# Mixing Rules for an Exact Determination of the Dielectric Properties of Engine Soot Using the Microwave Cavity Perturbation Method and Its Application in Gasoline Particulate Filters

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## Abstract

**:**

## 1. Introduction

## 2. Fundamentals of the MCP and Their Application to Soot-Loaded Particulate Filters

#### 2.1. Determination of Dielectric Properties Using the MCP

- -
- Changes in the electromagnetic field due to depolarization inside the sample.
- -
- Deviation of the field distribution in the cavity due to its non-ideal cylindrical shape.
- -
- Necessity to apply mixing rules for porous samples or samples with multiple species.

_{010}mode and occurs at a frequency of approx. 2.48 GHz. Compared to the larger resonator, this one is not capable of adjusting the sample temperature or gas atmosphere, allowing a simpler design of the sample tube. Hence, the cavity has smaller openings, which results in smaller deviations of the electromagnetic field in comparison to a perfectly cylindrical resonator. Additionally, the larger resonator can only be filled with small sample heights to allow an unhindered gas flow to set defined gas atmospheres. To nevertheless position the material sample centrically in the cavity, a porous quartz glass frit is mounted inside the sample tube. In contrast, in the smaller resonator, the sample can completely traverse the cavity, resulting in no depolarization effects due to the field distribution of the observed resonant mode. Thus, using the smaller resonator, the simplified MCP can be applied without the adjustments described in [27], and applicable mixing rules can be determined with less possible interferences. Due to the smaller sample diameter as a result of the generally smaller resonator dimensions, no more sample volume than for the measurements in the large resonator is required, despite the completely filled quartz tube.

#### 2.2. Influence of Mixing Rules on the Material Property Determination

#### 2.3. Possible Mixing Rules for Soot-Loaded Filter

## 3. Determination of the Mixing Rules for Soot-Loaded Particulate Filters

#### 3.1. Soot-Loaded Filter-Air Mixing Rule

^{®}5.6.

_{010}mode, which has no azimuthal dependence of the electromagnetic field. This allows for calculation of the resonant parameters in a two-dimensional model. Similar to the real measurement setup, the cylindrical sample (10 mm in diameter) is located in the center of the resonator, but passes completely through the resonator. This prevents depolarization effects, as described in [27]. Despite the two-dimensional approach, an exact replication of the particles is not possible without great computational effort. Therefore, the substrate–air mixture of the sample will be modeled in simplified form. Since, in the resonator setup only coarsely crushed filter substrates are measured to keep the sample as similar as possible to an intact particulate filter, the sample bulk in the simulation model is assumed as multiple cylindrical layers of filter substrate stacked upon each other separated by air. The substrate fraction ${\nu}_{\mathrm{bulk}}$ is adjusted by the height of the layers. In addition, the influence of GPFs with different degrees of soot loading is accounted by varying the substrate conductivity, which is the only source of dielectric losses (${\epsilon}_{\mathrm{r},\mathrm{pol}}^{\u2033}$ << $\frac{\sigma}{{\epsilon}_{0}\omega}$). The parameter values that varied in the simulation are listed in Table 1. The substrate permittivity, meanwhile, is not varied and has a value of ${\epsilon}_{\mathrm{r}}^{\prime}$ = 1.5, corresponding to the effective properties of soot-loaded GPF samples measured in this work by the MCP method.

_{111}mode, which has the lowest possible resonant frequency for this geometry and is often used in RF sensor applications.

^{−}

^{1}. Only the latter is relevant for the application of mixing rules concerning the dielectric losses ${\epsilon}_{\mathrm{r}}^{\u2033}$ (cf. Equation (3)). Since they are primarily relevant for signal changes of the RF sensor, in this work, only the quality factor will be evaluated in further detail.

#### 3.2. Soot-Substrate Mixing Rule

_{soot}/L

_{GPF}, was measured in the larger resonator. The hereby measured dielectric losses agree well with those from the smaller resonator for the same amount of soot. The effective dielectric losses of PrintexU-loaded cordierite substrate ${\epsilon}_{\mathrm{eff}}^{\u2033}$, show an almost linear relationship with the soot volume fraction ${\nu}_{\mathrm{soot}}$. Thus, Wiener’s mixing rule can be applied for mixtures of soot and filter substrate. Along with the simulation-based determined mixing rule to account the air content, it is now possible to deduce the dielectric losses of soot from effective sample properties.

## 4. Validation of Mixing Rules by Engine Test Bench Measurements

_{21}using the vector network analyzer (VNA) Anritsu MS2025B. Due to the low exciting electrical fields, a linear system regarding the interaction with the dielectric properties can be assumed and was used in the simulations, as in the determination of the mixing rules before. This setup, as constructed in the simulation model, can be seen in Figure 9.

_{GPF}before the first regeneration and third at 730 mg/L

_{GPF}after three partial regenerations and a subsequent soot loading. The spectra simulated with the mixing rules determined in Section 3 (Figure 12c) agree well with the measured spectra both for the unloaded and for the two loaded state. For comparison, the simulations were also performed with deviating mixing rules. Thus, for the spectra shown in Figure 12b, the air content of the sample respective to the filter was considered using Wiener’s rule instead of the rule determined in Section 3.1.

_{GPF}too low. The RF signal, on the other hand, deviates more and predicts a slightly higher attenuation. This may be not only due to a slightly incorrect prediction of the filter permittivity, but also caused by modeling antennas and canning not fully in detail.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Hennig, F.; Quass, U.; Hellack, B.; Küpper, M.; Kuhlbusch, T.A.J.; Stafoggia, M.; Hoffmann, B. Ultrafine and Fine Particle Number and Surface Area Concentrations and Daily Cause-Specific Mortality in the Ruhr Area, Germany, 2009–2014. Environ. Health Perspect.
**2018**, 126, 27008. [Google Scholar] [CrossRef] [PubMed] - Soppa, V.J.; Shinnawi, S.; Hennig, F.; Sasse, B.; Hellack, B.; Kaminski, H.; Quass, U.; Schins, R.P.F.; Kuhlbusch, T.A.J.; Hoffmann, B. Effects of short-term exposure to fine and ultrafine particles from indoor sources on arterial stiffness—A randomized sham-controlled exposure study. Int. J. Hyg. Environ. Health
**2019**, 222, 1115–1132. [Google Scholar] [CrossRef] [PubMed] - Kittelson, D.B. Engines and nanoparticles. J. Aerosol Sci.
**1998**, 29, 575–588. [Google Scholar] [CrossRef] - Chen, L.; Liang, Z.; Zhang, X.; Shuai, S. Characterizing particulate matter emissions from GDI and PFI vehicles under transient and cold start conditions. Fuel
**2017**, 189, 131–140. [Google Scholar] [CrossRef] - Schoenhaber, J.; Kuehn, N.; Bradler, B.; Richter, J.M.; Bauer, S.; Lenzen, B.; Beidl, C. Impact of European Real-Driving-Emissions Legislation on Exhaust Gas Aftertreatment Systems of Turbocharged Direct Injected Gasoline Vehicles. SAE Tech. Pap.
**2017**. [Google Scholar] [CrossRef] - Demuynck, J.; Favre, C.; Bosteels, D.; Hamje, H.; Andersson, J. Real-World Emissions Measurements of a Gasoline Direct Injection Vehicle without and with a Gasoline Particulate Filter. SAE Tech. Pap.
**2017**. [Google Scholar] [CrossRef][Green Version] - Rose, D.; Boger, T. Different Approaches to Soot Estimation as Key Requirement for DPF Applications. SAE Tech. Pap.
**2009**. [Google Scholar] [CrossRef][Green Version] - Walter, S.; Schwanzer, P.; Hagen, G.; Haft, G.; Dietrich, M.; Rabl, H.-P.; Moos, R. Hochfrequenzsensorik zur direkten Beladungserkennung von Benzinpartikelfiltern. In Automobil-Sensorik 3; Tille, T., Ed.; Springer: Berlin, Germany, 2020; pp. 185–208. [Google Scholar] [CrossRef]
- Gaiser, G.; Mucha, P. Prediction of Pressure Drop in Diesel Particulate Filters Considering Ash Deposit and Partial Regenerations. SAE Tech. Pap.
**2004**. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Liu, X.; Chanko, T.; Lambert, C.; Maricq, M. Gasoline Particulate Filter Efficiency and Backpressure at Very Low Mileage. SAE Tech. Pap.
**2018**. [Google Scholar] [CrossRef] - Lambert, C.; Chanko, T.; Dobson, D.; Liu, X.; Pakko, J. Gasoline Particle Filter Development. Emiss. Control Sci. Technol.
**2017**, 3, 105–111. [Google Scholar] [CrossRef][Green Version] - Saito, C.; Nakatani, T.; Miyairi, Y.; Yuuki, K.; Makino, M.; Kurachi, H.; Heuss, W.; Kuki, T.; Furuta, Y.; Kattouah, P.; et al. New Particulate Filter Concept to Reduce Particle Number Emissions. SAE Tech. Pap.
**2011**. [Google Scholar] [CrossRef] - Chan, T.W.; Meloche, E.; Kubsh, J.; Rosenblatt, D.; Brezny, R.; Rideout, G. Evaluation of a Gasoline Particulate Filter to Reduce Particle Emissions from a Gasoline Direct Injection Vehicle. SAE Int. J. Fuels Lubr.
**2012**, 5, 1277–1290. [Google Scholar] [CrossRef] - Wang-Hansen, C.; Ericsson, P.; Lundberg, B.; Skoglundh, M.; Carlsson, P.-A.; Andersson, B. Characterization of Particulate Matter from Direct Injected Gasoline Engines. Top. Catal.
**2013**, 56, 446–451. [Google Scholar] [CrossRef] - Suresh, A.; Khan, A.; Johnson, J.H. An Experimental and Modeling Study of Cordierite Traps—Pressure Drop and Permeability of Clean and Particulate Loaded Traps. SAE Tech. Pap.
**2000**. [Google Scholar] [CrossRef] - Schwanzer, P.; Mieslinger, J.; Dietrich, M.; Haft, G.; Walter, S.; Hagen, G.; Moos, R.; Gaderer, M.; Rabl, H.-P. Monitoring of a Particulate Filter for Gasoline Direct Injection Engines with a Radio-Frequency-Sensor. In Proceedings of the 11th Internationales Symposium für Abgasund Partikelemissionen, Ludwigsburg, Germany, 3–4 March 2020. [Google Scholar]
- Sethia, S.; Kubinski, D.; Nerlich, H.; Naber, J. RF Studies of Soot and Ammonia Loadings on a Combined Particulate Filter and SCR Catalyst. J. Electrochem. Soc.
**2020**, 167, 147516. [Google Scholar] [CrossRef] - Moos, R. Microwave-Based Catalyst State Diagnosis—State of the Art and Future Perspectives. SAE Int. J. Engines
**2015**, 8, 1240–1245. [Google Scholar] [CrossRef] - Sappok, A.; Bromberg, L.; Parks, J.E.; Prikhodko, V. Loading and Regeneration Analysis of a Diesel Particulate Filter with a Radio Frequency-Based Sensor. SAE Tech. Pap.
**2010**. [Google Scholar] [CrossRef] - Sappok, A.; Bromberg, L. Radio Frequency Diesel Particulate Filter Soot and Ash Level Sensors: Enabling Adaptive Controls for Heavy-Duty Diesel Applications. SAE Int. J. Commer. Veh.
**2014**, 7, 468–477. [Google Scholar] [CrossRef] - Feulner, M.; Hagen, G.; Hottner, K.; Redel, S.; Müller, A.; Moos, R. Comparative Study of Different Methods for Soot Sensing and Filter Monitoring in Diesel Exhausts. Sensors
**2017**, 17, 400. [Google Scholar] [CrossRef][Green Version] - Feulner, M.; Hagen, G.; Piontkowski, A.; Müller, A.; Fischerauer, G.; Brüggemann, D.; Moos, R. In-Operation Monitoring of the Soot Load of Diesel Particulate Filters: Initial Tests. Top. Catal.
**2013**, 56, 483–488. [Google Scholar] [CrossRef] - Nicolin, P.; Boger, T.; Dietrich, M.; Haft, G.; Bachurina, A. Soot Load Monitoring in Gasoline Particulate Filter Applications with RF-Sensors. SAE Tech. Pap.
**2020**. [Google Scholar] [CrossRef] - Dietrich, M.; Jahn, C.; Lanzerath, P.; Moos, R. Microwave-Based Oxidation State and Soot Loading Determination on Gasoline Particulate Filters with Three-Way Catalyst Coating for Homogenously Operated Gasoline Engines. Sensors
**2015**, 15, 21971–21988. [Google Scholar] [CrossRef] [PubMed][Green Version] - Beulertz, G.; Fritsch, M.; Fischerauer, G.; Herbst, F.; Gieshoff, J.; Votsmeier, M.; Hagen, G.; Moos, R. Microwave Cavity Perturbation as a Tool for Laboratory In Situ Measurement of the Oxidation State of Three Way Catalysts. Top. Catal.
**2013**, 56, 405–409. [Google Scholar] [CrossRef] - Steiner, C.; Walter, S.; Malashchuk, V.; Hagen, G.; Kogut, I.; Fritze, H.; Moos, R. Determination of the Dielectric Properties of Storage Materials for Exhaust Gas Aftertreatment Using the Microwave Cavity Perturbation Method. Sensors
**2020**, 20, 6024. [Google Scholar] [CrossRef] - Dietrich, M.; Rauch, D.; Porch, A.; Moos, R. A Laboratory Test Setup for in Situ Measurements of the Dielectric Properties of Catalyst Powder Samples under Reaction Conditions by Microwave Cavity Perturbation: Set up and Initial Tests. Sensors
**2014**, 14, 16856–16868. [Google Scholar] [CrossRef][Green Version] - Sihvola, A. Mixing Rules with Complex Dielectric Coefficients. Subsurf. Sens. Technol. Appl.
**2000**, 1, 393–415. [Google Scholar] [CrossRef] - Pozar, D.M. Microwave Engineering, 4th ed.; Wiley: Hoboken, NJ, USA, 2012. [Google Scholar]
- Chen, L. Microwave Electronics: Measurement and Materials Characterization; Wiley: Chichester, UK, 2005. [Google Scholar]
- Parkash, A.; Vaid, J.K.; Mansingh, A. Measurement of Dielectric Parameters at Microwave Frequencies by Cavity-Perturbation Technique. IEEE Trans. Microw. Theory Tech.
**1979**, 27, 791–795. [Google Scholar] [CrossRef] - Venermo, J.; Sihvola, A. Dielectric polarizability of circular cylinder. J. Electrost.
**2005**, 63, 101–117. [Google Scholar] [CrossRef] - Jylhä, L.; Sihvola, A. Equation for the effective permittivity of particle-filled composites for material design applications. J. Phys. D Appl. Phys.
**2007**, 40, 4966–4973. [Google Scholar] [CrossRef] - Cheng, E.M.; Malek, M.F.b.A.; Ahmed, M.; You, K.Y.; Lee, K.Y.; Nornikman, H. The Use of Dielectric Mixture Equations to Analyze the Dielectric Properties of a Mixture of Rubber Tire Dust and Rice Husks in a Microwave Absorber. Prog. Electromagn. Res.
**2012**, 129, 559–578. [Google Scholar] [CrossRef][Green Version] - Marquardt, P. Quantum-size affected conductivity of mesoscopic metal particles. Phys. Lett. A
**1987**, 123, 365–368. [Google Scholar] [CrossRef] - El Bouazzaoui, S.; Achour, M.E.; Brosseau, C. Microwave effective permittivity of carbon black filled polymers: Comparison of mixing law and effective medium equation predictions. J. Appl. Phys.
**2011**, 110, 74105. [Google Scholar] [CrossRef] - Jusoh, M.; Abbas, Z.; Hassan, J.; Azmi, B.; Ahmad, A. A Simple Procedure to Determine Complex Permittivity of Moist Materials Using Standard Commercial Coaxial Sensor. Meas. Sci. Rev.
**2011**, 11, 19–22. [Google Scholar] [CrossRef][Green Version] - Camacho Hernandez, J.N.; Link, G.; Soldatov, S.; Füssel, A.; Schubert, M.; Hampel, U. Experimental and numerical analysis of the complex permittivity of open-cell ceramic foams. Ceram. Int.
**2020**, 46, 26829–26840. [Google Scholar] [CrossRef] - Camerucci, M.A.; Urretavizcaya, G.; Castro, M.S.; Cavalieri, A.L. Electrical properties and thermal expansion of cordierite and cordierite-mullite materials. J. Eur. Ceram. Soc.
**2001**, 21, 2917–2923. [Google Scholar] [CrossRef] - Nelson, S.O. Density-Permittivity Relationships for Powdered and Granular Materials. IEEE Trans. Instrum. Meas.
**2005**, 54, 2033–2040. [Google Scholar] [CrossRef] - Tuhkala, M.; Juuti, J.; Jantunen, H. An indirectly coupled open-ended resonator applied to characterize dielectric properties of MgTiO
_{3}–CaTiO_{3}powders. J. Appl. Phys.**2014**, 115, 184101. [Google Scholar] [CrossRef] - Dube, D.C. Study of Landau-Lifshitz-Looyenga’s formula for dielectric correlation between powder and bulk. J. Phys. D Appl. Phys.
**1970**, 3, 1648–1652. [Google Scholar] [CrossRef] - Looyenga, H. Dielectric constants of heterogeneous mixtures. Physica
**1965**, 31, 401–406. [Google Scholar] [CrossRef] - Birchak, J.R.; Gardner, C.G.; Hipp, J.E.; Victor, J.M. High dielectric constant microwave probes for sensing soil moisture. Proc. IEEE
**1974**, 62, 93–98. [Google Scholar] [CrossRef] - Goncharenko, A.V. Generalizations of the Bruggeman equation and a concept of shape-distributed particle composites. Phys. Review. E Stat. Nonlinear Soft Matter Phys.
**2003**, 68, 41108. [Google Scholar] [CrossRef] - Maxwell Garnett, J.C. Colours in Metal Glasses, in Metallic Films, and in Metallic Solutions. II. Philos. Trans. R. Soc. Lond. Ser. A Contain. Pap. A Math. Phys. Character
**1906**, 205, 237–288. [Google Scholar] - Lichtenecker, K.; Rother, K. Die Herleitung des logarithmischen Mischungsgesetzes aus allgemeinen Prinzipien der stationären Strömung. Phys. Zeitschr.
**1931**, 32, 255–260. [Google Scholar] - Wiener, O. Zur Theorie der Refraktionskonstanten. Math. Phys. Kl.
**1910**, 62, 256–277. [Google Scholar] - Karkkainen, K.K.; Sihvola, A.H.; Nikoskinen, K.I. Effective permittivity of mixtures: Numerical validation by the FDTD method. IEEE Trans. Geosci. Remote Sens.
**2000**, 38, 1303–1308. [Google Scholar] [CrossRef][Green Version] - Seong, H.; Choi, S.; Lee, K. Examination of nanoparticles from gasoline direct-injection (GDI) engines using transmission electron microscopy (TEM). Int. J. Automot. Technol.
**2014**, 15, 175–181. [Google Scholar] [CrossRef] - Boger, T.; Rose, D.; Nicolin, P.; Gunasekaran, N.; Glasson, T. Oxidation of Soot (Printex
^{®}U) in Particulate Filters Operated on Gasoline Engines. Emiss. Control Sci. Technol.**2015**, 1, 49–63. [Google Scholar] [CrossRef][Green Version] - Choi, S.; Seong, H. Oxidation characteristics of gasoline direct-injection (GDI) engine soot: Catalytic effects of ash and modified kinetic correlation. Combust. Flame
**2015**, 162, 2371–2389. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Images of the resonators used in this work. (

**a**) Resonator with possibility to heat the sample; described in more detail in [28]. (

**b**) Non-heatable resonator with smaller cavity compared to (

**a**).

**Figure 2.**Example for application of different mixing rules on the effective permittivity ${\epsilon}_{\mathrm{r},\mathrm{eff}}^{\prime}$ (

**a**) and the dielectric losses ${\epsilon}_{\mathrm{r},\mathrm{eff}}^{\u2033}$ (

**b**) depending on the bulk fraction ${\nu}_{\mathrm{bulk}}$ for mixtures of air (${\epsilon}_{\mathrm{r},\mathrm{air}}^{\prime}$ = 1; ${\epsilon}_{\mathrm{r},\mathrm{air}}^{\u2033}$ = 0) and lossy material (exemplarily values: ${\epsilon}_{\mathrm{r},\mathrm{bulk}}^{\prime}$ = 10; ${\epsilon}_{\mathrm{r},\mathrm{bulk}}^{\u2033}$ = 10). Mixing rules according to Looyenga [44], Birchak [45], Bruggeman [46], Maxwell-Garnett [47], Lichtenecker [48] and Wiener for a series (Wiener I) as well as for a parallel circuit (Wiener II) of the mixture components [49].

**Figure 3.**Simulated electric field (red: high field strength; blue: low field strength) of the TM

_{010}resonant mode in a rotationally symmetric resonator setup with a centered sample split into several layers (${\nu}_{\mathrm{bulk}}$ = 60%; ${\sigma}_{\mathrm{bulk}}$ = 40 mS/m): (

**a**) 5 layers, (

**b**) 10 layers and (

**c**) 20 layers.

**Figure 4.**Simulated electric field (red: high field strength; blue: low field strength) of the TE

_{111}resonant mode in a monolith filter structure (${\nu}_{\mathrm{bulk}}$= 30%; ${\sigma}_{\mathrm{bulk}}$ = 10 mS/m) with different cell densities: (

**a**) 58 cpsi; (

**b**) 114 cpsi.

**Figure 5.**Inverse quality factor ${Q}^{-1}$ depending on the bulk fraction ${\nu}_{\mathrm{bulk}}$ at a substrate conductivity ${\sigma}_{\mathrm{bulk}}$ of 100 mS/m for (

**a**) different numbers of substrate layers in the resonator setup, respectively, and (

**b**) different cell densities in the filter setup according to Table 1 and as indicated by the color-coded labels within the figure. Dashed line: Wiener’s mixing rule for a series circuit (k = 1) in Equation (5).

**Figure 6.**Exponent of the applicable mixing rule $k$ depending on the substrate conductivity ${\sigma}_{\mathrm{bulk}}$ for (

**a**) a different number of substrate layers in the resonator setup, respectively, and (

**b**) different cell densities in the filter setup, according to Table 1. Please note: The ordinate axes scaling is enlarged to highlight the differences.

**Figure 7.**Various PrintexU–cordierite mixtures with a volume fraction ${\nu}_{\mathrm{soot}}$ of PrintexU of: (

**a**) 0.5%, (

**b**) 1.5%, (

**c**) 5.0%, (

**d**) 20%.

**Figure 8.**Effective dielectric losses of PrintexU–cordierite mixture ${\epsilon}_{\mathrm{eff}}^{\u2033}$ depending on the volume fraction of PrintexU ${\nu}_{\mathrm{soot}}$; the influence of the air fraction was eliminated (

**a**) by the mixing rule determined in Section 3.1 and (

**b**), for comparison, by Wiener’s mixing rule. The data were measured with the smaller resonator (black triangles) as well as with the larger resonator under consideration of depolarization effects (red triangle); possible linear correlation (i.e., Wiener’s mixing rule) for mixtures between substrate and soot illustrated by the dotted regression line.

**Figure 9.**Geometry of the simulation model; the grids before and after the GPF were not considered in the flow simulation.

**Figure 10.**Dielectric properties ((

**a**) permittivity; (

**b**) dielectric losses) of soot generated on the engine test bench at load corresponding to a velocity at 160 km/h depending on its temperature.

**Figure 11.**Exhaust gas data during soot loading with multiple partial regenerations caused by switching to lean operation; amount of soot accumulated in the GPF ${\Delta m}_{\mathrm{GPF}}$ (black), temperature ${T}_{\mathrm{upstream}}$ (red) and air-fuel ratio λ (blue) upstream of the GPF.

**Figure 12.**Frequency spectra of the transmission parameter $\left|{S}_{21}\right|$ at different times during filter loading: soot free filter at t = 0 s (black), before first regeneration at t = 3300 s and 472 mg/L

_{GPF}stored soot (red) and after three regenerations and subsequent soot loading at t = 10,000 s and 730 mg/L

_{GPF}stored soot (blue); (

**a**) measured data; (

**b**) simulated data using Wiener’s mixing rule to consider the air content; (

**c**) simulated data using the mixing rule determined in Section 3.1.

**Figure 13.**(

**a**) Measured (see text to Figure 11) and simulated amount of soot stored in the GPF ${\mathsf{\Delta}m}_{\mathrm{GPF}}$ as well as RF signal attenuation ${S}_{21,\mathrm{mean}}$ (averaged in the frequency range from 1.68 to 1.78 GHz) during filter loading; (

**b**) measured and simulated change in the mean RF signal attenuation ${\mathsf{\Delta}S}_{21,\mathrm{mean}}$ depending on the stored soot ${\mathsf{\Delta}m}_{\mathrm{GPF}}$.

**Table 1.**Parameter variation in the simulation models. Besides the bulk volume fraction of the sample ${\nu}_{\mathrm{bulk}}$ and also its conductivity ${\sigma}_{\mathrm{bulk}}$, (a) in case of the resonator model, the layer number and (b) for the filter structure, the cell density was varied (cpsi stands for cells per square inch).

${\nu}_{\mathrm{bulk}}$/% | 0/20/30/40/50/60/70/80/100 |

${\sigma}_{\mathrm{bulk}}$/mS/m | 1/10/20/40/60/80/100 |

(a) substrate layers | 5/10/20/200 |

(b) cell density/cpsi | 21/58/114/188 |

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**MDPI and ACS Style**

Walter, S.; Schwanzer, P.; Steiner, C.; Hagen, G.; Rabl, H.-P.; Dietrich, M.; Moos, R. Mixing Rules for an Exact Determination of the Dielectric Properties of Engine Soot Using the Microwave Cavity Perturbation Method and Its Application in Gasoline Particulate Filters. *Sensors* **2022**, *22*, 3311.
https://doi.org/10.3390/s22093311

**AMA Style**

Walter S, Schwanzer P, Steiner C, Hagen G, Rabl H-P, Dietrich M, Moos R. Mixing Rules for an Exact Determination of the Dielectric Properties of Engine Soot Using the Microwave Cavity Perturbation Method and Its Application in Gasoline Particulate Filters. *Sensors*. 2022; 22(9):3311.
https://doi.org/10.3390/s22093311

**Chicago/Turabian Style**

Walter, Stefanie, Peter Schwanzer, Carsten Steiner, Gunter Hagen, Hans-Peter Rabl, Markus Dietrich, and Ralf Moos. 2022. "Mixing Rules for an Exact Determination of the Dielectric Properties of Engine Soot Using the Microwave Cavity Perturbation Method and Its Application in Gasoline Particulate Filters" *Sensors* 22, no. 9: 3311.
https://doi.org/10.3390/s22093311