# Numerical Comparison of Prediction Models for Aerosol Filtration Efficiency Applied on a Hollow-Fiber Membrane Pore Structure

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Prediction Models for Air Filtration Efficiency

_{down}and C

_{up}are the number of particles downstream and upstream of the filter, respectively.

#### 2.1. Efficiency Prediction of Fibrous Filters

_{f}, Z and d

_{f}are the filter solidity, SCE, filter thickness and average collector diameter, respectively. The total SCE is a sum of contributions from different collection mechanisms and can be written as follows:

_{I}, η

_{R}, η

_{D}and η

_{A}are the single collector efficiencies due to inertial impaction, interception, diffusion and adhesion, respectively. The filtration theory, which is based on three main mechanisms, inertial impaction, interception and diffusion (Figure 2), does not take into account particle-fiber interaction, i.e., the particle rebound and re-entrainment. Therefore, we used Equation (3) to calculate the SCE based on collision efficiency (sum of collection efficiencies due to impaction interception and diffusion) multiplied by the collection efficiency caused by adhesion effects [46,47].

#### 2.1.1. SCE Due to Brownian Motion

_{B}, T, µ and d

_{p}are the Boltzmann constant, absolute temperature, air dynamic viscosity and particle diameter, respectively and C

_{s}is the Cunningham slip correction factor:

_{D}). For nanoparticles that have high diffusion coefficient, hence smaller Peclet number, Wang et al. [48] gave the following relationship:

_{f}≥ 2 µm as follows:

_{1}is a constant calculated as follows:

_{2}is calculated as follows:

#### 2.1.2. SCE Due to Interception

_{f}< 1) as follows:

_{f}as fiber Reynolds number characterizing flow field around a fiber calculated as follows:

#### 2.1.3. SCE Due to Inertial Impaction

_{p}is the particle density. If the Stokes’ number is higher than unity, the particles separate from streamlines and hit the collector. On the other hand, for Stokes’ number lower than one, the inertia effect will not take place. Several formulae have been derived for SCE due to inertial impaction. The most often used relationship is that proposed by Stechkina et al. [50]:

_{f}> 10. However, as suggested by Saleh et al. [65], this equation may also be used for Re

_{f}< 2. The relationship is as follows:

_{f}< 0.25 and 0.5 < Stk < 4.1 and Re

_{f}< 1, 0.8 < Stk < 2 and R < 0.2, respectively. Zhu et al. [69] derived a relationship with no restrictions concerning Stk, Re

_{f}and α as follows:

_{f}< 60 and 1 < Stk < 20 as follows:

_{f}< 40,000 and 0.07 < Stk < 5 as follows:

#### 2.1.4. SCE Due to Adhesion

_{p}is the particle Reynolds number calculated as follows:

_{pp}is not the standard fluid dynamics Reynolds number, it uses the particle density ρ

_{p}for the calculation [73]. Equation (41) was accurate for 1 < Stk < 120 and 0.4 < Re

_{f}< 5.75.

#### 2.2. Efficiency Prediction of CPM

_{I}for Nuclepore filters can be calculated using the model proposed by Pich [74] as follows:

_{c}as the slip correction factor and calculated as follows [66]:

_{D}can be calculated as follows [75]:

_{D}< 0.01 or

_{D}> 0.01, where N

_{D}is:

_{o}is the pore diameter. The interception efficiency on pore opening η

_{R}can be calculated using the model suggested by Spurny et al. [75]:

_{o}is the interception parameter for capillary pore filters calculated as follows:

_{DS}can be calculated using the expression proposed by Manton [76]:

_{2}= 4.5 and β

_{1}and δ are coefficients that are calculated as follows:

## 3. Materials and Methods

#### Hollow-Fiber Membranes

_{f(o)}was calculated as follows:

_{f(o)i}is an individual collector (pore) size, n

_{i}is the number of collectors/pores with a given size d

_{f(o)i}and N is the number of all measured collectors/pores, i.e., number of measurements obtained from the SEM pictures. The average collector/pore size is thus a weighted average of 125 values. The weighted average pore size was calculated using pore dimensions of the elliptical shape (the major and minor axes). The largest particle able to penetrate through the membrane is mostly given by the smaller pore dimension (i.e., that of minor axe). However, due to the random motion and shape of particles, some particles larger than the minor axe length can penetrate through the membrane. Therefore, the weighted average was calculated using both axes’ dimensions, giving a larger average pore size. This step ensures that the results of the predicted efficiencies will not be overrated. The main parameters of the membrane structure and conditions for which the models were compared are shown in Table 1. For the model comparison, we also used the standard deviation of pore and collector average diameter to depict uncertainty bounds. For the sake of brevity, this was done for final results only, i.e., overall efficiency.

## 4. Results and Discussion

#### 4.1. Fibrous Filters

_{0}is the adhesion distance. Adhesion energy is directly proportional to the particle size, therefore, higher energy is necessary to keep a larger particle attached to the fiber. It is similar for face velocity, which is mostly assumed the same as the impact velocity of the particle colliding with the fiber surface. The impact velocity should be less than the critical velocity ν derived from the adhesion energy given as follows [46]:

_{e}is the main electronic absorption frequency typically around 3 × 10

^{15}s

^{−1}. The subscript notation 1, 2 and 3 of ε and n indicate the particle, membrane surface and fluid, respectively. The typical value of Hamaker constant ranges between 10

^{−19}and 10

^{−20}[83]. However, significant influence will also have particle surface charges, which can cause the membrane to act as an electret filter, so the particles may be captured due to electrostatic forces. In this work however, we focus on the mechanical means of filtration only, so this effect is not considered.

#### Overall SCE and overall Filtration Efficiency

^{−66}which is practically equaled to zero. The results shown in Figure 9a are single collector efficiencies calculated using models for diffusion (Equation (18)), interception (Equation (21)), impaction (Equation (35)) and adhesion (Equation (41)). So it is an example of one selected combination of models for individual mechanism. The other was not calculated as it was assumed that the result would be the same or would vary somewhere in the order of 10

^{−70}, which is negligible.

#### 4.2. CPM

## 5. Conclusions

## Supplementary Materials

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Evaluation of collector diameter (

**a**) and pore size (

**b**) from SEM images using Stream Motion software.

**Figure 5.**Comparison of impaction efficiency based on different models and airflow velocity of 5 cm/s (

**a**) and 20 cm/s (

**b**) in relation to Stokes number.

**Figure 7.**Comparison of SCE due to diffusion mechanisms based on different models for an airflow velocity of 5 cm/s (

**a**) and 20 cm/s (

**b**) in relation to the Peclet number.

**Figure 9.**Single collector efficiency (

**a**), overall filter efficiency (

**b**) and overall penetration (

**c**).

**Figure 13.**Overall efficiency in relation to particle size based on CPM model for a velocity of 5 and 10 cm/s (

**a**) and 15 and 20 cm/s (

**b**).

Fiber wall thickness, Z (µm) | 36 |

Average pore size, d_{o} (nm) | 205 ± 157 |

Average collector diameter, d_{f} (nm) | 90 ± 83 |

Solidity, α (%) | 48 |

Porosity, ε (%) | 52 |

Temperature, T (K) | 296.15 |

Air density, ρ (kg m^{−3}) | 1.21 |

Air dynamic viscosity, µ (Pa s) | 1.83 × 10^{−5} |

Particle density, ρ_{p} (kg m^{−3}) | 1060 |

Mean free path of air molecules, λ (nm) | 67.3 |

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Bulejko, P. Numerical Comparison of Prediction Models for Aerosol Filtration Efficiency Applied on a Hollow-Fiber Membrane Pore Structure. *Nanomaterials* **2018**, *8*, 447.
https://doi.org/10.3390/nano8060447

**AMA Style**

Bulejko P. Numerical Comparison of Prediction Models for Aerosol Filtration Efficiency Applied on a Hollow-Fiber Membrane Pore Structure. *Nanomaterials*. 2018; 8(6):447.
https://doi.org/10.3390/nano8060447

**Chicago/Turabian Style**

Bulejko, Pavel. 2018. "Numerical Comparison of Prediction Models for Aerosol Filtration Efficiency Applied on a Hollow-Fiber Membrane Pore Structure" *Nanomaterials* 8, no. 6: 447.
https://doi.org/10.3390/nano8060447