# Numerical Model of Filtration Efficiency Based on Fractal Characteristics of Particulate Matter and Particle Filter

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## Abstract

**:**

_{f}) was introduced to redefine the particle size. The porous medium inside the particle filter was a solid phase fractal. The pore tortuosity fractal dimension (D

_{t}) and the pore area fractal dimension (D

_{a}) were introduced to define the fiber length of the trap. The Brownian diffusion coefficient and permeability were modified. A new fractal numerical model of GPF filtration efficiency was proposed based on the classical filtration theory. The results show that the fractal expansion model of filtration efficiency has good applicability. The influence of GPF structural parameters on filtration efficiency and pressure drop was analyzed. In this study, two performance metrics, trapping efficiency and pressure drop, were considered by fractal expansion filtration modeling. It is possible to increase or decrease filtration efficiency by adjusting the porosity and pore diameter.

## 1. Introduction

_{f}) is introduced to redefine the particle size. The fractal dimension of pore tortuosity (D

_{t}) and the dimension of pore surface integral (D

_{a}) were introduced to define the fiber length of the trap, and the Brownian diffusion coefficient and permeability were modified. It shows that the fractal expansion model of filtration efficiency has good applicability and good representation among one-dimensional numerical models.

## 2. Materials and Methods

#### 2.1. Testing Method

#### 2.1.1. High-Resolution Transmission Electron Microscope

#### 2.1.2. Scanning Electron Microscope

#### 2.1.3. Mercury Intrusion Porosimetry

#### 2.2. Fractal Analysis

#### 2.2.1. Particle Fractal

_{f}of aggregates indicates the compactness of particles. The fractal dimension is related to the structure of aggregates. The larger the dimension, the denser the structure of aggregates, and the higher the overlap between primary particles [18,19].

_{p}of the primary particles that are easy to identify. According to the study of Farias [20], the calculation formula containing the fractal dimension D

_{f}is shown in Equation (1):

_{f}and kg are determined by the slope and intercept of the fitted line, and the fitted line is obtained by plotting the logarithmic coordinate diagram of Np − Dg/Dp.

#### 2.2.2. Fractal of Porous Media

_{a}[24]:

_{E}is the Euclidean geometric dimension. In this paper, the Brownian diffusion of particles in the two-dimensional plane, D

_{E}= 2. The total pore area on the cross-section of porous media A

_{D}:

_{t}of tortuosity of pore channels in porous media:

#### 2.3. Model Proposal

#### Model Assumptions

_{c}) are regarded as average by using the spherical packed bed theory:

_{c}is the trap unit diameter, and pore is related to the internal pore model of GPF. In some studies, the pore model is cylindrical. When the GPF porous wall is loaded, PM accumulates around the collector unit until the pore unit is blocked [27]. The diameter pore of the aperture unit is expressed as follows:

_{c}of the collector unit can be obtained as a function of the pore, and the expression is shown in (15):

#### 2.4. Filtering Scheme

#### 2.4.1. Brownian Diffusion

_{D}of a single trap unit with a Brownian diffusion mechanism depends on the Péclet number (P

_{e}), which is related to the diffusion and advection processes. The calculation formula of Brownian diffusion capture efficiency E

_{D}is shown in Equation (16):

_{D}is the particle diffusion coefficient [27]. The calculation formula is as follows:

_{B}is the Boltzmann constant, d

_{p}is the particle diameter, T is the exhaust gas temperature flowing through the wall, and ug is the dynamic viscosity of the exhaust gas. The Stokes–Cunningham factor (C

_{c}) explains the slip fluid dynamics. In this case, C

_{c}is calculated from the Knudsen number Kn (λ refers to the average free path of gas) and the average pore size of GPF porous media [25]. Cc is denoted by,

#### 2.4.2. Direct Interception

_{R}(direct interception) of a single capture is calculated as Equation (22) [28].

_{R}is the interception coefficient: the ratio of particle diameter to fiber diameter.

#### 2.4.3. Inertial Impaction

_{I}of a single trap unit is a function of the Stokes number [27]:

_{I}determines that the coefficients a and b are in the same form, where a = 2.4363 and b = 0.7817. ρ

_{p}describes the density of ideal spherical particles with a particle size as the flow diameter. The calculation formula of the Stokes number Stk is shown in Equation (26):

_{p}: The density of ideal spherical particles with a particle size as flow diameter is described.

#### 2.5. Modified Model

#### 2.5.1. Particle Parameter Correction

_{f}can represent the agglomeration length of primary particles. Previous studies have shown that the most important components of inhalable particles are soot and SOF, and each particle can be considered as an aggregate of primary particles. As shown in Figure 5, the particles can be equivalent to a spherical structure; the core is soot, and the shell is the soluble organic fraction (SOF).

_{g},

_{mob}is the migration diameter of fractal aggregates, d

_{g}

_{,gyr}is the rotational diameter of particles, D

_{f}is the fractal dimension of particle aggregates, and d

_{p}is the primary particle diameter of aggregates.

_{p}is the primary particle, r

_{i}and r

_{G}define the position of the center and the center of gravity of the principal particle of a cluster composed of N identical particles, respectively.

#### 2.5.2. Non-Uniform Porous Media

_{dc}: Root of average area:

_{m}is the diffusion pressure of particles; M is the mass of particles, kg/mol; r is gas constant, R = 8.314 J/(mol·K); T is ambient temperature; and P is atmospheric pressure.

#### 2.6. Modified Model

#### 2.6.1. Filtering Effectiveness

_{D}), direct interception (E

_{R}), and inertial deposition (E

_{I}). Assuming that all mechanisms are independent of each other and the efficiency is the combined effect of multiple filtration mechanisms, the filtration efficiency of a single capture unit is [11],

_{h}and porosity is,

_{P}) with the fractal dimension (D

_{f}) of particle aggregates and filter fiber diameter (dc) with the fractal dimension (D

_{t}) of filter solid phase pore size:

#### 2.6.2. Pressure Drop

## 3. Results and Discussion

#### 3.1. Model Verification

#### 3.2. Effect of Microstructure Parameters on Filtration Efficiency

_{h}). The filtration performance of the trap model is closely related to the unit trap, and the diameter of the trap unit is calculated by porosity and pore size.

#### 3.2.1. Porosity

#### 3.2.2. Pore Diameter

#### 3.2.3. Media Thickness

#### 3.3. Simulation of Filtration Efficiency and Pressure Drop

^{3}, the temperature is 400 degrees Celsius, and the pressure drop is calculated according to Darcy’s law. Considering the influence of porosity, pore diameter, and medium thickness of GPF porous media on filtration efficiency, we find that these three parameters have a great influence on the two performance indexes of capture efficiency and pressure drop. It can be seen from Figure 13a,b that the larger the medium thickness and the smaller the porosity, the higher the capture efficiency and the lower the pressure drop. At the same pore size, as the thickness of the medium increases, the porosity is small, and the filtration efficiency will increase. Because the smaller the porosity, the more particles exist in the pores, and the particles find it more difficult to pass through the porous medium, resulting in increased filtration efficiency. But at the same time, low porosity also leads to less pressure drop because the fluid will encounter less resistance when passing through porous media.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Microstructure of GPF (

**a**) GPF structure simulation diagram, (

**b**) GPF CT scan image, (

**c**) GPF SEM scan image.

**Figure 10.**Effect of porosity on filtration efficiency (

**a**) Impact of porosity and particle size, (

**b**) Different porosity levels.

**Figure 11.**Effect of pore size on filtration efficiency (

**a**) Effect of filtration efficiency at different pore and particle sizes, (

**b**) Variation of filtration efficiency at different particle sizes.

**Figure 12.**Effect of wall thickness on filtration efficiency. (

**a**) Filtration efficiency at different media thicknesses and particle sizes, (

**b**) Variation of filtration efficiency at different particle sizes.

**Figure 13.**Analysis of filtration efficiency and pressure drop (

**a**) Filtration efficiency at different media thicknesses and particle sizes, (

**b**) Pressure drop at different media thicknesses and particle sizes, (

**c**) Effect of filtration efficiency at different pore and particle sizes, (

**d**) Effect of pressure drop at different pore and particle sizes, (

**e**) Filtration efficiency at different media thicknesses and pore sizes, (

**f**) Pressure drop at different media thicknesses and pore sizes.

Parameter | Alphabet | Numerical Value |
---|---|---|

Average pore size [um] | dpore | 16.64 |

Maximum pore size [nm] | Sportmax | 361,016 |

Minimum pore size [nm] | dpore_{min} | 5.48 |

Tortuosity | t | 9.7028 |

Porosity | α | 64.5914% |

Permeability [md] | K | 923.1625 |

Characteristic length [nm] | L_{0} | 17,786.93 |

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**MDPI and ACS Style**

Liu, Y.; Wang, H.; Yu, H.
Numerical Model of Filtration Efficiency Based on Fractal Characteristics of Particulate Matter and Particle Filter. *Atmosphere* **2023**, *14*, 1689.
https://doi.org/10.3390/atmos14111689

**AMA Style**

Liu Y, Wang H, Yu H.
Numerical Model of Filtration Efficiency Based on Fractal Characteristics of Particulate Matter and Particle Filter. *Atmosphere*. 2023; 14(11):1689.
https://doi.org/10.3390/atmos14111689

**Chicago/Turabian Style**

Liu, Yiqing, Hao Wang, and Haisheng Yu.
2023. "Numerical Model of Filtration Efficiency Based on Fractal Characteristics of Particulate Matter and Particle Filter" *Atmosphere* 14, no. 11: 1689.
https://doi.org/10.3390/atmos14111689