It is well known that anode and cathode pressures and cell temperature are the effective parameters in performance of Direct Methanol Fuel Cell (DMFC). In the present study, the genetic algorithm as one of the most powerful optimization tools is applied to determine operating conditions which result in the maximum power density of a DMFC. A quasitwodimensional, isothermal model is presented to determine maximum power of a DMFC. For validation of this model, the results of the model are compared to experimental results and shown to be in good agreement with them.
Direct methanol fuel cell is one of the most promising transportable power sources which can be used in mobiles, laptops, and small power generation [
The fuel cell performance has been the subject of several research papers. Xu et al. [
The main important parameters which affect the performance of the DMFC are operating parameters such as operating temperature and pressure in both sides of the cathode and anode. More recent research has focused on the values of these parameters as derived by experimental approaches, but, in current research, these parameters are optimized using genetic algorithm (GA) and a DMFC analytical model for creating the maximum power as the fitness function of the GA [
Genetic algorithm is an artificial system and it is similar to human genetic system. GA 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 [
The basic concepts of a genetic algorithm can be conveniently described as follows.
First generation: the first generation is created randomly at definite domain. The size of population is based on the number of variables.
Evaluation of each solution: each solution or chromosome is evaluated using fitness function.
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.
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 relation in which
Mutation: for finding the absolute maximum value and escaping from the local maximum value, mutation is applied with certain probability [
A modified genetic algorithm with an elitist concept is used to ensure that the preferential part in each population is not missing the best solutions. This genetic algorithm code is written in MATLAB software. The selected values of the various parameters of the genetic algorithm are presented in Table
Values of genetic algorithm parameters.
Number of generations  200 
Population size  20 
Number of parameters  3 
Crossover rate  95% 
Mutation rate  5% 
Upper and lower values of the parameters to be optimized (search space).
Parameter  Lower limit  Upper limit 

Cathode pressure  0.5 bar  3 bar 
Anode pressure  0.5 bar  2.5 bar 
Cell temperature  50°C  90°C 
The trend of genetic code to obtain the fuel cell optimal parameters.
The trend of this flow chart 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—as defined for the present work—is the maximum power density, and the goal of this work is to find the best values of the 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.
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, a fast and accurate model is needed. The present work uses a novel technique by combining some techniques from the literature to estimate the behavior of DMFC. For simulating the flow behavior in channels and Membrane Electrode Assembly (MEA), two onedimensional models are used to solve the governing equations in the direction of channels to calculate concentration, velocity, and pressure distribution and also to solve the equations in the direction normal to the channel to find fuel cell polarization curve [
Methanol from anode’s channel is transferred to MEA by diffusion phenomena, and then, in anode catalyst layer by an electrochemical reaction, 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: onedimensional flow through the channels and onedimensional flow across the polymer membrane. Also, MEA is subdivided into three parts: gas diffusion layer, catalyst layer, and membrane.
Variations in velocity and concentration through the channel are assumed to be one dimensional, and so for the single phase flow the equation is
Among the various models published until now, Garcia’s model [
in the diffusion layer:
in the membrane:
in the catalyst layer:
where
Values of fuel cell parameters.
Parameter  Symbol  Unit  Value  Reference 

Diffusion coefficient of methanol in liquid 

cm^{2} s^{−1} 

[ 
Diffusion coefficient of methanol in membrane 

cm^{2} s^{−1} 

[ 
Diffusion coefficient of oxygen  —  cm^{2} s^{−1} 

[ 
Reference current density of methanol 

A cm^{−2} 

[ 
Reference current density of oxygen 

A cm^{−2} 

[ 
The modeling in this paper is based on two onedimensional models linked together. Since for calculating current density,
After dividing the two solution domains (channels and PEM), a guessed anode’s channel and GDL interface concentration
Analytic solution algorithm for the DMFC.
For this purpose, an experimental setup for direct methanol fuel cell was developed (Figure
Values of DMFC parameters.
Parameter  Unit  Value 

Number of cells in stack  —  1 
Anode and cathode width of channel  cm  0.1 
Anode and cathode height of channel  cm  0.1 
Active area  cm^{2}  625 
Total area  cm^{2}  900 
Anode and cathode flow pattern  —  Serpentine 
Anode loading  mg/cm^{2} PtRu  4.0 
Cathode loading  mg/cm^{2} Pt  4.0 
Methanol concentration at anode inlet  Molar  1 
CH_{3}OH  mL/min  30 
Air  SL/min  2.5 
Cell pressure  Bar  1.0 
Type of MEA  —  Nafion 117 
Constant parameters of model.
Condition and parameters in use  Value  Reference 

Anode and cathode diffusion layer thicknesses (cm)  0.03  [ 
Anode catalyst layer thickness (cm)  0.005  [ 
Cathode catalyst layer thickness (cm)  0.003  [ 
Anode and cathode diffusion layer porosity  0.7  [ 
Anode and cathode catalyst layer porosity  0.3  — 
Anodic transfer coefficient ( 
0.8  [ 
Cathodic transfer coefficient ( 
0.8  [ 
Membrane layer thickness (cm)  0.02  [ 
MEA ionic conductivity (S/cm)  0.036  [ 
MEA porosity  0.3  [ 
Experimental setup of DMFC.
Validation of result by experimental data at constant pressure of 1 bar.
By comparing the presented analytical model with experimental results, 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 channels was assumed for optimization.
By executing this MATLAB optimization code, solutions are generated for maximum power densities (Table
Optimal values for parameters.
Cell temperature (°C)  90 
Anode pressure (bar)  2.5 
Cathode pressure (bar)  3 
Maximum fitness during the generations.
It also results in better proton transportation through the Nafion membrane and much faster mass transfer processes inside the cell. The cell performance in either anode’s or cathode’s side would be improved, and it can be justified as mentioned in the following.
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.
Unfortunately the present experimental setup cannot tolerate the obtained optimum loading condition of the 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 [
In this study, the genetic algorithm was applied to determine the optimal parameters for maximum power of a monocell Direct Methanol Fuel Cell. A quasitwodimensional (1D1D), isothermal model was presented for the DMFC. For validation of the model, the result of the model was compared to experimental data and the literature and shown to be in 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, optimal values for DMFC’s cell temperature and anode and cathode pressures were obtained: 90°C and 2.5 and 3 bars, respectively.