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Dynamic Heterogeneous Multiscale Filtration Model: Probing Micro- and Macro-scopic Filtration Characteristics of Gasoline Particulate Filters Jian Gong, Sandeep Viswanathan, David Rothamer, David E Foster, and Christopher J. Rutland Environ. Sci. Technol., Just Accepted Manuscript • DOI: 10.1021/acs.est.7b02535 • Publication Date (Web): 31 Aug 2017 Downloaded from http://pubs.acs.org on September 2, 2017

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Environmental Science & Technology

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Dynamic Heterogeneous Multiscale Filtration

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Model: Probing Micro- and Macro-scopic Filtration

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Characteristics of Gasoline Particulate Filters

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Jian Gong*, †, Sandeep Viswanathan†, David A. Rothamer†, David E. Foster†, Christopher J.

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Rutland† †

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Engine Research Center, University of Wisconsin-Madison,1008 Engineering Research Building,1500 Engineering Drive, Madison, WI 53706, United States

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*

Corresponding author e-mail: [email protected]

Present address: 1900 McKinley Avenue, MC 50183, Columbus, IN 47201, USA

ACS Paragon Plus Environment

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ABSTRACT – Motivated by high filtration efficiency (mass- and number-based) and low

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pressure drop requirements for gasoline particulate filters (GPFs), a previously developed

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heterogeneous multiscale filtration (HMF) model is extended to simulate dynamic filtration

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characteristics of GPFs. This dynamic HMF model is based on a probability density function

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(PDF) description of the pore size distribution and classical filtration theory. The microstructure

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of the porous substrate in a GPF is resolved and included in the model. Fundamental particulate

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filtration experiments were conducted using an exhaust filtration analysis (EFA) system for

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model validation. The particulate in the filtration experiments was sampled from a spark-ignition

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direct-injection (SIDI) gasoline engine. With the dynamic HMF model, evolution of the

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microscopic characteristics of the substrate (pore size distribution, porosity, permeability, and

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deposited particulate inside the porous substrate) during filtration can be probed. Also, predicted

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macroscopic filtration characteristics including particle number concentration and normalized

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pressure drop show good agreement with the experimental data. The resulting dynamic HMF

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model can be used to study the dynamic particulate filtration process in GPFs with distinct

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microstructures, serving as a powerful tool for GPF design and optimization.

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TOC/Abstract art

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Introduction

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Diesel particulate filters (DPFs) have been widely and successfully applied on diesel engines

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as standard aftertreatment devices to control particulate emission since 2007.1 In the last decade,

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there has been significant progress on the development of spark-ignition direct-injection (SIDI)

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gasoline engines, which show benefits for fuel economy and CO2 emission reduction.2,3

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However, SIDI engines were found to generate more ultrafine particulate (90%).13,14 Furthermore, fuel economy and power of SIDI engines are more sensitive to the

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exhaust backpressure introduced by GPFs13. The need to achieve high filtration efficiency and

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low pressure drop poses a significant challenge for the development of advanced GPFs for

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particulate emission control. On the other hand, in addition to mass-based particulate emission

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regulations throughout the world, a particle number limit of 6 x 1011#/km is being enforced in

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Europe for all SIDI engines from 2017.15 These challenges and stringent particulate emission

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regulations require a fundamental understanding of particulate filtration process in GPFs, as well

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as, detailed mathematic models that assist GPF development.


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Particulate filtration is a dynamic process, in which the microstructure of the porous GPF

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substrate varies continuously as filtration proceeds. In the literature, there are limited

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experimental and modeling studies on the dynamic filtration process in GPFs. It is critical to

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understand how the microscopic (e.g., collector size, local porosity, and local permeability) and

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macroscopic filtration characteristics (e.g., number-based filtration efficiency and pressure drop)

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vary with time. One widely adopted approach for simulating the dynamic filtration process is

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“unit-cell” method.16,17,18 This approach was developed on the basis of the flow field solutions

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initially derived by Kuwabara16 and Happel17 for low-speed laminar flow in a randomly packed

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bed of spheres. Lee et al.19 theoretically derived the equations of filtration efficiency and

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pressure drop for a clean filter on the basis of Kuwabara flow.16 Later, these equations were

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extended to soot loaded DPFs and used to simulate the filtration process.20,21,22,23 In the “unit-

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cell” approach, a representative mean pore size and a mean porosity were selected to represent

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the porous substrate by assuming a homogeneous porous substrate24. This assumption works well

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for filtration modeling in DPFs, which are less sensitive to the microstructure of the porous

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substrate. GPFs are exposed to a larger fraction of ultrafine particulate that can penetrate through

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the filter due to the absence of a “soot cake” layer as the primary filtration medium.25,26,27 As a

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result, the microstructure of the porous substrate is critical and the interaction between the

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particulate and the porous substrate has a significant impact on the filtration performance.

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Moreover, in a realistic filter, the porous substrate has an inherently heterogeneous

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microstructure, which spans a wide range of length scales. Recently, a PDF-based heterogeneous

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multiscale filtration (HMF) model was developed to calculate the filtration efficiency of clean

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GPFs by considering the heterogeneous microstructure of the substrate.28 Rather than using the

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mean pore size and mean porosity, a pore size probability density function (PDF) established


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from experimental observations29,30 was introduced to represent the heterogeneous multiscale

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porous substrate. The HMF model was validated on three particulate filters with distinct mean

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pore sizes at various flow rates28. In the present work, the PDF-based HMF model is extended to

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model the dynamic filtration characteristic of a GPF, which allows us to probe the dynamic

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evolution of the microstructure and the macroscopic filtration characteristics.

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Experiments

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Particulate emissions. Particulates in the filtration study were generated under steady-state

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operation of a single-cylinder SIDI gasoline engine fueled with EPA Tier II EEE gasoline

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(Haltermann Solutions). The SIDI engine specifications are summarized in Table S1 in the

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Supporting Information. The engine was operated at a fuel-rich condition to generate a particle-

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size distribution (PSD) representative of emissions during cold-start operation where emissions

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are expected to be at their highest.31 The specific engine operating parameters for the fuel-rich

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condition are defined in Table S2. A TSI scanning mobility particle sizer (SMPS) with a TSI

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3081 long differential mobility analyzer (DMA) and a TSI 3010 condensation particle counter,

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were used to measure the PSD in the diluted exhaust stream during the initial characterization.

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The dilution was achieved using a two-stage partial flow dilution system, with a heated primary

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porous tube dilution stage (350°C) followed by a room temperature secondary ejector dilution

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stage. Once the engine operating conditions were established, micro-scale filtration experiments

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on a GPF sample were performed using the EFA system.

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Exhaust Filtration Analysis (EFA) System. The EFA system was initially developed at the

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University of Wisconsin-Madison Engine Research Center (ERC) for diesel particulate filtration

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studies32,33. It was later modified by Viswanathan et al.34,35 for low particulate mass

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concentration filtration studies using SIDI engine particulate. A schematic view of the EFA

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system is depicted in Figure S1. A thin, rectangular section of the filter wall, referred to as a


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wafer, was placed in a stainless steel holder and sealed using silicone O-rings capable of

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handling temperatures up to 200°C. The wafer holder was designed to have near uniform

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filtration velocities across the inlet face of the wafer.32

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Raw exhaust was sampled downstream of the exhaust surge tank and sent through the wafer

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holder which was placed in an oven to enable precise control of the filtration temperature. A

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dilution system including an ejector diluter (Dekati® DI 1000), which was supplied with heated

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dilution air at 175°C, was adopted in the EFA system as shown in Figure S1. The flow rate (or

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filtration velocity) through the wafer was controlled by adjusting the dilution air flow at the

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ejector diluter downstream of the oven with respect to the total flow into the SMPS, CO2

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analyzer and flow orifice. Real time dilution ratio was determined using the measured CO2

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concentrations in the raw exhaust and downstream of the EFA.

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The specifications of the wafer sample used in this study are shown in Table S3. The pressure

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drop across the wafer holder was measured using a differential pressure transducer (Omega, PX-

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138) with a range of 5 psi and an accuracy of ±1% of full scale. In the EFA system, the absence

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of alternatively plugged channels allows us to study the fundamental filtration process within the

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filter walls without considering the effects of flow contraction and expansion at the entrances and

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exits of filter channels as in a typical full-scale filter. As a result, only the particulate filtration

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process, in which the exhaust gas with particulate passed through the wafer, was simulated. A

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constant face or wall velocity (reported in Table S3) calculated from measured volumetric flow

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rate was used in the simulation.

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A TSI engine exhaust particle sizer (EEPS) was used alongside the SMPS to monitor the PSD

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upstream and downstream of the EFA system. The EEPS is capable of measuring real-time PSD

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in the range of 5.6 to 560 nm with 1-s resolution. More details about the EFA system and


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experimental procedures can be found in Viswanathan et al.35 It should be noted that the gas

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temperature was held at 175 °C in order to prevent any potential particulate oxidation. It is

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widely reported that soot oxidation typically occurs at relatively high temperatures (passive NO2

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soot oxidation can be initiated around 250 °C36,37 and active thermal soot oxidation occurs

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around 550 °C38,39). During the filtration experiments, size-resolved particle number

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concentrations and pressure drop across of the EFA system were continuously recorded.

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Multiscale Filtration Modeling

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The state-of-art particulate filtration modeling involves several coexisting length scales:40

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entire filter brick scale (dm), single channel scale (cm), wall scale (mm) and pore scale (µm). At

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the filter brick scale, component and system level modeling studies such as particulate

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loading,41,42,43 regeneration control,44,45 and interactions between particulate filter and other

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emission control devices,46,47,48,49 have been extensively conducted. A complete mathematical

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description of the filter inlet and outlet channels can be found in Bissett50. In modeling the

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dynamic filtration process in the EFA system, in which filter inlet and outlet channels do not

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exist, description of the channel scales in a filter can be neglected and replaced by a single face

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velocity. In this section, a dynamic filtration model at the wall and pore scale is presented. A

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graphical depiction of the dynamic HMF model is shown in Figure 1. There are several

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assumptions in modeling the filtration process:1) axisymmetric geometry of the porous substrate;

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2) all particles with different mobility diameters sampled from the engine exhaust are spherical;

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3) a constant effective density (1000 kg/m3) for all particulate51.

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Wall Scale. At the substrate wall scale, a porous substrate with a thickness of ℎ was

discretized into a number of slabs with a uniform thickness of ∆ along the depth of the substrate

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wall. Each slab was discretized into a number of cells with a constant width of ∆ along the

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channel direction as shown in Figure 1. Moreover, a porosity distribution profile was imposed to


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resolve the heterogeneous wall structure. From recent TEM studies30,52 of the microstructure of

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particulate filters, porosity is inhomogeneous across the filter wall. The porosity increases

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drastically from a nominal mean value at the center of a filter to a higher value near the filter

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surfaces. Furthermore, the porosity distribution across the filter wall was found to have a

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significant effect on local flow distribution, filtration efficiency and pressure drop26,53,54. Our

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previous simulation studies25 showed that most of particles were deposited at the top slab with a

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constant or homogeneous porosity, resulting in a high particle concentration gradient near the top

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surface of the substrate. With an inhomogeneous porosity distribution, the highest concentration

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of particles was observed at the transition region of the porosity distribution rather than at the top

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slab. Also, particles are readily to penetrate deeper into the substrate due to the high porosities of

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the top slabs. Therefore, it is critical and necessary to resolve the porosity distribution to capture

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the dynamic interactions between particulate and the local microstructure of the porous substrate.

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As shown in Figure 1, each slab at different locations has a specific porosity. Due to lack of

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experimental porosity distribution data for this filter sample, a porosity distribution was imposed

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with a high porosity of 0.68 at the top and bottom surfaces and a nominal mean porosity of 0.48

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at the center of the filter wall on the basis of experimental porosity distributions of particulate

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filters30. An exponential function was imposed to model the transition between the high surface

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porosity and the nominal wall porosity, which is shown in eq S1. It should be noted that it is

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possible to have a porosity distribution, in which the porosity decreases near the substrate surface

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for coated particulate filters. The impact of various porosity distributions in GPF’s filtration

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performance will be studied in our future work55.


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Figure 1. Schematics of multiscale modeling of particulate filtration in a GPF.

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Mass conservation of particulate at each slab can be achieved by realizing that the mass of

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particulate flowing into the slab is equal to the sum of the mass of particulate flowing out of the

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slab and the mass of particulate deposited in the slab. By assuming that all the particles are

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spherical with a constant density, mass conservation equations for the particulate can be

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translated into number-based conservation equations. In other words, the number of particles

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breaking through the jth slab ( ) can be calculated from the number of particles exiting

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the (j-1)th slab ( ) and the filtration efficiency of the jth slab. PNd 1 ηd ∙ PNd

(1)

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At the first slab, a partition coefficient was employed to determine the fraction of upstream

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particulate deposited on the surface of the first slab, which has been widely used to model the

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partition fraction between surface “cake” filtration and “deep-bed” filtration.20,56,57 With the

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partition coefficient, the number of particles flowing through the first slab can be calculated as PNd

1 ϕ ∙ PNd

(2)


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where

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defined as

is the number of particles upstream of the filter. The partition coefficient is

ϕ

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d#!, d#!$ ψb # d#!$

(3)

where ' is a percolation factor with a value of 0.95. It determines the maximum collector size

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that can be reached before the end of “deep-bed” filtration. Here ( is the maximum diameter of a

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collector, which can be calculated from the initial collector size ()$ ) and mean porosity (*), i.e.,

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,( .

//1 .

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+

Eventually, the number of particles exiting the filter (at the last slab 2 3456 ) can be calculated. The total particle number filtration efficiency is given by eq 4. ηd 1

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PNd 789:;< PNd

Pressure drop is another critical parameter of particulate filtration. According to Darcy’s law58, the total pressure drop across the entire filter wall can be calculated as ∆P=>? ∆P!>@A + ∆PC>??

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(4)

μvC wG + ∆PC>?? kG

(5)

where I3 represents the permeability of a “soot cake” with a thickness of J3 . The wall velocity

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KL along the channel can be calculated from one-dimensional incompressible Navier Stokes

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equations.50,56 In simulating the EFA system, the wall velocity is a constant. The “soot cake”

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accumulates only above the substrate surface (the first slab), therefore there is only one pressure

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drop term on the right side of eq 5. On the other hand, the pressure drop across the entire wall is


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a summation of the pressure drop across each slab. The total pressure across the entire wall is

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then calculated as 89:;
40

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µm) does not change significantly. The initial population of large collectors is smaller and

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filtration efficiency of the large collectors (>40 µm) is also lower compared to the smaller

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collectors (10-20 µm). Therefore, there is less particulate depositing on large collectors and the

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increase in the collector diameter of the large collectors is negligible. Consequently, the collector

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size PDF shifts to the right with a sharp peak of 25 µm, which is slightly larger than the initial

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mean collector diameter of 18 µm as filtration proceeds. At the middle slab, as a result of the

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reduced amount of particulate being trapped, the increase in mean diameter of the collector

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cluster is milder and the collector size PDFs shows only slight differences at 20 min compared to

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that at 10 min. At the last slab, there is negligible increase in the mean diameter of the collector

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cluster. Detailed progressive evolutions of the collector size PDFs at the first slab and middle

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slab are shown in Figure S2.

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Figure 2. Evolution of mean collector sizes and PDFs (inset graphs) at first, middle, and last slab

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of porous substrate.

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(b) Particulate mass, porosity and permeability distributions. Snapshots of the particulate

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mass distributions at 4 min, 8 min, 12 min, and 20 min are shown in Figure 3. At the beginning of

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filtration (4 min), the filter is almost clean as there is only a small amount of particulate trapped

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in the filter. At 8 min, the amount of particulate in the filter starts to increase and there is a

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gradient of particulate mass trapped across the filter. Even though a high concentration of

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particulate is observed at the top 1/4 of the filter, there is considerable particulate penetrating into

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the middle of the filter as is evident from the higher particulate concentrations in the bottom 1/2

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of the filter at 8 min compared to 4 min. At 12 min, a clear gradient of particulate mass across

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the top 1/4 of the filter can be observed. A larger gradient of particulate mass is found at 20 min

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with a higher concentration of particulate in the top portion of the wall. There is little increase in

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the particulate in the bottom 1/2 of the filter. The local porosity distributions of the filter at

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various times are described in Figure 4 with a local minimum in porosity at the 3rd slab of the

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filter. The location corresponding to the minimum porosity actually correlates to a transition

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position in the imposed porosity distribution as shown in Figure 1. At this transition position, the

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porosity is neither too high nor too low resulting in a relatively high particulate trapping capacity

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as well as a high filtration efficiency. These observations indicate that the porosity distribution

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plays a significant role in altering the local microstructure and performance of the filter. Similar

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to the local porosity distribution of the filter wall, the local permeability is shown in Figure S3

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with a local minimum permeability of 0.8E-13 m2 at the 3rd slab, which is about 22.9% of the

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initial permeability of the clean filter.


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The progressive particulate mass, local porosity, and permeability distributions in the filter

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are consistent with each other, as well as, the mean cluster diameter evolutions in Figure 2. A

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majority of the incoming particulate is trapped in the very top portion of the filter, where a higher

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gradient of particulate mass, and a lower porosity and permeability are observed. There is a small

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fraction of particulate breaking through the top half of the filter and depositing in the bottom half

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of the filter. The particulate mass and porosity distributions in the bottom region of the filter are

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very similar between 20 min and 4 min. This indicates that the bottom 1/3 of the filter makes

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little contribution to the overall filtration performance for this loading case.

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Figure 3. Local particulate mass distribution across the filter wall at t=4 min, 8 min, 12 min and

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20 min.


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Figure 4. Local porosity across the filter wall at t=4 min, 8 min, 12 min and 20 min.

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Macroscopic filtration characteristics

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Pressure drop and filtration efficiency (mass- and number-based) are widely used to represent

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the macroscopic filtration characteristics of a GPF. An ideal GPF should have a low pressure

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drop and high filtration efficiency. During filtration, the macroscopic state of a filter can be

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characterized by the apparent permeability of the filter, which is directly correlated to the total

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pressure drop. On the other hand, contributions from local and individual collectors at various

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scales on trapping particulate lead to an overall filtration efficiency, which can be experimentally

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measured.

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(a) Normalized pressure drop.

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In the EFA system, the face velocity towards the filter slightly decreased with time during

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filtration. This decreased face velocity is assigned to the increased backpressure as a result of

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particulate deposition. To isolate the effect of face velocity in evaluating the apparent

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permeability, a normalized pressure drop is defined in eq 13 according to Darcy’s law. On the

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right side of eq 12, gas viscosity | and wall thickness ℎ are constants. Accordingly, the

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normalized pressure drop was employed to represent the apparent permeability.


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∆ | ∙ ℎ } IL,~

(12)

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The particulate packing density vw in eq 9 was calibrated to be 40 kg/m3 in order to match the

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experimental normalized pressure drop. With the calibration, the model is capable of describing

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the dynamic macroscopic state of the filter. The normalized pressure drops of individual slabs

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are shown in Figure 5. It is interesting to see that slabs at distinct positions behave differently.

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Even though the face velocity slightly decreases with time, the normalized pressure drops of the

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slabs at the top 1/3 of the filter increase exponentially with time. In contrast, the normalized

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pressure drops of the slabs at the bottom 1/2 of the filter increase marginally after 20 min. These

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different behaviors are the consequences of distinct contributions of various slabs during

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particulate filtration. Overall, the normalized total pressure drop of the filter in Figure 5 increases

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approximately linearly with time. This linear relationship is quite different compared to the non-

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linear relationship in traditional DPFs, in which normalized pressure drop shows two distinct

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slopes with time in the early stage of filtration.

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absence of complete transition to “soot cake” filtration regime in the current experiment.

20,60

The linear correlation is attributed to the

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Figure 5. Normalized individual pressure drop across each slab (1 to 36) and the total normalized

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pressure drop (inset graph).

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(b) Particle number filtration efficiency. In SIDI engines, there is a significant amount of

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ultrafine particulate below 100 nm. The filtration efficiencies for these ultrafine particles are

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critical for an effective GPF. During the rich operating mode, more than 95% of the particles in

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the SIDI engine exhaust are below 200 nm in mobility diameter as shown in Table S2. On the

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other hand, current particle number regulations count all particles with a mobility diameter above

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23 nm15. Moreover, evolution of the number concentrations at specific diameters is important to

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understand the complicated filtration process. Thus, the changes in number concentrations of

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various particle diameters (30, 50, 100, and 200 nm) are displayed in Figure 6. Also,

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experimental and simulated number-based filtration efficiencies for each particle size at 20 min

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are given in Figure 6.

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As can be seen from Figure 6, both experiments and simulations show very high number

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filtration efficiency close to 100% for particulate with diameters below 50 nm. The dynamic

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evolutions of particle number concentrations over the full spectrum of the particle size in 20 min

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are shown in Figure S4, in which the filtration efficiency decreases as particle size increases. The

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most penetrating particle size (MPS), which is defined as the particle size with the lowest capture

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efficiency, is the signature characteristic of a specific substrate under a specific operating

364

condition. From Figure S4, the most penetrating particle size for this filter substrate is found to

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be around 100 nm with a filtration efficiency of 45.7%. This observation is the consequence of

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interactions between diffusion collection and interception collection mechanisms in particle

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filtration. Diffusion collection efficiency decreases as particle size increases. In contrast,

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interception collection efficiency increases with particle size. For small particles, diffusion


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collection is dominant. As the particle size increases, interception collection starts to play a role

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in total particle filtration. The resulting consequence is the total filtration efficiency shows a “V”

371

shape in the particle size domain.

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The filtration efficiency of 100 nm particle is under-predicted by the model. For large

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particulate with a diameter of 200 nm, the filtration efficiency is above 97%, which is in a good

374

agreement with the model. Although there are some differences in the filtration efficiencies

375

between the experiment (45.71%) and simulation (33.31%) for 100 nm particle as shown Figure

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6, it should be noted that the experimental and simulated filtration efficiencies for comparison

377

were evaluated at 20 min. What is more important from Figure 6 is the dynamic evolution of the

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filtration efficiency with time. From this perspective, the experimental and simulated filtration

379

efficiencies are quite comparable over the full 20 min. Also, this discrepancy could be resulted

380

from the sensitivity of the particulate measurement system on the increased backpressure across

381

the wafer due to particulate deposition. As it can be seen from Figure 6, the particle

382

concentration starts to oscillate after 15 min. Overall, the dynamic evolution of the particle

383

concentration is correctly predicted by the model across the whole range of particle sizes.

Conc. [#/ cm 3]

10 10 10 10

Conc. [#/ cm 3]

10 10 10 10

384

6

30 nm 4

2

0

0

10 ηexp=100 %

10

ηsim=99.94 % 10

20

8

100 nm 6

4

2

0

10

10

6

50 nm 4

2

10

10

ηsim=33.31 % 10 Time [min]

20

ηsim=93.91 %

0

10 ηexp=45.71 %

ηexp=100 %

10

0

10

20

6

200 nm 4

2

0

0

ηexp=97.21 % ηsim=97.61% 10 Time [min]

20


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385

Figure 6. Evolution of particle number concentrations of mono-sized particles penetrating

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through the filter sample (dash: exp; solid: sim).

387

The role of GPF’s microstructure in filtration modeling. The microstructure of a GPF’s

388

substrate is critical to the filtration performance. The heterogeneous porosity across the substrate

389

has a direct effect on the local particulate distribution61. As shown in Figure 3, the highest

390

concentration of particulate is observed at the top 1/4 of the filter, which corresponds to the

391

transition position of the porosity profile across the substrate (between the high porosity at the

392

surfaces and the nominal low porosity at the center of the substrate). The resulting low porosity

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at this transition position prevents particulate from penetrating further into the substrate.

394

Furthermore, the pore size distribution of the GPF’s substrate is critical and determines the

395

filter’s initial filtration efficiency28. A filter with a narrow pore size distribution was found to

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have higher filtration efficiency62. During the filtration, the dynamic change of the pore size

397

distribution is directly related to the microscopic filtration characteristics as described in Figure 2

398

and Table 1. It is therefore necessary to resolve the microstructure of the GPF’s substrate in the

399

filtration model.

400

With the GPF’s microstructure resolved in the dynamic HMF model, the dynamic interaction

401

between the substrate’s microstructure and particulate can be studied. Specifically, the evolution

402

of microscopic characteristics of the porous substrate such as porosity, permeability, and the

403

amount of deposited particulate can be probed. The dynamic changes of the substrate’s

404

microscopic properties lead to the change of macroscopic filtration characteristics such as

405

pressure drop and filtration efficiency. As a result, the gap between the microscopic filter

406

properties and the macroscopic filtration characteristics during particulate filtration in GPFs is


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407

bridged, which is beneficial to better comprehend the filtration dynamics in GPFs. Moreover,

408

this dynamic HMF model can be applicable to explore optimum or ideal structure for

409

microscopic filter design.

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Acknowledgement

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The authors also would like to thank General Motors Research and Development for their

412

funding and support of this research through the GM-UW collaborative research laboratory

413

(CRL) program.

414

Supporting Information Available

415

Details of SIDI engine specifications and filtration experiments. Derivations of porosity,

416

permeability in the dynamic HMF model. Progressive evolutions of collector size PDF and

417

porosity distribution during filtration. This material is available free of charge via the Internet at

418

http://pubs.acs.org.

419

Nomenclature

420

Abbreviations

PDF

probability density function

HMF

heterogeneous multi-scale filtration

GPF

gasoline particulate filter

EFA

exhaust filtration analysis

SIDI

Spark-ignition direct-injection

DPF

diesel particulate filter

PFI

port fuel injection

DEFA

diesel exhaust filtration analysis


23


421

SEM

scanning election microscopy

LBM

lattice boltzmann method

Page 24 of 29

Symbols

PN

U

' (

G?> U)c ,

,c ∆ |

ℎ

KL J3

JL I3

IL

)c

particle number number filtration efficiency mobility particulate diameter partition coefficient percolation factor maximum diameter of a collector number of slabs number filtration efficiency of a collector with diameter of )c at particulate diameter of total single collector efficiency pressure drop across a filter gas viscosity wall thickness wall velocity cake thickness slab thickness soot cake permeability slab permeability individual collector diameter in a collector cluster


24

Page 25 of 29


`a+,b * ℎ

^+) v u

),

a*

pdf of a collectors cluster porosity of the jth slab thickness of filter wall length scale of the filter collector particulate density mass of deposited particulate mean diameter of collector cluster hydrodynamic term

422

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423

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Dynamic Heterogeneous Multiscale Filtration Model - ACS Publications

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