International Journal of Heat and Mass Transfer 55 (2012) 133–139

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International Journal of Heat and Mass Transfer

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Capillary pressures in carbon paper gas diffusion layers having hydrophilicand hydrophobic pores

Liang Hao, Ping Cheng ⇑MOE Key Laboratory of Power Machinery and Engineering, School of Mechanical and Power Engineering, Shanghai Jiaotong University, Shanghai 200240, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 June 2011Received in revised form 22 August 2011Accepted 22 August 2011Available online 28 September 2011

Keywords:Gas diffusion layerHydrophilic and hydrophobic poresCapillary pressurePore-scale simulationSaturationTwo-phase transport

0017-9310/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2011.08.049

⇑ Corresponding author. Tel./fax: +86 21 34206337E-mail address: pingcheng@sjtu.edu.cn (P. Cheng).

Capillary pressures in a carbon paper gas diffusion layer (GDL) having hydrophilic and hydrophobic poresof a polymer electrolyte membrane fuel cell (PEMFC) are investigated by both lattice Boltzmann simula-tions and experimental measurements. The simulated and measured capillary pressures as a function ofwater saturation for water drainage and imbibition processes in the GDL are presented and compared. Itis shown that the pore-scale simulated drainage and imbibition capillary pressure curves are in goodagreement with that obtained by experiment, both indicating the coexistence of hydrophilic and hydro-phobic properties in the polytetrafluoroethylene (PTFE) treated carbon paper GDLs. The fitted capillarypressure curves, obtained from this paper, can provide more accurate predictions of the capillary pressurein carbon paper GDLs with non-uniform porosity and wettability than the standard Leverett–Udell rela-tionship which was obtained for soil with more uniform porosity and wettability.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction rafluoroethylene (PTFE) to enhance its hydrophobic characteristics

Water management is regarded as a critical issue, which isgreatly related to the performance of the polymer electrolytemembrane fuel cells (PEMFCs) for their commercial reality [1]. InPEMFCs, the gas diffusion layer (GDL) is located between the gasflow channel and the catalyst layer (CL), playing an important roleon the transport of water produced in CL. According to the opera-tion condition of PEMFCs and the pore size of GDL, it has been sug-gested that water transport in GDL is strongly dominated bycapillary force, while viscous and inertial forces are negligible [2].Based on the Young–Laplace equation, the capillary pressure pc, de-fined by the difference of liquid pressure pl and gas pressure pg inporous medium, is related to the average pore radius r and the con-tact angle hc of the porous material by,

pc ¼ pl � pg ¼ �2r cos hc

rð1Þ

where r is the surface tension. Eq. (1) implies that the wettability isan important parameter to control the gas/liquid two-phase trans-port process in a porous GDL. For hydrophilic GDLs (hc < 90�), the li-quid phase is easy to fill the pores due to the lower liquid pressurethan the gas phase, thus leading to flooding problems. For hydropho-bic GDLs (hc > 90�), the liquid phase is difficult to fill the small poresdue to the requirement of high liquid pressure. Therefore, in order tofacilitate water removal in PEMFCs, GDL is usually treated by polytet-

ll rights reserved.

.

[3,4].Recently, much attention has been given to the study of water

transport in PEMFCs [5–8] because of flooding problems deterio-rate their performance. The unsaturated flow model and multi-phase mixture model are two widely used macroscopicapproaches [9] for study of gas and water two-phase transport inGDLs. In these models, the GDL is assumed as a homogeneousand isotropic porous medium, and macroscopic conservation equa-tions are obtained based on volumetric averaging theories for anelementary volume [10]. These macroscopic conservation equa-tions, coupled with empirical relationships of relevant macroscopicphysical properties, have been used for numerical simulations oftwo-phase flows in the GDL of PEMFCs. These macroscopic physicalproperties include, for example, the capillary pressure in GDLwhich is related to Leverett function J(s) by

pc ¼ rj cosðhcÞjUK

� �1=2

JðsÞ ð2Þ

where U and K are the average porosity and permeability of GDL,respectively, and s is the water saturation (defined as the ratio ofthe pore volume occupied by water to the entire pore volume in aporous medium). The standard Leverett–Udell function J(s) forhydrophilic and hydrophobic porous media is,

JðsÞ¼ �1:417ð1� sÞþ2:120ð1� sÞ2�1:263ð1� sÞ3; if hc >90�

1:417ð1� sÞ�2:120s2þ1:263s3; if hc <90�

(

ð3Þ

Nomenclature

A, B free energy parameterf, g distribution functionJ Leverett J-functionK permeabilitypc capillary pressurer pore radiuss saturationu velocity vectorx location vector

Greek symbolsH wetting potentialhc contact anglej surface tension parameter

l chemical potentialq densityr surface tensionsf, sg relaxation parameteru order parameterU porosityw bulk free energy density

Superscriptsg gasl liquids solid

134 L. Hao, P. Cheng / International Journal of Heat and Mass Transfer 55 (2012) 133–139

where s has a value of 0 6 s 6 1. It is important to note that Eq. (3) as-sumes that the contact angle is uniform in the porous medium underconsideration. Eq. (3) with constant hydrophobic contact angles(hc > 90�), has been widely adapted in PEMFC two-phase macroscopicmodels to describe the water transport in the porous GDL. However,it is known that numerical simulations obtained based on these mac-roscopic models had underpredicted water saturations in GDL ascompared to those obtained by neutrons imaging experiments[11,12]. It should be noted that Eq. (3) was obtained based on theexperimental data of soil bed, which can be considered as a ratherhomogeneous and isotropic porous medium with small porosity (lessthan 0.5) and uniform wettability distribution. Thus, the accuracy ofEq. (3) for application to typically anisotropic GDL with fiber-basedpore structure and generally high porosity (higher than 0.7) is ques-tionable. In addition, as mentioned by Ye and Nguyen [13], a GDLtreated by PTFE is not a porous material with uniform wettability,but with both hydrophilic and hydrophobic pores. The macroscopiccontact angle in Eq. (3) does not have a clear definition in this typeof porous material with non-uniform wettability distributions.

In the present work, both pore-scale simulation and experimen-tal measurements were performed to investigate the relationshipof capillary pressure and water saturation in PTFE treated carbonpaper GDL. It is shown that the Leverett function, obtained directlyfrom lattice Boltzmann simulations and measurement of the GDLporous materials in the present paper, is more accurate for descrip-tion of capillary pressures in fibrous GDLs of PEMFCs.

2. Experimental measurements and pore-scale simulationmethods

2.1. Experimental system

The experimental system for capillary pressure measurement ofa carbon paper GDL is shown in Fig. 1. This experimental setup in-cluded a test section (in which the GDL sample was placed), a dataacquisition system to collect the pressure of gas and water phases,a precision electronic balance to measure the amount of water fill-ing into the sample, and a syringe pump to drive the water into andout of the sample. The method using a micro-fluidic device to mea-sure the capillary pressure of GDL was also performed by Fair-weather et al. [14]. The PTFE treated Toray TGP-090 carbonpapers were used as the samples in the present experiment. Thecapillary pressure of the drainage (water filling into the porousmedium) and imbibition (water removing from the porous med-ium) processes were determined separately. In a drainage process,the gas in the test section was sucked by a syringe pump while thedeionized water in the container was filling into the test section

and test sample gradually as the gas pressure decreased, and direc-tions of water and gas flowing were indicated by arrows in Fig. 1.For each run, the syringe pump worked for 30 s with a flux of100 ll/min and then paused for 3 min to reach a stable water dis-tribution in the sample. Then, gas and water phase pressures in thetest section and the corresponding water mass change in the con-tainer were recorded by pressure transducers and by an electronicbalance, respectively. The above steps were repeated until thepressure difference of gas and liquid phase reached �15 kPa. Inan imbibition process, the syringe pump was set at injection modeat the end of the drainage process, and directions of water and gasflowing indicated by arrows in Fig. 1 were reversed. The succeed-ing steps, same as in the drainage process, were performed untilthe pressure difference of gas and liquid phase reached +15 kPa.In order to calculate the water content filling into the sample fromthe container, the reading of the electronic balance at the startingpoint that the water began entering the sample was needed as areference. The starting point was obtained by observing the pointthat the pressure difference between gas and water had a sharplychange as water began to enter into the GDL in the first drainageprocess. This method was also adapted by Nguyen et al. [15]. Inaddition, the container was covered by a plastic film during theexperiment to prevent the evaporation of the water and the con-nection in whole system was well sealed to prevent leakage.

Fig. 2 shows the cross sectional view of the test section from theside. In the test section, the GDL sample having a size of40 � 40 mm was sandwiched between a hydrophobic (Satorius,11807-050N) and a hydrophilic membrane (Millipore, HVLP04700),and then a distributor on each side. The distributors were used todistribute the water and gas more uniformly. The sandwich struc-ture was placed between two PMMA covers having a cross-sectionof 60 � 60 mm. Each of the cover had a rectangular cavity with across-section of 30 � 30 mm and a depth of 9 mm. Thus, two cham-bers were formed by the two cavities in the covers. The use ofhydrophobic and hydrophilic membranes in the test section wasto extend measurements of the capillary pressure and saturationin a wider range. This was because the hydrophobic membranecould prevent water breaking through the GDL in the drainage pro-cess while the hydrophilic membrane could prevent gas break-through in the imbibition process [14,15]. The test section waswell sealed and the leakage examination was performed beforeplacing it into the measurement system.

2.2. Pore scale simulation by free-energy lattice Boltzmann method

The lattice Boltzmann method (LBM), which is a promising toolto simulate multiphase flow in porous media, can give more

Chamber

Chamber

Gas Outlet

WaterInlet

GDL Sample

Teflon Gasket

Water Distributor

Seal Gasket

Hydrophilic Membrane

Hydrophobic Membrane

Chamber

Teflon Gasket

Fig. 2. Schematic of test section.

LabVIEW

Filter

Cut offValve

ControlValve

ControlValve

P

P, T

Test Section

Container

Balance

P2 P1, T1

data acquisition system

Filter

Syringe Pump

Deionized water

Sample

Fig. 1. Schematic of experimental system for capillary pressure-saturation relationship measurement of carbon paper GDL.

L. Hao, P. Cheng / International Journal of Heat and Mass Transfer 55 (2012) 133–139 135

realistic pore-scale simulations of water transport in GDL withaccurate pore geometry taken into consideration. Several latticeBoltzmann methods have been developed for simulating multi-phase flow problems, in which the intermolecular potential andfree-energy model are two most popular approaches. In the inter-molecular potential model, the microscopic interaction betweenfluid particles is introduced to describe the intermolecular poten-tial. However, as pointed out in [16], the surface tension in thismodel is actually a numerical artifact. In contrast, the free energymodel has a firm theoretical basis. The surface tension as well asthe contact angle when solid is present can be directly derivedfrom the free-energy function based on Cahn–Hilliard theory.Therefore, the free-energy LBM was adopted for numerical simula-tion in present work.

In the free-energy LBM model, the binary fluids flow is de-scribed by the evolution of distribution functions fa(x, t) and ga(x, t)as follows:

faðxþ eadt; t þ dtÞ � faðx; tÞ ¼ ½f eqa ðx; tÞ � faðx; tÞ�=sf ð4aÞ

gaðxþ eadt; t þ dtÞ � gaðx; tÞ ¼ geqa ðx; tÞ � gaðx; tÞ

� �=sg ð4bÞ

where sf and sg are two independent relaxation parameters, whichare related to the fluid viscosity and phase interface mobility.

The surface tension between different phases and the contactangle when solid boundary is presented can be derived from thefree-energy function G. For the Landau free-energy function [17],

G ¼Z

VdVðwðuÞ þ jðruÞ2=2Þ þ

ZS

dSwsðuÞ ð5Þ

where the bulk free energy density w(u) can be chosen in the formof a double-well potential [18],

wðuÞ ¼ �A2u2 þ B

4u4 ð6Þ

where A and B are two parameters, which are chosen by positivevalues for phase separation in this study. The chemical potentiall is derived from the free energy density function by [18],

l ¼ �Auþ Bu3 � jr2u ð7Þ

The minimization of the chemical potential in bulk results in,

l ¼ �Auþ Bu3 � jr2u ¼ 0 ð8Þ

Thus, the equilibrium phase parameter u0 in the bulk of the two dif-ferent fluids follows u0 = ±(A/B)1/2 = ±1 when restricting the valueA = B. The surface tension can be determined from the interfacewidth n and the equilibrium phase parameter in the bulk phaseu0 as [18],

r ¼ 4j3n

u20 ð9Þ

The macroscopic local density q, momentum qu and phase param-eter u are obtained from the distribution functions fa(x, t) andga(x, t) for three-dimension 19-velocity (D3Q19) model [19],

0.0 0.1 20. 0.3 0.4 0.5 0.6 0.7 80. 0.9 1.0-10000

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000 Experimental Data (Primary Drainage) LBM Results (Primary Drainage) LBM Results (Drainage) LBM Results (Imbibition)

Cap

illary

Pre

ssur

e (P

a)

Saturation (a) LBM simulated capillary pressure curve for carbon paper GDL with 10wt. %

PTFE

10000

Drainage Imbibition

136 L. Hao, P. Cheng / International Journal of Heat and Mass Transfer 55 (2012) 133–139

q ¼X18

a¼0

fa; qu ¼X18

a¼0

faea; u ¼X18

a¼0

ga ð10Þ

Furthermore, the surface energy between the solid wall and fluid isdescribed by the second term on the right hand side in Eq. (5) withus being the phase parameter of the wall. Based on the minimum offree-energy at equilibrium condition, the phase parameter of thewall is given by [17],

wsðuÞ ¼ �juH

ffiffiffiffiffiffiffiA

2j

rð11Þ

where H is the wetting potential which is related to the contact an-gle hc as [17],

cosðhcÞ ¼ð1þHÞ3=2 � ð1�HÞ3=2

2ð12Þ

More detailed description of the free-energy LBM and the boundaryconditions implementation can be found in our previous work onnumerical simulation of droplet dynamics on a hydrophobic wallof a micro-channel [20].

3. Results and discussion

The grid effect on the capillary pressure was firstly tested to val-idate the pore-scale LBM used in this paper. Based on the recon-structed pore structure of the carbon paper GDL [21], theprimary drainage process, which is defined as the first time non-wetting phase water entering into the porous medium, was firstsimulated. The fiber diameter of the carbon paper was assumedto be 7.5 lm and the contact angle of water was assumed to havea uniform value of 115�. Simulations were performed by imposingdifferent inlet pressures of non-wetting phase water, while keep-ing the same outlet pressure of the wetting phase. The non-wettingphase saturations were obtained at different capillary pressures byincreasing the non-wetting phase pressures gradually. To check thegrid independency of the numerical results, simulations for theGDL with uniform wettability were carried out for three differentgrid sizes of 3.8 lm, 2.5 lm and 1.5 lm respectively. The resultsof the capillary pressure for these three grid sizes are presentedin Fig. 3. It can be seen that the capillary pressure-saturationcurves were nearly the same for the grid resolutions of 2.5 lmand 1.5 lm, while the curve for 3.8 lm case has obvious deviation.Therefore, the grid size of 2.5 lm was chosen in this study for thetradeoff between the simulation efficiency and accuracy. Also, theprimary drainage capillary pressure curve, presented in Fig. 3 for

0.0 .10 0.2 0.3 0.4 0.5 .60 0.7 0.8 9.0 1.00

2000

4000

6000

8000

10000

12000 3.8um 2.5um 1.5um

Cap

illary

Pre

ssur

e (P

a)

Saturation

Fig. 3. Comparison of primary drainage capillary pressure-saturation curves withdifferent grid resolutions.

grid size of 2.5 lm with uniform wettability distribution, was quiteclose to the results presented by Schulz et al. [22] using full mor-phology simulation method. This further confirms the validity ofthe LBM simulated results obtained in this paper.

3.1. Capillary pressure curve of carbon paper GDL with PTFE

It is well known that the purpose of adding PTFE in the GDL is toincrease its hydrophobicity. In practice, the PTFE content added inGDL can easily be controlled during the treatment process [4]. Thisprocess can also be simulated by a numerical method based on thereconstructed microstructure of a carbon paper GDL [21]. Assum-ing the contact angle is 80� for carbon fiber and 115� for pure PTFE,the simulated capillary pressure curves of the drainage and imbibi-tion processes for a carbon paper GDL with 10 wt.% PTFE are shownin Fig. 4(a). The experimental and simulated results of the primarydrainage capillary pressure curves are also presented for compari-son purpose. As reported by Anderson [23], for materials with non-wetting phase contact angle larger than 50�, higher non-wettingphase pressure than wetting phase is required to force the non-wetting phase entering. The simulated and experimental resultsof the capillary pressure for primary drainage process confirm thatthe capillary pressures are positive in the whole saturation range.However, the primary drainage is not an important process under

0.0 0.1 20. 0.3 0.4 0.5 0.6 0.7 80. 0.9 1.0-10000

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000 Drainage (10% PTFE) Imbibition (10% PTFE) Drainage (30% PTFE) Imbibition (30% PTFE)

Cap

illary

Pre

ssur

e (P

a)

Saturation(b) Comparison of simulated capillary pressure curves for carbon paper with PTFE contents of

10wt % and 30wt. %

Fig. 4. Capillary pressure curve versus water saturation for carbon paper GDL.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

-15000

-10000

-5000

0

5000

10000

15000 Experimental Data (Drainage) Experimental Data (Imbibition) LBM Results (Drainage) LBM Results (Imbibition) Fairweather et al. (Exp. Drainage) [14] Fairweather et al. (Exp. Imbibition) [14] Nguyen et al. Exp. [15]

Cap

illary

Pre

ssur

e (P

a)

Saturation(a) 10wt. % PTFE

-15000

-10000

-5000

0

5000

10000

15000 Experimental Data (Drainage) Experimental Data (Imibition) LBM Results (Drainage) LBM Results (Imibition)

Cap

illary

Pre

ssur

e (P

a)

Saturation(b) 30wt. % PTFE

Fig. 5. Comparison of capillary pressure-saturation curves obtained from LBMsimulation and experiments with different PTFE contents.

L. Hao, P. Cheng / International Journal of Heat and Mass Transfer 55 (2012) 133–139 137

the fuel cell operation conditions, because the pores in GDL are al-ways wetted by the saturated water vapor before the accumulationof liquid water, as they were recovered from a filling condition.Therefore, the present work is focused mainly on the secondarydrainage and imbibitions processes.

Fig. 4(a) shows that the simulated capillary pressure curves ofdrainage and imbibition processes have obvious hysteresis, andthe solid fitting lines are plotted for accessorial presentation. It isnoted that the drainage capillary pressure is higher than that inthe imbibition process at the same water saturation. Theoretically,the capillary pressure hysteresis of porous medium can be attrib-uted to contact angle hysteresis of fluid. In GDL, the static contact an-gles of water in contact with carbon fiber and PTFE do not deviatemuch from the neutral contact angle of 90�. Thus, when liquid wateris in contact with these materials, it can result in an advancing con-tact angle larger than 90� and the receding contact angle less than90� as shown in inserted schematic of Fig. 4(a). This implies thatthe hydrophobic property is dominant when water entering thepores (drainage process) while the hydrophilic property is dominantwhen water leaving the pores (imbibition process). Fig. 4(a) showsthat the drainage process presents hydrophobic characteristic (i.e.,the capillary pressure is positive) within the most saturation range(0.2–1.0), but the imbibition process presents hydrophilic character-istic (i.e., the capillary pressure is negative) within the most satura-tion range (0.0–0.9). Meanwhile, the simulated capillary pressurecurves of drainage and imbibition processes (represented by thetwo solid lines) crossing the zero capillary pressure line, implyingboth hydrophilicity and hydrophobicity properties of the PTFE trea-ted GDL. In addition, the imbibition capillary pressure curve inFig. 4(a) implies that it is very difficult to empty the water filled incarbon paper GDL completely. There still has a large amount of liquidwater trapped in GDL pores at the liquid and gas phase pressureequilibrium, even though the GDL is hydrophobically modified. Thatis the reason why dry air is required to evaporate the undrainedwater in the GDL after the shutdown of PEMFCs.

Fig. 4(b) shows the effects of PTFE contents of 10 wt.% and30 wt.% in carbon paper on the simulated drainage and imbibitioncapillary pressure curves. It can be seen that the increase of thePTFE from 10 wt.% to 30 wt.% in the GDL does not affect muchthe capillary pressure. Thus, many hydrophilic pores still exist inthe carbon paper GDL even if it has a high PTFE content of30 wt.%. These simulated results are consistent with the experi-mental studies that the performance of the PEMFCs did not im-proved much if the PTFE content of the GDL is increased from10 wt.% to 30 wt.% [3,4].

Fig. 5 shows a comparison of the capillary pressure-saturationcurves of carbon paper GDL obtained from both LBM simulationsand experiments for a sample with (a) 10 wt.% PTFE content and(b) with 30 wt.% PTFE. The experimental data obtained by Fair-weather et al. [14] and Nguyen et al. [15] for Toray carbon paper with10 wt.% PTFE are also presented in Fig. 5(a). Due to the absence ofother published capillary pressure data for carbon paper with30 wt.% PTFE at present, only the experimental data from this workare presented in Fig. 5(b). It is shown from Fig. 5(a) that the capillarypressure curves from LBM simulations are in a reasonable agree-ment with the experimental data not only obtained in present workbut also obtained by Fairweather et al. [14] and Nguyen et al. [15].The capillary pressure hysteresis was not observed by Nguyenet al. [15], so only the drainage capillary pressure is presented. Itshould be noted that the microstructures of the GDL reconstructedbased on a stochastically method in this work may not be exactlythe same with the samples used in our experiments, though theyhave the same macroscopic parameters such as porosity. In the re-gions of saturation less than 0.2 or larger than 0.9 (i.e., when mostof the pores are filled either by gas or liquid water), the microstruc-tures of the residual pores begin to play an important role on the cap-

illary pressure. For this reason, larger errors between the simulationand measurements of the capillary pressure are found in these satu-ration ranges. Similarly, larger errors are also observed at smallerand higher saturations between Fairweather et al. [14] and Nguyenet al. [15]’s experimental data because of the different microstruc-ture of carbon paper samples used in their experiments.

Based on the Young–Laplace equation given by Eq. (1), it can beestimated that the threshold size of the pore to be filled in thedrainage process and emptied in the imbibition process are 6 lmand 3 lm, respectively, at the capillary pressure of ±10 kPa. Onthe other hand, the measured pore size distribution showed thatcarbon paper GDL only had a very small proportion of the poreswhich is smaller than 6 lm [24]. Fig. 5 shows that the water satu-ration changes very little after the capillary pressure higher than10 kPa or lower than �10 kPa at the drainage and imbibition pro-cesses, respectively. In addition, Gostich et al. [25] estimated theresidual saturation of about 0.04 for Toray carbon paper at the cap-illary pressure of �15 kPa based on their experiment, which is alsoclose to the results obtained in the present work.

3.2. Leverett function

As mentioned previously, the dimensionless Leverett functionhas been used to describe the capillary pressure for porous

Table 1Fitting value of the parameters in Eq. (15) for Leverett function.

Processtype

J0 A1 A2 B1 B2 RMS(%)

Drainage 0.13363 0.00498 0.00397 9.404 �11.19 7.8Imbibition �0.11178 0.00139 0.01422 12.80 �7.614 8.4

138 L. Hao, P. Cheng / International Journal of Heat and Mass Transfer 55 (2012) 133–139

medium with similar pore structures. According to Eq. (2), the Lev-erett J-function is defined as:

JðsÞ ¼ pc

r cosðhcÞKU

� �1=2

ð13Þ

The above Leverett function, with Udell’s correlation given by Eq.(3), has been used for macroscopic modeling of two-phase flow inporous medium with uniform wettability distribution. However,the assumption of uniform porosity and contact angle is invalid inthe PTFE treated fibrous GDL. Since it is difficult to give a clear valueof the contact angle for a PTFE-treated fibrous GDL, the cosine termin Eq. (13) can be ignored to give the modified J-function as [25],

JðsÞ ¼ pc

rKU

� �1=2

ð14Þ

which is also a dimensionless function. Using the permeability andporosity of the carbon paper GDL, as well as the surface tension of0.0725 N m�1, the capillary pressure data obtained by simulationand experiment in present work are scaled to the Leverett J-func-tion by Eq. (14) as shown in Fig. 6. The permeability and porosityare chosen to be K = 7.34 � 10�12 m2, U = 0.775 for 10 wt.% PTFEand K = 5.61 � 10�12 m2, U = 0.708 for 30 wt.% PTFE according toour previous work [21].

The Leverett function of the carbon paper with 10 wt.% and30 wt.% PTFE can be approximately scaled to a single curve fordrainage or imbibition capillary pressures, respectively. For com-parison purposes, the standard Leverett–Udell correlation givenby Eq. (3) is also plotted in Fig. 6. It is shown that the standard Lev-erett–Udell function, represented as dashed lines, is positive forhydrophobic porous media while is negative for hydrophilic por-ous media, which do not fit well with our pore-scale simulationand experiment results. Nguyen et al. [15] proposed the followingexponential function for the modified J-function for a GDL withPTFE treatment,

JðsÞ ¼ J0 þ A1 expðB1ðs� 0:5ÞÞ � A2 expðB2ðs� 0:5ÞÞ ð15Þ

where J0, A1, A2, B1 and B2 are five fitting parameters and the num-ber of 0.5 indicating the mean value of the saturation becauseð0 6 s 6 1Þ. The modified J-function data for drainage and imbibi-tion capillary pressures can be fitted well by Eq. (15) using properfitting parameters with fitting errors (RMS) listed in Table 1. The fit-ted results are presented in Fig. 6 by two solid lines. It is shown thatthe two solid lines obtained in the present work are more accuratefor the prediction of capillary pressures in GDLs with hydrophilic

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0 Exp. LBM (Drainage,30% PTFE) Exp. LBM (Drainage,10% PTFE) Exp. LBM (Imbibition,30% PTFE) Exp. LBM (Imbibition,10% PTFE) Udell's Correlation Fitting Lines

Leve

rett

Func

tion

Saturation

Fig. 6. Leverett function given by Eq. (14) for carbon paper GDLs with 10 wt.% and30 wt.% PTFE contents.

and hydrophobic pores than those given by the Leverett–Udell func-tion represented by the dashed lines.

4. Concluding remarks

Both the pore scale simulations and experiments were per-formed in this paper to study the capillary pressure in the carbonpaper GDL with different PTFE contents. The LBM simulated capil-lary pressure-saturation relationships of the carbon paper areshown in good agreement with experimental results, indicatingthe coexistence of both hydrophilic and hydrophobic pores in thePTFE treated GDL. The different capillary pressure curves for drain-age and imbibition processes also indicate the significant hystere-sis of the porous GDL. A modified Leverett J-function with properfitting parameters is fitted based on the capillary pressure data ob-tained from the fibrous GDL in present work. It is expected that themacroscopic models with the capillary pressure obtained from thispaper could provide more accurate numerical predictions of watertransport in the carbon paper GDL of a PEMFC.

Acknowledgement

This work was supported by the National Natural Science Foun-dation of China through Grant No. 51036005.

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