Progressive topology-curvature optimization of flow channel for PEMFC and performance assessment

Abstract

The curved bending regions of serpentine flow channels play a crucial role in mass transfer and the overall performance of the flow field in proton exchange membrane fuel cells (PEMFCs). This paper proposes a “2D Topology-Curvature Optimization” progressive design method to optimize the bend area structures, aiming to enhance PEMFC performance. Through numerical simulations, it compares the topology-curvature optimization model with both the algorithm-based optimization model and a validation model, and analyzes the mass transfer, heat transfer characteristics, and output performance of PEMFC under different flow fields. The results indicate that the optimized structures improve convection and diffusion within the flow field, effectively enhancing the transport and distribution of oxygen and water within the PEMFC. Performance improvements, ranked from highest to lowest, are TS-III > MD-G (Model-GA) > MD-P (Model-PSO) > TS-II > TS-I. Among the optimized models, TS-III (Topology Structure-III) exhibits the greatest increases in peak current density and peak power density, with improvement of 4.72% and 3.12%, respectively. When considering the relationship between performance improvement and pressure drop using the efficiency evaluation criterion (EEC), TS-II demonstrates the best overall performance.

Graphical abstract

Keywords

proton exchange membrane fuel cell (PEMFC) / structural topology / curvature design / progressive optimization / output performance / computational fluid dynamics (CFD)

1 Introduction

In the pursuit of carbon neutrality, the world is undergoing a third energy revolution, primarily driven by renewable energy sources and supported by various types of batteries as the main power source. Energy carriers for this transformation will predominantly be electricity and hydrogen [1]. Hydrogen energy, as a zero-carbon energy carrier, is an ideal medium for addressing global climate change and achieving the decarbonization of the energy system. Additionally, with its high-energy-density, green characteristics, and efficiency, hydrogen is playing an increasingly important role in the restructuring of global energy systems and industrial ecology. It is widely used in critical sectors such as industrial manufacturing, transportation, power generation, and heating [2,3].

Hydrogen fuel cells, as a new form of electrochemical power generation, operate on the basic principle of hydrogen reacting with oxygen in the presence of a catalyst to produce water and release electrical energy. This process is clean, pollution-free, and highly efficient in energy conversion, making hydrogen fuel cells a promising green energy power generation technology. Efforts to improve energy efficiency, optimize structures, and foster various forms of collaboration are actively promoting the adoption of hydrogen fuel cells in sectors such as aerospace, vehicles, ships, and distributed power stations [4]. Especially, the transportation sector has benefited from the use of batteries, which has not only effectively reduced greenhouse gas emissions from vehicles but also facilitated the utilization of renewable energy sources [5].

Currently, hydrogen fuel cells are mainly categorized into proton exchange membrane fuel cells (PEMFCs) and solid oxide fuel cells (SOFCs) [6]. Among these, PEMFCs are the most commonly used in various transportation and industrial engineering applications, especially in new energy vehicles due to their advantages of low emissions, high efficiency, rapid start-up, and quiet operation [7]. However, the performance of fuel cells is limited by factors such as kinetic characteristics, power density, and cost. As a key technology for new energy vehicles, PEMFCs still face significant challenges [8].

As the core component of PEMFC, the bipolar plates in PEMFCs play a crucial role in determining the distribution of reactants and products in the flow field, as well as in charge and heat transfer. As a comparatively ideal and widely used design structure, the serpentine flow field faces challenges in the bend areas of the flow channels. These areas experience complex mass transfer processes and low temperature-pressure-electric conversion efficiency, which directly impact the long-term operational performance and stability of the fuel cell.

Numerous studies have been conducted to optimize and improve the serpentine flow field structure and features for PEMFC. Researchers have employed both experimental or simulation methods to develop innovative bionic flow field designs and improve conventional flow channels. Novel and bionic flow channels, such as waves [9], lotuses [10], and honeycombs [11], have been proposed for PEMFCs. These studies analyze various factors, including internal complex flows, gas concentration distributions, current densities, and pressure drops. The results show that new flow field structures have various impacts on fuel cell performance, offering both improvements and drawbacks in related performance.

For instance, Shen et al. [12] proposed a novel three-dimensional optimized flow field that enhances the mass transfer and water transport properties, thereby improving the performance of the battery. The superiority of this design in strengthening mass transfer was evaluated using the electromagnetic transport coefficient (EMTC). Mahmoudimehr and Daryadel [13] and Mohammedi et al. [14] studied the impact of flow channel cross-sectional shapes on the performance of PEMFC. Rahimi-Esbo et al. [15] and Zhang et al. [16] investigated the effects of various channel width-to-rib ratios on mass concentration and current density distribution. Their findings indicated that an appropriate rib-width ratio can significantly enhance mass transfer performance.

Additionally, Ebrahimzadeh et al. [17] discovered that the highest current density in the flow channel occurred when the obstacle had a rectangular shape. These studies underscore the importance of both the overall design of the flow field and the change of local parameters in influencing mass transfer characteristics. A well-designed configuration can improve the performance of PEMFCs across multiple parameters.

Research on PEMFC channels often relies on traditional design methods, such as experimental testing and simulation parameter analysis. However, these approaches are typically unable to predict the impact of complex subjective and objective factors, leading to higher trial-and-error costs and longer development times. To address these challenges, researchers have increasingly employed numerical optimization algorithms and computational models to optimize the design of fuel cells and their associated systems. Methods such as genetic algorithms, particle swarm optimization (PSO), and simulated annealing have been used to solve optimization problems. These approaches are often combined with machine learning-based predictive regression models to achieve improved optimization and prediction accuracy [18,19].

For example, Chang [20] successfully applied genetic algorithms to optimize the geometric structure of fuel cells and developed a genetic algorithm neural network (GANN) model that accurately predicts the output voltage of PEMFCs. Seyhan et al. [21] utilized an artificial neural network (ANN) to optimize the parameters of serpentine flow channels with sinusoidal ribs in predicting the output current. Their predictions were found to align closely with experimental results. Eteiba et al. [22] conducted a comparative study involving multiple optimization techniques, including flower pollination algorithm (FPA) and artificial bee colony (ABC), to determine the optimal method for quickly sizing battery hybrid systems. Samy et al. [23] proposed the FPA optimization algorithm and compared it with other algorithms, such as ABC and PSO for optimizing a fuel cell hybrid system, demonstrating the effectiveness of their design.

Furthermore, the authors propose a multi-objective particle swarm optimization (MOPSO) algorithm to determine the optimal sizing of fuel cell systems [24]. Superior optimization algorithms are capable of automatically handling high-dimensional datasets and performing predictive optimization for target problems, minimizing the need for human intervention. However, these algorithms require validation data for support, are highly dependent on parameter selection and function design, and can consume significant computational resources during the optimization process.

Topology optimization is an advanced design technique that generates topological configurations without prior knowledge [25]. By offering greater design freedom, topology optimization can push the boundaries of traditional design methods [26], creating innovative solutions that can be realized through techniques like additive manufacturing [27]. Common fluid topology optimization methods include the density-based method, the level set method, the moving morphable component (MMC), bidirectional progressive structure optimization (BESO), among others. Fluid topology modeling generally includes 3D, 2D, and simplified models [28].

3D models provide high precision, but come with substantial computational demands and costs, and their complex microstructures may pose challenges for manufacturability. 2D models, on the other hand, require less computation, enabling quick design iterations, but tend to overlook changes in the thickness direction, reducing accuracy. Simplified models make appropriate assumptions based on structural characteristics to simplify the computational process. However, these models can introduce errors by neglecting important optimization details, potentially affecting the final design performance.

Currently, both academic and industrial sectors are increasingly focusing on optimizing the topology of fluid-solid, fluid-heat, and fluid-solid-heat couplings, especially in the fields of electronic chips, aerospace, and advanced equipment, with research primarily centering on optimizing radiators, heat exchangers, and moving components. Borrvall and Petersson [29] innovatively incorporated the Brinkman penalty term into the Stokes equation, significantly increasing the flow velocity in the solid domain relative to the fluid domain, enabling topology optimization for fluid flow channels. Olesen et al. [30] focused on minimizing energy loss by optimizing the topology of flow channels with varying Reynolds numbers. Dede and Liu [31] utilized the solid isotropic material penalty (SIMP) method for enhancing the microchannel configuration, achieving significant improvements in heat transfer and pressure reduction.

Yu et al. [32] adopted the MMC method to optimize channel configurations for better heat transfer and reduced fluid energy usage. Feppon et al. [33] utilized the level set method to perform topology optimization for radiator channels. Yao et al. [34] and Zhao et al. [35] developed a two-dimensional phase-change heat storage unit fin and flow model based on the topology optimization theory, confirming the reliability of topology optimization in structural design and its ability to optimize fin and flow structures with a high degree of freedom. Wang et al. [36] developed an adversarial neural network (GAN) prediction model based on a small sample dataset, enabling quick and accurate optimization of the coolant channel structure in PEMFCs for superior cooling performance. This approach was compared with traditional thermal-fluid structure topology methods.

Although many studies have focused on the design of flow fields in PEMFCs, few have utilized topology optimization methods to enhance flow field performance, particularly in the curved bending regions of serpentine flow channels, where effects are notable. To rapidly and accurately develop a bend structure model that enhances PEMFC performance, this paper proposes a new “2D topology-curvature optimization” method for progressive design. For comparison, Kriging surrogate models were integrated with genetic algorithms and PSO algorithms to determine the optimal parameters for the bend structures. Through numerical simulations, the optimized models obtained from both methods were compared and evaluated against a verification model. The results demonstrate that the “2D topology-curvature optimization” method can quickly and accurately generate predictive, optimized structural models, providing valuable insights for the design and intelligent manufacturing of PEMFCs.

2 Model and verification

2.1 Geometric model

In this paper, the serpentine flow field PEMFC experimental model (Fig.1(a)) from Wang et al. [37] is used to better evaluate the structural advantages and disadvantages of topologically derived bend area structures compared to conventional designs. The single-cell model of the PEMFC used in this paper is depicted in Fig.1(b). The cell system consists of the bipolar plate (BP), gas channel (GC), gas diffusion layer (GDL), catalytic layer (CL), and proton exchange membrane (PEM). The geometric and material parameters of the PEMFC are provided in Tab.1 and Tab.2.

2.2 Mathematical model

2.2.1 Model assumptions and boundary conditions

To ensure the consistency of physical property conditions and simplify the complex simulation process, it is assumed that the PEMFC operates under constant temperature, steady-state conditions, with electrical insulation on the outer shell; all gas components in the model are treated as incompressible ideal gases, and the flow is laminar flow; the porous properties of GDL, CL, and PEM are isotropic in the X-Y plane; and gravity effects are neglected.

The inlet temperature of the model flow field is maintained at room temperature, with natural convection heat transfer occurring on the outer wall. The gas velocity at the inlet is kept constant, and the flow rate is independent of other physical properties. Variables such as temperature, pressure, velocity, and voltage within the system are solved iteratively using first-order boundary conditions. Voltage-assisted scanning is performed at intervals of 0.1 V, with an operating voltage range from 0.9 to 0.2 V. The specific operational parameters are listed in Tab.3.

2.2.2 Governing equations of PEMFC

The primary elements of the transport mechanisms, along with the coupled heat and mass transfer processes in the model, are governed by the conservation equations for mass, momentum, charge, and component transport. As detailed in Tab.4, the mass conservation equation applies mainly to the two-phase flow of gas and liquid in the system. The anode gas consists mainly of hydrogen and water vapor, while the cathode gas consists mainly of oxygen, nitrogen, and water vapor, as shown in Eq. (1), where the source term S_m is related to mass. The momentum conservation equation for material transport in porous media is described by the Brinkman equation, as shown in Eq. (2), where the source term S_u is related to velocity. The energy conservation equation considers the effective heat transfer in mixed cold-hot fluid media, as shown in Eq. (3), where the source term S_q is related to the quantity of heat. The component conservation equation in porous media is provided in Eq. (4), while the gas diffusion equation is presented in Eq. (5), where, the subscript i indicates a specific component or species, representing the ith component. Water transport within the system significantly impacts the heat transfer process and efficiency of the PEMFC. Considering effects such as capillary pressure and diffusion permeation following gas–liquid phase change, the formulas for liquid water transport in channels and porous media are shown in Eqs. (6) and (7). The charge conservation equations are given in Eqs. (8) and (9). The Butler-Volmer equation is used to describe the electrode reaction on the surface of the catalytic layer, with the relevant equations shown in Eqs. (10) and (11). The parameters for these control equations are tabulated in Tab.5.

2.3 Grid independence test and model validation

To eliminate the influence of grid resolution on the simulation results, five grid division schemes with different densities were adopted, and the grid independence of the model was verified under the same operating conditions. Based on the number of Grid 4, the relative error (δ) of the polarization curve at high current density was calculated, as shown in Fig.2(a), where N is the number of grids. The results show that as the number of grids increases, the relative error gradually decreases, with minimal fluctuations. Considering both computational efficiency and accuracy, Grid 4 was selected for the numerical simulation, with the total number of grids in the entire PEMFC model estimated to be approximately 723270.

To ensure the reliability of the simulation calculation method and the accuracy of the results, the simulated polarization curve was compared with the experimental data recorded by Wang et al. [37]. As shown in Fig.2(b), the simulation results are basically consistent with the experimental data, confirming the feasibility and effectiveness of the geometric model and the simulation method used in this paper. This indicates that the model proposed can accurately assess the performance of the PEMFC.

3 Structure optimization

3.1 2-D topology and curvature optimization

Using the rational approximation of material properties (RAMPs) method in the variable density approach, under the constraint of material distribution volume, a fluid topology optimization model is established based on the steady-state N-S equation, with the goal of minimizing energy dissipation in the curved flow channel area. By optimizing the boundary morphology through curve curvature control, advanced surface modeling techniques, such as computer-aided geometric design (CAGD) , are applied to construct the 3D flow channel structure and the PEMFC single-cell model.

3.1.1 2-D corner domain model

To expedite computations and achieve a rapid and accurate forecast of the corner configuration, the 3D layout of the corner area (shown in red wireframe in Fig.3(a)) was simplified into a 2D model, as depicted in Fig.3(b). The width of the flow channel is 1 mm, with the corner area designated as the design region, spanning 2 mm × 5 mm, aligning with the structural dimensions and material characteristics of the 3D model flow channel, with water as the fluid. The inlet boundary conditions are set to fully developed flow with a velocity of u_in, the outlet pressure is 0 Pa, and all fluid–solid interfaces are assigned to no-slip boundary conditions.

3.1.2 Governing equations of topology

The flow problem for 2D topology optimization is conducted in steady-state incompressible laminar flow, governed by the steady-state incompressible N-S equation. Based on the RAMP interpolation method, the relationship between the model material properties of the model and the topological design variables is established, enabling the optimal distribution in the design domain. As outlined in Tab.6, the continuity equation is presented in Eq. (12), and the momentum equation is formulated using the Brinkman equation, incorporating the reverse osmosis coefficient term as shown in Eq. (13). The interpolation function is provided in Eq. (15). Here q_a is the Darcy penalty strength constant, and the curvature of α(γ) can be adjusted by modifying the value of q_a. The topological design variable, γ ∈ [0,1]. When γ = 1, α_min represents the fluid domain; γ = 0, α_max represents the solid domain. When γ falls between 0 and 1, the microelement region behaves as a porous medium. In the design domain, the volume force f acting on the element region is simulated to perform the topology optimization calculation. The optimization process incorporates the relationship between α_max and the dimensionless Darcy number Da, as shown in Eq. (17). The relevant boundary conditions are described by Eqs. (18) to (20), with a no-slip velocity condition (velocity = 0). The energy equation, shown in Eq. (21), adopts the steady-state heat conduction equation, where the heat transfer process is influenced by the heat capacity and flow rate of the fluid. To optimize the energy dissipation in the design domain, an effective thermal conductivity difference function containing the design variable γ is introduced in Eq. (22), where q_k is the penalty factor for the function k(γ), and it has the same effect as q_a. Furthermore, to avoid ill-conditioned numerical issues in the optimization framework, the Helmholtz filtering equation is used to filter the topology optimization results, as shown in Eq. (23). The design domain is then filtered using a hyperbolic tangent projection to reduce the fuzzy region between the fluid and solid phases, resulting in a more clearly defined topological layout, as shown in Eq. (24). The parameters for the governing equation are provided in Tab.7.

3.1.3 Curvature optimization control method

The structure of the corner area after topology optimization exhibits an irregular shape with rough, uneven boundaries that are not smooth. This irregularity poses challenges for traditional processing and additive manufacturing techniques to form flow channel entities. Therefore, it is necessary to further optimize the topology boundary through a curvature design method to achieve a more reasonable and manufacturing-friendly configuration. This is accomplished by generating Bezier interpolation spline curves and adjusting the shape of the curves to create smoother, rounded corner. The Bezier curve, defined with p_i as the control point and B_j(t) as the basis function, is

(1)p_i\left(t{;\alpha }_i,{\alpha }_{i+1}\right)={\sum }_{j=0}^n{B}_j\left(t\right)V_j,\text{t}\in \left[0,1\right].

The curve can achieve G2 continuity and has global or local parameter tunability [38]. The control form of boundary curve is shown in Fig.4.

3.1.4 Topology optimization forming process

To avoid the local differential solutions, a parameter continuity strategy is used to regulate the nonlinearity of the optimization objective. The optimization process is divided into three continuous stages [39,40], with the result of stage each previous stage serving as the initial value for the subsequent stage. Each stage is defined by the convergence of the Karush-Kuhn-Tucker (KKT) conditions, with the residual vector norm set to be less than 0.001. The values of the parameters q_a, q_k, and β vary in each stage, and their specific settings are depicted in Tab.8. The gradual increase in these parameter values ensures the penalty on the intermediate density field and sharpens the fluid–solid interface, leading to a more convex optimization problem [30]. The topology optimization forming process for each continuation stage is shown in Fig.5, and the final configuration of the optimized topology is displayed in Fig.6(a).

Then the curvature control method is used to optimize the boundary of the topological configuration. The irregular, sawtooth-like points are smoothed and optimized to achieve a continuous, smooth structure. Finally, advanced surface modeling (CAGD) is used to construct the curved, rounded 3D structure for all stages of topology optimization. The whole parallel optimization process and its results are shown in Fig.6.

3.2 Kriging agent model with GA and PSO optimization

3.2.1 Optimization routine

Using the Latin hypercube sampling (LHS) method, sample points were generated for mathematical modeling and simulation. The results of these simulations were then used to construct a Kriging surrogate model, which was further integrated with GA and PSO methods to identify the structural parameters of the bend area that maximize PEMFC performance. The specific steps of the optimization process are shown in Fig.7.

The power output can be used to measure the performance of a PEMFC. At a constant voltage, a higher current results in higher power, which signifies better cell performance. Therefore, output power is adopted as the objective function and can be expressed as

(2)P_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}=U_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}{I}_{\mathrm{a}\mathrm{v}\mathrm{e}}A,

where U_cell​ is the voltage, I_ave​ is the current density, and A is the active area of the cell.

Based on the varying central heights of the lower and upper boundaries of the curved area, the central height of the lower boundary (L1) and that of the upper boundary (L2) are selected as variables, as shown in Fig.8. Subject to the constraints imposed by the outer edge of the battery boundary, the variables are constrained as L1, L2 ∈ [0, 2] [mm], with the condition that L1 < L2.

3.2.2 Optimization result

Fig.9(a) shows the distribution of 23 extracted sample points. The computational model of each sample point is numerically simulated, and power calculations are performed at a voltage of 0.2 V. Of these, 20 sample points are used to train the Kriging surrogate model, while the remaining sample points are used to assess prediction accuracy. The results are shown in Fig.9(b). It can be observed that the power fluctuates in a bell-shaped distribution, with the lowest power occurring when the values of L1 and L2 are similar. A noticeable power peak occurs when L1 is approximately 0.8 and L2 is around 1.8.

Tab.9 shows the predicted values from the surrogate model alongside the simulation results. As shown, the predicted and actual values are in close agreement, with minimal error, and the correlation approaches 1, indicating the high prediction accuracy of the surrogate model. Further optimization using algorithms yields the following boundary parameters for the bending structure with maximum output power. The GA optimization results in L1 = 0.752 mm, L2 = 1.844 mm, while the PSO algorithm results in L1 = 0.858 mm, L2 = 1.907 mm. Mathematical modeling is performed for both sets of parameters, as shown in Fig.10.

3.3 Optimized flow channel models

The 3D curved bending region structures obtained were assembled into the complete flow channel model of the PEMFC. Three different curved channel models, namely TS-I, TS-II, and TS-III, were developed, each based on a topology-enhanced framework, as shown in Fig.11(a)–Fig.11(c). Additionally, the MD-G (Model-GA) and MD-P (Model-PSO) flow channel models were built by optimizing the structural parameters using the Kriging surrogate model in combination with GAs and PSO, as shown in Fig.11(d) and Fig.11(e). A comparative analysis was conducted to assess the performance of these optimized flow channels against validation channels.

4 Scheme evaluation and discussion

4.1 Output performance of optimized models

Fig.12 shows a comparison of the output performance of the PEMFC. The overall trend of the polarization curves for different flow channels remains consistent, as shown in Fig.12(a). When the current density is below 1 A/cm², the voltage loss is minimal and dominated by activation polarization. As the current density increases, the curve undergoes a significant change. At the same voltage output, the current densities of the topology-optimized and algorithm-optimized flow channels exceed those of the validation flow channels. Power density, which is positively correlated with current density, follows a similar trend, as shown in Fig.12(b).

As shown in Fig.12(c), the optimized flow channels show significant improvements in peak output performance compared to the straight flow channels. Moreover, the output performance of the topology-optimized flow channels increases progressively with the degree of topology optimization. The optimal flow channel models derived from GA and PSO demonstrate better output performance compared to TS-I and TS-II, although they still fall short of TS-II. In terms of optimization effectiveness, the models rank from best to worst as follows: peak current density—TS-III > MD-G > TS-II > MD-P > TS-I, and peak power density—TS-III > MD-G > MD-P > TS-II > TS-I. Notably, the TS-III flow channel model demonstrates the most significant improvement, with a peak current density increase of 4.72% and a peak power density increase of 3.12%.

4.2 Transfer performance of optimized models

The heat and mass transfer processes on the cathode side play an important role in determining the output performance of the PEMFCs [41]. In this paper, the interface between GDL and CL, as well as the bend areas of cathode, are selected to explore the influence of different structural configurations on the heat and mass transfer processes at an operating voltage of 0.5 V.

4.2.1 Oxygen distribution of cathode

The transport and distribution of oxygen at the cathode are critical factors influencing the rate of the electrochemical reaction in the PEMFCs. Enhancing local oxygen transport is crucial for significantly improving the performance of the PEMFC [42]. Fig.13(a) shows the distribution of the oxygen molar fraction at the interface between the CL and the GDL. It is observed that, as oxygen moves through the system, its molar fraction gradually decreases from upstream to downstream due to the ongoing electrochemical reaction. In the enlarged view of the rounded corner regions, it is evident that the optimized rounded corners facilitate better diffusion and transfer of oxygen beyond the flow channels. This results in a more uniform and extensive distribution of oxygen within these regions, ensuring a steady supply of oxygen to the rib regions, allowing for a more widespread distribution of oxygen within the catalyst layer, thereby promoting a more efficient and thorough electrochemical reaction.

Fig.13(b) shows the difference in the molar fraction of oxygen and the proportion of high oxygen content within the catalyst layer, providing a clearer picture of oxygen distribution. It can be seen that, under conditions where the lowest molar fractions are almost identical, the optimized flow channel exhibits a significantly larger maximum molar fraction, resulting in a greater difference compared to the conventional flow channel. By using a molar fraction of 0.5 as the benchmark, the proportion of high oxygen concentration within the catalyst layer was calculated. The verified flow channel accounts for 75.5%, while the optimized flow channels—TS-I, TS-II, TS-III, MD-G, and MD-P—account for 76.8%, 77%, 77.9%, 77.4%, and 77.1%, respectively. These results indicate an increase in the proportion of high oxygen content, highlighting the effectiveness of the optimized designs in improving oxygen distribution.

Fig.14 shows the distribution characteristics of oxygen within the rounded corner regions of the cathode flow channel. Fig.14(a) provides an overall view of the three-dimensional structure of the rounded corner area. It can be seen that, due to the adequate oxygen supply within the flow channel, there is minimal variation in oxygen content in the front half of the region. However, the differences in oxygen transport become more pronounced in the middle and rear segments of the region. Fig.14(b) further details the horizontal and vertical distribution of oxygen at the x−y and x−z interfaces. It is evident that, at the same positions, the optimized rounded corner structures have a higher oxygen content, facilitating a smoother oxygen transfer process. Under the influence of centrifugal force and fluid dynamics, oxygen tends to concentrate toward the outer edge of the bend, resulting in lower concentrations at the center and inner edge, leading to an uneven distribution of oxygen. The TS-III structure significantly mitigates this issue. Additionally, in the vertical direction, oxygen transport is significantly improved, with an evident increase in oxygen content. This suggests that the optimized structure enhances the oxygen transfer, enabling more efficient vertical transport of oxygen.

The distribution of oxygen in the right-angle flow channel primarily remains in the upper layer, which hinders the convection and diffusion of oxygen to the reaction sites within the fuel cell. This could lead to a significant amount of oxygen being underutilized, leading to energy wastage. The optimized structure has addressed this issue to a certain extent, improving the conservation and more efficient use of the oxygen supply. The data indicates that the improvement in oxygen transport capability follows the sequence from best to worst: TS-III > MD-G > MD-P > TS-II > TS-I.

4.2.2 Water distribution of cathode

Liquid water mainly accumulates at the cathode, hindering oxygen diffusion and causing a sharp decline in fuel cell performance at high current densities [43]. The distribution of the concentration gradient of water at the interface shows an inverse trend to that of oxygen, as shown in Fig.15. Liquid water generated by the electrochemical reaction upstream accumulates downstream due to air flushing, resulting in a gradual increase in water concentration from the inlet to the outlet. It is evident that the optimized topological structure enables better transport, facilitating the easier removal of liquid water from the bend areas of the flow channels and significantly reducing the accumulation of water in and around the flow channels. The transport performance of gas and water follows the order of TS-III > TS-II > TS-I > verified flow channel, from high to low. A high water mole fraction is defined as 0.6 or above. Compared to the verified flow channel, the high water content distribution ratios for TS-I, TS-II, TS-III, MD-G, and MD-P decreased by 9.4%, 10.1%, 11.7%, 10.5%, and 9.7%, respectively.

There are 10 flow channels within the flow field, numbered sequentially from 1 to 10, from the inlet to the outlet. Fig.16 shows the water transport within these flow channels. Optimizing the water transport capability of GDL is crucial for enhancing the electrochemical performance of the PEMFC [44]. Fig.16(a) shows the water distribution at the interface between the GDL and the flow channels. The color chart highlights the variations in water content distribution within each flow channel. Compared to the verified flow channels, the optimized flow field shows a significant reduction in water content. Fig.16(b) displays the average water mole fraction within each flow channel. It can be seen that the water content gradually increases from the inlet flow channel to the outlet flow channel. As the flow distance increases, the difference in water content between the optimized and conventional flow channels becomes more pronounced. The water content in each optimized flow channel is lower than that in the verified flow channel, with the water content in TS-III being the lowest. This indicates that the optimized bent-angle structure improves the water transport and removal capabilities within the flow channels to varying degrees. It can be concluded from the data that, from best to worst, the improvement in water transport capability follows the sequence of TS-III > MD-G > MD-P > TS-II > TS-I.

4.2.3 Pressure drop and temperature

From the inlet to the outlet, the temperature in the PEMFC gradually increases, while the pressure gradually decreases. Fig.17(a) shows the temperature distribution across the battery. Compared to the validation flow channels, the optimized model has a higher overall temperature with a smaller temperature rise, indicating that the electrochemical reactions are more thorough and efficient. Fig.17(b) displays the pressure drop along the flow channels. Compared with the validation model, the optimized model exhibits a higher pressure drop within each flow channel, accelerating the transport and transfer of substances within the channels. The average flow velocity in the validated flow channel is 23.1 m/s, while that in the optimized flow channel reaches approximately 26 m/s. However, this increase in velocity also results in a slight rise in parasitic power consumption.

According to the research conducted by Shen et al. [45,46], the efficiency evaluation criterion (EEC) is used to assess the enhancement of comprehensive mass transfer performance in the flow field, considering the relationship between performance improvement and pressure drop, expressed as

(3)EEC=\frac{Sh}{Eu},

where Sh and Eu represent the Sherwood and the Euler numbers, respectively.

(4)Sh=\frac{h_{{\mathrm{O}}_2}L}{D_{{\mathrm{O}}_2}},

where h_{{\mathrm{O}}_2} is the mass transfer coefficient, m/s; L is the characteristic length, m; and D_{{\mathrm{O}}_2} is the diffusion coefficient, m²/s.

(5)Eu=\frac{\mathrm{\Delta }p}{\rho u^2},

where in ρ is the density of the fluid, kg/m³; u is the velocity of the fluid, m/s; and ∆p is the pressure drop across the fluid, Pa.

Tab.10 shows the EEC values for different flow channel models at a voltage of 0.5 V. It can be seen that various bend structures have distinct impacts on mass transfer performance and pressure drop. Compared to the verified flow channel, the optimized model has a higher Sherwood number, a lower Euler number, and a higher EEC value. This indicates that the optimized flow channels offer smoother flow, with reduced pressure loss, improved mass transfer efficiency, and more uniform distribution of gases. TS-III has the highest Sherwood number, indicating the best mass transfer performance; however, the higher pressure drop leads to a higher Euler number. MD-G has a lower Sherwood number and, consequently, poorer mass transfer performance than TS-II, but it also has a lower Euler number and a smaller pressure drop. TS-II has the highest EEC value, indicating the best overall performance in balancing mass transfer efficiency and pressure loss.

5 Conclusions

In this paper, a progressive design method combining 2D topology combined optimization with curvature optimization is proposed to optimize and predict the configuration for the curved bending regions in the flow channel. The optimized models are compared with algorithm-optimized models and verified model through simulations. Key analysis, including I–V and power density curves, oxygen and water distribution at the cathode, and temperature-pressure relationships, are conducted to validate the accuracy of the optimization process. The main conclusions are as follows:

(1) Compared with verified model, the bent corner structure channels optimized through topology-curvature progression effectively enhance the output performance of PEMFCs. In terms of improvement in peak current density, the order from highest to lowest is: TS-III > MD-G > TS-II > MD-P > TS-I. A similar trend is observed for peak power density: TS-III > MD-G > MD-P > TS-II > TS-I. Among these, TS-III has increased peak current density by 4.72% and peak power density by 3.12%.

(2) The optimized structures of curved bending region accelerate convection and diffusion within the flow fields, improving the transport and distribution of oxygen and liquid water, reducing water accumulation in the system, enhancing the extent and efficiency of electrochemical reactions within the battery, compensating for mass transfer polarization losses in PEMFCs. The order of improvement in oxygen and water transport performance is: TS-III > MD-G > MD-P > TS-II > TS-I.

(3) The optimized models have both a higher pressure drop and flow velocity compared to the verified model, resulting in smoother fluid flow, reduced pressure loss, and better mass transfer efficiency. Using the EEC to consider the relationship between performance improvement and pressure drop, TS-II has the best overall performance.

(4) The 2D topology optimization method based on RAMP enables quick generation of reasonable flow channel configurations. The curvature fitting optimization design is more conducive to the manufacturing processes. The combination of these two approaches provides a more accurate and efficient design method. The predicted configuration can provide a reference for related optimization research, reducing time and trial-and-error costs in the design process.

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