# Effect of Pore Shape and Spacing on Water Droplet Dynamics in Flow Channels of Proton Exchange Membrane Fuel Cells

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}to 100 cm

^{2}. The results showed that the newly developed flow field was more beneficial to water removal than a serpentine flow field.

## 2. Models and Methods

#### 2.1. The VOF Model

^{3}), P is the pressure (Pa), μ is the viscosity (Pa·s), t is time (s), v is the velocity (m/s), k

_{eff}is effective thermal conductivity (W·m

^{−2}·K

^{−1}), and E is the energy (J).

_{1}and p

_{2}are the pressures in the two fluids on either side of the interface and R (m) is the radii in the orthogonal direction. The forces from the CSF and CSS model are:

#### 2.2. Model Settings and Boundary Conditions

^{3}, and the cross section was a regular quadrilateral. The water inlet micropore(s) were located at the center of the GDL, 20 mm from the gas inlet. In the case of two water inlets, the downstream micropore was located at a certain distance along the x-direction of the upstream micropore. GAMBIT software was used to divide the computing area. The grid diagram is shown in Figure 2.

_{1}, Y

_{1}, and Z

_{1}, the channel bottom was set with a hydrophobic GDL (135°), other surfaces were set as hydrophilic walls (45°).

#### 2.3. Solution Procedure

^{−5}m), 1 × 5 × 10 (10

^{−5}m), 10 × 10 × 10 (10

^{−5}m), and 20 × 20 × 20 (10

^{−5}m). The corresponding number of grids was 2.53 million, 1.55 million, 550,000, and 94,000, respectively. For the four mesh sizes, liquid phase fraction cloud maps were compared with different mesh precision. To obtain a clear phase interface and a suitable flow pattern and save computing resources as much as possible, the mesh size of 1 × 5 × 5 (10

^{−5}m) was selected for this work. Under the unsteady state condition, the convergence of simulation can be obtained by selecting the time step that satisfies the calculation. According to the time step mentioned by Wu et al. [26], the maximum time step is 1.5 × 10

^{−6}s under the condition that the maximum Courant number is 0.25 and the minimum grid size is 6.189 × 10

^{−6}m. In the simulation process, we chose a time step of 10

^{−6}s to meet the accuracy and time requirements of the simulation.

## 3. Results and Discussion

#### 3.1. Effect of the Pore Shape

_{h}. The following dimensionless numbers were obtained: Ca = 2.46 × 10

^{−4}, We = 1.68 × 10

^{−3}, Bo = 1.34 × 10

^{−3}. From these results, the effect of surface tension is much greater than the other forces, which explains why liquid water forms near-spherical droplets on the surface of hydrophobic GDLs.

#### 3.1.1. Effect of the Pore Shape on Droplet Volume and Cycle Time

_{G}and U

_{L}represent the inlet rate of gas and liquid water; L

_{w}represents the length of windward side of the pore; θ is the angle between the airflow and L

_{w}).

_{G}, U

_{G}, σ, ρ

_{G}, ρ

_{L}, g, D

_{h}are gas viscosity, gas inlet velocity (equal to gas apparent velocity and gas flow rate), surface tension coefficient, gas density, liquid density, acceleration of gravity, and the hydraulic diameter, using the dimensionless number to correlate these forces. Regarding the pore diameter 2R as the hydraulic diameter Dh, the calculation results in the table confirm that the surface tension in the smaller channels and GDL micropores plays a major role in the droplet size change. Since the critical droplet size is different in the above cases, it is concluded that the pore shape has a great effect on the droplet growth.

_{w}of the water inlet pore decreases in the order of case 6 > case 1 > case 3 > case 5 > case 2 > case 4 > case 7. The angle $\theta $ between the windward side of the pore and the direction of the airflow, which is defined as the windward side slope, has a different increasing trend in the order of case 6 = case 4 = case 7 > case 1 > case 5 = case 2 > case 3. The results show that although the length of the windward side is the smallest in case 7, the critical volume of the droplet is larger than the non-quadrangular pore condition due to the larger slope of the windward side. This result indicates that the increasing trend of the critical volume of the droplet follows the law of windward slope and the important influence of the windward slope on droplet movement.

#### 3.1.2. Effect of the Pore Shape on Downstream Liquid Flow Pattern

#### 3.1.3. Effect of the Pore Shape on Channel Pressure Drop

#### 3.2. Effect of the Pore Distance

_{a}= R

_{b}= 0.05 mm, U

_{G}= 1.0 m/s, U

_{L}= 0.1273 m/s (R

_{a}represents the upstream pore radius; R

_{b}represents the downstream pore radius; U

_{G}and U

_{L}represent the inlet rate of gas and liquid water). The contact angles for the GDL surface and the channel wall was set at 135° and 45°, respectively. Cases 8, 9, 10, 11, and 12 correspond to the pore spacing values of 0.3, 0.4, 0.6, 0.8, 1.2 mm, respectively.

#### 3.2.1. Effect of the Pore Distance on Droplet Interactions

#### 3.2.2. Effect of the Pore Distance on Gas Velocity Profile, Flow Regime, and Pressure Drop

#### 3.3. Implications in PEMFC Applications

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | The pore area, mm^{2} |

E | Energy, J |

k_{eff} | Effective thermal conductivity |

n | The surface normal |

L | The pore distance, mm |

L_{W} | The windward side length of micropores, mm |

${\widehat{n}}_{w}{\widehat{n}}_{w}$ | The unit vectors normal to the wall |

P | The pore circumference, mm |

p_{1},p_{2} | The pressures in the two fluids on either side of the interface, Pa |

R | Radii in the orthogonal direction, m |

R_{a} | The upstream pore, mm |

R_{b} | The downstream pore, mm |

S | Roundness of the pore |

T | Time, s |

${\widehat{t}}_{w}$ | The unit vectors tangential to the wall |

U_{G} | Gas velocity, m/s |

U_{L} | Liquid velocity, m/s |

v | Velocity, m/s |

V_{L-inlet} | Liquid velocity at the inlet, m/s |

Greek Letters | |

α | Volume fraction |

θ_{w} | The contact angle at the wall |

κ | Surface curvature |

μ | The viscosity Pa·s, |

ρ | Density, kg/m^{3} |

σ | Surface tension |

Subscripts | |

a | Symbol of upstream pore |

b | Symbol of downstream pore |

G | Gas |

L | Liquid |

w | Wall |

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**Figure 3.**Morphology of critical droplets under different cases ((

**a**) front view, (

**b**) top view; the airflow direction is from

**left**to

**right**).

**Figure 6.**Effect of the pore shape on the averaged liquid water coverage on the gas diffusion layer (GDL) surface.

**Figure 7.**Effect of the pore shape on the pressure drop between the inlet and outlet of the channel.

**Figure 8.**Effect of the pore distance on the morphology of two droplets. Case8–9: (

**a**)Fusion of droplets, (

**b**,

**c**) Continuous growth of droplets, (

**d**) Breaking of droplets; case10: (

**a**) Appearance of droplets, (

**b**) Fusion of droplets, (

**c**) Fragmentation of fusion droplets, (

**d**) Fusion of new droplets; case11–12: (

**a**–

**c**) Upstream and downstream droplets grow and break separately, (

**d**) Emergence and fusion of new droplets).

**Figure 9.**The flow lines and velocity vectors of the air around the droplets ((

**a**) X-Z plane, (

**b**) X-Y plane).

**Figure 10.**Effect of the pore distance on the liquid flow pattern in the channel. ((

**a**) case8: L = 0.3 mm, (

**b**) case9: L = 0.4 mm, (

**c**) case10: L = 0.6 mm, (

**d**) case11: L = 0.8 mm, (

**e**) case12: L = 1.2 mm).

**Figure 11.**Effect of the pore distance on the pressure drop between the inlet and outlet of the channel.

Case | Shape of Pore | A (mm^{2}) | P (mm) | $\mathit{S}=4\mathit{\pi}\mathit{A}/{\mathit{P}}^{2}$ | ${\mathit{L}}_{\mathit{w}}\text{}\left(\mathbf{mm}\right)$ | U_{G} (m/s) | U_{L} (m/s) | $\mathit{\theta}\text{}(\xb0)$ |
---|---|---|---|---|---|---|---|---|

1 | 0.00785 | 0.3142 | 1.000 | 0.1571 | 1.0 | 0.1237 | >90 | |

2 | 0.00785 | 0.3300 | 0.906 | 0.1100 | 1.0 | 0.1273 | 120 | |

3 | 0.00785 | 0.3380 | 0.863 | 0.1352 | 1.0 | 0.1273 | 126 | |

4 | 0.00785 | 0.3544 | 0.785 | 0.0886 | 1.0 | 0.1273 | 90 | |

5 | 0.00785 | 0.4038 | 0.604 | 0.1346 | 1.0 | 0.1273 | 120 | |

6 | 0.00785 | 0.4430 | 0.502 | 0.1772 | 1.0 | 0.1273 | 90 | |

7 | 0.00785 | 0.4430 | 0.502 | 0.0443 | 1.0 | 0.1273 | 90 |

Name | Meaning | Formula | Value |
---|---|---|---|

Capillary number | Viscous force/surface tension | $Ca={\mu}_{G}{U}_{G}/\sigma $ | $2.46\times {10}^{-4}$ |

Weber number | Inertial force/surface tension | $W{e}_{G}={\rho}_{G}{U}_{G}^{2}{D}_{h}/\sigma $ | $1.68\times {10}^{-3}$ |

Bond number | Gravity/surface tension | $Bo={\rho}_{L}g{D}_{h}^{2}/\sigma $ | $1.34\times {10}^{-3}$ |

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**MDPI and ACS Style**

Fan, M.; Duan, F.; Wang, T.; Kang, M.; Zeng, B.; Xu, J.; Anderson, R.; Du, W.; Zhang, L.
Effect of Pore Shape and Spacing on Water Droplet Dynamics in Flow Channels of Proton Exchange Membrane Fuel Cells. *Energies* **2021**, *14*, 1250.
https://doi.org/10.3390/en14051250

**AMA Style**

Fan M, Duan F, Wang T, Kang M, Zeng B, Xu J, Anderson R, Du W, Zhang L.
Effect of Pore Shape and Spacing on Water Droplet Dynamics in Flow Channels of Proton Exchange Membrane Fuel Cells. *Energies*. 2021; 14(5):1250.
https://doi.org/10.3390/en14051250

**Chicago/Turabian Style**

Fan, Mengying, Fengyun Duan, Tianqi Wang, Mingming Kang, Bin Zeng, Jian Xu, Ryan Anderson, Wei Du, and Lifeng Zhang.
2021. "Effect of Pore Shape and Spacing on Water Droplet Dynamics in Flow Channels of Proton Exchange Membrane Fuel Cells" *Energies* 14, no. 5: 1250.
https://doi.org/10.3390/en14051250