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The effects of the gravity on transient responses and cathode flooding in a proton exchange membrane fuel cell
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 8 ( 2 0 1 3 ) 3 3 3 0 e3 3 3 7
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The effects of the gravity on transient responses and cathode flooding in a proton exchange membrane fuel cell Mustapha Najjari a,*, Faycel Khemili b, Sassi Ben Nasrallah b a b
Research cell, Photovoltaic, Wind-driven and Geothermal Systems, University of Gabes, 6072 Zirig, Gabes, Tunisia Laboratoire d’Etudes des Syste`mes Thermiques et Energe´tiques, ENIM, Av. Ibn Eljazzar, Monasir 5019, Tunisia
article info
abstract
Article history:
The work presented in this paper includes contributions that provide insight into liquid
Received 6 November 2012
water transport in the proton exchange membrane fuel cell (PEMFC).
Received in revised form 21 December 2012
The purpose of the present study is to investigate the influence of gravity on the cathode flooding by accumulated liquid water at the interface between catalyst layer and
Accepted 4 January 2013
gas diffusion layer (CLeGDL interface) in a PEMFC. We introduce a one-dimensional and
Available online 4 February 2013
transient model based on numerical resolution of mass transport of liquid water and oxygen in the porous GDL domain. Obtained results indicate that gravity has significant ef-
Keywords:
fects on the transient time and on the cell voltage at the steady regime. In this work, both
Proton exchange membrane fuel cell
opposing and aiding gravity cases, as reference to capillary actions, are considered.
Gas diffusion layer
Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
Flooding Transient regime Gravity
1.
Introduction
The proton exchange membrane fuel cell (PEMFC) is one of the most promising technologies (stationary power generation, small portable applications, powering cars, .) and will compete with many other types of energy devices. PEMFC is an electrochemical energy conversion device that converts hydrogen and oxygen into water and, within this process, it produces heat and electricity. The oxygen required for a fuel cell comes from the air pumped into the cathode. One of the most critical aspects of a PEMFC is the water management. There is a delicate balance between the membrane hydration and avoiding cathode from flooding. Cathode flooding occurs when the rate of the water produced exceeds the rate of the water removed. Flooding in the cathode reduces the oxygen transport to the reaction sites and decreases the
effective catalyst area. In modern fuel cells, the cathode (or the anode) consists of: a catalyst layer (CL), gas diffusion layer (GDL), gas flow channel (GFC) and bipolar plates. The CL is a thin layer that speeds up the reaction of oxygen and hydrogen. It is usually made of platinum (Pt) powder very thinly coated onto carbon paper. The CL is rough and porous so the maximum surface area of the Pt can be exposed to oxygen and hydrogen. The water management in the CL has been studied by Eikerling [1]. It has concluded from this study that the CL is a critical fuel cell component in view of excessive flooding and there’s a limitation of current behaviour under certain conditions. Franco et al. [2,3] have explored the possible impact of water and ionomer on the CL performances. It has demonstrated that the morphology and microstructure changes in the CL are dependent on the carbon corrosion.
* Corresponding author. E-mail address: [email protected] (M. Najjari). 0360-3199/$ e see front matter Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijhydene.2013.01.024
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Nomenclature CO Cwl Cref Dc D0 Er F g hl i i0 JO K krl lm p R
3
oxygen concentration, mol m liquid water concentration, mol m3 reference oxygen concentration, mol m3 capillary diffusion coefficient, m2 s1 diffusion coefficient in pure gas phase, m2 s1 reversible cell potential, V Faraday constant gravity acceleration, m s2 GDL thickness, m local current density, A m2 reference current density, A m2 molar diffusion flux of oxygen absolute permeability, m2 relative permeability membrane thickness, m operating pressure, Pa gas constant
The GDL is a porous medium electrical conductor. Liquid water that builds up at the cathode of a fuel cell decreases the performances and inhibits robust operation. Recently, only few published models deal with how water is transported between the CL and GFC in the GDL domain [4e7]. The displacement of liquid water in the GDL is the result of the capillary action. However, gravity may affect this displacement depending on the orientation of the fuel cell with respect to gravity. In the literature, a number of experimental studies focusing on the effects of gravity on the PEMFC, have been reported. Morin et al. [8] have used an experimental technique to determine the effects of gravity on the water concentration profile through the thickness of the membrane (Nafion 117). It has found that the gravity retains liquid if the cathode inlet is at the bottom, which leads to a better membrane hydration. Chen and Wu [9] designed a number of experiments highlighting the effect of gravity on water management in the cathode of the PEMFC by changing the cathode and anode placed position. Experiments show that gravity is advantageous to discharging the liquid water in the PEMFC when the cell is placed anode-upward. Kimball et al. [10] examined a single-channel cell that permits a direct observation of liquid water motion and local density. Experiments show that flooding in PEMFC is gravity dependent and the local current densities depend on the dynamics of the flow in the GFC. The effect of gravity on the performance of a PEMFC, has been also experimentally investigated by Yi et al. [11] with different gas-intakes modes. Results indicate that the output power of the PEMFC stack, can be greatly enhanced for an optimized gravitational angle. In previous studies [12,13], we were interested in the effects of the flooding of the gas diffusion layer (GDL), as a result of liquid water accumulation. Changes of the molar concentration of oxygen gas phase with respect to the time and the liquid water rate production have been derived in terms of an effective GDL porosity.
Rcell re s T t ul Vcell
ohmic resistance, U electrical resistivity, U/m liquid water saturation temperature, K time, s liquid water velocity, m s1 cell voltage, V
Greek symbols the cathode transfer coefficient ac ε initial porosity of the GDL cathode activation overpotential, V hc ohmic overpotential, V hohm m dynamic viscosity, kg m1 s1 nair air stoichiometry liquid water density, kg m3 rl sm protonic conductivity of the membrane, U1 m1 q gravitational angle
The objective of the present paper is to provide a comprehensive model that describes the evolution of the cell voltage at a fixed working current density, under transient conditions and with a partial flooding of the porous cathode. For this purpose, a one-dimensional, transient model is developed. This study is devoted to numerically investigate the influences of gravity on liquid water transport by capillary and gravity effects in the GDL domain.
2.
Model development
The aim of this work is to study liquid water transport, under transient conditions that takes place in the cathode of a PEMFC. The problem domain under consideration (Fig. 1) is confined to the porous GDL where the major mass transport limitations occur. The oxygen phase diffuses into the GDL from the humidified air supplied at the inlet of the gas flow channel. The model employs the following basic assumptions: i Homogenous porous media with uniform porosity.
Membrane + catalyst layer
Porous GDL
lm
0
Gas flow channel
hl
x
Fig. 1 e Schematic of the problem showing coordinates and dimensions of the porous GDL. q [ 0 CL upward (aiding gravity). q [ 180 GDL upward (opposing gravity). q [ 90 vertical position (no gravity effects).
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ii At the beginning, the vapour pressure in the GDL exceeds its saturated value. The product water condenses in the liquid phase. iii The oxygen model flow-rate is related to the gradient of its molar concentration. We assume constant oxygen diffusivity. iv Isothermal model for the PEMFC cathode. v Liquid water may transport in the GDL under the combined actions of capillarity and gravity effects.
2.1.
! g is the gravity vector which has zero value in the horizontal direction and 9.8 m s2 in the vertical direction. The relative permeability krl increases with the saturation s. krl is expressed as [15,16]: krl ¼
s sirr 1 sirr
Water transport in the cathode of a PEMFC can originate from capillary actions and gravity forces. The continuity equation of liquid water can be written as: v u lÞ ¼ 0 ðεsMw Cw;l Þ þ div ðrl ! vt
(1)
where ε is the GDL porosity, rl is the liquid water density, s is the liquid water saturation (fraction of void volume occupied by liquid water), ! u l is the liquid water velocity and Cw,l is the liquid water density (mol m3). In this equation, phase change of water is neglected because this process is much slower compared to the other transport actions considered in this work such as diffusive flows of gaseous and liquid water [14]. The liquid water velocity is given by Darcy’s law: Kkrl / ! g V pl rl ! ul ¼ ml
(2)
(5)
where sirr is the irreducible saturation or immobile saturation. It can be considered as a threshold point below which liquid remains immobile (krl ¼ 0 if s < sirr).
2.2.
Water transport in the cathode
3
Oxygen transport: diffusional approach
In this model, we determine the molar diffusion flux of oxygen JO according to Fick’s law: / ! JO ¼ DO V CO
(6)
where CO is the molar concentration of the oxygen and DO its diffusion coefficient. The basic transport equation for oxygen molar concentration can be stated as: εð1 sÞ
vCO DO DCO ¼ 0 vt
(7)
In this paper, a dynamic, one-dimensional, isothermal PEMFC model is considered. Governing equations of mass transport are then given by the following equations, where positive x-direction is taken as the direction from CL to GFC: v v vs Kkrl rl gcosq ¼ 0 εs Dc ml vt vx vx
where K is the absolute permeability of the GDL, krl is the relative permeability of liquid phase, ml and pl are respectively the dynamic viscosity and the pressure of liquid phase, and g is the gravity acceleration. Capillary pressure is defined as the difference between liquid pressure and gas pressure:
v v vCO ¼0 ðεð1 sÞCO Þ DO vx vt vx
pc ¼ pg pl
2.3.
(3)
Assuming constant gas pressure, the governing transport equation in the porous GDL, can be expressed as [15]: / Kkrl / Kkrl ! ! u lz g z Dc V s þ r g V pc rl ! ml ml l
(4)
where Dc is the capillary diffusion coefficient of liquid water.
(8)
and
(9)
where q is the gravitational angle (Fig. 2).
Boundaries conditions
At the GDLeGFC, oxygen concentration is a function of the inlet concentration Cin O and stoichiometrie nair [14]. Therefore: 1 1 ðCO ÞGDLGFC ¼ Cin O 2nair
(10)
GDL GFC
CL GDL
0
GFC x
g
x
g
GFC GDL CL
g 0
= 0° CL upward
= 180° GDL upward
(aiding gravity)
(opposing gravity)
CL 0
x
= 90° vertical position (no gravity effects)
Fig. 2 e Schematic of different fuel cell orientations with gravity.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 8 ( 2 0 1 3 ) 3 3 3 0 e3 3 3 7
The oxygen consumption rate by the electrochemical reaction is given by: DO
vCO i ¼ vx CLGDL 4F
3333
time step Dt in CO and s fields, are less than 105. Dt ¼ 0.001 s is chosen for all numerical resolutions.
(11)
3.
where i is the working current density and F is the Faraday constant. The produced water by electrochemical reaction is related to the current density i and liquid water saturation at the CLeGDL interface, by:
Results and discussions
Due to the convective flow, water exists only in the vapour phase, in the gas flow channel:
The developed one-dimensional transient model for the cathode GDL is used to examine the effects of many important factors on the transient phenomena in PEMFC system. Calculations were conducted when the fuel cell was operated at constant current and the temperature is at a fixed value 80 C, so that isothermal condition is assumed. Inlet humidity and air flow-rate are adjusted by the cathode stoichiometry. Geometrical dimensions and operating parameters are listed in Table 1.
ðsÞGDLGFC ¼ 0
(13)
3.1.
(14)
Before discussing the results, the model must be validated by comparing the numerical predictions with experimental data from the literature provided by Bajpai et al. [19]. Fig. 3 shows a comparison concerning the evolution of the polarization curve (Eq. (15)). It can be found that calculated values agree well with the experimental results in the operating conditions. The cell potential falls as the current supplied by the cell increases. Thus, in the first stage, up to a certain current value, activation overvoltage drop prevails. At a later stage, as current density rises, ohmic losses prevail. They are derived from membrane resistance to protons transfer. When current is very high, at maximum power level, concentration overvoltage produces a quick drop of the voltage due to internal inefficiencies at high levels of reactive consumption. As the polarization curve only includes the steady state and no gravity effects, later results have focussed on the
Dc
vs vx
At t ¼ 0;
2.4.
þ CLGDL
Kkrl Mw i r gcosq ¼ ml l 2rl F
CO ¼ ðCO ÞGDLGFC
and s ¼ 0
(12)
Cell voltage
The output fuel cell voltage Vcell, can be defined as: Vcell ¼ Er hc hohm
(15)
In this relation, Er is the reversible cell potential (Nernst equation), hc is the activation overpotential corresponding to losses that are associated with the kinetics of the electrochemical reaction and hohm is the ohmic overpotential due to losses from the protonic and the electronic resistances in the cell. hohm is determined by the following equation: hohm ¼ i Rcell
(16)
where Rcell is the ohmic resistance which includes the electronic and the membrane resistances: Rcell ¼ Rmembrane þ Relectrode ¼
lm þ re hl sm
(17)
where lm and sm are, respectively, the thickness and the protonic conductivity of the membrane and re is the electrical resistivity of the cathode paper. Tafel equation [17] is used to describe the electrochemical reaction along the membrane/cathode interface; namely, hc ¼
RT Cref i ln i0 ð1 sÞðCO Þx¼0 ac F
(18)
where hc is the cathode overpotential, i0 is the exchange current density, ac is the cathode transfer coefficient, F is the Faraday constant, R is the universal gas constant, T the absolute temperature, i the local current density. Here the term (1 s) is used to account for the reduction of active surface due to the liquid water coverage of the catalyst particles.
2.5.
Model validation
Numerical resolution
The numerical resolution of equations (8) and (9) is based on the finite volume method [18]. A Fortran program is used for this purpose. For the present computations, 201 uniform mesh has been used. Convergence is considered to be reached when the relative changes between successive iterations, at each
Table 1 e Physical properties and parameters. Parameter
Value 1
1
Universal gas constant, R (J mol K ) Water molar weight, M (kg mol1) Liquid water density, rl (kg m3) Dynamic viscosity, ml (kg m1 s1) Faraday constant, F (C mol1) Cell temperature, T ( C) Absolute permeability, K (m2) Irreducible liquid water saturation, sirr 3 Inlet molar fraction of oxygen, Cin O (mol m ) Capillary diffusion coefficient (m2 s1) Diffusion coefficient in pure gas phase (m2 s1) Air stoichiometry, x Initial GDL porosity, ε0 GDL thickness, hl (m) Reference current density, i0 (A m2) Cathode transfer coefficient, ac Protonic conductivity of the membrane (U1 m1) Electrical resistivity of the cathode, re (mU cm) Membrane thickness, lm (cm) Reversible cell potential, Er (V) Reference oxygen concentration, Cref (mol m3)
8.314 0.018 971.8 4.47 $ 104 96,487 80 3 $ 109 0.01 0.8 7.35 $ 105 3.23 $ 105 1 0.8 0.34 $ 103 2 0.9 4 80 38 $ 106 1.23 40
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0.040
1.20
0.030
Experiments Bajpai et al. [15]
0.80
0.40
Liquid water saturation
Cell voltage (V)
Present model
No gravity 0.020
0.010
0.00 0.00
0.40
0.80
1.20
1.60
0.000
2.00
0
Current density (A/cm²)
5
10
15
20
25
Time (s)
Fig. 3 e Polarization curves from Bajpai [19] and the present model.
Fig. 5 e The transient evolution of the liquid water saturation at the GDLemembrane interface (no gravity case).
dynamic responses of the fuel cell, taking transient behaviour and gravity effects into account.
3.2.
Dynamic processes with zero gravity
Fig. 4 shows the evolution of the cell voltage with time at the operating current density of 1.4 A cm2, in the case of zero gravity (vertical position). Figs. 5 and 6 show the transient plots of liquid saturation and oxygen concentration at the GDLemembrane interface. It could be seen that the rate of the cell voltage decrease, was high then slowly reduced until about 15 s when the fuel cell reached its steady state and the cell voltage became constant.
It can be seen from Figs. 5 and 6 that during the initial response, liquid water saturation produced by the electrochemical reaction at highly current density, accumulates at the GDLemembrane interface. This accumulation of liquid water can lead to cathode flooding, which inhibits reactant transport. Produced liquid water at the GDLemembrane interface is transported by capillary action across the GDL toward the cathode gas flow channel to be removed. Liquid water saturation profiles against x-axis (along the GDL) are plotted in Fig. 7. Liquid water saturation gradually 0.400
Oxygen concentration
0.593
Cell potential (V)
No gravity 0.592
0.396
No gravity
0.392
0.591
0.388 0.590
0 0
5
10
15
20
Time (s)
Fig. 4 e The transient evolution of the cell voltage (no gravity case).
25
1
2
3
4
5
Time (s)
Fig. 6 e The transient evolution of the oxygen concentration at the GDLemembrane interface (no gravity case).
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Liquid water saturation at the GDL-membrane interface
0.040
No gravity Cathode gas-flow channel
Liquid water saturation
0.030
0.020
0.010
0.000
0.100
0.080
0.060
0.040
GDL upward - opposing gravity
0.020
Vertical position - no gravity CL upward - aiding gravity
0.000 0
x-axis
10
20
30
40
Time (s)
Fig. 7 e Liquid water saturation along the GDL domain (no gravity case).
increases from the membrane side to the gas channel side and finally reaches its steady state when liquid water produced by the electrochemical reaction, is balanced by diffusive flowrate due to capillary effects.
3.3.
Dynamic processes under gravity effects
In this section we will demonstrate that the response of the fuel cell is highly modified when considering gravity effects. Two distinct vertical configurations of the fuel cell were examined: i- the gravity force in the direction of the capillarydriven flow. In this case, anode is upward (CL upward). ii- the gravity force in the opposite direction of the capillary-driven flow where the cathode is upward (GDL upward).
Fig. 9 e Transient evolution of the liquid water saturation for different vertical orientations of the fuel cell.
The first case will be called aiding gravity and the second case the opposing gravity. The PEM fuel cell response with time was studied for both aiding and opposing gravity cases. Fig. 8 shows the evolution of the cell voltage for aiding and opposing gravity. A comparison between considering and neglecting gravity, indicates that the transient time is highly dependent on gravity effects. This may be interpreted based on the fact that the capillary action in the GDL on liquid water is dominated by gravity forces. Because of the importance of gravity, we observe different steady-states characteristics, with different vertical orientations of the fuel cell. In the case of aiding gravity, liquid water produced by the electrochemical reaction, may fall away from GDLemembrane interface, and be pushed out toward the gas
0.593
0.040
GDL upward - opposing gravity Vertical position - no gravity
0.592
CL upward - aiding gravity
No gravity
0.591
0.590
0.589
Steady state
0.020 t=4s
0.010
Cathode gas-flow channel
Liquid water saturation
Cell potential (V)
0.030
t=2s
t = 0.5 s
0.588 0
10
20
30
40
Time (s)
Fig. 8 e Transient evolution of the cell voltage for different vertical orientations of the fuel cell.
0.000
x-axis
Fig. 10 e Liquid water saturation along the GDL domain at different times (no gravity case).
3336
0.040
0.5908
0.10
0.5904
Steady state
0.020
t=6s t=3s
Cathode gas-flow channel
Liquid water saturation
CL upward - aiding gravity
Cell potential (V)
0.08
0.030
0.5900 0.06 0.5896
0.04 0.5892
0.010 t=1s
0.5888
t = 0.5 s
x-axis
Fig. 11 e Liquid water saturation along the GDL domain at different times (aiding gravity case).
flow channel rapidly and with minimal effects on the fuel cell voltage. However, for the opposing gravity case, gravity force overcomes the capillary action and allows liquid to accumulate at the GDLemembrane interface which causes the flooding of the GDL. As a result, the cell voltage is highly decreased at the steady state (Fig. 8). Fig. 9 shows that liquid water saturation at the GDLemembrane interface, is highly reduced in the aiding gravity case, due to the assistance from gravity in removing liquid water from the GDLemembrane interface. The dynamic profiles of liquid water content along the GDL, are presented in Figs. 10e12. In Fig. 10, the distribution of liquid water saturation, is plotted for three times and the steady state, in the non-gravity case (Fig. 13).
0.100
GDL upward - opposing gravity 0.060
Steady state t=6s
Cathode gas-flow channel
Liquid water saturation
80
120
160
200
Fig. 13 e The values of the cell voltage and the liquid saturation, at the steady state, for different gravitational angles (0 £ q £ 180 ).
At t ¼ 0 s, no liquid water in the GDL. Water accumulates at the GDLemembrane interface and diffuses gradually toward the cathode gas flow channel by capillary transport. At the steady state, the liquid water saturation has a linear profile (steady-state solution of a diffusive equation). The effect of gravity on the transient profiles of liquid saturation is illustrated in Fig. 11 and 12. When s < sirr, the shape of x-profiles are similar to the non-gravity case, which implies that the liquid transport is still dominated by capillarity. Thereafter, when s exceeds sirr, liquid accumulation at the GDLemembrane interface, is largest for the opposing gravity case and smallest for the aiding gravity case. A smaller liquid saturation variation along the GDL, can be observed in the case of aiding gravity.
3.4.
0.080
0.020
40
Gravitational angle (theta °)
0.000
0.040
0.02 0
Liquid water saturation at the GDL-membrane interface
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 8 ( 2 0 1 3 ) 3 3 3 0 e3 3 3 7
Cell voltage under varied gravitational angles
In order to investigate the fuel cell performance for varied gravitational angles, the following calculations were performed. If the cell is placed at a vertical position, the gravitational angle q is fixed at 90 . However, q equals 0 and 180 respectively for aiding and opposing cases (Fig. 2). The cell voltage and the liquid saturation, at the steady state, have been plotted for different gravitational angles (0 q 180 ). Dramatic drops in the cell voltage are detected when q is higher than 90 . This is mainly caused by the drops of the cell performance with increasing liquid flooding effects at the GDLemembrane interface.
t=3s
4.
t = 0.5 s
0.000
x-axis
Fig. 12 e Liquid water saturation along the GDL domain at different times (opposing gravity case).
Conclusions
The effects of gravity forces on liquid water transport in the GDL and the performance of a PEMFC, are presented in this paper. For this purpose, we develop a transient onedimensional model, based on the resolution of the diffusion
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 8 ( 2 0 1 3 ) 3 3 3 0 e3 3 3 7
equations for oxygen and liquid water saturation. We found that: * Gravity forces can not be neglected when studying the performance of a PEMFC. * When gravity is considered, the time needed to reach the steady state is reduced. * If gravity forces oppose capillary actions in the porous GDL, liquid water accumulates at the CLeGDL interface and the performance (cell voltage) of the PEMFC is highly decreased. * If gravity forces assist capillary actions, accumulated liquid water is rapidly evacuated to the GFC and a smaller drop of the cell voltage at the steady state is detected.
references
[1] Eikerling M. Water management in cathode catalyst layers of PEM fuel cells: a structure-based model. Journal of the Electrochemical Society 2006;153:E58e70. [2] Malek K, Franco AA. Microstructure-based modeling of aging mechanisms in catalyst layers of polymer electrolyte fuel cells. Journal of Physical Chemistry B 2011;115(25):8088e101. [3] Ferreira de Morais R, Sautet Ph, Loffreda D, Alejandro A, Franco AA. A multiscale theoretical methodology for the calculation of electrochemical observables from ab initio data: application to the oxygen reduction reaction in a Pt(1 1 1)-based polymer electrolyte membrane fuel cell. Electrochimica Acta 2011;56(28):10842e56. [4] Wong KH, Loo KH, Lai YM, Tan SC, Tse CK. Treatment of twophase flow in cathode gas channel for an improved onedimensional proton exchange membrane fuel cell model. International Journal of Hydrogen Energy 2011;36:3941e55. [5] Haddad A, Bouyekhf R, Moudni A. Dynamic modelling and water management in proton exchange membrane fuel cell. International Journal of Hydrogen Energy 2008;33: 6239e52. [6] Hussaini IS, Wang CY. Measurement of relative permeability of fuel cell diffusion media. Journal of Power Sources 2010; 195:3830e40.
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[7] Ramousse J, Deseure J, Lottin O, Didierjean S, Maillet D. Modelling of heat, mass and charge transfer in a PEMFC single cell. Journal of Power Sources 2005;145:416e27. [8] Morin A, Xu F, Gebel G, Diat O. Influence of PEMFC gas flow configuration on performance and water distribution studied by SANS: evidence of the effect of gravity. International Journal of Hydrogen Energy 2011;36:3096e109. [9] Chen S, Wu Y. Gravity effect on water discharged in PEM fuel cell cathode. International Journal of Hydrogen Energy 2010; 35:2888e93. [10] Kimball E, Whitaker T, Kevrekidis YG, Benziger JB. Drops, slugs, and flooding in polymer electrolyte membrane fuel cells. AIChE Journal 2008;54:1313e32. [11] Yi Y, Zheng-kai T, Zhi-gang Zhan, Mu Pan. Gravity effects on the performance of PEM fuel cell stack with different gas manifold positions. International Journal of Energy Researches 2012;36(7):845e55. [12] Najjari M, Khemili F, Ben Nasrallah S. The effects of the cathode flooding on transient responses of a PEM fuel cell. Renewable Energy International Journal 2008;33:1824e31. [13] Khemili F, Najjari M, Ben Nasrallah S. A comprehensive model for evaluating the performance of proton exchange membrane fuel cells. International Journal of Heat and Technology 2012;30(1):1e9. [14] Loo KH, Wong KH, Tan SC, Lai YM, Tse CK. Characterization of the dynamic response of proton exchange membrane fuel cells e a numerical study. International Journal of Hydrogen Energy 2010;35:11861e77. [15] Kumbur EC, Sharp KV, Mench MM. On the effectiveness of Leverett approach for describing the water transport in fuel cell diffusion media. Journal of Power Sources 2007;168: 356e68. [16] Song D, Wang Q, Liu Z, Huang Cheng. Transient analysis for the cathode gas diffusion layer of PEM fuel cells. Journal of Power Sources 2006;159:928e42. [17] Pasaogullari U, Wang CY. Liquid water transport in gas diffusion layer of polymer electrolyte fuel cells. Journal of the Electrochemical Society 2004;151(3):399e406. [18] Patankar V. Numerical heat transfer fluid flow. New York: Hemisphere/MacGraw-Hill; 1980. [19] Bajpai H, Khandelwal M, Kumbur EC, Mench MM. A computational model for assessing impact of interfacial morphology on polymer electrolyte fuel cell performance. Journal of Power Sources 2010;195:4196e205.
Available online at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/he
The effects of the gravity on transient responses and cathode flooding in a proton exchange membrane fuel cell Mustapha Najjari a,*, Faycel Khemili b, Sassi Ben Nasrallah b a b
Research cell, Photovoltaic, Wind-driven and Geothermal Systems, University of Gabes, 6072 Zirig, Gabes, Tunisia Laboratoire d’Etudes des Syste`mes Thermiques et Energe´tiques, ENIM, Av. Ibn Eljazzar, Monasir 5019, Tunisia
article info
abstract
Article history:
The work presented in this paper includes contributions that provide insight into liquid
Received 6 November 2012
water transport in the proton exchange membrane fuel cell (PEMFC).
Received in revised form 21 December 2012
The purpose of the present study is to investigate the influence of gravity on the cathode flooding by accumulated liquid water at the interface between catalyst layer and
Accepted 4 January 2013
gas diffusion layer (CLeGDL interface) in a PEMFC. We introduce a one-dimensional and
Available online 4 February 2013
transient model based on numerical resolution of mass transport of liquid water and oxygen in the porous GDL domain. Obtained results indicate that gravity has significant ef-
Keywords:
fects on the transient time and on the cell voltage at the steady regime. In this work, both
Proton exchange membrane fuel cell
opposing and aiding gravity cases, as reference to capillary actions, are considered.
Gas diffusion layer
Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
Flooding Transient regime Gravity
1.
Introduction
The proton exchange membrane fuel cell (PEMFC) is one of the most promising technologies (stationary power generation, small portable applications, powering cars, .) and will compete with many other types of energy devices. PEMFC is an electrochemical energy conversion device that converts hydrogen and oxygen into water and, within this process, it produces heat and electricity. The oxygen required for a fuel cell comes from the air pumped into the cathode. One of the most critical aspects of a PEMFC is the water management. There is a delicate balance between the membrane hydration and avoiding cathode from flooding. Cathode flooding occurs when the rate of the water produced exceeds the rate of the water removed. Flooding in the cathode reduces the oxygen transport to the reaction sites and decreases the
effective catalyst area. In modern fuel cells, the cathode (or the anode) consists of: a catalyst layer (CL), gas diffusion layer (GDL), gas flow channel (GFC) and bipolar plates. The CL is a thin layer that speeds up the reaction of oxygen and hydrogen. It is usually made of platinum (Pt) powder very thinly coated onto carbon paper. The CL is rough and porous so the maximum surface area of the Pt can be exposed to oxygen and hydrogen. The water management in the CL has been studied by Eikerling [1]. It has concluded from this study that the CL is a critical fuel cell component in view of excessive flooding and there’s a limitation of current behaviour under certain conditions. Franco et al. [2,3] have explored the possible impact of water and ionomer on the CL performances. It has demonstrated that the morphology and microstructure changes in the CL are dependent on the carbon corrosion.
* Corresponding author. E-mail address: [email protected] (M. Najjari). 0360-3199/$ e see front matter Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijhydene.2013.01.024
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Nomenclature CO Cwl Cref Dc D0 Er F g hl i i0 JO K krl lm p R
3
oxygen concentration, mol m liquid water concentration, mol m3 reference oxygen concentration, mol m3 capillary diffusion coefficient, m2 s1 diffusion coefficient in pure gas phase, m2 s1 reversible cell potential, V Faraday constant gravity acceleration, m s2 GDL thickness, m local current density, A m2 reference current density, A m2 molar diffusion flux of oxygen absolute permeability, m2 relative permeability membrane thickness, m operating pressure, Pa gas constant
The GDL is a porous medium electrical conductor. Liquid water that builds up at the cathode of a fuel cell decreases the performances and inhibits robust operation. Recently, only few published models deal with how water is transported between the CL and GFC in the GDL domain [4e7]. The displacement of liquid water in the GDL is the result of the capillary action. However, gravity may affect this displacement depending on the orientation of the fuel cell with respect to gravity. In the literature, a number of experimental studies focusing on the effects of gravity on the PEMFC, have been reported. Morin et al. [8] have used an experimental technique to determine the effects of gravity on the water concentration profile through the thickness of the membrane (Nafion 117). It has found that the gravity retains liquid if the cathode inlet is at the bottom, which leads to a better membrane hydration. Chen and Wu [9] designed a number of experiments highlighting the effect of gravity on water management in the cathode of the PEMFC by changing the cathode and anode placed position. Experiments show that gravity is advantageous to discharging the liquid water in the PEMFC when the cell is placed anode-upward. Kimball et al. [10] examined a single-channel cell that permits a direct observation of liquid water motion and local density. Experiments show that flooding in PEMFC is gravity dependent and the local current densities depend on the dynamics of the flow in the GFC. The effect of gravity on the performance of a PEMFC, has been also experimentally investigated by Yi et al. [11] with different gas-intakes modes. Results indicate that the output power of the PEMFC stack, can be greatly enhanced for an optimized gravitational angle. In previous studies [12,13], we were interested in the effects of the flooding of the gas diffusion layer (GDL), as a result of liquid water accumulation. Changes of the molar concentration of oxygen gas phase with respect to the time and the liquid water rate production have been derived in terms of an effective GDL porosity.
Rcell re s T t ul Vcell
ohmic resistance, U electrical resistivity, U/m liquid water saturation temperature, K time, s liquid water velocity, m s1 cell voltage, V
Greek symbols the cathode transfer coefficient ac ε initial porosity of the GDL cathode activation overpotential, V hc ohmic overpotential, V hohm m dynamic viscosity, kg m1 s1 nair air stoichiometry liquid water density, kg m3 rl sm protonic conductivity of the membrane, U1 m1 q gravitational angle
The objective of the present paper is to provide a comprehensive model that describes the evolution of the cell voltage at a fixed working current density, under transient conditions and with a partial flooding of the porous cathode. For this purpose, a one-dimensional, transient model is developed. This study is devoted to numerically investigate the influences of gravity on liquid water transport by capillary and gravity effects in the GDL domain.
2.
Model development
The aim of this work is to study liquid water transport, under transient conditions that takes place in the cathode of a PEMFC. The problem domain under consideration (Fig. 1) is confined to the porous GDL where the major mass transport limitations occur. The oxygen phase diffuses into the GDL from the humidified air supplied at the inlet of the gas flow channel. The model employs the following basic assumptions: i Homogenous porous media with uniform porosity.
Membrane + catalyst layer
Porous GDL
lm
0
Gas flow channel
hl
x
Fig. 1 e Schematic of the problem showing coordinates and dimensions of the porous GDL. q [ 0 CL upward (aiding gravity). q [ 180 GDL upward (opposing gravity). q [ 90 vertical position (no gravity effects).
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ii At the beginning, the vapour pressure in the GDL exceeds its saturated value. The product water condenses in the liquid phase. iii The oxygen model flow-rate is related to the gradient of its molar concentration. We assume constant oxygen diffusivity. iv Isothermal model for the PEMFC cathode. v Liquid water may transport in the GDL under the combined actions of capillarity and gravity effects.
2.1.
! g is the gravity vector which has zero value in the horizontal direction and 9.8 m s2 in the vertical direction. The relative permeability krl increases with the saturation s. krl is expressed as [15,16]: krl ¼
s sirr 1 sirr
Water transport in the cathode of a PEMFC can originate from capillary actions and gravity forces. The continuity equation of liquid water can be written as: v u lÞ ¼ 0 ðεsMw Cw;l Þ þ div ðrl ! vt
(1)
where ε is the GDL porosity, rl is the liquid water density, s is the liquid water saturation (fraction of void volume occupied by liquid water), ! u l is the liquid water velocity and Cw,l is the liquid water density (mol m3). In this equation, phase change of water is neglected because this process is much slower compared to the other transport actions considered in this work such as diffusive flows of gaseous and liquid water [14]. The liquid water velocity is given by Darcy’s law: Kkrl / ! g V pl rl ! ul ¼ ml
(2)
(5)
where sirr is the irreducible saturation or immobile saturation. It can be considered as a threshold point below which liquid remains immobile (krl ¼ 0 if s < sirr).
2.2.
Water transport in the cathode
3
Oxygen transport: diffusional approach
In this model, we determine the molar diffusion flux of oxygen JO according to Fick’s law: / ! JO ¼ DO V CO
(6)
where CO is the molar concentration of the oxygen and DO its diffusion coefficient. The basic transport equation for oxygen molar concentration can be stated as: εð1 sÞ
vCO DO DCO ¼ 0 vt
(7)
In this paper, a dynamic, one-dimensional, isothermal PEMFC model is considered. Governing equations of mass transport are then given by the following equations, where positive x-direction is taken as the direction from CL to GFC: v v vs Kkrl rl gcosq ¼ 0 εs Dc ml vt vx vx
where K is the absolute permeability of the GDL, krl is the relative permeability of liquid phase, ml and pl are respectively the dynamic viscosity and the pressure of liquid phase, and g is the gravity acceleration. Capillary pressure is defined as the difference between liquid pressure and gas pressure:
v v vCO ¼0 ðεð1 sÞCO Þ DO vx vt vx
pc ¼ pg pl
2.3.
(3)
Assuming constant gas pressure, the governing transport equation in the porous GDL, can be expressed as [15]: / Kkrl / Kkrl ! ! u lz g z Dc V s þ r g V pc rl ! ml ml l
(4)
where Dc is the capillary diffusion coefficient of liquid water.
(8)
and
(9)
where q is the gravitational angle (Fig. 2).
Boundaries conditions
At the GDLeGFC, oxygen concentration is a function of the inlet concentration Cin O and stoichiometrie nair [14]. Therefore: 1 1 ðCO ÞGDLGFC ¼ Cin O 2nair
(10)
GDL GFC
CL GDL
0
GFC x
g
x
g
GFC GDL CL
g 0
= 0° CL upward
= 180° GDL upward
(aiding gravity)
(opposing gravity)
CL 0
x
= 90° vertical position (no gravity effects)
Fig. 2 e Schematic of different fuel cell orientations with gravity.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 8 ( 2 0 1 3 ) 3 3 3 0 e3 3 3 7
The oxygen consumption rate by the electrochemical reaction is given by: DO
vCO i ¼ vx CLGDL 4F
3333
time step Dt in CO and s fields, are less than 105. Dt ¼ 0.001 s is chosen for all numerical resolutions.
(11)
3.
where i is the working current density and F is the Faraday constant. The produced water by electrochemical reaction is related to the current density i and liquid water saturation at the CLeGDL interface, by:
Results and discussions
Due to the convective flow, water exists only in the vapour phase, in the gas flow channel:
The developed one-dimensional transient model for the cathode GDL is used to examine the effects of many important factors on the transient phenomena in PEMFC system. Calculations were conducted when the fuel cell was operated at constant current and the temperature is at a fixed value 80 C, so that isothermal condition is assumed. Inlet humidity and air flow-rate are adjusted by the cathode stoichiometry. Geometrical dimensions and operating parameters are listed in Table 1.
ðsÞGDLGFC ¼ 0
(13)
3.1.
(14)
Before discussing the results, the model must be validated by comparing the numerical predictions with experimental data from the literature provided by Bajpai et al. [19]. Fig. 3 shows a comparison concerning the evolution of the polarization curve (Eq. (15)). It can be found that calculated values agree well with the experimental results in the operating conditions. The cell potential falls as the current supplied by the cell increases. Thus, in the first stage, up to a certain current value, activation overvoltage drop prevails. At a later stage, as current density rises, ohmic losses prevail. They are derived from membrane resistance to protons transfer. When current is very high, at maximum power level, concentration overvoltage produces a quick drop of the voltage due to internal inefficiencies at high levels of reactive consumption. As the polarization curve only includes the steady state and no gravity effects, later results have focussed on the
Dc
vs vx
At t ¼ 0;
2.4.
þ CLGDL
Kkrl Mw i r gcosq ¼ ml l 2rl F
CO ¼ ðCO ÞGDLGFC
and s ¼ 0
(12)
Cell voltage
The output fuel cell voltage Vcell, can be defined as: Vcell ¼ Er hc hohm
(15)
In this relation, Er is the reversible cell potential (Nernst equation), hc is the activation overpotential corresponding to losses that are associated with the kinetics of the electrochemical reaction and hohm is the ohmic overpotential due to losses from the protonic and the electronic resistances in the cell. hohm is determined by the following equation: hohm ¼ i Rcell
(16)
where Rcell is the ohmic resistance which includes the electronic and the membrane resistances: Rcell ¼ Rmembrane þ Relectrode ¼
lm þ re hl sm
(17)
where lm and sm are, respectively, the thickness and the protonic conductivity of the membrane and re is the electrical resistivity of the cathode paper. Tafel equation [17] is used to describe the electrochemical reaction along the membrane/cathode interface; namely, hc ¼
RT Cref i ln i0 ð1 sÞðCO Þx¼0 ac F
(18)
where hc is the cathode overpotential, i0 is the exchange current density, ac is the cathode transfer coefficient, F is the Faraday constant, R is the universal gas constant, T the absolute temperature, i the local current density. Here the term (1 s) is used to account for the reduction of active surface due to the liquid water coverage of the catalyst particles.
2.5.
Model validation
Numerical resolution
The numerical resolution of equations (8) and (9) is based on the finite volume method [18]. A Fortran program is used for this purpose. For the present computations, 201 uniform mesh has been used. Convergence is considered to be reached when the relative changes between successive iterations, at each
Table 1 e Physical properties and parameters. Parameter
Value 1
1
Universal gas constant, R (J mol K ) Water molar weight, M (kg mol1) Liquid water density, rl (kg m3) Dynamic viscosity, ml (kg m1 s1) Faraday constant, F (C mol1) Cell temperature, T ( C) Absolute permeability, K (m2) Irreducible liquid water saturation, sirr 3 Inlet molar fraction of oxygen, Cin O (mol m ) Capillary diffusion coefficient (m2 s1) Diffusion coefficient in pure gas phase (m2 s1) Air stoichiometry, x Initial GDL porosity, ε0 GDL thickness, hl (m) Reference current density, i0 (A m2) Cathode transfer coefficient, ac Protonic conductivity of the membrane (U1 m1) Electrical resistivity of the cathode, re (mU cm) Membrane thickness, lm (cm) Reversible cell potential, Er (V) Reference oxygen concentration, Cref (mol m3)
8.314 0.018 971.8 4.47 $ 104 96,487 80 3 $ 109 0.01 0.8 7.35 $ 105 3.23 $ 105 1 0.8 0.34 $ 103 2 0.9 4 80 38 $ 106 1.23 40
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0.040
1.20
0.030
Experiments Bajpai et al. [15]
0.80
0.40
Liquid water saturation
Cell voltage (V)
Present model
No gravity 0.020
0.010
0.00 0.00
0.40
0.80
1.20
1.60
0.000
2.00
0
Current density (A/cm²)
5
10
15
20
25
Time (s)
Fig. 3 e Polarization curves from Bajpai [19] and the present model.
Fig. 5 e The transient evolution of the liquid water saturation at the GDLemembrane interface (no gravity case).
dynamic responses of the fuel cell, taking transient behaviour and gravity effects into account.
3.2.
Dynamic processes with zero gravity
Fig. 4 shows the evolution of the cell voltage with time at the operating current density of 1.4 A cm2, in the case of zero gravity (vertical position). Figs. 5 and 6 show the transient plots of liquid saturation and oxygen concentration at the GDLemembrane interface. It could be seen that the rate of the cell voltage decrease, was high then slowly reduced until about 15 s when the fuel cell reached its steady state and the cell voltage became constant.
It can be seen from Figs. 5 and 6 that during the initial response, liquid water saturation produced by the electrochemical reaction at highly current density, accumulates at the GDLemembrane interface. This accumulation of liquid water can lead to cathode flooding, which inhibits reactant transport. Produced liquid water at the GDLemembrane interface is transported by capillary action across the GDL toward the cathode gas flow channel to be removed. Liquid water saturation profiles against x-axis (along the GDL) are plotted in Fig. 7. Liquid water saturation gradually 0.400
Oxygen concentration
0.593
Cell potential (V)
No gravity 0.592
0.396
No gravity
0.392
0.591
0.388 0.590
0 0
5
10
15
20
Time (s)
Fig. 4 e The transient evolution of the cell voltage (no gravity case).
25
1
2
3
4
5
Time (s)
Fig. 6 e The transient evolution of the oxygen concentration at the GDLemembrane interface (no gravity case).
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Liquid water saturation at the GDL-membrane interface
0.040
No gravity Cathode gas-flow channel
Liquid water saturation
0.030
0.020
0.010
0.000
0.100
0.080
0.060
0.040
GDL upward - opposing gravity
0.020
Vertical position - no gravity CL upward - aiding gravity
0.000 0
x-axis
10
20
30
40
Time (s)
Fig. 7 e Liquid water saturation along the GDL domain (no gravity case).
increases from the membrane side to the gas channel side and finally reaches its steady state when liquid water produced by the electrochemical reaction, is balanced by diffusive flowrate due to capillary effects.
3.3.
Dynamic processes under gravity effects
In this section we will demonstrate that the response of the fuel cell is highly modified when considering gravity effects. Two distinct vertical configurations of the fuel cell were examined: i- the gravity force in the direction of the capillarydriven flow. In this case, anode is upward (CL upward). ii- the gravity force in the opposite direction of the capillary-driven flow where the cathode is upward (GDL upward).
Fig. 9 e Transient evolution of the liquid water saturation for different vertical orientations of the fuel cell.
The first case will be called aiding gravity and the second case the opposing gravity. The PEM fuel cell response with time was studied for both aiding and opposing gravity cases. Fig. 8 shows the evolution of the cell voltage for aiding and opposing gravity. A comparison between considering and neglecting gravity, indicates that the transient time is highly dependent on gravity effects. This may be interpreted based on the fact that the capillary action in the GDL on liquid water is dominated by gravity forces. Because of the importance of gravity, we observe different steady-states characteristics, with different vertical orientations of the fuel cell. In the case of aiding gravity, liquid water produced by the electrochemical reaction, may fall away from GDLemembrane interface, and be pushed out toward the gas
0.593
0.040
GDL upward - opposing gravity Vertical position - no gravity
0.592
CL upward - aiding gravity
No gravity
0.591
0.590
0.589
Steady state
0.020 t=4s
0.010
Cathode gas-flow channel
Liquid water saturation
Cell potential (V)
0.030
t=2s
t = 0.5 s
0.588 0
10
20
30
40
Time (s)
Fig. 8 e Transient evolution of the cell voltage for different vertical orientations of the fuel cell.
0.000
x-axis
Fig. 10 e Liquid water saturation along the GDL domain at different times (no gravity case).
3336
0.040
0.5908
0.10
0.5904
Steady state
0.020
t=6s t=3s
Cathode gas-flow channel
Liquid water saturation
CL upward - aiding gravity
Cell potential (V)
0.08
0.030
0.5900 0.06 0.5896
0.04 0.5892
0.010 t=1s
0.5888
t = 0.5 s
x-axis
Fig. 11 e Liquid water saturation along the GDL domain at different times (aiding gravity case).
flow channel rapidly and with minimal effects on the fuel cell voltage. However, for the opposing gravity case, gravity force overcomes the capillary action and allows liquid to accumulate at the GDLemembrane interface which causes the flooding of the GDL. As a result, the cell voltage is highly decreased at the steady state (Fig. 8). Fig. 9 shows that liquid water saturation at the GDLemembrane interface, is highly reduced in the aiding gravity case, due to the assistance from gravity in removing liquid water from the GDLemembrane interface. The dynamic profiles of liquid water content along the GDL, are presented in Figs. 10e12. In Fig. 10, the distribution of liquid water saturation, is plotted for three times and the steady state, in the non-gravity case (Fig. 13).
0.100
GDL upward - opposing gravity 0.060
Steady state t=6s
Cathode gas-flow channel
Liquid water saturation
80
120
160
200
Fig. 13 e The values of the cell voltage and the liquid saturation, at the steady state, for different gravitational angles (0 £ q £ 180 ).
At t ¼ 0 s, no liquid water in the GDL. Water accumulates at the GDLemembrane interface and diffuses gradually toward the cathode gas flow channel by capillary transport. At the steady state, the liquid water saturation has a linear profile (steady-state solution of a diffusive equation). The effect of gravity on the transient profiles of liquid saturation is illustrated in Fig. 11 and 12. When s < sirr, the shape of x-profiles are similar to the non-gravity case, which implies that the liquid transport is still dominated by capillarity. Thereafter, when s exceeds sirr, liquid accumulation at the GDLemembrane interface, is largest for the opposing gravity case and smallest for the aiding gravity case. A smaller liquid saturation variation along the GDL, can be observed in the case of aiding gravity.
3.4.
0.080
0.020
40
Gravitational angle (theta °)
0.000
0.040
0.02 0
Liquid water saturation at the GDL-membrane interface
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 8 ( 2 0 1 3 ) 3 3 3 0 e3 3 3 7
Cell voltage under varied gravitational angles
In order to investigate the fuel cell performance for varied gravitational angles, the following calculations were performed. If the cell is placed at a vertical position, the gravitational angle q is fixed at 90 . However, q equals 0 and 180 respectively for aiding and opposing cases (Fig. 2). The cell voltage and the liquid saturation, at the steady state, have been plotted for different gravitational angles (0 q 180 ). Dramatic drops in the cell voltage are detected when q is higher than 90 . This is mainly caused by the drops of the cell performance with increasing liquid flooding effects at the GDLemembrane interface.
t=3s
4.
t = 0.5 s
0.000
x-axis
Fig. 12 e Liquid water saturation along the GDL domain at different times (opposing gravity case).
Conclusions
The effects of gravity forces on liquid water transport in the GDL and the performance of a PEMFC, are presented in this paper. For this purpose, we develop a transient onedimensional model, based on the resolution of the diffusion
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 8 ( 2 0 1 3 ) 3 3 3 0 e3 3 3 7
equations for oxygen and liquid water saturation. We found that: * Gravity forces can not be neglected when studying the performance of a PEMFC. * When gravity is considered, the time needed to reach the steady state is reduced. * If gravity forces oppose capillary actions in the porous GDL, liquid water accumulates at the CLeGDL interface and the performance (cell voltage) of the PEMFC is highly decreased. * If gravity forces assist capillary actions, accumulated liquid water is rapidly evacuated to the GFC and a smaller drop of the cell voltage at the steady state is detected.
references
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[7] Ramousse J, Deseure J, Lottin O, Didierjean S, Maillet D. Modelling of heat, mass and charge transfer in a PEMFC single cell. Journal of Power Sources 2005;145:416e27. [8] Morin A, Xu F, Gebel G, Diat O. Influence of PEMFC gas flow configuration on performance and water distribution studied by SANS: evidence of the effect of gravity. International Journal of Hydrogen Energy 2011;36:3096e109. [9] Chen S, Wu Y. Gravity effect on water discharged in PEM fuel cell cathode. International Journal of Hydrogen Energy 2010; 35:2888e93. [10] Kimball E, Whitaker T, Kevrekidis YG, Benziger JB. Drops, slugs, and flooding in polymer electrolyte membrane fuel cells. AIChE Journal 2008;54:1313e32. [11] Yi Y, Zheng-kai T, Zhi-gang Zhan, Mu Pan. Gravity effects on the performance of PEM fuel cell stack with different gas manifold positions. International Journal of Energy Researches 2012;36(7):845e55. [12] Najjari M, Khemili F, Ben Nasrallah S. The effects of the cathode flooding on transient responses of a PEM fuel cell. Renewable Energy International Journal 2008;33:1824e31. [13] Khemili F, Najjari M, Ben Nasrallah S. A comprehensive model for evaluating the performance of proton exchange membrane fuel cells. International Journal of Heat and Technology 2012;30(1):1e9. [14] Loo KH, Wong KH, Tan SC, Lai YM, Tse CK. Characterization of the dynamic response of proton exchange membrane fuel cells e a numerical study. International Journal of Hydrogen Energy 2010;35:11861e77. [15] Kumbur EC, Sharp KV, Mench MM. On the effectiveness of Leverett approach for describing the water transport in fuel cell diffusion media. Journal of Power Sources 2007;168: 356e68. [16] Song D, Wang Q, Liu Z, Huang Cheng. Transient analysis for the cathode gas diffusion layer of PEM fuel cells. Journal of Power Sources 2006;159:928e42. [17] Pasaogullari U, Wang CY. Liquid water transport in gas diffusion layer of polymer electrolyte fuel cells. Journal of the Electrochemical Society 2004;151(3):399e406. [18] Patankar V. Numerical heat transfer fluid flow. New York: Hemisphere/MacGraw-Hill; 1980. [19] Bajpai H, Khandelwal M, Kumbur EC, Mench MM. A computational model for assessing impact of interfacial morphology on polymer electrolyte fuel cell performance. Journal of Power Sources 2010;195:4196e205.