Optimization of key parameters in the proton exchange membrane fuel cell
Hung-Hsiang Lin
a, Chin-Hsiang Cheng
a,∗, Chyi-Yeou Soong
b, Falin Chen
c, Wei-Mon Yan
daDepartment of Mechanical Engineering, Tatung University, 40 Chungshan N. Road, Sec. 3, Taipei 10451, Taiwan, ROC bDepartment of Aerospace and Systems Engineering, Feng Chia University, Taichung 40724, Taiwan, ROC
cInstitute of Applied Mechanics, National Taiwan University, Taipei 107, Taiwan, ROC dDepartment of Mechatronic Engineering, Huafan University, Taipei 223, Taiwan, ROC
Received 2 May 2006; received in revised form 19 June 2006; accepted 19 June 2006 Available online 1 August 2006
Abstract
The present study is focused on the optimization of the proton exchange membrane fuel cell (PEMFC). A simplified conjugate-gradient method (SCGM) is employed to seek the optimal combination of the design parameters, including the channel width ratio (λ), the porosity of GDL (εGDL),
and the porosity of the catalyst layer (εCat). In this report, the concept of the method is described in detail and the results of the optimization process
are discussed. The results show that the present approach is capable of providing an optimal design under specific operation conditions. © 2006 Elsevier B.V. All rights reserved.
Keywords: PEM fuel cell; Optimization; Channel width ratio; Porosity; Model
1. Introduction
In the past several decades, fossil fuels have caused environ-mental pollution problems, such as the greenhouse effect and acid rain. Due to the increasing rate of consumption of global petroleum, finding other more efficient and cleaner substitutive power generation systems has become an issue of great concern. Among the possible alternative power generation systems, pro-ton exchange membrane (PEM) fuel cells have received much attention from energy-related researchers and industries world-wide. Since in PEM fuel cells electric power is generated by directly transforming the chemical fuel energy into electricity in electrochemical reactions, these devices exhibit a higher effi-ciency compared to traditional power generators. Also, without ignition and combustion during electricity generation, PEM fuel cells are free of severe pollution and noise. In addition, PEM fuel cells do not have the flaws of corrosion of the mechani-cal elements and material problems associated with the working medium. Recently, PEM fuel cell technology has been developed more actively than ever in the automobile and 3C (computer, communication, consumer) electronics industries. The com-mercialisation of PEM fuel cells into mass production can be
∗Corresponding author. Tel.: +886 2 25925252 3410; fax: +886 2 25997142.
E-mail address:[email protected](C.-H. Cheng).
accelerated. However, more efforts are still needed to explore the physical characteristic features as well as to improve the performance of PEM fuel cells in the coming years.
A number of reports have been presented in the past several decades; however, only those closely related to the present study are mentioned in the following.
Hental et al.[1]considered the effects of carbon plate calcina-tions at high temperature on the PEMFC efficiency. The authors claimed that, with a flow channel width of 1 mm and the thick-ness of the collector ribs of around 2 mm, the performance of their fuel cell was obviously elevated.
Honta˜n´on et al.[2]used CFD software to study the effects of GDL permeability and the ratio of the flow channel to the rib widths on the PEMFC performance. The results showed that a decrease in the permeability of the gas diffusion layer (GDL) reduces cell performance. It was also found that the influence of the GDL permeability on the cell performance is stronger than that of the flow channel width ratio.
Yoon et al.[3] similarly studied the influences of the flow channel width on the PEMFC efficiency. Their research showed that the gas diffusion effect plays a more important role than the electric conductivity effect. As the flow channel width is kept constant, the performance of fuel cell is higher with a smaller rib width.
Gurau et al. [4] developed a two-dimensional numerical model to study the changes in the gas concentration and 0378-7753/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
Nomenclature
D mass diffusivity (m2s−1)
F Faraday constant (96,487 C mol−1) I current density (A m−2)
j transfer current density (A m−3) J objective function
lb base width of the flow distributor (m) lc gas channel width (m)
ls shoulder width (m)
MO2 molar mass of oxygen (32.0 kg kmol−1)
n iteration step ne number of electrons
P power output per unit area (W m−2) R universal gas constant (8.314 J (mol K)−1) tGDL thickness of gas diffusion layer (m) tCat thickness of catalyst layer (m)
T temperature (K)
V potential
Greek letters
α transfer coefficient for the reaction
β step size
γ conjugate-gradient coefficient εCat porosity of catalyst layer εGDL porosity of gas diffusion layer η over-potential (V)
κ reaction rate constant λ channel width ratio, lc/lb ρ density of oxygen (kg m−3) σ electric conductivity ( −1m−1) τCat tortuosity of catalyst layer τGDL tortuosity of gas diffusion layer φ electric potential (V)
ω oxygen mass fraction ω0 oxygen mass fraction at inlet
electrochemical reaction rate by including the electrochemical kinetics of the catalyst layer and the convection–diffusion effect in the GDL. Results show that as the porosity of the GDL is increased, the entry velocity is elevated. They also found that the cell performance increases with the cell temperature.
Wang and Savinell[5]studied the effects of porosity, thick-ness, Pt-loading, CO poisoning of catalyst layer on the cell performance. In this report, it was observed that the catalyst layer porosity of an efficient fuel cell is between 0.4 and 0.7.
Yan et al.[6] studied the influence of flow channel geom-etry and GDL porosity on gas transport. Results provided in this report showed that a larger flow channel width ratio or a higher GDL porosity leads to a better cell performance. They also pointed out that an increase in the number of flow channels results in a more uniform current density distribution and thus a better cell performance.
Zhou and Liu [7] proposed a three-dimensional fuel cell model to study the influence of CO toxins and the variation
of hydrogen concentration in anode. Authors claimed that some phenomena, which were not found by using the one- and two-dimensional models, are worthy of investigation by the three-dimensional model. They also carried out a parametric analysis of the effects of the entry velocity, the porosity of the GDL, and the channel width ratio on the cell performance. Based on this report, it was observed that the channel width ratio and the GDL porosity both have their own optimal values.
Kumar and Reddy[8]studied the improvement of the perfor-mance of the PEMFC based on optimization of the dimensions and shape of the bipolar plates. It was found that the optimal dimensions for the channel width, the land width and the chan-nel depth were close to 1.5, 0.5 and 1.5 mm, respectively.
Hsu’s work[9]was concerned with the analysis of the trans-port phenomenon and the parametric optimization of a PEM fuel cell. This study was focused on the influence of conductiv-ity, channel width ratio, and porosity and thickness of the GDL. A two-dimensional half-cell model is utilized in simulation of the transport phenomenon.
Ying et al.[10]investigated the effects of channel configu-ration on an air-breathing fuel cell’s performance. The authors found that the best performance can be obtained in the cell with a 3 mm wide cathode channel and an open ratio of 75.9%.
Lum and McGuirk [11] carried out a series of parametric studies of the effects of variation of the electrode thickness, shoulder width, degree of permeability and oxidant concentra-tion, so as to provide a clearer understanding of how changes in these parameters affect the cell performance.
Berning and Djilali[12]presented the results of a parametric study based on a three-dimensional, non-isothermal model of a PEM fuel cell. The authors evaluated the influence of the channel width ratio and the thickness and porosity of the GDL.
From the above survey of related literature, it is noted that most of the studies were focused on numerical simulation of the transport phenomenon in a PEM fuel cell, or at most on the parametric study of the effects of physical variables. Only a few reports dealt with the optimization of a set of design parameters. In practice, the task of optimization searches for a combi-nation of the most appropriate values of a number of design parameters with a minimized objective function. Nowadays, several useful optimization methods are available, for exam-ple, the genetic algorithm[13], the conjugate-gradient method (CGM) [14], the Newton–Raphson method[15], the adaptive weighting input estimation method[16], etc. These methods are all applicable to minimize the prescribed objective functions. Among them, the CGM method, which stems from perturbation principles, transforms the optimal problem into three separate problems, namely, the direct problem, the sensitivity problem, and the adjoining problem. The solutions of the three problems give the updated conjugate-gradient direction as well as the step size towards convergence. Cheng and Chang[17,18]used the concept of the conjugate-gradient method to develop an inverse method for predicting the internal temperature distribution of PEM fuel cells. The results show that the outer surface tem-perature data can be used to predict the internal temtem-perature distribution at the interface between the carbon plate and the MEA.
Under these circumstances, in the present study, a multi-parametric optimization for the PEMFC design is investigated. Among the possible influential parameters, the channel width ratio, and the porosities of the GDL and the catalyst layer are rec-ognized as the influential factors that may affect the behaviour of proton exchange and electron conduction; therefore, these three parameters are selected as the design parameters to be optimized in this work. The simplified conjugate-gradient method (SCGM) proposed by Cheng and Chang[14]is employed herein as the optimization search scheme. This study is aimed at optimiza-tion of these three parameters so that the best performance of the PEMFC under some specified geometry and operating con-ditions can be obtained. However, it is noted that the present optimization approach is not limited to the present group of designed parameters. Where necessary, more designed parame-ters may be readily applied.
2. Optimization method
The present optimization method consists of two major numerical solutions: (1) the direct problem solver and (2) the simplified conjugate-gradient method (SCGM). The two solvers are briefly described as follows.
2.1. Direct problem solver
Fig. 1 shows a schematic of a two-dimensional half-cell model for the cathode side employed for simulation of the transport phenomena involved. From top to bottom, the model consists of the catalyst layer, the gas diffusion layer, the flow channel, and the carbon plate which is developed with the fol-lowing simplifications and assumptions:
Fig. 1. Schematic of the two-dimensional simulation domain of half-cell PEMFC (cathode).
1. The fuel cell operates steadily.
2. The total pressure and temperature are uniform and constant in the solution domain.
3. The gases are assumed to be compressible ideal gases, fully saturated with water vapor.
4. The oxygen flux is zero at the interface between the catalyst layer and the membrane.
5. The gas transport from the flow channel to the catalyst layer is only by diffusion.
6. The gas diffusion layer and the catalyst layer are considered to be isotropic porous media.
7. Owing to high ohmic resistance of the membrane, the inter-face between the catalyst layer and the membrane is assumed to be insulated against electrons.
The channel width ratio (λ) is defined by a ratio of the channel width lcto the carbon plate width lbas
λ = lc lb
(1) 2.1.1. Catalyst layer
In the catalyst layer, the oxygen is consumed due to the electro-chemical reaction therefore, a two-dimensional trans-port equation for the oxygen mass fraction (ωCat) in the catalyst layer can be expressed as
∂ ∂x ρO2D eff Cat ∂ωCat ∂x + ∂ ∂y ρO2D eff Cat ∂ωCat ∂y − j 2neFMO2 = 0 (2) Deff
Cat= ετCatCatDO2 (3)
whereρO2stands for the oxygen density,DO2the oxygen
diffu-sivity, F the Faraday constant, j the transfer current density, ne the number of electrons in transfer current density relation, and εCatandτCatrepresent the porosity and the tortuosity factor of the catalyst layer, respectively.
The cathode transfer current density can be expressed as j = neFκω exp αneFη RT (4) whereκ is the reaction rate constant, α the transfer coefficient for the reaction,η the over-potential, R the universal gas constant, and T is the cell temperature.
Due to the electro-chemical reactions activated in the catalyst layer, there exists a source term in the Poisson equation for the electric potential. That is
∂ ∂x σeff Cat ∂φCat ∂x + ∂ ∂y σeff Cat ∂φCat ∂y = j (5) σeff
Cat= (1 − εCat)τCatσCat (6)
whereσCatdenotes the electric conductivity of the catalyst layer andφCatthe electric potential of the catalyst layer.
2.1.2. Gas diffusion layer (GDL)
In the GDL, only the oxygen diffusion takes place without chemical reactions. The two-dimensional transport equation for the oxygen mass fractionωGDL in the GDL can be expressed as ∂ ∂x ρO2D eff GDL ∂ωGDL ∂x + ∂ ∂y ρO2D eff GDL ∂ωGDL ∂y = 0 (7) with Deff GDL= ετGDLGDLDO2 (8)
whereεGDLandτGDLare the porosity and the tortuosity factor of the GDL, respectively.
Since in the GDL there is no electrochemical reaction taking place, the electric potential equation possesses no electric source term and is expressed as
∂ ∂x στGDL GDL ∂φGDL ∂x +∂y∂ στGDL GDL ∂φGDL ∂y = 0 (9) σeff GDL= (1 − εGDL)τCatσGDL (10)
whereσGDLis the electric conductivity andφGDLis the electric potential in the GDL.
2.1.3. Carbon plate
It is essential to investigate the effects of gas channel width on the transport phenomena of the cathode reactants and on the ohmic resistance of the whole module.
The carbon plate serves as a conductor for the electric cur-rent. The electric potential equation in the carbon plate can be expressed as ∂ ∂x σp∂φ p ∂x + ∂ ∂y σp∂φ p ∂y = 0 (11)
whereσpis the electric conductivity andφpis the electric poten-tial in the carbon plate.
2.1.4. Boundary conditions
The physical conditions at the boundaries numbered B1 to B9 inFig. 1are described as follows:
The boundary conditions for oxygen mass fraction (ω) are • Catalyst layer: ∂ωCat ∂y = 0 at boundary BC6; ωCat= ωGDL at boundary BC5; ∂ωCat ∂x = 0 at boundaries BC7 and BC9 • GDL: Deff GDL ∂ωGDL ∂y = D eff Cat ∂ωCat ∂y at boundary BC5; ωGDL= ω0 at boundary B3; ∂ωGDL ∂y = 0 at boundaries BC4; ∂ωGDL ∂x = 0 at boundaries BC7 and BC9
The boundary conditions for electric potential (φ) are: • Catalyst layer: ∂φCat ∂y = 0 at boundary BC6; φCat= φGDL at boundary BC5; ∂φCat ∂x = 0 at boundaries BC7 and BC9 • GDL: σeff GDL ∂φGDL ∂y = σ eff Cat ∂φCat ∂y at boundary BC5; ∂φGDL ∂y = 0 at boundary BC3; φGDL = φp at boundary BC4; ∂φGDL ∂x = 0 at boundaries BC7 and BC9 • Carbon plate: φp= 0 at boundary BC1; σp∂φp ∂y = σeffGDL ∂φGDL ∂y at boundary BC4; ∂φp ∂x = 0 at boundaries BC7, BC8, and BC9; ∂φp ∂y = 0 at boundary BC2 2.1.5. Operation potential
With the solutions for the electric potential distribution yielded using the above model, the cell potential (V) can be calculated by the following equation:
V = VOC− η − φ (12)
where VOC denotes the ideal cell potential, η is for the over-potential of the cathode, and
φ = lp 0 (φ6− φ1) dx lp (13) denoting the average ohmic loss in terms of the difference between the electric potentials at boundaries B6 and B1, respectively. During the optimization, the cell voltage V and cell current I are varied accompanying iterative changes in the design parameters.
It is noted that the present two-dimensional model is rather straightforward and not able to include the effects of heat and liq-uid water accumulation. A more powerful model will certainly be required. However, the simplicity of the model is acceptable,
given that only the channel width ratio and the porosities of the GDL and the catalyst layer are taken into account and the opti-mization is emphasized rather than the complexity of the model. 2.2. Simplified conjugate-gradient method (SCGM)
In this study, the simplified conjugate-gradient method (SCGM)[10]is employed to optimize the values of the design parameters in search of optimal performance. The objective function in conjunction with the optimization process is defined in the following:
J(Ak)= 1
P, k = 1, 2, 3 (14)
where Ak(k = 1, 2, and 3) represent the channel width ratio (λ), the porosity of the GDL (εGDL), and the porosity of the catalyst layer(εCat). The cell power output (P) is expected to reach a max-imum as the magnitude of the objective function J is minimized. In the SCGM method, the values of the step sizeβkduring the searching procedure are fixed. In this manner, the complexities in the mathematical manipulation and the constraint of the form of the objective function can be alleviated. The initial guess for each design variable is made first, and in the successive steps the conjugate-gradient coefficients and the searching directions are evaluated. The new design variables are continuously improved. When the objective function reaches a minimum, the goal of the optimization is reached and the searching procedure is then ter-minated.
The procedure for the iterative optimization process with the SCGM method is presented as follows:
(1) Make the initial guess for the designed variables Ak, assign the values to the step sizesβk.
(2) Specify all boundary conditions, and then solve the direct problem.
(3) Calculate the objective function J(Ak, k = 1–3). If the conver-gence criterion is satisfied as the objective function reaches a minimum in the iterating process, the solution process is terminated. Otherwise, proceed to step (4).
(4) Calculate the gradient functions of the function J: ∂Jn
∂Ak = Jn
Ak, k = 1–3 (15)
(5) Calculate the conjugate-gradient coefficientsγkn(k = 1–3): γkn= (∂J/∂Ak)n (∂J/∂Ak)n−1 2 , k = 1 − 3 (16)
For the first step with n = 0,γk0= 0.
(6) Calculate the searching directionsξnk (k = 1–3): ξn+1k = ∂J∂An
k + γ n
kξnk, k = 1 − 3 (17)
(7) Update new design variables by
An+1k = Ank− βkξkn+1, k = 1 − 3 (18) and then return to step (2).
Table 1
Variables used in the present analysis
Parameters Notation Value
Temperature (K) T 333
Reaction rate constant (mol (m3s)−1) κ 1.5× 10−5
Oxygen mass fraction at inlet ω0 0.17
Gas channel width (m) lc 1.0× 10−3
Channel shoulder width (m) ls 1.0× 10−3
Thickness of gas diffusion layer (m) tGDL 5× 10−4
Thickness of catalyst layer (m) tCat 5× 10−5
Porosity of GDL εGDL 0.5
Tortuosity of GDL τGDL 1.5
Porosity of catalyst layer εCat 0.3
Tortuosity of catalyst layer τCat 1.5
Faraday constant (C mol−1) F 96487
Universal gas constant (J (mol K)−1) R 8.314 Molar mass of oxygen (kg kmol−1) MO2 32.0
Number of electrons ne 2
Transfer coefficient for reaction α 1 Electric conductivity of catalyst layer ( −1m−1) σCat 190
Electric conductivity of GDL ( −1m−1) σGDL 537
Electric conductivity in carbon plate ( −1m−1) σp 4000
Fig. 2. A typical case at conditions of σCat= 190 −1m−1, σGDL= 537
−1m−1,σ
p= 4000 −1m−1, tGDL= 5× 10−4m, tCat= 5× 10−5m,λ = 0.5,
εGDL= 0.5, andεCat= 0.3. (a) Polarization curve and (b) power curve of a PEM
3. Results and discussion 3.1. Preliminary study
In the computation, the activated over-potential acts as an independent variable to replace the external loads. By doing so, the polarization diagram of the fuel cell can be determined. The power density of the fuel cell is the product of the potential V and the current density I as
P = VI (19)
The model described in the preceding section is solved by a finite volume method with a uniformly distributed grid having 101 and 511 nodes in the x- and y- directions, respectively. The values of the geometrical and physical variables considered in the present study are listed inTable 1.
Fig. 2 shows the polarization and the power curves for a typical case at the conditions of σCat= 190 −1m−1,
σGDL= 537 −1m−1,σp= 4000 −1m−1, tGDL= 5× 10−4m, tCat= 5× 10−5m,λ = 0.5, εGDL= 0.5, andεCat= 0.3. For a lower over-potential, the weaker electrochemical reaction leads to a lower power density. However, when the over-potential is increased, the reaction is stronger so that the power density reaches a maximum at I = 7400 A m−2.
Sensitivity analysis of the three optimized design param-eters with respect to the cell voltage has been carried out and Fig. 3(a)–(c) are plotted to show the results. The cases shown in Fig. 3(a)–(c) are at σCat= 190 −1m−1, σGDL= 537 −1m−1,σp= 4000 −1m−1, tGDL= 5× 10−4m, tCat= 5× 10−5m. Fig. 3(a) shows the effects of the channel width ratio (λ) for εGDL= 0.5 and εCat= 0.3, Fig. 3(b) shows the effects of porosity of GDL (εGDL) forλ = 0.5 and εCat= 0.3, andFig. 3(c) conveys the effects of porosity of catalyst layer (εCat) forλ = 0.5 and εGDL= 0.5. It is found that in the ranges of parameters considered, the cell voltage is more sensitive to a change in the channel width ratio or the porosity of the GDL.
Fig. 3. Dependence of cell voltage on designed parameters atσCat= 190 −1m−1,σGDL= 537 −1m−1,σp= 4000 −1m−1, tGDL= 5× 10−4m, tCat= 5× 10−5m.
(a) Effects of channel width ratio (λ) for εGDL= 0.5 andεCat= 0.3. (b) Effects of porosity of GDL (εGDL) forλ = 0.5 and εCat= 0.3. (c) Effects of porosity of catalyst
However, it is relatively less sensitive to the porosity of the cat-alyst layer.
Fig. 4 further shows the effects of the channel width ratio (λ) on the performance of a PEM fuel cell at η = 0.25 V, tGDL= 5× 10−4m, tCat= 5× 10−5m,εGDL= 0.5, andεCat= 0.3. It is observed that indeed the channel width ratio has a subtle influence on the cell performance. Larger channel width ratios may enhance the fuel gas transport as well as the chemical reac-tion in the catalyst layer; however, a larger channel width leads to smaller ribs and hence results in an increase in the ohmic impedance. With these two opposing effects, there must be an optimal value for the channel width ratio (λ), at which the perfor-mance reaches a peak value. InFig. 4, the results show that the largest power of 6572 W m−2is obtained at the optimal channel width ratio ofλ = 0.65 under the conditions considered in this case.
Fig. 5shows the effects of the GDL porosity (εGDL) on the performance of a PEM fuel cell at η = 0.25 V. A higher GDL porosity implies a higher ohmic impedance and results in a reduction in the electric current. However, on the other hand, a higher GDL porosity also implies a stronger diffusion of reactant gas in the GDL.Fig. 5shows that the optimal value ofεGDLis 0.6 at η = 0.25 V, εCat= 0.3, tGDL= 5× 10−4m, tCat= 5× 10−5m, andλ = 0.5. At the optimal value of εGDL, the power density approaches 6843 W m−2.
The effects of the catalyst layer porosity (εCat) on the performance of a PEM fuel cell at η = 0.25 V, εGDL= 0.5, tGDL= 5× 10−4m, tCat= 5× 10−5m, andλ = 0.5 are plotted in
Fig. 6. Except for the case at very low or very high porosity εCat, the effects of the catalyst layer porosity are not as obvi-ous as those of the GDL porosity. This is attributed to the fact that the catalyst layer is relatively thin, and hence the effects of εCat on the performance are limited. The variation of the power output is only less than 1% as the porosityεCatis in the range of 0.3–0.8. Even so, the optimal porosity of the catalyst layerεCatis also obtained. For the case atη = 0.25 V, εGDL= 0.5,
Fig. 4. Effects of channel width ratioλ on performance of a PEM fuel cell at η = 0.25 V, tGDL= 5× 10−4m, tCat= 5× 10−5m,εGDL= 0.5, andεCat= 0.3.
Fig. 5. Effects of GDL porosityεGDLon performance of a PEM fuel cell at
η = 0.25 V, εCat= 0.3, tGDL= 5× 10−4m, tCat= 5× 10−5m, andλ = 0.5.
tGDL= 5× 10−4m, tCat= 5× 10−5m, and λ = 0.5, the optimal εCatis 0.5 and the corresponding power of 6099 W m−2can be observed.
3.2. Optimization of PEMFC
In the parametric studies considered above, only one param-eter is changed while others are fixed. In fact, multi-paramparam-eter optimization is more meaningful from the design viewpoint. Fig. 7shows the trajectory of the multi-parameter optimization process approaching the optimal design. The optimal combi-nation ofλ, εGDL, andεCat are sought in the parameter space (λ, εGDL,εCat). The optimization process is started with the initial guess of λ = 0.65, εGDL= 0.5, andεCat= 0.3. It can be observed inFig. 7that the SCGM method effectively brings the searching process to approach an optimal state. In this case,
Fig. 6. Effects of catalyst layer porosityεCaton performance of a PEM fuel cell
Fig. 7. Trajectory of the optimization process for a PEM fuel cell seeking an optimal set of designed parametersλ, εGDL, andεCat.
it is found that the iteration converges after about 400 iter-ations. The computation time required is approximately 80 h on a personal computer with an AMD-3GMHz CPU. Based on the results shown in Fig. 7, it is found that at λ = 0.55, εGDL= 0.5913 andεCat= 0.307, the optimization performance can be generated with a maximum power of 6866.53 W m−2at η = 0.25 V, tGDL= 5× 10−4m, tCat= 5× 10−5m. The variation of the power density during the optimization process is shown inFig. 8. Note that the power density associated with the initial design is approximately 2500 W m−2. The power density then approaches its maximum value when the optimal parameters are obtained.
To check the robustness of the searching method used in the present work,Fig. 9shows the convergence of the optimization process from three different initial guesses for the three param-etersλ, εGDL, andεCat, atη = 0.25 V. It can be observed inFig. 9
Fig. 8. Variation in power output during the optimization process.
Fig. 9. Trajectories of the optimization processes from different sets of initial guesses forλ, εGDL, andεCat.
that the present optimization method leads to the same optimal set of parameters with different initial guesses.
4. Conclusions
In this study, an optimization analysis employing the SCGM method[10]combined with a two-dimensional half-cell model for the transport phenomena has been successfully developed. The optimization of design parametersλ, εGDL,εCat, has been performed. The results show that, at the conditions ofλ = 0.55, εGDL= 0.5913 andεCat= 0.307, the optimal cell performance in terms of power is P = 6866.53 W m−2. Extension of the present method to one with more parameters is a worthwhile work for the future, and a more comprehensive model would also be inter-esting. It is noted that the present optimization approach is not limited to the present group of designed parameters. Where nec-essary, more designed parameters may be readily applied. References
[1] P.L. Hental, J.B. Lakeman, G.O. Mesped, P.L. Adock, J.M. Moor, New materials for polymer electrolyte membrane fuel cell current collectors, J. Power Sources 80 (1999) 235–241.
[2] E. Honta˜n´on, M.J. Escudero, C. Bautista, P.L. Garc´ıa-Ybarra, L. Daza, Optimization of flow-field in polymer electrolyte membrane fuel cells using computational fluid dynamics techniques, J. Power Sources 86 (2000) 363–368.
[3] Y.G. Yoon, W.Y. Lee, G.G. Park, T.H. Yang, C.S. Kim, Effects of chan-nel configurations of flow field plates on the performance of a PEMFC, Electrochim. Acta 50 (2004) 709–712.
[4] V. Gurau, H. Liu, S. Kakac, Two-dimensional model for proton exchange membrane fuel cells, AICHE J. 137 (11) (1998) 2410–2422.
[5] J.T. Wang, R.F. Savinell, Simulation studies on the fuel electrode of a H2–O2 polymer electrolyte fuel cell, Electrochim. Acta 37 (1992)
2737–2745.
[6] W.M. Yan, C.Y. Soong, F.L. Chen, H.S. Chu, Effects of flow distributor geometry and diffusion layer porosity on reactant gas transport and per-formance of proton exchange membrane fuel cells, J. Power Sources 125 (2004) 27–39.
[7] T. Zhou, H. Liu, A 3D model for PEM fuel cells operated on reformate, J. Power Source 138 (2004) 101–110.
[8] A. Kumar, R.G. Reddy, Effect of channel dimensions and sharp in the flow-field distributor on the performance of polymer electrolyte membrane fuel cell, J. Power Sources 113 (2003) 11–18.
[9] H.C. Hsu, Study of transport phenomena and optimal design for proton exchange membrane fuel cells, Master of Science (Advisor: C. H Cheng), Department of Mechanical Engineering, Tatung University, Taipei, Taiwan, 2004.
[10] W. Ying, T.H. Yang, W.Y. Lee, J. Ke, C.S. Kim, Three-dimensional analysis for effect of channel configuration on the performance of a small air-breathing proton exchange membrane fuel cell (PEMFC), J. Power Sources 145 (2005) 572–581.
[11] K.W. Lum, J.J. McGuirk, Three-dimensional model of a complete poly-mer electrolyte membrane fuel cell—model formulation, validation and parametric studies, J. Power Source 143 (2005) 103–124.
[12] T. Berning, N. Djilali, Three-dimensional computational analysis of trans-port phenomena in a PEM fuel cell—a parametric study, J. Power Source 124 (2003) 440–452.
[13] I. Mohamed, N. Jenkins, Proton exchange membrane (PEM) fuel cell stack configuration using genetic algorithms, J. Power Source 131 (2004) 142–146.
[14] C.H. Cheng, M.H. Chang, A simplified conjugate-gradient method for shape identification based on thermal data, Numer. Heat Transfer, Part B 43 (2003) 489–507.
[15] Y. Hung, N.G. Zamani, Optimum beam design using finite elements, Appl. Math. Comput. 80 (1996) 155–179.
[16] P.C. Tuan, M.C. Ju, The validation of the robust input estimation approach to two-dimensional inverse heat conduction problems, Numer. Heat Transfer, Part B 37 (2000) 247–265.
[17] C.H. Cheng, M.H. Chang, Predictions of internal temperature distribution of PEMFC by non-destructive inverse method, J. Power Source 139 (2005) 115–125.
[18] M.H. Chang, C.H. Cheng, Non-destructive inverse method for determi-nation of irregular internal temperature distribution in PEMFCs, J. Power Source 142 (2005) 200–210.
