# Impact of Membrane Phosphoric Acid Doping Level on Transport Phenomena and Performance in High Temperature PEM Fuel Cells

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Physical Model

#### 2.2. Governing Equations

_{mass}is the source term of the mass equation.

_{mom}is the source term of the momentum equation.

_{i}and D

_{eff,i}are the mass fraction and effective diffusivity, respectively. S

_{i}is the source term of the species equation.

_{p}and k

_{eff}are the specific heat and effective thermal conductivity, respectively. S

_{T}is the source term of the energy equation. The effective thermal conductivity can be determined by the following expression:

_{s}is the thermal conductivity of the solid phase, and k

_{f}is the thermal conductivity of the gas phase.

_{eff,s}is the effective electrical conductivity, σ

_{eff,m}the effective proton conductivity, ϕ

_{s}the electrical potential, ϕ

_{m}the proton potential. S

_{s/m}is the source term of the charge equation. The Butler–Volmer equation is applied to describe the hydrogen oxidation reaction and oxygen reduction reaction in the anode and cathode CLs, respectively.

#### 2.3. Numerical Implementation

^{−7}kg/s and 2.073 × 10

^{−8}kg/s, respectively. The detailed boundary conditions are illustrated in Table 4.

## 3. Results and Discussion

_{3}PO

_{4}) per repeat unit of the PBI polymer [20]. The effect of the membrane phosphoric acid doping level on cell performance is given in Figure 3. The current density-cell voltage and current density-power density curves are presented and compared. The polarization curves of four cases are illustrated in Figure 3a. It is clearly seen that the cell performance is significantly affected by the DL and it is improved with an increase in DL. This is attributed to the decrease in the proton transport resistance. The proton conductivity is a function of temperature and DL. The proton conductivity is increased with increasing DL. This means that the proton transport process is enhanced and the ohmic loss is decreased. The current densities of four cases at the cell voltage 0.3 V are 1.011 A/cm

^{2}, 1.155 A/cm

^{2}, 1.278 A/cm

^{2}and 1.372 A/cm

^{2}, respectively. The current density–power density curves are shown in Figure 3b. It can be observed that the power densities of four cases at the cell voltage 0.3 V are 0.303 W/cm

^{2}, 0.346 W/cm

^{2}, 0.383 W/cm

^{2}and 0.412 W/cm

^{2}, respectively.

## 4. Conclusions

^{2}to 1.372 A/cm

^{2}when the doping level is increased from 5 to 11. The maximum temperature is observed at the cathode CL and increased with the increasing doping level. The distributions of oxygen and water mass fraction are also affected by the doping level. A higher water mass fraction and a lower oxygen mass fraction are observed with an increase in the doping level. The proton conductivity is also significantly increased with the increasing doping level. In addition, the trend and magnitude of local current density is strongly affected by the doping level.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**The cell performance of fuel cells with different membrane phosphoric acid doping levels (Case A: DL = 5; Case B: DL = 7; Case C: DL = 9; Case D: DL = 11): (

**a**) cell voltage-current density curve; (

**b**) power density-current density curve.

**Figure 4.**Temperature distributions at the middle plane of fuel cells (Case A: DL = 5; Case B: DL = 7; Case C: DL = 9; Case D: DL = 11).

**Figure 5.**Species mass fraction at the middle plane of fuel cells (Case A: DL = 5; Case B: DL = 7; Case C: DL = 9; Case D: DL = 11): (

**a**) oxygen mass fraction: (

**b**) water mass fraction.

**Figure 6.**Species mass fraction profile distributions at y = 0.001 m (Case A: DL = 5; Case B: DL = 7; Case C: DL = 9; Case D: DL = 11): (

**a**) oxygen mass fraction; (

**b**) water mass fraction.

**Figure 7.**Species mass fraction at the cathode GDL and CL interface of fuel cells (Case A: DL = 5; Case B: DL = 7; Case C: DL = 9; Case D: DL = 11): (

**a**) oxygen mass fraction; (

**b**) water mass fraction.

**Figure 8.**Species mass fraction profile distributions at y = 0.001 m (Case A: DL = 5; Case B: DL = 7; Case C: DL = 9; Case D: DL = 11): (

**a**) oxygen mass fraction; (

**b**) water mass fraction.

**Figure 9.**Proton conductivity of the membrane of four cases (Case A: DL = 5; Case B: DL = 7; Case C: DL = 9; Case D: DL = 11).

**Figure 10.**Local current density distribution of fuel cells at the middle plane of membrane (Case A: DL = 5; Case B: DL = 7; Case C: DL = 9; Case D: DL = 11): (

**a**) x = 0.005 m; (

**b**) x = 0.025 m; (

**c**) x = 0.045 m.

Parameter | Value | Unit |
---|---|---|

Cell length/width | 50/2 | mm |

GFC height/width | 1.0/1.0 | mm |

Anode/Cathode GDL thickness | 0.2 | mm |

Anode/Cathode CL thickness | 0.01 | mm |

Membrane thickness | 0.05 | mm |

Description | Units |
---|---|

${\mathrm{S}}_{\mathrm{mass}}={\mathrm{S}}_{{\mathrm{H}}_{2}}$ (Anode CL) ${\mathrm{S}}_{\mathrm{mass}}={\mathrm{S}}_{{\mathrm{O}}_{2}}$ (Cathode CL) ${\mathrm{S}}_{\mathrm{mass}}={\mathrm{S}}_{{\mathrm{H}}_{2}\mathrm{O}}$ (Cathode CL) | kg m^{−3} s^{−1} |

${\mathrm{S}}_{\mathrm{mom}}=-\frac{\mathsf{\mu}}{\mathrm{K}}\overrightarrow{\mathrm{u}}$ (GDLs and CLs) | kg m^{−2} s^{−2} |

${\mathrm{S}}_{{\mathrm{H}}_{2}}=-\frac{{\mathrm{j}}_{\mathrm{a}}}{2\mathrm{F}}{\mathrm{M}}_{{\mathrm{H}}_{2}}$ (Anode CL) | kg m^{−3} s^{−1} |

${\mathrm{S}}_{{\mathrm{O}}_{2}}=-\frac{{\mathrm{j}}_{\mathrm{c}}}{4\mathrm{F}}{\mathrm{M}}_{{\mathrm{O}}_{2}}$ (Cathode CL) | kg m^{−3} s^{−1} |

${\mathrm{S}}_{{\mathrm{H}}_{2}\mathrm{O}}=\frac{{\mathrm{j}}_{\mathrm{c}}}{2\mathrm{F}}{\mathrm{M}}_{{\mathrm{H}}_{2}\mathrm{O}}$(Cathode CL) | kg m^{−3} s^{−1} |

${\mathrm{S}}_{\mathrm{T}}={\mathrm{j}}_{\mathrm{a}}\left|{\mathsf{\eta}}_{\mathrm{a}}\right|+{\mathsf{\sigma}}_{\mathrm{eff},\mathrm{m}}{\Vert \nabla {\mathsf{\varphi}}_{\mathrm{m}}\Vert}^{2}+{\mathsf{\sigma}}_{\mathrm{eff},\mathrm{s}}\Vert \nabla {\mathsf{\varphi}}_{\mathrm{s}}\Vert {}^{2}$(Anode CL) ${\mathrm{S}}_{\mathrm{T}}={\mathrm{j}}_{\mathrm{c}}\left|{\mathsf{\eta}}_{\mathrm{c}}\right|-{\mathrm{j}}_{\mathrm{c}}\frac{{\mathrm{dV}}_{0}}{\mathrm{dT}}\mathrm{T}+{\mathsf{\sigma}}_{\mathrm{eff},\mathrm{m}}{\left|\right|\nabla {\mathsf{\varphi}}_{\mathrm{m}}\left|\right|}^{2}+{\mathsf{\sigma}}_{\mathrm{eff},\mathrm{s}}{\left|\right|\nabla {\mathsf{\varphi}}_{\mathrm{s}}\left|\right|}^{2}$ (Cathode CL) ${\mathrm{S}}_{\mathrm{T}}={\mathsf{\sigma}}_{\mathrm{eff},\mathrm{m}}{\left|\right|\nabla {\mathsf{\varphi}}_{\mathrm{m}}\left|\right|}^{2}$ (Membrane) ${\mathrm{S}}_{\mathrm{T}}={\mathsf{\sigma}}_{\mathrm{eff},\mathrm{s}}{\left|\right|\nabla {\varphi}_{\mathrm{s}}\left|\right|}^{2}$ (GDLs and CCs) | W m^{−3} |

${\mathrm{S}}_{\mathrm{s}}=-{\mathrm{j}}_{\mathrm{a}}$(Anode CL) ${\mathrm{S}}_{\mathrm{s}}=+{\mathrm{j}}_{\mathrm{c}}$(Cathode CL) | A m^{−3} |

${\mathrm{S}}_{\mathrm{m}}=+{\mathrm{j}}_{\mathrm{a}}$(Anode CL) ${\mathrm{S}}_{\mathrm{m}}=-{\mathrm{j}}_{\mathrm{c}}$(Cathode CL) | A m^{−3} |

Parameter | Value | Units |
---|---|---|

Porosity of GDL/CL | 0.6/0.4 | - |

Volume fraction of membrane in the CL | 0.3 | - |

Anode/cathode reference exchange current density | 1 × 10^{9}/1 × 10^{4} | A m^{−3} |

Anode transfer coefficient | 0.5/1 | - |

Reference hydrogen concentration | 40.88 | mol m^{−3} |

Reference oxygen concentration | 40.88 | mol m^{−3} |

Thermal conductivity of CC/GDL/CL/membrane | 20/1.2/1.5/0.95 | W m^{−1} K^{−1} |

Electrical conductivity of CC/GDL/CL | 10,000/1250/300 | S m^{−1} |

Permeability of GDL/CL | 1.18 × 10^{−11}/2.36 × 10^{−12} | m^{2} |

Hydrogen diffusivity | 1.055 × 10^{−4} (T/333.15)^{1.5} (101,325/P) | m^{2} s^{−1} |

Oxygen diffusivity | 2.652 × 10^{−5} (T/333.15)^{1.5} (101,325/P) | m^{2} s^{−1} |

Water diffusivity | 2.982 × 10^{−5} (T/333.15)^{1.5} (101,325/P) | m^{2} s^{−1} |

Description | Conditions | Value | Units |
---|---|---|---|

Anode terminal | ϕ_{s} | 0 | V |

Cathode terminal | ϕ_{s} | V_{cell} | V |

Anode GFC inlet | Y | H_{2} = 1 | - |

T | 453.15 | K | |

Cathode GFC inlet | Y | O_{2}:N_{2} = 0.233:0.767 | - |

T | 453.15 | K |

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**MDPI and ACS Style**

Li, S.; Peng, C.; Shen, Q.; Wang, C.; Cheng, Y.; Yang, G.
Impact of Membrane Phosphoric Acid Doping Level on Transport Phenomena and Performance in High Temperature PEM Fuel Cells. *Membranes* **2021**, *11*, 817.
https://doi.org/10.3390/membranes11110817

**AMA Style**

Li S, Peng C, Shen Q, Wang C, Cheng Y, Yang G.
Impact of Membrane Phosphoric Acid Doping Level on Transport Phenomena and Performance in High Temperature PEM Fuel Cells. *Membranes*. 2021; 11(11):817.
https://doi.org/10.3390/membranes11110817

**Chicago/Turabian Style**

Li, Shian, Chengdong Peng, Qiuwan Shen, Chongyang Wang, Yuanzhe Cheng, and Guogang Yang.
2021. "Impact of Membrane Phosphoric Acid Doping Level on Transport Phenomena and Performance in High Temperature PEM Fuel Cells" *Membranes* 11, no. 11: 817.
https://doi.org/10.3390/membranes11110817