UNCORRECTED PROOF
3
Analysis of thermal and water management with temperature-dependent
diffusion effects in membrane of proton exchange membrane fuel cells
4
5
Wei-Mon Yan
a,∗, Falin Chen
b, Hung-Yi Wu
b, Chyi-Yeou Soong
c, Hsin-Shen Chu
d 6aDepartment of Mechatronic Engineering, Huafan University, Shih Ting, Taipei 223, Taiwan, ROC
7
bInstitute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan, ROC
8
cDepartment of Aeronautical Engineering, Feng Chia University, Seatwen, Taichung 407, Taiwan, ROC
9
dDepartment of Mechanical Engineering, Chiao Tung University, Hsin-Chu 300, Taiwan, ROC
10
Received 12 August 2003; accepted 21 November 2003 11
Abstract
12
In the present work, the detailed thermal and water management in the membrane of proton exchange membrane fuel cells (PEMFC) is investigated numerically. The coupling effects of mass diffusion and temperature gradient on the water distribution in the membrane are taken into account with consideration of the temperature-dependent diffusivity. Thermal and water transport equations with various boundary conditions are solved by the control volume finite difference method. Predictions show that under the conditions of fixed water concentration at the cathode side, the effect of cathode temperature, Tc, on the water concentration is significant. Increases in Tcmay lead
to an increase in membrane dehydration. At the water-flux condition on the cathode side, the influence of the operating temperature on the water distribution in the membrane shows a similar trend. The effects of the anode temperature, Ta, on the water management in the
membrane are also examined. It is found that Tahas considerable impact on the water content in the membrane. In addition, high current
density may cause non-uniformity of the temperature distribution in the membrane.
13 14 15 16 17 18 19 20 21 © 2003 Published by Elsevier B.V. 22
Keywords: Diffusion effects; Proton exchange membrane fuel cells; Water management 23
1. Introduction
24
Recent interests in proton exchange membrane fuel cell
25
(PEMFC) systems have caused extensive studies on thermal
26
and water management. During (PEMFC) operation, water
27
molecules can be carried from the anode side to the cathode
28
side of the membrane by electro-osmosis, and if the
trans-29
port rate of water is higher than the back-diffusion rate from
30
the anode to the cathode, the membrane will become
dehy-31
drated and too resistive to conduct high current. At the
cath-32
ode side of the membrane, where water molecules are not
33
only transported from anode side but also generated by the
34
cathodic reaction, electrode flooding occurs when the water
35
removal rate fails to keep up with its transport rate out of
36
the electrode. On the other hand, the temperature gradient in
37
the membrane may influence the fuel-cell performance by
38
affecting the transport of water and gaseous species as well
39
as the electrochemical reactions in the electrode. Therefore,
40
∗Corresponding author. Tel.:+886-2-2663-3847;
fax:+886-2-2663-3847.
E-mail address: [email protected] (W.-M. Yan).
it is appealing to have a theoretical model which can pro- 41
vide detailed understanding of the governing phenomena in- 42
side the membrane. This motivates the present study, which 43
examines the water concentration and temperature within 44
membrane of PEMFCs. 45
In past decades, there have been numerous studies of 46
transport phenomena in PEMFCs. Bernardi[1] proposed a 47
one-dimensional model of water management with consid- 48
eration of the membrane thickness. By using this model, it 49
was found that the diffusion in the water production and 50
evaporation rate in the PEMFC can result in the flooding of 51
the electrode or the membrane dehydration, and therefore af- 52
fect the performance of the fuel cells. In addition, the effects 53
of the humidification on the current–voltage curves of the 54
fuel cells under various operating conditions were presented. 55
Springer et al.[2]developed an isothermal, one-dimensional, 56
steady-state model for the PEMFC with Nafion® 117 [2]. 57
Diffusion, electro-osmotic drag and membrane conduction 58
were all taken into account. The results showed that the net 59
water-flux ratio under a typical operating condition is much 60
less than that within a fully-hydrated membrane. It was also 61
found that the membrane resistance is significantly enhanced 62
1 0378-7753/$ – see front matter © 2003 Published by Elsevier B.V. 2 doi:10.1016/j.jpowsour.2003.11.028
UNCORRECTED PROOF
NomenclatureCa water concentration per unit volume at
the anode side (mol cm−3)
Cc water concentration per unit volume at
the cathode side (mol cm−3)
CH2O water concentration in the membrane per
unit volume (mol cm−3)
Cp,l specific heat of liquid water (J kg−1K−1)
d density of the membrane (g cm−3)
D diffusion coefficient of water in the membrane (cm2s−1)
Da diffusion coefficient of water at the
anode side (cm2s−1)
Dc diffusion coefficient of water at the
cathode side (cm2s−1)
F Faraday’s constant 96487 (C mol−1)
i operating current density (A cm−2)
K thermal conductivity (W cm−1K−1)
˙mH2O molecular flux of water (mol cm−1s−1)
M molecular weight of water (kg mol−1)
R ohmic resistance per unit volume ( cm−1)
T temperature (◦C)
V volume of the membrane (cm3)
w water transfer coefficient
Greek letters
κ flux of water into membrane by concentration gradient (m s−1)
λ membrane hydration or water content (moles water/moles charge sites)
ν rate of water entry the membrane proportional to the current density
Subscripts
a anode side of the membrane c cathode side of the membrane m membrane
as the current density is increased. By comparison, the
re-63
sistance is reduced for a thin membrane.
64
Fuller and Newman[3]examined experimentally the
wa-65
ter transport number in Nafion® 117. The relationship
be-66
tween transport number and electro-osmotic coefficient was
67
presented. It was demonstrated that the transport number
68
decreases slowly as the membrane is dehydrated, but falls
69
quickly to zero when the water concentration approaches to
70
zero. Nguyen and White[4]performed modelling of the
wa-71
ter and heat management in PEMFC. The model included
72
the effect of electro-osmosis, diffusion of water; heat
trans-73
fer from solid phase to gas phase and latent heat as water
74
evaporation and condensation. It was found that the ohmic
75
loss is noticeable at high current density. The voltage loss is
76
twice amount of that at the cathode electrode. The reactant
77
gas at the anode needs to be humidified since the membrane 78
is dehydrated at high current densities. Fuller and Newman 79 [5]proposed a two-dimensional mathematical model for the 80
water and thermal management and the utilization of the fuel 81
of a PEMFC. Due to the water sorption depending strongly 82
on the temperature, the waste heat is a critical parameter in 83
the design of the proton exchange membrane fuel cells. 84
In the numerical analysis of Mosdale and Srinivasan[6], 85
it was clearly seen that the large current density limit of fuel 86
cell is more for pure oxygen than for air used at the cathode 87
side. Voss et al.[7]proposed a new technique for water man- 88
agement, by which it was found that if the back-diffusion 89
rate and the water concentration are increased, the water at 90
the cathode could be removed via the anode stream. Xie 91
and Okada [8] showed that the water transfer coefficient 92
of Nafion® 117 membrane in the H+ form was 2.6. The 93
Nafion®117 membrane has good performance for HCl so- 94
lutions with a concentration that ranges from 0.003 to 1 N. 95
Additionally, it was also shown that the water transport be- 96
haviour is related to the surface-change density, the hydra- 97
tion enthalpy and the water content in the membrane. 98
By using a linear transport equation for water in the 99
PEMFC, detailed transport phenomena of the PEMFC, in- 100
cluding diffusion and electro-osmotic drag effects, were an- 101
alytically solved by Okada et al. [9,10]. In these studies, 102
both semi-finite and finite boundaries were considered. The 103
predicted results showed that the current density, the wa- 104
ter penetration parameters, the membrane thickness and the 105
diffusion coefficient of water are the key factors in determi- 106
nation of the water content in the membrane. Foreign im- 107
purities such as NaCl will cause a serious impact on the 108
water depletion at the anode side. Water supplied from the 109
anode side of the membrane is needed. Okada extended 110
the modelling to account of the effect of impurity ions 111
at both the anode and the cathode side of the membrane 112 [11,12]. The results indicated that both the current density 113
and the membrane thickness are important parameters in 114
the water management of the membrane, especially when 115
the membrane surface has impurity ions. The distribution 116
of contaminant ions degrades the membrane and the per- 117
formance of the fuel cell. Deterioration of cell performance 118
in the presence of non-uniform impurities in the membrane 119
is more serious than in the case of non-uniform impurities 120
distribution. 121
Thermal management in the direct methanol fuel cell 122
(DMFC) was investigated by Argyropoulos et al.[13,14]. A 123
model was developed to investigate the effects of various op- 124
erating parameters (feed and oxidant temperatures, flow rate 125
and pressure, operating current density) and system design 126
(active area, material properties and geometry) on the per- 127
formance of the DMFC. The mathematical model includes 128
the gas-diffusion layer, the catalyst layer and the membrane. 129
It can also be used to predict the steady-state performance of 130
the DMFC stacks. The diffusion flux across a Nafion®mem- 131
brane can be accurately predicted by using Fick’s diffusion 132
coefficient. Motupally et al.[15]showed that increasing the 133
UNCORRECTED PROOF
cell pressure will decrease the water activity and reduce the
134
diffusion coefficient.
135
Baschuk and Li [16] developed a mathematical model
136
with variable degrees of water flooding in the PEMFC.
137
Physical and electrochemical processes occurring in the
138
membrane electrolyte, the cathode catalyst layer, the
elec-139
trode backing layer and the flow channel were considered.
140
Compared with experimental results, it was found that when
141
air is used as the cathode fuel, the flooding phenomena are
142
similar for different operating conditions of the pressures
143
and temperatures. When the cell pressure is increased
signif-144
icantly, the water flooding in the electrode becomes serious.
145
This will significantly reduce the power output. Recently,
146
Rowe and Li[17]carried out a two-dimensional simulation
147
of water transport in the PEMFC without external
humidifi-148
cation. This model calculated the fraction of product water
149
leaving the anode side of the fuel cell. The results indicated
150
that the amount of water leaving the anode depends on the
151
hydrogen stoichiometry, oxygen stoichiometry, current
den-152
sity, and cell temperature. One of the most recent PEMFC
153
models was proposed by Djilali and Lu[18]for analysis of
154
fuel-cell performance and water transport. The
thermody-155
namic equation was determined by the Nernst equation and
156
the reaction kinetics were calculated by the Butler–Volmer
157
equation. Analysis showed that the water requirement to
158
prevent the membrane from dehydrating or flooding is
159
important.
160
From the literature reviews presented above, it is
con-161
cluded that the effects of the temperature gradient on water
162
management in the PEMFCs are not well defined. In fact,
163
the water content in the membrane can be influenced by the
164
local temperature distribution since the diffusivity in water
165
transport is temperature-dependent. On the other hand, the
166
energy balance is also closely related to the water content
167
or local water concentration in the membrane. The objective
168
of the present study is to explore the coupling mechanisms
169
of thermal–mass-transport phenomena in the membrane of
170
PEMFC systems.
171
2. Analysis
172
Consideration is given to a PEMFC in which the polymer
173
electrolyte membrane made from Nafion®. Its thickness is
174
smaller than its length and width, as shown schematically
175
in Fig. 1. Therefore, it can treat it as a one-dimensional
176
problem. To simplify the analysis, the following assumptions
177
are made.
178
(i) The transports are steady-state and one-dimensional.
179
(ii) The pressure is constant.
180
(iii) An ideal gas mixture is assumed.
181
(iv) Liquid water flux is only determined in the membrane.
182
(v) The volume of the membrane is constant.
183
(vi) The convective effects are negligible for a small
184 Reynolds number. 185
x
y z x=o x=d Anode catalyst layer Cathode catalyst layer Membrane H+ H2O(drag) H2O(diffusion)Fig. 1. Schematic diagram of physical system.
(vii) Heat loss to the surrounding environment is small and 186
can be neglected. 187
(viii) Joule-heating is considered to be to the membrane 188
ohmic resistance. 189
With the above assumptions, the governing equations for the 190
water balance can then be formulated as follows. 191
2.1. Water transfer equation 192
In the membrane of a PEMFC, the water flux is com- 193
posed of two components, namely, a diffusion flux and an 194
electro-osmosis flux[4,5]. The latter is proportional to the 195
current density, i. The total water flux can then be described 196
by: 197 ˙mH2O= −DH2O dCH2O dx + i FwH2O , (1) 198
where: ˙mH2Ois the molar flux of the water;DH2Ois the dif- 199
fusion coefficient of water in the membrane;CH2Ois the wa- 200
ter concentration in the membrane; i is the current density; 201 F is the Faraday constant;wH2Ois the water transfer coeffi- 202
cient. Therefore, the rate of water concentration is given by: 203
∂CH2O ∂t = − ∂ ˙mH2O ∂x = ∂ ∂x DH2O ∂CH2O ∂x − i FwH2O (2) 204
For steady-state conditions, the above equation becomes: 205
d dx DH2O dCH2O dx − i FwH2O = 0 (3) 206 dDH2O dx dCH2O dx + DH2O d2CH2O dx2 − i FwH2O= 0 (4) 207
Generally, the water transfer coefficient is a function of water 208
concentration, for example: 209
wH2O= w (0) 1 + w(1)1 CH2O+ w 2 1C 2 H2O+ · · · (5) 210
To simplify the analysis, only the first two terms,w(0)1 and 211
w(1)1 , are taken to represent the zero-order and first-order 212
UNCORRECTED PROOF
coefficients with respect toCH2O. The water transfer
coeffi-213
cient can then be expressed as:
214 wH2O= w (0) 1 + w (1) 1 CH2O (6) 215
The water transfer coefficient for Nafion®membrane is
cal-216
culated by the following equation[3,9]:
217 wH2O= 1100wmVwet 22dVdry (7) 218
where: the volume ratio for dry to wet, Vwet/Vdry, is 16.2, and 219
the density of the membrane, d, is 2.02 g cm−3. In addition,
220
the water transfer coefficientwmis 3.2 at 80◦C. 221
The diffusion coefficient for liquid water in the membrane
222
is determined as a function of temperature (in K) and
mem-223
brane hydration[2], i.e.,
224 225 DH2O= exp 2416 1 303− 1 T (2.563 − 0.33λ 226 + 0.0264λ2− 0.000671λ3) × 10−10 (8a) 227
If the membrane hydration parameterλ is taken to be 14, as
228
given in[17], then the above equation reduces to:
229 DH2O= G exp −ξ T (8b) 230 Here: 231 ξ = 2416 (8c) 232 G = 2.903 × 10−7f(λ) (8d) 233 f(λ) = 2.563 − 0.33λ + 0.0264λ2− 0.000671λ3 (8e) 234 2.2. Energy equation 235
The energy equation is based on Fourier’s law of heat
236 conduction; i.e., 237 Km d2T dx2 + d dx( ˙mH2OCp,lT) + i 2R = 0 (9) 238
where: Kmis the membrane thermal conductivity, Cp,lis the 239
specific heat of liquid water, and R is the ohmic resistance
240
per unit volume. The first term represents the diffusion term
241
of the heat, the second term expresses the energy flux due to
242
the convection, and the third term stands for the joule-heating
243
owing to the membrane ohmic resistance.
244
2.3. Combination of water transport and energy equations 245
At first, the molar flux of water is changed into the mass
246
flux of water. ThenEq. (1)becomes:
247 ˙mH2O = −DH2O dCH2O dx + i FwH2O M (10) 248
where M is the molecular weight of water. Substituting the
249
above equation intoEq. (9)gives:
250 Km d2T dx2 + −2DH2O dCH2O dx MCp,l+ 2i Fw(1)1 CH2OMCp,l 251 ×dT dx + i 2R + −dDH2O dx dCH2O dx MCp,l 252 − DH2O d2CH2O dx2 MCp,l+ i Fw(1)1 dCH2O dx MCp,l T = 0 253 (11) 254
By combiningEqs. (8) and (11), the above equation can be 255
simply expressed as: 256
d2T dx2 + A exp −ξ T dCH2O dx + 2BCH2O dT dx = H (12a) 257
Here the constant, A, B, and H are: 258
A = −2GMCp,l Km (12b) 259 B = iw(1)1 MCp,l FKm (12c) 260 H = −i2R Km (12d) 261
Similarly, the water transfer equation,Eq. (4), can be sim- 262
plified as: 263 d2CH2O dx + ξ T2 dT dx − N exp ξ T dCH2O dx = 0 (13a) 264 where: 265 N = iw(1)1 FG (13b) 266 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.2 0.4 0.6 0.8 1
x/d
C/Co
i=0.1A/cm2 0.5 Present Predictions Okada et al. [9] 0.3Fig. 2. Comparison of present predictions with those of Okada et al.[15]
under conditions ofi = 0.1 A m−2,Ta= 60◦C,Tc= 60◦C and constant
cathode concentrationCc= 1.59 × 10−4mol m−3.
UNCORRECTED PROOF
Table 1
Physical parameters and corresponding values used in this work
Parameter Symbol Value
Constant term of water transference coefficient at anode side of membrane as expressed by a series expansion ofCH2O
w(0)a 0
Constant term of water transference coefficient at cathode side of membrane as expressed by a series expansion ofCH2O
w(0)c 0
First order term of water transfer coefficient at anode side of membrane as expressed by a series expansionCH2O
w(1)a 1.28 × 10−4
First order term of water transfer coefficient at cathode side of membrane as expressed by a series expansionCH2O
w(1)c 1.28 × 10−4
Current density (A cm−2) i 0–3.1
Coefficient characterizing water flux into anode side of membrane νa 0–1.0
Coefficient characterizing water flux into cathode side of membrane νc 0–1.0
Specific conductivity at anode side of membrane (cm s−1) κa 1× 10−3 to 1
Specific conductivity at cathode side of membrane (cm s−1) κc 1× 10−3 to 1
Thickness of membrane (cm) d l× 10−2
Thermal conductivity of membrane (W cm−1K−1) Km 0.0014
Specific heat of liquid water (J kg−1K−1) Cp,l 4180
Faraday constant (A s mol−1) F 96487
Molecular weight (kg mol−1) M 0.018
Ohmic resistance per unit length ( cm−1) R 0.000945
0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1
C/Co
x/d
60 70 80 90 Ta=100oC 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1C/Co
x/d
Ta=100oC 60 (a) (b)Fig. 3. Water concentration distribution in membrane withi = 0.1 A cm−2,Tc= 100◦C, and constant cathode concentrationCc= 1.59 × 10−4mol cm−3:
(a) constant diffusion coefficient; (b) variable diffusion coefficient.
UNCORRECTED PROOF
2.4. Boundary conditions 267
To solve the governing equations formulated in the
268
last section, the following boundary conditions are
speci-269
fied.
270
2.4.1. Concentration conditions at anode-membrane 271
interface 272
At the anode-side membrane interface, the condition of
273
water-flux balance[9–12]is imposed, namely:
274 275 νai F + κa[Ca− CH2O(0)] 276 = −D(0)a ∂CH2O(0) ∂x + i F[w(0)a + w(1)a CH2O(0)] (14) 277 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Tc=60oC 100 70 80 90 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Tc=60oC 100 70 80 90 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Tc=60oC 100 70 80 90 (a) (b) (c)
Fig. 4. Effect of Tc on water concentration distribution withi = 0.1 A
cm−2and constant cathode concentrationCc= 1.59 × 104mol cm−3: (a)
Ta= 60◦C; (b)Ta= 80◦C; (c)Ta= 100◦C.
where:νais a factor expressing the rate of water entry at the 278
anode side of the membrane and is proportional to the cur- 279
rent density;κais a factor characterizing the concentration- 280
gradient-driven water flux into or out of the membrane; Ca 281
is the concentration of water at the anode-membrane inter- 282
face;CH2O(0) is the water concentration in the membrane 283
atx = 0. 284
2.4.2. Concentration conditions at membrane-cathode 285
interface 286
Two types of boundary condition for the water concen- 287
tration at the membrane–cathode interface are studied. One 288
is the constant water concentration: 289
CH2O(d) = C0 (15) 290 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 x/d C/C(d) Tc=60oC 100 70 80 90 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 x/d C/C(d) Tc=60oC 100 70 80 90 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 x/d C/C(d) Tc=60oC 100 70 80 90 (a) (b) (c)
Fig. 5. Effect of Tc on water concentration distribution withi = 0.1 A
cm−2 and water-flux condition at cathode side: (a) Ta = 60◦C; (b)
Ta= 80◦C; (c)Ta= 100◦C.
UNCORRECTED PROOF
the other is a water-flux condition:
291 292 νci F + κc[Cc− CH2O(d)] 293 = D(0) c ∂CH2O(d) ∂x − i F[w(0)c + w(1)c CH2O(d)] (16) 294
where νc is a factor expressing the rate of water entry at 295
cathode side of the membrane proportional to the current
296
density; CH2O(d) is the water concentration at x = d in
297
Eq. (16);D(0)c is the diffusion coefficient of water at cathode 298
side of the membrane.
299
2.4.3. Thermal conditions at anode and cathode sides 300
In this study, the thermal conditions at the anode and
301
cathode sides of the membrane are constant temperatures,
302
Ta and Tc, respectively, i.e., 303 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Ta=60oC 100 70 80 90 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Ta=60oC 100 70 80 90 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Ta=60oC 70 80 90 100 (a) (b) (c)
Fig. 6. Effect of Ta on water concentration distributions withi = 0.1 A
cm−2 and constant cathode concentration Cc= 1.59 × 10−4mol cm−3:
(a)Tc= 60◦C; (b)Tc= 80◦C; (c)Tc= 100◦C.
T(0) = Ta (17) 304
T(d) = Tc (18) 305
3. Numerical method 306
The system of the governing equations mentioned above 307 is non-linear and is difficult to obtain an analytical solution. 308 In this work, the control volume finite difference method is 309
adopted to solve the non-linear, coupled ordinary differential 310
equations. The detailed solution scheme has been published 311
elsewhere[19]. To check the grid independence, solutions on 312
various grid systems are examined. In the separate numerical 313
runs, it is found that there are no differences among the 314
solutions with three grid arrangements of 1000, 2000 and 315
0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Tc=60oC 100 80 70 90 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Tc=60oC 100 70 80 90 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Tc=60oC 100 70 80 90 (a) (b) (c)
Fig. 7. Effect of Tcon water concentration distribution withTa= 60◦C
and constant cathode concentration Cc = 1.59 × 10−4mol cm−3: (a)
i = 0.1 A cm−2; (b)i = 0.5 A cm−2; (c)i = 1.1 A cm−2.
UNCORRECTED PROOF
3000 points. In order minimize the calculating time, 1000
316
grids are adopted for the present problem. Additionally, it
317
is important to compare the predicted results with existing
318
numerical or experimental data. In the comparison shown in
319
Fig. 2, it is apparent that the present predictions agree well
320
with those of Okada et al. [9]. Through these preliminary
321
tests, it is found that the numerical method is suitable for
322
the present study.
323
4. Results and discussion
324
In Section 2, several parameters appear in the
formula-325
tion. The physical parameters and their corresponding
val-326
ues are presented in Table 1. To disclose the effects of
327
the temperature-dependent diffusion coefficient on the
wa-328 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Ta=60oC 100 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Ta=60oC 100 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Ta=60oC 100 (a) (b) (c)
Fig. 8. Effect of Ta on water concentration distribution withTc= 60◦C
and constant cathode concentration Cc = 1.59 × 104mol cm−3: (a)
i = 0.1 A cm−2; (b)i = 0.5 A cm−2; (c)i = 1.1 A cm−2.
ter concentration distribution,Fig. 3(a) and (b) shows, re- 329 spectively, the distribution of water concentration with or 330
without consideration of a variable diffusion coefficient. It 331
is seen that the water concentration increases with x/d. In 332
addition, a large water concentration is noted for a system 333
with a lower anode temperature Ta. It is also found that 334
these are noticeable differences between the results with or 335
without consideration of variable diffusion coefficient. This 336
implies that the effects of a variable diffusion coefficient on 337
the water content in the membrane are of importance. 338
For thermal and water management in PEMFCs, the ther- 339
mal effects of the anode and cathode temperatures (Ta and 340 Tc) on the water concentration in the membrane may be im- 341
portant. The effects of Taand Tcon the water concentration 342
at a current density i = 0.1 A cm−2 and a water concen- 343 tration on cathode side ofCc = 1.59 × 10−4mol cm−3are 344
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 x/d C/Co 1.1 0.3 0.5 0.7 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d 1.1 0.3 0.5 0.7 0.9 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 x/d C/Co 1.1 0.3 0.5 0.7 0.9 (a) (b) (c) i=0.1A/cm2 i=0.1A/cm2 i=0.1A/cm2
Fig. 9. Effect of current density i on water concentration distribution with Ta= 60◦C and constant cathode concentrationCc= 1.59×104mol cm−3:
(a)Tc= 60◦C; (b)Tc= 80◦C; (c)Tc= 100◦C.
UNCORRECTED PROOF
shown inFig. 4. That data show that the water
concentra-345
tion at the anode side of the membrane decreases with
in-346
crease in Ta. This can be explained by the fact that, as Ta 347
is increased, the diffusion coefficient becomes larger (see
348
Eq. (8a)). Therefore, water diffusion from the anode side of
349
the membrane is enhanced. This means that an increase in
350
Ta causes dehydration of the anode. At a fixed Ta, a higher 351
water concentration within the membrane can be found in a
352
system with a higher cathode temperature Tc due to strong 353
back-diffusion from the cathode to the anode.
354
The effects of cathode temperature on the water
concen-355
tration distribution with water-flux conditions are shown in
356
Fig. 5. As inFig. 4, three sub-plots with different anode
tem-357
peratures Taare presented. It is noteworthy that the dimen-358
sionless water concentration, C/C(d), is presented, where
359
C(d) is the water concentration at the cathode side of the 360 60 60.2 60.4 60.6 60.8 61 0 0.2 0.4 0.6 0.8 1 x/d T( o C) 0.1 1.1 60 60.5 61 61.5 62 0 0.2 0.4 0.6 0.8 1 x/d T( o C) 0.1 1.1 60 60.5 61 61.5 62 62.5 63 0 0.2 0.4 0.6 0.8 1 x/d T( o C) (a) (b) (c) i=2.1A/cm2 i=2.1A/cm2 i=2.1A/cm2 1.1 0.1
Fig. 10. Effect of current density i on temperature distribution with Ta= 60◦C, constant cathode concentration Cc= 1.59 × 104mol cm−3:
(a)Tc= 61◦C; (b)Tc= 62◦C; (c)Tc= 63◦C.
membrane. An overall inspection inFig. 5indicates that, for 361 water-flux conditions at the cathode side, the water concen- 362
tration increases with the normalized depth from the anode 363
side. In the region near the anode side (i.e., at small values 364
of x/d), a larger normalized water concentration, C/C(d), is 365
noted for a system with a lower Tc. By contrast, in the region 366
away from the anode side (i.e., at large values of x/d), C/C(d) 367
increases with an increase in Tc. In fact, the local water con- 368
centration, C(x), is a function of the operating temperatures, 369 Tcand Ta. As Tcis raised, membrane dehydration occurs at 370
the anode side, but hydration occurs at the cathode side. 371
The dependence of the water concentration profiles on 372
the temperature at cathode side of the membrane (Tc = 60 373
to 100◦C) is shown inFig. 6. Here the water concentration 374
at the cathode side of the membrane is kept constant. The 375 results show that at fixed Taa higher water concentration at 376
the anode side of the membrane is found in a system with 377
a higher Tc. This is due to the fact that increasing Tc will 378
markedly enhance the membrane hydration. That is, the back 379
diffusion of water to the anode side is significant at a high Tc. 380
In order to realize how the current density affects the wa- 381
ter content in the membrane,Fig. 7 presents the effects of 382
the current density i on the water concentration distribution 383
1.5 1.52 1.54 1.56 1.58 1.6 10-3 0.01 0.1 1 κa (cm/s) C(mol/cm 3 ) Ta=60oC 70 80 90 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 10-3 0.01 0.1 1 κa (cm/s) C(mol/cm 3 ) Ta=60oC 70 80 90 (a) (b) ( 104 ) ( 104) × ×
Fig. 11. Effect of humidification factor κa on water concentration
dis-tribution withTa = 60◦C, i = 0.1 A cm−2 and water-flux condition at
cathode side under Tc: (a) concentration at anode side; (b) concentration
at cathode side.
UNCORRECTED PROOF
with Ta = 60◦C and a constant cathode concentration of 384
Cc= 1.59×104mol cm−3. The influence of i on water
con-385
centration at the anode side is similar for different Tc. Care-386
ful inspection of the data shows that there is a smaller water
387
concentration at the anode side at a large current density.
388
This can be explained by noting that an increase in current
389
density causes the membrane to be seriously dehydrated due
390
to water drag by electro-osmosis. As for the results
men-391
tioned above, at a fixed x/d and i, the water concentration
392
increases with an increase in Tc. 393
The effect of Ta on the water concentration distribution 394
is shown in Fig. 8 with Ta = 60◦C and Cc = 1.59 × 395
104mol cm−3 under different i. The water concentration
396
profile has a parabolic form. As the current density is
in-397
creased, however, the deviation in the water concentration
398
distribution at different Ta becomes small. Therefore, the 399
temperature at the anode side, Ta has only a small impact 400
on the water concentration in the membrane at high current
401
density.
402
The influence of current density i on the water
concen-403
tration distributions at different anode operating
tempera-404
tures are presented in Fig. 9. By comparing the results in
405
Fig. 9(a), it is found that the anode side of the membrane
406 1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 0 0.2 0.4 0.6 0.8 1 ν C(mol/cm 3 ) Tc=60oC 100 70 80 90 a 1.62 1.64 1.66 1.68 1.7 1.72 1.74 1.76 0 0.2 0.4 0.6 0.8 1 ν Tc=60oC 100 C(mol/cm 3 ) a 70 80 90 ( 104) ( 104 ) × × (a) (b)
Fig. 12. Effect of humidification factorνaon water concentration
distri-bution withTa= 60◦C,i = 0.1 A cm−2 and water flux at cathode side
under different Tc: (a) concentration at anode side; (b) concentration at
cathode side.
tends to become dehydrated as the current density is raised. 407
This is because that the electro-osmotic drag effect becomes 408
stronger as the current density is higher. It is also found in 409
the separate numerical runs that the membrane is much wet- 410
ter for the system with a higher Tcthan that with a lower Tc. 411
This is due to the temperature-dependence of the diffusion 412
coefficient. 413
The relationship between the current density and the tem- 414
perature distribution is shown inFig. 10. It is clearly shown 415
inFig. 10(a)that when the current density is raised, the tem- 416
perature changes sharply at the anode side of membrane. For 417
example, when it is necessary to speed up a car, the current 418
density must go up. This will cause dehydration of the mem- 419
brane, which, in turns, causes the temperature to increase 420
and become more non-uniform. Thermal expansion of the 421
membrane may become serious and lead to the breakdown 422
of the membrane. Therefore, the strength of the membrane 423
is a key factor for fuel cells operating under high current 424
density conditions. 425
The effect of the humidification parameterκaon the wa- 426
ter concentration at the anode and cathode sides withTa = 427
60◦C and i = 0.1 A cm−2 are shown in Fig. 11. When 428
κa is increased, water vapour enters the membrane more 429
freely from the anode gas-diffusion electrode through the 430
anode-membrane interface which, in turn, results in an in- 431
crease in the water content. A careful inspection ofFig. 11 432
indicates that the water concentration changes sharply when 433
κa is increased from 10−3 to 10−1cm s−1. But, for κa > 434
10−1, the effect ofκaon the water content in the membrane 435
is insignificant. 436
The influence of the parameters of the electro-osmotic 437
drag at anode side (νa) on the water concentration at the an- 438
ode and cathode sides is presented inFig. 12. It is observed 439
that the water concentration increases linearly with increase 440
inνa. Whenνais increased, the water enters easily the mem- 441
brane from the anode gas-diffusion electrode through the 442
anode–membrane interface and thus results in an increase 443
in the water content within the membrane. 444
5. Conclusions 445
A detailed analysis of the thermal and water manage- 446
ment in the PEMFC membrane with coupling effects of 447
mass diffusion and temperature gradient have been per- 448
formed by using a one-dimensional mathematical model. 449
The thermal–mass diffusion coupling effects are taken into 450
account with consideration of the temperature-dependent 451
diffusivity. The model can predict the water distribution in 452
the membrane under different operating conditions. This is 453
useful for selecting the optimal membrane material and es- 454
timating the gas-inlet temperature or working density in de- 455
signing a PEMFC. The major findings in this study are sum- 456
marized as follows. 457
(i) Increasing the temperature at the anode side of the 458
membrane can cause dehydration of the membrane. 459
UNCORRECTED PROOF
(ii) Increasing the current density will increase dehydration
460
of the anode side of the membrane. This is attributed
461
to the strong electro-osmotic drag effect under the
op-462
erating conditions of high current density.
463
(iii) At high current density, the temperature effect on the
464
water concentration becomes smaller. The current
den-465
sity effect dominates the water concentration
distribu-466
tion.
467
(iv) Temperature distribution changes sharply in the
mem-468
brane at high current densities. This can damage the
469
membrane.
470
(v) Increasing the humidification factor κa augments the 471
water concentration at both the anode and the
cath-472
ode sides of the membrane. Never the less, increase in
473
κa above 10−1cm s−1has little influence on the water 474
concentration.
475
(vi) At fixed current density, the effects of the parameters
476
of electro-osmotic drag, ν, on the water concentration
477
is considerable. The dependence of the water content
478
onν is almost linear.
479
Acknowledgements
480
The authors are grateful for financial support from the
481
National Science Council of Taiwan, NSC 92-2212-E-21
482
1-001 and NSC 92-2623-7-002-006-ET.
483
References 484
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![Fig. 2. Comparison of present predictions with those of Okada et al. [15]](https://thumb-ap.123doks.com/thumbv2/9libinfo/8839829.238532/4.892.468.859.104.242/fig-comparison-present-predictions-okada-et-al.webp)
