# Experiment and Prediction of Pressure Drop in a Fiber–Powder Composite Material with Porous Structure for Energy Wheels and Air Cleaners

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Pressure Drop Experimental Setup

#### 2.2. Pressure Drop Prediction Model for the Composite Material

- 1.
- In the $\frac{1-\epsilon}{{D}_{p}{\epsilon}^{3}}$ term in Equation (7), ${D}_{p}$ and $\epsilon $ are mainly affected by ${m}_{d}$, and the term is positively correlated with ${m}_{d}$. To reduce computing expenses, the term is simplified to the form of an exponential function containing ${m}_{d}$.
- 2.
- The ${D}_{p}$ in the $22\frac{\mu}{{D}_{p}\rho u}$ term is simplified to the form of an exponential function containing ${m}_{d}$.
- 3.
- Referring to Equation (7), power functions are used to describe the relationship between L and $\Delta {p}_{d}$ as well as the relationship between u and $\Delta {p}_{d}$.

#### 2.3. Pressure Drop Prediction Procedure

- A.
- Obtain the tested data for prediction ($u,L,{m}_{d},\Delta {p}_{s},\Delta {p}_{t}$) and calculate $\Delta {p}_{i}$.
- B.
- Train the substrate material pressure drop prediction model and obtain the coefficient ${j}_{i}$s.
- C.
- Calculate $\Delta {p}_{s}$ with the trained model.
- D.
- Train the adsorption material pressure drop prediction model and obtain the coefficients ${k}_{i}$s.
- E.
- Calculate $\Delta {p}_{i}$ with the trained model and calculate the predicted $\Delta {p}_{t}$.

## 3. Results and Discussion

#### 3.1. Experimental Results of Pressure Drop in Fibrous Core Materials

#### 3.2. Analysis of the Prediction Results

#### 3.3. Further Discussion of the Pressure Drop Prediction Method

- A.
- Obtain the tested data for prediction ($u,L,{m}_{d},\Delta {p}_{s},\Delta {p}_{t}$).
- B.
- Train the substrate material pressure drop prediction model and obtain the coefficient ${j}_{i}$s.
- C.
- Calculate $\Delta {p}_{s}$ with the trained model.
- D.
- Train the total pressure drop prediction model and obtain the coefficient ${k}_{i}$s.
- E.
- Calculate the predicted $\Delta {p}_{t}$ with the trained model.

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$C$ | Nozzle flow coefficient |

$D$ | Nozzle diameter (m) |

${D}_{p}$ | Equivalent spherical diameter of porous media (m) |

${f}_{f}$ | Friction factor |

F | Cross-sectional area of nozzle (m^{2}) |

$L$ | Length of material (m) |

$m$ | Material content (kg/m^{2}) |

$p$ | Air pressure (Pa) |

$Q$ | Air volume flowrate (m^{3}/s) |

$Re$ | Reynolds number |

$u$ | Airflow velocity (m/s) |

$V$ | Material volume (m^{3}) |

$Y$ | Expansion coefficient |

Greek symbols | |

$\epsilon $ | Porosity |

$\mu $ | Viscosity (Pa.s) |

$\rho $ | Fluid density (kg/m^{3}) |

Subscripts | |

$d$ | Adsorption material |

$e$ | Tested data |

$p$ | Predicted data |

$s$ | Substrate material |

$t$ | Tested material |

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**Figure 4.**The tested material: (

**a**) substrate material; (

**b**) substrate material with activated carbon.

**Figure 9.**Comparison of the predicted and tested $\Delta {p}_{i}$ (

**Left**: pressure drop range of 0–50 Pa;

**Right**: pressure drop range of 50–200 Pa).

**Figure 10.**Comparison of the predicted and tested $\Delta {p}_{t}$ (

**Left**: pressure drop range of 0–50 Pa;

**Right**: pressure drop range of 50–300 Pa).

**Figure 11.**Schematic illustration of the prediction method proposed in Section 3.3.

**Figure 12.**Comparison of the predicted and tested $\Delta {p}_{t}$ using the method proposed in Section 3.3 with Equation (16) (

**Left**: pressure drop range of 0–50 Pa;

**Right**: pressure drop range of 50–300 Pa).

**Figure 13.**Comparison of the predicted and tested $\Delta {p}_{t}$ using the method proposed in Section 3.3 with Equation (17) (

**Left**: pressure drop range of 0–50 Pa;

**Right**: pressure drop range of 50–300 Pa).

**Figure 14.**Comparison of the absolute prediction error of two different prediction methods (M1: the method proposed in Section 2.3; M2: the method proposed in Section 3.3 using Equation (16); M2M: the method proposed in Section 3.3 using Equation (17)).

Parameter | Acquisition Method | Use |
---|---|---|

Length of material (L) | Measured | Train the $\Delta {p}_{s}$ and $\Delta {p}_{i}$ prediction model |

Air velocity (u) | Measured | Train the $\Delta {p}_{s}$ and $\Delta {p}_{i}$ prediction model |

Desiccant material content (${m}_{d}$) | Measured | Train the $\Delta {p}_{i}$ prediction model |

Substrate material pressure drop ($\Delta {p}_{s}$) | Measured | Train the $\Delta {p}_{s}$ and $\Delta {p}_{i}$ prediction model |

Material total pressure drop ($\Delta {p}_{t}$) | Measured | Train the $\Delta {p}_{i}$ prediction model |

Material pressure drop increase ($\Delta {p}_{i}$) | Calculated | Train the $\Delta {p}_{i}$ prediction model |

**Table 2.**Test material parameters in Figure 7.

Test Material ID (4 Layers) | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Layer number of material | 4 | 4 | 4 | 4 |

Material width W (cm) | 40 | 40 | 40 | 40 |

Material height H (cm) | 40 | 40 | 40 | 40 |

Substrate material content (kg) | 0.108 | 0.108 | 0.108 | 0.108 |

Total content (kg) | 0.186 | 0.289 | 0.343 | 0.409 |

**Table 3.**Test material parameters in Figure 8.

Layer ID (Single Layer) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

Material width W (cm) | 40 | 40 | 40 | 40 | 40 | 40 | 40 |

Material height H (cm) | 40 | 40 | 40 | 40 | 40 | 40 | 40 |

Substrate material content (kg) | 0.027 | 0.027 | 0.027 | 0.027 | 0.027 | 0.027 | 0.027 |

Total content (kg) | 0.098 | 0.106 | 0.103 | 0.102 | 0.111 | 0.106 | 0.104 |

Data Point ID | $\mathit{u}\left(\mathbf{m}/\mathbf{s}\right)$ | $\Delta {\mathit{p}}_{\mathit{i}}\left(\mathbf{P}\mathbf{a}\right)$ | $\Delta {\mathit{p}}_{\mathit{t}}\left(\mathbf{P}\mathbf{a}\right)$ | Data Point ID | $\mathit{u}\left(\mathbf{m}/\mathbf{s}\right)$ | $\Delta {\mathit{p}}_{\mathit{i}}\left(\mathbf{P}\mathbf{a}\right)$ | $\Delta {\mathit{p}}_{\mathit{t}}\left(\mathbf{P}\mathbf{a}\right)$ |
---|---|---|---|---|---|---|---|

1 | 0.2 | 8.1 | 11.6 | 15 | 0.2 | 6.1 | 18.5 |

2 | 0.3 | 11 | 17 | 16 | 0.3 | 10.7 | 30.3 |

3 | 0.4 | 14.9 | 23.2 | 17 | 0.4 | 16.3 | 44.4 |

4 | 0.5 | 20.4 | 31.3 | 18 | 0.5 | 22.4 | 59.2 |

5 | 0.6 | 25.5 | 38.9 | 19 | 0.6 | 30.1 | 75.2 |

6 | 0.7 | 32.4 | 48.6 | 20 | 0.7 | 37.6 | 92.9 |

7 | 0.8 | 38.8 | 57.6 | 21 | 0.8 | 45.3 | 110.1 |

8 | 0.9 | 47.1 | 68.8 | 22 | 0.9 | 55.4 | 131.2 |

9 | 1.0 | 54.2 | 79.1 | 23 | 1.0 | 65.7 | 152.6 |

10 | 1.1 | 64.7 | 92.6 | 24 | 1.1 | 75.4 | 174.2 |

11 | 1.2 | 72.8 | 103.8 | 25 | 1.2 | 88.7 | 197.9 |

12 | 1.3 | 84.5 | 118.7 | 26 | 1.3 | 100 | 222.6 |

13 | 1.4 | 93.6 | 130.9 | 27 | 1.4 | 115.5 | 248.7 |

14 | 1.5 | 108.5 | 150 | 28 | 1.5 | 128.3 | 275.2 |

Data Point ID | 1–14 | 15–28 |
---|---|---|

Layer number of material | 2 | 8 |

Material width W (cm) | 40 | 40 |

Material height H (cm) | 40 | 40 |

Substrate material content (kg) | 0.054 | 0.216 |

Total content (kg) | 0.204 | 0.622 |

Parameters | $\Delta {\mathit{p}}_{\mathit{t}}$ |
---|---|

$RMSE$ training sets | 2.7 Pa |

$RMSE$ testing sets | 4.0 Pa |

$MAPE$ training sets | 5.2% |

$MAPE$ testing sets | 6.6% |

Parameters | $\Delta {\mathit{p}}_{\mathit{t}}$ | $\Delta {\mathit{p}}_{\mathit{t}}$ (After Modification) |
---|---|---|

$RMSE$ training sets | 6.3 Pa | 2.8 Pa |

$RMSE$ testing sets | 6.0 Pa | 3.6 Pa |

$MAPE$ training sets | 8.2% | 5.9% |

$MAPE$ testing sets | 7.1% | 6.1% |

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## Share and Cite

**MDPI and ACS Style**

Gao, H.; Li, Z.; Zhou, X.; Yin, X.; Shan, M.
Experiment and Prediction of Pressure Drop in a Fiber–Powder Composite Material with Porous Structure for Energy Wheels and Air Cleaners. *Buildings* **2023**, *13*, 2196.
https://doi.org/10.3390/buildings13092196

**AMA Style**

Gao H, Li Z, Zhou X, Yin X, Shan M.
Experiment and Prediction of Pressure Drop in a Fiber–Powder Composite Material with Porous Structure for Energy Wheels and Air Cleaners. *Buildings*. 2023; 13(9):2196.
https://doi.org/10.3390/buildings13092196

**Chicago/Turabian Style**

Gao, Han, Zhenhai Li, Xigang Zhou, Xiaolong Yin, and Mengmeng Shan.
2023. "Experiment and Prediction of Pressure Drop in a Fiber–Powder Composite Material with Porous Structure for Energy Wheels and Air Cleaners" *Buildings* 13, no. 9: 2196.
https://doi.org/10.3390/buildings13092196