# Effect of Pore Structure on Soot Deposition in Diesel Particulate Filter

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Soot Deposition Model

_{D}, has been proposed previously [9,11]. Our collaborators, including an automobile company and a Japanese national institute, estimated the value of the soot deposition probability, and it was 0.002 in the case of the cordierite filter [29]. It should be noted that, in the soot deposition region, the soot has many micro-pores. The flow can pass through the soot region. In our soot deposition model, the pressure rise across the soot region was included by the friction force in the fluid [8,25].

_{1}), the soot arrives around the filter substrate, corresponding to one lattice node in the Figure 1a. As mentioned before, only the soot mass fraction of P

_{D}is deposited, while that at (1 − P

_{D}) is not trapped and is rebounded into the original gas flow. As the calculation proceeds, the sum of the soot concentration at the lattice point is increased. At some point, the mass fraction of soot becomes united at the time step, IT

_{2}. Then, this node is treated as the solid phase of the soot layer, instead of the gas phase. At the next time step, IT

_{2}+ 1, the soot deposition region is moved to one node next to the original boundary. In this way, the layer of the soot region is thickened.

#### 2.2. Porous Structure of DPF

_{1}) was set to be 28 μm and the pore diameter of downstream region (D

_{2}) was set to be 15 μm was Filter 6. A porous material in which the pore diameter of upstream region (D

_{1}) was 15 μm and the pore diameter of downstream region (D

_{2}) was 28 μm was Filter 7.

#### 2.3. Numerical Domain

_{in}) was 1 cm/s, which was the value in experiments [29]. The temperature of exhaust gas at the inlet (T

_{in}) was set at 350 °C. The exhaust gas was composed of soot, oxygen, and nitrogen. The oxygen concentration was 10% in the volume fraction (X

_{O2}), and the soot mass fraction was 0.1 (Y

_{c,in}). The velocities were set at u = U

_{in}, v = 0, and w = 0. At the four side walls, the slip boundary was adopted based on the assumption of symmetry. At the outlet, the pressure was constant (atmospheric pressure), and the gradient of scalars, such as temperature and mass fraction, were set to be zero. The non-slip boundary was adopted at the surface of the filter substrate, corresponding to the wall boundary condition [31].

## 3. Results and Discussion

#### 3.1. Flow Field and Pressure Drop without Soot Deposition

#### 3.2. Soot Deposition for Filtration

_{s}> 2.7 g/L. Therefore, to suppress the large pressure rise during the depth filtration, it is better to make the pore size smaller, especially at the filter upstream region.

## 4. Conclusions

- (1)
- Even in cold flow, a complex flow pattern is observed, with a number of flow paths. If the porous structure is uniform, the linear pressure drops along the flow direction is observed across the filter wall. As the porosity is lower, or pore size is smaller, the initial pressure drop increases. In the case of non-uniform filters, the gradient of the pressure drop changes, but simply, the initial pressure drop is almost the same if the averaged porosity and pore size are the same.
- (2)
- When the flow path inside the filter is plugged with soot, the filter backpressure increases. Gradually, pores at the filter surface are clogged with soot, forming a soot deposition layer on the filter wall surface (called a soot cake). Once, the soot cake is formed, all the soot is trapped by this soot layer. Then, a transition between depth filtration and surface filtration is observed. By comparing the pressure drop of seven filters, Filter 5 with high porosity could be appropriate, because the pressure rise during the depth filtration is suppressed.

## Author Contributions

## Conflicts of Interest

## Abbreviations

## Notation

c | advection speed in LB coordinate |

D | pore size |

e | unit vector for advection speed in LB coordinate |

f | external force |

F_{i} | external force term |

f_{p,α} | distribution function of pressure |

f_{s,α} | distribution function of soot in gas phase |

IT | time step |

p | pressure |

P_{D} | soot deposition probability |

t | time |

u | velocity vector of (u, v, w) |

U_{in} | inlet velocity |

x | direction normal to the filter |

y | direction perpendicular to x |

z | direction perpendicular to x |

Y_{c} | mass fraction of soot |

ɛ | porosity |

κ | permeability |

ν | kinematic viscosity |

ρ | density |

τ | relaxation time |

## Subscripts

0 | reference condition |

1 | value of upstream region |

2 | value of downstream region |

C | properties of soot in gas phase |

in | value at inlet |

out | value at outlet |

s | properties of soot per total filter volume |

α | number of advection speed in LB coordinate |

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**Figure 1.**Soot deposition model. (

**a**) Time step of IT

_{1}; (

**b**) Time step of IT

_{2}; (

**c**) Time step of IT

_{2}+ 1.

**Figure 7.**Three-dimensional distribution of soot mass fraction in gas phase in Filter 1 at t = 15 s.

**Figure 8.**Three-dimensional soot layer at t = 15 s (

**left**); 20 s (

**middle**); 35 s (

**right**) in case of Filter 1.

**Figure 9.**Slice image of soot layer in x-y plane at t = 15 s (

**left**), 20 s (

**middle**), 35 s (

**right**) in case of Filter 1.

**Figure 11.**Relationship between deposited soot mass and pressure drop in Filters 1 to 7. (

**a**) Filters 1–5; (

**b**) Filters 1, 6, 7.

Filter No. | Porosity, ɛ (−) | Upstream Pore Size D_{1} (μm) | Downstream Pore Size, D_{2} (μm) | Average Pore Size, D (μm) | Initial Pressure Drop, p_{0} (Pa) |
---|---|---|---|---|---|

1 | 0.49 | 18 | 18 | 18 | 33.1 |

2 | 0.49 | 11 | 11 | 11 | 82.4 |

3 | 0.49 | 31 | 31 | 31 | 12.2 |

4 | 0.42 | 18 | 18 | 18 | 41.7 |

5 | 0.61 | 18 | 18 | 18 | 24.6 |

6 | 0.49 | 28 | 15 | 19 | 34.5 |

7 | 0.49 | 15 | 28 | 20 | 36.5 |

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Yamamoto, K.; Sakai, T.
Effect of Pore Structure on Soot Deposition in Diesel Particulate Filter. *Computation* **2016**, *4*, 46.
https://doi.org/10.3390/computation4040046

**AMA Style**

Yamamoto K, Sakai T.
Effect of Pore Structure on Soot Deposition in Diesel Particulate Filter. *Computation*. 2016; 4(4):46.
https://doi.org/10.3390/computation4040046

**Chicago/Turabian Style**

Yamamoto, Kazuhiro, and Tatsuya Sakai.
2016. "Effect of Pore Structure on Soot Deposition in Diesel Particulate Filter" *Computation* 4, no. 4: 46.
https://doi.org/10.3390/computation4040046