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Electrochimica Acta 55 (2010) 5332–5341

Contents lists available at ScienceDirect

Electrochimica Acta

journa l homepage: www.e lsev ier .com/ locate /e lec tac ta

odeling liquid water transport in gas diffusion layers byopologically equivalent pore network

ang Luo, Yan Ji, Chao-Yang Wang ∗, Puneet K. Sinha1

lectrochemical Engine Center (ECEC), and Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, PA 16802, USA

r t i c l e i n f o

rticle history:eceived 8 February 2010eceived in revised form 19 April 2010ccepted 21 April 2010vailable online 29 April 2010

a b s t r a c t

A topologically equivalent pore network (TEPN) model is developed for the first time to extract porenetworks directly from gas diffusion layer (GDL) microstructures and thus account for all structuralfeatures of a GDL material. A generic framework of TEPN modeling is presented to design GDL structuresthat enable improved water management. With TEPNs used as input to a two-phase flow simulator,constitutive relations and steady-state liquid saturation profiles for carbon paper and carbon cloth are

eywords:olymer electrolyte fuel cells (PEFC)opologically equivalent pore networkTEPN)as diffusion layer (GDL)

obtained and reported in this work. The results indicate a strong influence of the GDL morphology onwater transport characteristics, which helps unravel the structure–performance relationship for GDLs.

© 2010 Elsevier Ltd. All rights reserved.

elative permeabilityiquid water distribution

. Introduction

A key performance/durability limitation in polymer electrolyteuel cells (PEFC) centers on liquid water transport and result-ng flooding in constituent components. The gas diffusion layerGDL), being a gateway for reactant and product water trans-ort between gas channel and catalyst layer (CL), plays a crucialole in water management which requires a delicate balanceetween membrane hydration and water removal from CL andDL. While considerable research [1,2] has been conducted onater transport in PEFCs, fundamental understanding of liquidater dynamics addressing the role of GDL microstructure and

urface wettability remains largely absent. Recently, use of poreetwork (PN) models to elucidate the pore-scale physics of liq-id water transport in GDL was pioneered by Sinha and Wang3–5] and Gostick et al. [6] and later followed by Djilali ando-workers [7,8], Koido et al. [9], Nam and co-workers [10,11],

nd Prat and co-workers [12–14]. Detailed investigations haveevealed the role of capillary fingering in liquid water transportor the first time [3]. Additionally, the effect of GDL mixed-ettability on liquid water transport is investigated in detail,ighlighting the existence of an optimum GDL hydrophilic frac-

∗ Corresponding author.E-mail address: cxw31@psu.edu (C.-Y. Wang).

1 Present address: Electrochemical Energy Research Lab, GM Global R&D, Honeoyealls, NY 14472, USA.

013-4686/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.oi:10.1016/j.electacta.2010.04.078

tion enabling least mass transport limitations to PEFC operation[4,5].

While the PN models developed to date provided substantialinsight into the pore-scale liquid water transport in GDL, they[3–14] have been employing randomly generated pore-networkstructures which are created to merely match average propertiesof GDL materials, such as porosity and permeability. These averageGDL properties are insufficient to define a unique pore structuresince different porous media with totally different microstructurescan have the same porosity and permeability. The essential func-tion of PN modeling to include the characteristics of microstructureis grossly approximated in random pore networks, thus preventingachievement of full potential to delineate the role of various car-bon paper- or cloth-based GDL materials in liquid water transport.In this work, we shall introduce, for the first time, a topologicallyequivalent pore network (TEPN) modeling approach in which thepore network is extracted directly from high-resolution, three-dimensional microstructures of GDL materials. A method will bedescribed that deploys computer models to generate a three-dimensional GDL microstructure [15] and its structure-equivalentpore network [16]. Such a generic TEPN modeling framework isinstrumental not only to elucidate physical differences in the liq-uid water behavior in carbon paper- and cloth-based GDL materials

but also to design novel GDL materials and structures for opti-mal PEFC operation. Herein we shall discuss the characteristicsof liquid water transport in carbon paper- and cloth-based GDLmaterials comparatively, and link the differences to their respectivemicrostructural features.

G. Luo et al. / Electrochimica Acta 55 (2010) 5332–5341 5333

Table 1Input parameters for microstructure generation.

Parameter Carbon paper Carbon cloth

Input Published Input Published

Fiber diameter (�m) 6 7 [21] 10 70.7–0.8 [21] 0.78 0.7–0.8– 5 –5–10 [21] 75.6 69.45–12 [21] 51.3 50–60

2

ompct

esbdecimcttTipftc

(

(

(

(

psfemitvwt

represents a domain of 360 �m × 1000 �m × 1000 �m, containing1067 pores and 5200 throats. The pore-size distributions of theextracted pore networks are shown in Fig. 7(a). A vast difference inthe pore-size distributions of carbon paper and carbon cloth can be

Porosity 0.80Voxel size (�m) 3Thru-plane permeability (×10−12 m2) 6.5In-plane permeability (×10−12 m2) 9.0

. Topologically equivalent pore network

In PN models, a porous medium is represented by a networkf pores connected by narrower regions called throats. Complexorphology of pore structures is incorporated through network

arameters, such as pore- and throat-size distribution and theironnectivity. Macroscopic properties related to network parame-ers can be inferred from percolation theory.

The construction of a realistic GDL pore morphology is thessential prerequisite for unveiling the influence of an underlyingtructure on the two-phase behavior. This can be achieved eithery three-dimensional (3D) volume imaging or by constructing aigital microstructure based on stochastic models. Noninvasivexperimental techniques, such as X-ray and magnetic resonanceomputed microtomography are the popular methods for 3D imag-ng of pore structures. Another alternative is reconstruction of a

icrostructure using stochastic simulation techniques. The lowost and high speed of data generation makes stochastic genera-ion methods the preferred choice over the experimental imagingechniques. Moreover, computer methods can not only constructEPN of existing GDLs but also virtually design new GDLs by vary-ng porosity, texture, fiber diameter, etc. The latter advantage isarticularly useful in the search for optimal GDL pore structuresor future generations of PEFCs. As such, we have developed a sys-ematic approach to design a virtual GDL and test its performance,onsisting of the following steps:

1) A stochastic modeling method [15] is used to generate high-resolution pore-scale 3D images of GDL.

2) An extensive imaging analysis [16] is employed to extract thecomputer-generated microstructure in the form of a network ofpores and throats. During this process, properties such as vol-ume, radius, and cross-sectional shape of each pore and throatin the network are determined. The coordination number, i.e.the number of independent throats linked to a pore, is assignedaccordingly.

3) The extracted pore network is used as input to a pore-networkflow simulator to compute macroscopic properties, such as cap-illary pressure and relative permeability.

4) Further, these macroscopic correlations can be plugged into atwo-phase continuum model, such as M2 model [17], to fore-cast the performance of a virtual GDL in an operational fuelcell.

In the present study, computer models of Toray-060 carbonaper and E-Tek carbon cloth GDLs are reconstructed using atochastic generation method [15] with structural inputs obtainedrom the literature and/or manufacturers. Relevant input param-ters and intrinsic permeability obtained from the generatedicrostructures are summarized along with the values of real GDLs

n Table 1. The input parameters, i.e. fiber diameter, fiber orien-ation and porosity, are chosen such that absolute permeabilityalues in both in-plane and thru-plane directions match closelyith those measured experimentally in the literature. Represen-

ative scanning electron microscopy (SEM) images of carbon paper

Fig. 1. SEM image of carbon paper (Toray 060).

and carbon cloth are shown in Figs. 1 and 2 respectively. Figs. 3 and 4display SEM-like images of carbon paper and carbon cloth gener-ated by computers. The striking similarity between original SEMand virtual images confirms that the characteristics of the originalmicrostructures are well represented in reconstructed computermodels.

Through extensive imaging analysis of computer-constructedmicrostructure models, extracted pore networks are stored in datafiles and then used as a direct input to a two-phase flow simulator. Itshould be noted that pores and throats are irregular in shape in ourextracted pore networks. However, for clarity of presentation poresand throats are depicted in Figs. 5 and 6 by spheres and cylinders,respectively, with radii of their inscribed spheres. The carbon papernetwork represents a domain of 192 �m × 750 �m × 750 �m, con-taining 2537 pores and 16,501 throats. The carbon cloth network

Fig. 2. SEM image of carbon cloth (E-Tek).

5334 G. Luo et al. / Electrochimica Acta 55 (2010) 5332–5341

Fig. 3. Carbon paper microstructure generated by stochastic method.

ture g

cptpa

Fdrt

peak value at 100 �m. The secondary mode associated with small

Fig. 4. Carbon cloth microstruc

learly seen. For carbon paper, the distribution is unimodal with a

eak pore size of ∼20 �m. For carbon cloth, a bimodal pore-size dis-ribution spanning over a few orders of magnitude is observed. Therimary mode associated with large pores (i.e. inter-thread pores)mong the fiber bundles occurs between 31 �m and 145 �m with

ig. 5. Topologically equivalent pore network of carbon paper where pores areepicted by red spheres and throats by white cylinders. (For interpretation of theeferences to color in this figure legend, the reader is referred to the web version ofhe article.)

enerated by stochastic method.

pores (i.e. intra-thread pores) between individual fibers appearsbetween 1 �m and 31 �m with the peak value at 20 �m. The dis-tribution patterns, peak values of pore diameters of the extractedpore networks are in accordance with the measured data shown in

Fig. 6. Topologically equivalent pore network of carbon cloth where pores aredepicted by red spheres and throats by white cylinders. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthe article.)

G. Luo et al. / Electrochimica Ac

Fcd

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aw

Fc

rate structural attributes of real GDLs is quite limited. It is difficultto generate random structures with high porosity and anisotropyas well as pore-size distribution with large span and average coor-dination number higher than 6.

ig. 7. Pore-size distributions of carbon paper and carbon cloth from: (a) topologi-ally equivalent pore networks, and (b) measurements [18]. Note: the unit of poreiameter should be �m instead of mm in (b).

ig. 7(b) [18], confirming the validity of the generated TEPN struc-ures. Large spread in throat diameters, especially for carbon clothith narrow throats linking small intra-thread pores and with wide

hroats linking large inter-thread pores, can be readily observedFigs. 5 and 6). As shown in Fig. 8, the throat diameter distribu-ion for carbon cloth is also bimodal. Large pores and throats ofhe primary mode distribute quite uniformly through the samplend form major flow paths. Small pores and throats are local-

zed between neighboring large pores and form local minor flowaths.

Fig. 9 contains images of throat distribution for carbon papernd carbon cloth, viewed from the thru-plane direction. Togetherith 3D views (Figs. 1–6), it shows that carbon paper and carbon

ig. 8. Throat diameter distributions of topologically equivalent pore networks forarbon paper and carbon cloth.

ta 55 (2010) 5332–5341 5335

cloth have distinctive fibrous structures. The microstructure of car-bon paper is shown to be high topology layers formed by randomlyarranged fibers along the in-plane direction. The throats are denselyarranged in a highly random order. Its coordination number variesfrom 2 to 51, with the averaged value of 13. On the other hand,the carbon cloth shows the well organized plain-weave structures.The throats are coarsely distributed in a highly ordered and struc-tured manner. Its coordination number varies from 2 to 102, withthe averaged value of 10. Both carbon paper and carbon cloth havevery complex microstructures, such as wide distributions of poreand throat size and high coordination numbers, implying that theability of cubic or randomly generated pore networks to incorpo-

Fig. 9. Topologically equivalent pore networks viewed along the thru-plane direc-tion: (a) carbon paper, and (b) carbon cloth. (For clarity, only half of the carbon papersample is shown.)

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336 G. Luo et al. / Electrochim

In summary, because TEPN has an accurate description of sizend position of pores and throats and interconnectedness of poresnd throats and eliminates all the aforementioned limitations ofhe random pore networks, it becomes finally possible to predicttructure-specific macroscopic transport properties pertinent toiquid water transport in fuel cell GDLs.

. Modeling of fluid flow

.1. Model assumptions

Two-phase flow in a pore network is modeled under the follow-ng assumptions:

. Wetting properties are assumed to be constant in the network.Contact angle between liquid water (non-wetting phase forhydrophobic GDL) and carbon fibers is fixed at 110◦.

. While the radius of a throat serves to define its hydraulic con-ductance, the volume contributed by all throats is assumed to benegligible. The small throat/pore volume ratios of reconstructedsamples, 6.14% for carbon paper and 8.64% for carbon cloth, con-firm the validity of this assumption.

. Fluid pressures are only defined in pores.

. A pore can only have single occupancy by liquid or gas. Physically,the non-wetting fluid is the bulk fluid and the wetting fluid staysas wetting films in corners, but only bulk fluid is considered forsimplicity.

. The flow is laminar everywhere and governed byHagen–Poiseuille law.

. The fluids are Newtonian, incompressible, and immiscible.

. The injecting fluid is the non-wetting fluid.

. The resistance offered by a pore to flow is assumed negligible.

. Water enters GDL and flows through GDL in the liquid phasewithout phase change (evaporation and condensation).

.2. Invasion

The invading non-wetting fluid located in a pore is flowing intothroat filled with wetting fluid. Due to the surface tension at

he interface, the capillary force exists as resistance preventing theon-wetting fluid from freely entering the throat. For invasion ofon-wetting fluid into the throat, the pressure difference betweenon-wetting and wetting phases must exceed the throat entry pres-ure. Once the throat is open and the non-wetting liquid will fillhe entire throat and the pore that is on the other end of the throat,ecause pores typically have much greater radius than throats.

In real porous media, pores and throats have complex and irreg-lar shapes. In the network extraction algorithm used herein [16],ores and throats are approximated as arbitrary cross-section withdimensionless shape factor [19].

= A

P2(1)

here A is the cross-sectional area and P is the correspondingerimeter. In the present work, cross-sections of most throats are ofriangular shape in the TEPN of carbon paper and carbon cloth. Fortriangular throat connected with pores i and j, the entry pressure,cij

is given by [20]

√

cij = � cos �

1 + 2 �Gij

rij(2)

here � is the contact angle of wetting phase and rij the minimumnscribed radius of the throat.

ta 55 (2010) 5332–5341

3.3. Two-phase flow algorithms

In pore network simulations two algorithms are commonlyused. The first one is the quasi-static algorithm, which is concep-tually simple and computationally efficient. It neglects the viscousand dynamic effects. The second one is dynamic algorithm, whichaccounts for all the relevant physics. The trade-off is a large increasein complexity and computational requirement for the dynamicalgorithm.

The quasi-static algorithm is applicable to flow through porousmedia at an infinitesimal flow rate (Ca < 10−6) where the viscouspressure drop across the network is negligible and capillary forcescompletely control the fluid configuration. For a typical fuel cellapplication, the capillary number is of the order of 10−8. Thus thequasi-static algorithm is used here to compute the constitutive rela-tions: capillary pressure and relative permeability as functions ofliquid water saturation.

3.4. Constitutive relations

Capillary pressure curve Pc(S) and relative permeability curveKr(S) are two key relations entering into a continuum model forPEFCs. Both relations depend strongly on the pore-level distribu-tions of the phases as well as the complexity of pore structures. Thevolume averaged liquid water saturation is defined as

S = 1V

∑i ∈ RV

ViSi (3)

where Vi and Si represent volume and liquid water saturation ofpore i respectively, and RV the representative volume. If RV rep-resents the whole modeling sample, S will be the total averagesaturation, which serves as a key parameter Pc(S) and Kr(S) curves.Otherwise, if the representative volume contains only a portion ofpores, S is regarded as the local saturation, which is used in plot-ting saturation profiles. Notice from the flow assumptions that eachpore can only be occupied by one fluid. Thus Si is an integer thatequals either 0 or 1.

To compute the capillary pressure curve, it is assumed that airpressure throughout the network is uniform. Initially, all the poresand throats are completely filled with air and the inlet throats areconnected to a reservoir of liquid water. For liquid water to invadeinto a throat, the pressure difference across the liquid–gas meniscusmust exceed the throat entry capillary pressure as calculated fromEq. (2). During each step of quasi-static displacement, a search isperformed over all the interfacial positions to determine the min-imum capillary pressure that will allow water to invade furtherinto the GDL. After increasing the capillary pressure to this crit-ical value, liquid water is allowed to invade the connecting poreand any subsequent throats and pores that can be invaded at thiscapillary pressure. At this time, total volume averaged saturation isupdated based on the liquid water distribution in each pore. Onceno further invasion can occur, the capillary pressure is increased bya next minimum increment.

To compute the absolute permeability and relative permeabil-ity, Hagen–Poiseuille flow through throats is assumed. The volumeflow rate of phase ˛ between two neighboring pores i and j can thusbe expressed as:

Q ˛ij = g˛

ij (Pi − Pj) (4)

where Pi and Pj are the pressures in the pores, g˛ij

is the hydraulicconductance of phases ˛ in the throat that connects pore i and pore j.For triangular throats, the cross-sectional areas open for both wet-ting and non-wetting phases. The area occupied by wetting and

ica Acta 55 (2010) 5332–5341 5337

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A

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wr

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w

g

wfw[

s∑w(se

G

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bsni

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wa

so

G. Luo et al. / Electrochim

on-wetting phases is given by [20]

wij =

r2ij

4Gij

[1 −(

�

rijPcij

)2

(1 − 4�Gij)

](5)

nw =r2ij

4Gij

(�

rijPcij

)2

(1 − 4�Gij) (6)

here nw and w stand for the non-wetting and wetting phases,espectively.

The flow conductance is modeled by the mean hydraulic radius,˛ij

and the equivalent volume radius, r˛ij

[20]

˛ij = 0.5(rij + r˛

ij ) (7)

˛ij =√

A˛ij

�(8)

The conductance of the non-wetting phase through a throatetween pore i and pore j is given by Poiseuille’s law:

nwij =

Rnw2

ijAnw

ij

8�nwlij(9)

here � is the viscosity and lij is the length of the throat.For the wetting phase, it follows that [20],

wij =

rw2

ijAw

ij

8ˇ�wlij(10)

here ˇ is a dimensionless flow resistance factor which accountsor the reduced conductance of the wetting phase close to the poreall. ˇ = 2.5 is used for a no stress interfacial boundary condition

20].Since the fluids are assumed to be incompressible, volume con-

ervation applies at each pore:

j

Q ˛ij = 0 (11)

here j runs over all the pores connected to pore i. Eqs. (4) and11), together with the appropriate boundary conditions, form aystem of linear algebraic equations to determine the pressure atach pore:

P = b (12)

here G is a n × n conductance matrix (n is the number of poreseing solved), P is a n × 1 column vector storing pore pressures,nd b is a n × 1 source vector accounting for boundary conditionsnd sources.

Liquid water distribution inside pores and throats is determinedy quasi-static displacement and the corresponding capillary pres-ure is applied at all the inlet throats. Thus Eq. (12) is solvedumerically at each pore with pressure boundary conditions at the

nlet and outlet faces of the network.To compute absolute permeability, the network is forced to be

ully saturated with a single fluid (liquid or air). By specifying aressure drop across the GDL, Eq. (12) is solved and the volumeow rate Q at the inlet or outlet surface is calculated. The absoluteermeability K is determined from Darcy’s law:

= Q�˛L

As�P˛(13)

here As is the area of inlet or outlet surface and L is the thicknesslong the direction where pressure drop occurs.

Similarly, relative permeability is calculated by specify a pres-ure drop across the GDL for phase ˛, but at various saturation levelsf the invasion process. Calculate the volume flow rate of phase ˛,

Fig. 10. Capillary pressure curves of carbon paper and carbon cloth predicted byTEPN.

Q˛ at the inlet or outlet boundary. The relative permeability of aphase ˛, k˛

r , is given by:

k˛r = Q ˛�˛L

KAs�P˛(14)

Combining (13) and (14) gives:

k˛r = Q ˛

Q(15)

Thus k˛r is the ratio of flow rates of phase ˛ at the same pressure

drop, but with two different phase conditions. The numerator isfor two-phase flow while the denominator is for single-phase flowonly.

3.5. Boundary conditions

The inlet surface of the GDL is assumed to be in contact with areservoir of the non-wetting fluid (i.e. liquid water), whereas theoutlet surface is connected to a reservoir of the wetting fluid (air).Constant pressure boundary conditions are imposed on the inletand outlet surfaces. All other surfaces are subjected to no-flowboundary conditions.

4. Results and discussion

Throughout this paper, x denotes the thru-plane direction, andy and z the in-plane directions. For current configurations, x, y andz are the principle axes. For the flow of a viscous fluid throughan anisotropic homogeneous porous medium, it is assumed thatDarcy’s law must be satisfied for each phase:(

u˛x

u˛y

u˛z

)= − 1

�˛

[K˛

r,xKx

K˛r,yKy

K˛r,zKz

]∇P˛ (16)

For typical fuel cell applications where the GDL is very thin, itis reasonable to assume a constant gas-phase pressure. Thus the

following expression can be derived for liquid water,(

u˛x

u˛y

u˛z

)= 1

�˛

[K˛

r,xKx

K˛r,yKy

K˛r,zKz

]dPc

ds∇s (17)

5338 G. Luo et al. / Electrochimica Acta 55 (2010) 5332–5341

throug

ttam“gc

4

acpdtli

characteristics of capillary pressure curve are strongly determinedby both throat- and pore-size distributions. The liquid water inva-sion starts with larger throats and pores and then moves towardsmaller ones in the pore- and throat-size distributions. Entry pres-

Table 2

Fig. 11. Liquid-saturated pores at the first break

Eq. (17) implies that the liquid water flux is proportional tohe liquid relative permeability, the absolute permeability, andhe slope of capillary pressure curve. All these three parametersre strongly tied to the GDL microstructure. With the detailedicrostructure described by TEPN, two groups of simulations,

thru-plane flow” and “in-plane flow”, are carried out to investi-ate the impact of different pore structures of carbon paper andarbon cloth on their macroscopic transport properties.

.1. Absolute permeability

Calculated values of thru-plane and in-plane absolute perme-bilities based on the extracted pore networks of carbon paper andloth are listed in Table 2. Both absolute permeabilities of carbon

aper and carbon cloth are in a good agreement with those obtainedirectly from the microstructures (Table 1). For carbon paper, thehru-plane absolute permeability is 6.31 × 10−12 m2, whereas thearge throats in the bimodal distribution for carbon cloth increasets value to 70.9 × 10−12 m2.

h for (a) thru-plane flow, and (b) in-plane flow.

4.2. Capillary pressure

For comparison, capillary pressure curves (primary drainagecurves) of both carbon paper and carbon cloth are plotted in Fig. 10.Capillary pressure plays an important role in the description ofwater flow in GDLs. For a given capillary pressure the amount ofliquid water entering inside GDLs depends on the slope of cap-illary pressure curve. Liquid water being the non-wetting phasepreferentially invades larger throats in the pore structure. Hence,

Absolute permeability (10−12 m2) predicted from TEPN.

Material Thru-plane In-plane

Carbon paper 6.31 7.68Carbon cloth 70.9 52.7

G. Luo et al. / Electrochimica Acta 55 (2010) 5332–5341 5339

Table 3Predicted critical saturation at the first breakthrough.

sttepb(

pthorTrnfps0w

pcrbr2t3ppoots

4

pttpacsptt

clibhtll

Material Thru-plane flow In-plane flow

Carbon paper 0.15 0.42Carbon cloth 0.295 0.39

ure at zero saturation is associated with the upper limit of thehroat-size distribution and determined by one of the largesthroats in the sample. As shown in Fig. 8, the maximum throat diam-ter of carbon cloth (89.2 �m) is much larger than that of carbonaper (40.6 �m). Consequently the predicted entry pressure of car-on cloth (0.92 kPa) is significantly lower than that of carbon paper2.15 kPa).

The slope of the capillary pressure curve is also determined byore- and throat-size distributions and how these two distribu-ions are associated. The capillary pressure curve of carbon paperas a flat slope between saturation of 0.05 and 0.8. At saturationf 0.05 and 0.8 the capillary pressures are 4.5 kPa and 6.0 kPa, cor-esponding to 16.5 �m and 12.4 �m diameter throats respectively.his means that the dominant throat diameter is in a very narrowange between 12.4 �m and 16.5 �m. 75% pore volume is con-ected by dominant throats. The small decrease in throat diameter

rom 16.5 �m to 12.4 �m leads to the small increase in capillaryressure, thus the flat slope of capillary pressure curve betweenaturation of 0.05 and 0.8. In the region beyond the saturation of.8, capillary pressure for carbon paper increases rapidly as liquidater invades further into smaller pores.

With a bimodal throat- and pore-size distribution, capillaryressure curve of carbon cloth differs significantly from that ofarbon paper. The capillary pressure curve exhibits a bimodalamp increase: a flat slope and slow increase of capillary pressureetween the saturation of zero and 0.8, and a steep slope and fastise beyond that. At the saturation of 0.8 the capillary pressure is.0 kPa, corresponding to 37.5 �m diameter throats. This meanshat the dominant throat diameter is in a wide range between7.5 �m and 81.5 �m. The flat slope region is associated with therimary mode in the pore-size distribution of carbon cloth. Theore volume in this mode represents about 80% of the total volumef all pores. The sharp sloped region is associated with the sec-ndary mode. Abrupt slope change around saturation 0.8 indicateshat the peaks in the bimodal pore and throat distributions are welleparated.

.3. Liquid water distribution

In Fig. 11, liquid-saturated pores at the first breakthrough arelotted. All throats are depicted by straight lines. All pores are plot-ed proportionally to the real dimensions of the GDL structure ashe inscribed spheres of the pore spaces for simplicity. The liquidore cluster in blue color represents the connected cluster withliquid breakthrough point at the outlet surface. Others in green

olor are all dead-end liquid pore clusters. Blue surfaces repre-ent the inlet plane or reservoir. Grey surfaces represent the outletlane. Carbon paper has much larger pore number density thanhat of carbon cloth owing to their distinctive pore structure andhickness.

The total volume saturations at the first breakthrough for botharbon paper and carbon cloth structures are listed in Table 3. Theiquid saturation marking breakthrough in the thru-plane directions lower than that in the in-plane direction, as expected. All of the

reakthrough saturation values are in the range of 0.1–0.4, a rangeighly relevant to typical fuel cell operation. As shown in Fig. 11,he size of open and dead-end clusters for in-plane flow is mucharger than those for thru-plane flow. This is due to the fact thatiquid in the in-plane flow needs to travel over a greater length and

Fig. 12. Predicted saturation profiles at the first breakthrough for (a) thru-planeflow, and (b) in-plane flow.

invade many more pores to reach the outlet plane compared to thethru-plane flow.

Saturation profiles at the first breakthrough for different struc-tures are plotted in Fig. 12. In order to calculate the localsaturation using Eq. (3), the total GDL is equally divided inton representative volumes between inlet and outlet surfaces.The number n is chosen such that enough number of poresfalls into each RV to obtain a meaningful averaged saturation.If n is too large, a representative volume without any poresmay exist, giving rise to zero local saturation. Based on thisrule, carbon cloth in the thru-plane direction can have at most3 RVs.

Distribution of liquid-saturated pores at the first breakthroughshows fractal fingering flow: among all clusters connected with theinlet plane, only one cluster breaks through (Fig. 11). This fractalnature of capillary fingering gives the saturation profile an over-all decreasing concave trend toward the outlet. However, localsaturation does not monotonically decrease toward the outlet insome cases. The complex shape of local saturation profiles can beexplained by scrutinizing the 3D liquid water distribution shownin Fig. 11. For instance, for in-plane flow through carbon cloth, the

local saturation increases from 0.05 to 0.15 near the outlet region.This is due to the fact that the representative volume near the outletcontains two large liquid-saturated pores, whereas the representa-tive volume next to it has only one liquid-saturated pore. The shapeof the saturation profile is strongly linked to the characteristics of

5 ica Ac

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340 G. Luo et al. / Electrochim

he GDL microstructure. As a result, the saturation profile locallyay have a decreasing convex trend or even an increasing trend.

.4. Relative permeability

In Fig. 13, the predicted thru-plane and in-plane relative per-eabilities are compared for both carbon paper and carbon cloth

espectively. Relative permeability is determined by the conduc-ance matrix of phase ˛ flowing in throats. As such, it is influencedy the throat length, throat-size distribution, and connectivity ofeighboring throats.

At the same liquid saturation, the connectivity of throats flowny a specific phase is strongly direction-dependent, caused by thenisotropy of the structures of carbon paper and carbon cloth. Thiseads to a significant difference between the in-plane and thru-lane relative permeability curves. As shown in Fig. 11, liquid water

annot penetrate in the GDLs to form a continuous flow path untilcritical water saturation is attained where water breaks throughn the outlet surface. Hence, the liquid relative permeability is zeroelow the breakthrough saturation in Fig. 13.

ig. 13. Predicted relative permeability curves of (a) carbon paper, and (b) carbonloth.

ta 55 (2010) 5332–5341

As shown in Fig. 13, the liquid relative permeability in thein-plane direction is much smaller than in the thru-plane direc-tion. For carbon paper, below liquid saturation of 0.72, thein-plane liquid relative permeability is very small (<0.04), therebyexacerbating flooding under the land region. Hence, for high-humidity operation, carbon paper is not the best choice. However,under the low-humidity operation, carbon paper has the advan-tage of retaining product water and hence improving membranehydration.

Between the breakthrough saturation and saturation of 0.8, theliquid relative permeability curves for carbon cloth are not smooth,exhibiting step-wise variations. This is reflective of the uniquetopological structure of carbon cloth. In the saturation range, theliquid relative permeability is associated with the primary modein the bimodal pore-size distribution. Although merely 8% in num-ber, pores in this mode are very large and occupy 80% of the totalvolume of all pores. Liquid volume saturation can jump by a fewpercent even when only one large pore is invaded. Flat regionsof the curves indicate that a large change in liquid saturationhardly impacts the relative permeability. This happens when inva-sion proceeds in the dead end clusters consisting of large pores.Since those clusters have no connection with the outlet, there isno contribution to the liquid permeability despite the liquid vol-ume saturation increases appreciably. Abrupt-change regions ofthe curves, where a slight increase in liquid saturation causingrapid rise in relative permeability, happen when a dead-end clus-ter with large pores finally breaks through or connects with anopen cluster. This dramatically changes the total conductance ofthe entire network, with a little change in the saturation. Beyondthe saturation of 0.8, both thru-plane and in-plane liquid rela-tive permeability curves are seen to be very flat. This is becausemajor liquid flow paths are already established throughout thenetwork at the high saturation of 0.8. Adding minor liquid flowpaths by invading smaller throats in the secondary mode hardlyincreases the total conductance of the network. Thus, beyond sat-uration of 0.8 the liquid relative permeability stays very close tounity.

The absolute permeability of carbon cloth is almost 15 timesof carbon paper. At the same saturation, both thru-plane andin-plane relative permeabilities of carbon cloth are much higherthan those of carbon paper. As indicated by Eq. (17), the abso-lute and relative permeability, together with the slope of capillarypressure–saturation curve determine the water flux across the GDL.Hence, carbon cloth has a much higher capability to transport liquidwater than carbon paper, leading to less retention of liquid water incarbon cloth and hence making it more suitable for high-humidityoperation. For the same reason, carbon cloth GDL is inferior to car-bon paper under low-humidity operation.

The gas relative permeability is calculated under the assumptionthat liquid-filled pores offer no conductivity to gas flow. For car-bon paper, the predicted gas relative permeability in the in-planedirection is much smaller than that in the thru-plane direction,which could cause severe mass transport limitation under theland region. For carbon cloth, once the liquid saturation exceeds0.55, the thru-plane and in-plane gas relative permeabilities reduceto nearly zero. This means that gas transport can be effectivelyblocked by filling a few large pores along the main flow path. Forcarbon paper, the saturation needs to reach 0.7 to block the gastransport.

It should be noted that the predicted correlations are based on asingle computer reconstructed sample of carbon paper and carbon

cloth. They are not as accurate and smooth as statistically averagedrelative permeability, for which more stochastic generated samplesare required. However the predicted results still reveal the dramaticimpact of the GDL microstructure on their macroscopic properties,and the relative permeability correlations presented herein exhibit

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ew characteristics that have not been captured by random poreetworks.

. Conclusion

In this work we have developed a topologically equivalentore network (TEPN) method to describe the dramatic impactf GDL pore morphology on water transport properties. A vir-ual GDL design approach is described through generation ofEPNs of two commonly used GDLs, carbon paper and carbonloth. With a detailed representation of the complex poroustructure, macroscopic properties are distinguished between car-on paper and carbon cloth and linked to their respectiveicrostructural features. Bimodal pore- and throat-size distri-

utions characteristic of carbon cloth lead to two distinctiveegions in capillary pressure and relative permeability curves,hile only one region is observed for carbon paper owing to

ts unimodal pore- and throat-size distributions. It is found thatoncave-shaped saturation profiles, typical of capillary finger-ng, are representative for homogenously hydrophobic paper andloth GDLs. Virtual GDL design by TEPN method described in thisaper will help explore new GDL structures for optimal wateremoval.

cknowledgements

We would like to thank Drs V. P. Schulz, A. Wiegmann and J.ecker from Fraunhofer ITWM, Germany for collaboration with GDLtructure generation, and Dr Hu Dong from Numerical Rocks, Nor-

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ta 55 (2010) 5332–5341 5341

way for collaboration with TEPN generation. Partial support of thiswork by DOE EERE Fuel Cell Technologies Program is also acknowl-edged.

References

[1] C.Y. Wang, in: W. Lietsich, A. Lamm, H.A. Gasteiger (Eds.), Handbook of FuelCells – Fundamentals, Technology and Applications, vol. 3, John Wiley & Sons,Chicester, 2003, p. 337 (Chapter 29).

[2] C.Y. Wang, Chem. Rev. 104 (2004) 4727.[3] P.K. Sinha, C.Y. Wang, Electrochim. Acta 52 (2007) 7936.[4] P.K. Sinha, C.Y. Wang, Chem. Eng. Sci. 63 (2008) 1081.[5] P.K. Sinha, P.P. Mukherjee, C.Y. Wang, J. Mater. Chem. 17 (2007) 3089.[6] J.T. Gostick, M.A. Ioannidis, M.W. Fowler, M.D. Pritzker, J. Power Sources 173

(2007) 277.[7] B. Markicevic, A. Bazylak, N. Djilali, J. Power Sources 171 (2007) 706.[8] A. Bazylak, V. Berejnov, B. Markicevic, D. Sinton, N. Djilali, Electrochim. Acta 53

(2008) 7630.[9] T. Koido, T. Furusawa, K. Moriyama, J. Power Sources 175 (2008) 127.10] K.J. Lee, J.H. Nam, C.J. Kim, Electrochim. Acta 54 (2009) 1166.11] K.J. Lee, J.H. Nam, C.J. Kim, J. Power Sources 195 (2010) 130.12] O. Chapuis, M. Prat, M. Quintard, E. Chane-Kane, O. Guillot, N. Mayer, J. Power

Sources 178 (2008) 258.13] M. Rebaia, M. Prat, J. Power Sources 192 (2009) 534.14] L. Ceballos, M. Prat, J. Power Sources 195 (2010) 825.15] V.P. Schulz, J. Becker, A. Weigmann, P.P. Mukherjee, C.Y. Wang, J. Electrochem.

Soc. 154 (2007) B419.16] H. Dong, M.J. Blunt, Phys. Rev. E 80 (2009) 036307.17] G. Luo, H. Ju, C.Y. Wang, J. Electrochem. Soc. 154 (2007) B316.18] K. Yoshizawa, K. Ikezoe, Y. Tasaki, D. Kramer, E.H. Lehmann, G.G. Schererb, J.

Electrochem. Soc. 155 (2008) B223.19] G. Mason, N.R. Morrow, J. Colloid Interface Sci. 141 (1991) 262.20] L.S. Nilsen, P.E. Oren, S. Bakke, SPE J. 2 (1997) 136.21] M.F. Mathias, J. Roth, J. Fleming, W. Lehnert, in: W. Lietsich, A. Lamm, H.A.

Gasteiger (Eds.), Handbook of Fuel Cells – Fundamentals, Technology and Appli-cations, vol. 3, John Wiley & Sons, Chicester, 2003.