# Turbulent Aggregation and Deposition Mechanism of Respirable Dust Pollutants under Wet Dedusting using a Two-Fluid Model with the Population Balance Method

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## Abstract

**:**

## 1. Introduction

## 2. The Numerical Model of Spray Dedusting

#### 2.1. Two-Fluid (Euler–Euler) Model

**g**is gravity term,

**g**is gravity acceleration, $\sum _{p=1}^{2}{R}_{p}$ is interphase drag term,

**F**is additional physical force, and

**F**

_{vm}is virtual mass force.

_{i}and x

_{j}are the components of coordinates, σ

_{k}, σ

_{ε}are turbulent Prandt numbers of turbulent kinetic energy k and dissipation rate ε, G

_{k}is the turbulent kinetic energy generated by the average velocity gradient, μ is the coefficient of molecular viscosity, μ

_{t}is turbulence viscosity coefficient, v is the coefficient of kinetic viscosity, C

_{2}is an empirical coefficient.

#### 2.2. Population Balance Model

^{3}; β(u, v-u) denotes the aggregation nucleus of particles with volume u and v-u, using the unit m

^{3}/s; the first term on the right side of the equation denotes the number of new particles with volume v generated by aggregation, and ½ denotes that two particles participate simultaneously in a single aggregation event; the second term denotes the number of particles whose volume vanishes as v as a result of aggregation into larger particles.

_{s}is the density of the secondary phase and α

_{i}is the volume fraction of particle size i, defined as

_{i}is the volume of the particle size i. Then a fraction of α

_{i}called f

_{i}is introduced as the solution variable. This fraction is defined as

_{ag}is the particle volume resulting from the aggregation of particles k and j, and is defined as

_{ag}is greater than or equal to the largest particle size V

_{N}, then the contribution to class N − 1 is

#### 2.3. Aggregation Kernel Model

_{T}is a pre-factor that takes the capture efficiency coefficient of turbulent collision into account; γ denotes the shear rate, and it can be written as

_{i}

^{2}denotes the mean square velocity for particles i. The expression of the empirical capture efficiency coefficient of turbulent collision can be written as [45]

_{T}is the ratio between the viscous force and the Van der Waals force:

#### 2.4. Deposition Kernel Model

_{e}is the coefficient of the turbulent diffusivity (which can be evaluated from the turbulent energy dissipation rate), L

_{2}is the turbulent boundary layer thickness [47], n is an experimentally-fitted parameter, S is the total surface area of the chamber, V is volume of the chamber, and H is the height of the chamber. D and u

_{t}depend on the particle size as follows:

_{B}is the Boltzmann constant, T is the absolute temperature, μ is the gas viscosity, ρ is the particle density, g is the gravity constant, and C is the slip correction factor, which is represented as [48]

_{i}, so the dust volume fraction source term of each bin in the PBE can be further rewritten as

_{i}is the volume fraction of particles in bin i, and ρ is the density of dust particles.

## 3. Model Validation

#### 3.1. Experimental Set

^{−6}.

#### 3.2. Results and Comparison

## 4. Discussion

#### 4.1. Effect of Droplet Size

#### 4.2. Effect of Droplet–Particle Volume Flow Rate Ratio

_{f}was adjusted with other parameters unchanged. Table 3 shows the effect of r

_{f}on the dedusting efficiency. When r

_{f}increased, the dedusting efficiency η

_{f}increased significantly (from 39.2% to 54.7%). However, its increase slowed down when the r

_{f}increased from 7 to 9. When r

_{f}is larger than 9, the effect of increasing r

_{f}can be neglected, and the value finally stabilizes at about 58.5%. This is because when the volume flow rate of spray increases, the number of droplets increases, resulting in a significant increase in the probability of collision between dust particles and droplets. Therefore, the aggregation efficiency of particles and droplets increases noticeably. However, considering that the number of dust particles and the size of the mixing area are certain, when the number of droplets increases to a certain extent, some droplets will become redundant. At this time, the increase of spray volume has little effect on the growth of dedusting efficiency. The fitted formula for the dedusting efficiency η

_{f}and the droplet–particle volume flow rate ratio r

_{f}is given in Equation (35).

#### 4.3. Effect of Droplet–Particle Velocity Ratio

_{v}was adjusted with other parameters unchanged. Table 4 shows the effect of r

_{v}on the dedusting efficiency. The dedusting efficiency η

_{v}is only 53.1% when r

_{v}is 1. As the r

_{v}increases to 6, the η

_{v}rapidly increases to 67.3%. However, when the r

_{v}continues to increase from 6 to 10, the improvement of η

_{v}is not obvious and finally stabilizes at about 70%. The main reason for this trend is that when the velocity of the spray increases, the mean square velocity of droplet granules in the turbulent aggregation kernel increases. According to Equation (19), so the particle aggregation rate a(d

_{i}, d

_{j}) increases with the increase of the particle mean square velocity U

_{i}

^{2}, thereby improving the dust removal efficiency. On the other hand, the greater the mist spray speed, the shorter the time that droplets are in the air. This will negatively affect the capture of dust.

_{v}and the droplet–particle velocity ratio r

_{v}can be obtained using Equation (36). Then Figure 10 shows the fitted curves of η with respect to r

_{f}and r

_{v}.

#### 4.4. Effect of Ventilation Velocity

## 5. Conclusions

- The proposed mathematical model is well validated by the detailed experimental results for spray dedusting in the existing literature, and the numerical results of dedusting efficiency agree well with the experimental values. The turbulent aggregation kernel and the deposition kernel proposed in this paper are accurate enough to describe the wet dedusting process dominated by respirable dust capture and deposition.
- Key parameters analysis shows that the smaller the droplet diameter of the water mist formed by the atomizing device, the higher the capture efficiency of respirable dust. In other words, in practical application the dedusting efficiency of the respirable dust can be increased by ensuring that the water mist diameter is between 15 μm and 70 μm. The volume–flow rate ratio r
_{f}and velocity ratio of droplets to dust r_{v}also have great influence on dedusting efficiency. - When r
_{f}increases from 2 to 12, the dedusting efficiency increases from 39.2% to 54.7%. However, when this value continues to increase, the dedusting efficiency is not significantly improved. The fitting formula is given as η_{f}= 11.846ln(r_{f}) + 30.375, R^{2}= 0.9745. - When r
_{v}increases 10 times, the dedusting efficiency increases from 53.1% to 70.2%. Increasing the ratio clearly has little effect on dedusting efficiency. The fitting formula is η_{v}= −0.0819r_{v}^{3}+ 1.2102r_{v}^{2}− 2.4142r_{v}+ 54.451, R^{2}= 0.9745. - An optimized air distribution should be considered in the dedusting efficiency. Under the ventilation condition with 4 m/s inlet air, spray dedusting reaches the highest efficiency.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Schematic diagram of experimental model for spray dedusting. 1-air pump; 2-dust generator; 3-YJB-2500-type compensatory micro-manometer; 4-AKFC-92A-type dust meter; 5- nozzle; 6-high-pressure pipe; 7-high-pressure pump; 8-pressure gauge; 9-water tank; 10-slippery course; 11-laser particle size analyzer; 12-YYT-2000-type inclined micro-manometer; 13-Pitot tube; 14-axial flow fan; 15-airflow expanding section; 16-airflow stationary section.

**Figure 3.**Schematic diagram of the volume fraction and its cumulative distribution of respirable dust in different particle sizes from the literature.

**Figure 11.**Distribution of dust particle volume fraction. (

**a**) ventilation velocity of 1 m/s (

**b**) ventilation velocity of 2 m/s. (

**c**) ventilation velocity of 3 m/s (

**d**) ventilation velocity of 4 m/s. (

**e**) ventilation velocity of 5 m/s (

**f**) ventilation velocity of 6 m/s.

Kernel | References | Expression | Comments |
---|---|---|---|

Free molecular aggregation | Smoluchowski (1917) [35] | ${a}_{f}=\frac{2{k}_{B}T}{3\mu}\frac{{({d}_{i}+{d}_{j})}^{2}}{{d}_{i}{d}_{j}}$ | Very small particles (up to 1 μm) aggregate because of collisions due to Brownian motions; the frequency of collision is size-dependent |

Coulomb aggregation | Williams and Loyalka (1991) [36] | ${a}_{c}=\frac{z({\mathrm{e}}^{z}+1)}{2({\mathrm{e}}^{z}-1)}{a}_{f}$ $z=\frac{{q}_{1}{q}_{2}}{2\pi {k}_{B}T{\epsilon}_{0}\left({d}_{i}+{d}_{j}\right)}$ | The precondition is that the particle itself is already charged, no matter it is the same charge or different charge |

Turbulence aggregation | Saffman and Turner (1956) [37] Zaichik and Alipchenkov (2001,2008) [38,39] | For the viscous subrange: ${a}_{t}={\varsigma}_{T}\sqrt{\frac{8\pi}{15}}\gamma \frac{{\left({d}_{i}+{d}_{j}\right)}^{3}}{8}$, for the inertial subrange: ${a}_{t}={\varsigma}_{T}{2}^{3/2}\sqrt{\pi}\frac{{\left({d}_{i}+{d}_{j}\right)}^{2}}{4}\sqrt{\left({U}_{i}^{2}+{U}_{j}^{2}\right)}$ | In the turbulent flow field, aggregation can occur by two mechanisms: viscous subrange mechanism and inertial subrange mechanism |

Bubble aggregation | Wang et al. (2019) [40] | ${a}_{b}=\{\begin{array}{ll}0.5658\pi {V}_{b}{\left(\frac{{d}_{i}}{2}+\frac{{d}_{j}}{2}\right)}^{2}{\mathrm{sin}}^{2}{\theta}_{w}& for\text{}{V}_{ij}\le {V}_{\mathrm{max}}\\ 0& otherwise\end{array}$ | Aggregation will occur during a collision of two bubbles provided that the contact time exceeds the aggregation time required for drainage of the liquid film between them to a critical rupture thickness |

Droplet aggregation | Williams et al. (2019) [41] | ${a}_{d}=\epsilon \frac{\dot{\gamma}}{\pi}{\left({v}_{i}^{1/3}+{v}_{j}^{1/3}\right)}^{3}$ | Droplet centers are assumed to move along streamlines, and aggregation occurs when the distance between the droplets is less than the sum of their radii |

Nozzles | D10 (μm) | D50 (μm) | D90 (μm) | Q (L/min) | Dedusting Efficiency (%) |
---|---|---|---|---|---|

1 | 34.93 | 70.62 | 117.67 | 6.46 | 53.1 |

2 | 41.10 | 70.62 | 109.22 | 7.92 | 32.3 |

3 | 38.57 | 74.67 | 122.81 | 7.36 | 61.2 |

4 | 32.94 | 65.15 | 112.39 | 8.15 | 45.5 |

5 | 37.01 | 69.03 | 118.36 | 5.63 | 48.6 |

6 | 40.03 | 75.14 | 123.85 | 4.30 | 39.7 |

Droplet-Particle Volume Flow Rate Ratio | Dedusting Efficiency (%) | Droplet-Particle Volume Flow Rate Ratio | Dedusting Efficiency (%) |
---|---|---|---|

2 | 39.2 | 8 | 56.1 |

3 | 41.7 | 9 | 57.2 |

4 | 45.7 | 10 | 57.5 |

5 | 50.4 | 11 | 58.0 |

6 | 52.1 | 12 | 58.3 |

7 | 54.7 | — | — |

Droplet-Particle Velocity Ratio | Dedusting Efficiency (%) | Droplet-Particle Velocity Ratio | Dedusting Efficiency (%) |
---|---|---|---|

1 | 53.1 | 6 | 67.3 |

2 | 54.3 | 6.5 | 68.6 |

3 | 55.9 | 7 | 69.1 |

3.5 | 57.0 | 7.5 | 69.5 |

4 | 58.7 | 8 | 69.8 |

4.5 | 59.5 | 9 | 70.0 |

5 | 62.3 | 10 | 70.2 |

5.5 | 64.0 | — | — |

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**MDPI and ACS Style**

Wang, P.; Shen, S.; Zhou, L.; Liu, D.
Turbulent Aggregation and Deposition Mechanism of Respirable Dust Pollutants under Wet Dedusting using a Two-Fluid Model with the Population Balance Method. *Int. J. Environ. Res. Public Health* **2019**, *16*, 3359.
https://doi.org/10.3390/ijerph16183359

**AMA Style**

Wang P, Shen S, Zhou L, Liu D.
Turbulent Aggregation and Deposition Mechanism of Respirable Dust Pollutants under Wet Dedusting using a Two-Fluid Model with the Population Balance Method. *International Journal of Environmental Research and Public Health*. 2019; 16(18):3359.
https://doi.org/10.3390/ijerph16183359

**Chicago/Turabian Style**

Wang, Pei, Shuai Shen, Ling Zhou, and Deyou Liu.
2019. "Turbulent Aggregation and Deposition Mechanism of Respirable Dust Pollutants under Wet Dedusting using a Two-Fluid Model with the Population Balance Method" *International Journal of Environmental Research and Public Health* 16, no. 18: 3359.
https://doi.org/10.3390/ijerph16183359