1 Introduction

Proton exchange membrane fuel cell (PEMFC) is a power equipment which converts chemical energy in fuel (generally is hydrogen) and oxidant (generally is oxygen in air) into electrical energy. It is considered to be very promising new energy equipment in the twenty first century since its favorable characteristics such as high efficiency, environmentally friendly, wide adaptation to working environment, fuel flexibility and higher power density. Many countries, enterprises and scientific research institutions are devoting themselves in the research and development. Today, the technology of PEMFC is more and more mature, and its utility also becomes more and more popular in our daily life [1–4].

As new energy equipment, compared with the traditional power equipments, PEMFC has obvious superiority. However, at present its development and utility are less popular than the traditional power equipments such as internal combustion engine, metal battery. The reasons are various, but with the deepening of the research and advances in technology related, it will become increasingly widespread. Current research of PEMFC has been centering on improving the output performance, increasing safety, improving longevity and stability [5, 6]. The performance of the PEMFC can be improved by optimizing its operating parameters, such as gas flow rate, operating cell temperature, inlet pressure, humidity condition, and flow channel type. This paper aims to study and improve the structure of PEMFC bipolar plates to achieve the purpose of enhance the performance of PEMFC.

The flow channel on the bipolar plate in PEMFC is the gas transmission channel and water removal channel, whose shape and size plays a critical role of its performance. The design of the flow channel on bipolar plates mainly focuses on three aspects: the flow channel distribution pattern [7–13], the cross-section shape [14–16] and the land and channel dimensions [17–20]. Researcher still continuously innovate and develop new flow field structure. Biomimetic flow field is a new flow field structure based on the biological structure in nature [21]. Through observing the features of the biological structures, a series of rules have been summarized and applied to the design of flow field structure so as to improve the PEMFC performance. The newly designed biomimetic flow field by Currie [22], compared to conventional flow field, has higher average current density and more uniform distribution of current density as well. Kjelstru et al. [23] finds that the current density of H type flow field, under the same condition (20 %PtC−1), increases by 10–20 % over conventional flow fields. Kloess et al. [24] design two different flow fields: the lung flow field and the biomimetic flow field. Within their research, both numerical simulation and experimental method are employed to compare the performances of the new structures with that of the conventional flow field structures. From the results, it is found that both the lung flow field and biomimetic flow field have lower pressure drop between inlet and outlet and more even velocity distribution in diffusion layer, which is beneficial to reacting evenly. Wang et al. [25] design a new biomimetic flow channel to improve the PEMFC performance. Two dimensional numerical simulation was committed to the new biomimetic flow channel and traditional flow channel at Re = 10, 100 respectively. Besides, 3-D numerical simulation concerns the channel ratio was done. The results show that the biomimetic flow field has more even velocity distribution and lower pressure loss. Biomimetic flow channels perform better when evaluated either by the Renault value or from the channel ratio aspect. After testing the performance of the parallel flow field, the serpentine flow field and two new bio-inspired flow fields, Wang et al. [26] find that the new biomimetic flow fields possess higher gas consumption efficiency and higher water removal efficiency. In order to test the performance of different flow fields, a 25 cm2 fuel cell was assembled and tested for four different flow channels: serpentine, interdigitated, lung and biomimetic flow channel. The results show that lung flow field and the biomimetic flow field possess better fuel cell performance, and that the current densities of them increase nearly by 30 % over the conventional flow fields.

In the present work, a novel biomimetic flow field designs by mimicking the structures of leaf veins has been proposed to achieve higher PEMFC performance. Computational fluid dynamics (CFD) based simulations were conducted to compare three different configurations (parallel, serpentine and biomimetic channel, respectively) at different operation variables. It was found that flow field designs with biomimetic flow channel perform better than that with convectional flow channel under the same operating conditions.

2 Nature inspired flow field designs

The flow field design of a bipolar plate greatly affects the performance of a PEMFC. It must uniformly distribute the reactant gases over the MEA and prevent product water flooding. The flow field design involves the layout of the flow field and channel cross-section geometries including shape and dimensions. The leaves found in nature are considered a natural optimization of the enhanced heat transfer and mass transfer of flow channel. They have a uniform distribution of microtubules and facilitate the energy and material delivery. The mass transfer and heat transfer of PEMFC bipolar plate is similar to leaves. The flow field distribution in PEMFC bipolar plate can be regarded as a self-growth process like leaf vein formation, thus flow field is started from a given “send” and is grown, branched off or degenerated according to a certain growing rule, to form the optimal distribution patterns. In this paper, the flow field design is inspired by the distribution of leaf veins, and the channel dimensions are calculated using Murray’s Law.

2.1 The structure of a leaf

Living things in nature are evolutionary, and survivors can adapt to environment well. Plant is one of the survivors, most of which rely on leaves to grow. Both leaves and their veins are in different shapes, as we can see in Fig. 1.

Fig. 1
figure 1

The schematic and physical diagram of leaves. a The schematic diagram of leaf, b the pinnately physical diagram of leaf

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Veins, the vascular bundles in leaves, are the branches of vascular stems. The vascular bundles always distribute along the petiole in some form of special shape. The large and obvious vein, locating in the center of leaf, is called the midrib or main vein (Mid vein). And the smaller veins distributed along main vein are called the lateral veins (Lateral veins). The small veins or veinlets (Minor veins) (shown as in Fig. 1b) are the smallest bundles on the leaf which distribute all over the leaves. Those veins are the transport channel of water, salts and the output of photosynthetic products in plants [27]. As outstanding product of nature, these different patterns of veins undoubtedly have good biological and physic-chemical properties, and also they can be good at transferring material and supporting blades. These characteristics meet the requirements of the flow channel in PEMFC which demand good circulation and mechanical properties.

2.2 Murray’s law

In 1926, Murray [28] put forward the famous Murray’s Law after his fully study and analysis of physiological bifurcation systems. Murray’s Law reveals that if the diameters of the mother pipe and branches of a bifurcate system are in charge of special proportion, the consumption of energy and the pressure drop can be relative small when the fluid flows through the system. Murray’s Law has been demonstrated in a series of natural bifurcate systems since it was advanced. Its development and derivation is still going on, and gradually used to guide the design of fluid flow channel.

$$d_{0}^{3} = d_{1}^{3} + d_{2}^{3}$$
(1)
$$d_{0}^{3} = \sum\limits_{k = 1}^{m} {d_{k}^{3} }$$
(2)

For bifurcate structure (shown as Fig. 2), the relationship between the diameter of parent tube and diameters of branches can be described by Eq. (1), in which d 0 represents the diameter of parent tube, d 1, d 2 represent the diameters of branches, respectively; And the further derivation of the relationship between the diameter of parent tube and diameters of branches of multi-branch systems can be described by Eq. (2), in which the subscript k indicates the kth pipe, and m means the number of pipes. While d 1 = d 2, the following Eq. (3) can be derived:

$$d_{1} = d_{2} = 2^{ - 1/3} d_{0}$$
(3)
Fig. 2
figure 2

The schematic diagram of single bifurcation structure

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The processes are the theoretical derivation of Murray’s Law, and Eqs. (1), (2) are all the manifestations of Murray’s Law. Since there are no restrictions of the symmetry of the system and numbers of branches of the bifurcate system for the derivation of Murray’s Law, so Murray’s Law is also applicable to asymmetric and multi-branch system [29].

2.3 Biomimetic flow channel

The prototype of the biomimetic flow channel in the article is the naturally pinnate vein (shown as in Fig. 1b), and the dimensions of the channel are calculated by Murray’s Law. Based on pinnate vein, the main vein and lateral veins have been simplified into rectangular cross-section mainstream channel and secondary channels. The symmetric sub-channels are distributed along the center axis of the main channel, and arranged by the main channel at an angle in parallel. Research shows that the symmetric flow channel can promote the internal distribution of pressure more evenly to improve performance of PEMFC [30].

The dimensions of flow channel generally consist of width, height and width of the rib, which are critical for the performance of PEMFC as well. This article bases on reference of previous studies and combines with Murray’s Law to get the dimensions. Generally in current researches the dimensions of the flow channel as: width is between 0.5 and 3 mm; rib width is between 0.5 and 3 mm; depth is between 0.2 and 2.5 mm. Also, considering that the cross-section of the channels is rectangular and the ratios of width to depth of the secondary channels are around 1, the design has taken the widths of the channels as equivalent diameter. The ratios of the width between the main channel and sub-channels are 2:1. In order to improve the design further to make the flow channels more similar to the distribution of the plant veins, the third-level channels have been added into the flow channel. And, the distances between the third-level channels and the symmetry axis are respectively 3.5 and 7.5 mm. Meanwhile, in order to increase the outlet pressure to facilitate the discharge of product, the shape of the outlet channel has been changed from rectangular cross-section to ladder-shaped. The angle between the sub-channels and the symmetry axis in flow field is 75 degrees based on Murray’s Law [31] (shown as Fig. 3).

Fig. 3
figure 3

Biomimetic flow channel. a 2-dimensional diagram, b physical diagram

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2.4 Conventional flow channels

In order to investigate the advantages of a newly proposed biomimetic flow channel, the comparison with conventional flow channels is conducted in this paper. The conventional flow channels are parallel channel and serpentine channel which are commonly used in PEMFC. The dimensions of the channels are as follow, parallel channel: width is 0.9 mm, rib of width is 1 mm, and depth is 1 mm (shown as in Fig. 4a); serpentine channel: width is 1 mm, width of rib is 1 mm, and depth is 1 mm (shown as in Fig. 4b).

Fig. 4
figure 4

Conventional flow channels. a Parallel-channel, b serpentine-channel

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3 Mathematic model

3.1 Model assumptions

The present model assumes:

  • ideal gas mixtures;

  • steady state conditions;

  • the flow is laminar;

  • system is isothermal;

  • fluid is incompressible;

  • fuel cell porous zones are isotropic.

3.2 Model equations

In the current three-dimensional mathematical model of PEMFC, the basic governing equations [32], which include mass conservation equation, mass transfer equation, momentum conservation equation, energy conversation equation, species conservation equation and current conservation equation, are the foundation of numerical simulation.

$$\nabla \left( {\varepsilon \rho \overrightarrow {u} } \right) = S_{m}$$
(4)
$$\nabla \left( {\varepsilon \rho \overrightarrow {u} C_{k} } \right) = - \nabla J_{k} + S_{k}$$
(5)
$$\nabla (\varepsilon \rho \overrightarrow {u} \overrightarrow {u} ) = - \varepsilon \nabla p + \nabla \left( {\phi \mu \nabla \overrightarrow {u} } \right) + S_{u}$$
(6)
$$\frac{{\partial \left( {\varepsilon \rho c_{p} T} \right)}}{\partial t} + \nabla \left( {\varepsilon \rho c_{p} \overrightarrow {u} T} \right) = \nabla \left( {k^{eff} \nabla T} \right) + S_{Q}$$
(7)
$$\frac{{\partial \left( {\varepsilon c_{k} } \right)}}{\partial t} + \nabla \left( {\varepsilon c_{k} \overrightarrow {u} } \right) = \nabla \left( {D_{k}^{eff} \nabla c_{k} } \right) + S_{k}$$
(8)
$$\nabla \left( {\sigma_{e} \nabla \varphi_{e} } \right) + S_{e} = 0\nabla \left( {\sigma_{m} \nabla \phi_{m} } \right) + S_{m} = 0$$
(9)

here ε is porosity, ρ is density, \(\overrightarrow {u}\) is velocity vector, S m is source item of the mass, C k is species concentration of species k, J k is the diffusion flux of species k, S k is source item of mass changes, μ is the viscosity; S u is the momentum source item; C p as specific heat at constant pressure, T is the temperature; S Q is energy source item, S k is source item of species, p is pressure; k eff is the effective thermal conductivity coefficient, \(D_{k}^{eff}\) is effective diffusion coefficient of species, σ is the conductivity. The water transfer in proton exchange membrane (PEM) is affected by electro-osmosis (J d ), diffusion transfer (J p ) and infiltration transfer (J p ). Therefore the net water content inside the PEM is decided by the algebraic sum of the above-mentioned phenomenon. Water flux of membrane (J w ) is as follow:

$$J_{w} = J_{d} + J_{c} + J_{p} = n_{d} \frac{I}{F} - D_{m} \frac{\partial c}{\partial y} + D_{p} \frac{dp}{dy}$$
(10)

Within the equation, I is current density, F is the Faraday constant, c is concentration, D m diffusion coefficient, D p as the pressure migration coefficient, n d is electric migration coefficient,the expression is as follow:

$$n_{d} = \frac{2.2}{25}\lambda$$
(11)

The relationship between λ and water saturation α is as follow:

$$\lambda = 0.043\,\left( {\alpha < 0} \right)$$
$$\lambda = 0.043 + 17.81\alpha - 39.85\alpha^{2} + 36\alpha^{3} \,\left( {0 \le \alpha < 1} \right)$$
$$\lambda = 14.0 + 1.4(\alpha - 1)\,\left( {1 \le \alpha \le 3} \right)$$
$$\lambda = 16.8\,\left( {\alpha > 3} \right)$$

α is the water saturation and described as:

$$\alpha = \frac{{p_{w} }}{{p_{sat} \left( T \right)}} + 2s$$
(12)

P w is the local water vapor pressure, P sat (T) is the saturation pressure which can be obtained through local saturation vapor pressure instrument, s is the saturation of liquid water.

$$D_{m} = D_{\lambda } \exp \left[ {2416\left( {\frac{1}{303} - \frac{1}{T}} \right)} \right]$$
(13)

where,

$$D_{\lambda } = 10^{ - 10} \quad (\lambda < 2)$$
$$D_{\lambda } = 10^{ - 10} \times \left[ {1 + 2\left( {\lambda - 2} \right)} \right] \quad \left( {2 \le \lambda \le 3} \right)$$
$$D_{\lambda } = 10^{ - 10} \times \left[ {3 - 1.67\left( {\lambda - 3} \right)} \right] \quad \left( {3 < \lambda < 4.5} \right)$$
$$D_{\lambda } = 1.25 \times 10^{ - 10} \quad \left( {\lambda \ge 4.5} \right)$$

3.3 Boundary conditions and modeling parameters

The computational domain of simulation includes bipolar plates, flow channel, anode and cathode diffusion layer, anode and cathode catalyst layer and proton exchange membrane. Electrode and membrane parameters are shown in Table 1, geometrical and operating parameters are shown in Table 2. The boundary conditions of the inlet and outlet of flow channel for calculation are specified with mass flow inlet and pressure outlet respectively. Anode and cathode mass flow rate m an and m cat can be calculated by the following equations:

$$m_{an} = \left( {\frac{{M_{{H_{2} }} \times I_{ref} \times A}}{2F}} \right) \times \zeta_{a} /m_{{H_{2} }}$$
(14)
$$m_{cat} = \left( {\frac{{M_{{o_{2} }} \times I_{ref} \times A}}{2F}} \right) \times \zeta_{c} /m_{{o_{2} }}$$
(15)
Table 1 Electrode and membrane parameters
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Table 2 Geometrical and operational parameters
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3.4 Numerical solution of equations

The governing equations along with their appropriate boundary conditions, are discretized by the finite volume method and solved in the commercial CFD software package, Fluent (version 6.3). The source terms and physical properties are implemented in a UDF user-defined function and the species and charge transport equations are solved through the software’s user-defined scalars. The software utilizes the well-known SIMPLE (semi-implicit pressure linked equation) algorithm, and an algebraic multigrid solver to efficiently solve the resulting set of discretized linear equations [33]. The model solving process tends to converge when the relative error between two iterations is <10−6.

4 Result and discussion

There are various factors that affect the performance of PEMFC, and this paper aims at improving its flow field structure to enhance the performance. In general, the performance of PEMFC can be judged by the characteristics of I~V curve, power density, mass transfer, dynamics and longevity.

In this paper, the simulation data are reprocessed by post-process software, i.e. Tecplot. To analyze the performance of PEMFC, this paper set output performance (including I~V curve, power density) and the distribution of reactants and products (including pressure/heat/water/hydrogen/current density distribution, as well as water content in membrane) to be the criterions for judging.

4.1 Output performance

The output performance is one of the most critical characteristics for PEMFCs. In theory, the output performance can be indicated by the current and voltage (I~V) polarization curves and the power density curves. Figure 5 has shown the I-V polarization curves and the power density curves of the three flow fields with different channels. In the figure, it can be seen that the output performance of the biomimetic flow channel is better than the conventional flow channel, especially in low operating voltage. For the biomimetic flow channel, it can be seen that when the operating voltage is 0.65 v, the PEMFC has the maximum output power. The following contrastive analysis is based on the simulation data in anode of PEMFC when operating voltage is 0.65, thus to analyze the role of the biomimetic flow channel in improving the performance of PEMFC.

Fig. 5
figure 5

Comparisons of polarization curves and power density curves for different flow field designs. a Polarization curves, b power density curves

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4.2 Distribution of reactants and products

By comparing (a), (b), (c), in Fig. 6, the pressure drop observed in the serpentine flow channel is larger than other flow channels under the operating voltage. While the biomimetic flow channel has the smallest pressure drop and the most uniform distribution of pressure, which can be conducive to the stability of the output of power and the discharge of products.

Fig. 6
figure 6

Pressure distributions for different flow field designs. a Pressure distributions for parallel-channel, b pressure distributions for serpentine-channel, c pressure distributions for biomimetic flow channel

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As it’s shown in Fig. 7: the distribution of temperature in (a) is the most uneven, the high temperature regions are also the widest; (b) and (c) show that the distribution of temperature is relatively lower and uniform and the regions with high temperatures are in the vicinity of the outlet. So the biomimetic flow channel is better than the conventional flow channel for PEMFC.

Fig. 7
figure 7

Temperature distributions for different flow field designs. a Temperature distributions for parallel-channel, b temperature distributions for serpentine-channel, c temperature distributions for biomimetic flow channel  

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Liquid water can build up and stagnate in the flow channels. This reduces the flow of oxidizer to the cathode and reduces the overall efficiency of the fuel cell. Water, a product of the fuel cell reaction, must also be removed from the cathode side to prevent congestion, or flooding, in the oxidizer flow channels. Water mass fractions in the channels for different flow field designs were shown in Fig. 8. The biomimetic flow channel appears to have the best water management capabilities. For the biomimetic flow channel Fig. 8 indicates that most of the liquid water is formed in the outlet channels which mean it will be easy to remove from the system.

Fig. 8
figure 8

Water mass fractions in the channels for different flow field designs. a Water mass fractions for parallel-channel, b water mass fractions for serpentine-channel, c water mass fractions for biomimetic flow channel

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Figure 9 presents hydrogen mass fraction distribution for different flow field designs. As it can be seen in this figure, the biomimetic flow channel has the higher hydrogen mass fraction and more uniform hydrogen distribution. The higher the hydrogen mass fraction the better because it leads to increased reaction rate, and the uniform hydrogen distribution leads to even current density across the fuel cell.

Fig. 9
figure 9

Hydrogen mass fraction distribution for different flow field designs. a Hydrogen mass fractions for parallel-channel, b hydrogen mass fractions for serpentine-channel, c hydrogen mass fractions for biomimetic flow channel

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5 Conclusion

The structure of the flow fields in PEMFC has a significant impact on its performance. This paper, a novel biomimetic flow channel has been proposed for the design of flow fields in order to achieve higher PEMFC performance. By comparing three different configurations (parallel, serpentine and biomimetic channel, respectively) in terms of current density distribution, power density distribution, pressure distribution, temperature distribution, and hydrogen mass fraction distribution leads to the conclusion that the proposed biomimetic flow channel can improve the output of power, and internally, the distribution of reaction gases is more uniform in the channel and the excess product of the chemical reaction is easier to remove from PEMFC.