- Email: [email protected]
Numerical modeling and investigation of gas crossover effects in high temperature proton exchange membrane (PEM) fuel cells
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 ) 7 7 0 4 e7 7 1 4
Available online at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/he
Numerical modeling and investigation of gas crossover effects in high temperature proton exchange membrane (PEM) fuel cells Purushothama Chippar, Hyunchul Ju* School of Mechanical Engineering, Inha University, 253 Yonghyun-Dong, Nam-Gu, Incheon 402-751, Republic of Korea
article info
abstract
Article history:
A gas crossover model is developed for a high temperature proton exchange membrane
Received 14 April 2012
fuel cell (HT-PEMFC) with a phosphoric acid-doped polybenzimidazole membrane. The
Received in revised form
model considers dissolution of reactants into electrolyte phase in the catalyst layers and
24 July 2012
subsequent crossover of reactant gases through the membrane. Furthermore, the model
Accepted 29 July 2012
accounts for a mixed potential on the cathode side resulting from hydrogen crossover and
Available online 16 August 2012
hydrogen/oxygen catalytic combustion on the anode side due to oxygen crossover, which were overlooked in the HT-PEMFC modeling works in the literature. Numerical simulations
Keywords:
are carried out to investigate the effects of gas crossover on HT-PEMFC performance by
High temperature proton exchange
varying three critical parameters, i.e. operating current density, operating temperature and
membrane fuel cell (HT-PEMFC)
gas crossover diffusivity to approximate the membrane degradation. The numerical results
Polybenzimidazole (PBI)
indicate that the effect of gas crossover on HT-PEMFC performance is insignificant in
Numerical modeling
a fresh membrane. However, as the membrane is degraded and hence gas crossover
Hydrogen crossover
diffusivities are raised, the model predicts non-uniform reactant and current density
Oxygen crossover
distributions as well as lower cell performance. In addition, the thermal analysis demonstrates that the amount of heat generated due to hydrogen/oxygen catalytic combustion is not appreciable compared to total waste heat released during HT-PEMFC operations. Copyright ª 2012, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
1.
Introduction
The application of perfluorosulfonic acid (PFSA) membranes, such as DuPont’s Nafion membranes typically used in proton exchange membrane fuel cells (PEMFCs), is limited to low temperatures (<90 C) due to the stringent requirement for membrane hydration to ensure good proton conductivity. Therefore, PFSA membrane-based fuel cells suffers from several issues raised by low temperature operation, such as complicated water management requirements, high external humidification, and cooling loads. Furthermore, PFSA membrane fuel cells, particularly for residential applications,
face another barrier, namely, the low tolerance of the anode platinum (Pt) catalyst to carbon monoxide (CO) which is inevitably present in reformate fuel. Recently, the operation of PEMFCs at elevated temperatures (100 Ce200 C) has received much attention because of several benefits, such as faster electrode kinetics, improved mass transport, simple water management, and higher tolerance to CO. Therefore, a high-temperature proton exchange fuel cell (HT-PEMFC) is well suited for most distributed energy or combined heat and power (CHP) applications in which a hydrogen rich reformate gas is often used instead of pure hydrogen. The main focus of HT-PEMFC research resides on the
* Corresponding author. Tel.: þ82 32 860 7312; fax: þ82 32 868 1716. E-mail address: [email protected] (H. Ju). 0360-3199/$ e see front matter Copyright ª 2012, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijhydene.2012.07.123
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 ) 7 7 0 4 e7 7 1 4
development of alternative membranes which can exhibit high proton conductivity under low humidity conditions at the elevated temperatures. One of the most promising candidates is believed to be a phosphoric acid-doped polybenzimidazole (PBI) membrane. Since Wainright et al. [1] first proposed the use of phosphoric acid-doped PBI membranes for a HT-PEMFC, considerable progress has been made in PBI membrane development. Studies have reported good proton conductivity [1], excellent thermal stability [2], and nearly zero electroosmotic drag [3]. In addition to membrane development, several experimental efforts have been undertaken to investigate the physiochemical properties of PBI membranes [4e8]. Also, several theoretical HT-PEMFC models have been developed and introduced in the literature for understanding, prediction, and optimization of key physical phenomena in HT-PEMFCs [9e15]. Among these models, Cheddie and Munroe [10] and Sousa et al. [15] account for the effects of gas solubility into the phosphoric acid/PBI electrolyte. Cheddie and Munroe [10] presented a two-dimensional (2D), isothermal HT-PEMFC model wherein hydrogen and oxygen dissolution into phosphoric acid of catalyst layers (CLs) were taken into account. However, their model assumed CLs to comprise only liquid phase electrolyte (phosphoric acid) and solid-phase electron conducting regions, neglecting gas-phase reactant transport through CLs. As a result, their numerical predictions significantly overestimated mass transport loss in the CLs. Sousa et al. [15] treated CLs as spherical agglomerate porous structures and applied the CL model to a 2D isothermal HT-PEMFC model. Their numerical predictions indicate that an optimum phosphoric acid volume fraction in a CL is around 30%e55%. Most recently, Chippar and Ju [16] developed a three-dimensional non-isothermal HT-PEMFC model and investigated the impact of a coolant flow rate on multidimensional distributions of species, temperature, and current density as well as overall cell performance. However, hydrogen and oxygen crossover through PBI membrane was not considered in their model. In this study, gas crossover phenomena in HT-PEMFCs are newly modeled and implemented into the previous HT-PEMFC model [16]. Previously, Nam et al. [17] developed the gas crossover model for low temperature- (LT-) PEMFCs and numerically studied the influences of hydrogen and oxygen crossover on two-phase transport and water accumulation inside cells as well as overall cell performance. They also simulated decaying polarization curves due to membrane degradation using gas crossover evolution data measured during long-term LT-PEMFC operations. We adopted the gas crossover model of Nam et al. [17] and modified it for HT-PEMFCs. The gas crossover model rigorously accounts for hydrogen/oxygen dissolution into the aqueous electrolyte phase and subsequent diffusion through phosphoric acid-doped PBI membranes in HT-PEMFCs. Note that, although Cheddie and Munroe [10] and Sousa et al. [15] modeled the dissolution of the reactant gas into aqueous phosphoric acid, the effects of gas crossover through the membrane on the thermal-electrochemical behavior of cells and overall cell performance were not taken into consideration in their models. Due to hydrogen crossover from the anode side, a mixed potential occurs on the cathode side, whereas oxygen crossover results in hydrogen/oxygen catalytic combustion on the anode side, which possibly redistributes species and charge profiles
7705
inside HT-PEMFCs and consequently downgrades overall cell performance. The gas crossover model presented in this paper entails a detailed account of these gas crossover impacts.
2.
Numerical model
The proposed three-dimensional, two-phase, non-isothermal, electrochemical-transport coupled HT-PEMFC model is based on our previous HT-PEMFC model [16], and it is further improved by accounting for the effects of gas dissolution and subsequent crossover through the PBI membrane. The model considers all sub-components of an HT-PEMFC: membrane, catalyst layers (CLs), gas diffusion layer (GDLs), gas channels (GCs), and bipolar plates (BPs). The governing equations of the HT-PEMFC model, relevant source terms, and electrochemical properties at the anode and cathode CLs are summarized in Tables 1e3, respectively. Readers are referred to our previous publication [16] for a more detailed description of the model.
2.1.
Model assumptions
The specific assumptions invoked in the present model are: (1) Incompressible and laminar flow due to small pressure gradient and flow velocities. (2) Ideal gas mixture due to low pressure and high temperature HT-PEMFC operation. (3) Isotropic and homogeneous porous layers (GDLs, CLs) characterized by effective porosity and permeability.
2.2.
Transport properties
The diffusivity of species i, in the gas mixture is defined as [18] 1=2 1 xi 1:013 107 T1:75 1 1 ; where D ¼ þ Di;M ¼ P i;j 2 xj j¼n Mi Mj 1=3 1=3 p ci þ cj j Di;j jsi
(22)
cH2 ¼ 7:07; cH2 O ¼ 12:7; cN2 ¼ 17:9; cO2 ¼ 16:6:
Table 1 e HT-PEMFC model: governing equations. Governing equations Mass Momentum
V$ðr! u Þ ¼ Sm
(1)
Flow ðNavier Stokes equationsÞ : 2 channels u! u Þ ¼ Vp þ V$s (2) 1=3 V$ðr! Porous media ðDarcys equationsÞ : r! u ¼ ðK=nÞVp
(3)
Species
V$ð! u Ci Þ ¼ V$ Deff i VCi þ Si
(4)
Charge
Proton transport : V$ keff VFe þ SF ¼ 0
(5)
Electron transport : V$ seff VFs SF ¼ 0 (6) Energy
V$ rCp ! u T ¼ V$ keff VT þ ST
(7)
7706
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 ) 7 7 0 4 e7 7 1 4
Table 2 e HT-PEMFC model: source/sink terms. Source terms
Anode CL
Cathode CL
H2
nxover ja nxover H O SH2 ¼ 2 2 2 2F dCL dCL
(8)
O2
SO2 ¼
nxover jc 1 nxover O H2 2 4F dCL 2 dCL
(9)
H2O
SH2 O ¼ 2 Mass
Sm ¼
X k
nxover O2 dCL
(10a)
MH2 nxover MO2 nxover O2 H2 ja Sk ¼ MH2 þ 2F dCL
(11a)
nxover jc H SH2 O ¼ þ 2 2F dCL
Sm ¼
X k
(10b)
MO2 nxover MH2 nxover H2 O2 jc jc Sk ¼ MO2 MH2 O þ 4F 2F dCL
(11b)
Charge
S F ¼ ja
(12a)
SF ¼ jc
(13a)
ST ¼ jc h þ
(12b)
Heat
S T ¼ ja h þ
2nxover I2 O2 þ DHH2 keff dCL
Note that, in addition to molecular diffusion as defined in the Eq. (22), species diffusion transport can also be controlled by the Knudsen diffusion effect due to molecular-to-wall collision. According to the kinetic theory, the Knudsen diffusivity can be expressed as sffiffiffiffiffiffiffiffiffi dp 8RT : (23) Di;K ¼ 3 pMi
Ixover I2 dUo dUo H2 h þ T T þ j c keff dT dCL dT
(13b)
Therefore, the effective diffusivity of species in porous media is obtained by combining both the molecular and Knudsen diffusion effects with the effects of porosity and tortuosity of the porous medium using the Bruggemann correlation [19]: Di ¼ 3 n
1 1 þ Di;M Di;K
1 :
(24)
Table 3 e HT-PEMFC model: electrochemical properties. Description
Anode CL
Exchange current density ratio of the reaction surface to the 3 CL volume, airef 0 (A/m ) Reference H2/O2 molar concentration, (mol/m3) Transfer coefficients, a Thermodynamic equilibrium potential, U0 (V)
Cathode CL
1.0 109
1.0 104
40.88
40.88
aa ¼ 0:5 0
ac ¼ 0:5
1:1669 0:24 103 ðT 373:15Þ
(14)
(15)
fs fe U0 ðwith fs ¼ Vcell Þ
(16)
(17)
jc ¼ airef 0;c
Surface overpotential,h (V)
fs fe U0 ðwith fs ¼ 0Þ Transfer current density, j (A/m3)
ja ¼ airef 0;a
Electrochemical reactions :
X k
si Mzi
CH2 CH2;ref
!1=2
aa þ ac Fh Ru T
8 < Mi ¼ chemical formula of species i ¼ ne ; where si ¼ stoichiometry coefficient : n ¼ number of electrons transferred
CO2 CO2;ref
!3=4
Ixover ac H Fh þ 2 exp Ru T dCL
(18)
(19)
Hydrogen oxidation reactionðHORÞat the anode side : H2 2Hþ ¼ 2e
(20)
Oxygen reduction reactionðORRÞat the cathode side : 2H2 O O2 4Hþ ¼ 4e
(21)
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 ) 7 7 0 4 e7 7 1 4
7707
The dissolution of species in the aqueous electrolyte phase and their subsequent diffusion are described below. According to Cheddie and Munroe [10], the solubility and diffusivity of oxygen in a concentrated phosphoric acid can be expressed in terms of the weight percentage of phosphoric acid, mPA , and temperature as
The proton conductivity of the phosphoric acid-doped PBI membrane is correlated to the doping level and temperature as follows [10]:
3 257:13ðm Þ 431:08ðm Þ þ 178:45 7 6 6 7 7; 6 HPA O2 ¼ 0:1exp6 PA 2 PA 93500ðm Þ þ156646ðm Þ 64288 7 5 4 þ T
The effective proton conductivity in the CLs is obtained by combining the effects of the volume fraction of the membrane phase and tortuosity of the porous medium by using Bruggemann’s correlation:
2
PA 2
PA
(25)
2.3. (26)
Henry’s constant of oxygen in the phosphoric acid-doped PBI membrane can be obtained as a function of the volume fraction of phosphoric acid in the membrane, 3 PA as [10] HPBI O2 ¼
3
PA 1:945
h
PA 1:8 i HPA : O2 þ 5:79 1 3
(27)
The 3 PA in turn depends on the doping level of phosphoric acid in the membrane, X as [10] 3
PA
¼
4:81 þ1 X2
1 :
(28)
The X in the above equation can be computed based on the phosphoric acid concentration, M as [10] X ¼ 0:012M3 0:2111M2 þ 1:2363M þ 0:7119:
(29)
On the other hand, the oxygen diffusion coefficient in the phosphoric acid-doped PBI membrane can be obtained by Bruggemann’s relation as [10] DPBI O2 ¼
3
PA 1:8
DPA O2 :
(30)
The dissolved concentration of oxygen at the gas/electrolyte interface in the cathode CL is determined from the partial pressure of oxygen pO2 using Henry’s law as follows: g PBI PBI CPBI O2 ¼ HO2 pO2 ¼ HO2 CO2 ;mem RT :
100 2605:6 70:1X exp 8:0219 : T T
keff ¼ 3 1:5 mc k:
2
3 PA 2 PA 6 192:55ðm Þ þ323:55ðm Þ 125:61 7 6 9 7 6 7: DPA O2 ¼ 10 exp6 PA 2 PA 7 4 62010ðm Þ 105503ðm Þ þ 40929 5 þ T
k¼
(31)
Due to lack of studies regarding hydrogen dissolution into concentrated phosphoric acid, the diffusivity and solubility of hydrogen are assumed to be two times and four times larger than those of oxygen, respectively, i.e. based on the transport behavior of hydrogen and oxygen in water systems [10]: PA DPA H2 ¼ 2DO2 ;
(32)
PA HPA H2 ¼ 4HO2 :
(33)
(34)
(35)
Gas crossover model and relevant source terms
The gas crossover model accounts for the influences of hydrogen crossover (from the anode to cathode) and oxygen crossover (from the cathode to anode) on electrochemical processes and the resultant overall cell performance. The gas crossover model has been described in detail in a previous study [17] and hence only a brief summary is repeated here. The hydrogen crossover through the membrane causes a mixed potential at the cathode CL due to facile hydrogen oxidation kinetic and large surface overpotential at the cathode. Therefore, the final form of the ORR kinetic expression can be determined as Eq. (18) in Table 3 where the second term in the right-hand side of Eq. (18) represents the effect of the hydrogen crossover. Under the assumption that crossed hydrogen is uniformly and completely oxidized in the cathode CL, the hydrogen crossover current density, can be defined as below:
¼ 2Fnxover ¼ 2FDPBI Ixover H2 H2 H2
CPBI H2
aCL
dmem
:
(36)
On the other hand, the oxygen crossover through the membrane leads to catalytic hydrogen/oxygen combustion in the anode CL due to small potential difference between the solid and electrolyte phases at the anode. The influences of the hydrogen and oxygen crossover on mass, species (hydrogen, oxygen, and water), and energy balance are seen in their source/sink terms in Table 2.
2.4. Numerical implementation, computational domain, and boundary conditions The HT-PEMFC model described in Section 2 is numerically implemented in a commercial computational fluid dynamics (CFD) program, FLUENT, basing on its user defined functions (UDF). The convergence criteria for all species and energy calculation residuals are set to 108. Fig. 1 shows the mesh configuration of the simple single-straight channel geometry. The physical properties and, cell dimensions and operating conditions are given in Tables 4 and 5, respectively. The isothermal boundary condition is applied to the anode and cathode wall of the computational cell for temperature calculations. In addition, the no-flux condition is applied to the outer faces for flow and species transport equations except for the channel inlets and outlets. The inlet velocities in the anode and cathode GCs can be expressed as functions of
7708
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 ) 7 7 0 4 e7 7 1 4
Table 5 e Cell dimensions and operating conditions. Description Cell length Anode/cathode channel/rib width Anode/cathode channel height Thickness of the anode/cathode GDLs Thickness of the anode/cathode CLs Thickness of the membrane Anode/cathode inlet pressure Anode stoichiometry Cathode stoichiometry Anode/cathode inlet temperature RH of the anode/cathode inlet Phosphoric acid doping level
Fig. 1 e Mesh configuration of three-dimensional, singlechannel HT-PEMFC geometry.
the anode and cathode stoichiometric ratios, the operating current density, the cross-sectional areas of the anode and cathode GCs, and the concentrations of hydrogen and oxygen, which are functions of the anode/cathode inlet pressure and temperature: uin;a ¼
3.
xa ðI=2FÞAmem CH2 Aa;chan
and uin;c ¼
xc ðI=4FÞAmem : CO2 Ac;chan
(37)
Results and discussion
To examine the effects of gas crossover on HT-PEMFCs, we assessed the effects of three critical parameters, namely, operating current density, operating temperature, and gas crossover diffusivity itself. Regarding the effect of operating current density, it is evident that the impact of gas crossover is
Table 4 e Physical properties. Description Porosity of GDL, CL Volume fraction of ionomers in CL Permeability of GDL, CL Electronic conductivity in the GDL, CL, BP Specific heat capacities of GDL, CL, membrane, BP Specific heat capacities of species e H2, O2, N2, H2O Thermal conductivities of GDL, CL, membrane, BP Thermal conductivities of species e H2, O2, N2, H2O
Value 0.6, 0.3 0.4 1 1012, 1.0 1013 m2 1250, 300, 14000 S m1 568, 3300, 1650, 2930 J kg1 K1 14430, 929, 1042, 1968 J kg1 K1 1.2, 1.5, 0.95, 20 W m1 K1 0.2040, 0.0296, 0.0293, 0.02378 W m1 K1
Value 0.1 m 1 103 m 0.7 103 m 250 106 m 10 106 m 70 106 m 1.0 atm 2.0 (Pure H2) 2.0 (Air) 373 K, 453 K 0.0% 6.2
more significant at lower current density operation due to the lower reactant consumption rate and resultant higher concentration in the CL that leads to a higher dissolution rate of the reactant gas into the aqueous electrolyte phase. The hydrogen and oxygen crossover diffusivities given by Cheddie and Munroe [10] imply that the degree of gas crossover through a phosphoric acid-doped PBI membrane is considerably altered by operating temperature. Therefore, the operating temperature is another critical factor to control the gas crossover behavior inside an HT-PEMFC. Finally, several degradation mechanisms of the PBI membrane have been reported in the literature, such as chemical degradation [20], thermal degradation [21] and phosphoric acid evaporation [22]. In particular, it should be noted that the phosphoric acid loss due to evaporation not only decreases the membrane conductivity but also increases the gas crossover diffusivities for hydrogen and oxygen [10]. Although all of these membrane degradation mechanisms appear to be highly localized, due to the lack of experimental data on degradations, we assume the membrane to be degraded uniformly; thus, the gas crossover diffusivity is also uniformly raised as a function of the degree of membrane degradation. The parametric study was carried out at two operating temperatures (100 C and 180 C) under various operating current densities and hydrogen/oxygen crossover diffusivities. Four different cases of gas crossover diffusivity are defined here. Case 1 assumes that the membrane is perfectly impermeable to hydrogen and oxygen; hence, their crossover diffusivities are set to zero. Case 2 represents the case of a fresh phosphoric acid-doped PBI membrane, and the crossover processes for hydrogen and oxygen are approximated using the crossover diffusivities given by Cheddie and Munroe [10]. To consider a degraded membrane, the hydrogen/oxygen crossover diffusivities were further raised by one and two orders of magnitude for cases 3 and 4, respectively. Fig. 2 shows the hydrogen and oxygen concentration profiles in the anode and cathode CLs for cases 1 to 4 at the operating current density of 0.2 A cm2 based on the operating temperatures of 100 C and 180 C. As shown in Fig. 2(a) and (b), the hydrogen concentration under the land region is lower than that under the channel region due to the longer transport path from the anode flow channel. In a comparison of cases 1 to 4 for each operation temperature, the hydrogen distributions for cases 1, 2, and 3 are almost identical, indicating that
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 ) 7 7 0 4 e7 7 1 4
7709
Fig. 2 e Hydrogen concentration contours (in mol mL3) in the anode CL and oxygen concentration contours (in mol mL3) in the cathode CLs at an operating current density of 0.2 A cmL2: (a) Hydrogen; T [ 100 C, (b) Hydrogen; T [ 180 C, (c) Oxygen; T [ 100 C and (d) Oxygen; T [ 180 C.
the effect of crossover though the fresh membrane is negligible (case 2), and even a ten-fold increase in the hydrogenpermeation coefficient due to membrane degradation is still acceptable for HT-PEMFC operations (case 3). However, more severe hydrogen depletion is observed in case 4, which can be attributed to a combined result of the higher degree of hydrogen crossover from the anode through the membrane and the higher rate of catalytic hydrogen/oxygen combustion at the anode CL driven by stronger oxygen crossover from the cathode in case 4. In addition, higher hydrogen depletion was predicted at the higher operating temperature because the amount of hydrogen crossover increases with temperature. The same trend is observed in the oxygen concentration contours in the cathode CL in Fig. 2(c) and (d) where oxygen depletion near the cathode outlet region is more severe with a higher degree of membrane degradation (case 4) and/or a higher operating temperature (180 C). Note that the severe oxygen depletion in case 4 is attributed to both its higher oxygen crossover rate from the cathode to anode as well as the higher hydrogen crossover from the anode to cathode that leads to additional ORR and mixed potential at the cathode CL.
Fig. 3(a) shows the local current density distributions in the membrane for cases 1 to 4 at the operating current density of 0.2 A cm2 and the operating temperature of 100 C. In all cases, the local current density near the land region is lower than near the channel region along the in-plane direction (Z ). Along the cathode flow direction (Y ), the local current density continuously decreases toward the cathode outlet. These trends indicate that oxygen depletion is the sole factor in determining the current density distribution for all the cases. In a comparison of cases 1 to 4, the current density distributions for cases 1, 2, and 3 are almost identical, indicating that the degree of gas crossover up to a ten-fold increase in the gaspermeation coefficient due to membrane degradation has a negligible influence on HT-PEMFC performance. However, spatial non-uniformity in the current density profile is clearly increased in case 4 due to the higher degree of gas crossover through the membrane. Fig. 3(b) displays the local current density contours at 180 C. As compared with Fig. 3(a), the local current densities near the cathode outlet are reduced due to the higher degree of hydrogen and oxygen crossover at the elevated temperature.
7710
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 ) 7 7 0 4 e7 7 1 4
Fig. 3 e Local current density distribution (in A mL2) in the membrane at an operating current density of 0.2 A cmL2: (a) T [ 100 C and (b) T [ 180 C.
Fig. 4 e Crossover current density distribution (in A mL2) in the membrane at an operating current density of 0.2 A cmL2: (a) T [ 100 C and (b) T [ 180 C.
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 ) 7 7 0 4 e7 7 1 4
7711
Fig. 5 e Cathode overpotential distribution (in volts) in the CL at an operating current density of 0.2 A cmL2: (a) T [ 100 C and (b) T [ 180 C.
To analyze the degree of hydrogen crossover, the hydrogen crossover current density was calculated by Eq. (36) and plotted in Fig. 4 for the same simulation cases (at 0.2 A cm2 under two operating temperatures of 100 C and 180 C). For all cases, the hydrogen crossover current density decreases toward the anode downstream because the hydrogen concentration in the anode CL is high near the anode inlet and continuously depleted along the flow direction by not only HOR but also hydrogen permeation through the membrane and the hydrogen/oxygen catalytic combustion due to the oxygen crossover. In a comparison of cases 2 to 4, the hydrogen crossover current densities of case 4 are roughly two orders of magnitude larger than those of case 2, which clearly indicates that the amount of hydrogen crossover flux is directly proportional to the considered hydrogen crossover diffusivity. In addition, much higher hydrogen crossover current density was predicted with the higher operating temperature (180 C) since the hydrogen crossover diffusivity is a function of increasing temperature. Fig. 5 shows the cathode overpotential distributions in the cathode CL for cases 1 to 4 at the operating temperatures of 100 C and 180 C. The cathode overpotential increases toward the land region and cathode downstream due to lower local oxygen concentrations there as seen in Fig. 2(c) and (d). More importantly, a comparison of the 100 C and 180 C cases indicates that operating the cell at the higher temperature significantly reduces the cathode overpotential, although the available oxygen concentration for ORR in the cathode CL is lower at 180 C (see Fig. 2(d)). This is mainly due to the enhanced electrochemical kinetics of ORR at the elevated operating temperature. Fig. 6 shows the effects of hydrogen and oxygen crossover through the membrane on cell polarization curves at two different operating temperatures (100 C and 180 C). First, superior cell performance is achieved at the higher operating temperature due to improved ORR kinetics, better proton conductivity, and more efficient mass transport with increasing temperature. Further, the polarization curves clearly demonstrate that the impact of gas crossover is more
significant at lower current densities, because the hydrogen and oxygen concentrations remaining in the CLs are higher under lower current density operations. In addition, the polarization curves for cases 1 to 3 are similar to each other at both temperatures (100 C and 180 C), which means that a ten-fold increase in the gas-permeation coefficient (case 3) is acceptable for HT-PEMFC operations under wide ranges of operating current density and temperature. However, a more pronounced effect of gas crossover is seen in case 4, particularly at the higher operating temperature (180 C). These results imply that careful attention to suppress gas crossover is required for low current density and/or high temperature operations. Tables 6 and 7 summarize the overall heat balance and the individual heat sources for cases 1 to 4 under the operating current density of 0.2 A cm2 at 100 C and 180 C, respectively. The simulation results show that the largest part of the total waste heat release is due to irreversible ORR reaction at the cathode, that is, roughly 70% of the total heat generation. In
Fig. 6 e Overall polarization curves for cases 1e4 at the operating temperatures of 100 C and 180 C.
7712
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 ) 7 7 0 4 e7 7 1 4
Table 6 e Summary of energy balance results under operating current density of 0.2 A cmL2 at 100 C. Case 1 (1) Anode CL
Irreversible reaction heat, [W] ST;irrev;a ¼
R
j:hdV
Case 2
Case 3
Case 4
0.00344 (0.985%) 0.00345 (0.985%)
0.00345 (0.977%)
0.00348 (0.903%)
0.00399 (1.141%) 0.00399 (1.139%)
0.00399 (1.129%)
0.00400 (1.038%)
V
Ohmic joule heating, [W] ST;joule;a ¼
R I dV keff 2
V
H2/O2 catalytic combustion heat [W] Sxover ¼ DH Sxover 0.0 T;a O2 R I2 Ohmic joule heating, [W] ST;joule;mem ¼ dV eff V k R (3) Cathode CL Irreversible reaction heat, [W] ST;irrev;c ¼ j:hdV (2) Membrane
0.0000376 (0.011%) 0.000374 (0.106%) 0.00352 (0.911%)
0.02678 (7.659%) 0.02675 (7.643%)
0.02675 (7.572%)
0.02675 (6.935%)
0.26724 (76.43%) 0.26740 (76.41%)
0.26883 (76.10%)
0.28304 (73.37%)
0.01232 (3.524%) 0.01231 (3.516%)
0.01232 (3.487%)
0.01245 (3.227%)
Mixed potential and entropic heat due to Ixover dU0 H hydrogen crossover, [W] Sxover ¼ 2 hþT T;c dCL dT R vU0 dV Entropy heat, [W] ST;rev;c ¼ j$ T vT V
0.0
0.00152 (0.428%)
0.01489 (3.859%)
0.03588 (10.26%) 0.03589 (10.26%)
0.03605 (10.21%)
0.03763 (9.755%)
(1)þ(2)þ(3), [W]
0.3497
0.3533
0.3858
V
Ohmic joule heating, [W] ST;joule;c ¼
R I dV keff 2
V
Sum
0.000152 (0.043%)
0.3499
Table 7 e Summary of energy balance results under operating current density of 0.2 A.cmL2 at 180 C.
(1) Anode CL
R
Irreversible reaction heat, [W] ST;irrev;a ¼ j:hdV R I2 V Ohmic joule heating, [W] ST;joule;a ¼ dV eff V k
Case 1
Case 2
0.00235 (0.846%) 0.00202 (0.727%)
0.00236 (0.844%) 0.00202 (0.722%)
H2/O2 catalytic combustion heat [W] Sxover ¼ DH Sxover 0.0 T;a O2 R I2 Ohmic joule heating, [W] ST;joule;mem ¼ dV eff V k R (3) Cathode CL Irreversible reaction heat, [W] ST;irrev;c ¼ j:hdV (2) Membrane
Case 3
Case 4
0.00238 (0.801%) 0.00253 (0.568%) 0.00202 (0.683%) 0.00208 (0.465%)
0.00023 (0.084%)
0.00226 (0.763%) 0.01562 (3.493%)
0.01165 (4.193%)
0.01164 (4.160%)
0.01165 (3.929%) 0.01178 (2.639%)
0.21214 (76.33%)
0.21292 (76.10%)
0.21996 (74.18%) 0.28598 (64.07%)
0.00623 (2.241%)
0.00623 (2.225%)
0.00629 (2.122%) 0.00725 (1.625%)
V
Ohmic joule heating, [W] ST;joule;c ¼
R I dV keff 2
V
Mixed potential and entropic heat due 0.0 0.00072 (0.258%) 0.00717 (2.420%) 0.06686 (14.98%) Ixover dU0 H2 xover to hydrogen crossover, [W] ST;c ¼ hþT dCL dT R 0.04354 (15.669%) 0.04367 (15.607%) 0.04476 (15.10%) 0.05426 (12.156%) vU0 Entropy heat, [W] ST;rev;c ¼ j$ T dV vT V Sum
(1)þ(2)þ(3), [W]
addition, it should be noted that the heat generated by hydrogen/oxygen catalytic combustion on the anode side is about 3.5% in the most severe gas crossover case in this study (case 4 at 180 C and 0.2 A cm2). Therefore, the contribution of the hydrogen/oxygen chemical reaction does not seem to be significant to the total heat release during HT-PEMFC operations, even when two orders of magnitude greater gas crossover diffusivities are considered.
4.
Conclusions
In this study, a gas crossover model that considers the dissolution of hydrogen/oxygen into the electrolyte phase and subsequent diffusion through a phosphoric acid-doped PBI membrane was developed and incorporated into a HT-PEMFC model developed in an earlier study [16]. The main interest of this study is to numerically assess the impact of gas crossover on HT-PEMFC performance. The gas crossover model rigorously accounts for the major outcomes of hydrogen and
0.2779
0.2798
0.2964
0.4463
oxygen crossover, i.e. a mixed potential at the cathode CL and the hydrogen/oxygen catalytic combustion at the anode CL. The numerical results show that the gas crossover has a negligible influence on overall cell performance in a fresh PBI membrane (case 2) and a moderately degraded membrane (case 3), which was assumed to have one order of magnitude higher crossover diffusivities than those of the fresh membrane. However, the effect of gas crossover begins to appear in more a severely degraded membrane with two orders of magnitude larger gas crossover diffusivities (case 4). A comparison of case 4 with cases 1 to 3 clearly shows that the increased effect of gas crossover increases the degree of nonuniformity in the hydrogen, oxygen, and current density distributions. In addition, the simulation results for case 4 indicate that gas crossover is more detrimental to cell operation at a higher operating temperature and/or lower current density due to more facile crossover of hydrogen and oxygen with elevated temperature and due to higher reactant concentration in the CL with lower current density. Finally, the thermal analysis carried out in this study demonstrated
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 ) 7 7 0 4 e7 7 1 4
that the heat generated via hydrogen/oxygen catalytic combustion at the anode CL is not significant, occupying only 3.5% of the total waste heat release in the worst gas crossover case in this study (case 4 at 180 C and 0.2 A cm2). This paper contributes to enhancing the fundamental understanding of the gas crossover phenomena occurring during HT-PEMFC operation. As an extension of this work, our efforts are underway to numerically study the effects of local heterogeneous gas crossover due to membrane pinhole formations.
Acknowledgment This work was supported by the New & Renewable Energy R&D program (grant no. 2010T100200501) of the Ministry of Knowledge Economy of the government of the Republic of Korea. The authors gratefully acknowledge this support.
Nomenclature
A c C Di F H i0 I j K M p Q R S T ! u U0 Vcell X
area, m2 specific heat, J kg1 K1 molar concentration, mol m3 mass diffusivity of species i, m2 s1 Faraday constant, 96487 C mol1 Henry’s constant, mol m3 atm1 exchange current density, A m2 operating current density, A m2 transfer current density, A m3 hydraulic permeability, m2 molecular weight, kg mol1 partial pressure, Pa heat, watt universal gas constant, 8.314 J mol1 K1 source term in the conservation equation temperature, K fluid velocity and superficial velocity in a porous medium, m s1 thermodynamic equilibrium potential, V cell potential, V doping level
Greek symbols a transfer coefficient 3 porosity 3 mc volume fraction of the ionomer phase in the CL f phase potential, V h overpotential, V m viscosity, kg m1 s1 r density, kg m3 s viscous shear stress, N m2 k ionic conductivity, S m1 x stoichiometry flow ratio
Superscripts e electrolyte eff effective value in the porous region
g ref
7713
gas reference value
Subscripts a anode c cathode CL catalyst layer GC gas channel GDL gas diffusion layer hydrogen H2 i species index in channel inlet m mass equation mem membrane oxygen O2 u momentum equation w water F potential equation 0 standard condition, viz., 298.15 K and 101.3 kPa (1 atm)
references
[1] Wainright JS, Wang JT, Weng D, Savinell RF, Litt MH. Aciddoped polybenzimidazoles: a new polymer electrolyte. J Electrochem Soc 1995;142(7):L121e3. [2] Samms SR, Wasmus S, Savinell RF. Thermal stability of proton conducting acid doped polybenzimidazole in simulated fuel cell environments. J Electrochem Soc 1996; 143(4):1225e32. [3] Weng D, Wainright JS, Landau U, Savinell RF. Electro-osmotic drag coefficient of water and methanol in polymer electrolytes at elevated temperatures. J Electrochem Soc 1996;143(4):1260e3. [4] Qingfeng Li, Hjuler HA, Bjerrum NJ. Oxygen reduction on carbon supported platinum catalysts in high temperature polymer electrolytes. Electrochim Acta 2000;45(25e26): 4219e26. [5] Qingfeng Li, Hjuler HA, Bjerrum NJ. Phosphoric acid doped polybenzimidazole membranes: physiological characterization and fuel cell application. J Appl Electrochem 2001;31(7):773e9. [6] Ma YL, Wainright JS, Litt MH, Savinell RF. Conductivity of PBI membranes for high-temperature polymer electrolyte fuel cells. J Electrochem Soc 2004;151(1):A8e16. [7] He RH, Qingfeng Li, Bach A, Jensen JO, Bjerrum NJ. Physiochemical properties of phosphoric acid doped polybenzimidazole membranes for fuel cells. J Membr Sci 2006;277:38e45. [8] Liu Z, Wainright JS, Litt MH, Savinell RF. Study of the oxygen reduction reaction (ORR) at Pt interfaced with phosphoric acid doped polybenzimidazole at elevated temperature and low relative humidity. Electrochim Acta 2006;51:3914e23. [9] Cheddie D, Munroe N. Mathematical model of a PEMFC using a PBI membrane. Energy Convers Manage 2006;47:1490e504. [10] Cheddie D, Munroe N. A two phase model of an intermediate temperature PEM fuel cell. Intl J Hydrogen Energy 2007;32: 832e41. [11] Cheddie D, Munroe N. Three dimensional modeling of high temperature PEM fuel cells. J Power Sources 2006;160:215e23.
7714
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 ) 7 7 0 4 e7 7 1 4
[12] Scott K, Pilditch S, Mamlouk M. Modeling and experimental validation of a high temperature polymer electrolyte fuel cell. J Appl Electrochem 2007;37:1245e59. [13] Jiao K, Li X. A three-dimensional non-isothermal model of high temperature proton exchange membrane fuel cells with phosphoric acid doped polybenzimidazole membranes. Fuel Cells 2010;10(3):351e62. [14] Siegel C, Bandlamudi G, Heinzel A. Systematic characterization of a PBI/H3PO4 sol-gel membranemodeling and simulation. J Power Sources 2011;196: 2735e49. [15] Sousa T, Mamlouk M, Scott K. An isothermal model of a laboratory intermediate temperature fuel cell using PBI doped phosphoric acid membranes. Chem Eng Sci 2010;65: 2513e30. [16] Chippar P, Ju H. Three-dimensional non-isothermal modeling of a phosphoric acid-doped polybenzimidazole (PBI) membrane fuel cell. Solid State Ionics, in press.
[17] Nam J, Chippar P, Kim W, Ju H. Numerical analysis of gascrossover effects in polymer electrolyte fuel cells (PEFCs). Appl Energy 2010;87:3699e709. [18] Bird RB, Stewart WE, Lightfoot EN. Transport phenomena. New York: John Wiley & Sons; 1960. [19] Meredith RE, Tobias CW. In: Tobia CW, editor. Advances in electrochemistry and electrochemical engineering, vol. 2. New York: Interscience Publishers; 1962. [20] Liao JH, Li QF, Rudbeck HC, Jensen JO, Chromik A, Bjerrum NJ, et al. Oxidative degradation of polybenzimidazole membranes as electrolytes for high temperature proton exchange membrane fuel cells. Fuel Cells 2011;10(3):351e62. [21] Zhang J, Song C, Zhang J. Accelerated lifetime testing for proton exchange membrane fuel cells using extremely high temperature and unusually high load. J Fuel Cells Sci Tech 2011;8(051006):1e5. [22] Yu S, Xiao L, Benicewicz BC. Durability studies of PBI-based high temperature PEMFCs. Fuel Cells 2008;8:165e74.
Available online at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/he
Numerical modeling and investigation of gas crossover effects in high temperature proton exchange membrane (PEM) fuel cells Purushothama Chippar, Hyunchul Ju* School of Mechanical Engineering, Inha University, 253 Yonghyun-Dong, Nam-Gu, Incheon 402-751, Republic of Korea
article info
abstract
Article history:
A gas crossover model is developed for a high temperature proton exchange membrane
Received 14 April 2012
fuel cell (HT-PEMFC) with a phosphoric acid-doped polybenzimidazole membrane. The
Received in revised form
model considers dissolution of reactants into electrolyte phase in the catalyst layers and
24 July 2012
subsequent crossover of reactant gases through the membrane. Furthermore, the model
Accepted 29 July 2012
accounts for a mixed potential on the cathode side resulting from hydrogen crossover and
Available online 16 August 2012
hydrogen/oxygen catalytic combustion on the anode side due to oxygen crossover, which were overlooked in the HT-PEMFC modeling works in the literature. Numerical simulations
Keywords:
are carried out to investigate the effects of gas crossover on HT-PEMFC performance by
High temperature proton exchange
varying three critical parameters, i.e. operating current density, operating temperature and
membrane fuel cell (HT-PEMFC)
gas crossover diffusivity to approximate the membrane degradation. The numerical results
Polybenzimidazole (PBI)
indicate that the effect of gas crossover on HT-PEMFC performance is insignificant in
Numerical modeling
a fresh membrane. However, as the membrane is degraded and hence gas crossover
Hydrogen crossover
diffusivities are raised, the model predicts non-uniform reactant and current density
Oxygen crossover
distributions as well as lower cell performance. In addition, the thermal analysis demonstrates that the amount of heat generated due to hydrogen/oxygen catalytic combustion is not appreciable compared to total waste heat released during HT-PEMFC operations. Copyright ª 2012, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
1.
Introduction
The application of perfluorosulfonic acid (PFSA) membranes, such as DuPont’s Nafion membranes typically used in proton exchange membrane fuel cells (PEMFCs), is limited to low temperatures (<90 C) due to the stringent requirement for membrane hydration to ensure good proton conductivity. Therefore, PFSA membrane-based fuel cells suffers from several issues raised by low temperature operation, such as complicated water management requirements, high external humidification, and cooling loads. Furthermore, PFSA membrane fuel cells, particularly for residential applications,
face another barrier, namely, the low tolerance of the anode platinum (Pt) catalyst to carbon monoxide (CO) which is inevitably present in reformate fuel. Recently, the operation of PEMFCs at elevated temperatures (100 Ce200 C) has received much attention because of several benefits, such as faster electrode kinetics, improved mass transport, simple water management, and higher tolerance to CO. Therefore, a high-temperature proton exchange fuel cell (HT-PEMFC) is well suited for most distributed energy or combined heat and power (CHP) applications in which a hydrogen rich reformate gas is often used instead of pure hydrogen. The main focus of HT-PEMFC research resides on the
* Corresponding author. Tel.: þ82 32 860 7312; fax: þ82 32 868 1716. E-mail address: [email protected] (H. Ju). 0360-3199/$ e see front matter Copyright ª 2012, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijhydene.2012.07.123
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 ) 7 7 0 4 e7 7 1 4
development of alternative membranes which can exhibit high proton conductivity under low humidity conditions at the elevated temperatures. One of the most promising candidates is believed to be a phosphoric acid-doped polybenzimidazole (PBI) membrane. Since Wainright et al. [1] first proposed the use of phosphoric acid-doped PBI membranes for a HT-PEMFC, considerable progress has been made in PBI membrane development. Studies have reported good proton conductivity [1], excellent thermal stability [2], and nearly zero electroosmotic drag [3]. In addition to membrane development, several experimental efforts have been undertaken to investigate the physiochemical properties of PBI membranes [4e8]. Also, several theoretical HT-PEMFC models have been developed and introduced in the literature for understanding, prediction, and optimization of key physical phenomena in HT-PEMFCs [9e15]. Among these models, Cheddie and Munroe [10] and Sousa et al. [15] account for the effects of gas solubility into the phosphoric acid/PBI electrolyte. Cheddie and Munroe [10] presented a two-dimensional (2D), isothermal HT-PEMFC model wherein hydrogen and oxygen dissolution into phosphoric acid of catalyst layers (CLs) were taken into account. However, their model assumed CLs to comprise only liquid phase electrolyte (phosphoric acid) and solid-phase electron conducting regions, neglecting gas-phase reactant transport through CLs. As a result, their numerical predictions significantly overestimated mass transport loss in the CLs. Sousa et al. [15] treated CLs as spherical agglomerate porous structures and applied the CL model to a 2D isothermal HT-PEMFC model. Their numerical predictions indicate that an optimum phosphoric acid volume fraction in a CL is around 30%e55%. Most recently, Chippar and Ju [16] developed a three-dimensional non-isothermal HT-PEMFC model and investigated the impact of a coolant flow rate on multidimensional distributions of species, temperature, and current density as well as overall cell performance. However, hydrogen and oxygen crossover through PBI membrane was not considered in their model. In this study, gas crossover phenomena in HT-PEMFCs are newly modeled and implemented into the previous HT-PEMFC model [16]. Previously, Nam et al. [17] developed the gas crossover model for low temperature- (LT-) PEMFCs and numerically studied the influences of hydrogen and oxygen crossover on two-phase transport and water accumulation inside cells as well as overall cell performance. They also simulated decaying polarization curves due to membrane degradation using gas crossover evolution data measured during long-term LT-PEMFC operations. We adopted the gas crossover model of Nam et al. [17] and modified it for HT-PEMFCs. The gas crossover model rigorously accounts for hydrogen/oxygen dissolution into the aqueous electrolyte phase and subsequent diffusion through phosphoric acid-doped PBI membranes in HT-PEMFCs. Note that, although Cheddie and Munroe [10] and Sousa et al. [15] modeled the dissolution of the reactant gas into aqueous phosphoric acid, the effects of gas crossover through the membrane on the thermal-electrochemical behavior of cells and overall cell performance were not taken into consideration in their models. Due to hydrogen crossover from the anode side, a mixed potential occurs on the cathode side, whereas oxygen crossover results in hydrogen/oxygen catalytic combustion on the anode side, which possibly redistributes species and charge profiles
7705
inside HT-PEMFCs and consequently downgrades overall cell performance. The gas crossover model presented in this paper entails a detailed account of these gas crossover impacts.
2.
Numerical model
The proposed three-dimensional, two-phase, non-isothermal, electrochemical-transport coupled HT-PEMFC model is based on our previous HT-PEMFC model [16], and it is further improved by accounting for the effects of gas dissolution and subsequent crossover through the PBI membrane. The model considers all sub-components of an HT-PEMFC: membrane, catalyst layers (CLs), gas diffusion layer (GDLs), gas channels (GCs), and bipolar plates (BPs). The governing equations of the HT-PEMFC model, relevant source terms, and electrochemical properties at the anode and cathode CLs are summarized in Tables 1e3, respectively. Readers are referred to our previous publication [16] for a more detailed description of the model.
2.1.
Model assumptions
The specific assumptions invoked in the present model are: (1) Incompressible and laminar flow due to small pressure gradient and flow velocities. (2) Ideal gas mixture due to low pressure and high temperature HT-PEMFC operation. (3) Isotropic and homogeneous porous layers (GDLs, CLs) characterized by effective porosity and permeability.
2.2.
Transport properties
The diffusivity of species i, in the gas mixture is defined as [18] 1=2 1 xi 1:013 107 T1:75 1 1 ; where D ¼ þ Di;M ¼ P i;j 2 xj j¼n Mi Mj 1=3 1=3 p ci þ cj j Di;j jsi
(22)
cH2 ¼ 7:07; cH2 O ¼ 12:7; cN2 ¼ 17:9; cO2 ¼ 16:6:
Table 1 e HT-PEMFC model: governing equations. Governing equations Mass Momentum
V$ðr! u Þ ¼ Sm
(1)
Flow ðNavier Stokes equationsÞ : 2 channels u! u Þ ¼ Vp þ V$s (2) 1=3 V$ðr! Porous media ðDarcys equationsÞ : r! u ¼ ðK=nÞVp
(3)
Species
V$ð! u Ci Þ ¼ V$ Deff i VCi þ Si
(4)
Charge
Proton transport : V$ keff VFe þ SF ¼ 0
(5)
Electron transport : V$ seff VFs SF ¼ 0 (6) Energy
V$ rCp ! u T ¼ V$ keff VT þ ST
(7)
7706
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 ) 7 7 0 4 e7 7 1 4
Table 2 e HT-PEMFC model: source/sink terms. Source terms
Anode CL
Cathode CL
H2
nxover ja nxover H O SH2 ¼ 2 2 2 2F dCL dCL
(8)
O2
SO2 ¼
nxover jc 1 nxover O H2 2 4F dCL 2 dCL
(9)
H2O
SH2 O ¼ 2 Mass
Sm ¼
X k
nxover O2 dCL
(10a)
MH2 nxover MO2 nxover O2 H2 ja Sk ¼ MH2 þ 2F dCL
(11a)
nxover jc H SH2 O ¼ þ 2 2F dCL
Sm ¼
X k
(10b)
MO2 nxover MH2 nxover H2 O2 jc jc Sk ¼ MO2 MH2 O þ 4F 2F dCL
(11b)
Charge
S F ¼ ja
(12a)
SF ¼ jc
(13a)
ST ¼ jc h þ
(12b)
Heat
S T ¼ ja h þ
2nxover I2 O2 þ DHH2 keff dCL
Note that, in addition to molecular diffusion as defined in the Eq. (22), species diffusion transport can also be controlled by the Knudsen diffusion effect due to molecular-to-wall collision. According to the kinetic theory, the Knudsen diffusivity can be expressed as sffiffiffiffiffiffiffiffiffi dp 8RT : (23) Di;K ¼ 3 pMi
Ixover I2 dUo dUo H2 h þ T T þ j c keff dT dCL dT
(13b)
Therefore, the effective diffusivity of species in porous media is obtained by combining both the molecular and Knudsen diffusion effects with the effects of porosity and tortuosity of the porous medium using the Bruggemann correlation [19]: Di ¼ 3 n
1 1 þ Di;M Di;K
1 :
(24)
Table 3 e HT-PEMFC model: electrochemical properties. Description
Anode CL
Exchange current density ratio of the reaction surface to the 3 CL volume, airef 0 (A/m ) Reference H2/O2 molar concentration, (mol/m3) Transfer coefficients, a Thermodynamic equilibrium potential, U0 (V)
Cathode CL
1.0 109
1.0 104
40.88
40.88
aa ¼ 0:5 0
ac ¼ 0:5
1:1669 0:24 103 ðT 373:15Þ
(14)
(15)
fs fe U0 ðwith fs ¼ Vcell Þ
(16)
(17)
jc ¼ airef 0;c
Surface overpotential,h (V)
fs fe U0 ðwith fs ¼ 0Þ Transfer current density, j (A/m3)
ja ¼ airef 0;a
Electrochemical reactions :
X k
si Mzi
CH2 CH2;ref
!1=2
aa þ ac Fh Ru T
8 < Mi ¼ chemical formula of species i ¼ ne ; where si ¼ stoichiometry coefficient : n ¼ number of electrons transferred
CO2 CO2;ref
!3=4
Ixover ac H Fh þ 2 exp Ru T dCL
(18)
(19)
Hydrogen oxidation reactionðHORÞat the anode side : H2 2Hþ ¼ 2e
(20)
Oxygen reduction reactionðORRÞat the cathode side : 2H2 O O2 4Hþ ¼ 4e
(21)
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 ) 7 7 0 4 e7 7 1 4
7707
The dissolution of species in the aqueous electrolyte phase and their subsequent diffusion are described below. According to Cheddie and Munroe [10], the solubility and diffusivity of oxygen in a concentrated phosphoric acid can be expressed in terms of the weight percentage of phosphoric acid, mPA , and temperature as
The proton conductivity of the phosphoric acid-doped PBI membrane is correlated to the doping level and temperature as follows [10]:
3 257:13ðm Þ 431:08ðm Þ þ 178:45 7 6 6 7 7; 6 HPA O2 ¼ 0:1exp6 PA 2 PA 93500ðm Þ þ156646ðm Þ 64288 7 5 4 þ T
The effective proton conductivity in the CLs is obtained by combining the effects of the volume fraction of the membrane phase and tortuosity of the porous medium by using Bruggemann’s correlation:
2
PA 2
PA
(25)
2.3. (26)
Henry’s constant of oxygen in the phosphoric acid-doped PBI membrane can be obtained as a function of the volume fraction of phosphoric acid in the membrane, 3 PA as [10] HPBI O2 ¼
3
PA 1:945
h
PA 1:8 i HPA : O2 þ 5:79 1 3
(27)
The 3 PA in turn depends on the doping level of phosphoric acid in the membrane, X as [10] 3
PA
¼
4:81 þ1 X2
1 :
(28)
The X in the above equation can be computed based on the phosphoric acid concentration, M as [10] X ¼ 0:012M3 0:2111M2 þ 1:2363M þ 0:7119:
(29)
On the other hand, the oxygen diffusion coefficient in the phosphoric acid-doped PBI membrane can be obtained by Bruggemann’s relation as [10] DPBI O2 ¼
3
PA 1:8
DPA O2 :
(30)
The dissolved concentration of oxygen at the gas/electrolyte interface in the cathode CL is determined from the partial pressure of oxygen pO2 using Henry’s law as follows: g PBI PBI CPBI O2 ¼ HO2 pO2 ¼ HO2 CO2 ;mem RT :
100 2605:6 70:1X exp 8:0219 : T T
keff ¼ 3 1:5 mc k:
2
3 PA 2 PA 6 192:55ðm Þ þ323:55ðm Þ 125:61 7 6 9 7 6 7: DPA O2 ¼ 10 exp6 PA 2 PA 7 4 62010ðm Þ 105503ðm Þ þ 40929 5 þ T
k¼
(31)
Due to lack of studies regarding hydrogen dissolution into concentrated phosphoric acid, the diffusivity and solubility of hydrogen are assumed to be two times and four times larger than those of oxygen, respectively, i.e. based on the transport behavior of hydrogen and oxygen in water systems [10]: PA DPA H2 ¼ 2DO2 ;
(32)
PA HPA H2 ¼ 4HO2 :
(33)
(34)
(35)
Gas crossover model and relevant source terms
The gas crossover model accounts for the influences of hydrogen crossover (from the anode to cathode) and oxygen crossover (from the cathode to anode) on electrochemical processes and the resultant overall cell performance. The gas crossover model has been described in detail in a previous study [17] and hence only a brief summary is repeated here. The hydrogen crossover through the membrane causes a mixed potential at the cathode CL due to facile hydrogen oxidation kinetic and large surface overpotential at the cathode. Therefore, the final form of the ORR kinetic expression can be determined as Eq. (18) in Table 3 where the second term in the right-hand side of Eq. (18) represents the effect of the hydrogen crossover. Under the assumption that crossed hydrogen is uniformly and completely oxidized in the cathode CL, the hydrogen crossover current density, can be defined as below:
¼ 2Fnxover ¼ 2FDPBI Ixover H2 H2 H2
CPBI H2
aCL
dmem
:
(36)
On the other hand, the oxygen crossover through the membrane leads to catalytic hydrogen/oxygen combustion in the anode CL due to small potential difference between the solid and electrolyte phases at the anode. The influences of the hydrogen and oxygen crossover on mass, species (hydrogen, oxygen, and water), and energy balance are seen in their source/sink terms in Table 2.
2.4. Numerical implementation, computational domain, and boundary conditions The HT-PEMFC model described in Section 2 is numerically implemented in a commercial computational fluid dynamics (CFD) program, FLUENT, basing on its user defined functions (UDF). The convergence criteria for all species and energy calculation residuals are set to 108. Fig. 1 shows the mesh configuration of the simple single-straight channel geometry. The physical properties and, cell dimensions and operating conditions are given in Tables 4 and 5, respectively. The isothermal boundary condition is applied to the anode and cathode wall of the computational cell for temperature calculations. In addition, the no-flux condition is applied to the outer faces for flow and species transport equations except for the channel inlets and outlets. The inlet velocities in the anode and cathode GCs can be expressed as functions of
7708
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 ) 7 7 0 4 e7 7 1 4
Table 5 e Cell dimensions and operating conditions. Description Cell length Anode/cathode channel/rib width Anode/cathode channel height Thickness of the anode/cathode GDLs Thickness of the anode/cathode CLs Thickness of the membrane Anode/cathode inlet pressure Anode stoichiometry Cathode stoichiometry Anode/cathode inlet temperature RH of the anode/cathode inlet Phosphoric acid doping level
Fig. 1 e Mesh configuration of three-dimensional, singlechannel HT-PEMFC geometry.
the anode and cathode stoichiometric ratios, the operating current density, the cross-sectional areas of the anode and cathode GCs, and the concentrations of hydrogen and oxygen, which are functions of the anode/cathode inlet pressure and temperature: uin;a ¼
3.
xa ðI=2FÞAmem CH2 Aa;chan
and uin;c ¼
xc ðI=4FÞAmem : CO2 Ac;chan
(37)
Results and discussion
To examine the effects of gas crossover on HT-PEMFCs, we assessed the effects of three critical parameters, namely, operating current density, operating temperature, and gas crossover diffusivity itself. Regarding the effect of operating current density, it is evident that the impact of gas crossover is
Table 4 e Physical properties. Description Porosity of GDL, CL Volume fraction of ionomers in CL Permeability of GDL, CL Electronic conductivity in the GDL, CL, BP Specific heat capacities of GDL, CL, membrane, BP Specific heat capacities of species e H2, O2, N2, H2O Thermal conductivities of GDL, CL, membrane, BP Thermal conductivities of species e H2, O2, N2, H2O
Value 0.6, 0.3 0.4 1 1012, 1.0 1013 m2 1250, 300, 14000 S m1 568, 3300, 1650, 2930 J kg1 K1 14430, 929, 1042, 1968 J kg1 K1 1.2, 1.5, 0.95, 20 W m1 K1 0.2040, 0.0296, 0.0293, 0.02378 W m1 K1
Value 0.1 m 1 103 m 0.7 103 m 250 106 m 10 106 m 70 106 m 1.0 atm 2.0 (Pure H2) 2.0 (Air) 373 K, 453 K 0.0% 6.2
more significant at lower current density operation due to the lower reactant consumption rate and resultant higher concentration in the CL that leads to a higher dissolution rate of the reactant gas into the aqueous electrolyte phase. The hydrogen and oxygen crossover diffusivities given by Cheddie and Munroe [10] imply that the degree of gas crossover through a phosphoric acid-doped PBI membrane is considerably altered by operating temperature. Therefore, the operating temperature is another critical factor to control the gas crossover behavior inside an HT-PEMFC. Finally, several degradation mechanisms of the PBI membrane have been reported in the literature, such as chemical degradation [20], thermal degradation [21] and phosphoric acid evaporation [22]. In particular, it should be noted that the phosphoric acid loss due to evaporation not only decreases the membrane conductivity but also increases the gas crossover diffusivities for hydrogen and oxygen [10]. Although all of these membrane degradation mechanisms appear to be highly localized, due to the lack of experimental data on degradations, we assume the membrane to be degraded uniformly; thus, the gas crossover diffusivity is also uniformly raised as a function of the degree of membrane degradation. The parametric study was carried out at two operating temperatures (100 C and 180 C) under various operating current densities and hydrogen/oxygen crossover diffusivities. Four different cases of gas crossover diffusivity are defined here. Case 1 assumes that the membrane is perfectly impermeable to hydrogen and oxygen; hence, their crossover diffusivities are set to zero. Case 2 represents the case of a fresh phosphoric acid-doped PBI membrane, and the crossover processes for hydrogen and oxygen are approximated using the crossover diffusivities given by Cheddie and Munroe [10]. To consider a degraded membrane, the hydrogen/oxygen crossover diffusivities were further raised by one and two orders of magnitude for cases 3 and 4, respectively. Fig. 2 shows the hydrogen and oxygen concentration profiles in the anode and cathode CLs for cases 1 to 4 at the operating current density of 0.2 A cm2 based on the operating temperatures of 100 C and 180 C. As shown in Fig. 2(a) and (b), the hydrogen concentration under the land region is lower than that under the channel region due to the longer transport path from the anode flow channel. In a comparison of cases 1 to 4 for each operation temperature, the hydrogen distributions for cases 1, 2, and 3 are almost identical, indicating that
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 ) 7 7 0 4 e7 7 1 4
7709
Fig. 2 e Hydrogen concentration contours (in mol mL3) in the anode CL and oxygen concentration contours (in mol mL3) in the cathode CLs at an operating current density of 0.2 A cmL2: (a) Hydrogen; T [ 100 C, (b) Hydrogen; T [ 180 C, (c) Oxygen; T [ 100 C and (d) Oxygen; T [ 180 C.
the effect of crossover though the fresh membrane is negligible (case 2), and even a ten-fold increase in the hydrogenpermeation coefficient due to membrane degradation is still acceptable for HT-PEMFC operations (case 3). However, more severe hydrogen depletion is observed in case 4, which can be attributed to a combined result of the higher degree of hydrogen crossover from the anode through the membrane and the higher rate of catalytic hydrogen/oxygen combustion at the anode CL driven by stronger oxygen crossover from the cathode in case 4. In addition, higher hydrogen depletion was predicted at the higher operating temperature because the amount of hydrogen crossover increases with temperature. The same trend is observed in the oxygen concentration contours in the cathode CL in Fig. 2(c) and (d) where oxygen depletion near the cathode outlet region is more severe with a higher degree of membrane degradation (case 4) and/or a higher operating temperature (180 C). Note that the severe oxygen depletion in case 4 is attributed to both its higher oxygen crossover rate from the cathode to anode as well as the higher hydrogen crossover from the anode to cathode that leads to additional ORR and mixed potential at the cathode CL.
Fig. 3(a) shows the local current density distributions in the membrane for cases 1 to 4 at the operating current density of 0.2 A cm2 and the operating temperature of 100 C. In all cases, the local current density near the land region is lower than near the channel region along the in-plane direction (Z ). Along the cathode flow direction (Y ), the local current density continuously decreases toward the cathode outlet. These trends indicate that oxygen depletion is the sole factor in determining the current density distribution for all the cases. In a comparison of cases 1 to 4, the current density distributions for cases 1, 2, and 3 are almost identical, indicating that the degree of gas crossover up to a ten-fold increase in the gaspermeation coefficient due to membrane degradation has a negligible influence on HT-PEMFC performance. However, spatial non-uniformity in the current density profile is clearly increased in case 4 due to the higher degree of gas crossover through the membrane. Fig. 3(b) displays the local current density contours at 180 C. As compared with Fig. 3(a), the local current densities near the cathode outlet are reduced due to the higher degree of hydrogen and oxygen crossover at the elevated temperature.
7710
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 ) 7 7 0 4 e7 7 1 4
Fig. 3 e Local current density distribution (in A mL2) in the membrane at an operating current density of 0.2 A cmL2: (a) T [ 100 C and (b) T [ 180 C.
Fig. 4 e Crossover current density distribution (in A mL2) in the membrane at an operating current density of 0.2 A cmL2: (a) T [ 100 C and (b) T [ 180 C.
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 ) 7 7 0 4 e7 7 1 4
7711
Fig. 5 e Cathode overpotential distribution (in volts) in the CL at an operating current density of 0.2 A cmL2: (a) T [ 100 C and (b) T [ 180 C.
To analyze the degree of hydrogen crossover, the hydrogen crossover current density was calculated by Eq. (36) and plotted in Fig. 4 for the same simulation cases (at 0.2 A cm2 under two operating temperatures of 100 C and 180 C). For all cases, the hydrogen crossover current density decreases toward the anode downstream because the hydrogen concentration in the anode CL is high near the anode inlet and continuously depleted along the flow direction by not only HOR but also hydrogen permeation through the membrane and the hydrogen/oxygen catalytic combustion due to the oxygen crossover. In a comparison of cases 2 to 4, the hydrogen crossover current densities of case 4 are roughly two orders of magnitude larger than those of case 2, which clearly indicates that the amount of hydrogen crossover flux is directly proportional to the considered hydrogen crossover diffusivity. In addition, much higher hydrogen crossover current density was predicted with the higher operating temperature (180 C) since the hydrogen crossover diffusivity is a function of increasing temperature. Fig. 5 shows the cathode overpotential distributions in the cathode CL for cases 1 to 4 at the operating temperatures of 100 C and 180 C. The cathode overpotential increases toward the land region and cathode downstream due to lower local oxygen concentrations there as seen in Fig. 2(c) and (d). More importantly, a comparison of the 100 C and 180 C cases indicates that operating the cell at the higher temperature significantly reduces the cathode overpotential, although the available oxygen concentration for ORR in the cathode CL is lower at 180 C (see Fig. 2(d)). This is mainly due to the enhanced electrochemical kinetics of ORR at the elevated operating temperature. Fig. 6 shows the effects of hydrogen and oxygen crossover through the membrane on cell polarization curves at two different operating temperatures (100 C and 180 C). First, superior cell performance is achieved at the higher operating temperature due to improved ORR kinetics, better proton conductivity, and more efficient mass transport with increasing temperature. Further, the polarization curves clearly demonstrate that the impact of gas crossover is more
significant at lower current densities, because the hydrogen and oxygen concentrations remaining in the CLs are higher under lower current density operations. In addition, the polarization curves for cases 1 to 3 are similar to each other at both temperatures (100 C and 180 C), which means that a ten-fold increase in the gas-permeation coefficient (case 3) is acceptable for HT-PEMFC operations under wide ranges of operating current density and temperature. However, a more pronounced effect of gas crossover is seen in case 4, particularly at the higher operating temperature (180 C). These results imply that careful attention to suppress gas crossover is required for low current density and/or high temperature operations. Tables 6 and 7 summarize the overall heat balance and the individual heat sources for cases 1 to 4 under the operating current density of 0.2 A cm2 at 100 C and 180 C, respectively. The simulation results show that the largest part of the total waste heat release is due to irreversible ORR reaction at the cathode, that is, roughly 70% of the total heat generation. In
Fig. 6 e Overall polarization curves for cases 1e4 at the operating temperatures of 100 C and 180 C.
7712
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 ) 7 7 0 4 e7 7 1 4
Table 6 e Summary of energy balance results under operating current density of 0.2 A cmL2 at 100 C. Case 1 (1) Anode CL
Irreversible reaction heat, [W] ST;irrev;a ¼
R
j:hdV
Case 2
Case 3
Case 4
0.00344 (0.985%) 0.00345 (0.985%)
0.00345 (0.977%)
0.00348 (0.903%)
0.00399 (1.141%) 0.00399 (1.139%)
0.00399 (1.129%)
0.00400 (1.038%)
V
Ohmic joule heating, [W] ST;joule;a ¼
R I dV keff 2
V
H2/O2 catalytic combustion heat [W] Sxover ¼ DH Sxover 0.0 T;a O2 R I2 Ohmic joule heating, [W] ST;joule;mem ¼ dV eff V k R (3) Cathode CL Irreversible reaction heat, [W] ST;irrev;c ¼ j:hdV (2) Membrane
0.0000376 (0.011%) 0.000374 (0.106%) 0.00352 (0.911%)
0.02678 (7.659%) 0.02675 (7.643%)
0.02675 (7.572%)
0.02675 (6.935%)
0.26724 (76.43%) 0.26740 (76.41%)
0.26883 (76.10%)
0.28304 (73.37%)
0.01232 (3.524%) 0.01231 (3.516%)
0.01232 (3.487%)
0.01245 (3.227%)
Mixed potential and entropic heat due to Ixover dU0 H hydrogen crossover, [W] Sxover ¼ 2 hþT T;c dCL dT R vU0 dV Entropy heat, [W] ST;rev;c ¼ j$ T vT V
0.0
0.00152 (0.428%)
0.01489 (3.859%)
0.03588 (10.26%) 0.03589 (10.26%)
0.03605 (10.21%)
0.03763 (9.755%)
(1)þ(2)þ(3), [W]
0.3497
0.3533
0.3858
V
Ohmic joule heating, [W] ST;joule;c ¼
R I dV keff 2
V
Sum
0.000152 (0.043%)
0.3499
Table 7 e Summary of energy balance results under operating current density of 0.2 A.cmL2 at 180 C.
(1) Anode CL
R
Irreversible reaction heat, [W] ST;irrev;a ¼ j:hdV R I2 V Ohmic joule heating, [W] ST;joule;a ¼ dV eff V k
Case 1
Case 2
0.00235 (0.846%) 0.00202 (0.727%)
0.00236 (0.844%) 0.00202 (0.722%)
H2/O2 catalytic combustion heat [W] Sxover ¼ DH Sxover 0.0 T;a O2 R I2 Ohmic joule heating, [W] ST;joule;mem ¼ dV eff V k R (3) Cathode CL Irreversible reaction heat, [W] ST;irrev;c ¼ j:hdV (2) Membrane
Case 3
Case 4
0.00238 (0.801%) 0.00253 (0.568%) 0.00202 (0.683%) 0.00208 (0.465%)
0.00023 (0.084%)
0.00226 (0.763%) 0.01562 (3.493%)
0.01165 (4.193%)
0.01164 (4.160%)
0.01165 (3.929%) 0.01178 (2.639%)
0.21214 (76.33%)
0.21292 (76.10%)
0.21996 (74.18%) 0.28598 (64.07%)
0.00623 (2.241%)
0.00623 (2.225%)
0.00629 (2.122%) 0.00725 (1.625%)
V
Ohmic joule heating, [W] ST;joule;c ¼
R I dV keff 2
V
Mixed potential and entropic heat due 0.0 0.00072 (0.258%) 0.00717 (2.420%) 0.06686 (14.98%) Ixover dU0 H2 xover to hydrogen crossover, [W] ST;c ¼ hþT dCL dT R 0.04354 (15.669%) 0.04367 (15.607%) 0.04476 (15.10%) 0.05426 (12.156%) vU0 Entropy heat, [W] ST;rev;c ¼ j$ T dV vT V Sum
(1)þ(2)þ(3), [W]
addition, it should be noted that the heat generated by hydrogen/oxygen catalytic combustion on the anode side is about 3.5% in the most severe gas crossover case in this study (case 4 at 180 C and 0.2 A cm2). Therefore, the contribution of the hydrogen/oxygen chemical reaction does not seem to be significant to the total heat release during HT-PEMFC operations, even when two orders of magnitude greater gas crossover diffusivities are considered.
4.
Conclusions
In this study, a gas crossover model that considers the dissolution of hydrogen/oxygen into the electrolyte phase and subsequent diffusion through a phosphoric acid-doped PBI membrane was developed and incorporated into a HT-PEMFC model developed in an earlier study [16]. The main interest of this study is to numerically assess the impact of gas crossover on HT-PEMFC performance. The gas crossover model rigorously accounts for the major outcomes of hydrogen and
0.2779
0.2798
0.2964
0.4463
oxygen crossover, i.e. a mixed potential at the cathode CL and the hydrogen/oxygen catalytic combustion at the anode CL. The numerical results show that the gas crossover has a negligible influence on overall cell performance in a fresh PBI membrane (case 2) and a moderately degraded membrane (case 3), which was assumed to have one order of magnitude higher crossover diffusivities than those of the fresh membrane. However, the effect of gas crossover begins to appear in more a severely degraded membrane with two orders of magnitude larger gas crossover diffusivities (case 4). A comparison of case 4 with cases 1 to 3 clearly shows that the increased effect of gas crossover increases the degree of nonuniformity in the hydrogen, oxygen, and current density distributions. In addition, the simulation results for case 4 indicate that gas crossover is more detrimental to cell operation at a higher operating temperature and/or lower current density due to more facile crossover of hydrogen and oxygen with elevated temperature and due to higher reactant concentration in the CL with lower current density. Finally, the thermal analysis carried out in this study demonstrated
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 ) 7 7 0 4 e7 7 1 4
that the heat generated via hydrogen/oxygen catalytic combustion at the anode CL is not significant, occupying only 3.5% of the total waste heat release in the worst gas crossover case in this study (case 4 at 180 C and 0.2 A cm2). This paper contributes to enhancing the fundamental understanding of the gas crossover phenomena occurring during HT-PEMFC operation. As an extension of this work, our efforts are underway to numerically study the effects of local heterogeneous gas crossover due to membrane pinhole formations.
Acknowledgment This work was supported by the New & Renewable Energy R&D program (grant no. 2010T100200501) of the Ministry of Knowledge Economy of the government of the Republic of Korea. The authors gratefully acknowledge this support.
Nomenclature
A c C Di F H i0 I j K M p Q R S T ! u U0 Vcell X
area, m2 specific heat, J kg1 K1 molar concentration, mol m3 mass diffusivity of species i, m2 s1 Faraday constant, 96487 C mol1 Henry’s constant, mol m3 atm1 exchange current density, A m2 operating current density, A m2 transfer current density, A m3 hydraulic permeability, m2 molecular weight, kg mol1 partial pressure, Pa heat, watt universal gas constant, 8.314 J mol1 K1 source term in the conservation equation temperature, K fluid velocity and superficial velocity in a porous medium, m s1 thermodynamic equilibrium potential, V cell potential, V doping level
Greek symbols a transfer coefficient 3 porosity 3 mc volume fraction of the ionomer phase in the CL f phase potential, V h overpotential, V m viscosity, kg m1 s1 r density, kg m3 s viscous shear stress, N m2 k ionic conductivity, S m1 x stoichiometry flow ratio
Superscripts e electrolyte eff effective value in the porous region
g ref
7713
gas reference value
Subscripts a anode c cathode CL catalyst layer GC gas channel GDL gas diffusion layer hydrogen H2 i species index in channel inlet m mass equation mem membrane oxygen O2 u momentum equation w water F potential equation 0 standard condition, viz., 298.15 K and 101.3 kPa (1 atm)
references
[1] Wainright JS, Wang JT, Weng D, Savinell RF, Litt MH. Aciddoped polybenzimidazoles: a new polymer electrolyte. J Electrochem Soc 1995;142(7):L121e3. [2] Samms SR, Wasmus S, Savinell RF. Thermal stability of proton conducting acid doped polybenzimidazole in simulated fuel cell environments. J Electrochem Soc 1996; 143(4):1225e32. [3] Weng D, Wainright JS, Landau U, Savinell RF. Electro-osmotic drag coefficient of water and methanol in polymer electrolytes at elevated temperatures. J Electrochem Soc 1996;143(4):1260e3. [4] Qingfeng Li, Hjuler HA, Bjerrum NJ. Oxygen reduction on carbon supported platinum catalysts in high temperature polymer electrolytes. Electrochim Acta 2000;45(25e26): 4219e26. [5] Qingfeng Li, Hjuler HA, Bjerrum NJ. Phosphoric acid doped polybenzimidazole membranes: physiological characterization and fuel cell application. J Appl Electrochem 2001;31(7):773e9. [6] Ma YL, Wainright JS, Litt MH, Savinell RF. Conductivity of PBI membranes for high-temperature polymer electrolyte fuel cells. J Electrochem Soc 2004;151(1):A8e16. [7] He RH, Qingfeng Li, Bach A, Jensen JO, Bjerrum NJ. Physiochemical properties of phosphoric acid doped polybenzimidazole membranes for fuel cells. J Membr Sci 2006;277:38e45. [8] Liu Z, Wainright JS, Litt MH, Savinell RF. Study of the oxygen reduction reaction (ORR) at Pt interfaced with phosphoric acid doped polybenzimidazole at elevated temperature and low relative humidity. Electrochim Acta 2006;51:3914e23. [9] Cheddie D, Munroe N. Mathematical model of a PEMFC using a PBI membrane. Energy Convers Manage 2006;47:1490e504. [10] Cheddie D, Munroe N. A two phase model of an intermediate temperature PEM fuel cell. Intl J Hydrogen Energy 2007;32: 832e41. [11] Cheddie D, Munroe N. Three dimensional modeling of high temperature PEM fuel cells. J Power Sources 2006;160:215e23.
7714
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 ) 7 7 0 4 e7 7 1 4
[12] Scott K, Pilditch S, Mamlouk M. Modeling and experimental validation of a high temperature polymer electrolyte fuel cell. J Appl Electrochem 2007;37:1245e59. [13] Jiao K, Li X. A three-dimensional non-isothermal model of high temperature proton exchange membrane fuel cells with phosphoric acid doped polybenzimidazole membranes. Fuel Cells 2010;10(3):351e62. [14] Siegel C, Bandlamudi G, Heinzel A. Systematic characterization of a PBI/H3PO4 sol-gel membranemodeling and simulation. J Power Sources 2011;196: 2735e49. [15] Sousa T, Mamlouk M, Scott K. An isothermal model of a laboratory intermediate temperature fuel cell using PBI doped phosphoric acid membranes. Chem Eng Sci 2010;65: 2513e30. [16] Chippar P, Ju H. Three-dimensional non-isothermal modeling of a phosphoric acid-doped polybenzimidazole (PBI) membrane fuel cell. Solid State Ionics, in press.
[17] Nam J, Chippar P, Kim W, Ju H. Numerical analysis of gascrossover effects in polymer electrolyte fuel cells (PEFCs). Appl Energy 2010;87:3699e709. [18] Bird RB, Stewart WE, Lightfoot EN. Transport phenomena. New York: John Wiley & Sons; 1960. [19] Meredith RE, Tobias CW. In: Tobia CW, editor. Advances in electrochemistry and electrochemical engineering, vol. 2. New York: Interscience Publishers; 1962. [20] Liao JH, Li QF, Rudbeck HC, Jensen JO, Chromik A, Bjerrum NJ, et al. Oxidative degradation of polybenzimidazole membranes as electrolytes for high temperature proton exchange membrane fuel cells. Fuel Cells 2011;10(3):351e62. [21] Zhang J, Song C, Zhang J. Accelerated lifetime testing for proton exchange membrane fuel cells using extremely high temperature and unusually high load. J Fuel Cells Sci Tech 2011;8(051006):1e5. [22] Yu S, Xiao L, Benicewicz BC. Durability studies of PBI-based high temperature PEMFCs. Fuel Cells 2008;8:165e74.