UNCORRECTED PROOF
3
Convenient two-dimensional model for design of fuel channels
4
for proton exchange membrane fuel cells
5
Falin Chen
a,∗, Ying-Zhi Wen
a, Hsin-Sen Chu
b, Wei-Mon Yan
c, Chyi-Yeou Soong
d6
aInstitute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan, ROC 7
bDepartment of Mechanical Engineering, National Chiao Tung University, Hsin-Chu 300, Taiwan, ROC 8
cDepartment of Mechatronic Engineering, Huafan University, Shih-Ting, Taipei 22305, Taiwan, ROC 9
dDepartment of Aeronautical Engineering, Feng Chia University, Seatwen, Taichung 40745, Taiwan, ROC 10
Received 6 August 2003; accepted 10 October 2003 11
Abstract 12
A theoretical, two-dimensional, along-the-channel model has been developed to design fuel channels for proton exchange membrane (PEM) fuel cells. This has been implemented by solving the resultant ordinary differential equation with a straightforward shooting computational scheme. With such a design tool, an analysis can be made of the effects due to some operation and design parameters, such as inlet velocity, inlet pressure, catalyst activity, height of channel, and porosity of gas-diffusion layer to obtain a fuel cell with high performance. Present results indicate that there is always a trade-off between higher power density and higher efficiency of the fuel cell. Namely, a design for higher power density (a better performance) is always accompanied with a higher fuel efficiency (or a larger fuel consumption rate and a higher fuel cost), and vice versa. When some relevant physical parameters are determined experimentally and applied in the present model, a quantitative design for a fuel cell of high efficiency or performance is feasible.
13 14 15 16 17 18 19 20 © 2003 Published by Elsevier B.V. 21
Keywords: Fuel channels; Proton exchange membrane fuel cells; Gas-diffusion layer 22
1. Introduction
23
The proton exchange membrane (PEM) fuel cell is an 24
electrochemical device which combines fuel (hydrogen) and 25
oxidant (oxygen) to produce electricity, and water and heat 26
are the major by-products. In past decades, substantial ef-27
forts[1–4]have been devoted to reducing the cost as well as 28
promoting the efficiency of the fuel cell. In this respect, the 29
design of high-efficiency fuel channels is one of the impor-30
tant issues [5–19]. Several different types of fuel channels 31
have been used in practical designs, such as straight chan-32
nels, surpentine channels, and interdigitate channels. The 33
morphology of the channel also varies, namely, meander, 34
spiral or straight types. For a meander or a spiral channel, 35
the length may be several meters long. In such a long chan-36
nel, the fuel may be entirely consumed before exit, which 37
implies that a certain portion of the cell may not have fuel 38
for chemical reactions. That is, a part of the cell may not 39
produce any electrons during operation and this reduces the 40
efficiency. Therefore, analysis of the channel flow becomes ∗Corresponding author. Tel.:+886-2-363-0979; fax: +886-2-363-9290.
E-mail address: [email protected] (F. Chen).
a necessity in fuel cell design and an efficient and conve- 41 nient theoretical model for channel analysis is essential. 42 To date, there have been two major approaches for the 43 analysis of channel flow. One uses computational fluid 44 dynamic (CFD) techniques to examine the two- or three- 45 dimensional flow in fuel channels [5–10], the other uses 46 a one-dimensional approximation approach to investi- 47 gate the variation of flow structure and includes the fuel 48 concentration as well as the current generated along the 49 channel [11–19]. In the CFD approaches, mathematical 50 models are usually developed for the whole PEM fuel cell, 51 which may consider the conservation of mass, momen- 52 tum and energy, equations governing the electrochemical 53 reaction, and various kinds of physical properties of the 54 components such as diffusivity of the gas-diffusion layer, 55 electro-osmosis in membrane, fuel convection across the 56 membrane, chemical reaction and activity in catalyst lay- 57 ers, and membrane hydration[20–23]. With this approach, 58 however, the computation is rather time-consuming and the 59 analysis procedure is so tedious that the computation of 60 the whole flow field in channels becomes inefficient and is 61 sometimes even an unnecessary step to obtain the fuel cell 62
design. 63
1 0378-7753/$ – see front matter © 2003 Published by Elsevier B.V. 2 doi:10.1016/j.jpowsour.2003.10.003
UNCORRECTED PROOF
Nomenclature
˜c dimensionless concentration
C concentration (mol m−3)
C0 inlet concentration (mol m−3)
F Faraday constant (96,500 C mol−1)
h channel height (m)
i local current density (A m−2)
i0 inlet current density (A m−2)
Ki i = 1–5, constants inEq. (15)
L channel length (m)
M molar weight (kg mol−3)
n transferred electron number
P pressure (Pa)
P0 inlet pressure (atm)
¯P pressure gradient (Pa m−1)
R universal gas constant (8.314 J mol−1K−1)
T gas temperature (K)
u velocity in x-direction (m s−1)
u0 inlet velocity in x-direction (m s−1)
¯u cross-sectional averaged velocity in x-direction
˜u dimensionless velocity in x-direction v velocity in y-direction (m s−1)
V0 suction velocity (m s−1)
Greek letters
α charge transfer coefficient γ reaction order
η over potential (V) µ viscosity (kg m−1s−1)
ξ empirical constant regarding the slip condition at the porous boundary
ρ density (kg m−3)
A more efficient and convenient scheme, which can to 64
some extent reach the goal of fuel cell design both quantita-65
tively and qualitatively, is the channel model[11–19]. Since 66
the flow passes the fuel channel very rapidly, cross-sectional 67
variations of flow structure and other physical parameters 68
such as fuel concentration, fuel density and fuel tempera-69
ture can be ignored. A major concern is the variation of 70
the relevant physical properties along the channel. It was 71
found that the along-the-channel model is much simplified 72
and that the equations can be solved more conveniently and 73
efficiently. Recently, several along-the-channel models have 74
been developed for the above-mentioned purpose. Nguyen 75
and co-workers [10,11] have proposed a set of governing 76
equations which include water and energy transport across 77
the membrane and have considered heat removal along the 78
channel. Other workers[12–14]have examined the gas dy-79
namics, concentration decay and current drop along the 80
channel. Argyropoulos et al.[15–19] have investigated the 81
pressure drop and the temperature variation along the chan-82
nel by considering the mass and energy conservation of 83
two-phase flows. In these studies, however, the channel flow 84 was assumed to have a constant velocity along the channel. 85 Under most circumstances, however, this assumption may 86 result in significant discrepancies with reality. 87 In the present work, a two-dimensional theoretical model 88 is developed and the includes a continuity equation, momen- 89 tum equations and the Tafel equation. At the bottom of the 90 channel, the consumption of fuel due to chemical reactions 91 is modeled by way of sucking the fuel through the porous 92 boundary. This model allows investigation of the variations 93 of the flow structure, the fuel concentration, and the current 94 density along the channel. A systematic parametric study 95 is implemented to examine the influence of relevant design 96 and operation parameters on the effective length of the fuel 97 channel. Specifically, the effects due to the inlet velocity 98 and pressure and the porosity of the gas-diffusion layer at 99 the bottom can be investigated. The activity of the catalyst 100 attached to the gas diffusion layer can also be analyzed. 101
2. The theoretical model 102 Two-dimensional horizontal channel flow shown schemat- 103 ically inFig. 1. The channel has a constant height and is 104 sufficiently long that the conditions at both the entrance and 105 the exit do not affect the flow of the domain under consid- 106 eration. The fluid, either hydrogen in the anode or oxygen 107 in the cathode, is assumed to be an ideal gas. Since the flow 108 velocity is so high, the variation of temperature along the 109 channel is assumed to be negligible. It is also assumed that 110 the bipolar plate in which the fuel channel is built is an ideal 111 collecting electrode with no ohmic loss. The overpotential 112 can then be maintained at a constant value along the chan- 113 nel[12]. As a result, the local current density is a function 114 of the fuel concentration at each specific position[7]. For 115 such a chemically active flow, the continuity equation is: 116
∂ρ ∂t + ∂(ρu) ∂x + ∂(ρv) ∂y = 0, (1) 117
in which u = u(x, y) and v = v(x, y) are the velocity in 118 the x-and y-directions, respectively. Due to the fact that the 119 channel height is very small compared with the channel 120 length, it may be assumed that the density distribution across 121
Fig. 1. Schematic description of flow considered. Two-dimensional viscous flow is bounded between bipolar plate (solid wall) at top and gas-diffusion layer (porous wall) at bottom. Velocity profile is essentially a parabolic curve. Because of the porous wall there is a velocity slip at bottom. Because of chemical reaction in the gas-diffusion layer, there is a suction flow across the porous wall, accounted for by suction velocity V0(x).
UNCORRECTED PROOF
the channel is uniform, i.e., ρ = ρ(x). Because ρ(x) =122
MC(x),Eq. (1)can be converted to: 123 ∂(Cu) ∂x + ∂(Cv) ∂y = 0. (2) 124
Integrating Eq. (2) along y and applying Leibnitz’s rule 125 gives: 126 C ∂ ∂x h 0 u dy +dC dx h 0 u dy + C h 0 ∂v ∂ydy = 0. (3) 127
By letting¯u(x) ≡ (1/h)0hu(x, y)dy be the averaged veloc-128
ity over the cross-section of the channel and by applying 129
the non-slip condition at the top u(x, h) = 0 and the slip 130
condition at the bottomu(x, 0) = ξ ¯u(x), we obtain: 131
hd[C(x) ¯u(x)]
dx = C(x) v(x, 0) = −C(x) V0(x), (4) 132
where V0is the suction velocity along the bottom of channel, 133
which is defined asV0= (1/nF)(i(x)/C(x)). Thus,Eq. (4) 134 becomes: 135 d[C(x) ¯u(x)] dx = − 1 h i(x) nF. (5) 136
The relation between the overpotential η and the electric 137
current density i is governed by the Tafel equation: 138 i(x) = i0 C(x) C0 γ exp αF RTη , (6) 139
so that the suction velocity V0can be expressed by: 140 V0(x) = 1 nF i0 Cγ0exp αF RTη Cγ−1(x). (7) 141
Note that, the slip boundary condition at the bot-142
tom u(x, 0) = ξ ¯u(x) is derived from the so-called 143
Beavers–Joseph boundary condition at the interface be-144
tween a fluid and a porous layer [24], and the parameter 145
ξ essentially accounts for the permeability of the porous
146
media since the factor due to the velocity gradient at the 147
bottom is absorbed into the averaged velocity ¯u(x). Given 148
that the velocity gradient does not change significantly 149
along the channel, a change inξ may be seen as a change in 150
the permeability or, equivalently, a change in the porosity 151
of the porous medium below. In the present study, it con-152
sidered thatξ varies from 0.1 to 1 [24]due to the fact that 153
the porosity of the gas-diffusion layer at the bottom is high. 154
For the momentum equation, it is assumed that, for the 155
present two-dimensional channel flow, the velocity in the 156
x-direction is much larger than that in the y-direction. After
157
applying order analysis on the momentum equations (or the 158
Navier–Stokes equations) in both x-and y-directions, a single 159
momentum equation results, as follows: 160 ρ(x) u∂u ∂x + v ∂u ∂y = −dP dx + µ ∂2u ∂y2. (8) 161
The pressure gradient along the channel is assumed to be 162
constant, i.e.,(dP/dx) = constant = ¯P. Under the normal 163
operating conditions of a 1 kW fuel cell, ¯P = 12 Pa m−1 164 [2,12–14], which is to be used in the present analysis. Inte- 165
gratingEq. (8)along the height yields: 166
MC h 0 u∂u ∂x + v ∂u ∂y dy = −h ¯P + µ ∂u ∂y h 0 . (9) 167
To simplifyEq. (9) further, it is assumed that u(x, y) is a 168 quasi-parabolic velocity profile [25] defined as u(x, y) = 169
A(x)y2+ B(x)y + G(x). At y = 0, u(x, 0) = ξ ¯u(x) and 170 this leads to G(x) = ξ ¯u(x); at y = h, u(x, y) = 0 and 171
B(x) = −ha(x) − (ξ/h)¯u(x). To obtain A(x), the non-slip 172 boundary condition is applied at the top u(x, h) = 0 and 173 this results inA(x) = −(6/h2)(1 − (1/2)ξ)¯u(x). As a re- 174 sult, the approximated velocity function in x-direction is 175
obtained as: 176177 u(x, y) = ¯u(x) −6 1−1 2ξ y2 h2 + 2(3 − 2ξ) y h+ ξ . 178 (10) 179 This equation implies that the horizontal velocity is a 180 second-order parabolic function of y, with a small slip at 181 the bottom of channel, as shown schematically in Fig. 1. 182 To obtainv(x, h), Eq. (10)is substitute inEq. (8)and the 183 resultant equation is integrated along y and the bound- 184 ary conditions ofv are applied at the top and the bottom 185 of the channel, i.e., v(x, h) = 0 and v(x, 0) = V0(x), 186
yielding: 187188 v(x, y) =[C(x) ¯u(x)] C(x) 2 1−1 2ξ y2 h2 − (3 − 2ξ) y2 h − ξy 189 − V0(x). (11) 190
Eqs. (10) and (11) are substituted in Eq. (9) to give the 191
following equation for the channel flow: 192193
2 15ξ 2−1 5ξ + 6 5 d[C(x) ¯u2(x)] dx + ξ hC(x) ¯u(x) V0(x) 194 + ¯P M + 12 1−1 2ξ µ Mh2¯u(x) = 0. (12) 195 Eqs. (5), (6) and (12) are the governing equations for the 196 two-dimensional flow along the channel, in which the vari- 197 ations of the fuel concentration C(x), the velocity u(x) and 198 the current density i(x) are to be solved. Since these equa- 199 tions have an initial value problem, the initial velocity at the 200 entrance, u0, and the initial concentration fed to the channel, 201
C0, are required. It is therefore assumed that˜u(x) = ¯u(x)/u0 202 and ¯c(x) = C(x)/C0 and two relationships are substituted 203
these into the equations to give: 204
d(˜c˜u) dx + K1˜c γ= 0, (13) 205 K2 d(˜c˜u2) dx + K3+ K4˜c γ˜u + K 5˜u = 0, (14) 206 POWER 5756 1–10
UNCORRECTED PROOF
where 207 K1= i0 nFhC0u0 exp αF RTη , K2=152ξ2−15ξ +65, K3= ¯P MC0u20 , K4= ξ i0 nFhC0u0 exp αF RTη , K5= 12 1−1 2ξ µ Mh2C 0u0. (15) 208The initial conditions become ˜u = 1 and ˜c = 1. These two 209
equations are solved by a fourth order Runge–Kutta scheme. 210
Eqs. (13) and (14) are the two simplified equations for the 211
averaged horizontal velocity ˜u and averaged fuel concentra-212
tion ˜c. After obtaining these two values, the Tafel equation 213
(Eq. (6)) can be applied to obtain the local current den-214
sity along the channel. The combination of these equations 215
and the shooting scheme becomes a convenient tool to de-216
termine various physical parameters relevant to the design 217
of fuel channels of PEM fuel cells. This scheme is differ-218
ent from those developed in other studies[11–19]in which 219
the velocity variations in both x- and y-directions were ig-220
nored and only the effective length was considered. Using 221
this simplified model, allows not only examination of the 222
effective length of the fuel channel under various operation 223
conditions, but also the effects due to relevant design and 224
operation parameters on the fuel cell performance. 225
3. Typical flow field: an example
226
Analyses in the present and the following sections are 227
made on the basis of the base case; the values of its relevant 228
physical parameters are shown inTable 1. This case essen-229
tially corresponds to a 1 kW PEM fuel cell under normal 230
operation conditions[2,12–14]. The results in terms of the 231
variations of velocity, fuel concentration and current den-232
sity along the channel are presented in Fig. 2. It is found 233
that the fuel concentration decays monotonically along the 234
channel due to the chemical reaction occurring at the bot-235
tom of channel, while the flow velocity increases along the 236
channel because of the depletion of fuel downstream. Both 237
effects are the direct consequence of fuel consumption along 238
the channel, which in turn leads to a decrease in generated 239
current (Fig. 2(b)) because of, again, the depletion of fuel 240
downstream. These findings suggest that variations of these 241
physical properties occur simultaneously, and therefore will 242
be considered as a whole instead of separately as in previ-243
ous studies [11–19] Also the velocity is taken as constant 244
while the fuel concentration changes along the channel. 245
The generation of current density decreases along the 246
channel because of the decay in fuel concentration (see 247
Fig. 2(b)). Note that, for the present base case, the fuel con-248
Table 1
Values of physical parameters of base case corresponding to 1 kW PEM fuel cell[2,12–14].
Channel side Cathode
Gas flow Oxygen
Half-reaction O2(g)+ 4H+
+ 4e− → 2H2O
Channel temperature, T (K) 353.15
Inlet gas velocity, u0(m s−1) 0.1
Inlet gas pressure, P0 (atm) 2
Inlet gas concentration, C0 (mol m−3) 69.00 Exchange current density, i0 (A m−2) 10−5
Activation overpotential,η (V) 0.3
Reaction order,γ 0.5
Electrons transferred in reaction, n 4
Charge-transfer coefficient,α 2.0
Molar weight, M (kg mol−1) 32× 10−3
Viscosity,µ (kg m s−1) 2× 10−5
Channel height, h (m) 10−3
Channel length, L (m) 0.1
Pressure gradient, ¯P (Pa m−1) 10 Universal gas constant, R (J mol−1K−1) 8.314
Faraday constant, F (C mol−1) 96500
Slip velocity fraction,ξ 0.1
K1 (m−1) 13.77
K2 1.181
K3 (m−1) 452.9
K4 (m−1) 1.377
K5 (m−1) 1032.6
centration decrease by about 50% and the local current den- 249 sity by about 25% within the 10 cm long channel. The vari- 250 ations in these two parameters are very significant for such 251 a short distance, and imply that an accurate calculation of 252 the effective channel length cannot be ignored in the design 253 of a high-efficiency fuel cell. In a practical sense, too long a 254 channel will result in a large dead region in the bipolar plate, 255
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 2 4 6 8 x (cm) 10 (a) u~ c u ~~ c~ 0.26 0.28 0.36 0.38 0 2 4 6 8 x (cm) (b) 0.3 0.32 0.34 i (A cm-2) 10
Fig. 2. Along-the-channel variations of major quantities of present prob-lem: (a) dimensionless velocity˜u and concentration ˜c; (b) generated cur-rent density. Results are calculated on basis of base case shown inTable 1.
UNCORRECTED PROOF
while too short a channel will cause a great part of the fuel256
to leave the cell without reaction. A long channel gives a low 257
local current density downstream (or a low performance of 258
the fuel cell) but, with respect of fuel consumption, a long 259
channel length may ensure that the fuel is consumed before 260
leaving the exit. The latter will result in better efficiency of 261
gas usage. 262
4. Physical parameter effects
263
Several physical parameters have significant effects on the 264
channel flow, which may in turn affect both the design and 265
operation conditions of the fuel cell. These physical param-266 1.00 1.20 1.40 1.60 1.80 2.00 (a) u0=0.07 ms-1
u~
0.1 0.2 0.5 1.0 0.50 0.60 0.70 0.80 0.90 1.00 (b) 1.0 0.5 0.2c~
0.1 u0=0.07 ms-1 0.26 0.28 0.30 0.32 0.34 0.36 0.38 (c) 1.0 0.5i
(A cm-2) 0.2 0.1 u0=0.07 ms-1 2 4 6 0 8 10 x (cm)Fig. 3. Effects of inlet velocity on along-the-channel variations of: (a) dimensionless velocity˜u; (b) dimensionless concentration ˜c; (c) local current density. eters include the flow velocity at the inlet, the fuel concen- 267 tration, the activity of the catalyst, the channel height, and 268 the porosity of the gas-diffusion layer. The following exam- 269 ines the physical effects due to these five parameters with 270 special attention to their influence on the effective length of 271 the channel. When the effect of a particular physical param- 272 eter is considered, this parameter will be change systemat- 273 ically while the other four physical parameters are fixed at 274
the values shown inTable 1. 275
4.1. Effect of inlet velocity 276 The variation of flow velocity, fuel concentration and local 277 current density along the channel are shown inFig. 3(a)–(c),
UNCORRECTED PROOF
respectively, for variation in inlet velocity from 0.07 to278
1 m s−1. For a lower inlet velocity, because the fuel has more 279
time to diffuse into the active layer at the bottom to react, 280
the fuel concentration decays more rapidly (Fig. 3(b)). De-281
crease in fuel concentration leads to an increase in flow ve-282
locity in accordance with the conservation of mass (Fig. 3(a)) 283
and results in a larger decrease in the local current density 284
due to the higher depletion of fuel or a lower reaction rate 285
(Fig. 3(c)). It is interesting to note that, for an inlet veloc-286
ity of 1 m s−1, the changes in flow velocity, fuel concentra-287
tion and local current density along the channel are all very 288
small. As a result, practically, if a uniformly high current 289 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 (a) P0=1.5 atm
u~
2.0 3.0 5.0 0.40 0.50 0.60 0.70 0.80 0.90 1.00 (b) 5.0 3.0c~
2.0 P0=1.5 atm (c) 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 5.0i
(A cm-2) 3.0 2.0 P0=1.5 atm 2 4 6 0 x (cm) 8 10Fig. 4. Effects of inlet fuel pressure on along-the-channel variations of: (a) dimensionless velocity ˜u; (b) dimensionless concentration ˜c; (c) local current density.
density along the channel (and thus a higher power density) 290 is required, then a larger inlet fuel velocity can be applied 291 to the cell, but at the expense of a higher fuel consumption 292 rate. On the other hand, if the fuel efficiency is the major 293 concern, it is necessary to apply a lower inlet velocity so 294 that the fuel consumption (and thus the reaction) along the 295
channel can be implemented more completely. 296
4.2. Effect of fuel concentration 297 From the equation of state of an ideal gas, i.e., C0 = 298
P0/RT, the fuel concentration is a function of the partial
UNCORRECTED PROOF
pressure of the fuel gas. Accordingly, the change in fuelcon-299
centration can be attributed to the change in partial pressure 300
of the fuel. Thus, to examine the effects of fuel concentra-301
tion, the inlet pressure P0of the fuel was changed from 1.5 302
to 5 atm. The results are presented inFig. 4(a)–(c)and show 303
that for a higher inlet pressure (or a higher fuel concentra-304
tion), and thus a larger pressure gradient along the channel 305
when the pressure at outlet is assumed to be constant, the 306
increase in flow velocity along the channel is smaller (but 307
the overall velocity is higher) (Fig. 4(a)) because the con-308
sumption rate of the fuel along the channel is smaller (and 309
the decrease of the fuel concentration along the channel is 310
smaller) (seeFig. 4(b)). Due to the smaller fuel consumption 311 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 (a)
u~
i0=10-4 Am-2, h h h h h h h h h =0.28 V i0=10-5 Am-2, =0.30 V i0=10-6 Am-2, =0.32 V 0.50 0.60 0.70 0.80 0.90 1.00 (b) i0=10-4 Am-2, =0.28 V i0=10-5 Am-2, =0.30 V i0=10-6 Am-2, =0.32 Vc~
(c) 0.00 0.20 0.60 0.80 1.00 i0=10-4 Am-2, =0.28 V 0.40 (A cmi
-2) i0=10-5 Am-2, =0.30 V i0=10-6 Am-2, =0.32 V 2 4 6 0 x (cm) 8 10Fig. 5. Effects imposed by different catalysts on along-the-channel variations of: (a) dimensionless velocity ˜u; (b) dimensionless concentration ˜c; (c) local current density.
rate, the decrease in current density along the channel is also 312 smaller. As a result, to ensure that the current density along 313 the channel can be uniformly high, or equivalently to have a 314 fuel cell of higher power density, the cell must be supplied 315 with a fuel of higher concentration (a higher inlet pressure) 316 but, again, at the expense of a higher fuel consumption rate. 317
4.3. Effect of catalyst activity 318 The activity of the catalyst can be indicated by the value of 319 the overpotentialη, which can be converted into the current 320 density i by the Tafel equation (Eq. (6)). The present analy- 321 sis considers three different activities, corresponding to three
UNCORRECTED PROOF
overpotentialsη = 0.28, 0.30 and 0.32 V. The corresponding322
current densities are i0= 10−4, 10−5and 10−6A cm−2. A 323
catalyst of higher activity leads to a faster reaction rate and 324
thus a faster speed of electron generation and, accordingly 325
corresponds to a lower overpotential loss and a higher cur-326
rent density. Note that, a different catalyst activity implies 327
a different fuel consumption rate, which can be reflected by 328
the change in suction velocity V0at the bottom of the chan-329
nel (seeEq. (7)). 330
The results shown inFig. 5(a)–(c)illustrate that the differ-331
ence in activity of the catalyst examined in the present study 332
does not result in an obvious difference in either the flow 333
velocity (Fig. 5(a)) or the fuel concentration (Fig. 5(b)), be-334
cause the curves of different activities virtually overlap each 335
other. On the other hand, it has a significant effect on the lo-336 1.00 2.00 3.00 4.00 5.00 6.00 0.00 0.20 0.40 0.60 0.80 1.00 0.15 0.20 0.25 0.30 0.35 0.40 (a) h=0.50 mm
u~
0.75 1.00 1.25 1.50 (b) 1.50 1.25 1.00c~
0.75 h=0.50 mm (c) 1.50 1.25 1.00i
(A cm-2) 0.75 h=0.50 mm 2 4 6 0 x (cm) 8 10Fig. 6. Effects of channel height on along-the-channel variations of: (a) dimensionless velocity ˜u; (b) dimensionless concentration ˜c; (c) local current density.
cal current density, as shown inFig. 5(c), namely: a higher 337 catalyst activity leads to a higher local current density and 338 results in a slightly more rapid decrease along the channel. 339 This indicates that an efficient scheme to raise the fuel cell 340 performance without consuming more fuel is to use a cata- 341 lyst of higher activity. This is a common scenario accepted 342 by fuel cell researchers, world-wide. Accordingly, the de- 343 velopment of a high-activity catalyst for PEM fuel cells has 344 been a major research issue, or may be the most important 345
one. 346
4.4. Effect of channel height 347 Since it is assumed that the fuel is well mixed across the 348 channel height, a larger height of channel is equivalent to
UNCORRECTED PROOF
a greater mass of the fuel per unit length, or a moresuffi-349
cient supply of the fuel. Five different heights of channel are 350
considered, viz.,h = 0.5, 0.75, 1.0, 1.25 and 1.5 mm; the 351
results are shown inFig. 6(a)–(c). It is seen that a smaller 352
height of channel leads to a larger increase in flow velocity 353
and a larger decrease in fuel concentration along the chan-354
nel. Alternatively, small height of channel results in a larger 355
decrease of local current density along the channel and thus 356
a smaller power density (or a worse performance) for the 357
fuel cell. Note that, since both the flow velocity and the fuel 358
concentration have been normalized by the corresponding 359
inlet conditions, a larger decrease of fact concentration in 360
fact accounts for a smaller amount of fuel consumed along 361
the channel. According to these results, the application can 362
be made as follows. To have a fuel cell of higher fuel ef-363
ficiency, it is necessary to have a larger height of channel 364
because the fuel can be consumed more efficiently. Corre-365
spondingly, to have a fuel cell of higher power density, it is 366 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 0.50 0.60 0.70 0.80 0.90 1.00 0.26 0.28 0.30 0.32 0.34 0.36 0.38
u~
c~
(a) ξ=0.1 0.2 0.3 0.5 1.0 (b) 1.0 0.5 0.3 0.2 ξ=0.1 (c) 1.0i
(A cm-2) 0.5 0.3 0.2 ξ=0.1 2 4 6 0 x (cm) 8 10Fig. 7. Effects of porosity of gas-diffusion layer accounted for by parameter ξ on along-the-channel variations of: (a) dimensionless velocity ˜u; (b) dimensionless concentration ˜c; (c) local current density.
necessary to have height of channel so that a larger amount 367
of fuel can be supplied. 368
4.5. Effect of porosity of gas-diffusion layer 369 The change in porosity of the gas-diffusion layer can be 370 reflected in the present model by changing the parameterξ 371 at the bottom boundary. Physically, as mentioned above, a 372 largerξ means a larger porosity of the gas-diffusion layer. 373 Five cases are considered, namely,ξ = 0.1, 0.2, 0.3, 0.5 and 374 1.0. The results show that a largerξ (or a larger porosity) 375 results in a smaller increase in flow velocity (Fig. 7(a)) as 376 well as a smaller decrease in fuel concentration (Fig. 7(b)) 377 because a larger portion of fuel is allowed to be sucked into 378 the porous boundary at the bottom due to the larger poros- 379 ity. Accordingly, a greater chemical reaction occurs in the 380 gas-diffusion layer and leads to a uniformly high current den- 381 sity along the channel (Fig. 7(c)). Accordingly, as discussed
UNCORRECTED PROOF
in above, a fuel cell with higher power density requires a382
larger porosity of the gas-diffusion layer to maintain a uni-383
formly high current density along the channel. This conclu-384
sion is consistent with previous analyses of the optimization 385
of the porosity of the gas-diffusion layer ([26],and references 386
wherein)] which show that a higher porosity gives better per-387
formance of the fuel cell because in the gas-diffusion layer 388
a larger space (or pores) is given to the water transportation 389
so that the flooding can be prevented in high power density 390
regimes. Thus, a larger porosity of the gas-diffusion layer is 391
preferred for a fuel cell with a high power density. 392
5. Concluding remarks
393
A theoretical model for two-dimensional flow in fuel 394
channels has been developed. This allows calculation of 395
variations in fuel velocity, fuel concentration, and current 396
density along the channel. The resultant ordinary equations 397
and the initial conditions at the inlet of the channel consist 398
of an initial value problem, which can be easily solved by 399
a straightforward shooting scheme. The combination of the 400
simplified equation model and the popular shooting scheme 401
becomes a convenient, as well as an efficient, scheme for 402
the design of fuel cell channels, such that several design and 403
operation parameters can be determined for a fuel cell with 404
either high power density or high fuel efficiency. 405
To have a fuel cell of high power density, results obtained 406
in the present study suggest that it is necessary to: (1) in-407
crease the inlet velocity of fuel; (ii) increase the inlet pres-408
sure of fuel; (iii) decrease the height of fuel channel; and/or 409
(iv) increase the porosity of the gas-diffusion layer. These 410
four schemes also apply to fuel cell with longer fuel chan-411
nels (or a larger size). Nevertheless, a high power density is 412
always followed with a high fuel consumption rate, which 413
results in a low fuel efficiency. In other words, to enhance 414
the fuel efficiency of a cell or to apply a fuel cell with shorter 415
fuel channels (or smaller size), it is necessary to apply a 416
lower inlet velocity and/or a lower inlet pressure so that the 417
fuel has a sufficiently long time to react with the catalyst in 418
the gas-diffusion layer at the bottom. With the same reason, 419
a channel with a larger height as well as a gas-diffusion layer 420
with a smaller porosity will also help enhance the fuel effi-421
ciency (or shorten the channel length). Results show further 422
that a catalyst of high activity is always a most desirable for 423
fuel cell design because it gives a high power density while 424
the accompanied efficiency loss is virtually negligible. 425
The above conclusions, nevertheless, are made only in 426
qualitative senses; namely, they give only a trend for the 427
design consideration. To obtain useful data for a quantita-428
tive design, experiments must be implemented to determine
some relevant parameters used in the simplified model. Par- 429 ticularly, the value ofξ cannot be determined without going 430 through a series of experiments in which the porosity of the 431
gas-diffusion layer is varied. 432
Acknowledgements 433 The authors are grateful for the financial support for this 434 research through the following research grants of National 435 Science Council of Taiwan: NSC 92-2623-7-002-006-ET 436
and NSC 91-2218-E-211-001. 437
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