Multi-Objective Optimization of Blockage Design Parameters Affecting the Performance of PEMFC by OEM-AHP-EWM Analysis

Abstract

Adding blockages to the gas flow channels in the bipolar plates has a significant effect on the performance of the proton exchange membrane fuel cell (PEMFC). The design parameters of the gas flow channels with blockages mainly include the blockage shape (S), blockage number (N), blockage height (H), and channel–rib width ratio (CRWR) value. This paper systematically examines the combined effects of S, N, H, and CRWR value on current density (I), pressure drop (ΔP), net output power (W_net), and non-uniformity of oxygen distribution (NU) of PEMFC through the application of the orthogonal experimental method (OEM). To provide a comprehensive optimization strategy, a novel multi-criteria decision framework is introduced, which integrates the analytic hierarchy process (AHP) and entropy weight method (EWM) to balance different evaluation objectives. Results from the AHP-EWM analysis reveal that the weight values of I, ΔP, W_net, and NU are 0.415, 0.08, 0.325, and 0.18, respectively. The CRWR value exhibits the greatest effect on the comprehensive performance of PEMFC, followed by H, N, and S. The optimal design parameter combination identified in this paper is a triangular blockage with nine blockages, a height of 0.8 mm, and a CRWR value of 0.25, corresponding to the highest comprehensive score of 31.8306 among the 25 groups of orthogonal experiments. This paper provides a new optimization perspective and certain guidance for the performance optimization direction of PEMFC.

1. Introduction

Since the 1960s, proton exchange membrane fuel cells (PEMFCs) have developed rapidly due to their advantages of high energy efficiency and zero pollution [1]. Currently, they are widely applied to automotive applications, distributed generation systems, and large vessels [2,3,4,5], and the installed power of PEMFCs has also increased significantly [5,6]. Therefore, they are considered to be the most promising energy conversion devices in the 21st century. However, the production of fuel cells with high performance, high reliability, and long-term durability at low cost remains a top priority for research and development [7,8]. For example, the limited lifespan of PEMFCs still hinders their commercialization. Over prolonged operation periods, the performance of PEMFCs gradually deteriorates due to intricate operational conditions and component aging, eventually reaching the minimum acceptable threshold [9]. Hence, Prognosis and Health Management (PHM) is vital for PEMFCs to enable proactive interventions to prevent fuel cell failures, consequently prolonging the life of the PEMFCs. To achieve this target, different investigations from various aspects of PEMFCs have been conducted. For example, researchers are committed to improving the original catalysts [10,11], developing novel proton exchange membranes (PEM) [12,13,14], and proposing bio-inspired flow channel designs [15,16].

Adding blockages in the gas flow channel is also an effective and popular method to further improve the performance of PEMFCs, which has a great influence on the transportation of reactants and drainage capacity during the operation of PEMFCs. Therefore, researchers have conducted a series of comparative analyses and optimization studies on different blockages [17]. Perng et al. [18] studied the performance of PEMFCs with different numbers of rectangular blockages in the anode and cathode flow channels and demonstrated that the net power of PEMFCs with five blockages was 8% higher than that of PEMFCs without blockages. They [19] also examined the influence of various angles and heights of trapezoid blockages on the performance and illustrated that the trapezoidal blockage with an angle of 60° and a height of 1.125 mm increased the net power of PEMFCs by about 90% compared with the conventional straight flow channel (CSFC). Similarly. Yin et al. [20] found that when the trapezoidal blockage leading angle and trailing angle both were 45°, the average current density was increased by 10.7% and 6.4% in comparison with the CSFC and the blockage with a trailing angle of 90°. In addition, when the number of trapezoidal blockages was five, the drainage capacity of the PEMFCs was remarkable. Wang et al. [21] proposed a parallel trapezoid blockage plate (PTBP) and a staggered trapezoid blockage plate (STBP), and revealed that compared with the CSFC and PTBP, the maximum net power of the STBP increased by 6.39% and 2.54%, respectively. Chen et al. [22] designed the wave flow channel and simulated the effects of different flow channel minimum depths and wavelengths on the performance of PEMFCs. The results showed that the minimum depth and wavelength of the optimal flow channel were 0.45 mm and 2 mm, and the current density was 23.8% higher than that of the CSFC at 0.4 V. Cai et al. [23] used a genetic algorithm to optimize the influence of channel center amplitude and the number of waves on current density and pressure drop of PEMFCs. The results demonstrated that the output power density of the optimal channel with a center amplitude of 0.305 mm and the number of waves of 3.52 was increased by 2.2% in comparison with the CSFC.

In addition to adding blockages to the gas flow channel mentioned above, changing the channel–rib width ratio (CRWR) is also one of the important factors affecting the performance of PEMFCs. Kerkoub et al. [24] studied the effects of six different CRWR values on the performance of PEMFCs based on parallel, serpentine, and interdigitated flow channels and found that reducing the CRWR value could improve the performance of PEMFCs; the serpentine flow field had the best performance under the same CRWR value conditions. At high operating current densities, the massive accumulation of liquid water in the gas diffusion layer (GDL) will lead to flooding and impede gas diffusion, resulting in rapid degradation of cell performance [25]. Jeon [26] investigated the effect of CRWR on water transport behavior using a multiphase lattice Boltzmann method and found that a large CRWR value would help enhance water removal in the GDL, but if the CRWR value was very large, it may lead to mass transport limitations. Xia et al. [27] found that increasing CRWR value was beneficial to gas diffusion in porous electrodes, but the longer electron motion paths led to higher ohmic losses and revealed that the optimal CRWR value was about 1, with a peak power density of 0.428 W/cm². Varghese et al. [28] also proved the above conclusion and demonstrated that the performance of PEMFCs decreased mainly due to the contact resistance between the bipolar plate (BP) and GDL surface as the CRWR value increased. Wang et al. [29] showed that increasing the rib area ratio to 30% could reduce the electrical resistance of fuel cells and improve performance. Compared with the rib area ratio of 70%, rib area ratios of 20% and 30% improved the performance of PEMFCs by 24.3%.

In conclusion, it can be seen that adding blockages and changing the CRWR value both have a significant influence on the performance of PEMFCs. However, although the above-mentioned literature has explored the effect of individual design parameters on the efficiency of PEMFC, there is still a lack of comprehensive research to examine the synergistic effects of these factors on the performance of PEMFCs systematically and efficiently. In this paper, the combined effects of blockage shape, blockage number, blockage height, and CRWR value are systematically studied using the orthogonal experimental method (OEM), and a new method for optimizing the performance of PEMFCs using a multi-criteria decision framework is introduced. The framework integrates the analytic hierarchy process (AHP) and the entropy weight method (EWM) to balance the performance objectives, providing a more comprehensive optimization strategy. There are three steps to this work: (i) the experimental matrix is designed using an L₂₅(5⁶) orthogonal table, effectively reducing the number of simulations while considering four design parameters for each of the five levels; (ii) multi-objective evaluation methods (AHP-EWM) are applied to determine the relative importance of performance objectives such as current density, pressure drop, net output power, and non-uniformity of oxygen distribution; and (iii) analysis of range (ANOR) and analysis of variance (ANOVA) are conducted to determine the most optimal design parameters to optimize the PEMFC configuration with the objective of comprehensive scores. This method not only reduces the computational complexity but also deepens the understanding of the interaction between different design parameters. This study provides certain guidance for the performance of PEMFC optimization.

2. Models

2.1. Physical Model

A three-dimensional physical model of a single-channel PEMFC unit with blockages is established in Figure 1. The PEMFCs mainly include an anode bipolar plate (ABP), anode gas diffusion layer (AGDL), anode microporous layer (AMPL), anode catalyst layer (ACL), proton exchange membrane (PEM), cathode bipolar plate (CBP), cathode gas diffusion layer (CGDL), cathode microporous layer (CMPL), and cathode catalyst layer (CCL). The length (L) of BP is 23 mm, the width (W) is 2 mm, and the height (H) is 1.5 mm; the height (h) of the gas flow channel is 1 mm; and the thicknesses of the GDL, the microporous layer (MPL), the catalyst layer (CL), and the PEM are 0.3 mm, 0.05 mm, 0.0129 mm, and 0.108 mm, respectively. The other relevant physical and operating parameters are listed in detail in

Table 1. Physical and operating parameters.

Parameters	Value	Unit
Porosity (GDL, MPL, CL)	0.6, 0.5, 0.3	/
Permeability (GDL, MPL, CL)	1 × 10−11, 1 × 10−12, 1 × 10−13	m2
Contact angle (GDL, MPL, CL)	120, 120, 100	°
Open circuit voltage	1.15	V
Operating temperature	343.15	K
Operating pressure	101,325	Pa
Relative humidity (anode, cathode)	30%, 100%	/
Reference current density	10,000	A/m2
Reference concentration (anode, cathode)	56.4, 3.39	mol/m3
Standard entropy change (anode, cathode)	130.68, 32.55	J/(mol·K)
Stoichiometric ratio (anode, cathode)	1.5, 2.0	/
Surface/volume ratio	200,000	m−1
Concentration exponent (anode, cathode)	1.0, 1.0	/
Proton conduction coefficient	1	/
Proton conduction exponent	1	/
Ionomer volume fraction in CL	0.3	/
Equivalent weight of PEM	1100	kg/kmol
Exchange coefficient (anode, cathode)	0.5, 0.5	/
Thermal conductivity (BP, GDL, MPL, CL)	20, 10, 1, 1	W/(m·K)
Electrical conductivity (BP, GDL, MPL, CL)	20,000, 8000, 5000, 5000	S/m
Specific heat capacity (BP, PEM, GDL, MPL, CL)	1580, 833, 568, 3300, 3300	J/(kg·K)

.

2.2. Numerical Model

2.2.1. Governing Equations

To simplify the calculation process for PEMFCs, several commonly accepted assumptions were employed, detailed as follows [30,31]:

(1)

The PEMFC model was considered to operate under non-isothermal, steady-state conditions.

(2)

The gas mixtures were treated as incompressible and conform to ideal gas.

(3)

The flow of inlet reactants within the channel was presumed to be laminar.

(4)

The impact of gravity was disregarded.

(5)

Porous materials were assumed to possess homogeneous and isotropic properties.

Based on the aforementioned assumptions, the governing equations for PEMFCs can be summarized as follows [32]:

Continuity equation:

$$\nabla \cdot \left(\rho \mathit{u}\right)=S_c$$

(1)

where ρ, u, and S_c are the density, velocity vector, and mass source.

Momentum equation:

$$\nabla \cdot \left(\epsilon \rho \mathit{u}\mathit{u}\right)=-\epsilon \nabla P+\nabla \cdot \left(\epsilon \mu \nabla \mathit{u}\right)+S_m$$

(2)

where ε, P, μ, and S_m are the porosity, pressure, viscosity, and momentum source. The S_m can be stated as:

Anode and cathode channels:

$$S_m=0$$

(3)

Porous zones:

$$S_m=-{\displaystyle \frac{\mu }{K}}{\epsilon }^2\mathit{u}$$

(4)

where K is the permeability.

Conservation of species:

$$\nabla \left\{-\rho w_i{\displaystyle \sum _{j=1}^N{D}_{ij\_eff}\left[\frac{M}{M_j}\left(\nabla w_j+w_j\frac{\nabla M}{M}\right)+\left(x_j-w_j\right)\frac{\nabla P}{P}\right]}+w_i\rho \mathit{u}\right\}=S_i$$

(5)

where w, D_{ij_eff}, x, M, and S_i are the mass fraction, effective binary diffusion coefficient, molar fraction, molecular mass, and source term of different species. The S_i can be stated as:

$$\mathrm{Hydrogen}: S_{H_2}=-{\displaystyle \frac{j_a}{2F}}{M}_{H_2}$$

(6)

$$\mathrm{Oxygen}: S_{O_2}=-{\displaystyle \frac{j_c}{4F}}{M}_{O_2}$$

(7)

$$\mathrm{Water}: S_{H_{2}O}={\displaystyle \frac{j_c}{2F}}{M}_{H_{2}O}$$

(8)

The D_{ij_eff} is described by Bruggeman’s correlation as follows:

$$D_{ij\_eff}={\epsilon }^{1.5}{D}_{ij}$$

(9)

$$D_{ij}=D_{ij\_ref}{\left({\displaystyle \frac{T}{T_0}}\right)}^{1.5}$$

(10)

where D_ij, D_{ij_ref}, T, and T₀ are the binary diffusivity, reference binary diffusivity, operating temperature, and reference temperature.

Conversation of charge:

$$\nabla \cdot \left(-{\sigma }_s\nabla {\phi }_s\right)=S_{\phi s}$$

(11)

$$\nabla \cdot \left(-{\sigma }_m\nabla {\phi }_m\right)=S_{\phi m}$$

(12)

where σ, φ, and S_φ are the electric conductivity, phase potential, and source term of the current. The subscripts ‘s’ and ‘m’ are solid and membrane. The S_φ can be stated as:

$$\mathrm{Anode} \mathrm{catalyst}: S_{\phi s}=-j_a S_{\phi m}=j_a$$

(13)

$$\mathrm{Cathode} \mathrm{catalyst}: S_{\phi s}=-j_c S_{\phi m}=j_c$$

(14)

where j is the transfer current density and subscripts ‘a’ and ‘c’ are anode and cathode, which can be obtained from the Bulter–Volmer function as follows:

$$j_a=a{i}_{0,a}^{ref}{\left({\displaystyle \frac{C_{H_2}}{C_{H_{2,ref}}}}\right)}^{0.5}\left({\displaystyle \frac{{\alpha }_a+{\alpha }_c}{RT}}F{\eta }_a\right)$$

(15)

$$j_c=a{i}_{0,c}^{ref}\left({\displaystyle \frac{C_{O_2}}{C_{O_{2,ref}}}}\right)\exp \left(-{\displaystyle \frac{{\alpha }_c}{RT}}F{\eta }_c\right)$$

(16)

where a, i₀, α, F, η, and R are the electrocatalytic surface area, exchange current density, transfer coefficient, Faraday’s constant, activation overvoltage, and universal gas constant.

2.2.2. Boundary Conditions

The boundary conditions for the numerical simulation were set as follows:

(1) The symmetric boundary conditions were employed on both sides of the x-axis;

(2) The no-flux condition was applied to the external boundaries and the no-slip condition was used to all flow channel walls;

(3) The anode current collector was set to zero and the cathode current collector was set to the operating voltage;

(4) The inlet and outlet of the gas flow channel were set as mass flow rate and pressure outlet. The value of the mass flow rate at the inlets can be obtained from the following equations [33,34]:

$$m_{\mathrm{a}}={\displaystyle \frac{{\rho }_{\mathrm{g}}^{\mathrm{a}}{\zeta }_{\mathrm{a}}{I}_{\mathrm{ref}}{A}_{\mathrm{act}}}{2F{C}_{{\mathrm{H}}_2}{A}_{\mathrm{in}}^{\mathrm{a}}}} m_{\mathrm{c}}={\displaystyle \frac{{\rho }_{\mathrm{g}}^{\mathrm{c}}{\zeta }_{\mathrm{c}}{I}_{\mathrm{ref}}{A}_{\mathrm{act}}}{4F{C}_{{\mathrm{O}}_2}{A}_{\mathrm{in}}^{\mathrm{c}}}}$$

(17)

$$C_{{\mathrm{H}}_2}={\displaystyle \frac{P_{\mathrm{g}}^{\mathrm{a}}+\mathrm{\Delta }{P}_{\mathrm{g}}^{\mathrm{a}}-R{H}_{\mathrm{a}}{P}_{\mathrm{sat}}}{RT}} C_{H_{2}O}^{\mathrm{a}}={\displaystyle \frac{R{H}_{\mathrm{a}}{P}_{\mathrm{sat}}}{RT}}$$

(18)

$${\rho }_{\mathrm{g}}^{\mathrm{a}}=C_{{\mathrm{H}}_2}^{\mathrm{a}}{M}_{{\mathrm{H}}_2}+C_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{a}}{M}_{{\mathrm{H}}_2\mathrm{O}}$$

(19)

$$C_{{\mathrm{O}}_2}=0.21{\displaystyle \frac{P_{\mathrm{g}}^{\mathrm{c}}+\mathrm{\Delta }{P}_{\mathrm{g}}^{\mathrm{c}}-R{H}_{\mathrm{c}}{P}_{\mathrm{sat}}}{RT}} C_{H_{2}O}^{\mathrm{c}}={\displaystyle \frac{R{H}_{\mathrm{c}}{P}_{\mathrm{sat}}}{RT}} C_{N_2}^{\mathrm{c}}=0.79{\displaystyle \frac{P_{\mathrm{g}}^{\mathrm{c}}+\mathrm{\Delta }{P}_{\mathrm{g}}^{\mathrm{c}}-R{H}_{\mathrm{c}}{P}_{\mathrm{sat}}}{RT}}$$

(20)

$${\rho }_{\mathrm{g}}^{\mathrm{c}}=C_{{\mathrm{H}}_2}^{\mathrm{a}}{M}_{{\mathrm{H}}_2}+C_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{a}}{M}_{{\mathrm{H}}_2\mathrm{O}}$$

(21)

$$Y_{\mathrm{i}}={\displaystyle \frac{M_{\mathrm{i}}{C}_{\mathrm{i}}}{{\displaystyle \sum M_{\mathrm{i}}{C}_{\mathrm{i}}}}}$$

(22)

where ζ, A_act, I_ref, A_in, and C are the stoichiometry ratio, active area, reference current density, inlet area, and molar concentrations, respectively.

2.3. Model Validation and Grid Independence Test

2.3.1. Model Validation

To ensure the reliability of the computational model results, the PFF with an active area of approximately 5.3 cm² from Zhang et al. [35] was selected. The operating pressure was 253,312.5 Pa, the operating temperature was 353.15 K, the stoichiometry ratios of oxygen and hydrogen were both 2.0, and their relative humidities were both 30%. The polarization curves obtained from both numerical simulation and experiment under identical operating conditions were compared, as presented in Figure 2. The root mean square error between the two sets of data was 0.048, demonstrating that the PEMFC model is validated and dependable.

2.3.2. Mesh Validation

The numerical simulation in this paper was conducted using the ICEM 2020 R1 software to mesh the three-dimensional model and the PEMFC module included with ANSYS Fluent 2020 R1 to simulate different operating conditions. The PEMFC unit with a single channel was established for grid independence to eliminate the influence of the grid on the simulation results.

Table 2. Influence of grid sizes on the numerical results.

	Grid Sizes	Current Density (A/cm2)	Relative Error
Grid 1	40,843	0.9902	0.99%
Grid 2	60,435	0.9945	0.58%
Grid 3	90,060	0.9976	0.26%
Grid 4	136,275	1.0003	-
Grid 5	203,899	1.0014	0.1%

shows the calculation results of current density for five different grid sizes when the operating voltage was 0.4 V. It can be seen from the table that the relative error of current density between Grid 4 and Grid 5 was only 0.1%. Therefore, taking into account the simulation accuracy and computational efficiency, Grid 4 with a grid size of 136,275 was chosen as the basic number of grids for all PEMFC models.

3. Research Methods

Figure 3 is a flowchart of the multi-criteria decision framework, which mainly combines the OEM-AHP-EWM analysis. A detailed introduction to these three methods will be provided.

3.1. OEM Introduction

For multi-objective optimization experiments, if the control variable method is used to individually study the effects of different design parameters and their combinations, numerous simulation cases are required, which will cost huge computing resources and time costs. For example, if there are four operation parameters and the five levels are selected for each operating parameter, a total of 5⁴ = 625 groups of simulation cases would be needed to comprehensively evaluate each combination. To address this issue, the orthogonal experimental method is usually used to arrange experiments due to its advantages of fewer experiments and higher efficiency. Therefore, in this section, an orthogonal table was used to arrange the simulation cases, thus minimizing the computational workload. This approach enables a systematic and efficient analysis of the effects of different parameter combinations, optimizing resource usage without compromising the integrity of the results.

The shapes (S), numbers (N), and heights (H) of blockages, along with the CRWR value, have a significant influence on the performance of PEMFC. Therefore, these factors with different design combinations were selected as the research object. As illustrated in Figure 4, the shapes of blockage S were triangular (Tri), wave (Wav), rectangular (Rec), trapezoidal (Tra), and sectorial (Sec); the numbers of blockage N were 3, 4, 5, 7, and 9; the heights of blockage H (mm) were 0.4, 0.5, 0.6, 0.7, and 0.8; the CRWR values were 4, 3, 1.5, 1, and 0.25. The orthogonal levels and factors are listed in

Table 3. Orthogonal levels and factors table.

Levels	Factors
	S	N	H (mm)	CRWR Value
1	Tri	3	0.4	4
2	Wav	4	0.5	3
3	Rec	5	0.6	1.5
4	Tra	7	0.7	1
5	Sec	9	0.8	0.25

. According to the principle of orthogonal experimental design, the L₂₅(5⁶) orthogonal table was adopted, as shown in

Table 4. L₂₅(5⁶) Orthogonal experiment table.

Case No.	Factors
	S		N		H		CRWR Value
	No.	Factor	No.	Factor	No.	Factor	No.	Factor
1	1	Tri	1	3	1	0.4	1	4
2	1	Tri	2	4	2	0.5	2	3
3	1	Tri	3	5	3	0.6	3	1.5
4	1	Tri	4	7	4	0.7	4	1
5	1	Tri	5	9	5	0.8	5	0.25
6	2	Wav	1	3	2	0.5	3	1.5
7	2	Wav	2	4	3	0.6	4	1
8	2	Wav	3	5	4	0.7	5	0.25
9	2	Wav	4	7	5	0.8	1	4
10	2	Wav	5	9	1	0.4	2	3
11	3	Rec	1	3	3	0.6	5	0.25
12	3	Rec	2	4	4	0.7	1	4
13	3	Rec	3	5	5	0.8	2	3
14	3	Rec	4	7	1	0.4	3	1.5
15	3	Rec	5	9	2	0.5	4	1
16	4	Tra	1	3	4	0.7	2	3
17	4	Tra	2	4	5	0.8	3	1.5
18	4	Tra	3	5	1	0.4	4	1
19	4	Tra	4	7	2	0.5	5	0.25
20	4	Tra	5	9	3	0.6	1	4
21	5	Sec	1	3	5	0.8	4	1
22	5	Sec	2	4	1	0.4	5	0.25
23	5	Sec	3	5	2	0.5	1	4
24	5	Sec	4	7	3	0.6	2	3
25	5	Sec	5	9	4	0.7	3	1.5

.

3.2. AHP Introduction

The AHP method is a decision-making method that integrates both qualitative and quantitative analysis to calculate the weights of different evaluation objectives. According to the nine-level scale method of AHP theory, experts are required to score the relative importance of different evaluation objectives within the same level to form a judgment matrix. The judgment matrix is then normalized to calculate the weights of the evaluation objective. To ensure the reliability of the results, the consistency index (CI) and consistency ratio (CR) are solved for consistency checking. The relevant equations are presented as follows:

$$CI={\displaystyle \frac{{\lambda }_{\max }-n}{n-1}}$$

(23)

$$CR={\displaystyle \frac{CI}{RI}}$$

(24)

where λ_max is maximum characteristic roots and n is the total number of influencing factors, n = 4. RI is random consistency index, which is selected from

Table 5. RI index value table.

n	1	2	3	4	5	6	7	8	9
RI index	0	0	0.52	0.89	1.12	1.26	1.36	1.41	1.46

. If CR < 0.1, the consistency of the judgment matrix is considered acceptable [36].

3.3. EWM Introduction

The EWM method is an objective weighting method used to determine the weights of each evaluation objective in decision analysis. Firstly, the evaluation matrix is constructed, and the matrix is next normalized to eliminate any scale differences between the criteria. After the normalization, the entropy value of the different evaluation objectives is calculated, which reflects the degree of dispersion or uncertainty associated with the objective. Finally, the weight value of each evaluation objective is calculated. The equations used in this method are as follows:

$$M=\left[\begin{array}{cccc}{m}_{11}& m_{12}& \cdots & m_{1b}\\ m_{21}& m_{22}& \cdots & m_{2b}\\ ⋮& ⋮& \ddots & ⋮\\ m_{a1}& m_{a2}& \cdots & m_{ab}\end{array}\right]\underset{\mathrm{Normalizing} \mathrm{process}}{\overset{\left\{\begin{array}{c}{n}_{ij}={\displaystyle \frac{m_{ij}-{\left(m_{ij}\right)}_{\min }^j}{{\left(m_{ij}\right)}_{\max }^j-{\left(m_{ij}\right)}_{\min }^j}}\hspace{1em}(\mathrm{Positive} \mathrm{objective})\\ n_{ij}={\displaystyle \frac{{\left(m_{ij}\right)}_{\max }^j-m_{ij}}{{\left(m_{ij}\right)}_{\max }^j-{\left(m_{ij}\right)}_{\min }^j}}\hspace{1em}(\mathrm{Negative} \mathrm{objective})\end{array}\right.}{\to }}N=\left[\begin{array}{cccc}{n}_{11}& n_{12}& \cdots & n_{1b}\\ n_{21}& n_{22}& \cdots & n_{2b}\\ ⋮& ⋮& \ddots & ⋮\\ n_{a1}& n_{a2}& \cdots & n_{ab}\end{array}\right]$$

(25)

$$E_j=-{\displaystyle \frac{1}{\ln a}}{\displaystyle {\sum }_{i=1}^a{p}_{ij}}\ln p_{ij}$$

(26)

$$p_{ij}=n_{ij}/{\displaystyle {\sum }_{i=1}^a{n}_{ij}}$$

(27)

$$d_j=1-E_j$$

(28)

$$W_j=d_j/{\displaystyle {\sum }_{i=1}^a{d}_j}$$

(29)

where M is the original matrix, N is the normalization matrix, a is number of nodes, b is number of factors, m_ij is value of each parameter, E_j is entropy, d_j is degree of diversification, and W_j is weight.

3.4. Evaluation Objectives

The performance of PEMFCs is described from four key evaluation objectives, namely current density (I), pressure drop (ΔP), net output power (W_net), and non-uniformity of oxygen distribution (NU). The solutions for the last three objectives mentioned above are as follows:

(1) Pressure drop (ΔP) is solved by the pressure difference between the inlet (P_in) and outlet (P_out) on the cathode side, and the equation is as follows:

$$\mathrm{\Delta }P=P_{\mathrm{in}}-P_{\mathrm{out}}$$

(30)

(2) Net output power (W_net) is defined as the difference between the gross power produced (W_{fuel cell}) and the energy consumed (W_p) in the PEMFC, and the equation is as follows:

$$W_{\mathrm{net}}=W_{\mathrm{fuel} \mathrm{cell}}-W_{\mathrm{p}}$$

(31)

$$W_{\mathrm{p}}=\mathrm{\Delta }P\times A_{\mathrm{ch}}\times u_{\mathrm{in}}$$

(32)

where A_ch is the area of the inlet and u_in is the inlet velocity.

(3) Non-uniformity of oxygen distribution (NU) is calculated as the uniformity of oxygen distribution at the surface of GDL and CL at the cathode side, and the equation is as follows:

$$N=\sqrt{{\displaystyle \frac{{\displaystyle \int {\left(f-\overline{f}\right)}^{2}dS}}{{\overline{f}}^2{\displaystyle \int dS}}}}$$

(33)

$$\overline{f}={\displaystyle \frac{{\displaystyle \int fdS}}{{\displaystyle \int dS}}}$$

(34)

where f is the value of oxygen molar concentration.

4. Results and Discussion

4.1. Data Acquisition by OEM

According to the sequence of cases designed in the orthogonal table, numerical simulations were conducted for each set of parameter combinations. The relevant data obtained from these simulations were systematically recorded and are presented in

Table 6. Numerical results of orthogonal table.

Case No.	Evaluation Objectives
	I (A/cm2)	ΔP (Pa)	Wnet (W)	NU
1	1.1101	3.31	0.2043	0.3439
2	1.1098	4.76	0.2042	0.3418
3	1.0841	10.52	0.1995	0.3368
4	1.0632	29.81	0.1956	0.3385
5	0.8875	391.83	0.1631	0.3491
6	1.0661	6.58	0.1962	0.3363
7	1.0366	13.34	0.1907	0.3342
8	0.8310	133.85	0.1528	0.3347
9	1.1448	30.71	0.2106	0.3411
10	1.1168	5.76	0.2055	0.3439
11	0.8025	93.49	0.1476	0.3356
12	1.1338	17.45	0.2086	0.3400
13	1.1440	62.36	0.2105	0.3385
14	1.0821	9.43	0.1991	0.3411
15	1.0561	20.04	0.1943	0.3444
16	1.1170	11.32	0.2055	0.3400
17	1.0960	45.24	0.2016	0.3369
18	1.0279	9.58	0.1891	0.3364
19	0.8157	90.31	0.1500	0.3379
20	1.1406	13.66	0.2099	0.3439
21	1.0439	31.64	0.1921	0.3313
22	0.7665	59.81	0.1410	0.3426
23	1.1202	4.89	0.2061	0.3412
24	1.1266	9.53	0.2073	0.3395
25	1.1078	27.60	0.2038	0.3408

.

4.2. Analyses of AHP and EWM

4.2.1. Weights Calculation Using AHP

(1)

Construct and normalize the data judgment matrix

Experts were invited to score the current density, pressure drop, net power, and non-uniformity of oxygen concentration according to the nine-level scale method. The scoring results are listed in

Table 7. Scoring results from experts.

Objectives	I (A/cm2)	ΔP (Pa)	Wnet (W)	NU
I (A/cm2)	1	7	2	5
ΔP (Pa)	1/7	1	1/6	1/3
Wnet (W)	1/2	6	1	4
NU	1/5	3	1/4	1

. The scoring results were formed into a data judgment matrix A. Then, each column of the scoring results in Table 7 was normalized to form the normalized data, as listed in

Table 8. Normalized data.

Objectives	I (A/cm2)	ΔP (Pa)	Wnet (W)	NU
I (A/cm2)	0.543	0.412	0.584	0.484
ΔP (Pa)	0.078	0.059	0.049	0.032
Wnet (W)	0.271	0.353	0.293	0.387
NU	0.109	0.176	0.073	0.0967

. The method of processing data was to divide the data in each column by the sum of the columns to obtain normalized data.

The original judgment matrix A was as follows:

$$A=\left[\begin{array}{cccc}1\hfill & 7& 2& 5\\ 1/7& 1& 1/6& 1/3\\ 1/2& 6& 1& 4\\ 1/5& 3& 1/4& 1\end{array}\right]$$

(35)

(2)

Calculate weights of evaluation objectives

The weight of each evaluation objective was obtained by taking the average of the values in each row. The corresponding weights of evaluation objectives obtained were I = 0.51, ΔP = 0.05, W_net = 0.33, and NU = 0.11, respectively. These weights provided a quantitative basis for the overall performance evaluation framework.

(3)

Consistency checking

The weights solved above were formed into a weight vector w. The weight vector w = [0.51, 0.05, 0.33, 0.11]. Multiplying judgment matrix A with weight vector w:

$$Aw=\left[0.51, 0.05, 0.33, 0.11\right]\left[\begin{array}{cccc}1\hfill & 7& 2& 5\\ 1/7& 1& 1/6& 1/3\\ 1/2& 6& 1& 4\\ 1/5& 3& 1/4& 1\end{array}\right]=\left[\begin{array}{c}2.107\\ 0.219\\ 1.360\\ 0.459\end{array}\right]$$

(36)

Further, solving for the maximum characteristic roots λ_max:

$${\lambda }_{\max }={\displaystyle \frac{1}{n}}{\displaystyle \sum _{n=1}^n\frac{{\left(Aw\right)}_i}{w_i}}={\displaystyle \frac{1}{4}}\left(4.165+4.027+4.1716+4.040\right)=4.101$$

(37)

According to Equations (23) and (24), the value of CR was calculated to be approximately 0.0374, which was well below the threshold of 0.1. This low CR value indicates that the judgment matrix performs a high level of internal consistency, and the calculated weights were acceptable.

4.2.2. Weights Calculation Using EWM

(1)

Construct and normalize the original data matrix

The original data matrix M was composed of 25 groups of orthogonal experimental data with four evaluation objects. Then the Min–Max normalization method was used to normalize the original matrix, rescaling the data to a uniform range, typically between 0 and 1, to obtain the normalized matrix N. The results are shown below (only partial normalized matrix data is displayed):

$$M=\left[\begin{array}{cccc}1.1101& 3.31& 0.2043& 0.3439\\ 1.1098& 4.76& 0.2042& 0.3418\\ ⋮& ⋮& ⋮& ⋮\\ 1.1266& 9.53& 0.2073& 0.3395\\ 1.1078& 27.60& 0.2038& 0.3408\end{array}\right]\to N=\left[\begin{array}{cccc}0.9083& 1& 0.9095& 0.2921\\ 0.9075& 0.9963& 0.9080& 0.4101\\ ⋮& ⋮& ⋮& ⋮\\ 0.9519& 0.9840& 0.9526& 0.5393\\ 0.9022& 0.9375& 0.9023& 0.4663\end{array}\right]$$

(38)

(2)

Calculate the E_j and d_j for each criterion

According to Equations (26)–(28), the entropy E_j and degree of diversification d_j of each evaluation object were calculated as shown in

Table 9. The values of E_j and d_j.

	I (A/cm2)	ΔP (Pa)	Wnet (W)	NU
Ej	0.9595	0.9859	0.9593	0.9688
dj	0.04054	0.0141	0.0407	0.0312

.

(3)

Calculate the W_j for each criterion

According to Equation (29), the value of weight for each criterion was calculated as I = 0.32, ΔP = 0.11, W_net = 0.32, and NU = 0.25, respectively.

4.2.3. Determination of Comprehensive Weights by AHP-EWM

The comprehensive weight (W_com) is solved by Equation (39), which is obtained from the weight data calculated by the AHP method (W_AHP) and EWM method (W_EWM), respectively. This method has been applied in multiple fields [33,37]. By integrating these two methods, the AHP-EWM method balances subjective expert judgment with objective data dispersion, resulting in a more robust weighting model. The weights solved by AHP, EWM, and AHP-EWM are listed in

Table 10. Weight of AHP, EWM, and AHP-EWM.

Methods	Weight Values
	I (A/cm2)	ΔP (Pa)	Wnet (W)	NU
AHP	0.51	0.05	0.33	0.11
EWM	0.32	0.11	0.32	0.25
AHP-EWM	0.415	0.08	0.325	0.18

.

$$W_{\mathrm{com}}=\alpha \cdot W_{\mathrm{AHP}}+\beta \cdot W_{\mathrm{EWM}}$$

(39)

$$\beta =1-\alpha$$

(40)

where α and β are the resolution factor, with α usually taken as 0.5 [37].

4.3. Analysis of ANOR and ANOVA

Based on the orthogonal experimental data in Section 4.1, the comprehensive score calculated by the AHP-EWM weight method was used as the main evaluation object of PEMFCs, as listed in

Table 11. AHP-EWM comprehensive score of orthogonal tables.

Case No.	Comprehensive Score	Case No.	Comprehensive Score
1	0.8538	14	1.3296
2	0.9693	15	2.1666
3	1.4170	16	1.4971
4	2.9505	17	4.2002
5	31.8306	18	1.3150
6	1.0931	19	7.6729
7	1.6195	20	1.6963
8	11.1628	21	3.0865
9	3.0617	22	5.2104
10	1.0530	23	0.9845
11	7.9206	24	1.3584
12	1.9955	25	2.7953
13	5.5929

and Figure 5. To further investigate the influence of different factors, the ANOR and ANOVA were used to analyze the above data by Minitab 21.

The range value R represents the difference between the maximum value and the minimum value of the average numerical simulation results at each level under single-factor conditions. The larger the range value R, the greater the influence weight of this factor on performance objectives. The results of ANOR for the comprehensive score are listed in

Table 12. Result of ANOR.

Objectives		Factors
		S	N	H	CRWR Value
Comprehensive score	k1	38.0212	14.4511	9.7618	8.5918
	k2	17.9901	13.9949	12.8864	10.4707
	k3	19.0052	20.4722	14.0118	10.8352
	k4	16.3815	16.3731	20.4012	11.1381
	k5	13.4351	39.5418	47.7719	63.7973
	k1/4	7.6042	2.8902	1.9524	1.7184
	k2/4	3.5980	2.7990	2.5773	2.0941
	k3/4	3.8010	4.0944	2.8024	2.1670
	k4/4	3.2763	3.2746	4.0802	2.2276
	k5/4	2.6870	7.9084	9.5544	12.7595
	R	4.9172	5.1094	7.6020	11.0411

. According to the data of ANOR, the influence ability of each factor was CRWR value > H > N > S. The CRWR value affects the performance of PEMFCs by influencing electron transport, reactant distribution, and pressure loss. A higher CRWR value will reduce contact area and increase Ohmic losses, but it will improve reactant distribution and reduce pressure losses, although this may lead to uneven flow distribution. In contrast, a lower CRWR value enhances electron conduction and water removal, but increases local resistance. Subsequently, the ANOVA was further conducted. With a degree of freedom (DOF) consideration, the critical F-value at α = 0.025 was F_{(0.025, 4, 8)} = 5.053, and the calculated ANOVA results are shown in

Table 13. Result of ANOVA.

Objectives	Variable	DOF	SS	MS	F Value	Significance
Comprehensive score	S	4	76.26	19.07	1.08
	N	4	91.49	22.87	1.30
	H	4	191.61	47.90	2.71
	CRWR value	4	459.40	114.85	6.51	*
	Error	8	141.24	17.66
	Total	24	960.00

“*” represents significance.

. According to the data of ANOVA, the influence ability of each factor was CRWR value > H > N > S. The influence orders of ANOR and ANOVA were consistent.

In addition, Figure 6 illustrates the effect curve of ANOR analysis, which identifies the optimal design parameter combination for the PEMFC configuration. This optimal design parameter combination is triangular blockage with a number of nine, a height of 0.8 mm, and a CRWR value of 0.25. The optimal combination corresponds to Case No. 5. Based on the comprehensive weighting calculated using the AHP-EWM method, this configuration achieved a comprehensive score of 31.8306, which is the highest score among the 25 groups of orthogonal experiments.

The OEM-AHP-EWM method was introduced in this paper, which was utilized to provide a multi-dimensional evaluation framework. This comprehensive evaluation capability makes it possible to fully consider the influence of different factors on the performance of PEMFCs, thereby leading to more accurate optimization conclusions.

5. Conclusions

In this paper, the combined effects of the shape, number, height of blockage, and CRWR value on the performance of PEMFCs were systematically investigated using OEM. Additionally, multi-objective optimization and performance evaluation of PEMFC were conducted by integrating the AHP and EWM. The main conclusions are as follows:

(1)

The AHP-EWM analysis revealed that current density (I) and net output power (W_net) are the most critical factors in PEMFC performance evaluation, with weights of 0.415 and 0.325, respectively. In contrast, the weights assigned to pressure drop (ΔP) and non-uniformity of oxygen distribution (NU) were 0.08 and 0.18, emphasizing the dominant role of current density in the overall performance evaluation.

(2)

Based on the determined weight distribution, the 25 groups of orthogonal experiments were comprehensively scored. These scores were then analyzed using ANOR and ANOVA methods. The significance ranking of the four design parameters was CRWR value > H > N > S, indicating that CRWR value yields the greatest impact on the comprehensive performance of PEMFC.

(3)

The optimal combination of design parameters was identified as a triangular blockage shape, with nine blockages, a height of 0.8 mm, and a CRWR value of 0.25. This configuration achieved the highest comprehensive score of 31.8306, demonstrating the best balance among current density, net output power, pressure drop, and oxygen distribution uniformity.