# Modelling the Influence of Different Soot Types on the Radio-Frequency-Based Load Detection of Gasoline Particulate Filters

^{1}

^{2}

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## Abstract

**:**

^{®}that not only allowed for calculating the loading and regeneration process of GPFs under different engine operating conditions but also determined the impact on both sensor systems. To simulate the regeneration behavior of gasoline soot accurately, an oxidation model was developed. To identify the influence of different engine operating points on the sensor behavior, various samples generated at an engine test bench were examined regarding their kinetic parameters using thermogravimetric analysis. Thus, this compared the accuracy of soot mass determination using the RF sensor with the differential pressure method. By simulating a typical driving condition with incomplete regenerations, the effects of the soot kinetics on sensor accuracy was demonstrated exemplarily. Thereby, the RF sensor showed an overall smaller mass determination error, as well as a lower dependence on the soot kinetics.

## 1. Introduction

_{x}trap (LNT) [27,28]. Thus, it would be possible to monitor GPFs with a TWC coating not only for soot loading but also concurrently for oxygen storage [21].

^{®}5.4 (COMSOL Inc., Stockholm, Sweden) to determine the signals of both sensor systems under different operating conditions. Thereby, the simulation model should not simulate all processes perfectly realistically, but rather help to recognize possible perturbations to the sensor signals. Thus, the here-given simulation model can help to develop methods to eliminate these influences during the processing of the measured signal. As an example, this study showed the effects of the different reaction kinetics of various soot types on the sensor accuracy during incomplete regenerations. Therefore, this simulation model allows for a direct comparison between the accuracy of the two sensor concepts under exactly defined operating conditions.

## 2. Simulation Model Design

#### 2.1. Flow Distribution in GPFs

#### 2.2. Determination of the Filter Temperature

#### 2.3. Soot Storage

#### 2.4. Reaction Kinetics of Soot

_{2}, as well as the incomplete oxidation to CO (Equations (12) and (13)):

_{2}to CO can be expressed using a factor ${f}_{\mathrm{CO}}$ that depends on the temperature and oxygen content in the exhaust gas (Equation (14)):

^{3}/kg·s [52]. For simplicity, the simulations and the corresponding reaction kinetic analyses assume a first-order reaction similar to several models in the literature [50,53]. An implementation of reaction orders different from one would be possible for complete regenerations. On the other hand, in the case of incomplete oxidation and a subsequent loading, a description of the effects by mixed partially regenerated and fresh soot on the reaction order would be necessary. In simulation models for diesel engines, NO

_{2}combustion is also taken into account in some cases [50]. Due to the stoichiometric operation of gasoline engines and the resulting lower number of nitrogen oxides after the TWC, this model does not include that reaction.

#### 2.5. RF Parameter Calculation

^{6}S/m has been assigned to them. The GPF temperature and soot loading are transferred from the soot storage model to the RF-model, whereby no distinction is made between the deep bed and the soot cake. With these two parameters, the location-dependent complex dielectric parameters of the filter ${\epsilon}_{\mathrm{filter}}^{\prime}$ and ${\epsilon}_{\mathrm{filter}}^{\u2033}$ (which correspond to the real and imaginary part, respectively, of a complex relative permittivity $\epsilon ={\epsilon}^{\prime}-\mathrm{j}{\epsilon}^{\u2033}$), averaged over filter substrate, soot, and air, are calculated.

_{111}). To help with modeling the influence of the temperature distribution in the GPF, the dielectric parameters are measured in a wide temperature range from room temperature to 600 °C. Due to too small amounts of real gasoline soot available, it was not possible to determine their permittivity using this method. For this reason, the dielectric properties of PrintexU were used in the simulations regardless of the type of soot. Since the gasoline soot used for the following investigations was deposited on cordierite substrates and the soot kinetics were determined using soot–cordierite mixtures, cordierite was applied as the substrate material in the simulation model.

^{2}that was above 0.98 for both linear approximations. On the other hand, for cordierite, almost constant dielectric parameters were measured over the entire temperature range (${\epsilon}_{\mathrm{substrate}}^{\prime}=7.25$ and ${\epsilon}_{\mathrm{substrate}}^{\u2033}=0.0009$).

^{3}, and for PrintexU, the density was ${\rho}_{\mathrm{soot}}$ = 1060 kg/m

^{3}. The density of the filter ${\rho}_{\mathrm{GPF}}$ was calculated using the mass and volume of the filter.

## 3. Kinetics of Realistic Soot

_{0}to 0.3 m/m

_{0}) to evaluate the parameters. The pre-exponential factor could then be determined using the y-axis intersection for this line, which corresponded to $\mathrm{ln}\left({k}_{\mathrm{c},0}{M}_{\mathrm{c}}\right)$. On the other hand, the negative slope was multiplied by the gas constant R to obtain the activation energy ${E}_{\mathrm{A}}$. This is shown exemplarily for the operating point “120 km/h” in Figure 5. Here, as well as for the other investigated soot types, high linearity of the measured data was found.

## 4. Application of the Model to Driving Cycles

_{111}were considered.

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

a | filter channel width |

cpsi | cells per square inch |

${c}_{\mathrm{CO}}$ | concentration of carbon monoxide |

${c}_{{\mathrm{CO}}_{2}}$ | concentration of carbon dioxide |

${c}_{{\mathrm{O}}_{2}}$ | oxygen concentration in the exhaust gas |

${c}_{\mathrm{p},\mathrm{exhaust}}$ | heat capacity of exhaust gas |

${c}_{\mathrm{soot}}$ | soot concentration in the exhaust gas |

DPF | diesel particulate filter |

${d}_{\mathrm{s}}$ | thickness of soot cake |

${d}_{\mathrm{s},\mathrm{deep}}$ | virtual thickness of soot deposited in deep bed |

${E}_{\mathrm{A}}$ | activation energy for soot oxidation |

F | factor equal to 28.454 |

${f}_{\mathrm{CO}}$ | factor describing the ratio of CO_{2} to CO during soot oxidation |

${f}_{\mathrm{res}}$ | resonant frequency |

GDI | gasoline direct injection |

GPF | gasoline particulate filter |

${\mathit{k}}_{\mathrm{c}}$ | reaction rate constant |

${k}_{\mathrm{c},0}$ | pre-exponential factor |

${M}_{\mathrm{c}}$ | molar mass of carbon |

${m}_{\mathrm{err}}$ | sensor error |

${m}_{\mathrm{soot}}$ | soot mass deposited in the filter |

${m}_{\mathrm{soot},\mathrm{bed}}$ | soot mass collected using deep bed filtration |

${m}_{\mathrm{soot},\mathrm{cake}}$ | soot mass deposited in the soot cake |

${\dot{\mathit{m}}}_{\mathrm{exhaust}}$ | exhaust gas mass flow rate |

${\dot{\mathit{m}}}_{\mathrm{soot},\mathrm{in}}$ | soot mass flow rate into the exhaust gas system |

${\dot{\mathit{m}}}_{\mathrm{soot},\mathrm{exhaust}}$ | soot mass flow through the filter walls |

n | reaction order of the soot oxidation |

${Q}_{\mathbf{0}}$ | quality factor |

${\dot{Q}}_{\mathrm{GPF}}$ | heat flow into GPF caused by the exhaust gas |

R | gas constant |

RF | radio frequency |

TE_{111} | resonance mode in the filter canning |

TGA | thermogravimetric analysis |

${T}_{\mathrm{GPF}}$ | temperature of the GPF substrate |

${T}_{\mathrm{inlet}}$ | gas temperature in the inlet channels |

${T}_{\mathrm{outlet}}$ | gas temperature in the outlet channels |

${v}_{\mathrm{w}}$ | gas velocity through the filter channel walls |

${w}_{\mathrm{s}}$ | thickness of the filter channel walls |

${\alpha}_{\mathrm{cake}}$ | filter penetrability caused by the soot cake |

${\alpha}_{\mathrm{deep}}$ | filter penetrability caused by the deep bed filtration |

${\alpha}_{\mathrm{clean}}$ | filter penetrability without the soot loading |

${\alpha}_{\mathrm{filter}}$ | filter penetrability |

$\Delta p$ | differential pressure |

${\epsilon}_{\mathrm{filter}}^{\prime}$ | real part of the filter permittivity |

${\epsilon}_{\mathrm{filter}}^{\u2033}$ | imaginary part of the filter permittivity |

${\epsilon}_{\mathrm{soot}}^{\prime}$ | real part of the soot permittivity |

${\epsilon}_{\mathrm{soot}}^{\u2033}$ | imaginary part of the soot permittivity |

${\epsilon}_{\mathrm{substrate}}^{\prime}$ | real part of the filter substrate permittivity |

${\epsilon}_{\mathrm{substrate}}^{\u2033}$ | imaginary part of the filter substrate permittivity |

$\eta $ | dynamic viscosity of the exhaust gas |

${\eta}_{\mathrm{bed}}$ | deep bed filtration efficiency |

${\eta}_{\mathrm{cake}}$ | soot cake filtration efficiency |

${\kappa}_{0}$ | filter wall permeability |

${\kappa}_{\mathrm{channel}}$ | filter channel permeability |

${\kappa}_{\mathrm{s}}$ | soot cake permeability |

${\kappa}_{\mathrm{s},\mathrm{deep}}$ | deep bed permeability |

λ | air–fuel ratio |

$\rho $ | exhaust gas density |

${\rho}_{\mathrm{GPF}}$ | filter density |

${\rho}_{\mathrm{soot}}$ | soot density |

${\rho}_{\mathrm{substrate}}$ | filter substrate density |

$\sigma $ | conductivity |

$\varphi $ | fraction of the soot in the deep bed contributing to the soot cake formation |

$\mu $ | magnetic permeability |

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**Figure 1.**Schematic layout of the simulation model. The wire screens are used to exactly define the cavity resonator in the radio-frequency (RF) simulation.

**Figure 2.**Schematic structure of an inlet (left) and an outlet filter channel (right) with width $a$, wall thickness ${w}_{\mathrm{s}}$, and soot cake thickness in the inlet channel ${d}_{\mathrm{s}}$ (adapted from Konstandopoulos et al. [32]).

**Figure 3.**Measured dielectric parameters of PrintexU (left: ${\epsilon}_{\mathrm{soot}}^{\prime}$; right: ${\epsilon}_{\mathrm{soot}}^{\u2033}$) from 20 °C to 600 °C.

**Figure 4.**Soot oxidation measured using thermogravimetric analysis (TGA) with a heating rate of 5 K/min for different soot types.

**Figure 5.**The oxidation rate of the “120 km/h, λ < 1” operation point of the temperature at a heating rate of 5 K/min in an Arrhenius plot with $n$ = 1. The dotted red line corresponds to a linear fit in a range of 10% to 70% mass conversion (marked in grey), The slope correlates with the activation energy, while the y-axis intersection point is equal to the logarithm of the reaction rate.

**Figure 6.**The reaction rate constant ${k}_{\mathrm{c}}$ as a function of the inverse temperature for different soot types.

**Figure 7.**Stored soot mass with the “120 km/h, λ = 1” soot. After reaching 1 g/L, regeneration (marked in red) was initiated by adding 2 vol.% O

_{2}until 0.5 g/L remained on the filter. After the third loading, permanent lean exhaust gas was present.

**Figure 8.**Sensor signals of the differential pressure sensor (Δp) and the RF sensor ($|{\Delta f}_{\mathrm{res}}/{f}_{\mathrm{res},0}|$ and 1000/${Q}_{0}$) as a function of the stored soot mass. The regeneration phases are marked in red.

**Figure 9.**Sensor error in g/L as a function of the stored soot mass in the pressure differential sensor and the RF sensor based on ${f}_{\mathrm{res}}$ relative to ${Q}_{0}$. The regeneration phases are marked in red.

**Figure 10.**The proportional error of the Δp and the RF sensors (evaluation of the resonant mode TE

_{111}) after the end of the first regeneration.

Soot Type | ${\mathit{E}}_{\mathbf{A}}$ (kJ/mol) | ${\mathit{k}}_{\mathbf{c},0}$ (m^{3}·kg^{−1}·s^{−1}) |
---|---|---|

120 km/h, λ = 1 | 232.6 | $1.2\times {10}^{14}$ |

160 km/h, λ = 1 | 239.8 | $3.9\times {10}^{14}$ |

120 km/h, λ < 1 | 206.7 | $9.7\times {10}^{11}$ |

160 km/h, λ < 1 | 211.7 | $2.5\times {10}^{12}$ |

2 g/L PrintexU | 271.9 | $1.7\times {10}^{15}$ |

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## Share and Cite

**MDPI and ACS Style**

Walter, S.; Schwanzer, P.; Hagen, G.; Haft, G.; Rabl, H.-P.; Dietrich, M.; Moos, R.
Modelling the Influence of Different Soot Types on the Radio-Frequency-Based Load Detection of Gasoline Particulate Filters. *Sensors* **2020**, *20*, 2659.
https://doi.org/10.3390/s20092659

**AMA Style**

Walter S, Schwanzer P, Hagen G, Haft G, Rabl H-P, Dietrich M, Moos R.
Modelling the Influence of Different Soot Types on the Radio-Frequency-Based Load Detection of Gasoline Particulate Filters. *Sensors*. 2020; 20(9):2659.
https://doi.org/10.3390/s20092659

**Chicago/Turabian Style**

Walter, Stefanie, Peter Schwanzer, Gunter Hagen, Gerhard Haft, Hans-Peter Rabl, Markus Dietrich, and Ralf Moos.
2020. "Modelling the Influence of Different Soot Types on the Radio-Frequency-Based Load Detection of Gasoline Particulate Filters" *Sensors* 20, no. 9: 2659.
https://doi.org/10.3390/s20092659