Energy 70 (2014) 643e652
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Optimum design parameters and operating condition for maximum power of a direct methanol fuel cell using analytical model and genetic algorithm M. Tafaoli-Masoule a, *, A. Bahrami b, E.M. Elsayed c, d a
Fuel Cell Research Technology Group, Department of Mechanical Engineering, Babol University of Technology, P. O. Box 484, Babol, Iran Department of Mechanical Engineering, University of Tehran, Tehran, Iran Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia d Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt b c
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 July 2013 Received in revised form 8 April 2014 Accepted 17 April 2014 Available online 11 May 2014
It is well known that anode and cathode pressures, cell temperature and channel geometry are the effective parameters in the performance of DMFC (direct methanol fuel cell). In the present paper, the GA (genetic algorithm) as one of the most powerful optimization tools is applied to determine the optimal values for these parameters which result in maximum power density of a DMFC. The predominant part of the genetic algorithm is the fitness function. For the fitness function calculation, calculation of more than one thousand cases is necessary. Unfortunately, large numbers of experiments are needed, which is very time-consuming and costly. To overcome this challenge, a quasi two dimensional, isothermal model is used to obtain the power of DMFC as the fitness function of GA. For validation of this model, the results of the model are compared with experimental results and literature and shown to be in good agreement with them. Ó 2014 Elsevier Ltd. All rights reserved.
Keywords: Direct methanol fuel cell Genetic algorithm Modeling Design parameters Operating conditions
1. Introduction Recently, developments in renewable energy technologies and high cost of energy from fossil fuels have increased the utilization of different type of fuel cells [1]. Fuel cells are devices that convert chemical energy directly into electrical energy without combustion [2]. In the recent years, polymer electrolyte membrane fuel cells which use hydrogen or methanol as fuel have received increasing attention for energy generation on small and portable scale, due to high fuel to energy conversion efficiency [3]. DMFC (direct methanol fuel cell) with liquid methanol is easy to store and apply. It is one of the most promising portable power sources and an alternative to Li-ion batteries which can be used in mobiles, laptops and urban transportation systems [4]. The basic structure and operation of DMFC are shown in Fig. 1. DMFC is well known to be influenced by large numbers of parameters such as flow rate, methanol concentration, operating temperatures, anode and cathode pressures and so forth. In order to improve the performance of the DMFC, it is necessary to determine the effects of various parameters on the performance
* Corresponding author. Tel.: þ98 21 5505 4367; fax: þ98 21 8807 9656. E-mail address:
[email protected] (M. Tafaoli-Masoule). http://dx.doi.org/10.1016/j.energy.2014.04.051 0360-5442/Ó 2014 Elsevier Ltd. All rights reserved.
of the fuel cell. Other parameters include: (1) type and thickness of membrane, (2) catalyst type, (3) geometrical parameters of the flow field, and (4) the type of the gas diffusion layer. However, while these parameters are very important, they are not variable during fuel cell use. Therefore, they are not considered as operating variables. The fuel cell performance has been the subject of several research papers. Notably, Okur et al. [5] determined the optimum operating conditions for the process of preparing anode electrocatalysts for DSBHFC (direct sodium borohydride fuel cell). In addition, Carton and Olabi [6] established some experiments for comparison between three different configurations of flow plates of a fuel cell, the manufacturer’s serpentine flow plate and two new configurations; the maze and the parallel design. Moreover, Oliveira et al. [7] presented a 1D mathematical model for a microbial fuel cell in order to predict the correct trends for the influence of current density on the cell voltage while a comprehensive numerical model was developed by Xu et al. [8] to predict the electrochemical performance of SOFC (solid oxide fuel cell). A onedimensional, steady-state, two-phase DMFC (direct methanol fuel cell) model was proposed by Johan et al. [9] to precisely investigate complex physiochemical phenomena inside DMFCs. Jeong et al. [10] developed a mathematical model to study optimum operating
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Fig. 1. Schematic presentation of the direct methanol fuel cell principle.
strategies for minimizing methanol consumption while Wang et al. [11] focused on the effect of cell temperature and oxygen flow rate on cell response using experimental directions. Bennett et al. [12] developed an analytical model to determine the oxygen concentration profile in the cathode backing layer and flow channel along a one-dimensional cross-section of the fuel cell. The model was then applied to examine the effects of new lowcrossover membranes and to suggest new design parameters for those membranes. Furthermore, the multi-objective optimization problem was addressed by Wu and Lin [13] to improve the fuel efficiency and total exergetic efficiency of a non-isothermal DMFC (direct methanol fuel cell) system. The main important parameters which affect the performance of the DMFC are operating parameters such as fuel cell temperature and pressure in both sides of cathode and anode as well as channel geometry. The main focuses of the mentioned works were based on evaluating the required objective function by means of simulation and experimental approaches for limited and certain values of DMFC parameters and then comparing their objective functions to find the optimal solution. This approach is quite time-consuming and the accuracy of the results depends on the selected values of the parameters in the experiments. To the best knowledge of authors, there is no complete work done on the optimization of DMFC by combining an analytic model with genetic algorithm to obtain the optimal solution. The present research work combines a valid analytic model with genetic algorithm in order to examine many different cases in the possible search domain of each parameters instead of testing limited cases which were considered in the previous studies [5e13]. The combination of genetic algorithm and a valid analytical model as fitness function is a powerful tool for the investigation of these operating parameters.
2. Genetic algorithm The GA (genetic algorithm) method creates an artificial system according to the Darwinian mechanisms of evolution which is based on high probability for powerful solutions and low probability for weak solutions. The properties of this artificial system are
similar to human genetic system and the base of the genetic algorithm method is a random search. Genetic algorithm is a parallel mathematical algorithm that transforms a set of population (namely chains of “chromosomes” using genetic operations) into a new population (namely a next-generation) based on the fitness of each “chromosome” [14]. The principle concepts of a genetic algorithm are quite simple and can be described as follows: 1. First generation: the first generation is created randomly at definite domain. The size of population is based on the number of variables. 2. Evaluation of each solution: Each solution or “chromosome” is evaluated using fitness function. 3. Tournament selection: Each of both cases is selected randomly and according to the fitness function, the most appropriate one is introduced into the mating pool. 4. Crossover: For the purpose of searching for the best solutions at definite domain, each of both cases is combined with a definitive probability. In the present work, the algorithm uses the real digits systems and the crossover is defined according to the following approach in which R is a random number between 0 and 1:
child1;2 ¼
parent1 þ parent2 2
jparent1 þ parent2 j R
5. Mutation: For finding the absolute maximum value and escaping from the local maximum value, mutation is applied with certain probability. Following relation is used for this operator:
1 0 n X 1 parent þ parent iC B n C B i¼1 C child ¼ B C B 2 A @
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3. Fitness function of genetic algorithm
Table 1 Values of genetic algorithm parameters. Number of generations Population size Number of parameters Crossover rate Mutation rate
645
200 25 4 90% 10%
Table 2 Upper and lower values of the parameters to be optimized (search space). Parameter
Lower limit
Upper limit
Cathode pressure Anode pressure Cell temperature Channel height
0.5 bar 0.5 bar 50 C 1 mm
5 bar 2.5 bar 130 C 3 mm
There are two terminating criteria for each genetic code. The first criterion is the maximum number of generations and the second criterion is the genetic convergence. The first criterion is used for the present code. The genetic algorithm has been used in the optimization of different fields [15e17]. A modified genetic algorithm with an elitist concept is applied to ensure that the preferential part in each population is not missing the best “chromosomes”. This program is written in MATLAB software. The selected values of the various parameters of the genetic algorithm are presented in Table 1 while the upper and lower limits of the optimized parameters are shown in Table 2. The flowchart of the genetic code used to obtain the optimal parameters is shown in Fig. 2. The trend of this flowchart can be summarized as follows: The first step is to begin with random generation of an initial population. The analytical model is applied and the maximum power density as the fitness function of GA is calculated for each “chromosome”. The best “chromosome” in each generation passes directly to the next generation as an elite “chromosome” (elitist concept). This is followed by selection between each two random “chromosomes” with elimination of the weak “chromosome” and preservation of the powerful “chromosome”. The crossover and mutation are then applied for further searches. These two important parameters have a controllable role in genetic codes. If the condition (i.e., number of generations) is satisfactory, the elite “chromosome” is shown as the best value of the parameters otherwise random “chromosomes” are created in lieu of the eliminated cases following the search program.
The predominant part of the genetic algorithm is the fitness function as gauged by the “chromosomes”. The fitness function e as defined for the present work e is the maximum power density and the goal of this work is to find the best values of these parameters which result in the highest output power. For the fitness function calculation, calculation of more than one thousand cases is necessary. Unfortunately, large numbers of experiments are needed, which is very time-consuming and costly. To overcome this challenge, the DMFC is modeled and used as the fitness function of GA. 3.1. DMFC modeling The modeling of fuel cells has been well studied [18e22]. Generally, fuel cells modeling can be classified into three methods: analytical, semi-empirical and mechanistic methods. Analytical methods use some simplified assumptions to reduce the number of governing equations. These methods give a good estimation of voltage losses, but very few of these methods have an acceptable accuracy [20e22]. Semi-empirical methods use some experimental results and create some simple equations, but they are often restricted to specific fuel cells and the relations cannot be generally [23e25]. Mechanistic methods use algebraic and differential equations to model fuel transport phenomena in the PEM (polymer electrolyte membrane) and electrokinetic reactions in the catalyst layer. These methods are very precise but they often require a long convergence time. Generally, numerical techniques for solving governing equations of DMFC are very accurate but they need much more time than other methods. Analytical methods are less accurate and more rapid. For optimization of operating parameters with genetic algorithm, it is needed a fast and accurate model. The present work uses a novel technique by combining some techniques from literature to estimate the behavior of DMFC. For this purpose, two one dimensional models are used to simulate the flow behavior in channels and MEA (membrane electrode assembly). These models solve the governing equations in the direction of channels to calculate concentration, velocity and pressure distribution and also, the equations in the direction normal to the channel to find fuel cell polarization curve. 3.2. Mathematical modeling Methanol from anode’s channel is transferred to MEA by diffusion phenomena and then by an electrochemical reaction in anode
Fig. 2. The trend of genetic code to obtain the fuel cell optimal parameters.
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catalyst layer, proton, water, carbon dioxide and electron are produced. In cathode catalyst layer, oxygen from cathode channel reacts with proton and electron. For modeling the fuel cell, two separated regions are considered: one dimensional flow through the channels and one dimensional flow across the polymer membrane. Also, MEA is subdivided to three parts: gas diffusion layer, catalyst layer and membrane. 3.2.1. Channel’s equations Variations in velocity and concentration through the channel are assumed one dimensional and so for single phase flow, the equation is:
d MN ½r U ¼ i i dx i i H
(1)
where U is average velocity in the channel and N is molar flux between the channel and MEA, H and r are channel’s height and fluid density, respectively; Subscript i shows gas and liquid phases in cathode and anode channels. By the conservation of species, equation of single phase flow for each component is:
hkm Cik Cik h i d Hch k UC ¼ dx i i H
(2)
where C is molar concentration and k is an identifier for the type of species in each side (water and methanol at anode and oxygen and water vapor at cathode side). The right hand side of Eq. (2) shows the effect of diffusion at channel/GDL interface. Coefficient hm can be estimated [26] by:
hkm ¼ Sh
Dkeff H
(3)
where H is channel’s height and Sh is Sherwood number; Dkeff is effective diffusion coefficient of fluid from channel to gas diffusion layer. 3.2.2. PEM modeling Among the various models published until now, Garcia’s model [20] as an analytical model is well known. The model is made based on some simplified assumptions which will be described in the following sections. 3.2.2.1. Anode. By using the above assumptions and suitable boundary conditions for three domains: gas diffusion layer, catalyst layer and the membrane, analytic relations for concentration distribution of methanol through them can be derived [20]: In the diffusion layer:
cBMeOH ¼
KI cA I cb
dB
þ cb
(4)
In the membrane:
A cM MeOH ¼ KII cII
dB þ dA Z þ1 dM
(5)
In the catalyst layer:
cA MeOH ¼
ICell Z 2 þ C1 Z þ C2 12F dA DA
(6)
where
C1 ¼
A cA II cI
dA
C2 ¼ cA I
ICell ð2dB þ dA Þ 12F dA DA
A cII cA I dB
dA
þ
(7)
ICell dB ðdB þ dA Þ 12F dA DA
(8)
Thus, the current density can be calculated by: dZ B þdA MeOH aI0;ref
ICell ¼ dB
kcA MeOH
l aA ha F=RT cA MeOH þ e
eaA ha F=RT dz
(9)
where dA and dB are catalyst and backing layers thickness, respectively, and I0,ref is the reference current density. Here a, aA, ha, F and T are the specific surface area of the anode, anode activation, anode overpotential, faraday constant and cell temperature, respectively. From the above, the methanol crossover through membrane can be estimated as: M NMeOH ¼ DM
dcM I MeOH þ xMeOH Cell dZ F
(10)
In the right hand side, the first term and the second term show the methanol crossover due to diffusion from membrane and the crossover by electro-osmotic drag respectively. Table 3 shows the values of parameters used. 3.2.2.2. Cathode. Assuming single phase flow, the continuity equation leads to:
dðruÞ ¼ 0
(11)
Thus, the mass flow rate per unit area is
ru ¼ constant
(12)
From Darcy’s equation:
m
Vp ¼ u k
(13)
where k and m are permeability of cathode gas diffusion layer and viscosity of gas, respectively. The distribution of species at cathode side (oxygen, water vapor and carbon dioxide) can be estimated by Maxwell’s equation of distribution as:
Table 3 Values of fuel cell parameters. Parameter
Symbol
Unit
Value
Ref.
Diffusion coefficient of methanol in liquid Diffusion coefficient of methanol in membrane Diffusion coefficient of oxygen Reference current density of methanol Reference current density of oxygen
Dm DB e MeOH I0;ref O2 I0;ref
cm2 s1 cm2 s1 cm2 s1 A cm2 A cm2
4.9 106e2436(1/3331/T) 2.8 105e2436(1/3531/T) 1 103 94.25 104e35570(1/3531/T) 0.04222 104e73200(1/3531/T)
[21] [21] [26] [26] [26]
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647
Fig. 4. Experimental setup of DMFC.
where x and N are the mole fraction and molar flow rate of each species, respectively, and Dij is the diffusion coefficient. By solving Eqs. (4)e(14), velocity and mole fraction of gases in cathode GDL (gas diffusion layer) can be obtained.
3.3. Methodology of solution The modeling in this paper is based on two one dimensional models linked together. Since for calculating current density, ICell, at a certain anode overpotential, ha, Garcia’s model [20] needs concentration of methanol at interface of the channel and GDL, so it can be a suitable parameter for making connection between channel and PEM. After dividing the two solution domains (channels and PEM), a guessed anode’s channel and GDL interface concentration (CMeOH) are used to solve Garcia’s model in anode side to estimate the resulted current density, methanol consumption and crossover. Then, the net penetration of each species in PEM from channel (Ni) is clear. The oxygen consumption at cathode side can also be obtained by solving Eqs. (11)e(14). Then, by using Eqs. (1)e(4), input velocity and concentration of methanol for the next cell can be estimated and CMeOH can be updated by Eq. (2). After some number of iterations and reaching a suitable solution error, next step is started. By following this process, Icell and then, cell voltage can be estimated. The flowchart of this procedure is shown in Fig. 3 which gives an overview of the solution procedure.
Table 4 Values of DMFC parameters.
Fig. 3. Analytic solution algorithm for the DMFC.
Vxi ¼
X xi Nj xj Ni j
CT Deff ij
(14)
Parameter
Unit
Value
Number of cells in stack Anode and cathode width of channel Anode and cathode height of channel Active area Total area Anode and cathode flow pattern Anode loading Cathode loading Methanol concentration at anode inlet CH3OH Air Cell pressure Type of MEA
e cm cm cm2 cm2 e mg/cm2 PteRu mg/cm2 Pt Molar mL/min SL/min bar e
1 0.1 0.1 625 900 Serpentine 4.0 4.0 1 30 2.5 1.0 Nafion 117
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Fig. 5. Test station of single cell DMFC.
Fig. 6. Software environmental of test station.
3.4. Validation of modeling 3.4.1. Developing the experimental setup For this purpose, an experimental setup for direct methanol fuel cell was developed (Fig. 4) using different values for the experimental parameters (Table 4). The test station consists of three parts. Each part will do a special task. The first part controls the feed rate and conduction duty in the anode side while the second part will do the same task in the cathode side and the third part will purge the fuel cell after the experiments. The schematics of single cell test station of DMFC and software environment of test station are shown in Figs. 5 and 6 respectively. More description about these two figures can be found in Ref. [27]. The initial constant values for fuel cell modeling are different in research papers. These fuel cell constant values for simulation and optimization are shown in Table 5. In Fig. 7, the polarization curves of the model and experimental results are compared in different cell temperatures. As shown in Fig. 7, the analytic polarization curves are in good agreement with experimental curves, but they are not exactly the
same as experimental curves due to some simplified assumptions for developing the analytical model. For instance, the current of DMFC was assumed one dimensional. So, the proposed curves have a little difference compared with their corresponding experimental curves especially at the end of curves due to concentration losses.
Table 5 Constant parameters of model. Condition and parameters in use
Value
Ref
Anode and cathode diffusion layer thickness (cm) Anode catalyst layer thickness (cm) Cathode catalyst layer thickness (cm) Anode and cathode diffusion layer porosity Anode and cathode catalyst layer porosity Anodic transfer coefficient (aa) Cathodic transfer coefficient (ac) Membrane layer thickness (cm) MEA Ionic conductivity (S/cm) MEA porosity
0.03 0.005 0.003 0.7 0.3 0.8 0.8 0.02 0.036 0.3
[21,26] [21] [21] [26] Assumed [21] [21] [21] [20] [26]
M. Tafaoli-Masoule et al. / Energy 70 (2014) 643e652 Table 7 Fuel cell geometrical parameters Guo and Ma [21].
1
Experimental T=60 Experimental T=70 Experimental T=80 Modeling T=60 Modeling T=70 Modeling T=80
0.9 0.8
Voltage (V)
649
0.7 0.6
Parameter
Symbol
Unit
Value
Length of anode channel Length of cathode channel Width of anode channel Width of cathode channel Height of anode channel Height of cathode channel Number of anode channel Number of cathode channel
La Lc Wa Wc ha hc na nc
cm cm cm cm cm cm e e
3 3 0.2 0.2 0.2 0.2 10 10
0.5 0.4
4. Optimization result and discussion
0.3 0.2
0
20
40
60
80
100
120
140
160
180
Current Density (mA/cm2) Fig. 7. Validation of result by experimental data at constant pressure of 1 bar.
Although the method for creating fitness function values is not very precise and other approaches such as CFD (computational fluid dynamics) method have an accurate response, the proposed analytical model is very useful for present aim and has the capability to merge with genetic code while the other approaches are so time-consuming and costly and it is not possible to apply them especially at numerous number of GA’s generations. Therefore, analytical approaches which have fast and acceptable responses are very suitable. 3.4.2. Comparing with literature For further validation of this analytical model, the results are compared with the data from a published analytical model [21]. In Tables 6 and 7, the operating parameters and geometrical parameters derived from the reference [21] are listed, respectively. Since some of the assumed parameters of Garcia’s model [20] could not be found in the model developed by Guo and Ma [21], these parameters will be computed by changing parameters and fitting polarization curve for methanol concentration of 0.25 M (Fig. 8). It is clear from Fig. 8 that the proposed modeling has an acceptable accuracy in the ohmic region, but it has some deviation from experimental data at the beginning and the end of curves due to activation and concentration drop. By comparing the presented analytical model with the experimental results and literature data, the proposed model is validated. The proposed analytical model also has the ability to be used in the calculation for fitness function of genetic algorithm. Since the results of parallel channels modeling are in good agreement with experimental approach, this type of channel was assumed for optimization. The assumed constant geometrical values for optimization are listed in Table 8. As it is shown in Fig. 8, the value of 0.5 M for methanol concentration has an acceptable agreement, so this value is used as methanol concentration for the optimization.
By executing this MATLAB optimization code, solutions are generated for maximum power densities as presented in Table 9. The obtained values for parameters create the maximum power density of 567 mW/cm2. The convergence of the genetic algorithm towards the best solution is also illustrated in Fig. 9. Fig 10 shows the convergence of GA at constant crossover of 90% and variable mutation rate e for the different cases, the maximum power density is 567 mW/cm2. The optimum value of channel height by attention to the definite domain for these parameters shows that the lower values of channel height create higher power density and it can be described as more diffusion flux. As the pressure increases, the crossover current density at OCV (open circuit voltage) decreases, so it can be concluded that the OCV is increased by decreases of methanol crossover with the increase of pressure. Seo and Lee [28] showed this trend with the experiments. Also, the higher cell temperature gives better fuel cell performance. Increasing the cell temperature can invigorate the activity of the catalysts and can increase the mass transfer of reactants. It also results in better proton transportation through the Nafion membrane and much faster mass transfer processes inside the cell [11]. The cell performance in either anode’s or cathode’s side would be improved and it can be justified as it is mentioned below:
Table 6 Fuel cell operating conditions. Parameter
Unit
Value
Temperature of the fuel cell Pressure of cathode channel Inlet flow rate of anode Inlet flow rate of cathode
C
90 2.5 2 800
bar cm3 min1 cm3 min1
Fig. 8. Polarization curves of 2D analytical model [22] (symbols) and present modeling result.
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Table 8 Fuel cell constant geometrical values for optimization. Symbol
Unit
Value
Length of anode channel Length of cathode channel Width of anode channel Width of cathode channel Number of anode channel Number of cathode channel
La Lc Wa Wc na nc
cm cm cm cm e e
3 3 0.2 0.2 10 10
Table 9 Optimal values for parameters. 1 130 2.5 5
0.6 0.5 0.4 0.3 0.2 0.1
0
20
40
60
80
100
120
140
160
180
200
Generations (Mutation = 5%)
Seo and Lee [28], Yang et al. [29] and Nakagawa et al. [30] set up some experiments and concluded that higher temperatures lead to a much higher power density. Also, an analytical model was developed by Rosenthal et al. [31] and Zou et al. [32] which shows that higher temperatures have more efficiency too. Unfortunately the present experimental setup cannot tolerate the obtained optimum loading condition of test due to incapability of the structure to pass this temperature condition but this optimum value for producing maximum power density is closely in good agreement with reference [33].
0.8 0.7 0.6
0.8
Maximum power (W/cm2)
The cell pressure increase would lead to an increase in cell reactants’ pressure. In regard to the pressure increase, the activated gases permeability in the gas diffusion layer would be increased, besides, the mass transfer resistance of cell would be decreased. The gases are stagnant in the extensive areas on GDL corners and current distribution channels, when the cell is working under the environmental pressure. These areas in MEA usually do not produce any current, but when the cell is working under the pressure, the reactants are forced to pass these areas; therefore, the effective active area in the electrochemical reactions would be increased.
Maximum power (W/cm2)
0.7
0
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
40
60
80
100
120
140
160
180
200
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.5
20
Generations (Mutation = 10 %)
Maximum Power (W/cm2)
Channel height (mm) Cell temperature ( C) Anode pressure (bar) Cathode pressure (bar)
Maximum power (W/cm2)
0.8
Parameter
0
20
40
60
80
100
120
140
160
180
200
Generations (Mutation = 20%)
0.4
Fig. 10. Maximum power during the generations with constant crossover of 90% and different mutation.
0.3 0.2
5. Conclusions 0.1 0
20
40
60
80
100
120
140
160
Generations Fig. 9. Maximum fitness during the generations.
180
200
In this study, the genetic algorithm was applied to determine the optimal parameters of process for maximum power of a mono cell DMFC. A quasi two dimensional (1D-1D), isothermal model was presented for the direct methanol fuel cell. For validation of model, the result of the model was compared to experimental data and
M. Tafaoli-Masoule et al. / Energy 70 (2014) 643e652
literature and shown to be in good agreement with them. This model was used for determining the maximum power density of a DMFC which was considered as the fitness function for GA. A genetic code was developed using MATLAB. Finally, the optimal values for DMFC’s cell temperature, anode and cathode pressure and channel height were obtained 130 C, 2.5 and 5 bar and 1 mm, respectively.
651
Superscripts A anode catalyst layer B anode diffusion layer MeOH methanol M membrane eff effective value k water and methanol at anode side, oxygen and water vapor at cathode side
Nomenclature
a C cA I cA II cb D F H I MeOH I0;ref L N M k R P
l T U V W x Z u
specific surface area of the anode (cm1) molar concentration (mol cm3) concentration at the interface of anode diffusion and catalyst layer (mol cm3) concentration at the interface of anode catalyst layer and membrane (mol cm3) bulk concentration of methanol in the flow channel (mol cm3) diffusion coefficient (cm2 s1) Faraday’s constant (96,487 C mol1) channel height (cm) current density (A cm2) reference current density (A cm2) channel length (cm) molar flow rate (kg cm2 s1) molecular weight (kg mol1) permeability, constant in the rate expression (Eq. (9)) (dimensionless) universal gas constant (8.314 J mol1 K1) pressure (Pa) constant in the rate expression (Eq. (9)) (mol m3) temperature (K) velocity (cm s1) voltage (v) channel width (cm) molar fraction, channel’s direction coordinate (cm) MEA normal direction coordinate (cm) velocity in channel’s (cm/s)
Greek symbols a transfer coefficient r density (kg m3) d thickness (cm) hc cathode overpotential (v) ha anode overpotential (v) x electro osmotic drag coefficient m viscosity (kg m1 s1) Subscripts A anode catalyst layer B anode diffusion layer MeOH methanol M membrane ref reference value eff effective value ch channel b bulk cell cell a anode side c cathode side i gas phases in cathode and anode channels, liquid phases in cathode and anode channels T total I interface of anode diffusion layer and anode catalyst layer II interface of anode catalyst layer and membrane
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