# Filtration Kinetics of Depth Filters—Modeling and Comparison with Tomographic Data of Particle Depositions

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## Abstract

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## 1. Introduction

## 2. Approach for Calculating Filtration Kinetics Considering Tomographic Data

## 3. Results and Discussion

#### 3.1. Application of Modeling Approach

_{p50,3}= 1.5 µm). Therefore, experimental data regarding pressure drop and filtration efficiency, as well as spatially and temporally resolved particle deposition within the filter material during the filtration process, were available. The averaged structural data of the used filter material is given in Table 2.

#### 3.2. Initial Filtration Efficiency

#### 3.3. Filtration Kinetics—Macroscopic

#### 3.4. Filtration Kinetics—Microscopic

_{F,i}/L

_{max}> 0.7). With the tomographic method, an accumulation of the deposited material was experimentally found in the same zone (Figure 10b,c). Calculations using the averaged porosity as an input parameter showed significant deviations with respect to loading behavior on a microscopic scale in comparison to the visually obtained experimental data. This emphasized the improvement in the calculation of the microscopic loading behavior by this new approach and the suggesting of incorporating the microstructure into one-dimensional calculations as presented herein.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## List of Abbreviations

Abbreviation | Meaning | |

CT | Computer tomography | |

XRM | X-ray computed microscopy |

## List of Latin Symbols

Symbol | Meaning | Unit |

${\mathrm{C}}_{\mathrm{D},\mathrm{i}}$ | Correction factor for flow slip for diffusional deposition in subfilter $\mathrm{i}$ | dimensionless |

${\mathrm{C}}_{\mathrm{D},\mathrm{i}}^{\prime}$ | Correction factor for diffusional deposition in subilfter $\mathrm{i}$ | dimensionless |

${\mathrm{C}}_{\mathrm{R},\mathrm{i}}$ | Correction factor for flow slip for deposition by interception mechanism in subfilter $\mathrm{i}$ | dimensionless |

$\mathrm{D}$ | Diffusivity | m^{2}/s |

${\mathrm{d}}_{\mathrm{F}}$ | Fiber diameter | µm |

${\mathrm{d}}_{\mathrm{F},50}$ | Mean fiber diameter | µm |

${\mathrm{d}}_{\mathrm{F},\mathrm{i}}$ | Fiber diameter in subfilter $\mathrm{i}$ | µm |

${\mathrm{d}}_{\mathrm{F},\mathrm{i},\mathrm{t}}$ | Fiber diameter in subfilter $\mathrm{i}$ at loading step t | µm |

${\mathrm{d}}_{\mathrm{F},\mathrm{i},\mathrm{t}+1}$ | Fiber diameter in subfilter $\mathrm{i}$ at loading step t + 1 | µm |

${\mathrm{d}}_{\mathrm{p}}$ | Particle diameter | µm |

${\mathrm{E}}_{\mathrm{i}}$ | Separation efficiency of subfilter | dimensionless |

${\mathrm{E}}_{\mathrm{total}}$ | Total separation efficiency of filter | dimensionless |

${\mathrm{k}}_{\mathrm{B}}$ | Boltzmann constant | m^{2}kg/s^{2}K |

${\mathrm{Kn}}_{\mathrm{f},\mathrm{i}}$ | Knudsen number at the fiber in subfilter $\mathrm{i}$ | dimensionless |

${\mathrm{Ku}}_{\mathrm{i}}$ | Kuwabara factor in subfilter $\mathrm{i}$ | dimensionless |

${\mathrm{L}}_{\mathrm{F},\mathrm{i}}$ | Depth of subfilter $\mathrm{i}$ | mm |

${\mathrm{L}}_{\mathrm{F},\mathrm{total}}$ | Overall depth of filter material | mm |

${\mathrm{N}}_{\mathrm{p},\mathrm{i},\mathrm{t}}$ | Number of particles deposited in subfilter i at timestep t | dimensionless |

${\mathrm{n}}_{\mathrm{i}}$ | Number of particles in inlet stream of the first subfilter | dimensionless |

${\mathrm{Pe}}_{\mathrm{i}}$ | Peclet-number in subfilter $\mathrm{i}$ | dimensionless |

${\mathrm{p}}_{\mathrm{i}}$ | Pressure loss of subfilter i | pa |

${\mathrm{R}}_{\mathrm{i}}$ | Interception parameter in subfilter $\mathrm{i}$ | dimensionless |

${\mathrm{Stk}}_{\mathrm{i}}$ | Stokes number in subfilter $\mathrm{i}$ | dimensionless |

${\mathrm{T}}_{\mathrm{i}}$ | Separation efficiency of subfilter $\mathrm{i}$ | dimensionless |

${\mathrm{u}}_{0,\mathrm{i}}$ | gas velocity in subfilter i | |

${\mathrm{u}}_{0}$ | Inlet gas velocity | m/s |

${\mathrm{V}}_{\mathrm{i},0}$ | Free volume of subfilter i | m^{3} |

${\mathrm{V}}_{\mathrm{p},\mathrm{i},\mathrm{t}}$ | Volume of particles deposited in subfilter i at loading step t | m^{3} |

## List of Greek Symbols

Symbol | Meaning | Unit |

$\mathsf{\alpha}$ | Packing density | dimensionless |

${\mathsf{\alpha}}_{\mathrm{i}}$ | Packing density of subfilter $\mathrm{i}$ | dimensionless |

$\Delta {\mathrm{F}}_{\mathrm{i},\mathrm{j}}$ | Fraction of fiber diameter j in subfilter i | dimensionless |

$\Delta {\mathrm{p}}_{\mathrm{i}}$ | Pressure difference in subfilter i | pa |

$\Delta {\mathrm{p}}_{\mathrm{total}}$ | Overall pressure difference of filter | pa |

$\mathsf{\epsilon}$ | Porosity | dimensionless |

${\mathsf{\epsilon}}_{\mathrm{av}}$ | Average porosity | dimensionless |

${\mathsf{\epsilon}}_{\mathrm{i},\mathrm{t}}$ | Porosity of subfilter $\mathrm{i}$ at timestep t | dimensionless |

${\mathsf{\epsilon}}_{\mathrm{i},\mathrm{t}+1}$ | Porosity of subfilter $\mathrm{i}$ at timestep t + 1 | dimensionless |

${\mathsf{\eta}}_{\mathrm{D}+\mathrm{R},\mathrm{i}}$ | Collection efficiency of single fiber through diffusional and interception mechanism in subfilter $\mathrm{i}$ | dimensionless |

${\mathsf{\eta}}_{\mathrm{R},\mathrm{i}}$ | Collection efficiency of single fiber through interception mechanism in subfilter $\mathrm{i}$ | dimensionless |

${\mathsf{\eta}}_{\mathrm{I},\mathrm{i}}$ | Collection efficiency of single fiber through inertial mechanism in subfilter $\mathrm{i}$ | dimensionless |

$\vartheta $ | Temperature | K |

$\mathsf{\mu}$ | Dynamic viscosity of air | Pas |

${\mathsf{\rho}}_{\mathrm{p}}$ | Density of dust particles | kg/m^{3} |

${\mathsf{\phi}}_{\mathrm{i}}$ | Single fiber efficiency in subfilter $\mathrm{i}$ | dimensionless |

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**Figure 1.**Illustration of the modeling approach for calculating the filtration processes in depth filter material under consideration of the filter microstructure.

**Figure 2.**Illustration of considered approaches for recalculating filter structure due to particle deposition (yellow dots) during filtration assuming constant (

**left**) and dynamic increasing fiber diameter marked as red dashed line (

**right**).

**Figure 4.**3D-reconstructed filter material (

**above**) and measured raw porosity distribution in axial flow direction of the test filter (

**below**) obtained via XRM. The axially resolved porosity was used as an input to the model approach and the averaged porosity was used as a reference.

**Figure 5.**SEM-image and histogram of fiber diameter distribution of filter material used as input for the calculations.

**Figure 6.**Comparison between experimentally measured (average ± mean deviation of the triplicate measurement) and modeled filtration efficiency of clean filter material using different input porosities.

**Figure 7.**Comparison between experimental and modeled pressure difference during filtration using different input porosities and applying different model assumptions when considering particle deposition.

**Figure 8.**Comparison between experimental (average ± mean deviation of the triplicate measurement) and modeled filtration efficiencies at different pressure differences during the filtration process.

**Figure 9.**Calculated porosity inside the filter material at different loading states applying the averaged porosity (

**left**) and axially resolved porosity (

**right**).

**Figure 10.**Qualitative comparison of calculated porosity curves using average and axially resolved porosity as input. Data are qualitatively compared with 3D-images of particle depositions visualized via X-ray microscopy at different exp. pressure differences: 52 Pa (

**a**), 168 Pa (

**b**) and 498 Pa (

**c**). Dashed lines approach the borders of the filter material.

**Figure 11.**Comparison between experimental (exp.) (average ± mean deviation of the triplicate measurement) and modeled (mod.) mass distribution inside the filter at three different loadings states. (

**a**) mod.: 40 Pa; exp.: 52.83 ± 20.44, (

**b**) mod.: 159 Pa; exp.: 168 ± 13.33 and (

**c**) mod.: 526 Pa; exp.: 498 ± 25.33.

**Table 1.**Models for calculating the individual separation mechanisms according to diffusion, inertial deposition, and interception applied in the calculations.

Mechanism | Model Equation | Reference |
---|---|---|

${\mathsf{\eta}}_{\mathrm{D}+\mathrm{R},\mathrm{i}}$ | ${\mathsf{\eta}}_{\mathrm{D}+\mathrm{R},\mathrm{i}}=1.6\xb7{\left(\frac{1-{\mathsf{\alpha}}_{\mathrm{i}}}{{\mathrm{Ku}}_{\mathrm{i}}}\right)}^{1/3}\xb7{\mathrm{Pe}}_{\mathrm{i}}^{-2/3}{\mathrm{C}}_{\mathrm{D}}\xb7{\mathrm{C}}_{\mathrm{D}}^{\prime}$ ${\mathrm{Ku}}_{\mathrm{i}}=-0.5\xb7\mathrm{ln}\left({\mathsf{\alpha}}_{\mathrm{i}}\right)-0.75+{\mathsf{\alpha}}_{\mathrm{i}}-0.25\xb7{\mathsf{\alpha}}_{\mathrm{i}}^{2}$ ${\mathsf{\alpha}}_{\mathrm{i}}=1-{\mathsf{\epsilon}}_{\mathrm{i}}$ ${\mathrm{Pe}}_{\mathrm{i}}=\frac{{\mathrm{u}}_{0,\mathrm{i}}\xb7{\mathrm{d}}_{\mathrm{F},\mathrm{i}}}{\mathrm{D}}$ $\mathrm{D}=\frac{{\mathrm{k}}_{\mathrm{B}}\xb7\vartheta}{3\xb7\mathsf{\pi}\xb7\mathsf{\mu}\xb7{\mathrm{d}}_{\mathrm{p}}}$ ${\mathrm{C}}_{\mathrm{D},\mathrm{i}}=1+0.388\xb7{\mathrm{Kn}}_{\mathrm{f},\mathrm{i}}\xb7{\left(\frac{\left(1-{\mathsf{\alpha}}_{\mathrm{i}}\right)\xb7{\mathrm{Pe}}_{\mathrm{i}}}{{\mathrm{Ku}}_{\mathrm{i}}}\right)}^{1/3}$ ${\mathrm{C}}_{\mathrm{D},\mathrm{i}}^{\prime}=\frac{1}{1+{\mathsf{\eta}}_{\mathrm{D},\mathrm{i}}}$ | [36] |

${\mathsf{\eta}}_{\mathrm{R},\mathrm{i}}$ | ${\mathsf{\eta}}_{\mathrm{R},\mathrm{i}}=0.6\xb7\frac{1-{\mathsf{\alpha}}_{\mathrm{i}}}{{\mathrm{Ku}}_{\mathrm{i}}}\xb7\frac{{\mathrm{R}}_{\mathrm{i}}^{2}}{1+{\mathrm{R}}_{\mathrm{i}}}\xb7{\mathrm{C}}_{\mathrm{R},\mathrm{i}}$ ${\mathrm{R}}_{\mathrm{i}}=\frac{{\mathrm{d}}_{\mathrm{p}}}{{\mathrm{d}}_{\mathrm{F},\mathrm{i}}}$ ${\mathrm{C}}_{\mathrm{R},\mathrm{i}}=1+\frac{1.996\xb7{\mathrm{Kn}}_{\mathrm{F},\mathrm{i}}}{\mathrm{R}}$ | [36] |

${\mathsf{\eta}}_{\mathrm{I},\mathrm{i}}$ | ${\mathsf{\eta}}_{\mathrm{I}.\mathrm{i}}=\frac{2\xb7\left(1-{\mathsf{\alpha}}_{\mathrm{i}}\right)\xb7\sqrt{{\mathsf{\alpha}}_{\mathrm{i}}}}{{\mathrm{Ku}}_{\mathrm{i}}}\xb7{\mathrm{Stk}}_{\mathrm{i}}\xb7{\mathrm{R}}_{\mathrm{i}}+\frac{\left(1-{\mathsf{\alpha}}_{\mathrm{i}}\right)\xb7{\mathsf{\alpha}}_{\mathrm{i}}}{{\mathrm{Ku}}_{\mathrm{i}}}\xb7{\mathrm{Stk}}_{\mathrm{i}}$ ${\mathrm{Stk}}_{\mathrm{i}}=\frac{{\mathsf{\rho}}_{\mathrm{p}}\xb7{\mathrm{d}}_{\mathrm{p}}^{2}\xb7{\mathrm{u}}_{0,\mathrm{i}}}{18\xb7{\mathrm{d}}_{\mathrm{F},\mathrm{i}}\xb7\mathsf{\mu}}$ | [37] |

${\mathsf{\epsilon}}_{\mathbf{Filter}}$ | ${\mathbf{L}}_{\mathbf{Filter}}$ | ${\mathbf{d}}_{\mathbf{F},50}$ |
---|---|---|

98.41% | 1.65 cm | 24.2 µm |

**Table 3.**Operating conditions during filtration measurements [43].

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

Mean particle size d_{p50,3} | 1.5 µm |

Concentration (mass) | 0.1 mg/L |

Particle density | 3.14 g/cm^{3} |

Face velocity | 0.43 m/s |

Filtration time | 60 min |

Dataset | Number Sub Filter | Axial Resolution |
---|---|---|

Raw porosity | 1173 | 12.9 µm |

Axial porosity | 42 | 370.4 µm |

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## Share and Cite

**MDPI and ACS Style**

Hoppe, K.; Wischemann, L.; Schaldach, G.; Zielke, R.; Tillmann, W.; Thommes, M.; Pieloth, D. Filtration Kinetics of Depth Filters—Modeling and Comparison with Tomographic Data of Particle Depositions. *Atmosphere* **2023**, *14*, 640.
https://doi.org/10.3390/atmos14040640

**AMA Style**

Hoppe K, Wischemann L, Schaldach G, Zielke R, Tillmann W, Thommes M, Pieloth D. Filtration Kinetics of Depth Filters—Modeling and Comparison with Tomographic Data of Particle Depositions. *Atmosphere*. 2023; 14(4):640.
https://doi.org/10.3390/atmos14040640

**Chicago/Turabian Style**

Hoppe, Kevin, Lukas Wischemann, Gerhard Schaldach, Reiner Zielke, Wolfgang Tillmann, Markus Thommes, and Damian Pieloth. 2023. "Filtration Kinetics of Depth Filters—Modeling and Comparison with Tomographic Data of Particle Depositions" *Atmosphere* 14, no. 4: 640.
https://doi.org/10.3390/atmos14040640