# Numerical Simulation on Flow Dynamics and Pressure Variation in Porous Ceramic Filter

^{*}

## Abstract

**:**

_{2}TiO

_{5}, SiC, and cordierite, we numerically realized the fluid dynamics in a diesel filter (diesel particulate filter, DPF). These inner structures were obtained by X-ray CT scanning to reproduce the flow field in the real product. The porosity as well as pore size was selected systematically. Inside the DPF, the complex flow pattern appears. The maximum filtration velocity is over ten times larger than the velocity at the inlet. When the flow forcibly needs to go through the consecutive small pores along the filter’s porous walls, the resultant pressure drop becomes large. The flow path length ratio to the filter wall thickness is almost the same for all samples, and its value is only 1.2. Then, the filter backpressure closely depends on the flow pattern inside the filter, which is due to the local substrate structure. In the modified filter substrate, by enlarging the pore and reducing the resistance for the net flow, the pressure drop is largely suppressed.

## 1. Introduction

_{2}TiO

_{5}, SiC, and cordierite, in the numerical simulation. To consider the fluid dynamics in the real product, we applied X-ray CT scanning for the filter inner structure [13,14,16,21,22]. We chose five samples (see Table 1) to set the porosities such as the pore size systematically. They have different porosity, pore size, cell density, and wall thickness. By comparing the filter properties of different materials, we discussed the flow field in detail, which could be deeply related with the filter backpressure.

## 2. Numerical Method

#### 2.1. Lattice Boltzmann Method

**c**

_{α}(α = 1 to 15) is the 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. 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/$\sqrt{3}$ 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}) were evaluated in terms of the low Mach number approximation [13], together with the ideal gas equation.

^{2}δ

_{t}, showing that Navier–Stokes equations are derived by the Chapman–Enskog procedure [24]. In the numerical code, all variables were converted to be dimensionless. Based on the similarity of the Reynolds number (Re = U

_{in}W/ν), the real values such as flow velocity, were obtained, where U

_{in}is the inflow velocity of the diesel exhaust and W is the inlet width of the 3D simulation area.

#### 2.2. Inner Structure of Filter Substrate

_{2}TiO

_{5}, SiC, and cordierite with different porosity and pore size were tested. Table 1 shows filter properties of five samples. The porosity and the mean pore size were independently selected. In order to evaluate the flow field and the filter backpressure in the real filter, we applied the X-ray CT scanning. An example is shown in Figure 1, with the direct photograph of sample 1. As seen in the photograph, a part of the filter wall shown by the yellow circle was detected by the X-ray CT, and the three-dimensional substrate structure was obtained. In this figure, three slice images of sample 1 are shown. The coordinate system will be described later. The measurement resolution of the X-ray CT scanning was 1.42 μm/pix, which was the corresponding grid spacing in the numerical domain. Using CT data, it was possible to conduct the three-dimensional simulation to visualize the flow field inside the DPF. Since a part of the CT data was used in the simulation, convergence studies were conducted to set the proper numerical domain, which is explained later.

#### 2.3. Numerical Domain

_{in}) was given at 1 cm/s [13]. Different from previous simulations [13,14,16,19,20,21,22,23], only the flow was simulated, and the component of the exhaust gas was nitrogen. The temperature at the inlet (T

_{in}) was 350 °C, which was used to determine the kinematic viscosity of the inflow. Each velocity component was set at u

_{x}= U

_{in}, u

_{y}= 0, and u

_{z}= 0. On four side walls, we assumed the periodic structure of the porous material. Then, the slip boundary was adopted. At the exit (outlet), the pressure was always the atmospheric value. At the surface of the filter substrate, the nonslip wall boundary was used [27].

## 3. Results

#### 3.1. Flow Field and Filter Backpressure

_{x}with the velocity vector in XY plane at the center of the numerical domain with sample 1. The inlet width was changed at W = 57, 114, 142, 189 μm. The magnitude of the velocity is shown by the color from blue to red. The flow direction and its magnitude were largely changed, with the flow recirculation of negative velocity. Note that the inflow velocity was the smaller value of 1 cm/s, while the maximum flow velocity in the whole numerical domain was about 15 cm/s. Resultantly, the flow was largely accelerated, because the flow forcibly needed to go through the narrow pores in the filter wall, which is also seen in Figure 3. By comparing four figures, the clear difference was not observed. Then, by changing the inlet width of the filter, we checked the pressure distribution.

_{x}

^{2}+ u

_{y}

^{2}+ u

_{z}

^{2})

^{1/2}

_{x}, u

_{y}, u

_{z}are velocity components. Surprisingly, the maximum velocity was over 15 cm/s in sample 4, which exhibited the highest filter backpressure. Accordingly, the large acceleration of the filtration velocity was observed in the case of the filter with the high filter backpressure, because it was expected that there was the region of very narrow space between the filter substrates. The variation of the filtration velocity was also very large, compared with other samples. Conversely, in sample 5, whose filter backpressure was the lowest, the maximum filtration velocity was much smaller than that in the other four cases. Then, it could be derived that the filter backpressure is particularly related with the flow field. In the next section, the flow path is further examined.

#### 3.2. Flow Path Length inside DPF

_{p}is the density of particles, d

_{p}is the particle diameter,

**V**and

**u**are the particle velocity and the flow velocity, respectively. In this formula, C

_{D}is the drag coefficient. By assuming the so-called Stokes’ Law, then C

_{D}= 24/Re, where Re is based on the particle diameter. Here, it was assumed that the density of particles was equal to the fluid density. The diameter of all particles was equivalent to 200 nm, which was small enough to follow the flow stream. The particle deposition by any interception effects was not included. In addition, since the flow was steady, it was easy to evaluate the flow path along the streamline by chasing the particle motion inside the filter wall. Before the simulation, initial positions of the particles were equally placed at all grids at the inlet.

_{w}), the flow path length was evaluated by the ratio to the filter wall thickness. We calculated the flow path length by using approximately 3600 particles. The ordinate was the value of particle numbers divided by the total particles in each range of the flow path length. Expectedly, the flow path could be longer when the particle moved along the complicated substrate structure. However, independent of the sample, the range of the flow path length was almost the same. Actually, the averaged value was 1.24, 1.20, 1.23, 1.24, and 1.23 for samples 1 to 5, respectively. That is, the flow path length was slightly longer than the wall thickness of the filter. Therefore, independent of the inner structure of the filter sample, the flow path length is almost constant.

## 4. Discussion

## 5. Conclusions

_{2}TiO

_{5}, SiC, and cordierite, we simulated the flow inside DPF. These inner structures were obtained by the X-ray CT scanning, in order to reproduce the real flow field in the porous filter. The following conclusions were drawn.

- (1)
- Inside DPFs, a complex flow pattern appears, with the large flow acceleration and the flow recirculation. In some areas, the maximum filtration velocity is over ten times larger than the inlet velocity of 1 cm/s. It is because the flow forcibly needs to go through the narrow pores along the porous filter wall, resulting in the large filter backpressure. By comparing 5 samples, it can be seen that the resultant pressure drop through the filter wall is smaller when the porosity or the pore size of the filter is larger.
- (2)
- To further discuss the flow field, the path length inside the DPF was estimated quantitatively. We followed the flow trajectory by tracing the particle motion in the fluid. The flow path length ratio to the filter wall thickness was almost the same for all samples, and its value was only 1.2. Therefore, independent of the porous material, the flow path length is almost constant under the present flow conditions.
- (3)
- The filter backpressure closely depends on the flow pattern inside the filter, which is due to the local substrate structure. In the modified filter substrate, by enlarging the pore size and reducing the resistance for the net flow, the pressure drop is largely suppressed. Therefore, when the local porosity is not varied much, the smooth and constant flow could be achieved, ensuring that the large velocity fluctuation is avoided. For the reduction of the filter backpressure, the uniform pore structure is suitable.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviation

Notation | |

c | advection speed in lattice Boltzmann spacing |

p_{α} | distribution function of pressure |

p | pressure |

t | time |

u | flow velocity of three components, u, v, w |

U_{in} | inlet velocity |

X | flow direction along the inflow |

Y | direction perpendicular to X |

Z | direction perpendicular to X |

ε | porosity |

ν | kinematic viscosity |

ρ | density |

τ | relaxation time |

Subscripts | |

0 | reference condition at the atmosphere |

in | value at inlet of numerical domain |

p | value of particle |

α | number of advection speed in LB coordinate |

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**Figure 3.**Distribution of velocity in X-direction at X = 92 μm (

**upper**), X = 255 μm (

**middle**), X = 411 μm (

**lower**) of sample 1.

**Figure 4.**Distributions of velocity in X-direction of sample 1; (

**a**) W = 57 μm; (

**b**) W = 114 μm; (

**c**) W = 142 μm; (

**d**) W = 189 μm.

**Figure 9.**Particle position transported by convection in sample 1 at t = (

**a**) 4 ms, (

**b**) 7 ms, (

**c**) 11 ms, (

**d**) 14 ms, (

**e**) 21 ms.

**Figure 10.**Particle number ratio of different flow path length ratios for five filters; (

**a**) No. 1, (

**b**) No. 2, (

**c**) No. 3, (

**d**) No. 4, (

**e**) No. 5.

**Figure 12.**Distribution of pressure of corrected substrates in samples 3 and 4, together with other samples.

Sample No. | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Material | Al_{2}TiO_{5} | Al_{2}TiO_{5} | Al_{2}TiO_{5} | SiC | Cordierite |

Porosity, ε (%) | 56 | 50 | 49 | 37 | 60 |

Mean pore size (μm) | 17 | 10 | 17 | 11 | 14 |

Cell density | 230 | 230 | 230 | 300 | 300 |

Wall thickness, δ_{w} (μm) | 372 | 351 | 374 | 361 | 265 |

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**MDPI and ACS Style**

Yamamoto, K.; Toda, Y.
Numerical Simulation on Flow Dynamics and Pressure Variation in Porous Ceramic Filter. *Computation* **2018**, *6*, 52.
https://doi.org/10.3390/computation6040052

**AMA Style**

Yamamoto K, Toda Y.
Numerical Simulation on Flow Dynamics and Pressure Variation in Porous Ceramic Filter. *Computation*. 2018; 6(4):52.
https://doi.org/10.3390/computation6040052

**Chicago/Turabian Style**

Yamamoto, Kazuhiro, and Yusuke Toda.
2018. "Numerical Simulation on Flow Dynamics and Pressure Variation in Porous Ceramic Filter" *Computation* 6, no. 4: 52.
https://doi.org/10.3390/computation6040052