# A Lumped-Mass Model of Membrane Humidifier for PEMFC

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## Abstract

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## 1. Introduction

## 2. Model Description

#### 2.1. Gas-to-Gas Membrane Humidifier

#### 2.2. Manifolds

#### 2.3. Heat and Mass Exchanger

#### 2.3.1. Heat Transfer

#### 2.3.2. Mass Transfer

#### 2.4. Simulation

## 3. Results and Discussion

#### 3.1. Effects of Operating Conditions

#### 3.2. Effects of Geometric Parameters

#### 3.3. Humidifier Performance under Transient Conditions

## 4. Conclusions

- The model can be used to study the transient responses of the humidifier, which would be beneficial for the water management of PEMFC systems. The changes in flow characteristics over time can be applied to the development of a control strategy for the humidifier.
- If the flow rates are fixed, the relative humidity of the moist air had the strongest effect on the relative humidity of the dry air at the outlet.
- Compared to the dry air flow rate, the moist air flow rate had a less significant influence on the relative humidity of the moist air. Adjusting the flow rate on the dry side is an effective approach to manage the water content in the supply air.
- The manifolds may become saturated with water during long-term operation, especially under low load conditions. This situation should be avoided to improve the PEMFC system performance.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$A$ | area |

$c$ | heat capacity |

$C$ | heat capacity ratio |

$\u2102$ | mass flow rate ratio |

$d$ | membrane thickness |

$D$ | diameter |

$Di$ | diffusion coefficient |

$\Delta {T}_{lm}$ | log mean temperature difference |

$f$ | friction factor |

$g$ | gravitational acceleration |

$h$ | convective heat transfer |

$\mathbb{h}$ | convection mass transfer |

$He$ | head loss |

$k$ | thermal conductivity |

$\mathbb{k}$ | mass diffusion coefficient |

$L$ | length |

$\dot{m}$ | mass flow rate |

$n$ | number of membranes |

$NTU$ | number of transfer units for heat transfer analysis |

$\mathbb{N}\mathbb{T}\mathbb{U}$ | number of transfer units for mass transfer analysis |

$P$ | pressure |

$Pr$ | Prandtl number |

$q$ | heat transfer rate |

$R$ | total thermal resistance |

$\mathbb{R}$ | total mass resistance |

$Re$ | Reynolds number |

$Sc$ | Schmidt number |

$Sh$ | Sherwood number |

$t$ | time |

$T$ | temperature |

$U$ | overall heat transfer coefficient |

$\mathbb{U}$ | overall mass transfer coefficient |

$\mathrm{V}$ | velocity |

Greek letters | |

$\gamma $ | packing fraction |

$\epsilon $ | effectiveness for heat transfer analysis |

$\u03f5$ | effectiveness for mass transfer analysis |

$\mu $ | dynamic viscosity |

$\rho $ | density |

$\upsilon $ | kinematic viscosity |

$\phi $ | relative humidity |

$\omega $ | specific humidity |

Subscripts and superscripts | |

$a$ | ambient |

$d$ | dry flow |

$f$ | manifold |

$h$ | hydraulic |

$i$ | inlet |

$is$ | inner side |

$m$ | membrane |

$min$ | minimum |

$max$ | maximum |

$o$ | outlet |

$os$ | outer side |

$sh$ | shell |

$w$ | wet flow |

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**Figure 2.**Effects of the operating temperature on the mass transfer rate and relative humidity of dry air.

**Figure 3.**Effects of operating relative humidity on the mass transfer rate and relative humidity of dry air.

**Figure 4.**Effects of temperature difference on the heat transfer rate and relative humidity of dry air.

**Figure 9.**Effects of long-term operating conditions and possible water saturation in the outlet manifold.

Parameters | Value |
---|---|

Shell diameter (mm) | 86 |

Inner tube diameter (mm) | 0.9 |

Membrane thickness (mm) | 0.1 |

Number of tubes | 1660 |

Membrane material | PTFE |

Length of tube (mm) | 254 |

Port diameter (mm) | 38 |

Length of manifold (mm) | 65 |

Housing/shell material | ABS |

Total length of humidifier module (mm) | 384 |

Case | Simulation Purposes | Fixed Parameters | Varied Parameters |
---|---|---|---|

1 | Effects of operating temperature and mass flow rate | ${\phi}_{w,i}=100\%,{\phi}_{d,i}=5\%$ | ${\dot{m}}_{w}={\dot{m}}_{d}=0.001\to 0.015\mathrm{kg}/\mathrm{s}$ ${T}_{w,i}=40\to 70\xb0\mathrm{C}$ ${T}_{d,i}=30\to 60\xb0\mathrm{C}$ |

2 | Effects of operating relative humidity and mass transfer rate | ${\phi}_{d,i}=5\%$ ${T}_{w,i}=60\xb0\mathrm{C},{T}_{d,i}=50\xb0\mathrm{C}$ | ${\dot{m}}_{w}={\dot{m}}_{d}=0.001\to 0.015\mathrm{kg}/\mathrm{s}$ ${\phi}_{w,i}=60\to 100\%$ |

3 | Effects of temperature difference on heat transfer rate | ${\dot{m}}_{w}={\dot{m}}_{d}=0.01\mathrm{kg}/\mathrm{s}$ ${\phi}_{d,i}=5\%$ ${T}_{d,i}=40\xb0\mathrm{C}$ | ${T}_{w,i}=50\to 70\xb0\mathrm{C}$ $\left(\Delta T=10\to 30\xb0\mathrm{C}\right)$ |

4 | Effects of geometric parameters: Membrane length Membrane thickness Number of membranes | ${\phi}_{w,i}=100\%,{\phi}_{d,i}=5\%$ T _{w,i} = 70 °C, T_{d,i} = 60 °C
| ${\dot{m}}_{w}={\dot{m}}_{d}=0.001\to 0.015\mathrm{kg}/\mathrm{s}$ $L=0.178\to 0.381\mathrm{m}$ $d=0.05\to 0.15\mathrm{mm}$ $n=800\to 2500$ |

5 | Responses of humidifier in long-term transient operation | ${\dot{m}}_{w}=0.015\mathrm{kg}/\mathrm{s}$ ${\dot{m}}_{d}=0.005\mathrm{kg}/\mathrm{s}$ ${\phi}_{w,i}=100\%,{\phi}_{d,i}=5\%$ ${T}_{w,i}=70\xb0\mathrm{C},{T}_{d,i}=60\xb0\mathrm{C}$ $t=1000\mathrm{s}$ | (All parameters are fixed) |

6 | Responses of humidifier in short-term transient operation | ${\phi}_{w,i}=100\%,{\phi}_{d,i}=5\%$ ${T}_{w,i}=60\xb0\mathrm{C},{T}_{d,i}=50\xb0\mathrm{C}$ $t=10\mathrm{s}$ | ${\dot{m}}_{w}={\dot{m}}_{d}=0.001\to 0.015\mathrm{kg}/\mathrm{s}$ |

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**MDPI and ACS Style**

Vu, H.N.; Nguyen, X.L.; Yu, S.
A Lumped-Mass Model of Membrane Humidifier for PEMFC. *Energies* **2022**, *15*, 2113.
https://doi.org/10.3390/en15062113

**AMA Style**

Vu HN, Nguyen XL, Yu S.
A Lumped-Mass Model of Membrane Humidifier for PEMFC. *Energies*. 2022; 15(6):2113.
https://doi.org/10.3390/en15062113

**Chicago/Turabian Style**

Vu, Hoang Nghia, Xuan Linh Nguyen, and Sangseok Yu.
2022. "A Lumped-Mass Model of Membrane Humidifier for PEMFC" *Energies* 15, no. 6: 2113.
https://doi.org/10.3390/en15062113