A New Filtration Model of a Particulate Filter for Accurate Estimation of Particle Number Emissions

Abstract

In the context of increasingly stringent vehicle emission regulations, computer-aided engineering has been indispensable for optimizing the design and the operational strategies of emission control systems. This paper proposes a new filtration model for particulate filters that enables the accurate estimation of solid particle number emissions above 10 and 23 nm in diameter (SPN10 and SPN23, respectively). The model incorporates a persistent slip factor and a linear filtration efficiency of cake layers into the unit collector model proposed by Konstandopoulos and Johnson. This enhancement captures PM escape phenomena, such as a passage through interconnected large pores in filter walls. Simulations using a 1D + 1D two-channel framework with the proposed model successfully reproduced experimental results of SPN10 and SPN23 emissions downstream of a miniature gasoline particulate filter (GPF) tested with a synthetic particle generator. The model was also able to represent the observed continuous emissions during a cake filtration mode. Additional simulations using the same model parameters showed good agreement with experimental data of SPN10 and SPN23 emissions downstream of a full-size GPF tested with a gasoline direct injection (G-DI) engine under 5 steady-state operating conditions. The simulations revealed that particles in the 10–100 nm size range dominated the downstream SPN emissions despite their high filtration efficiency, whereas particles in the 100–200 nm size range were less significant. The proposed model is expected to contribute to the GPF developments to comply with the stringent emission regulations of the upcoming Euro 7.

1. Introduction

Ambient air quality is a critical factor for public health. The industrial revolution, particularly in the 19th century, significantly deteriorated air quality through the emissions of primary pollutants [1]. A notable example is the Great Smog of London in 1952, which resulted in approximately 12,000 excess deaths over two weeks due to severe air pollution [2]. The term “smog”, a contraction of “smoke” and “fog”, refers to the mixture of particulate matter (PM) and sulfur oxides produced primarily by coal and fossil fuel combustion [3]. This event highlighted the urgent need for air quality regulation, ultimately leading to the UK Clean Air Act in 1956. Subsequent research has shown that PM was a complex mixture of volatile and non-volatile components varying in size and chemical composition [4]. PM exposure is linked to cardiovascular and respiratory diseases [5] and has been classified as a Group 1 carcinogen by the World Health Organization (WHO). Consequently, the ambient air quality standards of PM₁₀ and PM_2.5 (particles in diameters less than 10 and 2.5 μm, respectively) have been established and are now widely monitored in each area. Furthermore, studies report that ultrafine particles can deposit in the lungs and penetrate the bloodstream, which leads to premature death [6,7]. Thus, controlling not only PM₁₀ and PM_2.5 but also ultrafine particles remains a major challenge for public health.

Fuel-powered engines and vehicles are considered to be one of the major sources of PM in the atmosphere [8]. In response, regulatory limits on PM emission from vehicle tailpipes have been introduced and progressively tightened over recent decades. Type-approval emission testing is conducted using a chassis dynamometer under defined driving cycles. The tailpipe exhaust is diluted, and PM in the sample gas is collected with filters, which are subsequently analyzed gravimetrically to mimic the PM evaluation in the atmosphere. The European Commission established the first PM emission regulation for diesel light-duty (LD) vehicles in 1992, with a limit of 140 mg/km under the urban driving cycle (UDC) [9]. The limit has been reduced to 4.5 mg/km under the worldwide harmonized light vehicles test cycle (WLTC), which better represents real driving conditions and is more stringent from a viewpoint of emission evaluation. For gasoline direct injection (G-DI) LD vehicles, PM regulations came into force in 2009 with a limit of 5 mg/km [10] and further reduced to 4.5 mg/km in 2011 [11].

As tailpipe PM emissions have decreased due to advancements in emission control technologies, such as diesel particulate filters (DPFs), a more accurate measurement methodology had been desired to quantify low particle concentrations near ambient air levels. In this circumstance, the Particle Measurement Programme (PMP) was founded under the United Nations Economic Commission for Europe (UNECE). It established a solid particle number (SPN) measurement method which is capable of detecting such low concentrations [12]. The SPN measurement system is required to equip a primary particle number diluter (PND1) heated to 150–200 °C and an evaporation tube (ET) or a catalytic stripper (CS) heated to 300–400 °C, in order to remove volatile components that can form a large number of particles and deteriorate measurement repeatability. A particle number counter (PNC), based on a condensation particle counter (CPC) principle, downstream of a second particle number diluter (PND2) counts particles, which is regarded as SPN, as the volatile components are eliminated. The PNC is calibrated to detect particles larger than 23 nm in diameter to ensure measurement repeatability. Hereinafter, the SPN larger than 23 nm in diameter measured by this method is referred to SPN23 in this paper.

In parallel with the PM mass regulations, SPN23 limits were first introduced for diesel LD vehicles in 2011 at 6 × 10¹¹ particles/km [11] followed by heavy-duty (HD) engines in 2013 [13], G-DI LD vehicles in 2014 [14], and non-road mobile machinery (NRMM) engines in 2019 [15]. In 2016, the SPN23 regulation was extended to real driving emissions (RDE) using a portable emissions measurement system (PEMS), which covers a broader range of conditions beyond laboratory testing [16,17,18]. Subsequent studies on SPN measurement technologies lowering its detection limit of particle diameter from 23 to 10 nm (hereinafter, SPN10) revealed that particles smaller than 23 nm are emitted from modern vehicles [19,20,21,22]. Research has shown that SPN10 emissions from the G-DI LD vehicles are approximately 50% higher than SPN23 levels [21,22]. Moreover, significantly increased PM emissions from G-DI vehicles have been observed during cold starts in a low ambient temperature [23,24]. The European Commission decided to introduce SPN10 regulations under the upcoming Euro 7 standard as a precautionary measure for public health. These will be more stringent than SPN23 regulations, as they additionally include particles in diameter between 10 and 23 nm [25].

In order to meet increasingly severe legislative requirements, DPFs and gasoline particulate filters (GPFs) have become indispensable as emission control technologies. Their developments have progressed since the 1980s, with substantial efforts devoted to particle filtration modeling, behavior of ceramic honeycomb-structure wall-flow particulate filters (PFs), and their optimal operational strategies in vehicles [26,27]. Filtration models have incorporated various factors, including PM and ash distributions inside the PF, the oxidation of captured PM, and the effects of catalytic reactions [28,29,30,31,32]. High-fidelity implementations such as three-dimensional computational fluid dynamics (CFD) and lattice Boltzmann simulations have been found [33]. On the other hand, computationally low-cost and simplified models can be utilized to express the filtration in an adequate manner. For example, Nicolin et al. [34] developed a zero-dimensional soot oxidation model, and it was in good agreement with experimental results. While simplified models reduce computational demands, they require increased parameters and calibrations to represent PF characteristics. A commonly accepted compromise between accuracy and computational costs is the use of one-dimensional simulations. They provide time-resolved analyses [35] and can be extended to include phenomena such as ash transport, along with PM [36].

A widely adopted filtration model was introduced by Konstandopoulos and Johnson [32], who proposed a so-called unit collector model to simulate particle filtration within the porous walls of the PF. This model assumes a uniform wall structure with a single pore diameter and porosity for each discrete volume. However, real PF walls exhibit a broad pore size distribution, and the limitations of the unit collector model are related to this heterogeneity. For instance, large pores may clog more slowly than small ones or even remain open, leading to a persistent slip of the particles through interconnected pathways. GPFs for the application to G-DI engines have been developed, and their characteristics can be summarized as thin walls as well as high porosity and large pore sizes in the porous walls, to minimize their pressure drops due to back pressure constraints in the G-DI engines [37,38,39,40,41]. In general, the GPFs exhibit low filtration efficiency because of a trade-off relationship with the low pressure drops. Accurate prediction of the persistent slip is particularly important for the GPF applications.

Although PM emissions from G-DI engines are generally low, SPN10 ones are high enough to exceed the regulatory limits. PM deposition within the PF increases filtration efficiency as PM initially accumulates in the walls and then forms cake layers on top of them. The former and the latter are regarded as deep-bed and cake filtration modes, respectively. In typical G-DI vehicle operations, GPF regeneration is performed passively through frequent fuel cut-offs, which removes deposited PM that contributes to the filtration. As a result, GPFs often operate in a clean or nearly clean state, despite their initially low filtration efficiency. Advanced modeling approaches that consider pore structure heterogeneity have been proposed [42,43,44,45,46,47]. However, these models require extensive pore structure information or parameter calibrations, resulting in increases in computational costs and efforts. Such requirements can hinder software-in-the-loop or hardware-in-the-loop applications aimed at developments for RDE legislation where various hardware designs and control unit calibrations are necessary. Although these advanced models have been validated in terms of GPF filtration efficiencies, they do not provide detailed analyses of SPN10 and SPN23 emissions downstream of the GPF, which represent current and future emission limits for type-approval testing.

The objective of this study is to develop a computationally efficient filtration model for PFs that can reproduce the persistent slip of PM. First, a previously reported experiment by Nakamura et al. [48], which characterizes SPN10 and SPN23 emissions downstream of a miniature GPF using a particle generator, is briefly reviewed. Second, a new filtration model is proposed with emphasis on predicting PM slip from the PF. The model introduces a persistent slip factor and a linear filtration efficiency of cake layers to reduce computational costs while maintaining accuracy. Third, simulation results by a 1D + 1D two-channel framework are presented to evaluate the capability of the model to reproduce the characteristics of SPN10 and SPN23 emissions observed in the miniature GPF experiment. Finally, the model is validated against experimental data from full-size GPF testing on a G-DI engine focusing on SPN10 and SPN23 emissions. Characteristics of the SPN emissions, as well as deposited PM distributions inside the GPFs, are also analyzed by the simulations in detail. The proposed model is expected to provide accurate predictions of SPN emissions and support the developments of optimized GPF operation strategies, in line with the forthcoming Euro 7.

2. Materials and Methods

2.1. Miniature GPF Testing with a Particle Generator

This subsection summarizes an experiment for characterizations of SPN10 and SPN23 emissions downstream of a miniature GPF test piece (

Table 1. GPF specifications [48].

Material	Cordierite
Catalyst	Uncoated
Cell structure	Square
Cell density	0.307 1/mm2 (198 1/inch2)
Inlet/outlet channel width	1.592 mm
Wall thickness	213 μm (8.4 mil)
Wall porosity	0.542
Wall pore diameter	7.66 μm

) in diameter and length of 1 inch, as reported by Nakamura et al. [48]. A combustion aerosol standard (CAST) manufactured by Matter Engineering AG, which is based on a laminar co-flow diffusion flame, was used to generate synthetic soot particles. These soot agglomerates, produced via propane oxidation, closely resemble PM emitted by internal combustion engines in both morphology and chemical composition. The synthetic particles were introduced into the GPF along with ambient air using a pump at a constant flow rate. The resulting pressure drop across the GPF was measured, and soot mass, SPN10, and SPN23 concentrations were monitored at both upstream and downstream locations. Soot mass concentrations were measured using photoacoustic sensor systems (AVL (Graz, Austria) micro soot sensor 2: MSS2 model 497) equipped before and after the GPF. MSS2 has a detection limit of 0.001 mg/m³, and their sensor signal amplitude is linearly proportional to soot mass independent of particle size and morphology with minimal interferences of the other components such as hydrocarbons [49]. Validation of the MSS2 measurements was demonstrated by Nakamura et al. [48], showing a linear correlation between sensor signals and the mass of elemental carbon and non-volatile particles emitted from CAST and diesel engines, respectively. Simultaneous SPN10 and SPN23 measurements were conducted using particle number counting systems (AVL particle counter: APC model 489) [50], upstream and downstream of the GPF. These systems comply with the technical requirements of UNECE Global Technical Regulation No. 15 (GTR15)—Amendment 6 [51]. They also include dual AVL CPCs, which are calibrated for SPN10 and SPN23 measurements, enabling their concurrent measurements. An additional advantage of the simultaneous measurements is the ability to provide a piece of information about a particle size distribution expressed as a fraction of particles in diameters between 10 and 23 nm in the total SPN emission. It was used to define one of the initial conditions in simulations described later. The volatile particle remover (VPR) consisting of PND1, CS, and PND2 was also calibrated. The obtaining particle number concentration reduction factors (PCRFs) which account for both dilution ratios and particle losses inside the systems were applied so that the SPN measurements were implemented within a single count mode range of AVL CPCs.

2.2. A New Filtration Model of a Particulate Filter

Exhaust containing PM first enters inlet channels of a honeycomb-structured PF, passes through the porous walls, flows into outlet channels, and ultimately exits the PF. PM is captured either within the porous walls or by cake layers formed on top of them while the remaining PM leaves the PF. The PM deposition within the walls is referred to as deep-bed filtration, whereas cake filtration describes the accumulation of PM on top of the walls after PM fills pores in the walls during the deep-bed filtration [52].

A fundamental of the PF model employed in this study is based on the work by Wurzenberger et al. [35], which describes a one-dimension + one-dimension (1D + 1D) two-channel framework. In this approach, two orthogonal coordinates are defined: the longitudinal direction along the inlet and outlet channels (denoted as z) and the wall-thickness direction across the porous wall (denoted as x) (Figure 1). The model solves mass, momentum, and energy balance equations for flow and PM transport across the inlet channel, porous wall, and outlet channel. It also introduces PM classes. Each of them has individual characteristics such as particle diameter and number concentration.

The transient PM mass conservation equations in the inlet and outlet channels as well as the wall are described by:

$$\begin{gathered} {\displaystyle \frac{\mathsf{\partial }\left(m_{PMj,in}\right)}{\mathsf{\partial }t}}=-{\displaystyle \frac{\mathsf{\partial }\left({\dot{m}}_{PMj,in}\right)}{\mathsf{\partial }z}}·dz-{\dot{m}}_{PMj,in,s} \\ {\displaystyle \frac{\mathsf{\partial }\left(m_{PMj,wall}\right)}{\mathsf{\partial }t}}=-{\displaystyle \frac{\mathsf{\partial }\left({\dot{m}}_{PMj,wall}\right)}{\mathsf{\partial }x}}·dx-{\dot{m}}_{PMj,wall,n}·E_{dep,j} \\ {\displaystyle \frac{\mathsf{\partial }\left(m_{PMj,out}\right)}{\mathsf{\partial }t}}=-{\displaystyle \frac{\mathsf{\partial }\left({\dot{m}}_{PMj,out}\right)}{\mathsf{\partial }z}}·dz+{\dot{m}}_{PMj,out,n} \end{gathered}$$

(1)

where m_PMj,in, m_PMj,wall, and m_PMj,out denote PM mass of the class j in the inlet channel, the wall, and the outlet channel, respectively. The terms of −$m·$_PMj,in,s, −$m·$_PMj,wall,n, and +$m·$_PMj,out,n represent PM mass fluxes of the class j exiting the inlet channel at its south side boundary, exiting the wall at its north side boundary, and entering the outlet channel at its north side boundary, respectively. The wall is modeled using the unit collector model [32], which assumes a structure composed of spherical collectors of uniform diameter arranged in slabs. These slabs are distributed in the x-direction with number l and thickness ∆x as illustrated in Figure 2. PM is deposited on the spherical collectors, and the collectors grow in dependence of the deposited PM mass, leading to an increase in filtration efficiency.

The deposition rate is governed by the overall filtration efficiency, E_dep,j, which is defined as:

$$E_{dep,j}=1-\mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{3}{2}}·\left(1-{\epsilon }_{wall}\right)·{\displaystyle \frac{{\eta }_{DR,j}}{d_c}}·{\displaystyle \frac{U_{\infty }}{u_{gw,sf}}}·∆x$$

(2)

where ε_wall, η_DR,j, d_c, U_∞, and u_gw,sf, are the wall porosity, the combined collection efficiency of the PM class j, the collector diameter, the collector approach velocity, and the superficial wall velocity, respectively. The combined collection efficiency η_DR,j for the PM class j is dominated by Brownian diffusion and interception mechanisms, and is expressed as:

$${\eta }_{DR,j}={\eta }_{D,j}+{\eta }_{R,j}-{\eta }_{D,j}·{\eta }_{R,j}$$

(3)

where η_D,j and η_R,j, are the collection efficiencies due to diffusion and interception, respectively. They are defined as:

$$\begin{gathered} {\eta }_{D,j}=C_D·{{Pe}_j}^{-2/3} \\ {\eta }_{R,j}=C_{R,A}·N_{R,j}+C_{R,B}·{N_{R,j}}^2 \end{gathered}$$

(4)

where Pe_j is the Peclet number for the PM class j in each wall slab, and N_R,j is the ratio of the particle diameter of the PM class j to the collector diameter. The constants C_D, C_R,A, and C_R,B are empirical coefficients. The collector diameter d_c of each slab grows as PM accumulates, and is given by:

$$d_c=2·{\left[{\displaystyle \frac{3}{4·\pi }}·{\displaystyle \frac{m_{PM,wall}}{{\rho }_{PM,wall}}}+{\left({\displaystyle \frac{d_{c,i}}{2}}\right)}^3\right]}^{1/3}$$

(5)

where m_PM,wall, ρ_PM,wall, and d_c,i are the PM mass captured in each slab, the density of the deposited PM, and the initial collector diameter, respectively. The initial collector diameter can geometrically be expressed by measurable wall porosity ε_wall,i and pore diameter d_pore,i in a clean state as following.

$$d_{c,i}={\displaystyle \frac{3}{2}}·{\displaystyle \frac{1-{\epsilon }_{wall,i}}{{\epsilon }_{wall,i}}}·d_{pore,i}$$

(6)

The transition from deep-bed to cake filtration is modeled using a partition factor Φ, which is a fraction of available free flow areas to maximum ones in the wall:

$$\mathsf{\Phi }={\displaystyle \frac{{d_c}^2-{d_{c,i}}^2}{{\left(\psi ·b\right)}^2-{d_{c,i}}^2}}$$

(7)

where b is the collector diameter, and it is scaled by a percolation factor ψ.

As described in Equations (5) and (6), the unit collector model used in PF simulations assumes a uniform wall structure characterized by a single collector diameter. However, real PF walls exhibit an inherently heterogeneous pore structure. Consequently, PM captures depend not only on diffusion and interception mechanisms, but also on the local pore size distribution within the walls. In particular, large pores may form interconnected networks that allow PM to bypass depositions, even when cake filtration appears to occur macroscopically. This phenomenon is frequently observed in PFs with high porosity, large pore diameter, and thin walls, which are typical designs for G-DI applications due to their low pressure drop requirements.

This study proposes a modified filtration model to account for such behavior by introducing a persistent slip factor ξ and a linear filtration efficiency of cake layers γ into the unit collector framework. The slip factor ξ represents a fraction of wall area, through which PM bypasses entirely, expressing analogous to flow paths formed by interconnected large pores. As a result, three pathways for incoming PM are considered: (i) deep-bed filtration, (ii) cake filtration, and (iii) persistent slip through the porous walls (Figure 2).

The fraction of PM captured by the cake layers is expressed by the modified partition factor Φ_m as following.

$${\mathsf{\Phi }}_m=\left(1-\xi \right)·{\displaystyle \frac{{d_c}^2-{d_{c,i}}^2}{{\left(\psi ·b\right)}^2-{d_{c,i}}^2}}$$

(8)

The fractions of PM deposited within the walls and persistently slipped are denoted as ζ_wall and ζ_slip, respectively, as follows.

$$\begin{gathered} {\zeta }_{wall}={\displaystyle \frac{\left(1-\xi \right)·\left(1-{\mathsf{\Phi }}_{\mathrm{m}}\right)}{1-\left(1-\xi \right)·{\mathsf{\Phi }}_{\mathrm{m}}}} \\ {\zeta }_{slip}={\displaystyle \frac{\xi }{1-\left(1-\xi \right)·{\mathsf{\Phi }}_{\mathrm{m}}}} \end{gathered}$$

(9)

The linear filtration efficiency γ is defined as a function of the PM mass accumulated in the cake layers [53]:

$$\gamma ={\displaystyle \frac{m_{PM,cake}}{q}}$$

(10)

where m_PM,cake is the captured PM mass in the cake layers, and q is the threshold mass corresponding to 100% filtration efficiency. Combining γ with Φ_m, the updated partition factor Φ_u is expressed as following.

$${\mathsf{\Phi }}_u={\mathsf{\Phi }}_m+\left(1-{\mathsf{\Phi }}_m\right)·\gamma$$

(11)

The new filtration model describes PM mass flow through the PF using a 1D + 1D two-channel framework in both the longitudinal direction z and the wall-thickness direction x, divided into l slabs (Figure 1). Within each slab, PM is partially captured based on the macroscopic filtration efficiency E_dep,j contributing to collector growth. The remaining PM flows to the next slab, and a portion may eventually leave the wall. Once the pores are filled, PM deposits on top of the wall, forming the cale layer with its fraction determined by the updated partition factor Φ_u. A fixed fraction of PM bypasses the walls entirely from the beginning as the persistent slip. Together with PM escaped from the wall, it flows to the outlet channel and exits the PF, which can be observable as tailpipe emissions.

The proposed model was implemented in AVL CRUISE M, a vehicle system simulation platform. Numerical calculations based on the 1D + 1D two-channel framework with the proposed model were conducted under experimental conditions to verify and validate the simulation results.

2.3. Full-Size GPF Testing with a G-DI Engine

Experiments were conducted using a full-size GPF with a diameter of 5.66 inches and a length of 6 inches installed in an engine test bench with a three-cylinder and 1.0 liter G-DI engine. The SPN10 and SPN23 emissions downstream of the GPF were measured under various operating conditions. The experimental setup is illustrated in Figure 3. The tested GPF had the same design as the miniature GPF previously characterized with the particle generator in Section 2.1., except for its dimensions (Table 1). A three-way catalyst (TWC) was positioned in a close-coupled location while the GPF was deployed at an under-floor position downstream of the TWC. To evaluate filtration performance, two units of APC and MSS2 were installed upstream and downstream of the GPF. This configuration enabled simultaneous measurements of SPN10 and SPN23 concentrations using the APC and soot mass concentrations with the MSS2 in the exhaust gas, thereby allowing for comprehensive evaluation of the filtration efficiency of the GPF.

It is well established that particle losses during transport in carrier gas can occur due to mechanisms such as electrophoresis, thermophoresis, diffusion, gravitational settling, and inertia deposition [54]. In order to ensure repeatability and reproducibility in particle measurements, the experimental layout must be configured to minimize such losses during exhaust and sample gas transport.

In this study, sample gas was extracted using probes installed at locations where the distance from the nearest pipe bend and between probes exceeded more than 5 × and 3 × inner pipe diameter, respectively, to ensure fully developed exhaust gas flow. The probes were oriented upward to allow condensed water to drain back into the pipe to minimize the particle losses due to moisture accumulation. A PND1 of the APC was installed immediately downstream of the probe, connected via a heated 2 m sampling line maintained at 200 °C. This setup aligns with the direct tailpipe SPN measurement configuration recommended for HD engine testing by Giechaskiel et al. [55]. A subsequent CS was maintained at 350 °C to remove volatile particles from the sample gas. PCRFs were selected to ensure that measured particle number concentrations by AVL CPCs downstream of the CS and a PND2 remained within a single count mode range (up to 30,000 particles/cm³) through the experiments. Both VPR and AVL CPCs of the APC were calibrated in accordance with technical requirements specified by UNECE GTR15 [51].

For the soot mass measurements using the MSS2, a heated dilutor operating at 120 °C was integrated immediately downstream of the probe. A dilution ratio of 10:1 was applied to prevent condensation in the following sampling line, which was heated at 52 °C. Thermophoretic losses within the probes, which are caused by temperature gradients and particles moving from the center toward the probe walls, were compensated using correction factors based on the temperature difference between exhaust gas and dilutor in accordance with ISO 8178-1 Annex C.1 [56]. The MSS2 sensors were calibrated against EC content of particles generated by a CAST source and analyzed using a thermal and optical transmittance (TOT) method. Span checks were implemented before the experiments using an absorber window that produces a specific acoustic resonance in the measurement cell of the sensor. The resonance is created by periodic irradiation of an infrared-red laser and was detected by a microphone to verify sensor responses.

Finally, exhaust mass flow rates were calculated from average fuel flow rates and an air-fuel ratio. The fuel flow rates were measured using a fuel flow meter (model: FP-214H) manufactured by Ono Sokki Co., Ltd. (Yokohama, Japan) for 30 s after the engine was stabilized at each operating condition for the filtration performance testing.

At the beginning of the experiments, the engine was stabilized to ensure repeatability of the test results. Prior to testing, the GPF was regenerated by operating the G-DI engine at a speed of 3000 rpm and torque of 100 Nm to elevate the GPF temperature. Additionally, compressed air was injected upstream of the GPF to increase the oxygen concentration for 15 min. After the regeneration process, the filtration performance testing was conducted under steady-state conditions. Following 5 discrete operating points within the typical operating range of the engine were selected in order to vary space velocity (SV), inlet gas temperature, and PM emission, which directly influence the filtration performance:

• 3000 rpm/100 Nm
• 3000 rpm/50 Nm
• 2000 rpm/100 Nm
• 2000 rpm/50 Nm
• 1000 rpm/50 Nm

3. Results

This study conducted one experiment using the miniature GPF test piece with CAST and five evaluations using the full-size GPF with the G-DI engine.

Table 2. Measured quantities at the GPF inlet.

Experimental Conditions	Space Velocity (/h)	Gas Temperature (°C)	Soot Mass (μg/s)	SPN10 (Particles/s)	SPN23 (Particles/s)	Fraction of SPN10-SPN23
CAST	83,630	R.T.	0.789	2.521 × 1010	1.368 × 1010	0.46
3000 rpm/100 Nm	146,961	750	5.098	3.413 × 1010	2.062 × 1010	0.40
3000 rpm/50 Nm	64,122	622	7.878	3.293 × 1010	2.249 × 1010	0.32
2000 rpm/100 Nm	84,689	654	7.198	3.656 × 1010	2.328 × 1010	0.36
2000 rpm/50 Nm	34,538	518	4.297	1.385 × 1010	0.987 × 1010	0.29
1000 rpm/50 Nm	16,024	389	3.870	1.586 × 1010	1.037 × 1010	0.35

summarizes the measured quantities at the GPF inlet for each experiment. All values were time-averaged under steady state conditions.

The SV in the CAST experiment was 83,630 1/h, which is a typical operation condition of the GPF and comparable to that of the engine test at 2000 rpm/100 Nm. Furthermore, the G-DI engine experiments covered broad ranges of SV (16,024–146,961 1/h) and exhaust gas temperatures (389–750 °C) as well as SPN10 and SPN23 emissions (1.385–3.656 × 10¹⁰ and 0.987–2.328 × 10¹⁰ particles/s, respectively). The CAST generated particles with high stability, showing the relative deviation of 4% in the SPN10 emission during the experiment. In contrast, the G-DI engine exhibited greater deviations of 8–23%.

The CAST soot mass emission rate was 0.789 μg/s, which is approximately 4.9 to 9.9 times lower than that of the G-DI engine (3.870–7.878 μg/s). However, the difference in SPN emissions between CAST and G-DI engine was less pronounced. The SPN10 and SPN23 emissions from CAST were 2.521 × 10¹⁰ and 1.368 × 10¹⁰ particles/s, respectively, while those from the G-DI engine ranged 1.385–3.656 × 10¹⁰ and 0.987–2.328 × 10¹⁰ particles/s, respectively. Notably, the SPN emissions at 2000 rpm/50 Nm and 1000 rpm/50 Nm were even lower than those from CAST despite the fact that the soot mass emissions were opposingly 5.4 and 4.9 times higher, respectively.

The fraction of “SPN10-SPN23” in total particle number for the G-DI experiments ranged from 0.29 to 0.40, whereas the corresponding fraction for CAST was 0.46. This means that the particles from CAST were the smallest among all the experiments. These characteristics were reflected in the reconstructed particle size distributions at the GPF inlet.

An additional advantage of simultaneous measurements of SPN10 and SPN23 is the ability to extract information about particle size distributions. In this study, particle size distributions upstream of the GPF were reconstructed using the measured fraction of “SPN10-SPN23” relative to the total SPN emission. A log-normal distribution was assumed with a fixed geometric standard deviation of 1.85, which falls within the reported range (1.7–2.0) for G-DI engine emissions in the literature [21,22,46]. Based on this assumption, the particle size distributions at the GPF inlet were calculated for each experiment so that the fraction of “SPN10-SPN23” matched the corresponding measurement. Figure 4 presents the calculated particle size distributions. Count median diameters (CMDs) of the G-DI engine emissions ranged from 25 to 31 nm whereas that of CAST was 22 nm, which was the smallest among all the experiments. These reconstructed particle size distributions were subsequently used as input data for the simulation analyses.

3.1. Pressure Drop of the Miniature GPF Test Piece in a Clean State

In order to characterize pressure drops across a GPF, relative pressures at the inlet and outlet of a miniature GPF test piece were measured as a function of ambient airflow rate without PM loading. The airflow was introduced using the same experimental setup employed for the testing with CAST. The corresponding space velocity ranged from 44,000 to 167,000 1/h, with flow rate increments applied stepwise. The pressure drops increased from approximately 76 to 311 Pa across this space velocity range. Simulations of the pressure drops were also performed using 1D + 1D two-channel framework in AVL CRUISE M. A wall permeability of 7.2 × 10⁻¹³ m² and contraction/expansion inertial loss coefficients of 1.0 were applied under the same experimental conditions. The comparison between experimental and simulated results is illustrated in Figure 5, showing good agreement with each other. Within the tested range of space velocities, the pressure drops were predominantly attributed to flow resistance through the clean porous walls. The inertial losses can be negligible, as the relationship between space velocities and pressure drops was linear, which is also discussed in [48].

3.2. Emissions Downstream of the Miniature GPF Test Piece with CAST

Figure 6 depicts comparisons between experimental and simulated results of SPN10 and SPN23 emissions downstream of the miniature GPF test piece, using CAST as well as associated pressure drops. Sharp drops in SPN10 and SPN23 emissions downstream of the GPF, accompanied by a steep rise in pressure drop in the beginning of the experiment, indicated a transition from deep-bed to cake filtration [47,48,49] as discussed in [48]. Additionally, a small quantity of particles continued to pass through the GPF throughout the experiment, and the fraction of particles between 10 and 23 nm in diameter (“SPN10-SPN23”) remained nearly constant.

In the experiment, SPN10 and SPN23 emissions behind the GPF rapidly decreased from 4.971 × 10⁹ and 4.589 × 10⁹ particles/s to 3.738 × 10⁸ and 2.571 × 10⁸ particles/s within 200 s, respectively. Then, they further declined to 1.021 × 10⁸ and 6.172 × 10⁷ particles/s, respectively, towards the end of the experiment. The fraction of “SPN10-SPN23” remained stable, increasing slightly from 0.31 to 0.40 over time. The initial pressure drop was 146 Pa, rising steeply around 200 s, followed by a linear increase towards the end of the experiment.

The simulation results using the 1D + 1D two-channel framework with the proposed filtration model reproduced these trends well in Figure 6. Parameters used in the simulation are shown in

Table 3. Parameters used for simulations in 1D + 1D two-channel framework with the proposed filtration model.

Diffusion efficiency coefficient CD	0.55
Interception efficiency coefficient CR,A	3.0
Interception efficiency coefficient CR,B	1 × 10−7
Slip factor ξ	0.0115
PM mass threshold for attaining 100% filtration efficiency of cake layers q	0.85 kg/m3
Percolation factor ψ	0.96

. In the simulation, the initial SPN10 and SPN23 emissions were 5.303 × 10⁹ and 4.115 × 10⁹ particles/s decreasing to 3.400 × 10⁸ and 2.145 × 10⁸ particles/s at 200 s, respectively. The initial peaks matched the corresponding experimental results in −7% and 10%, respectively, and the differences at 200 s were 9% and 17%, respectively. The observed continuous escapes of the particles in the experiment were successfully expressed by the simulation. The simulated “SPN10-SPN23” fraction stayed at a constant value of 0.43, differing by only 0.03 from the experimental result. The simulated pressure drop began at 141 Pa and reached 974 Pa, showing two phases: an initial steep rise and a subsequent linear increase, which are both consistent with experimental behavior.

3.3. Emissions Downstream of the Full-Size GPF with G-DI Engine

Figure 7 compares the experimental and simulated SPN10 and SPN23 emissions downstream of the full-size GPF under various G-DI engine operating conditions: 3000 rpm/100 Nm, 3000 rpm/50 Nm, 2000 rpm/100 Nm, 2000 rpm/50 Nm, and 1000 rpm/50 Nm. The inlet SPN10 and SPN23 emissions for each condition are also shown as references. Additional parameters including space velocity, gas temperature, and soot mass emissions at the GPF inlet are listed in Table 2 and the simulations were conducted using the parameters shown in Table 3. Soot mass emissions downstream of the GPF were below the detection limit of MSS2 and are omitted from Figure 7.

Before filtration performance testing, the GPF was regenerated by operating the G-DI engine at 3000 rpm/100 Nm and injecting compressed air upstream of the GPF for 15 min. During this process, exhaust gas temperature and oxygen concentration reached 709 °C and 1.5%, respectively. Consequently, the SPN10 and SPN23 emissions at the GPF outlet rose from 4.113 × 10⁷ and 2.496 × 10⁷ particles/s to 3.682 × 10⁹ and 2.878 × 10⁹ particles/s, respectively, and remained stable for approximately 10 min. It is indicated that PM deposited in the GPF were oxidized completely. Following the regeneration, the compressed air supply was stopped, and the filtration performance was evaluated under the specified engine conditions, assuming the GPF was in a clean state.

The simulated SPN10 and SPN23 emissions downstream of the GPF showed good agreement with experimental data. Averaged values over an eight min duration at each operating point are plotted in Figure 7. In simulations following the initial test at 3000 rpm/50 Nm, the PM mass accumulated in prior conditions was used as the initial loading state. Across conditions of 3000 rpm/100 Nm, 3000 rpm/50 Nm, and 2000 rpm/100 Nm, the experimental SPN10 and SPN23 emissions ranged from 2.342 to 3.509 × 10⁹ and 2.121–3.040 × 10⁹ particles/s, respectively, closely matching the simulated values of 2.012–3.343 × 10⁹ and 1.895–3.077 × 10⁹ particles/s. Similarly, under 2000 rpm/50 Nm and 1000 rpm/50 Nm conditions, the experimental SPN10 and SPN23 emissions ranged from 4.218 to 5.936 × 10⁸ and 3.960–5.694 × 10⁸ particles/s, respectively, while the simulated values ranged from 3.294 to 5.229 × 10⁸ and 2.749–4.831 × 10⁸ particles/s, respectively. The relative errors between experimental and simulated results were −2 to 31% with the largest discrepancies observed at the condition of 1000 rpm/50 Nm. Specifically, the discrepancy at the condition of 1000 rpm/50 Nm was likely due to increased signal variability caused by exhaust pulsation effects. Since the exhaust flow rate was not measured in a time-resolved manner, these pulsations were not accounted for in the simulation, which resulted in large differences.

4. Discussion

Experimental and numerical investigations were conducted to validate the proposed filtration model by comparing SPN10 and SPN23 emissions downstream of a GPF under controlled conditions. Two test configurations were employed: a miniature GPF test piece (1 inch in diameter and length) tested with CAST and a full-size GPF (5.66 inches in diameter and 6 inches in length) tested with a G-DI engine. Corresponding simulations were carried out using the proposed filtration model in a 1D + 1D two-channel framework under identical conditions.

Prior to PM loading experiments, pressure drops across the miniature GPF with ambient air without PM were evaluated. Experiments were conducted over an SV range of 44,000 to 167,000 1/h. The pressure drop exhibited a linear relationship with SV, as shown in Figure 4, indicating that inertial losses could be negligible. Simulated results closely matched the experimental data when wall permeability and contraction/expansion inertial loss coefficients were set to 7.2 × 10⁻¹³ m² and 1.0, respectively. The adopted wall permeability value lies within the range of 6.9 ± 1.5 × 10⁻¹³ m² which was derived by Nakamura et al. [48] using an analytical approach based on flow resistance descriptors [57]. Therefore, the pressure drop simulations were valid, and the corresponding parameters were subsequently used in the PM loading simulations for the miniature GPF with CAST.

Simultaneous measurements of SPN10 and SPN23 provided insights into particle size distributions. In the exhaust upstream of the GPFs, the fraction of “SPN10-SPN23” to the total SPN emission ranged from 0.29 to 0.40 in the G-DI engine experiments and was 0.46 in the CAST experiment (Table 2). These values are comparable to the reported range of 0.35 to 0.50 for G-DI engine emissions in Giechaskiel et al. [58] which collected the data measured through a constant volume sampler (CVS) over a WLTC. The fractions observed at 3000 rpm/50 Nm and 2000 rpm/50 Nm (0.32 and 0.29, respectively) suggest relatively larger particles compared to the reported range, while the other conditions aligned well with expected G-DI emission characteristics. Particle size distributions were reconstructed assuming log-normal distributions with a fixed geometric standard deviation of 1.85, which is consistent with the literature-reported value, and used as input for the simulations (Figure 4). The CMDs ranged from 25 to 31 nm for the G-DI engine and 22 nm for the CAST, of which the latter is the smallest among all tested conditions.

Simulations of the miniature GPF testing using the proposed model demonstrated good agreement with experimental data. The model parameters listed in Table 3 were the results of the optimization to match the experimental SPN10 and SPN23 emissions. The simulation accurately reproduced the initial peaks of the emissions out of the GPF in a clean state, their subsequent decreases, and the slight escapes observed towards the end of the experiment. During the deep-bed filtration phase, PM removal was predominantly governed by Brownian diffusion and interception mechanisms in the walls. As the filtration transited to the cake regime (around 200 s), the filtration efficiency became high and a small amount of the particles passed through the GPF, which was successfully expressed in the simulation. The nearly constant value of “SPN10-SPN23” fraction over time suggests that persistent slip through interconnected large pores allowed smaller particles to bypass the GPF, because such a small particle in diameter of 10–23 nm is efficiently captured due to Brownian diffusion in theory. Additionally, the simulation accurately reflected the observed pressure drop behavior, including the initial steep rise followed by the gradual growth. Thus, the proposed model was successfully verified using the experimental result of the miniature GPF testing with CAST.

A parameter study was conducted to analyze the effects of newly introduced parameters in this research—a slip factor (ξ), a PM mass threshold for attaining 100% filtration efficiency of cake layers (q), and a percolation factor (ψ)—on the filtration behavior of the miniature GPF tested with CAST. Figure 8 depicts the SPN10 and SPN23 emissions downstream of the GPF under variation in these parameters. The results after 1000 s are excluded to improve graphical clarity, since the emission decline remained unchanged. The slip factor ξ which represents a fraction of persistent slip of PM reproduces the continuous leakage of PM from the GPF throughout the experiment (Figure 8(a-1,a-2)). The SPN emissions are particularly influenced by the parameter q, which governs the linear filtration efficiency of the cake layers once the cake filtration mode becomes dominant after approximately 200 s (Figure 8(b-1,b-2)). The percolation factor ψ, which determines the PM mass captured in the walls, results in the differences in the SPN emissions during the transition from deep-bed to cake filtration modes (0–400 s) (Figure 8(c-1,c-2)). Similar tendencies are observed between SPN10 and SPN23 emissions, despite variation in all three parameters (Figure 8). The combination of the parameters successfully reproduces the continuous PM leakage observed experimentally even after the cake filtration mode became dominant. Additionally, it is notable that the particles in diameter between 10 and 23 nm, which are captured by the wall with high filtration efficiency due to Brownian diffusion, remained downstream of the GPF as indicated by their nearly constant fraction to the total SPN emission at around 0.43. Consequently, the simulated SPN10 and SPN23 emissions reproduce the experimental results with the parameter set listed in Table 3.

The filtration behavior of the miniature GPF observed in the experiment using CAST as a synthetic particle source can represent its performance for the application to G-DI engines. CAST has been widely employed in fundamental studies of soot from diesel engines because of its similarities in morphology, chemical composition, and optical properties [48,49,59,60]. The calibration methodology of particle number-counting systems has been extensively discussed over the past decade, and the VPR and CPCs of APC used in the experiments were calibrated with particles generated by CAST [61]. Oo et al. [62] reported that morphology of PM emitted from G-DI engines was similar to that from diesel engines using TEM, although Wang et al. [63] noted that G-DI PM consisted of a more complex mixture of aggregates compared to diesel PM, depending on engine operating conditions. Despite morphological differences, it has been confirmed that filtration efficiency of GPFs was independent of the morphology but dependent on particle diameter and loaded PM mass inside a PF, due to the Brownian diffusion mechanism [64,65]. Thus, the proposed model verified by the miniature GPF experiment using CAST can be applicable to the ones with the G-DI engine.

Simulations of the full-size GPF under various G-DI engine operating conditions using the same model parameters (Table 3) further confirmed the applicability of the proposed model. The average SPN10 and SPN23 emissions downstream of the GPF showed good correlation with experimental data, with deviations ranging from −2% to 31% (Figure 7). These discrepancies may stem from uncertainties in exhaust flow rate measurements and PM loading during intervals of each filtration performance evaluation. The exhaust flow rates were calculated from average fuel flow rates measured for 30 s during each evaluation with an air-fuel ratio. PM was especially emitted at intervals of the evaluation when the engine operating condition changed. Due to the lack of transient exhaust flow rate data, the PM emissions during the intervals could not be included in the simulations, which led to the discrepancies. The largest error appeared at 1000 rpm/50 Nm likely due to exhaust flow pulsation, which was observed in the PM emission measurements. This effect was not considered in the simulations as well, due to the lack of transient exhaust flow rate measurements. Despite these limitations, the proposed filtration model effectively expressed the key features of GPF filtration performance and demonstrated its validity for G-DI engine applications.

The simulation framework enables detailed analyses of accumulated PM distributions inside the PF, contributions of each pressure drop component to the overall one, and particle number emissions of each particle size downstream of the PF. This capability supports a deeper understanding of filtration mechanisms and facilitates the optimization of GPF design and operational strategies.

The simulations provided quantitative distributions of PM deposited inside the GPF. Figure 9 illustrates the temporal evolution of PM mass deposited in the walls and of cake layers during the experiments with both miniature and full-size GPFs. For the miniature GPF, results beyond 420 s are omitted to enhance graphical clarity as the cake layer accumulation became linear in the following phase. The result of full-size GPF tested at the engine operation conditions of 2000 rpm/100 Nm is depicted for the comparison as the SV is close to each other. In the initial stages of the miniature GPF experiment, PM was primarily deposited within the walls exhibiting an exponential increase up to approximately 200 s attaining to approximately its mass of 12 g/m³. Simultaneously, the formation of cake layers was observed at the beginning of the experiment around 20 s with a quadratic increase in their mass during early stages, followed by a linear growth after roughly 100 s. In contrast, the full-size GPF experiment exhibited minimal PM accumulation with only a slight increase in the walls reaching its mass of only 1.6 g/m³ and the cake layers were negligible. These findings indicate that the miniature GPF underwent a transition from deep-bed to cake filtration modes, which the latter was dominated after approximately 200 s. Conversely, the full-size GPF operated in the deep-bed regime throughout the testing due to the lower PM emission at the GPF inlet compared to its overall filtration area.

Figure 10 illustrates the PM mass distributions within both miniature and full-size GPFs. To enhance graphical clarity during the transition from deep-bed to cake filtration modes observed in the miniature GPF testing, the results beyond 200 s are excluded. The result of full-size GPF tested at the engine operation conditions of 2000 rpm/100 Nm is chosen for the comparison, as the SV is similar to that of the miniature GPF experiment. In the miniature GPF, PM initially deposited in the walls uniformly along a longitudinal axis and increased exponentially to 200 s (Figure 10(a-1)). During the early phase, the PM distribution across the wall thickness was also homogeneous, but the deposition shifted towards the upper region of the walls from approximately 100 s onward (Figure 10(b-1)). Cake layer formation began at around 20 s and developed uniformly along the longitudinal direction (Figure 10(c-1)). In contrast, the full-size GPF showed a non-uniform PM distribution in the walls along the longitudinal axis with the deposition minimized at the center and maximized at both ends (Figure 10(a-2)). Additionally, PM accumulated preferentially in the upper part of the wall cross-section (Figure 10(b-2)). Although the amount of PM mass of cake layers was limited, its longitudinal distribution trend resembled that of PM deposited in the walls (Figure 10(c-2)).

Since PM mass distributions inside the GPFs are governed by exhaust flow dynamics, relative wall velocities along normalized longitudinal positions for both miniature and full-size GPFs at the beginning of each testing are analyzed by the simulations and shown in Figure 11. The operating condition of 2000 rpm/100 Nm for the G-DI engine is selected for the comparison. The relative wall velocity of the miniature GPF was nearly constant along its length whereas that of the full-size GPF exhibited a local minimum near the center increasing towards both inlet and outlet sides. The distribution in the full-size GPF aligns with the result reported by Konstandopoulos and Johnson [32], who also employed a 1D + 1D two-channel model with mass and momentum balance equations. This behavior can be explained by the flow dynamics inside the GPF. As the gas enters the inlet channels, a contraction effect leads to increased wall velocity at the inlet side. The flow continues along the inlet channels due to its inertia, while pressure decreases because of wall friction. Near the end of the inlet channels, the pressure begins to rise due to plugs at the outlet side forcing gas through the permeable walls into the outlet channels. Consequently, the wall velocity increases again towards the end of the inlet channels. In contrast, the miniature GPF whose length is one-sixth compared to the full-size one exhibited an almost uniform relative wall velocity due to the lower wall friction. These flow characteristics of both miniature and full-size GPF are clearly reproduced in the relative wall velocity profiles illustrated in Figure 11.

These trends are consistent with the PM distributions inside the GPFs shown in Figure 10. The miniature GPF demonstrated uniform PM mass distributions in the walls and of the cake layers along the longitudinal direction, which became more balanced across the wall thickness after the transition to the cake filtration mode (Figure 10(a-1,b-1,c-1)). Conversely, the full-size GPF exhibited the minimal PM depositions in the walls at the middle of the length during the deep-bed filtration through the experiment (Figure 10(a-2,b-2,c-2)). As a PM flux results from a flow rate and a PM concentration, the PM distributions along the longitudinal locations are dominated by the wall velocity profile governed by the flow dynamics. Thus, the clear differences in the PM distributions inside the miniature and full-size GPFs are explained by the flow dynamics resulting in the wall velocity profiles.

Figure 12 shows the breakdown of pressure drop components during the experiments of the miniature GPF with CAST and of the full-size GPF with the G-DI engine. It is categorized into contributions from clean walls, PM-loaded walls, cake layers, contraction/expansion at the GPF inlet/outlet, and frictions by inlet and outlet channels. For the comparison between miniature and full-size GPFs, the engine operation condition of 2000 rpm/100 Nm was selected as it provided SV comparable to that of the miniature GPF testing. In the miniature GPF, the pressure drop of the loaded walls was dominant in the beginning of the experiment, which increased sharply until approximately 200 s and stabilized at around 330 Pa. The contribution from the cake layers began at approximately 20 s and exhibited a linear increase throughout the experiment. The pressure drop of the clean walls remained constant at approximately 130 Pa from the start while the losses due to the contraction/expansion and the channel frictions were negligible over the testing. In contrast, the pressure drops of the loaded walls and the cake layers were negligible in the case of the full-size GPF, due to the limited PM loading. The pressure drop contributions from the frictions in the inlet and outlet channels were more pronounced. These characteristics align with the trends of the PM mass loading profiles in the walls and of the cake layers (Figure 9), the PM mass distributions in the walls and of the cake layers (Figure 10), and the flow dynamics inside the GPFs, described above, based on the wall velocity profiles along the longitudinal direction (Figure 11).

Analyzing the contributions of particles by diameter to the total SPN emissions in simulations offers valuable insights for not only the emissions themselves but also developments of GPF designs and their operational strategies. Figure 13 presents the SPN emissions for each particle diameter class upstream and downstream of the full-size GPF along with their corresponding filtration efficiencies at the end of the testing under the G-DI engine operating at 2000 rpm/100 Nm. The SPN emission at the GPF inlet was based on the particle size distribution reconstructed from the fraction of “SPN10-SPN23” to the total one as shown in Figure 4, which was used in the simulation as the input data. The measured SPN10 and SPN23 emissions at the GPF inlet were 3.287 × 10¹⁰ and 2.126 × 10¹⁰ particles/s, respectively, whereas those at the GPF outlet were 2.060 × 10⁹ a 1.853 × 10⁹ particles/s, respectively. The SPN distribution at the GPF inlet peaks at 3.018 × 10⁹ particles/s with a particle diameter of 21 nm. The emissions are reduced due to GPF filtration with a shifted peak of 3.907 × 10⁸ particles/s at a particle diameter of 39 nm downstream of the GPF. The emission of the particles larger than 100 nm ranges from 6.391 × 10⁵ to 3.850 × 10⁷ particles/s and contribute minimally to the total SPN emission. The total filtration efficiency is approximately 0.84 although it varies with the particle size. Higher filtration efficiencies are observed for smaller particles, reaching 0.99 for diameters up to 18 nm and declining with increasing diameter. The minimum filtration efficiency of 0.80 is seen at a particle diameter of 93 nm, and the absolute difference in the SPN emissions between inlet and outlet for these larger particles is small.

This size-dependent behavior is attributed to the dominant filtration mechanisms: Brownian diffusion is more effective for smaller particles, while interception becomes significant for larger particles as discussed in Nakamura et al. [48]. The overall filtration efficiency is the combined result of these mechanisms and reaches a minimum in the 100–200 nm diameter range where both effects are relatively weak. The observed shift in the SPN distribution peak from 21 nm at the GPF inlet to 39 nm at the outlet is due to the higher filtration efficiency for the smaller particles via Brownian diffusion. The deviation from perfect filtration (i.e., an efficiency of 1.00) in several 10 s nm diameter is attributed to the persistent slip factor introduced in the proposed filtration model. This factor accounts for the bypassing of particles through interconnected larger pores in the walls, resulting in an increase in the SPN emissions downstream of the GPF. In combination with the linear filtration efficiency of the cake layers, the proposed model successfully reproduces both SPN10 and SPN23 emissions downstream of the GPF, even though filtration efficiency attains nearly 1.00 in theory in the particle size range of 10–23 nm due to Brownian diffusion. It is noteworthy that particles within the 10–100 nm range remain the dominant contributions to downstream SPN emissions despite their high filtration efficiency. In contrast, emissions from the 100–200 nm range are less significant, although the filtration efficiency is lower. These findings demonstrate the capability of the proposed model to quantify the size-dependent contributions of particles to the SPN emissions downstream of the GPF, thereby enabling detailed analyses of the SPN emissions.

In the proposed model, the persistent slip factor and the linear filtration efficiency of cake layers have been incorporated into the unit collector framework. The model successfully reproduces particle emissions in the 10–23 nm diameter range downstream of the GPF, which are efficiently captured by the walls due to Brownian diffusion but were observed in the experiments as a fraction of “SPN10-SPN23” to the total SPN emission. For further validation of the PM leakage mechanism, detailed observations of PM pathways and cake layer growth using techniques such as scanning electron microscopy (SEM) in combination with filtration efficiency measurements may be necessary in future work.

In this study, it was confirmed that the proposed filtration model successfully reproduced the SPN emissions downstream of the miniature GPF with CAST under a controlled constant exhaust flow rate. Furthermore, the model was validated using the experiments with the full-size GPF on the G-DI engine under the steady-state conditions with the variations in the SV, gas temperature, and inlet SPN emissions. However, type-approval emission testing is conducted over a WLTC in the laboratory, and RDE testing is performed using a PEMS where exhaust conditions dynamically change during the test. Future work should validate the applicability of the proposed model to such dynamic transient modes, including the WLTC and RDE cycles, in order to fully support GPF developments for the upcoming Euro 7 target.

5. Conclusions

A new filtration model for a PF, which enables the accurate estimation of SPN emissions by representing PM slip through its walls, is proposed, and its verification and validation were conducted in this study.

The model introduces a persistent slip factor into an updated partition factor which determines a transition from deep-bed to cake filtration modes. As a result, three possible pathways for PM inside the PF are considered: a deposition in a wall, a capture by a cake layer, and a bypass through the wall. By integrating the linear filtration efficiency of the cake layer, the model represents PM escape phenomena like its flow-through from interconnected large pores in the walls. As a result, the model is capable of the accurate estimation of SPN emissions downstream of the PF.

Simulations using a 1D + 1D two-channel framework with the proposed model showed good agreement with the experimental data of a miniature GPF (one inch in diameter and length) testing with a synthetic particle generator (CAST). The simulations reproduced the initial peaks of SPN10 and SPN23 emissions downstream of the clean-state GPF within differences of −7% and 10% from the experimental results, respectively. Moreover, the simulations successfully demonstrated the observed subsequent decreases and continuous emission behavior in the cake filtration mode towards the end of the experiment. The simulated fraction of particles between 10 and 23 nm in diameter relative to the total SPN emission remained constant at 0.43, differing by only 0.03 from the experimental value.

Using the same model parameters, additional simulations were conducted for a full-size GPF (5.66 inches in diameter and 6 inches in length) tested with a G-DI engine under five steady-state operating conditions, varying SVs, exhaust gas temperatures, and SPN concentrations. The average SPN10 and SPN23 emissions that were downstream of the GPF from the simulations matched experimental results with deviations ranging from −2% to 31%. Although the discrepancies may be caused by the lack of transient exhaust flow rate measurements, the proposed filtration model demonstrated its validity for G-DI engine applications.

The simulations also revealed spatial distributions of deposited PM inside the GPF, contributions of each pressure drop component, and SPN emissions by particle size. In the miniature GPF testing with CAST, the PM deposition in the walls followed by the cake layer formation produced the observed temporal profiles of SPN10 and SPN23 emissions. In contrast, deep-bed filtration remained dominant in the full-size GPF testing with the G-DI engine, due to relatively low PM loading. The spatial distributions of deposited PM were influenced by flow dynamics inside the GPF: uniform and parabolic patterns along the longitudinal axis in the miniature and full-size GPFs, respectively, due to length differences. In the full-size GPF testing with the G-DI engine, particles between 10 and 100 nm dominated the downstream SPN emissions despite high filtration efficiency due to Brownian diffusion. In contrast, particles in the 100–200 nm range were less significant even though the filtration efficiency was low.

The proposed model has been verified using the miniature GPF with CAST and validated with the full-size GPF under the G-DI engine operations. It enables the accurate estimation of SPN emissions downstream of PFs and supports deeper insights into filtration mechanisms. The model is expected to contribute to the optimization of PF designs and operational strategies accelerating their developments to meet the stringent emission regulations of the forthcoming Euro 7.