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Gas diffusion through differently structured gas diffusion layers of PEM fuel cells Zhigang Zhan a,b,∗ , Jinsheng Xiao a,c , Yongsheng Zhang a,b , Mu Pan a , Runzhang Yuan a a State Key Laboratory of Advanced Technology for Materials Synthesis and Processing, Wuhan University of Technology, Hubei 430070, China b School of Energy and Power Engineering, Wuhan University of Technology, Hubei 430070, China c School of Automotive Engineering, Wuhan University of Technology, Hubei 430070, China

Received 13 August 2006; received in revised form 25 January 2007; accepted 31 March 2007 Available online 24 May 2007

Abstract Proton exchange membrane fuel cell (PEMFC) gas diffusion layers (GDLs) play important parts in diffusing gas, discharging liquid water, and conducting electricity, etc. When liquid water is discharged through GDL to gas channel, there will be some pores of GDLs occupied by liquid water. In this study, based on a one-dimensional model, the distribution of liquid water phase saturation is analyzed for different GDL structures including GDL with uniform porosity, GDL with sudden change porosity (GDL with microporous layer (MPL)) and GDL with gradient porosity distribution. The effect on gas diffusion of the changes of porosity and liquid saturation due to water remaining in GDL pores is calculated. The conclusions are that for uniform porosity GDL, the gas diffusion increases with the increase of porosity and contact angle and increases with the decrease of the thickness of GDL; for GDL with MPL, the larger the MPL porosity and the thinner the MPL thickness are, the stronger the gas diffusion is; for gradient change porosity GDL with the same average equivalent porosity, the larger the porosity gradient is, the more easily the gas diffuses. The optimization for GDL gradient structure shows that the GDL with a linear porosity distribution of 0.4x + 0.4 is the best of the computed cases. 䉷 2007 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. Keywords: Proton exchange membrane; Fuel cell; Gas diffusion layer; Porosity; Saturation; Gradient

1. Introduction Gas diffusion layer (GDL) of proton exchange membrane fuel cell (PEMFC) is of porous media such as carbon ﬁber or carbon cloth, etc. and plays an important part in gas diffusion, liquid water discharge and electricity conduct, etc. Properties of GDL such as its thickness, porosity, etc. have great effects on the performance of PEMFC. When fuel cells are operating, liquid water forms and is discharged from catalyst layers to gas channels through GDLs, occupation of some pores of GDLs by liquid water will occur, then decreasing the gas diffusion. This phenomenon is more serious when under high current density. A large amount of research has been done on water transfer and gas diffusion in GDLs. However, most studies simply treat the ∗ Corresponding author. State Key Laboratory of Advanced Technology for Materials Synthesis and Processing, Hubei 430070, China. Tel.: +86 027 62061240. E-mail address: [email protected] (Z. Zhan).

porosity of GDLs as constant, which certainly cannot reﬂect the importance of GDL porosity on PEM fuel cell performance [1–4]. Sub-layer or micro-porous layer (MPL) inserted between catalyst layer and GDL is one of the widely used structures to improve water management and increase gas diffusion capacity. Using “unsaturated ﬂow theory” and “multi-phases mixture model”, Pasaogullari et al. [5] studied the liquid saturation distribution of GDL and MPL, compared the differences between the two models and concluded that the gas pressure difference between the two sides of GDL has an important effect on liquid water transport and gas diffusion; Jin et al. correlated gas diffusion coefﬁcient in GDL with liquid saturation and porosity, and set up a porosity model for the porous medium made of ﬁbers; using the theory of capillary pressure, they studied the distribution of liquid saturation in GDLs with uniform porosity and with MPL, and studied the effect on fuel cell performance of the ﬁber diameter, porosity, etc. [6]. Qi et al. tested the effects on fuel cell performance of MPL of different thicknesses and different contents of PTFE, and concluded that MPL is

0360-3199/$ - see front matter 䉷 2007 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2007.03.041

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Z. Zhan et al. / International Journal of Hydrogen Energy 32 (2007) 4443 – 4451

Nomenclature d D F J K krl kk M p qH2 O Q s T r V x z e ε

diameter of ﬁber, um mass diffusivity, m2 /s Faraday constant, C/mol current density, A/m2 permeability relative permeability Kozeny constant mole molecular weight, g/mol pressure, Pa liquid water ﬂux, kg/m2 s liquid water volume remaining in GDL, m3 saturation temperature radius of pore, um volume of liquid or porosity, m3 dimensionless thickness of GDL and MPL effective diffusion factor effective diffusion coefﬁcient of water through membrane surface tension, N/m thickness of GDL or MPL, um porosity

very helpful for preventing electrode ﬂooding [7]. Kong et al. studied the effects of the distribution of pore diameter on gas transport and cell’s performance, and they thought that pore diameter of GDL should be of bimodal, water discharged through large pores and gas diffused through small pores [8]. Sun [9], Jeng et al. [10] studied the effects of the thickness of GDL on gas reaching the catalyst layer and electron transport, electrical conductivity and the ratio of channel/bank width, and thought that the effects are very signiﬁcant. Wilkinson et al. put forward the concept of gradient, and thought both the operation parameter and the MEA structure should be gradiently optimized [11]. Chu et al. suggested gradient porosity GDL and investigated the effect of average porosity on the oxygen transport and then on the variation of limit current density, i.e. the inﬂuence of porosity on the concentration polarization, where the effect of phase change and liquid water was not considered [12]. Roshandel et al. revealed the effects on PEM fuel cell performance of porosity distribution variation resulting from the compression pressure corresponding to assembly process and presence of liquid water [13]. Considering the porosity distribution variation, the present authors studied the liquid water ﬂux through differently structured GDLs, but without studying further the effects of porosity and liquid saturation on gas diffusion [14]. This paper focuses on the effects of the porosity distribution variation and the liquid saturation on gas diffusion through differently structured GDL of PEMFCs, which include GDL with uniform porosity, GDL with MPL, and GDL with gradient porosity distribution. In the following, the mathematical model based on capillary theory and multi-gas diffusion theory

contact angle dynamic viscosity, Pa s kinematic viscosity, m2 /s density, kg/m3

Subscript c f g H2 O i K l p rl w nw eff

capillary or contact ﬁber gas water the ith species of a mixture Kozeny constant liquid pore relatively wetting phase non-wetting phase effective

is discussed in Section 2, then the solution procedure, results, and analyses are followed in Sections 3 and 4, and ﬁnally some interesting conclusions are drawn. 2. Model development 2.1. Capillary pressure and saturation distribution Capillary pressure is the pressure difference between the nonwetting phase and the wetting phase; for the hydrophobic GDL, it is expressed as [15]. pc = pnw − pw = pl − pg , pc =

cos c (K/ε)0.5

(1.417s − 2.12s 2 + 1.26s 3 )m

(1) c > 90◦ ,

(2)

where s = Vl /Vp is liquid saturation, K = ε3 df2 /(16kK (1 − ε)2 ) is the absolute permeability, and kr1 = s 3 is the relative permeability. Liquid water transport in GDL is caused by capillary pressure difference. Suppose under steady condition, the water produced (qH2 O ) in electrochemical reaction in catalyst layer and the water transferred effectively from the membrane are changed into liquid fully and discharged through GDL to gas channel, hence Kk r1 Kk r1 dpc ds qH2 O = − 1 ∇pc = − 1 , (3) 1 1 ds dx

Z. Zhan et al. / International Journal of Hydrogen Energy 32 (2007) 4443 – 4451

or

2.3. Efﬁcient diffusion factor of GDL

qH2 O = −

cos c ε 2 df √ 4(1 − ε) kK

× s 3 (1.417 − 4.24s + 3.789s 2 )

ds . dx

(4)

2.1.1. GDL with uniform porosity For GDL with uniform porosity, there is no change of the porosity along the GDL thickness. Integrating Eq. (4) one gets cos c ε 2 df √ 4(1 − ε) kK 1.417s 4 4.24s 5 3.789s 6 × − + + C. 4 5 6

qH2 O x = −

(5)

2.1.2. GDL with gradient porosity Suppose the porosity varies along the thickness following the polynomial rules, a linear type and a parabola type are chosen. Porosity varying along the thickness follows linear rules, i.e. ε = a1 x + a0 , ε 3 s (1.417 − 4.24s + 3.789s 2 ) qH2 O = − 1−ε √ 4 kK a1 ε + s3 cos c df (1 − ε)2 ds 2 3 × (1.417s − 2.12s + 1.263s ) . dx

(6)

Porosity varying along the thickness follows parabola rule, i.e., ε = a2 x 2 + a1 x + a0 , then qH2 O = −

√

Gas diffusion ﬂux through porous media is described by General Fick’s Law: eff jci , (9) qm,i = −Dm,i jx 1.5 T p0 eff Dm,i = Dm,i f1 (ε)f2 (s) , (10) p T0 where jci /jx is the species concentration gradient; Dm,i is the multi-component mass diffusivity of species i in plain medium when temperature is T0 and pressure is p0 , p and T are the pressure and the temperature, and is pressure coefﬁcient. eff of porous diffusion medium is The effective diffusivity Dm,i correlated with porosity ε and liquid saturation s. According to the effective medium theory [2,6,16,17], f1 (ε)=ε 1.5 ; for multilength scale, particle-based porous medium such as catalytic converter wash-coat [18], f1 (ε) = 1 − (1 − ε)0.46 ; For random ﬁbrous porous medium [19], f1 (ε) = ε(ε − εp )/(1 − εp ) ; εp is a percolation threshold and is an empirical constant. The liquid saturation restricts the gas diffusion by reducing the diffusion area and creating tortuous diffusion path, which is usually modeled as f2 (s) = (1 − s)m . According to the effective medium theory [2,16,17], m = 1.5, according to [18] m = 0.71, but according to Martys [20], the ideal wetting/non-wetting system the 1-s dependence has a power of around 2, and for contact angle of 90◦ , this is much larger than 2. This paper mainly discusses the effects on gas diffusion due to the changes of porosity and liquid saturation, without considering the changes of temperature and pressure; therefore, we

ε/(1 − ε)s 3 (1.417 − 4.24s + 3.789s 2 )

(/ cos c )(4 kK /df ) + ε(2a2 x + a1

)/(1 − ε)2 s 3 (1.417s

− 2.12s 2

+ 1.263s 3 )

·

ds dx

2.2. Liquid water volume remaining in GDL

deﬁne the effective diffusion factor as

GDL is made up of carbon ﬁber or of other porous materials, which functions in diffusing gas, draining water, conducting electrons, etc. The electric conductivity of GDL has never been an obstacle to the effective and steady operation of PEMFC, but ﬂooding and then gas diffusion decrease cause serious problems frequently. Therefore, GDL structure should be optimally designed to increase liquid water ﬂux through GDL and to decrease liquid water volume remaining in GDL so as to improve oxygen diffusion from gas channel to catalyst layer. Considering the saturation and porosity variation along the thickness of GDL, liquid water volume remaining in GDL under steady conditions can be gotten from

z(x) = f1 (ε)f2 (s)

Ql =

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0

Vl (x) dx =

ε(x)s(x) dx.

(8)

0

This equation presents the volume occupied by liquid water when a ﬁxed liquid water ﬂux ﬂows through GDL, hence Ql should be as small as possible.

(7)

(11)

and choose f1 (ε)=ε 1.5 , f2 (s)=(1−s)2

to correct the diffusivity Dm,i in GDLs of PEM fuel cells [21–25], i.e.,

z(x) = ε 1.5 (1 − s)2 ,

(12)

The average diffusion factor in GDL is expressed as

z¯ = z(x) dx .

(13)

0

It directly expresses the change of gas diffusion caused by the volume of the liquid water remaining in GDL and by the change of GDL porosity; z¯ should be as large as possible. 3. Solution procedure A software is developed by the authors with Visual Basic to solve the equations. The values of the main variables chosen are shown as in Table 1.

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Z. Zhan et al. / International Journal of Hydrogen Energy 32 (2007) 4443 – 4451

Table 1 Values of the variables

GDL increases and blocks some gas diffusion paths, but the total gas diffusion paths still increase, therefore, the diffusion factor increases with porosity (Table 2).

Variable

Units

Values

Current density Effective diffusion factor of membrane Kozeny constant Faraday constant Surface tension Viscosity of liquid water Mole molecular weight of liquid

A/cm2 m2 /s

1 0.5 6 96 487 0.0625 3.65 × 10−7 18

Base case GDL thickness Base contact angle Base porosity Base ﬁber diameter

m

Base case MPL thickness Base contact angle Base ﬁber diameter

m

C/mol N/m m2 /s g/mol ◦

m ◦

m

250 140 0.5 2 50 140 1

The liquid saturation distribution in GDL is analyzed based on such an assumption that the water produced in electrochemical reaction with the current density being 10 000 A/m2 in the catalyst layer and the water transferred from the membrane with effective diffusion factor of 0.5 are all changed into liquid and discharged out of GDL, i.e., qH2 O = MH2 O

J (1 + 2e ). 2F

(14)

4. Results and discussions 4.1. A GDL with uniform porosity For GDLs with uniform porosity, the saturation distribution in GDLs is calculated from Eq. (5), while the saturation on the interface of GDL/gas channel is chosen as 0.01 to determine the integral constant C. In Fig. 1 (a, c, and e), y-coordinate is the liquid saturation, x-coordinate is the dimensionless thickness of GDL; in Fig. 1 (b, d, and f), y-coordinate is the diffusion factor, x-coordinate is the dimensionless thickness of GDL, the following ﬁgures have similar coordinates. Fig. 1 (a and b) shows the liquid water saturation and diffusion factor for different porosity GDLs with ﬁxed thickness of 250 um and ﬁxed contact angle of 140◦ ; Fig. 1 (c and d) for different contact angle GDLs with ﬁxed thickness of 250 um and porosity of 0.5; Fig. 1 (e and f) for different thickness GDLs with ﬁxed porosity of 0.5 and ﬁxed contact angle of 140◦ , respectively. x = 0 is the catalyst layer/GDL interface, x = 1 is the GDL/ﬂow channel interface. We can see clearly that the liquid saturation in a certain position decreases with the increase of porosity and contact angle, and it increases with the increase of the thickness of GDL. As the liquid water volume remaining in GDL is an integral of the saturation in GDL, its change rule is therefore the same as that of saturation for different GDLs (Table 2). As for the diffusion factor, the change trend is the opposite: it increases with the increase of porosity and contact angle and decreases with the increase of the GDL thickness (Fig. 1 (b, d, and f)). As porosity increases, the liquid water volume remaining in

4.2. A GDL with an MPL In order to increase the liquid ﬂux of GDL and to alleviate MEA ﬂooding, many PEMFCs are inserted with MPLs between the catalyst and the normal GDL. The saturation distribution in MPL is also governed by Eq. (5), but some material or structure properties, such as porosity, particle diameter, etc. are changed. The capillary pressure at the interface of MPL/GDL is continuous while the saturation has a sudden change; thus the saturation distribution in GDL + MPL is different from a single GDL, and the liquid water volume remaining in GDL is improved; as a result, it improves the diffusion factor. 4.2.1. A GDL with a different porosity for the MPL Fig. 2 shows the distribution of liquid saturation and diffusion factor along the thickness of GDL and MPL. The thickness of GDL = 0.2 mm, ﬁber diameter d = 2 um, contact angle =140◦ , porosity ε =0.5; thickness of MPL =0.05 mm, ﬁber diameter d = 1 um, contact angle = 140◦ , porosity ε = 0.2, 0.3, 0.4, 0.5, respectively. x-coordinate in Fig. 2 (a and b) is the dimensionless thickness of GDL + MPL, y-coordinate in Fig. 2(a) is liquid saturation but in Fig. 2(b) is diffusion factor. x = 0 is the catalyst layer/MPL interface, x = 0.2 is the catalyst layer/GDL interface, x = 1 is the GDL/ﬂow channel interface. It can be seen from Fig. 2(a) that the saturation at the interface of catalyst layer/MPL decreases with the increase of the MPL porosity; when porosity of MPL is 0.4 and 0.5, the saturation value at this interface is less than that of a single GDL, which means there is more porosity remaining for oxygen diffusion. When the porosity of MPL is 0.5, the diffusion factor is 0.320578, larger than 0.31116 of a single uniform GDL with 0.5 porosity, which helps the oxygen to diffuse from the ﬂow channel to the catalyst layer. 4.2.2. A GDL with different thickness of MPL Fig. 3 shows the distribution of liquid saturation and diffusion factor along thickness direction in MPL + GDL with different thickness of MPL, the jump in the curve is the MPL/GDL interface. The total thickness of MPL and GDL maintains 0.25 mm, porosity in GDL is 0.5, contact angle is 140◦ , ﬁber diameter d = 2 m, thickness = 0.2, 0.18, 0.16 mm, respectively; In MPL, contact angle = 140◦ , ﬁber diameter d = 1 m, thickness is 0.05, 0.07, 0.09 mm, respectively, porosity ε = 0.5 (Fig. 3 (a and b)) and ε = 0.3 (Fig. 3 (c and d)). It can be seen that liquid saturation at catalyst layer/MPL interface decreases when thickness of MPL decreases, but the diffusion factor at the same place increases. When porosity in MPL is 0.5, compared with the single uniform porosity GDL, the liquid saturation at catalyst layer/MPL interface is the least when thickness of MPL is 0.05 mm; the total liquid water volume remaining in GDL and MPL is the most, but the diffusion factor at catalyst layer/MPL interface is still the largest, so the effect is better.

Z. Zhan et al. / International Journal of Hydrogen Energy 32 (2007) 4443 – 4451

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Fig. 1. Liquid saturation and diffusion factor in GDL with uniform porosity: (a) Liquid saturation (GDL thickness 250 m, contact angle 140◦ ); (b) diffusion factor (GDL thickness 250 m, contact angle 140◦ ); (c) liquid saturation (GDL thickness 250 m, porosity 0.5); (d) diffusion factor (GDL thickness 250 m, porosity 0.5); (e) liquid saturation (Porosity 0.5, contact angle 140◦ ); (f) diffusion factor (porosity 0.5, contact angle 140◦ ).

Table 2 Liquid water remaining and average effective diffusion factor in uniform porosity GDL

0.3

0.4

0.5

0.6

Ql (m3 ) z¯

9.61267E − 06 0.13312

1.05183E − 05 0.21523

1.11179E − 05 0.31116

1.1425E − 05 0.41993

It can be seen from Fig. 3(c) that when MPL porosity is 0.3, the liquid water saturation at catalyst layer/MPL interface for all different thickness MPLs is higher than that of a

uniform porosity with 0.5 GDL and the diffusion factor is much lower Fig. 3(d); of course such kind of MPL is not what we need.

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Fig. 2. Liquid saturation and diffusion factors in GDL + MPL with different porosities: (a) liquid saturation; (b) diffusion factor.

4.3. A GDL with a gradient in porosity When some kind of MPL is inserted between catalyst layer and GDL, the saturation distribution is changed, hence the liquid ﬂux through GDL or MPL is improved and MEA ﬂooding alleviated. The basic reason is that the capillary pressure difference driving the liquid out of GDL increases. It is speculated that if a GDL with gradient porosity is used the performance should be optimized. In the following computed cases, the thickness, contact angle, and ﬁber diameter of GDL are ﬁxed as 250 um, 140◦ and 2 um, respectively. 4.3.1. A GDL porosity varying along the thickness following a linear rule Suppose that the porosity of GDL is of linear variation, namely ε = a1 x + a0 , values of a1 and a0 should ensure that ε is between 0 and 1. a1 chosen as 0.2, 0.3, and 0.4, a0 as 0.2, 0.3, 0.4, and 0.5, totaling 12 cases, the average porosity changes from 0.3 to 0.7. Fig. 4 and Table 3 show part of the results. Fig. 4(a) is about the liquid volume remaining in GDL vs average porosity; Fig. 4(b) is about the effective diffusion factor through GDL vs average porosity. Comparing the uniform with the linear variation GDLs for the same average porosity, as shown in Fig. 4, it can be seen that the liquid water remaining in GDL reduces, and effective diffusion factor increases along

Fig. 3. Liquid saturation and diffusion factor in GDL + MPL with different thicknesses. (a) Liquid saturation (porosity of MPL is 0.5); (b) diffusion factor (porosity of MPL is 0.5); (c) liquid saturation (porosity of MPL is 0.3); (d) diffusion factor (porosity of MPL is 0.3).

Z. Zhan et al. / International Journal of Hydrogen Energy 32 (2007) 4443 – 4451

with the increase of a1 . For example, when the average porosity is 0.5, compared with uniform porosity GDL, the liquid water remaining in GDL of ε=0.2x+0.4 and ε=0.4x+0.3 reduces by 23.73% and 35.42%, respectively; the diffusion factor increases by 3.5% and 6%, respectively. The effect is obvious. The values in Table 3 are porosity change rule (a kind of linear function), the average porosity, liquid water quantity remaining in GDL, and the effective diffusion factor, respectively. The results show that when the gradient of the porosity a1 is ﬁxed, the average porosity increases with the increases of a0 , the liquid water remaining in GDL, and the effective diffusion factor increases with the increase of average porosity, which is the same as that of a single GDL (a1 = 0). Considering that enough oxygen is needed for electrochemical reaction, low resistance of electron transfer and easy manufacture of GDL, GDL with linear porosity 0.4x + 0.4 is the

Fig. 4. Comparison of GDL with linear porosity: (a) Liquid water remaining in GDL; (b) effective diffusion factor.

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best among the computed cases, i.e., its average porosity is 0.6, liquid water quantity remaining in GDL is 7.39412E-06 kg, and the effective diffusion factor is 0.43958.

Fig. 5. Performance comparison of GDL with different porosity structures: (a) GDL with different porosity structure; (b) liquid water remaining in GDL in GDL; (c) effective diffusion factor. 1: = 0.6; 2: = 0.4x + 0.4; 3: = 0.2x + 0.5; 4: = 0.15x 2 + 0.3x + 0.4.

Table 3 Liquid water remaining and average effective diffusion factor in gradient porosity GDL

0.2x + 0.2

0.2x + 0.3

0.2x + 0.4

0.2x + 0.5

Average porosity Ql (m3 ) z¯

0.3 6.8802E − 06 0.14251

0.4 7.8659E − 06 0.22521

0.5 8.4792E − 06 0.32192

0.6 8.7299E − 06 0.43181

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4.3.2. A GDL porosity varying along the thickness following a parabolic rule Suppose the porosity varies along the thickness following the parabola rule, i.e. ε = a2 x 2 + a1 x + a0 . Values of a2 , a1 and a0 should ensure that ε is between 0 and 1. a2 chosen as 0.1 and 0.2, a1 as 0.1, 0.2, and 0.3, a0 as 0.2, 0.3, and 0.4, totaling seven cases. The average porosity changes from 0.42 to 0.62. The results show that the liquid water remaining in GDL and the effective diffusion factor increase with the increase of average porosity, which is the same as that of a single GDL. Considering the difﬁculties to prepare a GDL with a parabola gradient porosity, it has no advantage over the linear type GDL. 4.4. Comparison of GDLs with the same average porosity but with a different porosity structure For GDLs with the same average porosity but different porosity structures a comparison is made to determine the one having the optimal performance. Fig. 5(a) shows the different porosity structures, including a single GDL, two linear GDLs and a parabola GDL; the average porosity is 0.6. Fig. 5(b and c) shows the liquid volume remaining in GDLs and the effective diffusion factor of GDLs, respectively, x-coordinate of the ﬁgures is the structure type. As for the liquid volume remaining in GDL, parabola type is the least, and linear type 0.4x + 0.4 is the second least, which decreases by 3.08% and 0.79% when compared with the single GDL (Fig. 5(b)). As for effective diffusion factor, GDL with linear porosity 0.4x + 0.4 is the best, parabola structure GDL is the second (Fig. 5(c)). Such a result is caused by the effects of different distributions of porosity and liquid saturation in GDLs. Considering it is more difﬁcult to prepare a parabola GDL than to prepare a linear one, 0.4x +0.4 should be the best among the computed cases.

5. Conclusion 1. Under steady conditions liquid water quantity remaining in GDLs increases with the increase of porosity and the thickness of GDL, and decreases with the increase of contact angle; Effective gas diffusion factor increases with the increase of porosity, contact angle and decreases with the increase of the thickness of GDL. 2. When an MPL is placed between the catalyst layer and GDL, the liquid saturation is redistributed across the MPL and GDL. The effective gas diffusion factor increases with the increase of the MPL porosity and the decrease of the MPL thickness. 3. GDL with gradient porosity is more favorable for the liquid water to be discharged out and less liquid remaining in GDL, which is beneﬁcial to the gas diffusion. For GDLs with the same porosity, the larger the gradient is, the less the liquid remains in GDLs, and the larger the gas diffusion factor is. Of the computed cases, GDL with a linear porosity 0.4x + 0.4 is the best.

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