# Numerical Simulation of Particle-Laden Flow and Soot Layer Formation in Porous Filter

^{*}

## Abstract

**:**

## 1. Introduction

_{2}to the atmosphere. For solving the global warming problem, CO

_{2}emissions should be reduced. In general, the transport sector is a significant contributor to CO

_{2}emission in the world [1,2]. Diesel engines have an advantage of lower fuel consumption, compared to gasoline engines [3]. However, there are drawbacks in terms of NOx and particulate emissions. It is known that particulates of diesel soot can penetrate into the lung, causing human carcinogenic effects [4,5]. Then, in many countries, stricter exhaust emission standards such as Euro VI have been set. Therefore, an after-treatment of diesel exhaust gas is needed [6,7,8,9].

## 2. Numerical Methods

_{α}(α = 1 to 15) is the gas velocity of each advection along the lattice coordinate. The evolution equation for the convection of the flow is

_{t}. The variable of τ is the relaxation time for controlling the rate to the equilibrium distribution due to collision between gas in the flow, which is related with the kinetic viscosity using ν = (2 τ − 1)/6 c

^{2}δ

_{t}, showing that the Navier–Stokes equations are derived by the Chapman–Enskog procedure [29]. The equilibrium distribution function, p

_{α}

^{eq}, is

_{α}= 1/9 (α = 1:6), w

_{α}= 1/72 (α = 7:14), and w

_{15}= 2/9. The sound speed, c

_{s}, is c/√3 with p

_{0}= ρ

_{0}RT

_{0}= ρ

_{0}c

_{s}

^{2}. Here, p

_{0}and ρ

_{0}are the pressure and density at the room temperature. In the simulation, the temperature was constant. The pressure and the velocity vector of u = (u

_{x}, u

_{y}, u

_{z}) are evaluated in terms of the low Mach number approximation [23], together with the ideal gas equation.

_{in}W/ν), the real values such as flow velocity were treated in the lattice space of the LBM, where U

_{in}is the inflow velocity of the diesel exhaust, W is the inlet width of the 3D numerical domain in Figure 1.

_{D}is the soot deposition probability, which is a model parameter. It corresponds to the deposition ratio which determines the local mass of deposited soot at each spatial grid. In order to consider the soot size, the soot deposition probability in the new model [31] is described by the Brownian diffusion and the interception effect [32], by which it is possible to investigate the dependence of the soot size. Moreover, we can discuss the contributions of the Brownian diffusion and the interception effect, separately.

**Figure 1.**Three-dimensional numerical domain used for the particle-laden flow across the porous SiC-DPF is shown. The total size is 450 μm (x) × 60 μm (y) × 60 μm (z), with the grid size of 1 μm.

## 3. Results and Discussion

#### 3.1. Soot Deposition Region and Pressure Drop

#### 3.2. Effect of Soot Size on Pressure Drop

_{soot}. Figure 7 shows the soot deposition regions at t = 90 s. It is the slice image in the x–y plane at z = 21 μm. These are the profiles of d

_{soot}= 75, 100, 125, 150 nm. Since they are the results obtained at the same period, it is expected that the soot mass supplied into the filter is the same. For all cases, the soot deposition region is observed relatively in the upstream region of the filter wall. As the soot size is larger, the soot layer forming on the filter wall surface becomes thinner. For further discussion, we evaluated the soot mass deposited on the filter wall. Figure 8a shows the time-variations of the deposited soot mass by changing the soot size. The resultant pressure drop is also shown in Figure 8b. As seen in Figure 8a, less soot is deposited with an increase in the soot size. It seems very reasonable, because the smaller soot is more efficiently deposited by the Brownian diffusion. On the other hand, it may not be easy to explain the resultant pressure drop in Figure 8b. It is expected that the smaller soot shows the larger pressure drop due to the efficient deposition. In fact, the pressure drop of d

_{s}

_{oot}= 75 nm is initially larger. However, when time elapses, the pressure drop of the larger soot is conversely superior, which is the same tendency reported by the engine test bench [36].

_{soot}= 75 nm is even higher than those of other three cases at t < 50 s. Then, we further discuss the soot deposition region.

_{soot}= 75, 100, 125, 150 nm. The abscissa is the x-coordinate of the flow direction of the exhaust gas. By considering that the soot layer has a three-dimensional structure, the deposited soot mass is integrated in the y–z plane. Then, we can discuss the difference between the soot deposition profiles. To focus on the soot layer formation, the profiles of t = 45 s are shown. As shown in Figure 2 and Figure 3, it is the time when all pores on the filter wall surface are covered with soot. As seen in this figure, dependent on the soot size, a clear difference is observed. That is, as the soot size is smaller, the soot is deposited more upstream. In other words, in the case of d

_{soot}= 150 nm, more soot is trapped by the depth filtration. Since the smaller soot is efficiently deposited due to the Brownian diffusion, it is understandable to consider that the smaller soot is trapped at the more upstream region. Resultantly, the surface filtration appears earlier with a linear pressure increase in Figure 8b. Hence, when we discuss the pressure drop during the filtration process, it is important to consider where the soot deposition occurs, together with the deposited soot mass.

## 4. Conclusions

- (1)
- The contributions of the Brownian diffusion and the interception effect were evaluated quantitatively. The soot deposition mainly occurs due to the Brownian diffusion. The soot deposition region is restricted to the area of the upstream of the filter wall surface.
- (2)
- Independent of the soot size, the shift from the depth filtration to the surface filtration is observed. By checking the soot deposition region, the pressure drop increases steeply during the depth filtration. Once all pores on the filter wall surface are covered with soot, the pressure rise is reduced, showing the linear increase during the surface filtration. As the soot size is smaller, the shift to the surface filtration appears earlier.
- (3)
- As the soot size is smaller, the soot layer forming on the filter wall surface becomes sparse. The resultant soot permeability of the smaller soot is larger. Then, due to the larger soot permeability, the pressure drop of the smaller soot is expectedly reduced. However, the smaller soot is trapped more efficiently by the Brownian diffusion. Then, only in the earlier stage of the filtration, the pressure drop of the smaller soot is larger. After that, the pressure drop is conversely smaller. Therefore, for discussing the pressure drop, it is important to consider where the soot deposition occurs as well as the deposited soot mass in the filter.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Sample Availability

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**Figure 5.**Slice images of the soot deposition region in the x–y plane obtained at z = 21 μm are shown for three cases (

**i**–

**iii**).

**Figure 6.**Time-variations of (

**a**) deposited soot mass and (

**b**) pressure drop for three cases (i) to (iii) are shown to discuss the soot deposition process.

**Figure 8.**Time-variations of (

**a**) deposited soot mass and (

**b**) pressure drop by changing the soot size.

**Figure 10.**Profiles of soot permeability in x−y plane by changing the soot size at z = 21 μm; t = 90 s.

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**MDPI and ACS Style**

Yamamoto, K.; Yagasaki, S.
Numerical Simulation of Particle-Laden Flow and Soot Layer Formation in Porous Filter. *Solids* **2022**, *3*, 282-294.
https://doi.org/10.3390/solids3020020

**AMA Style**

Yamamoto K, Yagasaki S.
Numerical Simulation of Particle-Laden Flow and Soot Layer Formation in Porous Filter. *Solids*. 2022; 3(2):282-294.
https://doi.org/10.3390/solids3020020

**Chicago/Turabian Style**

Yamamoto, Kazuhiro, and Shota Yagasaki.
2022. "Numerical Simulation of Particle-Laden Flow and Soot Layer Formation in Porous Filter" *Solids* 3, no. 2: 282-294.
https://doi.org/10.3390/solids3020020