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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING Asia-Pac. J. Chem. Eng. 2013; 8: 104–114 Published online 17 April 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/apj.1635

Research article

Numerical modeling of liquid water transport inside and across membrane in PEM fuel cells Hua Meng,* Bo Han, and Bo Ruan School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, Zhejiang 310027, China Received 1 September 2011; Revised 7 February 2012; Accepted 13 February 2012

ABSTRACT: Liquid water transport inside a proton exchange membrane (PEM) fuel cell plays a crucial role in cell performance and durability. In this paper, a two-phase multi-dimensional PEM fuel cell model has been developed to properly handle water transport inside and across the polymer membrane in both absorbed water phase and free liquid phase. The numerical model has been employed to investigate effects of the cell operating current and cell temperature on liquid water distributions on both anode and cathode sides, as well as inside the membrane. Numerical results reveal that liquid water transport inside and across the polymer membrane plays an important role in PEM fuel cell water distributions. Increasing the fuel cell operating current density would result in more liquid water accumulation on both anode and cathode sides, a phenomenon dictated mainly by more water production. However, liquid water evaporation on the anode side also makes a profound impact at an increased operating current density. Decreasing the cell operating temperature, particularly from 80 to 60 C, would result in more liquid water inside the fuel cell. This phenomenon is mainly controlled by the decreased liquid water evaporation process on both anode and cathode sides. Further decreasing the cell boundary temperature from 60 to 40 C would only lead to very slight liquid water increase. The calculated results show qualitative agreement with available neutron imaging data. © 2012 Curtin University of Technology and John Wiley & Sons, Ltd. KEYWORDS: PEM fuel cell; water management; liquid water transport; membrane; numerical modeling

INTRODUCTION Water management plays a crucial role in proton exchange membrane (PEM) fuel cell performance and durability, and is a key practical issue in cell design and optimization. In order to fully comprehend water transport mechanisms, particularly liquid water transport behaviors, in PEM fuel cells, extensive research efforts have been expended in this area in the past years, including both experimental[1–10] and numerical studies.[11–35] Because of the existing limitations and difficulties in experimentally detecting and observing liquid water transport and distributions in porous materials in PEM fuel cells, numerical modeling and simulation becomes an indispensable tool for PEM fuel cell water management. Many multi-dimensional two-phase PEM fuel cell models have been developed for fundamental understanding of the liquid water transport phenomena in PEM fuel cells,[11–35] and good progress has been achieved in recent years. Because of water production

*Correspondence to: Dr. Hua Meng, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, Zhejiang 310027, China. E-mail: [email protected] © 2012 Curtin University of Technology and John Wiley & Sons, Ltd. Curtin University is a trademark of Curtin University of Technology

and oxygen transport limitation on the cathode side of a PEM fuel cell, a majority of these two-phase numerical models focused their attentions mainly on liquid water transport processes in the fuel cell cathode. However, experimental results[9,10] from the neutron imaging technique clearly illustrated that liquid water could transport through the polymer membrane and penetrate into the anode side of a PEM fuel cell. Since the hydrogen transport process on the anode side would not be severely hindered by liquid water in anode porous materials, comparing with its oxygen transport counterpart on the cell cathode, this liquid water transport pathway could therefore play an important role in PEM fuel cell water management. For example, a PEM fuel cell might be designed in a way that a large amount of liquid water produced in the fuel cell cathode could be expelled into and drained from the cell anode. In this way, the liquid water flooding problem in the PEM fuel cell cathode would be alleviated. Among the many existing multi-dimensional twophase PEM fuel cell models for studying liquid water transport phenomena in PEM fuel cells, only limited ones considered liquid water transport and distributions in the fuel cell anode.[14,20,34] Berning and Djilali[14] developed a three-dimensional two-phase PEM fuel cell model based on the unsaturated multi-fluid theory,

Asia-Pacific Journal of Chemical Engineering

LIQUID WATER TRANSPORT INSIDE AND ACROSS MEMBRANE

considering water phase change and heat transfer on the anode and cathode sides, but neglecting the entire membrane region. The liquid water transport and distribution on both anode and cathode sides of the fuel cell were carefully analyzed. Ju et al.[20] built a threedimensional two-phase PEM fuel cell model based on the multi-phase mixture approach,[35] accounting for liquid water transport across the polymer membrane using the liquid pressure difference on the cathode and anode sides of a PEM fuel cell. A related source term was added in the liquid water transport equation, and liquid water distributions on both anode and cathode sides were discussed. A similar treatment was recently applied by Wang and Chen[34] for comparing the calculated water variations in the throughmembrane direction of a two-dimensional PEM fuel cell cross section with the available experimental data, showing reasonably good agreement. However, in these existing numerical models, physical details concerning liquid water absorption into and transport inside the fuel cell membrane were neglected. In the open literature, Weber and Newman[36] presented a physical model for describing water transport inside the polymer membrane. This physical model was a further refinement from early models proposed by other researchers[37,38] to treat the polymer membrane as a porous material for liquid water transport. In this model, the membrane structure was assumed to change with water content; once the membrane was in contact with liquid water because of the slightly hydrophobic polymer matrix, liquid water would be able to open up and expand liquidfilled channels inside the membrane. As a result, the polymer membrane would act as a porous material to allow free liquid water transport inside and across it driven by a pressure difference. They later on further developed a mathematical formulation[39] based on this physical model, applied this method for numerical studies, and compared the numerically calculated liquid water variations in the through-membrane direction with experimental data.[40] This mathematical formulation is only one-dimensional and is thus insufficient to paint a complete picture of liquid water transport and distributions in PEM fuel cells. Furthermore, the mathematical model is quite different from the now well-developed multi-dimensional two-phase PEM fuel cell models built in the conventional computational fluid dynamics (CFD) framework.[11–35] In this paper, building on a multi-phase multidimensional PEM fuel cell model, which has been developed and used successfully for investigating liquid water transport phenomena[22,23,31,32] and cold-start processes[41,42] in PEM fuel cells, the physical model proposed by Weber and Newman[36] for describing free liquid water transport inside and across the polymer membrane driven by liquid pressure difference is theoretically formulated and numerically accommodated into a © 2012 Curtin University of Technology and John Wiley & Sons, Ltd.

conventional multi-dimensional CFD framework. Moreover, a previously developed interfacial numerical treatment[31,32] is further incorporated to capture the correct physics of liquid water transport across the multi-layer porous materials in a PEM fuel cell. For example, the discontinuous liquid saturation jump and continuous liquid pressure variation at an interface between two different porous materials, including at an interface between the catalyst layer and membrane, are correctly handled. Therefore, a two-phase multidimensional PEM fuel cell model is established in this paper for considering liquid water transport in both the fuel cell cathode and anode, as well as inside and across the cell membrane. This PEM fuel cell model is then applied to study the effects of the operating current and cell temperature on liquid water distributions in a PEM fuel cell. The calculated numerical results are shown to agree qualitatively with the available neutron imaging data.[10,40]

THEORETICAL FORMULATION AND NUMERICAL APPROACH The present PEM fuel cell model solves a complete set of conservation equations of mass, momentum, energy, water content, liquid water, electron, and proton transport to obtain the flow-field, temperature variation, liquid saturation distribution, and the electronic and ionic phase-potential fields. These conservation equations have been summarized in Table 1, along with the relevant electrochemical and physical relationships listed in Table 2. More details concerning this PEM fuel cell model could be found in Refs[22,23,31,32] and are thus not repeated in this paper. In this section, we focus our attention solely on presenting the new modeling features in the liquid water transport equation, namely the liquid water transport phenomena inside and across the PEM fuel cell membrane. As described by Weber and Newman,[36] as well as by other early researchers,[37,38] once in contact with liquid water, the polymer membrane would undergo structural change and subsequently become a porous material, similar to the catalyst and gas diffusion layer, for the liquid water transport process. This physical phenomenon becomes possible because of the slightly hydrophobic polymer matrix (the contact angle is calculated to be around 90.02o[39]) in the membrane material, which is basically teflon. This would allow liquid water to obtain sufficient pressure to open up and expand liquid-filled channels in the membrane. The free liquid water transport process inside and across the porous membrane material would then be driven by liquid pressure difference.[36,39] Therefore, the following liquid water transport equation would be applicable not only in the anode and cathode porous materials but also inside the membrane: Asia-Pac. J. Chem. Eng. 2013; 8: 104–114 DOI: 10.1002/apj

105

106


H. MENG, B. HAN AND B. RUAN

Table 1. Conservation equations.

Description

Equation

Mass

rðr uÞ ¼ Sm

Momentum

1 e2 ð1sÞ2

!

rðru uÞ ¼ rp þ rt þ Su ! rðuci Þ ¼ r Deff i rci þ Si

Species

!!

Liquid water

r (Dl r l) + Sl + Sld = 0 ! r rCp uT þ r rl Cp;l u!l T ¼ r k eff rT þ ST h i r½Dc rs r rl Kmrll K rp ¼ Svl Ww þ Sld Ww

Proton

r (keff r fe) + Se = 0

Electron

r (seff r fs) + Ss = 0

Water content in membrane Energy

Table 2. Electrochemical and physical relationships.

Description Transfer current density

Expression aa þac cH2 j ¼ aeff jref 0;a cH2;ref RT F in anode ac cO2 j ¼ aeff jref 0;c cO2;ref exp RT F in cathode

Unit

1=2

A m3

Over potential

= fs fe in anode side = fs fe Uo in cathode side

V

Theoretical open-circuit potential

Uo = 1.23 0.9 10 3(T 298) 1:0 for l≤14 nd ¼ 1:5=8ðl-14Þ þ 1:0 otherwise

V

Electro-osmotic drag coefficient Water activity Water saturation pressure Partial pressure of water vapor Membrane water diffusivity Water content diffusivity

a ¼ CwpRsatu T log10 psat ¼ 2:1794 þ 0:02953ðT 273:15Þ 9:1837105 ðT 273:15Þ2 þ 1:4454107 ðT 273:15Þ3 v p = CwRuT ( 0:28l ½2346=T 7 3:110 l e 1 e 0 < l≤3 Dm w ¼ 8 4:1710 l 1 þ 161el e½2346=T otherwise rm Dl ¼ EW Dm w

atm Pa m2 s1 Mol m1 s1

Capillary pressure

1 k ¼ ð0:5139l 0:326Þ exp 1268 303 T1 0:043 þ 17:81a 39:85a2 þ 36:0a3 s≤0 l¼ l ¼ 14 þ 2:8s s>0 e 1=2 pc ¼ K s cosθc J ðsÞ

Liquid pressure

pl = p pc

Pa

Capillary liquid water diffusivity

c Dc ¼ rl Kmrll K @p @s

Kg m1 s1

Condensation or evaporation rate

Svl = hpc(pv psat)

Mol m3

mem Sld ¼ hld rEW ðl le Þ

Mol m3

Krl = s3, Krg = (1 s)3

m2

Effective reaction area

aeff = a(1 s)

m2

Effective diffusivity

1:5 1:5 Deff i ¼ D i e ð1 sÞ

m2 s1

Proton conductivity Equilibrium water content

Liquid water transition rate in membrane Relative permeability

© 2012 Curtin University of Technology and John Wiley & Sons, Ltd.

S m1

Pa

Asia-Pac. J. Chem. Eng. 2013; 8: 104–114 DOI: 10.1002/apj


rl Krl K r½Dc rs r rp ¼ Svl Ww þ Sld Ww ml


(1)

The detailed mathematical derivation can be found in Ref.[22] In this equation, the liquid saturation is defined as s¼

Vl Vp

(2)

which denotes the ratio of the liquid volume to the pore volume. The parameter, Dc, is the capillary liquid water diffusivity, expressed as Dc ¼

rl Krl K @pc ml @s

(3)

This capillary liquid water diffusivity in porous materials can be calculated using a relationship between the capillary pressure and liquid saturation: pc ¼

e 1=2 K

for water transport processes inside and across the polymer membrane in both the absorbed water phase and the free liquid phase. The two water transport equations are closely coupled through a source term, Sld, which considers a transition process between the free liquid water and the bonded water content, l, inside the cell membrane. In this way, the water content and free liquid water inside the membrane could exist in thermodynamic non-equilibrium. Effects of the electro-osmotic drag and back diffusion on water transport inside and across the membrane are taken into account in the water content transport equation. It should be emphasized that the pressure-driven water transport process inside and across the polymer membrane only occurs once the membrane is in contact with liquid water. According to existing numerical methods proposed for the phase transition between the free liquid water and bonded water content in the cathode catalyst layer [15,21,30,43] , the following expression is employed to calculate this additional source term: Sld ¼ hld

s cosθc J ðsÞ

(7)

(4)

In the present numerical studies, the well-known Leverett’s function, J(s), has been used for calculating liquid water transport processes in the anode and cathode porous materials, including the catalyst and gas diffusion layers J ðsÞ ¼ 1:417s 2:120s2 þ 1:263s3

rmem ðl le Þ EW

θc > 90o

(5)

Inside the cell membrane, however, the applicability of this J(s) function for calculating the capillary liquid water diffusivity is found to be very poor. Because of the lack of experimental data at the present time, we choose a constant capillary diffusivity inside the polymer membrane for present qualitative studies. Once liquid water exists, liquid-filled channels would be opened up in the cell membrane, and the liquid water capillary diffusivity is defined as Dc ¼ 2:0 105 kg m1 s1

where the transition rate coefficient, hld, is chosen as 1.0 s1.[30] In Eqn (7), the parameter, le, is the equilibrium membrane water content in contact with liquid water, and it is defined in the following form: le ¼ 14 þ 2:8s

(8)

In order to incorporate the above-derived liquid water transport equations in the membrane into our previously developed multi-dimensional two-phase PEM fuel cell model,[22,23,31,32] we need to correctly handle the interfacial liquid water transport phenomena between the membrane and two catalyst layers on the anode and cathode sides, respectively, as shown in Fig. 1. This can be accomplished using the following simple treatment:[31,32] qlint ¼ K int plcl plmem

(9)

(6)

This value is an order of magnitude lower than the typical capillary liquid water diffusivity in catalyst layers and is found to be able to provide reasonably good numerical results. More fundamental research is certainly required to obtain an appropriate function to correlate the liquid pressure with liquid saturation inside the membrane material in the future. The water content transport equation, as listed in Table 1, together with the pressure-driven liquid water transport equation, Eqn (1), could properly account © 2012 Curtin University of Technology and John Wiley & Sons, Ltd.

In Eqn (9), the interfacial liquid water transport parameter, Kint, can be chosen to be a sufficiently large value to simultaneously satisfy equality of the liquid pressure and liquid water flux across the interface. In our previous modeling and numerical studies,[31,32] Kint = 102 mol pa1 m2 s1 is found to work well. Using this simple numerical treatment, a discontinuous liquid saturation jump and continuous liquid pressure variation across an interface can be appropriately captured. It should be pointed out that in Eqn (9), an expression for calculating the liquid water capillary Asia-Pac. J. Chem. Eng. 2013; 8: 104–114 DOI: 10.1002/apj

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108



Interface

l

treatment, this PEM fuel cell model is able to handle liquid water transport across different porous materials, including the catalyst layers and gas diffusion media on both anode and cathode sides, and the polymer membrane, once it is in contact with liquid water.

l

pmem

pcl

RESULTS AND DISCUSSION

CL

MEM qlmem

The multi-dimensional two-phase PEM fuel cell model developed in the early section has been built into a commercial CFD package, FLUENT (Ansys, Inc, Canonsburg, PA, USA), through its user coding capabilities and is employed herein for investigating liquid water transport and distributions in the cathode, anode, and membrane in a PEM fuel cell. The numerical studies are conducted in a two-dimensional cross section, as shown in Fig. 2, with intent to elucidate the fundamental physics and clearly distinguish the parametric effects without involving further geometric complications. The relevant fuel cell geometric parameters are listed in Table 3. The computational mesh for the present parametric investigation is generated based on our previous gridindependence studies,[22,31] resulting in a non-uniform grid system with 10 computational cells in the membrane, 10 in each of the gas diffusion layers, and five in each of the catalyst layers in the throughmembrane direction. A total of 2 800 computational cells have been used in the present numerical studies.

l

qcl

Figure 1. Schematic of an interface between the PEM fuel cell membrane and catalyst layer.

pressure at the interface on the membrane side is still needed. Given the lack of experimental information at the present time, Eqns (4) and (5) have been used in the following preliminary numerical studies. Further investigations in this area are needed in the future. The same numerical approach for solving the interfacial liquid water transport is applied at interfaces between the catalyst layers and gas diffusion media on both anode and cathode sides in a PEM fuel cell, thus correctly capturing the interfacial liquid water transport phenomena in the multi-layer porous materials. In summary, a multi-dimensional two-phase PEM fuel cell model, which is capable of handling water transport inside and across the polymer membrane in both the absorbed water phase and free liquid phase, has been built in a conventional CFD framework. Combining with a simple interfacial numerical

Table 3. Cell geometric parameters.

Fuel cell geometry (mm) Layer thickness

Diffusion Catalyst Membrane

Land width Channel width Computational cell numbers

0.3 0.01 0.025 0.5 1.0 2800

5

P L A T E

1

G C

3

G D L

C L

M E M

C L

G D L

G

4 C

P L A T E

2

z x

Anode

5

Cathode

Figure 2. A two-dimensional PEM fuel cell cross section (GC, gas channel; GDL, gas diffusion layer; CL, catalyst layer; MEM, membrane). This figure is available in colour online at www.apjChemEng.com. © 2012 Curtin University of Technology and John Wiley & Sons, Ltd.



Figure 3 shows liquid saturation variations across the anode gas diffusion layer, anode catalyst layer, membrane, cathode catalyst layer, and cathode gas diffusion layer directly under the fuel cell currentcollecting land. The numerical result is obtained at a

0.35

aGDL

aCL

M E M

cGDL

cCL

1600

0.30

1400

0.25 1200 0.20 0.15

1000

0.10

800

Capillary Pressure (Pa)

The PEM fuel cell simulated is set at 2 atm on both anode and cathode sides. The cell stoichiometry number is 2 on both sides with a reference current density at 1 A cm2. Hydrogen and air with fullyhumidified inlet conditions are fed into the anode and cathode sides, respectively. Based on these operation conditions, the species concentrations at boundaries 3 and 4 can be easily determined. A constant cell operation current density is defined at boundary 2 on the cathode side, whereas a constant cell temperature is prescribed at outside boundaries 1 and 2. The other related boundary condition definitions can be found in our prior publications.[22,31] In this paper, liquid water transport and distributions in the fuel cell anode, cathode, and membrane is numerically investigated. The effects of the PEM fuel cell operating current and outside boundary temperature on liquid water transport behaviors are numerically studied. The calculated numerical results are found to be consistent with neutron imaging data published in the open literature.[10,40] The relevant physicochemical parameters used in present numerical studies are provided in Table 4.

Liquid Saturation


0.05 600 0.00 1.0

1.1

1.2

1.3

1.4

1.5

1.6

Cross-Membrane Distance (mm)

Figure 3. Liquid saturation and the corresponding capillary

pressure variations in the through-membrane (x-) direction under the current-collecting land.

Table 4. Physicochemical parameters.

Parameter Anode volumetric exchange current density, aj0 (A m3) Cathode volumetric exchange current density, aj0 (A m3) Reference hydrogen concentration, CH2 (mol m3) Reference oxygen concentration, CO2 (mol m3) Anode transfer coefficients Cathode transfer coefficient Faraday constant, F (C mol1) GDL porosity Porosity of catalyst layer Membrane porosity for liquid water Volume fraction of ionomer in catalyst layer GDL permeability (m2) Catalyst layer permeability (m2) Membrane permeability for liquid water Equivalent weight of ionomer (kg mol1) Dry membrane density (kg m3) Effective electronic conductivity in CL/GDL (S m1) Operation pressure (atm) Condensation rate coefficient (s1) Evaporation rate coefficient (s1 Pa1) Liquid water transition rate coefficient (s1) Liquid water density (kg m3) Liquid water viscosity (N s m2) Surface tension (N m1) Contact angle in GDL Contact angle in CL Contact angle in membrane Thermal conductivity of CL/GDL (W m1 K1) Thermal conductivity of the membrane (W m1 K1) Heat of vaporization (J kg1) Density of carbon material (kg m3) Heat capacity of carbon material (J kg1 K1) Heat capacity of liquid water (J kg1 K1) Heat capacity of membrane material (J kg1 K1)

Value 1.0E+9 1.0E+4 40 40 aa = ac = 1 ac = 1 96487 0.6 0.12 0.1 0.4 1.0E12 1.0E13 1.8E18 1.1 1980 5000 2 5000 1.0E4 1.0 1000 3.5E4 6.25E2 110 95 90.02 1.5 0.5 2.3E+6 2200 1050 4200 1050

CL, catalyst layers; GDL, gas diffusion layers. © 2012 Curtin University of Technology and John Wiley & Sons, Ltd.


109

110


cell boundary temperature at 80 C and a constant cell operating current density at 1 A cm2, with fullyhumidified conditions set on both anode and cathode sides. Discontinuous jumps of liquid saturation across various interfaces between different porous materials are correctly captured in Fig. 3, whereas the corresponding liquid capillary pressure exhibits continuous variations. The discontinuous liquid saturation variation across an interface between two porous materials is recently confirmed by numerical results from a detailed porenetwork simulation[44] In Fig. 3, liquid saturation possesses the highest values inside the polymer membrane, decreases in the two catalyst layers, and further decreases in the two gas diffusion media on the anode and cathode sides. In addition, more liquid water exists on the cathode side than on the anode side because of water production in the fuel cell cathode catalyst layer and the electro-osmotic drag acting from anode to cathode. This general liquid water variation trend agrees with the available neutron imaging data.[10,40] It should be emphasized that because of the constant liquid water capillary diffusivity applied in Eqn (6), the validation can only be qualitative in the present work. More fundamental research is needed to obtain an appropriate functional form for this parameter in the future. Detailed liquid water distributions in different porous materials in the PEM fuel cell are further illustrated in Fig. 4. It is found that more liquid water would accumulate in the corner regions directly under the current-collecting land on both anode and cathode sides because the cell temperature is relatively low in these regions, and furthermore, it is difficult for liquid water to escape into gas channels from these regions. Detailed cell temperature variations will be shown and discussed later in this section.


Figure 5 illuminates the liquid water condensation/ evaporation phenomena, related to the source term Svl in Eqn (1), in the gas diffusion layers on both anode and cathode sides. The positive values indicate water condensation whereas the negative values represent liquid water evaporation. Results in this figure confirm that liquid water mainly condenses at the two corners under the current-collecting land on the cathode side because of the relative low cell temperature in these regions. This is also consistent with the liquid saturation distributions presented in Fig. 4. In this particular case, a small amount of liquid water would evaporate near gas channel on the cathode side, whereas a large amount of liquid water would be transported through the cell membrane into the anode side and become totally evaporated.

Figure 5. Liquid water condensation/evaporation source

term in the anode and cathode gas diffusion media at a cell boundary temperature of 80 C and a cell operating current density of 1 A cm2 (unit: mol m3 s1). This figure is available in colour online at www.apjChemEng.com.

Figure 4. Liquid saturation distributions in different porous materials. This figure is available

in colour online at www.apjChemEng.com.





Effects of the cell operating current on liquid water variations are investigated in Fig. 6. In this figure, the liquid saturation variation in the through-membrane direction is averaged in the lateral (z-) direction in order to make qualitative comparisons with experimental data. In general, increasing the operating current density from 0.5 to 1.5 A cm2, with all the other operation conditions maintained the same, would lead to more liquid water inside the PEM fuel cell because more water would be produced at a higher cell operating current. At a high operating current, the cell temperature would also increase, owing to extra waste heat production, as shown in Fig. 7. This figure clearly illustrates the cell temperature distributions in different porous materials in the PEM fuel cell under three different cell operating currents. A higher cell temperature would lead to more liquid water evaporation. Therefore, effects of the fuel cell operating current on liquid water distributions in a

PEM fuel cell is dictated by two counter-acting factors: water production, which would result in more liquid water condensation, and cell temperature increase, which could lead to more liquid water evaporation. Figure 8 shows the water condensation/evaporation phenomena at an operating current density of 1.5 A cm2. Comparing with the numerical results in Fig. 5, it is clear that at a higher operating current density, more liquid water condenses under the current-collecting land in the cathode gas diffusion layer simply because of more water production. On the other hand, owing to cell temperature increase at a higher operating current as discussed earlier, more liquid water could get easily evaporated near the cathode gas channel region and also in the anode gas diffusion layer. The evaporation

0.35

MEM +CL

aGDL 0.30

cGDL

Liquid Saturation

0.25 0.20 0.15

0.5 A/cm2 1.0 A/cm22 1.5 A/cm

0.10 0.05 0.00 1.0

1.1

1.2

1.3

1.4

1.5

1.6


Figure 6. Averaged liquid saturation variations in the

through-membrane (x-) direction at a cell boundary temperature of 80 C and three different cell operating currents.

Figure 8. Liquid water condensation/evaporation source term in the anode and cathode gas diffusion media at a cell boundary temperature of 80 C and a cell operating current density of 1.5 A cm2 (unit: mol m3 s1). This figure is available in colour online at www.apjChemEng.com.

Figure 7. Cell temperature distributions in the porous materials at a cell boundary

temperature of 80 C and three different cell operating currents (unit: K). This figure is available in colour online at www.apjChemEng.com.



111



phenomenon is especially profound on the anode side, as clearly elucidated in Fig. 8. Effects of the cell temperature on liquid water distributions in the PEM fuel cell are displayed in Fig. 9. Again, the liquid saturation variation in this figure in the through-membrane direction is averaged in the lateral direction. It is found that decreasing the fuel cell boundary temperature from 80 to 60 C would result in more liquid water inside the fuel cell, particularly inside the two catalyst layers and the polymer membrane. However, further decreasing the cell boundary temperature from 60 to 40 C would only lead to very slight increase of liquid water inside the PEM fuel cell. This variation trend is in qualitative agreement with the experimental observations [10,40]. 0.30

MEM +CL

aGDL

cGDL

Figure 10 shows the water condensation/evaporation phenomena at a cell boundary temperature of 60 C, with a cell operating current density set at 1 A cm2. Comparing with the results in Fig. 5, it can be clearly seen that decreasing the operating cell temperature would lead to slightly more water condensation, particularly deep inside the gas diffusion layer directly under the gas channel on cathode side. However, the dictating mechanism for liquid water increase at a decreased cell boundary temperature is the decreased liquid water evaporation process. Comparing the results in Figs. 5 and 10, it is found that liquid water evaporation decreases on both anode and cathode sides of the PEM fuel cell as the cell operating temperature decreases.

CONCLUSIONS

0.25

Liquid Saturation

112

0.20 o

0.15

40 C o 60 C o 80 C

0.10

0.05

0.00 1.0

1.1

1.2

1.3

1.4

1.5

1.6


Figure 9. Averaged liquid saturation variations in the

through-membrane (x-) direction at three different cell boundary temperatures and a cell operating current density of 1 A cm2.

Figure 10. Liquid water condensation/evaporation source term in the anode and cathode gas diffusion media at a cell boundary temperature of 60 C and a cell operating current density of 1 A cm2 (unit: mol m3 s1). This figure is available in colour online at www.apjChemEng.com. © 2012 Curtin University of Technology and John Wiley & Sons, Ltd.

A two-phase multi-dimensional PEM fuel cell model has been developed in this paper for considering liquid water transport on both anode and cathode sides, as well as inside and across the polymer membrane. A physical model for describing liquid water transport across the cell membrane driven by a pressure difference is theoretically formulated and numerically incorporated in this PEM fuel cell model. Together with the water content transport equation, the two water transport equations are capable of handling water transport inside and across the cell membrane in both the absorbed water phase and free liquid phase. A simple interfacial numerical treatment for liquid water transport across multi-layer porous materials inside a PEM fuel cell is further integrated to capture the liquid saturation jump at an interface between two different porous layers and simultaneously maintain a continuous liquid pressure variation across the interface. This PEM fuel cell model is applied herein for investigating effects of the cell operating current and cell temperature on liquid water distributions. Numerical results illustrate liquid water distributions in the anode, cathode, as well as inside the membrane in a PEM fuel cell. It is found that liquid water transport inside and across the polymer membrane plays an important role in PEM fuel cell water distributions. Increasing the fuel cell operating current density results in more liquid water accumulated in different porous materials on both anode and cathode sides, a phenomenon dictated mainly by more water production. However, liquid water evaporation, especially on the anode side, also exerts its profound influence at an increased operating current density. Decreasing the cell operating temperature, particularly from 80 to 60 C, results in more liquid water inside the PEM fuel cell. This phenomenon is mainly controlled by the decreased liquid water evaporation process on both anode and cathode sides of the PEM fuel cell. Further decreasing the cell boundary temperature Asia-Pac. J. Chem. Eng. 2013; 8: 104–114 DOI: 10.1002/apj



from 60 to 40 C only leads to very slight liquid water increase. The calculated results show qualitative agreement with available neutron imaging data. Acknowledgement This research work is financially supported by the National Natural Science Foundation of China (No. 10972197).

NOMENCLATURE a c Cp D Dc Dl EW F hld hpc i j k kc ke K Kint nd p pc q Ru s S T u Uo V Vcell W

Reacting surface area or water activity Molar concentration, mol m3 Constant-pressure heat capacity, J kg1 K1 Mass diffusivity, m2 s1 Capillary liquid water diffusivity, kg m1 s1 Water content diffusivity, mol m1 s1 Equivalent weight of dry membrane, kg mol1 Faraday constant, 96487 C mol1 Liquid water transition rate coefficient Condensation/evaporation parameter Current density vector, A m2 Transfer current density, A m3 Thermal conductivity, W m1 K1 Condensation rate coefficient, s1 Evaporation rate coefficient, s1 Pa1 Permeability, m2 Interfacial liquid water transport parameter, mol Pa1 m2 s1 Electro-osmotic drag coefficient Gas-phase pressure, Pa Capillary pressure, Pa Liquid water flux, mol m2 s1 Universal gas constant, J mol1 K1 Liquid saturation Source term Temperature, K Gas-phase velocity, m s1 Open-circuit potential, V Volume Cell voltage, V Molecular weight, kg mol1

Greek: e em Φ θc k l m r

Porosity Fraction of the membrane phase in the catalyst layer Phase potential, V Over-potential, V Contact angle Proton conductivity, S m1 Water content Viscosity, kg m1 s1 Gaseous density, kg m3


s Electronic conductivity, S m1, or surface tension, N m1 t Viscous stress tensor w Mole fraction Superscripts: e eff l m sat v

Equilibrium value Effective value Liquid phase Membrane Saturation value Vapor phase

Subscripts: cl e i int l ld m mem p s T u vl w

Catalyst layer Electrolyte Species Interfacial value Liquid Liquid to dissolved water phase Mass Membrane Pore Electron Temperature Velocity Water vapor to liquid phase Water

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