## Abstract

To reduce particulate matter (PM) including soot in diesel exhaust gas, a diesel particulate filter (DPF) has been developed. Since it is difficult to observe the phenomena in a DPF experimentally, we have conducted a lattice Boltzmann simulation. In this study, we simulated the flow in a metallic filter. An X-ray computed tomography (CT) technique was applied to obtain its inner structure. The processes of soot deposition and oxidation were included for a continuously regenerating diesel filter. By comparing experimental data, a parameter of soot deposition probability in the numerical model was determined.

## 1. Introduction

Diesel vehicles, such as buses and trucks, are widely used in commerce. However, particulate matter (PM) contained in the diesel exhaust gas is considered to pose health risks and contribute to diseases and conditions such as cancer and pollenosis [1,2]. Therefore, PM emission regulations have been markedly strengthened, particularly in Europe, North America and Japan. For the after-treatment of diesel exhaust gas, the diesel particulate filter (DPF) has been developed to remove PM [3,4].

As a DPF removes PM, it will slowly become clogged with the PM it has adsorbed. This will cause the exhaust pressure to gradually increase and, consequently, fuel consumption will worsen. In the worst possible outcome, the diesel engine will stop working entirely. Therefore, when PM accumulates to some extent, it is necessary to replace the DPF with a new one or to incinerate and remove the PM from the existing filter (filter regeneration) [5]. The future generation of this technology aims to develop a continuously regenerating DPF that continuously oxidizes adsorbed PM using a catalyst or a heater during operation of the vehicle. However, since the principal element of PM is carbon, the possibility exists that PM can spontaneously ignite under certain conditions, which causes cracking or erosion in the filter as a result of soot oxidation.

In order to solve these problems and to develop a filter that allows continuous regeneration, the processes of adsorption and oxidation of PM must be sufficiently understood. In the development of existing filters, experiments primarily consist of injecting exhaust gas through filter samples and then estimating the amount of fine particles adsorbed inside the filter. In this scenario, usually only the pressure difference across the filter and the temperature and composition of the exhaust gas can be measured. Therefore, a numerical simulation of the phenomena would be useful.

In the present study, a metallic fibre filter was analysed to obtain its internal structure, using an X-ray computed tomography (CT) method, and a numerical simulation was performed, using the obtained internal structure of the filter. We simulated a continuously regenerating DPF while simultaneously considering the deposition and combustion processes of soot particles.

## 2. Analytical model and X-ray computed tomography measurement

In this section, the lattice Boltzmann method (LBM), used for the numerical model, is briefly explained [6,7]. The flow is described by the lattice Bhatnagar–Gross–Krook (BGK) equation in terms of the distribution function. The evolution equation using the pressure distribution function is
2.1where *c*=*δ*_{x}/*δ*_{t}, *δ*_{x} and *δ*_{t} are the lattice constant and the time step, and *τ* is the relaxation time, which controls the rate of approach to equilibrium. The equilibrium distribution function, , is given by
2.2The sound speed, *c*_{s}, is , with . Here, *p*_{0} and *ρ*_{0} are the pressure and density in the reference conditions. The pressure and local velocity ** u** (=(

*u*

_{x},

*u*

_{y},

*u*

_{z})) are obtained using the ideal gas equation: 2.3and 2.4In the above equation, to consider the variable density, we adopted the low Mach number approximation [7]. To perform the three-dimensional calculation, a d3q15 model was applied.

To analyse the combustion field for soot oxidation, the distribution functions of temperature and chemical species are required, in addition to the above distribution function of pressure. Similar to the flow field, the scalar quantities of temperature, soot and oxygen concentration are obtained using the distribution function corresponding to each scalar quantity. The relaxation time is related to transport coefficients, such as kinetic viscosity, using *ν*=(2*τ*−1)/6*c*^{2}*δ*_{t}. The LBM formula for temperature and concentration fields is
2.5where *s* is the temperature, *T*, or the mass fraction of species, *Y* _{i}, and *Q*_{s} is the source term, due to the chemical reaction [7]. The equilibrium distribution function, , is given by
2.6The temperature and the mass fraction of species are determined by these distribution functions:
2.7and
2.8

In a continuously regenerating DPF, the following three processes are included: (i) soot is reacted with oxygen in its gas phase; (ii) the unburned soot in the first process is deposited onto the filter surface; and (iii) the deposited soot burns on the filter surface. The deposition process of soot particles is explained by Brownian diffusion, interception, inertial impaction and gravity [8]. When the actual size of PM is taken into consideration, the analysis must be performed on the nanometre scale and, consequently, a large computational cost will be needed for the calculation. Therefore, it is impracticable to consider the complex geometry of nano-sized soot particles. Instead, the soot concentration is monitored at the surface of the filter or the soot layer [9]. The mass fraction of the deposited soot is given by
2.9where *Y* _{C,surface} is the mass fraction of soot on the filter surface or the deposited soot layer at each time step. It should be noted that *P*_{D} is the soot deposition probability: soot at *P*_{D} is deposited; soot at (1−*P*_{D}) is not deposited and is bounced back into the flow. Exfoliation of the soot layer, due to the flow, is not taken into consideration. As expected, *Y*_{C,surface} is increased when the soot, in its gas phase, accumulates on the filter surface. As the soot deposition continues, the soot concentration becomes unity at some time. When this limit is reached, the solid site is piled up. The deposited soot region is treated as a non-slip wall, which implies a dynamic change in the boundary condition for the fluid. In this way, the soot deposition layer grows. When a continuously regenerating DPF is simulated, both the soot oxidation and the soot deposition are taken into consideration. In the present simulation, the soot oxidation was set as a one-step irreversible reaction, and the reaction rate proposed by Lee *et al*. [10] was applied.

To analyse the flow inside an actual filter, the internal structure of the filter was obtained using a three-dimensional X-ray CT measurement [11]. Figure 1 shows the image of the filter obtained by this measurement. The image area was 6.65×6.65×3.46 mm, the pixel number in the cross section was 576×576, and the resolution in both horizontal (cross section) and vertical (deposition layer) directions was 11.5 μm pixel^{−1}. Based on projection data from the images, three-dimensional matrix data of the filter with 576×576×300 points were obtained. The grid size in the simulation was the same value as the resolution in the CT measurement. The porosity of the filter is 0.8. The fibre structure of the filter is well observed.

## 3. Results and discussion

### (a) Evaluation of soot deposition probability

As mentioned above, the model was used to analyse the deposition phenomenon on the filter wall surface and the soot layer using the soot deposition probability, *P*_{D}. In this study, *P*_{D} was experimentally evaluated. The experiment consisted of the following procedures. A 10 mm square of metallic fibre filter was cut out, using a diamond cutter, and the square was inserted into a sample holder with an 8 mm diameter hole. Exhaust gas from a diesel vehicle, whose mass concentration and PM particle-size distribution were already known [12], was flowed into the sample holder and the weight of the PM deposited onto the filter surface was measured. The mass of trapped soot inside the filter was counted using a scanning mobility particle sizer (SMPS), and was also evaluated by a CT image of the soot deposited inside the filter. The temperature of the filter was the same as that of the exhaust gas and the soot particles were deposited onto the filter without performing regeneration.

A field consisting of the same conditions as the experiment was reproduced as numerical calculations, and deposition calculations with different values of *P*_{D} were conducted. Just as occurred in the experiment, soot oxidation was not considered in the simulation and only soot deposition occurred. Figure 2 shows the time variations of the amounts of soot deposited on the filter surface when *P*_{D}=0.0001, 0.001, 0.01, 0.1 and 1.0. In figure 2, the experimental data are indicated by the dotted line. Furthermore, it can be seen that the amount of deposited soot was almost proportional to the passage of time. The amount of deposited soot increased with *P*_{D}. However, at *P*_{D}>0.01, the total mass of deposited soot did not increase, because most of the soot was trapped inside the filter. When the experimental results were compared with those obtained by the simulation, the temporal changes in the amount of deposited soot were close to the experimental values when *P*_{D}=0.001. In the next section, a continuously regenerating DPF was simulated using *P*_{D}=0.001.

### (b) Simulation of a continuously regenerating diesel particulate filter

Figure 3*a* shows the coordinate system and calculation domain used for the simulation. In the coordinate system, the direction of the exhaust gas flow passing through the filter is *X*, and the directions perpendicular to the flow direction are *Y* and *Z*. The size of the calculation domain was 3.695 mm in the *X* direction, and 0.589 mm in the *Y* and *Z* directions. The number of grid points was 320 (*X*) × 51 (*Y*) × 51 (*Z*). The filter was set at the centre of the calculation domain, and entrance and exit zones were set in front of the filter region and at the rear of the filter region. Referring to the experiment, the conditions of the exhaust gas were set as follows: temperature, 400^{°}C; mass fraction of soot, 4.54167×10^{−5}; and velocity, 0.497 m s^{−1}. The oxygen concentration of the exhaust gas and the filter temperature (*T*_{w}) were varied. Typically, the volumetric oxygen concentration of the exhaust gas was 10 per cent.

Figure 3*b* shows an example of the three-dimensional profiles of soot at *T*_{w}=800^{°}C, which was obtained 20 ms after the initiation of the calculation. In figure 3*b*, the *XY* section expresses the distribution of the soot concentration in the gas phase, while the three-dimensionally displayed grey region expresses the solid phase of the deposited soot. As shown, the amount of soot in the exhaust gas decreased owing to soot deposition and combustion. Soot was primarily deposited onto a location near the inlet of the filter.

Figure 4 shows the soot concentration in the gas phase (*Y* _{c,g}), the temperature (*T*), the accumulated soot mass fraction in the solid phase (*Y* _{c,s}) and the distribution of the reaction rate (*W*_{c}) on the *XY* section. As shown by the soot concentration in the gas phase, the soot concentration was not zero at the outlet of the filter. This means that the after-treatment of the filter was insufficient. For comparison, a calculation without considering soot deposition was performed. Consequently, the amount of burned soot was found to be very small at *T*_{w}=800^{°}C. Therefore, the decrease in the soot concentration was considered to be primarily due to soot deposition. The temperature distribution shown in figure 4*b* indicates that the temperature increase due to soot combustion was very small. The comparison between figure 4*c* and *d* revealed that the reaction rate was particularly large on the filter surface. The reason for this is that, since the soot concentration increased with the soot deposition on the filter surface, the reaction rate increased on the filter surface. As mentioned above, when the filter temperature was 800^{°}C, the after-treatment of the filter for the exhaust gas was insufficient and, consequently, unreacted soot, which had not been incinerated and removed, was emitted from the filter outlet.

Next, the effects of the filter temperature and oxygen concentration on the after-treatment of the exhaust gas were investigated, using different filter temperatures and oxygen concentrations. For quantitative investigation of the change in the soot concentration inside the filter, the soot concentrations in the gas phase on the *YZ* section were averaged, and the change in the soot concentration in the flow direction (*X* direction) was investigated. The filter temperature was set at 800^{°}C, 1000^{°}C and 1200^{°}C, and the volumetric oxygen concentration was set at 10 and 21 per cent. Figure 5 shows the soot concentration distribution obtained when the soot concentration became stable 20 ms after the initiation of the calculation. As shown in figure 5, when the filter temperature was 800^{°}C, independent of oxygen concentration, the soot concentration was not zero at the filter outlet; i.e. the after-treatment for the exhaust gas was insufficient. However, when the filter temperature was 1000^{°}C or higher, soot in the exhaust gas was almost completely removed. When the oxygen concentration was higher, the decrease in the soot concentration in the exhaust gas was more substantial; i.e. the after-treatment for the exhaust gas was promoted. Based on these simulation results, it is possible to find the conditions of complete after-treatment in a continuously regenerating diesel filter.

## 4. Conclusion

In this study, using the internal structure of a metallic fibre filter obtained by an X-ray CT technique, the flow and reaction inside a DPF were simulated using the LBM. By simultaneously calculating soot deposition and oxidation processes, a continuously regenerating DPF was investigated. The following results were obtained:

— A comparison experiment showed that when the soot deposition probability of

*P*_{D}was 0.001, the time variation of deposited soot in the simulation significantly corresponded to the experimental data.— Using the soot deposition probability evaluated by the experiments, a continuously regenerating DPF was simulated. Unburned soot in the gas phase was deposited onto the filter surface and then the deposited soot was oxidized.

— The amount of oxidized soot increased with the filter temperature. When the filter temperature was 1000

^{°}C or higher, soot in the exhaust gas could be almost completely eliminated in a continuously regenerating DPF.— When the volumetric oxygen concentration was higher, soot was oxidized more significantly and, consequently, the after-treatment process in a continuously regenerating DPF was promoted.

## Footnotes

One contribution of 25 to a Theme Issue ‘Discrete simulation of fluid dynamics: applications’.

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