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Effects of porosity gradient in gas diffusion layers on performance of proton exchange membrane fuel cells
Energy 35 (2010) 4786e4794
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Energy journal homepage: www.elsevier.com/locate/energy
Effects of porosity gradient in gas diffusion layers on performance of proton exchange membrane fuel cells Yu-Xian Huang a, Chin-Hsiang Cheng b, *, Xiao-Dong Wang c, Jiin-Yuh Jang a a
Dept. of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan Dept. of Aeronautics and Astronautics, National Cheng Kung University, No.1, Ta- Shieh Road, Tainan 70101, Taiwan c Dept. of Thermal Engineering, School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, PR China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 April 2010 Received in revised form 7 September 2010 Accepted 8 September 2010 Available online 13 October 2010
A three-dimensional, two-phase, non-isothermal model has been developed to explore the interaction between heat and water transport in proton exchange membrane fuel cells (PEMFCs). Water condensate produced from the electrochemical reaction may accumulate in the open pores of the gas diffusion layer (GDL) and retard the oxygen transport to the catalyst sites. This study predicts the enhancement of the water transport for linear porosity gradient in the cathode GDL of a PEMFC. An optimal porosity distribution was found based on a parametric study. Results show that a optimal linear porosity gradient with 31 ¼ 0.7 and 32 ¼ 0.3 for the parallel and z-serpentine channel design leads to a maximum increase in the limiting current density from 10,696 Am2 to 13,136 Am2 and 14,053 Am2 to 16,616 Am2 at 0.49 V, respectively. On the other hand, the oxygen usage also increases from 36% to 46% for the parallel channel design and from 55% to 67% for the z-serpentine channel design. The formation of a porosity gradient in the GDL enhances the capillary diffusivity, increases the electrical conductivity, and hence, benefits the oxygen transport throughout the GDL. The present study provides a theoretical support for existing reports that a GDL with a gradient porosity improves cell performance. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Fuel cell Porosity gradient Water management Gas diffusion layer
1. Introduction The proton exchange membrane fuel cell (PEMFC) with the clean fuel sources of hydrogen and oxygen supply is considered to be a promising power source [1]. This fuel cell has many important attributes such as high efficiency, cleanness, quietness, low temperature operation, quick start-up, no liquid electrolyte, and simple cell design. However, before this system becomes competitive with conventional internal combustion engines, its performance and costs must be further optimized. Numerous numerical optimization investigations have been published on PEMFC in recent years [2e6]. The performance of a PEMFC is critically dependent on the electrocatalytic activity of the precious metal catalyst such as platinum or a platinum alloy and the quality of the various components including the gas diffusion layer (GDL). The porous GDL, as a component of the membrane electrode assembly (MEA), plays an important role in the PEMFC through affecting the diffusion of reactants and water as well as the conduction of
* Corresponding author. Tel.: þ886 6 2757575x63657; fax: 886 6 2389940. E-mail address: [email protected] (C.-H. Cheng). 0360-5442/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2010.09.011
electrons [7e12]. The micro-porous layer (MPL) has been extensively studied due to the critical role in achieving high performance of the PEMFC. For example, Pasaogullari and Wang [8] applied an MPL to the catalyst layer (CL) and observed an appreciable increase in the performance. In this study, the porosities within the CL, MPL, and GDL are constant, and the difference between these layers helps remove the water condensate from the CL. Furthermore, there have been various studies of the effects of the PTFE and carbon loadings in the MPL [13e19]. It is expected that reducing the porosity of the GDL generally leads to an increase in the electronic conductivity of the GDL; however, it may retard the supply of the reactant gases in the GDL. On the contrary, increasing the porosity of the GDL is favorable for sufficient gas supply; however, it may reduce the electronic conduction. Besides, the porosity of the GDL is one of the influential factors significantly affecting the uniformity of the distributions of the reactant gases and the current density over the catalyst layer. Therefore, concerning improvement in gases transport without severely reducing the electronic conduction in the GDL, forming a porosity gradient in the GDL could be an approach worthy of consideration. Wang et al. [20] studied the effects of carbon black in the MPL and examined the physical properties of the GDL including
Y.-X. Huang et al. / Energy 35 (2010) 4786e4794
Nomenclature C D DC Dl F h hfg i im j kl krl kp kc ke M Mm nd Pg Pc Su Sc Sj SL s t T
concentration (mol m3) mass diffusivity (m2 s1) capillary diffusivity (m2 s1) water diffusivity in the membrane (m2 s1) Faraday constant 96,487 (C mol1) gas mixture enthalpy(J kg1) latent heat of vaporization (J kg1) current density (A cm2) ionic current density (A cm2) transfer current density (A cm3) permeability of the liquid water (m2) relative permeability of the liquid water (m2) Permeability (m2) condensation rate (s1) evaporation rate (s1) molecular weight (kg kmol1) membrane equivalent weight (kg kmol1) electro-osmotic drag coefficient gas pressure (Pa) capillary pressure head (Pa) sources term in the momemtum equation sources term in the species equation source term in the phase potential equation production rate of liquid water for phase change (kg s1) liquid water saturation thinkness (m) temperature (K)
surface morphology, gas permeability, hydrophobic character, porosity and conductivity. Authors prepared MPLs with different carbon materials and presented a novel GDL with a porosity gradient formed by coating MPLs with different carbon loadings on the catalyst layer side and on the flow field side to improve the liquid water removal [21]. Recently, Zhan et al. [22] used a twophase flow model of the fuel cell electrode to show that a GDL with a porosity gradient improves the liquid water removal from the catalyst layer. Tang et al. [23] investigated the effect of a porositygraded MPL to obtain better cell performance by experiments. They also concluded that the improved performance is probably due to the improved liquid water transport through the large pores and with gas diffusion in the small pores. To maintain the membrane electrode assembly (MEA) with a satisfactory water content and distribution, a novel gas diffusion layer was designed by inserting a water management layer (WML) between the traditional GDL and the catalyst layer of the PEMFC by Chen et al. [24]. Their testing results indicated that the water management in the MEA could be significantly improved by using the WML. Chu et al. [25] analyzed the non-uniform porosity of GDL in PEMFC, which is a necessity because the presence of liquid water in the GDL actually leads to a non-uniformly distributed porosity in the GDL. In addition, Kong et al. [26] also found that the pore-size distribution is an important structural parameter affecting the cell performance as well as the total porosity. The existing studies have recognized that a porosity gradient in the GDL, especially in the GDL on the cathode side, will enhance the transport of the gas reactants and increase the capillary diffusion in a PEMFC. However, the related information is still insufficient, and hence, the physical insight regarding the effects of porosity gradient in the GDL on the performance of the fuel cells are not discussed in depth. In these circumstances, the present study is aimed at providing evidences by means of numerical method to
U V x, y, z
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gas velocity vector (m s1) cell voltage (V) Cartesian coordinates (m)
Greek symbols 3 porosity h electrode over-potential (V) qc contact angle (degree) 1 l membrane water content [kmol H2O(kmol SO 3) ] 1 1 leff effective thermal conductivity (W m K ) m viscosity coefficient (kg m1s1) rg mass density of gas (kg m3) rl mass density of liquid water (kg m3) ss electronic conductivity (S m1) sm proton conductivity (S m1) s* surface tension (N m1) F electric potential (V) Subscripts CL catalyst layer eff effective g gas GDL gas diffusion layer water H2 O in inlet l liquid m membrane out outlet
show the effects of the porosity gradient on the performance of the fuel cells. Therefore, a three-dimensional, two-phase numerical simulation model is developed to investigate the transport phenomena in the PEMFC with a porosity gradient formed in the cathode GDL. Numerical analysis of the effects of the porosity gradient in the GDL on the performance of a PEMFC is evaluated, and the dependence of the capillary diffusivity on the porosity of the GDL is also examined so as to provide a thorough understanding for the porosity gradient effects.
2. Theory and mathematical model 2.1. Theoretical analysis The two-fluid method used in the present work was refined from that adopted in Wang et al. [27] to incorporate the heat effects using energy equations for entire cells. The model assumes that the system is steady; the inlet reactants are ideal gases; the flow is laminar; and the porous layers such as the diffusion layer, catalyst layer and PEM are isotropic. The model includes continuity, momentum and species equations for gaseous species, liquid water transport equations in the channels, gas diffusion layers, and catalyst layers, water transport equation in the membrane, electron and proton transport equations. The ButlereVolmer equation was used to describe electrochemical reactions in the catalyst layers. The main governing equations and relevant physicochemical parameters are clearly listed in Appendix I and Table 1, respectively. The thermal conductivities of solid matrix in Eq. (A6) are 150 Wm1K1 for the gas diffusion layers, catalyst layers and membrane. The source terms (Su, Sj, and others) and other relevant physicochemical parameters in eqs. (A1) to (A13) were well calibrated and listed in [27].
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2.2. Boundary conditions
Table 1 Fixed base case parameters. Description
Symbol
Value
Cell design parameters Anode GDL thickness (m) Anode catalyst layer thickness (m) Membrane thickness (m) Cathode catalyst layer thickness (m) Cathode GDL thickness (m)
tGDLa tCLa tM tCLc tGDLc
0.3 103 0.005 103 0.35 103 0.005 103 0.3 103
Tcell
323 0.5 0.4 0.28 0.4 0.5 100
Simulation conditions Cell temperature (K) Porosity of anode GDL Porosity of anode catalyst layer Porosity of membrane Porosity of cathode catalyst layer Porosity of cathode GDL Relative humidity of inlet fuel stream (%) Cathode inlet mass fraction ratio Anode inlet mass fraction ratio Cathode inlet air flow rate (cc min1) Anode inlet hydrogen flow rate (cc min1) Anodic exchange current density (Am3) Cathodic exchange current density (Am3) Anode charge transfer coefficients Anode charge transfer coefficients Cathode charge transfer coefficients Cathode charge transfer coefficients Bruggemann coefficient of gas diffusion layer Bruggemann coefficient of catalyst layer Condensation rate (s1) Evaporation rate (s1) Dynamic viscosity of water (pa s) Surface tension (N m1)
3GDLc 3V,Cat,a 3M 3V,Cat,c 3GDLc
RHa and RHc
2.3. Numerical methods
mN2 : mO2 : mH2O 0.71 : 0.21 : 0.08 0.56 : 0.44 mH2 : mH2O Q_ c 295.4 Q_ a
74.5
ref I0;a
9.227 108
ref
I0;c
1.05 106
aa,a aa,c ac,a ac,c sGDL sCat
0.5 0.5 1.5 1.5 1.5 1.5 100 100 3.65 104 0.07
kc ke
ml s*
The volumetric flow rates of the entering liquid mixtures, Q_ , on the anode and cathode sides were set to be 74.5 cc min1 and 295.4 cc min1 (corresponding to stoichiometries 1.5 and 3.5 on the anode and cathode sides) respectively [28]. The thermal conditions on the channels walls and the surfaces of the ribs are maintained at constant temperature of 323 K. The edges of the all the components of the fuel cell are assumed to be adiabatic. All the fixed parameters used in the present study are calibrated in [27] and listed in Table 1.
The equations were solved by using the SIMPLEC algorithm and the finite-volume method. A structured multi-grid solver was used with a convergence criterion of 1 106 for all quantities. The meshes typically have 410,000 grid points in the computational domain. Computations were performed on a personal computer with an Intel CoreÔ 2 Quad 3.0G CPU. Typical computational time for a single case is more than 25 h. The models considered in the present study include parallel, zserpentine, and interdigitated channel patterns. Schematic diagrams of the PEMFCs for the three patterns are given in Figs. 1 and 2. In Fig. 2, the geometry is coarsely meshed here so that the reader can easily see the structure of the PEMFC. The current densities and capillary diffusivities for these three patterns are compared when a porosity gradient is formed in the cathode GDL. 3. Results and discussion Fig. 3 shows the performance curves for the fuel cells having parallel, z-serpentine, and interdigitated channels and various porosity gradients formed in the cathode GDL. In this figure, four combination cases for 31 and 32 are considered. The case with constant porosities at 31 ¼ 0.5 and 32 ¼ 0.5 is referred to as the original case which actually has a constant porosity in the GDL. Then, three cases with porosity gradients specified with (31,
Fig. 1. Schematic of PEMFC.
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Fig. 2. Coarsely meshed channel pattern for each model.
32) ¼ (0.2, 0.8), (0.3, 0.7), and (0.4, 0.6) are investigated. Note that the average value of the porosity in these cases is fixed at 0.5 such that the effects of porosity gradient could be fairly assessed. It is found that the original constant-porosity case leads to a limiting current density of 10,696 Am2 at around 0.49 V with the parallel channels as shown in Fig. 3a. Among the three variable-porosity cases, the porosity gradient with 31 ¼ 0.7 and 32 ¼ 0.3 leads to the highest limiting current density of 13,136 Am2 at 0.49 V with the same channel pattern, whereas the case with the highest porosity gradient at 31 ¼ 0.8 and 32 ¼ 0.2 yields the lowest limiting current
Fig. 3. IV curves for the parallel, z-serpentine, and interdigitated channel designs.
density of 12,528 Am2 at 0.49 V. However, all the three variableporosity cases exhibit appreciable improvement in performance as compared to the constant-porosity one. The case with a porosity gradient at 31 ¼ 0.7 and 32 ¼ 0.3 achieves an increase in the limiting
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Table 2 Dependence of pressure drop on the porosity gradient for interdigitated channel design. Case
31 ¼ 0.5,32 ¼ 0.5
31 ¼ 0.6,32 ¼ 0.4
31 ¼ 0.7,32 ¼ 0.3
31 ¼ 0.8,32 ¼ 0.2
Pressure drop (Pi Pe)
75.8 Pa
63.5 Pa
54.8 Pa
47.16 Pa
current density by 23% in comparison with the constant-porosity case. The above finding agrees with the experimental observation of Tang et al. [23], who addressed that the low porosity region near the catalyst results in higher capillary force, and on the other hand, the high porosity region near the channels results in lower capillary force. Therefore, a net capillary force reduced by a porosity gradient in the GDL forces the water to move from the catalyst toward the channels through the GDL. By improving water transport in the GDL with a porosity gradient, the current density can be increased. However, when the porosity near the catalyst is too small, the concentration of supplied fuel will be decreased, or when the porosity near the channels is too large, the electric conductivity of the GDL will be decreased. In either case, the performance of the fuel cell will definitely be reduced. Therefore, one can then expect
that an optimal porosity gradient may exist for a better performance. This probably is a reason for the finding that the porosity gradient with 31 ¼ 0.7 and 32 ¼ 0.3 leads to the highest cell performance and the porosity gradient with 31 ¼ 0.8 and 32 ¼ 0.2 leads to a poorer performance. The z-serpentine design shown in Fig. 3b exhibits the same trends in the dependence of performance curves on the porosity gradients in the cathode GDL. In this figure, it is found that the current density can be increased from 14,053 to 16,616 Am2 (18% increase) when the porosity gradient at 31 ¼ 0.7 and 32 ¼ 0.3 is formed. However, for the interdigitated channel pattern shown in Fig. 3c, it is observed that the improvement in the current density by forming a porosity gradient is not as significant as in the cases
Fig. 4. Current density distributions for constant and variable cathode GDL porosities in the parallel channel design.
Fig. 5. Current density distributions for constant and variable cathode GDL porosities in the z-serpentine channel design.
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Fig. 6. Liquid water saturation distributions on the interface between the cathode GDL and flow channels for constant and variable cathode GDL porosities in the parallel channel design.
shown in Fig. 3 a and b. This probably is because the interdigitated channel design can force the gas fuel to flow through the gas diffusion and the catalyst layers effectively for electrochemical reaction. The porosity gradient effect helps remove the water in these layers and resolves the flooding problem. The improvement in the current density with the interdigitated channel design is not significant because this design itself already produces a relatively high current density and hence, a porosity gradient formed in GDL leads to only a very small improvement. On the other hand, one of the major disadvantages of the interdigitated channel design is that this design generally causes a greater pressure drop [11]. Nevertheless, in this study it is observed that the pressure drop can be remarkably reduced by forming a porosity gradient in the GDL. Table 2 provides numerical results for the pressure drop between the inlet and the outlet under various porosity gradients with the interdigitated channel design. The numerical results for the pressure drop are obtained simply by calculating Pi Pe, the difference between the inlet and the outlet pressures, which are carried out from the momentum equation for the gaseous species, Eq. (A2). The equation is already provided in
Fig. 7. Liquid water saturation distributions on the interface between the cathode GDL and flow channels for constant and variable cathode GDL porosities in the z-serpentine channel designs.
Appendix I. It is observed that the pressure drops with the constantporosity case is 75.8 Pa. The pressure drop is 63.5 Pa for the case with 31 ¼ 0.6 and 32 ¼ 0.4, 54.8 Pa for case with 31 ¼ 0.7 and 32 ¼ 0.3, and 47.16 Pa for 31 ¼ 0.8 and 32 ¼ 0.2. It clearly reveals that by forming a porosity gradient in the GDL the disadvantage of high-pressuredrop with the interdigitated channel design is possible to be improved even though the improvement in the current density is not significant. Figs. 4 and 5 show the detailed information regarding the current density distributions within the membranes for the parallel and the Z-serpentine channel designs with or without porosity gradient. A comparison between Fig. 4 a and b shows that the current density is improved, particularly in the areas close to the inlet manifold. A relatively low current density zone in the downstream region of the parallel channels flow field is observed with
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Fig. 8. Capillary diffusivity distributions on the interface between the cathode GDL and flow channels for constant and variable cathode GDL porosities in the parallel channel designs.
constant porosity. The low current density zone may be attributed to higher liquid water saturation occurring downstream, which will be discussed later. By forming a porosity gradient in the cathode GDL, the local current density in the entire flow field can be significantly increased for the case with the parallel channels. In addition, the current density distribution can also be improved by the porosity gradient for the z-serpentine channel design. Plotted in Fig. 5 a and b are the current distributions for the z-serpentine channel design with and without porosity gradient, respectively. Again, the current density is greatly elevated in the entire flow field due to variable-porosity effects. Fig. 6 a and b show the liquid water saturation distribution at an interface between the cathode GDL and the flow channels for the parallel design. The case with 31 ¼ 0.7 and 32 ¼ 0.3 has higher liquid water saturations downstream in the middle of the model, as shown in Fig. 6b. The intense chemical reaction in the catalyst layer for 31 ¼ 0.7 and 32 ¼ 0.3 at the cathode GDL increases the current density and the liquid water saturation. Fig. 7a shows the liquid water saturation distribution for the zserpentine design at 31 ¼ 0.5 and 32 ¼ 0.5. This figure illustrates that the liquid water saturation is relatively higher in the downstream zone of the flow field. With 31 ¼ 0.7 and 32 ¼ 0.3, the liquid water
Fig. 9. Capillary diffusivity distributions on the interface between the cathode GDL and flow channels for constant and variable cathode GDL porosities in the z-serpentine channel designs.
saturation can be greatly increased in the entire flow field, as shown in Fig. 7b. This implies that due to an increased supply of gases, the electrochemical reaction becomes stronger and hence more water is produced and the liquid water saturation is increased. It is found that in the downstream zone, the value of the liquid water saturation reaches a maximum of 0.2698. Fig. 8 illustrates the capillary diffusivity at the interface between the cathode GDL and the flow channels for the parallel design. The capillary diffusivity is calculated by equation (A12). Comparison in the capillary diffusivity distributions with and without porosity gradient, based on Fig. 8 a and b, shows that the porosity gradient effectively improves the capillary diffusivity which increases the removal of water from the gas diffusion layers. The magnitude of capillary diffusivity is essential to the water removal. In Fig. 9, distribution of capillary diffusivity for the zserpentine design is shown for the constant and variable-porosity cases. In a special case as the GDL is fully saturated with water, the
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mass fraction is apparently lower than that without a porosity gradient. As compared with the cases without porosity gradient, when the porosity gradient is applied, the oxygen usage [(mO2 inlet mO2 outlet)/mO2 inlet] rises from 36% to 46% for the parallel channel design and from 55% to 67% for the z-serpentine channel design. 4. Conclusions A porosity gradient is formed in the cathode gas diffusion layer to enhance gas reactant transport and increase the capillary diffusivity in a PEMFC. This study is aimed at providing evidences by means of numerical method to show the effects of the porosity gradient in the cathode GDL on the performance of the fuel cells. Results show that a linear porosity gradient with 31 ¼ 0.7 and 32 ¼ 0.3 for the parallel channel design leads to a maximum increase in the limiting current density from 10,696 Am2 to 13,136 Am2 at 0.49 V. For z-serpentine channel design, an appreciable increase in the limiting current density is also observed. When the porosity gradient is applied with the zserpentine channel design, the limiting current density is increased from 14,053 Am2 to 16,616 Am2 at 31 ¼ 0.7 and 32 ¼ 0.3. In addition, the porosity gradient leads to an increase in the capillary diffusivity so as to help remove the water condensate, and also increases the oxygen usage from 36% to 46% for the parallel channel design and from 55% to 67% for the z-serpentine channel design. However, for the interdigitated channel design, the improvement in the current density is not significant. Nevertheless, it is observed that the pressure drop caused by the interdigitated channel design can be remarkably reduced from 75.8 Pa to 47.16 Pa, by forming a porosity gradient in the cathode GDL. Therefore, the disadvantage of high-pressure-drop with the interdigitated channel design is possible to be improved. Acknowledgements Authors acknowledge the financial support from Department of Industrial Technology, Ministry of Economic Affairs, Taiwan, under Grant 98-EC-17-A-13-S2-0064. Appendix I The PEM fuel cell model includes continuity, momentum and species equations for gaseous species, liquid water transport equations in the channels, gas diffusion layers, and catalyst layers, water transport equation in the membrane, electron and proton transport equations. They include: Fig. 10. Oxygen mass fraction distributions for parallel and z-serpentine channel designs with constant and variable porosities.
gas/fluid interface will cease to exist and then the surface tension effects will vanish. That is, the surface tension force should reach a maximum at some critical liquid water saturation and vanish at 100% liquid water saturation. According to the information presented by Wang and Cheng [29], the capillary diffusivity reaches a maximum at a liquid water saturation of 0.2. Here, it is indeed observed that in the downstream zone the capillary diffusivity is greatly increased by the porosity gradient. Fig. 10 shows the oxygen mass fraction distributions in different channel designs for 31 ¼ 0.5 and 32 ¼ 0.5 and 31 ¼ 0.7 and 32 ¼ 0.3. The positions of the measured points are indicated by the blue and the red circles plotted in the figures. Firstly, it is observed that with a porosity gradient formed in cathode GDL, at the same gas flow rate more oxygen is consumed in the channels so that the oxygen
(1) Continuity equation for the gaseous species:
V$ 3rg Ug ¼ SL
(A1)
(2) Momentum equation for the gaseous species:
3 ð1 sÞ
r ¼ 3Vpg þ V$ U U g g g 2
V$ mg VUg þ Su ð1 sÞ
3
(A2)
(3) Species equation for the gaseous species:
V$ 3rg Ug Ck ¼ V$ rg Dk;eff VCk þ Sc SL
(4) Proton and electron transport equations:
(A3)
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V$ðsm VFm Þ ¼ Sj
(A4)
V$ðss VFs Þ ¼ Sj
(A5)
(5) The energy equation for the PEM fuel cell:
V$ 3ð1 sÞrg Ug Cp;g T þ V$ 3srl Ul Cp;l T i2 ¼ V$ leff VT þ jh þ þ hfg SL
s
(A6)
where j is the transfer current density calculated by ButlereVolmer equation, leff is effective thermal conductivity, accounting for contribution of solid matrix and fluids in the pores of porous media, and can be expressed as:
leff ¼ 2ls þ
3
1
2ls þ lf
þ
13 3ls
(A7)
(6) Liquid water transport equation in the membrane:
MH2 O rdry ad MH2 O ! V$ Dl Vl ¼ 0 i m l F Mm
(A8)
(7) Liquid water transport equation in the flow channels, gas diffusion layers and catalyst layers:
r kp krl vpc r kp krl n MH2 O ! V$ l Vpg þ V$ d Vs V$ l i m ¼ SL ml vs ml F (A9) where pc is the capillary pressure head,
q < 90 0:5 3 1:417ð1 sÞ 2:12ð1 sÞ2 þ1:262ð1 sÞ3 pc ¼ s*cosqc kðzÞ p (A10)
q > 90 0:5 3 pc ¼ s*cosqc kðzÞ 1:417s 2:12s2 þ 1:262s3 p
(A11)
where the capillary diffusivity can be written in terms of the capillary pressure head as follows:
DC ¼
dpc kl ds ml
(A12)
A higher capillary diffusivity in the catalyst and the gas diffusion layers favors the removal of the water condensate from these porous layers. In this study, the porosity in the cathode GDL is treated as a function of z as
3ðzÞ ¼ az þ b
(A13)
where a ¼ (31 32) / tGDLc and b ¼ 32. 31 and 32 are the local porosities of cathode GDL at z ¼ tGDLc (contact with cathode channels) and z ¼ 0 (contact with catalyst layer), respectively. Note that in this study, to enhance the performance of the PEMFC, one has 31 S 32.
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Effects of the cathode gas diffusion layer characteristics on the performance of polymer electrolyte fuel cells. Journal of Appied Electrochemistry 2002;32:383e8. [18] Giorgi L, Antolini E, Pozio A, Passalacqua E. Influence of the PTFE content in the diffusion layer of low Pt loading electrodes for polymer electrolyte fuel cells. Electrochimica Acta 1998;43:3675e80. [19] Moreira J, Ocampo AL. Influence of the hydrophobic material content in the gas diffusion electrodes on the performance of a PEM fuel cell. International Journal of Hydrogen Energy 2003;28:625e7. [20] Wang XL, Zhang HM, Zhang JL, Xu HF, Tian ZQ, Chen J, et al. Micro-porous layer with composite carbon black for PEM fuel cells. Electrochimica Acta 2006;51:4909e15. [21] Wang X, Zhang H, Zhang J, Xu H, Zhu X, Chen J, et al. A bi-functional microporous layer with composite carbon black for PEM fuel cells. Journal of Power Sources 2006;162:474e9. [22] Zhan ZG, Xiao JS, Li DY, Pan M, Yuan R. Effects of porosity distribution variation on the liquid water flux through gas diffusion layers of PEM fuel cells. Journal of Power Sources 2006;160:1041e8. [23] Tang H, Wang S, Pan M, Yuan R. Porosity-graded micro-porous layers for polymer electrolyte membrane fuel cells. Journal of Power Sources 2007;166:41e6. [24] Chen J, Matsuura T, Hori M. Novel gas diffusion layer with water management function for PEMFC. Journal of Power Sources 2004;131:155e61. [25] Chu HS, Yeh C, Chen F. Effects of porosity change of gas diffuser on performance of proton exchange membrane fuel cell. Journal of Power Sources 2003;123:1e9. [26] Kong CS, Kim DY, Lee HK, Shul YG, Lee TH. Influence of pore-size distribution of diffusion layer on mass-transport problems of proton exchange membrane fuel cells. Journal of Power Sources 2002;108:185e91. [27] Wang XD, Zhang XX, Yan WM, Lee DJ, Su A. An inverse geometry design problem for optimization of single serpentine flow field of PEM fuel cell. International Journal of Hydrogen Energy 2009;34:3823e32. [28] Um S, Wang CY. Three-dimensional analysis of transport and electrochemical reactions in polymer electrolyte fuel cells. 2004; 125:40e51. [29] Wang CY, Cheng P. Multiphase flow and heat transfer in porous media. Advances in Heat Transfer 1997;30:93e196.
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Effects of porosity gradient in gas diffusion layers on performance of proton exchange membrane fuel cells Yu-Xian Huang a, Chin-Hsiang Cheng b, *, Xiao-Dong Wang c, Jiin-Yuh Jang a a
Dept. of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan Dept. of Aeronautics and Astronautics, National Cheng Kung University, No.1, Ta- Shieh Road, Tainan 70101, Taiwan c Dept. of Thermal Engineering, School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, PR China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 April 2010 Received in revised form 7 September 2010 Accepted 8 September 2010 Available online 13 October 2010
A three-dimensional, two-phase, non-isothermal model has been developed to explore the interaction between heat and water transport in proton exchange membrane fuel cells (PEMFCs). Water condensate produced from the electrochemical reaction may accumulate in the open pores of the gas diffusion layer (GDL) and retard the oxygen transport to the catalyst sites. This study predicts the enhancement of the water transport for linear porosity gradient in the cathode GDL of a PEMFC. An optimal porosity distribution was found based on a parametric study. Results show that a optimal linear porosity gradient with 31 ¼ 0.7 and 32 ¼ 0.3 for the parallel and z-serpentine channel design leads to a maximum increase in the limiting current density from 10,696 Am2 to 13,136 Am2 and 14,053 Am2 to 16,616 Am2 at 0.49 V, respectively. On the other hand, the oxygen usage also increases from 36% to 46% for the parallel channel design and from 55% to 67% for the z-serpentine channel design. The formation of a porosity gradient in the GDL enhances the capillary diffusivity, increases the electrical conductivity, and hence, benefits the oxygen transport throughout the GDL. The present study provides a theoretical support for existing reports that a GDL with a gradient porosity improves cell performance. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Fuel cell Porosity gradient Water management Gas diffusion layer
1. Introduction The proton exchange membrane fuel cell (PEMFC) with the clean fuel sources of hydrogen and oxygen supply is considered to be a promising power source [1]. This fuel cell has many important attributes such as high efficiency, cleanness, quietness, low temperature operation, quick start-up, no liquid electrolyte, and simple cell design. However, before this system becomes competitive with conventional internal combustion engines, its performance and costs must be further optimized. Numerous numerical optimization investigations have been published on PEMFC in recent years [2e6]. The performance of a PEMFC is critically dependent on the electrocatalytic activity of the precious metal catalyst such as platinum or a platinum alloy and the quality of the various components including the gas diffusion layer (GDL). The porous GDL, as a component of the membrane electrode assembly (MEA), plays an important role in the PEMFC through affecting the diffusion of reactants and water as well as the conduction of
* Corresponding author. Tel.: þ886 6 2757575x63657; fax: 886 6 2389940. E-mail address: [email protected] (C.-H. Cheng). 0360-5442/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2010.09.011
electrons [7e12]. The micro-porous layer (MPL) has been extensively studied due to the critical role in achieving high performance of the PEMFC. For example, Pasaogullari and Wang [8] applied an MPL to the catalyst layer (CL) and observed an appreciable increase in the performance. In this study, the porosities within the CL, MPL, and GDL are constant, and the difference between these layers helps remove the water condensate from the CL. Furthermore, there have been various studies of the effects of the PTFE and carbon loadings in the MPL [13e19]. It is expected that reducing the porosity of the GDL generally leads to an increase in the electronic conductivity of the GDL; however, it may retard the supply of the reactant gases in the GDL. On the contrary, increasing the porosity of the GDL is favorable for sufficient gas supply; however, it may reduce the electronic conduction. Besides, the porosity of the GDL is one of the influential factors significantly affecting the uniformity of the distributions of the reactant gases and the current density over the catalyst layer. Therefore, concerning improvement in gases transport without severely reducing the electronic conduction in the GDL, forming a porosity gradient in the GDL could be an approach worthy of consideration. Wang et al. [20] studied the effects of carbon black in the MPL and examined the physical properties of the GDL including
Y.-X. Huang et al. / Energy 35 (2010) 4786e4794
Nomenclature C D DC Dl F h hfg i im j kl krl kp kc ke M Mm nd Pg Pc Su Sc Sj SL s t T
concentration (mol m3) mass diffusivity (m2 s1) capillary diffusivity (m2 s1) water diffusivity in the membrane (m2 s1) Faraday constant 96,487 (C mol1) gas mixture enthalpy(J kg1) latent heat of vaporization (J kg1) current density (A cm2) ionic current density (A cm2) transfer current density (A cm3) permeability of the liquid water (m2) relative permeability of the liquid water (m2) Permeability (m2) condensation rate (s1) evaporation rate (s1) molecular weight (kg kmol1) membrane equivalent weight (kg kmol1) electro-osmotic drag coefficient gas pressure (Pa) capillary pressure head (Pa) sources term in the momemtum equation sources term in the species equation source term in the phase potential equation production rate of liquid water for phase change (kg s1) liquid water saturation thinkness (m) temperature (K)
surface morphology, gas permeability, hydrophobic character, porosity and conductivity. Authors prepared MPLs with different carbon materials and presented a novel GDL with a porosity gradient formed by coating MPLs with different carbon loadings on the catalyst layer side and on the flow field side to improve the liquid water removal [21]. Recently, Zhan et al. [22] used a twophase flow model of the fuel cell electrode to show that a GDL with a porosity gradient improves the liquid water removal from the catalyst layer. Tang et al. [23] investigated the effect of a porositygraded MPL to obtain better cell performance by experiments. They also concluded that the improved performance is probably due to the improved liquid water transport through the large pores and with gas diffusion in the small pores. To maintain the membrane electrode assembly (MEA) with a satisfactory water content and distribution, a novel gas diffusion layer was designed by inserting a water management layer (WML) between the traditional GDL and the catalyst layer of the PEMFC by Chen et al. [24]. Their testing results indicated that the water management in the MEA could be significantly improved by using the WML. Chu et al. [25] analyzed the non-uniform porosity of GDL in PEMFC, which is a necessity because the presence of liquid water in the GDL actually leads to a non-uniformly distributed porosity in the GDL. In addition, Kong et al. [26] also found that the pore-size distribution is an important structural parameter affecting the cell performance as well as the total porosity. The existing studies have recognized that a porosity gradient in the GDL, especially in the GDL on the cathode side, will enhance the transport of the gas reactants and increase the capillary diffusion in a PEMFC. However, the related information is still insufficient, and hence, the physical insight regarding the effects of porosity gradient in the GDL on the performance of the fuel cells are not discussed in depth. In these circumstances, the present study is aimed at providing evidences by means of numerical method to
U V x, y, z
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gas velocity vector (m s1) cell voltage (V) Cartesian coordinates (m)
Greek symbols 3 porosity h electrode over-potential (V) qc contact angle (degree) 1 l membrane water content [kmol H2O(kmol SO 3) ] 1 1 leff effective thermal conductivity (W m K ) m viscosity coefficient (kg m1s1) rg mass density of gas (kg m3) rl mass density of liquid water (kg m3) ss electronic conductivity (S m1) sm proton conductivity (S m1) s* surface tension (N m1) F electric potential (V) Subscripts CL catalyst layer eff effective g gas GDL gas diffusion layer water H2 O in inlet l liquid m membrane out outlet
show the effects of the porosity gradient on the performance of the fuel cells. Therefore, a three-dimensional, two-phase numerical simulation model is developed to investigate the transport phenomena in the PEMFC with a porosity gradient formed in the cathode GDL. Numerical analysis of the effects of the porosity gradient in the GDL on the performance of a PEMFC is evaluated, and the dependence of the capillary diffusivity on the porosity of the GDL is also examined so as to provide a thorough understanding for the porosity gradient effects.
2. Theory and mathematical model 2.1. Theoretical analysis The two-fluid method used in the present work was refined from that adopted in Wang et al. [27] to incorporate the heat effects using energy equations for entire cells. The model assumes that the system is steady; the inlet reactants are ideal gases; the flow is laminar; and the porous layers such as the diffusion layer, catalyst layer and PEM are isotropic. The model includes continuity, momentum and species equations for gaseous species, liquid water transport equations in the channels, gas diffusion layers, and catalyst layers, water transport equation in the membrane, electron and proton transport equations. The ButlereVolmer equation was used to describe electrochemical reactions in the catalyst layers. The main governing equations and relevant physicochemical parameters are clearly listed in Appendix I and Table 1, respectively. The thermal conductivities of solid matrix in Eq. (A6) are 150 Wm1K1 for the gas diffusion layers, catalyst layers and membrane. The source terms (Su, Sj, and others) and other relevant physicochemical parameters in eqs. (A1) to (A13) were well calibrated and listed in [27].
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2.2. Boundary conditions
Table 1 Fixed base case parameters. Description
Symbol
Value
Cell design parameters Anode GDL thickness (m) Anode catalyst layer thickness (m) Membrane thickness (m) Cathode catalyst layer thickness (m) Cathode GDL thickness (m)
tGDLa tCLa tM tCLc tGDLc
0.3 103 0.005 103 0.35 103 0.005 103 0.3 103
Tcell
323 0.5 0.4 0.28 0.4 0.5 100
Simulation conditions Cell temperature (K) Porosity of anode GDL Porosity of anode catalyst layer Porosity of membrane Porosity of cathode catalyst layer Porosity of cathode GDL Relative humidity of inlet fuel stream (%) Cathode inlet mass fraction ratio Anode inlet mass fraction ratio Cathode inlet air flow rate (cc min1) Anode inlet hydrogen flow rate (cc min1) Anodic exchange current density (Am3) Cathodic exchange current density (Am3) Anode charge transfer coefficients Anode charge transfer coefficients Cathode charge transfer coefficients Cathode charge transfer coefficients Bruggemann coefficient of gas diffusion layer Bruggemann coefficient of catalyst layer Condensation rate (s1) Evaporation rate (s1) Dynamic viscosity of water (pa s) Surface tension (N m1)
3GDLc 3V,Cat,a 3M 3V,Cat,c 3GDLc
RHa and RHc
2.3. Numerical methods
mN2 : mO2 : mH2O 0.71 : 0.21 : 0.08 0.56 : 0.44 mH2 : mH2O Q_ c 295.4 Q_ a
74.5
ref I0;a
9.227 108
ref
I0;c
1.05 106
aa,a aa,c ac,a ac,c sGDL sCat
0.5 0.5 1.5 1.5 1.5 1.5 100 100 3.65 104 0.07
kc ke
ml s*
The volumetric flow rates of the entering liquid mixtures, Q_ , on the anode and cathode sides were set to be 74.5 cc min1 and 295.4 cc min1 (corresponding to stoichiometries 1.5 and 3.5 on the anode and cathode sides) respectively [28]. The thermal conditions on the channels walls and the surfaces of the ribs are maintained at constant temperature of 323 K. The edges of the all the components of the fuel cell are assumed to be adiabatic. All the fixed parameters used in the present study are calibrated in [27] and listed in Table 1.
The equations were solved by using the SIMPLEC algorithm and the finite-volume method. A structured multi-grid solver was used with a convergence criterion of 1 106 for all quantities. The meshes typically have 410,000 grid points in the computational domain. Computations were performed on a personal computer with an Intel CoreÔ 2 Quad 3.0G CPU. Typical computational time for a single case is more than 25 h. The models considered in the present study include parallel, zserpentine, and interdigitated channel patterns. Schematic diagrams of the PEMFCs for the three patterns are given in Figs. 1 and 2. In Fig. 2, the geometry is coarsely meshed here so that the reader can easily see the structure of the PEMFC. The current densities and capillary diffusivities for these three patterns are compared when a porosity gradient is formed in the cathode GDL. 3. Results and discussion Fig. 3 shows the performance curves for the fuel cells having parallel, z-serpentine, and interdigitated channels and various porosity gradients formed in the cathode GDL. In this figure, four combination cases for 31 and 32 are considered. The case with constant porosities at 31 ¼ 0.5 and 32 ¼ 0.5 is referred to as the original case which actually has a constant porosity in the GDL. Then, three cases with porosity gradients specified with (31,
Fig. 1. Schematic of PEMFC.
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Fig. 2. Coarsely meshed channel pattern for each model.
32) ¼ (0.2, 0.8), (0.3, 0.7), and (0.4, 0.6) are investigated. Note that the average value of the porosity in these cases is fixed at 0.5 such that the effects of porosity gradient could be fairly assessed. It is found that the original constant-porosity case leads to a limiting current density of 10,696 Am2 at around 0.49 V with the parallel channels as shown in Fig. 3a. Among the three variable-porosity cases, the porosity gradient with 31 ¼ 0.7 and 32 ¼ 0.3 leads to the highest limiting current density of 13,136 Am2 at 0.49 V with the same channel pattern, whereas the case with the highest porosity gradient at 31 ¼ 0.8 and 32 ¼ 0.2 yields the lowest limiting current
Fig. 3. IV curves for the parallel, z-serpentine, and interdigitated channel designs.
density of 12,528 Am2 at 0.49 V. However, all the three variableporosity cases exhibit appreciable improvement in performance as compared to the constant-porosity one. The case with a porosity gradient at 31 ¼ 0.7 and 32 ¼ 0.3 achieves an increase in the limiting
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Table 2 Dependence of pressure drop on the porosity gradient for interdigitated channel design. Case
31 ¼ 0.5,32 ¼ 0.5
31 ¼ 0.6,32 ¼ 0.4
31 ¼ 0.7,32 ¼ 0.3
31 ¼ 0.8,32 ¼ 0.2
Pressure drop (Pi Pe)
75.8 Pa
63.5 Pa
54.8 Pa
47.16 Pa
current density by 23% in comparison with the constant-porosity case. The above finding agrees with the experimental observation of Tang et al. [23], who addressed that the low porosity region near the catalyst results in higher capillary force, and on the other hand, the high porosity region near the channels results in lower capillary force. Therefore, a net capillary force reduced by a porosity gradient in the GDL forces the water to move from the catalyst toward the channels through the GDL. By improving water transport in the GDL with a porosity gradient, the current density can be increased. However, when the porosity near the catalyst is too small, the concentration of supplied fuel will be decreased, or when the porosity near the channels is too large, the electric conductivity of the GDL will be decreased. In either case, the performance of the fuel cell will definitely be reduced. Therefore, one can then expect
that an optimal porosity gradient may exist for a better performance. This probably is a reason for the finding that the porosity gradient with 31 ¼ 0.7 and 32 ¼ 0.3 leads to the highest cell performance and the porosity gradient with 31 ¼ 0.8 and 32 ¼ 0.2 leads to a poorer performance. The z-serpentine design shown in Fig. 3b exhibits the same trends in the dependence of performance curves on the porosity gradients in the cathode GDL. In this figure, it is found that the current density can be increased from 14,053 to 16,616 Am2 (18% increase) when the porosity gradient at 31 ¼ 0.7 and 32 ¼ 0.3 is formed. However, for the interdigitated channel pattern shown in Fig. 3c, it is observed that the improvement in the current density by forming a porosity gradient is not as significant as in the cases
Fig. 4. Current density distributions for constant and variable cathode GDL porosities in the parallel channel design.
Fig. 5. Current density distributions for constant and variable cathode GDL porosities in the z-serpentine channel design.
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Fig. 6. Liquid water saturation distributions on the interface between the cathode GDL and flow channels for constant and variable cathode GDL porosities in the parallel channel design.
shown in Fig. 3 a and b. This probably is because the interdigitated channel design can force the gas fuel to flow through the gas diffusion and the catalyst layers effectively for electrochemical reaction. The porosity gradient effect helps remove the water in these layers and resolves the flooding problem. The improvement in the current density with the interdigitated channel design is not significant because this design itself already produces a relatively high current density and hence, a porosity gradient formed in GDL leads to only a very small improvement. On the other hand, one of the major disadvantages of the interdigitated channel design is that this design generally causes a greater pressure drop [11]. Nevertheless, in this study it is observed that the pressure drop can be remarkably reduced by forming a porosity gradient in the GDL. Table 2 provides numerical results for the pressure drop between the inlet and the outlet under various porosity gradients with the interdigitated channel design. The numerical results for the pressure drop are obtained simply by calculating Pi Pe, the difference between the inlet and the outlet pressures, which are carried out from the momentum equation for the gaseous species, Eq. (A2). The equation is already provided in
Fig. 7. Liquid water saturation distributions on the interface between the cathode GDL and flow channels for constant and variable cathode GDL porosities in the z-serpentine channel designs.
Appendix I. It is observed that the pressure drops with the constantporosity case is 75.8 Pa. The pressure drop is 63.5 Pa for the case with 31 ¼ 0.6 and 32 ¼ 0.4, 54.8 Pa for case with 31 ¼ 0.7 and 32 ¼ 0.3, and 47.16 Pa for 31 ¼ 0.8 and 32 ¼ 0.2. It clearly reveals that by forming a porosity gradient in the GDL the disadvantage of high-pressuredrop with the interdigitated channel design is possible to be improved even though the improvement in the current density is not significant. Figs. 4 and 5 show the detailed information regarding the current density distributions within the membranes for the parallel and the Z-serpentine channel designs with or without porosity gradient. A comparison between Fig. 4 a and b shows that the current density is improved, particularly in the areas close to the inlet manifold. A relatively low current density zone in the downstream region of the parallel channels flow field is observed with
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Fig. 8. Capillary diffusivity distributions on the interface between the cathode GDL and flow channels for constant and variable cathode GDL porosities in the parallel channel designs.
constant porosity. The low current density zone may be attributed to higher liquid water saturation occurring downstream, which will be discussed later. By forming a porosity gradient in the cathode GDL, the local current density in the entire flow field can be significantly increased for the case with the parallel channels. In addition, the current density distribution can also be improved by the porosity gradient for the z-serpentine channel design. Plotted in Fig. 5 a and b are the current distributions for the z-serpentine channel design with and without porosity gradient, respectively. Again, the current density is greatly elevated in the entire flow field due to variable-porosity effects. Fig. 6 a and b show the liquid water saturation distribution at an interface between the cathode GDL and the flow channels for the parallel design. The case with 31 ¼ 0.7 and 32 ¼ 0.3 has higher liquid water saturations downstream in the middle of the model, as shown in Fig. 6b. The intense chemical reaction in the catalyst layer for 31 ¼ 0.7 and 32 ¼ 0.3 at the cathode GDL increases the current density and the liquid water saturation. Fig. 7a shows the liquid water saturation distribution for the zserpentine design at 31 ¼ 0.5 and 32 ¼ 0.5. This figure illustrates that the liquid water saturation is relatively higher in the downstream zone of the flow field. With 31 ¼ 0.7 and 32 ¼ 0.3, the liquid water
Fig. 9. Capillary diffusivity distributions on the interface between the cathode GDL and flow channels for constant and variable cathode GDL porosities in the z-serpentine channel designs.
saturation can be greatly increased in the entire flow field, as shown in Fig. 7b. This implies that due to an increased supply of gases, the electrochemical reaction becomes stronger and hence more water is produced and the liquid water saturation is increased. It is found that in the downstream zone, the value of the liquid water saturation reaches a maximum of 0.2698. Fig. 8 illustrates the capillary diffusivity at the interface between the cathode GDL and the flow channels for the parallel design. The capillary diffusivity is calculated by equation (A12). Comparison in the capillary diffusivity distributions with and without porosity gradient, based on Fig. 8 a and b, shows that the porosity gradient effectively improves the capillary diffusivity which increases the removal of water from the gas diffusion layers. The magnitude of capillary diffusivity is essential to the water removal. In Fig. 9, distribution of capillary diffusivity for the zserpentine design is shown for the constant and variable-porosity cases. In a special case as the GDL is fully saturated with water, the
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mass fraction is apparently lower than that without a porosity gradient. As compared with the cases without porosity gradient, when the porosity gradient is applied, the oxygen usage [(mO2 inlet mO2 outlet)/mO2 inlet] rises from 36% to 46% for the parallel channel design and from 55% to 67% for the z-serpentine channel design. 4. Conclusions A porosity gradient is formed in the cathode gas diffusion layer to enhance gas reactant transport and increase the capillary diffusivity in a PEMFC. This study is aimed at providing evidences by means of numerical method to show the effects of the porosity gradient in the cathode GDL on the performance of the fuel cells. Results show that a linear porosity gradient with 31 ¼ 0.7 and 32 ¼ 0.3 for the parallel channel design leads to a maximum increase in the limiting current density from 10,696 Am2 to 13,136 Am2 at 0.49 V. For z-serpentine channel design, an appreciable increase in the limiting current density is also observed. When the porosity gradient is applied with the zserpentine channel design, the limiting current density is increased from 14,053 Am2 to 16,616 Am2 at 31 ¼ 0.7 and 32 ¼ 0.3. In addition, the porosity gradient leads to an increase in the capillary diffusivity so as to help remove the water condensate, and also increases the oxygen usage from 36% to 46% for the parallel channel design and from 55% to 67% for the z-serpentine channel design. However, for the interdigitated channel design, the improvement in the current density is not significant. Nevertheless, it is observed that the pressure drop caused by the interdigitated channel design can be remarkably reduced from 75.8 Pa to 47.16 Pa, by forming a porosity gradient in the cathode GDL. Therefore, the disadvantage of high-pressure-drop with the interdigitated channel design is possible to be improved. Acknowledgements Authors acknowledge the financial support from Department of Industrial Technology, Ministry of Economic Affairs, Taiwan, under Grant 98-EC-17-A-13-S2-0064. Appendix I The PEM fuel cell model includes continuity, momentum and species equations for gaseous species, liquid water transport equations in the channels, gas diffusion layers, and catalyst layers, water transport equation in the membrane, electron and proton transport equations. They include: Fig. 10. Oxygen mass fraction distributions for parallel and z-serpentine channel designs with constant and variable porosities.
gas/fluid interface will cease to exist and then the surface tension effects will vanish. That is, the surface tension force should reach a maximum at some critical liquid water saturation and vanish at 100% liquid water saturation. According to the information presented by Wang and Cheng [29], the capillary diffusivity reaches a maximum at a liquid water saturation of 0.2. Here, it is indeed observed that in the downstream zone the capillary diffusivity is greatly increased by the porosity gradient. Fig. 10 shows the oxygen mass fraction distributions in different channel designs for 31 ¼ 0.5 and 32 ¼ 0.5 and 31 ¼ 0.7 and 32 ¼ 0.3. The positions of the measured points are indicated by the blue and the red circles plotted in the figures. Firstly, it is observed that with a porosity gradient formed in cathode GDL, at the same gas flow rate more oxygen is consumed in the channels so that the oxygen
(1) Continuity equation for the gaseous species:
V$ 3rg Ug ¼ SL
(A1)
(2) Momentum equation for the gaseous species:
3 ð1 sÞ
r ¼ 3Vpg þ V$ U U g g g 2
V$ mg VUg þ Su ð1 sÞ
3
(A2)
(3) Species equation for the gaseous species:
V$ 3rg Ug Ck ¼ V$ rg Dk;eff VCk þ Sc SL
(4) Proton and electron transport equations:
(A3)
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V$ðsm VFm Þ ¼ Sj
(A4)
V$ðss VFs Þ ¼ Sj
(A5)
(5) The energy equation for the PEM fuel cell:
V$ 3ð1 sÞrg Ug Cp;g T þ V$ 3srl Ul Cp;l T i2 ¼ V$ leff VT þ jh þ þ hfg SL
s
(A6)
where j is the transfer current density calculated by ButlereVolmer equation, leff is effective thermal conductivity, accounting for contribution of solid matrix and fluids in the pores of porous media, and can be expressed as:
leff ¼ 2ls þ
3
1
2ls þ lf
þ
13 3ls
(A7)
(6) Liquid water transport equation in the membrane:
MH2 O rdry ad MH2 O ! V$ Dl Vl ¼ 0 i m l F Mm
(A8)
(7) Liquid water transport equation in the flow channels, gas diffusion layers and catalyst layers:
r kp krl vpc r kp krl n MH2 O ! V$ l Vpg þ V$ d Vs V$ l i m ¼ SL ml vs ml F (A9) where pc is the capillary pressure head,
q < 90 0:5 3 1:417ð1 sÞ 2:12ð1 sÞ2 þ1:262ð1 sÞ3 pc ¼ s*cosqc kðzÞ p (A10)
q > 90 0:5 3 pc ¼ s*cosqc kðzÞ 1:417s 2:12s2 þ 1:262s3 p
(A11)
where the capillary diffusivity can be written in terms of the capillary pressure head as follows:
DC ¼
dpc kl ds ml
(A12)
A higher capillary diffusivity in the catalyst and the gas diffusion layers favors the removal of the water condensate from these porous layers. In this study, the porosity in the cathode GDL is treated as a function of z as
3ðzÞ ¼ az þ b
(A13)
where a ¼ (31 32) / tGDLc and b ¼ 32. 31 and 32 are the local porosities of cathode GDL at z ¼ tGDLc (contact with cathode channels) and z ¼ 0 (contact with catalyst layer), respectively. Note that in this study, to enhance the performance of the PEMFC, one has 31 S 32.
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