Fuel cells with conventional channel and tapered channel are schematically shown in Fig. 1. The basic geometric parameters are listed in Table 1. The inlet geometric parameters of the PEMFC with conventional channel and tapered channel are identical, while the outlet geometric parameters remain unchanged. The ratio of the outlet cross-sectional height to the inlet cross-sectional height for the tapered flow channel is defined as π=H_out/H_in. The PEMFC primarily consists of seven parts: the anode and cathode gas channels, the anode and cathode gas diffusion electrodes (GDEs), the anode and cathode catalyst layers (CLs), and the proton exchange membrane. H₂ enters the anode, passes through the gas diffusion layer to reach the catalyst layer, where catalyzed by the anode catalyst, a hydrogen molecule is oxidized into two protons (H⁺), releasing two electrons. Protons are transported through the electrolyte membrane to the cathode via hydration, while electrons travel through the diffusion layer and bipolar plate to the external circuit, eventually reaching the cathode catalyst layer. Oxygen or air enters the cathode, passing through the cathode diffusion layer, undergoes a reduction reaction at the cathode catalyst layer, reacting with protons and electrons to form H₂O (liquid or vapor). Compared to the anode, the slower cathode reaction is the main factor limiting PEMFC performance [14]. Therefore, the cathode side of a high-temperature PEMFC is selected as the research subject in the paper. Moreover, this paper focuses on high-temperature proton exchange membrane fuel cells (HT-PEMFCs), which operate at a working temperature of 373 K. It is assumed that the generated water remains in a gaseous state throughout the process, and the issue of water phase change does not arise.

The mass, heat, and charge transport processes within the PEMFC are governed by the following conservation equations: mass conservation, momentum conservation, species conservation, energy conservation, liquid water and dissolved water conservation, and proton and electron transport equations, as detailed in Refs. [15–17].

Mass conservation equation: (1)∇⋅(ερu→)=Smwhere ρ is gas density, u→ is gas velocity, and S_m is the mass source term.

Momentum conservation equation: (2)∇⋅(ερu→u→)=−ε∇p+∇⋅(εμ∇u→)+Smomwhere ε is porosity, p is pressure, μ is gas dynamic viscosity, and S_mom is the momentum source term.

Energy conservation equation: (3)∇⋅(ερcpu→T)=∇⋅(keff∇T)+SEwhere c_p is specific heat capacity, T is temperature, k_eff is effective thermal conductivity, and S_E is the energy source term.

Species transport equation: (4)∇⋅(u→εci)=∇⋅(Dieff∇ci)+Siwhere c_i is species volume fraction, Dieff is effective species diffusion coefficient, and S_i is the species source term.

Charge conservation equation: (5)∇⋅(σsol,mem∇φsol,mem)+R¯sol,mem=0where the subscripts sol and mem represent the solid phase and membrane phase, respectively; σ, φ and R¯ represent conductivity, potential, and volumetric transfer current, respectively.

Water saturation equation: (6)∇⋅[ρlKsγμldpcds∇⋅s]=rwwhere s is water saturation, r_w is condensation rate, and P_c is capillary pressure. Table 2 shows the Source terms for governing equations in the model.

The local current density is defined by the Butler-Volmer equation: (7)iloc=i0CO2CO2ref(exp⁡(αcRgTFηc)−exp⁡(−1−αcRgTFηc))where i₀ represents the exchange current density, CO2 and CO2ref represent the oxygen molar concentration and reference oxygen molar concentration, respectively. ηc represents the cathode overpotential.

The cathode overpotential is calculated as follows: (8)ηc=ENernst−Vcell−R0ilocwhere E_Nernst is the Nernst potential, R₀ is the resistance, and V_cell is the actual operating voltage. The Nernst potential is calculated as follows: (9)ENernst=1.229−0.87×10−3(T−298.15)+RT2F[ln⁡pH2+12ln⁡pO2]where pH2 and pO2 are the partial pressures of hydrogen and oxygen, respectively, R is the universal gas constant, and F is the Faraday constant.

The average current density of the fuel cell i¯loc, and the standard deviation of the current density distribution, θiloc, are described as follows: (10)i¯loc=∫ilocds∫ds (11)θiloc=∫(iloc−i¯loc)2ds∫ds

The model operating conditions and inlet parameters are shown in Table 3. Mass flow rate is used at the inlet, and the gas outlet is set to a pressure boundary condition, fixed at 1 atm. The channel walls are no-slip boundaries, while slip boundary conditions are considered at the electrodes. Moreover, the mass fractions of oxygen, water, and nitrogen being 0.228, 0.023, and 0.749, respectively. The outlet condition is atmospheric pressure, and the manifold effect is not considered. Thermal convection occurs on the wall surfaces of the flow channels, while the influence of thermal radiation is neglected.

The computational grid for the PEMFC model with tapered channel is shown in Fig. 2. A structured grid was used. Since the diffusion layer is thin, the grid was refined at the interface between the diffusion layer and the channel to improve computational accuracy. Besides, when the PEMFC operating voltage is 0.4 V, the relative errors of current density between grid systems with 1120, 22,400 and 51,200 elements are limited within 3%. Therefore, the grid system with 22,400 elements is dense enough and adopted in subsequent study to save computational resources and time while ensuring result accuracy.

To ensure the reliability of the numerical model, the model geometry, operating conditions, and other key parameters were kept consistent with the experiment of Haghayagh et al. [18]. The polarization curve obtained from the numerical simulation in this study and the experimental data from Haghayagh is shown in Fig. 3. From Fig. 3, it can be observed that the simulation results closely match the experimental data, with a maximum error not exceeding 6%. Therefore, the model is suitable for simulating the performance of PEMFC.

This section investigates the effects of varying mass flow rates and height ratios (as defined in Table 1) on the performance of the PEMFC with a tapered channel, including current density, standard deviation, and channel pressure drop. In this study, the values of height ratio of channel inlet and outlet π were set to 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0, while maintaining a constant channel inlet height. The values of inlet mass flow rate m were set to 2.5e−7, 5.0e−7, 7.5e−7, 1.0e−6, and 1.25e−6 kg/s, respectively. The description of cases is provided in Table 4, labeled A~E for increasing values of m, and 1~10 for increasing values of π. Unless otherwise specified, the cell operating voltage in this paper is 0.4 V.

This section aims to develop a high-precision prediction model for the average current density, current density standard deviation, and pressure drop of the PEMFC, which is a multi-input multi-output problem. The BP neural network is a multi-layer feedforward network with high self-learning and adaptive capabilities. It typically consists of an input layer, hidden layers, and an output layer. The feedforward network sequentially transmits information from the input layer to the hidden layers and subsequently to the output layer. Different layers are interconnected via neurons, where each neuron structure is used for information reception and transmission and can be regarded as a specific output function—the activation function (AF). A commonly used activation function is defined as follows: (13)f(x)=1/(1+e−x)

Based on the 50 cases from Section 3.1, the height ratio and mass flow rate are selected as input parameters, and average current density, standard deviation, and channel pressure drop of the PEMFC are selected as output parameters. Therefore, the number of neurons in the input and output layers is 2 and 3, respectively. The number of neurons in the hidden layer is determined using the following empirical formula: (14)i′=2s′+1 or i′=s′+t+awhere i′ is the number of input neurons, s′ is the number of hidden neurons, t is the number of output neurons, and a is an integer between [0, 10].

The 50 datasets obtained in Section 3.1 are randomly divided for training, testing, and validation sets, accounting for 80%, 10%, and 10%, respectively. The validation set is utilized to prevent model overfitting. The topology of the BP neural network is illustrated in Fig. 10.

The accuracy of the neural network is evaluated using the Mean Relative Error (MRE) and the Coefficient of Determination (R²), defined as follows: (15)MRE=1N∑i˙=1N|yi−yi−yi|×100%

(16)R2=1−∑i=1n(yi−yi−)2∑i=1n(y¯−yi)2where N represents the number of test samples, y_i represents the true value, yi− represents the model predicted value, and y¯ represents the average of the true values. MRE indicates the deviation between the true and predicted values, and R₂ is used to measure the degree to which the model’s predicted output explains the actual value. In general, the closer the MRE is to 0 and the closer R² is to 1, the higher the reliability of the model.

The selection of the neural network architecture is mainly based on three factors: topology, training algorithm, and activation function. The topology primarily considers the number of hidden layers and neurons. Since there are 2 input variables and 3 output parameters, the input layer and output layer are designed with 2 and 3 neurons, respectively. The activation function enables the neural network to handle nonlinear problems and enhances its function approximation capability. Both the training algorithm and hyperparameters also exert an impact on the convergence and accuracy of the model. Referring to relevant studies and preliminary work, the hyperparameter settings of the BP neural network in this study are as follows: the number of hidden layer neurons is set to 12, which achieves a good balance between network complexity and generalization ability. The tansig function is adopted as the activation function, the number of training epochs is set to 2000, and the learning rate is 0.001. Fig. 11 presents a comparison between the numerical simulation results and the predictions of the surrogate model. Based on the calculation, the MRE of current density, standard deviation, and pressure drop are 0.84%, 1.44%, and 1.77%, respectively. And R² are all above 0.999. Therefore, the prediction model proposed in this study is considered capable of providing design guidance for PEMFC performance optimization.

This section combines the BP neural network and NSGA-II to optimize the performance of PEMFC. Based on the well-trained neural network model, the genetic algorithm is employed to perform online optimization of the Pareto optimal solutions for the PEMFC current density, standard deviation, and pressure drop.

During the search process for various possible optimal designs of the PEMFC using the genetic algorithm, a large number of possible solutions are generated. The set of non-dominated solutions is referred to as the Pareto solution set. Since different weights can be assigned to each objective parameter under different scenarios, the genetic algorithm solving the comprehensive objective function can obtain a set of Pareto solutions and one optimal solution that minimizes the comprehensive objective function for each scenario. The objective functions in this study are maximum average current density, minimum standard deviation, and minimum channel pressure drop.

The Pareto solution set obtained in this study is presented in Fig. 12. To decide an optimal tapered channel solution among the three objectives, the LINMAP method is employed to perform a trade-off between current density, standard deviation and pressure drop. The LINMAP method identifies the optimal solution by calculating the Euclidean distance between the Pareto and ideal solutions [20]. The optimal tapered channel solution in this study (marked by a star in Fig. 12) is 3161.28 A/m² for current density, 53.31 A/m² for current density standard deviation, 5.65 Pa for pressure drop when π = 0.385, m = 12.46e−7 kg/s. The CFD simulation results for the optimal model (with a height ratio of 0.385) were also obtained using the input parameters corresponding to the Pareto optimal solution. As shown in Table 5, the CFD numerical simulation of the PEMFC at the optimal height ratio yielded the average current density of 3141.41 A/m², the standard deviation of 53.58 A/m², and the pressure drop of 5.49 Pa. Additionally, the CFD numerical simulation of the PEMFC at π = 1 (for the conventional channel) were also obtained, which yielded the average current density of 2935.31 A/m², the current density standard deviation of 55.63 A/m², and the pressure drop of 1.54 Pa. Fig. 13 illustrates the current density distribution of the PEMFC with conventional channel and optimal tapered channel at an operating voltage of 0.4 V. It can be seen that during the reaction, the current density reaches its maximum at the channel inlet and gradually decreases along the gas flow direction. Compared with the conventional channel, although the pressure drop of the PEMFC with the optimal tapered channel increases, the magnitude of the increase is relatively small. Therefore, the increase in channel pressure drop due to height ratio reduction remains within a reasonable range. Compared to the results of the conventional channel, the current density increased by 7.02% and the current density standard deviation decreased by 3.7%, while the pressure drop increased 4.11 Pa. Therefore, the channel with the optimal height ratio can achieve higher current density and power density while enhancing the uniformity of the cell current distribution.