based numerical models.

2.5. Electrical conductivity

The effective electrical conductivity has been correlated to theexperimental data reported by Flückiger [47] and Reum [68] forcarbon paper TGP-H-060 with 20 wt.% PTFE over the porosity range0.6<ε < 0.8:

seff ;ipðεÞ ¼ 48221� 46729ε ½S=m� (6a)

seff ;tpðεÞ ¼ 6582� 7229ε ½S=m� (6b)

The degree of anisotropy of the electrical conductivity is evenhigher than that observed for the dry diffusivity and the absolutepermeability. The in-plane values are about one order of magnitudehigher than the values in the through-plane direction. As oppositeto mass transport properties, the electrical conductivity increasesas the GDL is compressed due to the larger number of contactpoints among fibers. In particular, the in-plane conductivity in-creases linearly with decreasing thickness so that the in-planeresistance remains almost constant [69].

2.6. Capillary pressureesaturation relationship

One key property affecting two-phase transport phenomenawhich deserves further attention is the capillary pressureesatura-tion relationship, pc�s. Most of the multiphase models developedfor PEM fuel cells, either PEMFCs or DMFCs, usually adopt the semi-empirical correlation originally proposed by Leverett [70], andsubsequently modified by Rose and Bruce [71] with the intent tocapture wettability effects:

pcðs; εÞ ¼ pg � pl ¼ s

ffiffiffiffiffiffiffiffiffiffiε

KðεÞr

cosðqcÞJðsÞ (7)

where s is the surface tension of the immiscible fluid pair, ε and Kare the porosity and absolute permeability of the porous media,respectively, qc is the contact angle between both phases, and J(s) isthe so-called Leverett J-function. Note that the capillary pressurehas been defined in this work as pc ¼ pg�pl. Since the seminal workof Wang et al. [72], a common practice in the modeling communityis to use as Leverett J-function the polynomial fit to Leverett’sexperimental data proposed by Udell [73]:

JðsÞ ¼�1:4ð1� sÞ � 2:1ð1� sÞ2 þ 1:3ð1� sÞ3 qc < 90�1:4s� 2:1s2 þ 1:3s3 qc > 90�

(8)

The contact angle in this expression is assumed to be around100� to account for the hydrophobic characteristics of PTFE-treatedGDLs. As a result, the capillary pressure is negative, i.e., pg is lessthan pl, over the entire range of liquid saturation.

However, the applicability of thismodeling approach toGDLshasbeen questioned many times [74e81], and may be especially erro-neous under the two-phase flow conditions existing in the anode ofa DMFC. GDL-specific experimental data for thewatereair fluid pairhave shown that gas diffusion layers present mixed-wettabilitycharacteristics and display permanent hysteresis between imbibi-tion (water displaces air) and drainage (air displaces water) pro-cesses. Based on these findings, it should be expected that capillarytransport of gas-phase bubbles in the anode of a DMFC is betterdescribed by the pc�s curve associated with the drainage process,i.e., gas phase displaces liquid phase, while capillary transport ofliquid-phase droplets in the cathode of a DMFC or a PEMFC is

P.A. García-Salaberri et al. / Journal of Power Sources 246 (2014) 239e252 243

represented more adequately by the pc�s curve associated with theimbibition process, i.e., liquid phase displaces gas phase.

In comparison with the hydrophobic Leverett J-functionapproach (qc > 90�), GDL-specific experimental data presentobvious differences in shape and magnitude. However, a morefundamental difference arises in the sign of the capillary pressure.Whereas experimental imbibition curves agree qualitatively withthe hydrophobic Leverett J-function and show negative capillarypressures in almost the entire range of liquid saturation, experi-mental drainage curves do not reflect this behavior, and the capil-lary pressure is positive (see, e.g., [75] and references therein).Moreover, the influence of compression is exactly the opposite.While the hydrophobic Leverett J-function approach predictsincreasing negative capillary pressures as the GDL is compressed,i.e., for lower porosity, GDL-specific drainage data are displaced tohigher, i.e., more positive, capillary pressures. This contradictorysituation can be observed in Fig. 2, where the pc�s curves predictedby the hydrophobic Leverett J-function approach (qc ¼ 100�) andGDL-specific drainage data [74] are shown for different GDL po-rosities. The opposite behavior of both approaches suggests that theapplicability of fully hydrophobic relationships may be physicallyincorrect to describe capillary transport in the anode of a DMFC. Adetailed analysis of this topic is presented below in Section 3.

The experimental drainage data used in this work to describethe capillary pressureesaturation relationship correspond to themeasurements reported by Gostick et al. [74] for TGP-H-120(10 wt.% PTFE) using the watereair fluid pair. In Ref. [74] the pc�scurves associated with an uncompressed GDL and a 30% com-pressed GDL were presented. Even though the measurements werenot exclusively adapted to the two-phase system present in theanode of a DMFC, these data can be considered as a good startingpoint for the exploratory purpose of this work. Future work shouldbe carried out to obtain a more detailed description of capillarypressure as a function of GDL compression. Also of interest wouldbe to investigate the influence of additional variables that may leadto Gibbs-Marangoni effects, such as the concentration of methanolor the temperature.

The van Genuchten-type equation proposed by Gostick et al.[74] to describe the experimental drainage curves is as follows:

pcðsÞ ¼ pg � pl ¼ pc;b

��s� sr1� sr

�1=n� 11=m

� pref (9)

Fig. 2. Capillary pressureesaturation relationship, pc�s, predicted by the hydrophobicLeverett J-function approach (qc ¼ 100�) and GDL-specific drainage data [74]. Thedifferent curves correspond to different values of porosity, ε. The dependence of thecapillary pressure on porosity was obtained following the procedure presented in thissection.

The value of the different parameters appearing in this equationcan be found in Ref. [74].

To account for the effect of GDL compression on capillarypressurewe performed a linear interpolation between the two pc�scurves reported by Gostick et al. [74]. The inverse of the charac-teristic pore radius, r�1

p , was considered as the representative var-iable to perform the interpolation, i.e.,

pc�s;r�1

p

�¼ bpuc ðsÞþð1�bÞpccðsÞ; b¼

r�1p �

�r�1p

�c�r�1p

�u��r�1p

�c (10)

where puc ðsÞ and pccðsÞ are the pc�s curves associated with the un-compressed and 30% compressed GDLs, respectively.

According to Leverett [70], the characteristic pore radius wascalculated as:

rpðεÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKavgðεÞ

ε

r(11)

where KavgðεÞ ¼ ½K ipðεÞ þ KtpðεÞ�=2 is the average value of the ab-solute permeability. Note that, as it should be expected, with theexpressions for Kip (ε) and Ktp (ε) given in Eq. (4) the characteristicpore radius decreases with decreasing porosity (that is, withincreasing GDL compression). In the simulations performed withthe Leverett J-function approach, Eqs. (7) and (8), the inverse of thecharacteristic pore radius, r�1

p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiε=KðεÞp

, was calculated in thesame way.

To compute the characteristic pore radius of the compressedPTFE-treated GDL, rcp, the porosity was calculated based on thethickness compression ratio, CR, using the following expression:

εc ¼ ε

u � CR1� CR

; CR ¼ Dd

du(12)

derived by assuming that the change of thickness experienced bythe GDL is due solely to the decrease of pore volume. Here, Dd is thereduction of the GDL thickness, and du and ε

u are the uncompressedGDL thickness and porosity, respectively. These values were re-ported by Gostick et al. [74]. The characteristic pore radius of theuncompressed GDL, rup, was directly obtained from the value of theuncompressed GDL porosity given in Ref. [74].

2.7. Phase change source terms

Most of the published models for DMFCs which consider non-equilibrium phase change phenomena assume different de-scriptions for the phase change rate of methanol and water (see,e.g., [3,29e31,82]). However, except for the fact that both expres-sions were developed by different authors, no arguments can befound in the literature to justify these differences.

Specifically, the evaporation/condensation rate of methanol isoften modeled according to the expression originally proposed byDivisek et al. [83]:

_Rm ¼ Alghlgsð1� sÞ�Csatm � Cmv

�(13)

where the phase change rate is proportional to the product of theliquid and the gas saturation, s and (1�s), respectively, and thedriving-force is the difference between the concentration ofmethanol vapor, Cmv, and the saturation concentration of methanol,Csatm . Alg and hlg denote the specific interfacial area and the inter-

facial transfer coefficient between the liquid and the gas phase inthe porous media, respectively. Both parameters are assumedconstant in numerical models.

P.A. García-Salaberri et al. / Journal of Power Sources 246 (2014) 239e252244

On the other hand, since the original work of Murgia et al. [84],the phase change rate of water is usually modeled using a formu-lation similar to that employed in PEMFC modeling studies [85]:

_Rw ¼8<:

kevpεsrlMw

psatw � pwv

�pwv < psatw

kconεð1�sÞXwv

RT

psatw � pwv

�pwv > psatw

(14)

where ε is the porosity, rl is the density of liquid water, Mw is themolecular weight of water, Xwv is the mole fraction of water vaporin the gas phase, T is the operating temperature, and R the uni-versal gas constant. The driving-force is the difference betweenthe partial pressure of water vapor, pwv, and the saturation pres-sure of water, psatw . In the above expressions, kevp and kcon denotethe evaporation and condensation rate constants, respectively. Thebasis of this formulation is physically more meaningful. The ex-pressions for the evaporation and condensation processes aredifferentiated, so that the interfacial mass-transfer associated witheach phenomenon is proportional to the amount of species perunit volume governing that process, i.e., liquid-phase species forthe evaporation process and gas-phase species for the condensa-tion process.

In the present work, the phase change rate of methanol andwater has been formulated under a common modeling framework,following the same physical principles considered in PEMFCmodeling studies. Thus, the expression proposed for the phasechange rate of methanol (i ¼ m) and water (i ¼ w) is as follows:

_Ri ¼8<:

kevpεsXilrlMw

psati � piv

�piv < psati

kconεð1�sÞXiv

RT

psati � piv

�piv > psati

(15)

where Xil is the mole fraction of methanol/water in the liquid phaseand Xiv is the mole fraction of methanol/water vapor in the gasphase.

In a first approximation, we have assumed that the evaporationand condensation rate constants of both methanol and water areequal. This hypothesis results from the assumption that the un-derlying physical characteristics of the heat and mass transferphenomena governing the phase change processes are similar forboth species. The only difference arising between the evaporationand condensation rate of methanol and water is attributed to thedifferent amount of each species present in the porous media. Forthe diluted methanol solutions typically employed in liquid-feedDMFCs, this modeling approach indeed reflects that water iscloser to thermodynamic equilibrium conditions, as it should beexpected [86]. The values of the evaporation and condensation rateconstants have been estimated from those previously presented inthe literature and are listed in Table A.1. It should be pointed outthat this modeling approach is only an alternative to previous for-mulations employed in numerical models to describe non-equilibrium phase change phenomena. As considered in othermodeling studies [1,86,87], the assumption that both methanol andwater are in thermodynamic equilibrium corresponds to the limitkevp, kcon / N.

2.8. Numerical implementation

As a first step in our multiphysics modeling approach, theinhomogeneous assembly compression of the GDL was simulatedusing a novel FEM model [11] implemented in the commercialcode ABAQUS�/Standard [88]. The assembly process was modeledby imposing a specified GDL compression ratio, CR, expressed as apercentage of its initial thickness. For further details, the reader is

referred to Ref. [11]. Then, the deformed mesh of the GDL and theassociated porosity distribution were imported into the commer-cial finite-volume-based code ANSYS� FLUENT 13.0 [89]. This CFDcode was used to solve the different conservation equations,complemented with numerous User Defined Functions (UDFs).The UDFs were employed to customize different aspects of themodel, including governing equations, boundary conditions,source terms, effective properties, model parameters, etc. Furtherdetails about the numerical procedure followed to couple the 2D/1D model can be found in Ref. [6]. The number of quadrilateralelements/cells employed in the simulations performed in bothABAQUS� and ANSYS� FLUENT was 7600, corresponding to anelement/cell size of 5 mm. Grid refinement studies confirmed thatthis level of refinement was appropriate to accurately describe theinhomogeneities induced by the assembly compression withoutexcessively increasing the computational cost.

2.9. Numerical validation

Before proceeding further, we checked the ability of the modelto reproduce the numerical results found in the literature. In a realfuel cell both reactant/product concentrations and liquid/gassaturation evolve along the channel, so that polarization curvesobtained with our 2D across-the-channel model could not bestrictly compared with any experimentally observed IeV curve.This is also the case in other 2D across-the-channel models, such asthe one proposed by Yang and Zhao [3], which will be used here forcomparative purposes. To mimic the modeling assumptions ofRef. [3], we considered an undeformed GDL with homogenousporosity and isotropic properties with a Bruggeman-type correc-tion for the dry diffusivity, i.e., f ip=tpl=g ðεÞ ¼ ε

1:5. In addition, theLeverett J-function approach, Eqs. (7) and (8), with qc ¼ 100� wasused to describe the pc�s relationship, and Eqs. (13) and (14) wereemployed to model the phase change rates of methanol and water.The value of the geometrical dimensions and operating conditionswere also adapted to that employed by Yang and Zhao [3]. Despiteall these modifications, the resulting baseline model still presentssome fundamental differences with the model by Yang and Zhao[3], such as the formulation of the MOR electrochemical kinetics,the consideration of a global electrical contact resistance or thedescription of mass transport processes in the cathode GDL.Nevertheless, the purpose of this validation was not to establish aone-to-one comparison, but to check the ability of our model toprovide similar results to those previously reported in the litera-ture. The spatial distributions of liquid methanol concentration,Cml, methanol vapor concentration, Cmv, and gas-void fraction, 1�s,predicted by the present model are shown in Fig. 3. As compared tothe numerical results reported by Yang and Zhao [3, Fig. 4], it can beseen that both models exhibit good qualitative agreement. Thenumerical discrepancies between them should be attributed to thedifferences discussed above.

3. Results and discussion

3.1. Hydrophobic Leverett J-function vs. GDL-specific experimentaldrainage data

An interesting finding emerges when the numerical resultsobtained with the hydrophobic Leverett J-function (qc > 90�) arecompared to those obtained using GDL-specific drainage data. Inthe presence of inhomogeneous compression effects, both ap-proaches lead to gas-phase velocities with completely differentorientations. This opposite behavior, illustrated in Fig. 4, isexplained based on Darcy’s law for the gas phase, Eq. (A.5). As it iswell-known, this law states that the gas-phase velocity is

Fig. 3. Molar concentration of liquid methanol, Cml, molar concentration of methanolvapor, Cmv, and gas-void fraction, 1�s, predicted by the present model as compared tothe numerical results presented by Yang and Zhao [3] (Fig. 4).

P.A. García-Salaberri et al. / Journal of Power Sources 246 (2014) 239e252 245

proportional to the pressure gradient of the gas phase (ugfVpg).However, since the pressure gradient that is necessary to drive theconvective flow of the liquid phase is rather small, the gas-flow isessentially dominated by the capillary pressure gradient,Vpg ¼ Vpl þ VpczVpc, so that ugfVpc.

This can be checked by comparing the order of magnitude of theliquid-phase pressure gradient, Vpl, with the characteristic capil-lary pressure gradient, Vpc, across the GDL. The order of magnitudeof the liquid-phase pressure gradient is estimated according toDarcy’s law for the liquid phase, Eq. (A.3):

Fig. 4. Gas-phase velocity, ug, capillary pressure, pc, and gas-void fraction, 1�s, as predicted bspecific drainage data [74] (right). The velocity field associated with the hydrophobic LeveretV ¼ 0.25 V, CR ¼ 20%, sin ¼ 0.95, Cin

ml ¼ 1 M, T ¼ 80 �C.

��Vpl��w mlKavgkrg

��ul�� (16)

On the other hand, the characteristic value of the capillarypressure gradient can be estimated from the capillary pressuredrop between the virtually uncompressed region under the chan-nel and the compressed region under the rib, Dpcha=ribc . Using thethickness of the GDL, dagdlwOð10�4Þ m, as characteristic length,this leads to

��Vpc��wDpcha=ribcdagdl

(17)

Thus, comparing the order of magnitude of both pressure gra-dients, it turns out that:

jDplj��Vpc��wO�10�4=10�3

�(18)

To get this result we estimated the different properties and pa-rameters byconsidering a liquid saturation level s¼ 0.75, an averageporosity ε ¼ 0.74, and an operating temperature T ¼ 80 �C. The ve-locity of the liquid phase, julj, was estimated from boundary con-dition (A.12a). In this expression, the fluxes of liquid methanol andwater at the GDL/CL interface, Nml and Nwl, respectively, werecalculated from Eqs. (A.13d), (A.13e) and (A.13f), assuming a typicalcurrent density, iwOð102=103Þ A=m2, and a liquid methanol con-centration at theGDL/CL interface, Cml;aclwOð10=102Þ mol=m3. Thisestimations provided a characteristic liquid-phase velocity,��ul��wOð10�7=10�6Þ m=s, that was consistent with the values ob-

tained in the numerical simulations. On the other hand, the capillary

pressure drop across the GDL, Dpcha=ribc , was computed from Eq. (10)considering a porosity ε¼ 0.76 for the region under the channel andε ¼ 0.72 for the region under the rib. The liquid saturation level,s¼ 0.75,was assumed equal in both regions as afirst approximation.

Due to the dominant role of the capillary pressure gradient, theincreasing negative capillary pressures predicted by the Leverett J-function as the GDL is compressed (i.e., for lower porosities, seeFig. 2) lead to an unrealistic gas-flow towards themore compressed

y considering the hydrophobic Leverett J-function approach (qc ¼ 100�) (left), and GDL-t J-function is scaled by a factor of 3 to facilitate its visualization. Operating conditions:

P.A. García-Salaberri et al. / Journal of Power Sources 246 (2014) 239e252246

region under the rib. Note that the disordered structure of the ve-locity field under the rib corner seen in Fig. 4 is due to the highporosity reduction and the corresponding large capillary pressuregradients that appear in this region of concentrated stresses.Accordingly, a local mesh refinement would be necessary to in-crease the spatial resolution in this area. By way of contrast, theincreasing positive capillary pressures associated with GDL-specificdrainage data as the GDL is compressed (see Fig. 2) lead to a morerealistic gas-flow towards the virtually uncompressed region underthe channel. This situation can also be observed in Fig. 4, where thedistributions of capillary pressure associated with the gas-phasevelocity resulting from both modeling approaches are presented.

Regarding the numerical implementation of themodel, it shouldbe noted that the simulations carried out with the hydrophobicLeverett J-function often diverged in the first few hundred itera-tions. Although several strategies were developed to increase thestability of the numerical method, they proved insufficient toestablish a robust model that worked for a wide range of operatingconditions. These numerical difficulties also evidence that the hy-drophobic Leverett J-function may provide an inadequate physicaldescription of capillary transport phenomena in the anode of aDMFC. Interestingly enough, all the problems disappeared whenGDL-specific drainage data were employed.

3.2. Inhomogeneous compression and liquid/gas saturation

Another aspect that deserves further attention is the dependenceof the capillary pressure on liquid/gas saturation. Even though theconservation equation for the liquid saturation (derived from themass conservation equation of the gas phase) is quite complex,

V$

26666664�

rgkrgmg

K|fflfflfflfflffl{zfflfflfflfflffl}L

0BBBBBB@vpcvs

Vs|fflffl{zfflffl}Ps

þ vpcvr�1

pVr�1

p|fflfflfflfflfflffl{zfflfflfflfflfflffl}Pr

þVpl|{z}Pl

1CCCCCCA

37777775

¼Mm _RmþMw _Rw|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}U

; ðA:4Þ

for the following qualitative discussion this equation may besimplified by dropping some terms that are typically small. Inparticular, the source term representing the phase change rate ofmethanol and water, U, and the term associated with the pressuregradient of the liquid phase, V$ðLPlÞ, can be neglected in mostoperating conditions. This can be checked by comparing the orderof magnitude of these terms to that of the capillary diffusion termassociated with the liquid saturation gradient, which shall bedenoted by Q to abbreviate the formulation:

V$ LPs

�w

rgkrgKavgDs

mgd2agdl

vpcvs

¼ Q (19)

Thus, straightforward order-of-magnitude estimates lead to:

V$ LPl

�V$ LPs

�wrgmlkrg���ul

���mgkrldagdl

1QwO

�10�4=10�3

�(20)

U

V$ LPs

�w�Mw=m_Rw=m

� 1QwO

�10�2=10�1

�(21)

On the other hand, the term associated with the gradient of theinverse of the pore radius, V$ðLPrÞ, is of the same order that the

capillary diffusion term associated with the liquid saturationgradient:

V$ LPr

�V$ LPs

�wrgkrgKavgDr�1p

mgd2agdl

vpcvr�1

p

1QwOð1Þ (22)

These estimationswere carried out under the same assumptionsstated above for the analysis of the pressure gradient of the gasphase, Vpg. However, in this case there are some additional vari-ables. The partial derivatives of the capillary pressure wherecalculated from Eq. (10) considering a liquid saturation levels¼ 0.75 and a porosity ε ¼ 0.74. The phase change rate of methanol(m) and water (w), _Rw=m, as well as the variation of the liquidsaturation, Ds, were directly estimated from the numerical results.They are typically Oð1=10Þ mol=ðm3$sÞ and Oð10�2Þ, respectively.Finally, the variation of the inverse of the pore radius, Dr�1

p , wasestimated in the order of 105/106 m�1.

According to the above estimations, Eq. (A.4) may be simplified,in first approximation, to:

V$

"rgkrgmg

K

vpcvs

Vsþ vpcvr�1

pVr�1

p

!#¼ 0: (23)

Since the inverse of the pore radius, r�1p , is a function of porosity,

ε, we shall alternatively rewrite the above equation in terms of ε,instead of r�1

p . Note that this change of variable is more meaningfulfor interpretation purposes, because saturation and porosity areboth dimensionless variables, and of the same order of magnitude,so that OðjVsjÞwOðjVεjÞ. In addition, we shall define the ratio Gbetween the partial derivative of the capillary pressure withrespect to porosity, ε, and with respect to liquid saturation, s, i.e.,

G ¼�vpcvε

��vpcvs

; (24)

which is in turn a function of ε and s. Then, introducing the porosityin Eq. (23) and considering the above definition of G, we rewrite theequation in the abbreviated form:

V$�DcðVsþ GVεÞ� ¼ 0 (25)

where Dc ¼ �ðrgkrg=mgÞðvpc=vsÞK is the so-called capillary diffu-sivity tensor. Note that all the terms in this tensor are positive sincecapillary pressure decreases with increasing liquid saturation, i.e.,ðvpc=vsÞ < 0, both for the hydrophobic Leverett J-function and theGDL-specific drainage data shown in Fig. 2.

It is interesting to note that the term associated with theporosity gradient, Rc ¼ V$ðDc G VεÞ, can alternatively be written asa resistance source term affecting liquid/gas capillary transport:

V$ DcVs

�þ Rc ¼ 0 (26)

This term, associated with inhomogeneous compression effects(Vε s 0), vanishes for the GDLs with uniform porosity (Vε ¼ 0)considered in all previous DMFC models accounting for multiphasetransport effects. Due to its novelty, we shall discuss in detail thequalitative effect of this term in the description of multiphasecapillary transport in the anode GDL of a DMFC.

First of all, it should be noted that if G � 1 the capillary resis-tance Rc is small and can be neglected, so that the inhomogeneitiescaused by the assembly compression of the GDL do not influencediffusive capillary transport. By way of contrast, if Gw1 the effect ofthe capillary resistance Rc becomes important, and it significantlyaffects the distribution of liquid/gas saturation. In this case, roughorder-of-magnitude estimates for the two terms appearing in

Fig. 5. Variation of G ¼ ðvpc=vεÞ=ðvpc=vsÞ with the gas-void fraction, 1�s, obtainedfrom GDL-specific drainage data [74]. The different curves correspond to differentvalues of the porosity, ε.

P.A. García-Salaberri et al. / Journal of Power Sources 246 (2014) 239e252 247

Eq. (25) provides the following relation between spatial variationsof saturation and porosity:

Vsx� GVε/DsDε

w� G (27)

where D denotes spatial variations. Roughly speaking, the role ofthe capillary resistance Rc is to induce spatial variations in thesaturation level that compensate the variations in capillary pres-sure imposed by the inhomogeneous compression (i.e., porositygradients). As can be inferred from Eq. (27), an important aspect inthe relation between the liquid saturation gradient, Vs, and theporosity gradient, Vε, is the sign of G. Both the hydrophobic LeverettJ-function and GDL-specific drainage data exhibit the samedependence on liquid/gas saturation, so that (vpc/vs) has the samesign in both cases. However, as previously discussed, the depen-dence on porosity is the opposite. Whereas the hydrophobic Lev-erett J-function predicts lower (i.e., more negative) capillarypressures as the GDL is compressed, GDL-specific drainage datapredict higher (i.e., more positive) capillary pressures. Thus, thesign of G obtained in both approaches is different:

Leverett : vpcvs < 0; vpc

vε > 0/G < 0

Drainage : vpcvs < 0; vpc

vε < 0/G > 0(28)

Using these results in Eq. (27), it can be concluded that in thecase of the hydrophobic Leverett J-function the capillary resistanceRc tends to decrease the liquid saturation level (i.e., increase the gassaturation level) in the regions where the porosity is lower, whilethe opposite situation results for GDL-specific drainage data. Thisbehavior implies that Rc acts effectively as a resistance to liquid-phase capillary transport for the hydrophobic Leverett J-function,whereas it acts as a resistance to gas-phase capillary transport forGDL-specific drainage data. Considering the different signs of Gassociated with both modeling approaches, Eq. (27) leads to thefollowing relations:

Leverett : G < 0/DsDε > 0; or Dð1�sÞ

Dε < 0

Drainage : G > 0/DsDε < 0; or Dð1�sÞ

Dε > 0(29)

Thus, the main effect of GDL compression (i.e., lower porosity)on the capillary transport in the anode of a DMFC can be summa-rized as:

Leverett : Yε/Ys/Rc hinders liquid transportDrainage : Yε/Y1� s/Rc hinders gas transport (30)

This situation is reflected by the gas saturation distributionshown in Fig. 4 for the hydrophobic Leverett J-function approach. Itcan be seen that the gas-void fraction is higher in the more com-pressed region under the rib due to the effect of the capillaryresistance Rc, which hinders liquid-phase capillary transport wherethe porosity is lower. In contrast, the influence of the capillaryresistance Rc cannot be appreciated for GDL-specific drainage data.This is due to the particular dependence that the experimentaldrainage curves exhibit on saturation and compression (i.e.,porosity), which for the lowgas-coverage factor at the GDL/Channelinterface considered in the simulations of Fig. 4, 1�sin ¼ 0.05, re-sults in values of G � 1. A detailed discussion of the influence of thegas coverage factor is given in the following section.

As a final remark, it should be noted that the unrealisticbehavior of the hydrophobic Leverett J-function discussed in theprevious section has not been observed in preceding modelingstudies [3,82,86,87], or even in the simulations performed herein tovalidate the model on an uncompressed GDL geometry. This is

because when porosity variations are ignored, Rc ¼ 0, capillarytransport depends only on saturation. In this situation, the gas-voidfraction distribution is exclusively dominated by the effect of thecapillary diffusivity Dc. In consequence, a higher gas saturationlevel results under the rib due to the larger capillary diffusion path(see Fig. 3). This, in turn, leads to higher (less negative) capillarypressures in the region under the rib and, therefore, a realistic gas-flow towards the channel. The potential effect of the inhomoge-neous compression cannot be appreciated.

Note also that, in the absence of inhomogeneous compressioneffects, both the hydrophobic Leverett J-function and GDL-specificdrainage data predict a similar behavior. The reason is that bothmodeling approaches present the same dependence on saturation(capillary pressure decreases with s) and thus the capillary diffu-sivity, Dcðfvpc=vsÞ, has the same sign. The only difference betweenthem is that the pressure of the liquid phase is higher than thepressure of the gas phase for the hydrophobic Leverett J-functionapproach (pc ¼ pg � pl < 0), while the situation is the opposite forGDL-specific drainage data (pc ¼ pg � pl > 0).

3.3. Influence of the gas coverage factor at the GDL/Channelinterface

As previously discussed, the capillary resistance,Rc ¼ V$ðDc G VεÞ, plays a key role in determining the liquid/gassaturation distribution in the presence of inhomogeneouscompression effects. However, the relative importance of thecapillary resistance and capillary diffusion terms in Eq. (26) de-pends largely on the gas coverage factor at the GDL/Channelinterface, 1� sin. Indeed, the above expression for Rc shows that thevariable that governs the gas-void fraction distribution is the ratioG, which is in turn a function of the porosity, ε, and the gas satu-ration level, 1�s.

Focusing hereafter our attention on empirical GDL-specificdrainage data, Fig. 5 shows that the influence of the capillaryresistance Rc results negligible, G � 1, when the gas saturation isclose to zero or approximates the value of the residual liquidsaturation, 1� srz0:9. In contrast, the effect of the capillaryresistance Rc becomes important, GwOð1Þ, for intermediate gassaturations. This behavior is a consequence of the shape of the pc�scurves associated with GDL-specific data. As can be seen in Fig. 2,the slope of the empirical drainage curves significantly increaseswhen the liquid saturation is near zero or close to its residual value,srz0.1. However, it presents moderate values for intermediate

P.A. García-Salaberri et al. / Journal of Power Sources 246 (2014) 239e252248

saturations. Note also that the ratio G increases with decreasingporosity. It should be pointed out that this behavior is not appre-ciated on the hydrophobic Leverett J-function approach. In thiscase, the capillary resistance Rc is always comparable to the capil-lary diffusion term, V$ð�DcVsÞ, so that GwOð1Þ regardless of thevalue of s.

The situation discussed above is illustrated in Fig. 6. This figureshows the gas-void fraction distributions obtained for threedifferent gas coverage factors at the GDL/Channel interface(1 � sin ¼ 0, 0.2, 0.4) on a 20% compressed GDL. Note that these gascoverage factors are in the range of those obtained numerically byYang et al. [82]. It can be seen that for small channel gas-voidfractions (1 � sin / 0) the influence of the capillary resistance Rcturns out to be negligible. In this case, the gas saturation distribu-tion is exclusively dominated by the effect of the capillary diffu-sivityDc. Accordingly, higher gas-void fractions are found under therib due to the larger capillary diffusion path. This type of solution issimilar to those previously presented in this study (see, e.g., Figs. 3and 4) and in the literature [3,82,86,87]. By way of contrast, theeffect of the capillary resistance Rc becomes important as soon asthe gas saturation level becomes larger. As previously discussed,when GDL-specific drainage data are considered the capillaryresistance Rc hinders gas capillary transport in the regions wherethe porosity decreases, so that lower gas-void fractions are foundunder the rib. In particular, the gas-void fraction decreases signif-icantly under the rib corner due to the sharp reduction of the poresize there. The strong gas-void fraction gradients established in thissingular region may indicate a preferential location for the evacu-ation of gas-phase bubbles from the GDL to the channel, consistentwith the in-situ observations made by Hartnig et al. [90] in anoperating DMFC by means of X-ray synchrotron radiography.

The effect induced by the inhomogeneous compression on thegas-void fraction distribution is physically sound. It should be ex-pected that the regions with larger pores (i.e., higher porosity) trapmore gas. Furthermore, this result agrees with other PEMFCmodeling studies focused on the influence of GDL spatially-varyingproperties on two-phase transport phenomena [24,25]. These

Fig. 6. Gas-void fraction, 1�s, for different gas coverage factors at the GDL/Channelinterface, 1�sin. Operating conditions: V ¼ 0.2 V, CR ¼ 20%, Cin

ml ¼ 1 M, T ¼ 80 �C.

works concluded that the spatial variations induced by the in-homogeneities of the GDL (PTFE treatment, manufacturing processand assembly compression) may justify the spatial patternsobserved experimentally in the water distribution throughout theGDL. However, the effect induced by the capillary resistance Rc in aPEMFC is the opposite to that observed here for the anode of aDMFC. In fact, these works employed the hydrophobic Leverett J-function which seems to be more appropriate to describe capillarytransport under the two-phase flow conditions present in PEMFCs.As a result, the capillary resistance Rc hinders liquid capillarytransport, and more liquid water is trapped in the regions withlarger pores (i.e., higher porosity).

From the experimental point of view, the potential effect of GDLinhomogeneous compression on capillary transport has beenexplored in a fewworks relatedwith PEMFCs. It isworthmentioningthe ex-situ experimentalworks carried out by Bazylak et al. [91] andGao et al. [92] by means of fluorescence microscopy and neutronradiography, respectively. Even though both works showed that theinhomogeneous compression has a significant impact on the waterdistribution level in the GDL (Toray� carbonpaper TGP-H series,10%wet-proof), opposite conclusions were derived from them. Bazylaket al. [91] found that the compressed regions of the GDL providedpreferential pathways for water transport and breakthrough. Thisbehavior was attributed to the irreversible damage of the PTFEcoating and fibers in the compressed regions, a physicalmechanismthat is not reproduced in our model. In contrast, Gao et al. [92]showed that the liquid water saturation level under the ribs wasless than that under the channels both before and after the break-through event. This phenomenon was attributed to the changes inthe morphological properties induced by the compression, lowervoid volume and smaller pore sizes. This contradictory situationevidences that the influence of compression on capillary transportshould be further analyzed in future work.

According to the investigation carried out here, the different re-sultspresented in these studiesmaybeexplainedqualitativelyby thedifferent signs of the relevantGammaparameter associatedwith thetested GDLs as they were compressed. In Ref. [91] the excessivedamage of the PTFE content under the rib seems to dominate overthe pore size reduction caused by the GDL compression. Therefore,the relevant Gamma parameter would be that associated with PTFEcontent, i.e., the ratio between the partial derivative of the capillarypressure with respect to PTFE content and with respect to liquidsaturation. This ratio is positive, since a lower PTFE content displacesthe pc � s imbibition curves, which are those of interest to describethe invasion process of water into the air-filled GDL, to higher (i.e.,less negative) capillary pressures [75]. Thus, the effect of reducingthe PTFE content of the GDL on the experimental imbibition pc � scurves is qualitatively the opposite to that of reducing the porosity(i.e., increasing the compression ratio) of the GDL according to thecapillary pressuremeasurements carried out byGostick et al. [74,75]whichwere considered in thiswork. Accordingly,morewater shouldbe expected under the rib due to the capillary resistance Rc inducedby the higher PTFE content under the channel.

By way of contrast, Gao et al. [92] attributed the higher watersaturation observed under the channel to the changes in themorphological properties of the GDL induced by compression,lower void volume and smaller pore sizes. In this case the experi-mental observations are consistent with the capillary pressuremeasurements considered here [74,75]. The relevant Gammaparameter would be that associated with the changes in porosity(pore size), as defined in Eq. (24). And consequently, the negativevalue of Gamma corresponding to the experimental imbibitioncurves will lead to a lower liquid saturation level under the rib.Thus, more water should be expected under the channel where thepore sizes are larger.

P.A. García-Salaberri et al. / Journal of Power Sources 246 (2014) 239e252 249

4. Conclusions

An isothermal two-phase 2D/1Dmodel for the anode of a liquid-feed DMFC is presented. The model takes into account the effects ofthe inhomogeneous assembly compression of the GDL, incorpo-rated through the non-uniform porosity distribution provided by aprevious analysis of the GDL compression process [11]. The effectiveanisotropic properties of the GDL (i.e., diffusivity, permeability, andelectrical conductivity) are correlated as a function of porositythrough experimental data reported in the literature correspondingto Toray� carbon paper TGP-H series. While a growing consensuswas found for dry effective properties, extensive experimental andnumerical work is still needed to characterize two-phase proper-ties. The model also incorporates experimental capillary pressurecurves reflecting the mixed-wettability characteristics of GDLs, as amajor improvement with respect to the traditional Leverett J-function approach. Due to the uncertainty in the experimentalcharacterization of the pc � s curves of CLs, the model assumes aconstant saturation profile at the GDL/CL interface. This capillaryboundary condition, although inconsistent with physical nature, isnot expected to affect the main results of this paper, concernedwith the effects of the inhomogeneous assembly compression onthe two-phase transport phenomena in the anode GDL of a DMFC.

Focusing on the numerical results, it is shown that the inho-mogeneous compression of the GDL has a large impact on two-phase transport phenomena in this key element of the cell. Thenumerical results evidence that the hydrophobic Leverett J-func-tion approach (qc > 90�) traditionally employed in DMFC modelingstudies is physically inconsistent to describe capillary transport inthe anode of a DMFC. In contrast, more realistic results are obtainedwhen GDL-specific drainage data (gas-phase displaces liquid-phase) are taken into account. Moreover, it is found that the gassaturation distribution in the GDL can be characterized by the ratioG, which measures the relative influence of the capillary resistanceinduced by the inhomogeneous compression, Rcðfvpc=vεÞ, to thewell established capillary diffusivity, Dcðfvpc=vsÞ. For small gas-void fractions G is also small, and the capillary diffusivity is thedominant two-phase transport mechanism, so that the gas satu-ration level is higher under the rib due to the larger capillarydiffusion path. On the other hand, for larger gas-void fractions G isof order unity, and the effect of the capillary resistance Rc becomesimportant, leading to higher gas-void fractions in the region underthe channel due to the larger pore sizes there.

A few ex-situ experimental works exploring the influence ofGDL inhomogeneous compression on capillary water transport andbreakthrough were found in the literature; however, oppositeconclusions were derived from them. Thus, further studies on theeffect of GDL compression on two-phase capillary transport arewarranted. Theseworks would surely benefit from the introductionof generalized G ratios, such as the one defined here, to characterizethe influence of inhomogeneous operating conditions (tempera-ture, methanol concentration, etc.) and GDL properties (pore size,interfacial contact angle, etc.) on capillary transport phenomena.Experiments reporting the invasion process of gaseous carbon di-oxide in saturated and partially saturated GDLs with methanolaqueous solutions, more relevant for the two-phase conditionspresent in the anode of a DMFC, would be of particular interest tovalidate the numerical findings presented here.

As a final remark, it is worth noting that the present 2D across-the-channel constitutes only a first step towards the developmentof a full 3D-model for DMFCs that takes into account the influenceof inhomogeneous compression effects on multiphase transportphenomena. Therefore, further work is still required to implementthe 3D-model and to validate its results by direct comparison withexperimentally observed quantities.

Acknowledgments

This work was supported by Project ENE2011-24574 of theSpanish Ministerio de Economía y Competitividad. The authorsthank Dr. Jeff Gostick (McGill University) for fruitful discussions andcomments concerning this work.

Appendix A. Model formulation

A.1. 2D model (anode GDL)

In the following sections, bold symbols denote two-dimensionalvectors, e.g., ul ¼ (ulx,uly), and overlined bold symbols denote two-dimensional orthotropic tensors, e.g.,

K ¼�Kxx 00 Kyy

¼�K ip 00 Ktp

(A.1)

A.1.1. Conservation equations

V$ðrlulÞ ¼ �Mm _Rm �Mw _Rw (A.2)

ul ¼ �krlmlKVpl

V$

�� rgkrg

mgK�vpcvsVsþ vpc

vr�1pVr�1

p þ Vpl

(A.3)

¼ Mm _Rm þMw _Rw (A.4)

ug ¼ �krgmg

KVpg (A.5)

V$ðulCmlÞ � V$�Deffml;lVCml

�¼ � _Rm (A.6)

V$ ugCmv

�� V$

"rgD

effmv;gV

Cmv

rg

!#¼ _Rm (A.7)

V$ ugCwv

�� V$

"rgD

effwv;gV

Cwv

rg

!#¼ _Rw (A.8)

V$�seff

Vf�

¼ 0 (A.9)

A.1.2. Boundary conditionsGDL/Channel interface (thermodynamic equilibrium):

pl ¼ pinl ; s ¼ sin; Cml ¼ Cinml;

Cmv ¼ Csatmv; Cwv ¼ Csat

wv;�seff

Vf�$n ¼ 0

(A.10)

GDL/Rib interface:

ul$n ¼ 0 (A.11a)

"� rgkrg

mgK

vpcvs

Vsþ vpcvr�1

pVr�1

p þ Vpl

!#$n ¼ 0 (A.11b)

�ulCml � D

effml;lVCml

�$n ¼ 0 (A.11c)

P.A. García-Salaberri et al. / Journal of Power Sources 246 (2014) 239e252250

"ugCmv � rgD

effmv;gV

Cmv

rg

!#$n ¼ 0 (A.11d)

"ugCwv � rgD

effwv;gV

Cwv

rg

!#$n ¼ 0 (A.11e)

f ¼ frib (A.11f)

GDL/CL interface:

ðrlulÞ$n¼ Mw

�Nwl þ _Rw;acldacl

�þMm

�Nml þ _Rm;acldacl

� (A.12a)

"� rgkrg

mgK

vpcvs

Vsþ vpcvr�1

pVr�1

p þ Vpl

!#$n

¼ �Mw _Rw;acldacl �Mm _Rm;acldacl �MCO2NCO2

(A.12b)

�ulCml � D

effml;lVCml

�$n ¼ Nml þ _Rm;acldacl (A.12c)

"ugCmv � rgD

effmv;gV

Cmv

rg

!#$n ¼ � _Rm;acldacl (A.12d)

"ugCwv � rgD

effwv;gV

Cwv

rg

!#$n ¼ � _Rw;acldacl (A.12e)

�� seff

Vf�$n ¼ �i (A.12f)

A.2. 1D model (CLs, membrane, and cathode GDL)

At the GDL/CL interface the 2D model is locally coupled to a 1Dmodel accounting for the electrochemical reactions in both theanode and the cathode, as well as the mixed potential associatedwith methanol crossover. The output variables provided by the 1Dmodel at each computational face along the GDL/CL interface arethe current density, i, the molar flux of liquid water, Nwl, the molarflux of liquid methanol, Nml, and the molar flux of carbon dioxide,NCO2

. The model assumes non-Tafel kinetics to describe the com-plex kinetics of the multi-step Methanol Oxidation Reaction (MOR)and first-order Tafel-like kinetics to model the Oxygen ReductionReaction (ORR) [6]. The dissolving process of oxygen in the tripleboundary points where the reaction takes place is incorporatedthrough the Henry constant kH;O2

. To describe the methanol-crossover flux across the membrane, Ncross, both diffusive trans-port and electro-osmotic drag phenomena are taken into account.Finally, the diffusive mass transport of oxygen at the cathode GDL ismodeled by considering a single-phase Bruggeman-type resistance.The equations describing the 1D model are as follows:

iþ 6FNcross ¼ dcclðai0Þc

CO2;ccl

kH;O2CO2;ref

!exp

�acFRT

hc

(A.13a)

i ¼ daclðai0ÞakCml;aclexp

�aaFRT ha

�Cml;acl þ lexp

�aaFRT ha

� (A.13b)

NO2¼ i

4Fþ 32Ncross ¼ Deff

O2

dcgdl

CO2;amb � CO2 ;ccl

�(A.13c)

Ncross ¼ nmdiFþ Deff

m;mem

dmemCml;acl; nmd ¼ Mw

rwnwd Cml;acl (A.13d)

Nml ¼i6F

þ Ncross (A.13e)

Nwl ¼�16þ nwd

iF

(A.13f)

NCO2¼ i

6F(A.13g)

The solution of the system of Eqs. (A.13a)-(A.13d) provides ex-pressions for the anode and cathode overpotentials, ha and hc,respectively, as a function of the local current density, i, and theliquid methanol concentration at the anode catalyst layer, Cml,acl.Introducing the resulting expressions in the equation for the cellvoltage, Vcell, we obtain the following nonlinear relation among i,Cml,acl, facl and Vcell:

f i;Cml;acl;facl

�hEcell � Vcell � ha

i;Cml;acl

�� hc i;Cml;acl

�� i�dmem

smem

� 2ðfacl � fribÞ ¼ 0 (A.14)

In this expression Ecell is the ideal thermodynamic cell voltage,and the terms i(dmem/smem) and 2ðfacl � fribÞ represent the ohmiclosses across the polymer membrane and the GDLs, respectively.Note that the voltage drop across the GDL is assumed equal for bothelectrodes. In addition, the almost equipotential surface of theanode rib is considered as the reference electronic potential; seeboundary condition (A.11f).

The solution of Eq. (A.14) determines the local current density iat each face along the GDL/CL interface, for given local values ofCml,acl and facl coming from the solution of the 2D model at everyiteration and for a given fixed value of Vcell. Along with expressions(A.13d)e(A.13g), this links the 2D and 1Dmodels through boundaryconditions (A.12a)e(A.12c) and (A.12f), thus closing the mathe-matical problem.

Nomenclature

SymbolsAlg specific liquid/gas interfacial area [m�1]a catalyst surface area per unit volume [m�1]C molar concentration [mol m�3]CR thickness compression ratio; see Eq. (12)Dc capillary diffusivity tensor; see Eq. (25) [Kg m�1 s�1]Ecell ideal thermodynamic cell voltage [V]F Faraday’s constant [C mol�1]f(ε) normalized porosity function; see Eq. (1)g(s) normalized saturation function; see Eq. (1)hlg liquid/gas mass transfer coefficient [m s�1]i current density [A m�2]i0 exchange current density [A m�2]J(s) Leverett J-function; see Eq. (8)K absolute permeability tensor [m2]kH Henry’s law constant (methanol [Pa], oxygen [-])kevp evaporation rate constant [Pa s�1]kcon condensation rate constant [s�1]kr relative permeabilityM molecular weight [kg mol�1]m Van Genuchten constant; see Eq. (9)N molar flux [mol m�2 s�1]

Table A.1Physical constants, transport, kinetic and design parameters.

Parameter Value Reference

F 96485 C mol�1 e

R 8.314 J mol�1 K�1 e

Mm 0.032 kg mol�1 e

Mw 0.018 kg mol�1 e

MCO20.044 kg mol�1 e

XO2 ;amb 0.21 (air) e

pambin 1.013 � 105 Pa e

pain 1.013 � 105 Pa e

pc 1.013 � 105 Pa e

εuagdl 0.78 [38]εacl 0.3 [3]εcgdl 0.78 [38]rl 1000 � 0.0178 (T � 277.15)1.7 kg m�3 [93]ml 0.458509 � 5.30474 � 10�3 T þ 2.31231 � 10�5 T2

� 4.49161 � 10�8 T3 þ 3.27681 � 10�11 T4 kg m�1 s�1[93]

mg 2.03 � 10�5 kg m�1 s�1 [3]

Dbulkml;water

10�5.4163�(999.787/T) m2 s�1 [94]

Dbulkmv;gas �6.954 � 10�6 þ 4.5986 � 10�8 T þ 9.4979 � 10

�11 T2 m2 s�1[3]

Dbulkwv;gas 2:56� 10�5 T

307:15

�2:334m2 s�1 [3]

DbulkO2 ;air 2:5� 10�5 T

298

�3=2�pamb

pinc

�m2 s�1 [95]

Deffm;mem 4:9� 10�10 exp

�2436

1333 � 1

T

��m2 s�1 [96]

nwd 2:9 exp�1029

1333 � 1

T

��[97]

kevp 10�6 Pa�1 s�1 e

kcon 103 s�1 e

log10ðpsatwvÞ �2.1794 þ 0.02953 (T � 273)�9.1837 � 10�5

(T � 273)2 þ 1.4454 � 10�7 (T � 273)3 atm[3]

kH;O2 exphð�666=TÞþ14:1

RT

i[3]

kH,m 0.096 exp [0.04511 (T � 273)] atm [3]smem 7:3 exp

�1268

1298� 1

T

��S m�1 [96]

aa 105 m�1 [98]ac 105 m�1 [98]i0,a 94:25 exp

�35570R

1353 � 1

T

��A m�2 [6]

i0,c 0:04222 exp�73200

R

1353 � 1

T

��A m�2 [1]

aa 0.5 [84]ac 1 [3]k 7.5 � 10�4 [98]l 2.8 � 10�3 mol m�3 [98]CO2 ;ref 0.52 mol m�3 [3]Ecell 1:213� 1:4� 10�4ðT � 298Þ � 0:5

2RTF log

�pinc

pamb

�V [96]

P.A. García-Salaberri et al. / Journal of Power Sources 246 (2014) 239e252 251

n Van Genuchten constant; see Eq. (9)n outward normal unit vectornid electro-osmotic drag coefficient of species ip static pressure [Pa]pc capillary pressure, defined as pc ¼ pg�pl [Pa]pc,b Van Genuchten constant; see Eq. (9) [Pa]R universal gas constant [J mol�1 K�1]Rc capillary resistance defined in Eq. (26) [Kg m�3 s�1]_R interfacial mass transfer rate [mol m�3 s�1]r fillet radius [m]rp characteristic pore radius; see Eq. (11) [m]s liquid saturationsr residual liquid saturation; see Eq. (9)1�s gas saturationT temperature [K]u (superficial) velocity vector [m s�1]Vcell output cell voltage [V]w half-width [m]Xi mole fraction of species ix in-plane coordinate [m]y through-plane coordinate [m]

Greek lettersa transfer coefficientb interpolation parameter defined in Eq. (10)G ratio defined in Eq. (24)d thickness [m]ε porosityh overpotential [V]k experimental constantqc contact angle [�]l experimental constant [mol m�3]m dynamic viscosity [kg m�1 s�1]r density [kg m�3]s interfacial tension [N m�1]; proton conductivity [S m�1]s electrical conductivity tensor [S m�1]4 electronic potential [V]

Subscriptsa anodeacl anode catalyst layeragdl anode gas diffusion layeramb ambientavg average valuec cathodeccl cathode catalyst layercgdl cathode gas diffusion layercross crossoverg gas phasei species ij phase j (¼l/g, or liquid/gas phase)l liquid phasem methanolmem membranemv methanol vaporref reference valuerib bipolar plate ribw waterwv water vapor

Superscriptsbulk bulk propertyc compressed

dry single-phase conditionseff effectivein inlet condition at the GDL/Channel interfaceip in-plane directionsat saturated conditionstp through-plane directionu uncompressedwet multiphase flow conditions

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