PDF-Based Heterogeneous Multiscale Filtration Model - American

Mar 30, 2015 - Based on the analysis of experimental porosimetry data, a pore size probability density function is introduced to represent heterogenei...
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PDF-Based Heterogeneous Multiscale Filtration Model Jian Gong*,† and Christopher J. Rutland‡ †

Cummins, Inc., 1900 McKinley Avenue, MC 50183, Columbus, Indiana 47201, United States Engine Research Center, University of WisconsinMadison, 1008 Engineering Research Building, 1500 Engineering Drive, Madison, Wisconsin 53706, United States

‡

S Supporting Information *

ABSTRACT: Motivated by modeling of gasoline particulate filters (GPFs), a probability density function (PDF) based heterogeneous multiscale filtration (HMF) model is developed to calculate filtration efficiency of clean particulate filters. A new methodology based on statistical theory and classic filtration theory is developed in the HMF model. Based on the analysis of experimental porosimetry data, a pore size probability density function is introduced to represent heterogeneity and multiscale characteristics of the porous wall. The filtration efficiency of a filter can be calculated as the sum of the contributions of individual collectors. The resulting HMF model overcomes the limitations of classic mean filtration models which rely on tuning of the mean collector size. Sensitivity analysis shows that the HMF model recovers the classical mean model when the pore size variance is very small. The HMF model is validated by fundamental filtration experimental data from different scales of filter samples. The model shows a good agreement with experimental data at various operating conditions. The effects of the microstructure of filters on filtration efficiency as well as the most penetrating particle size are correctly predicted by the model.

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INTRODUCTION Particulate filters were initially developed in the 1980s for controlling particulate emission from diesel engines in order to meet tightening particulate emission regulations.1 Diesel particulate filters (DPFs) are widely used on diesel engines as standard aftertreatment devices and give very high efficiency (>90%) for removing diesel particulates.2 For DPF, the filtration is dominated by the “soot cake” at most of the time. However, the particulates from combustion engines fueled with gasoline or gasoline/ethanol blends are smaller.3 There is significantly less particulate emission generated for advanced fuel-neutral combustion engines. Therefore, there is less likelihood of forming a “soot cake”, which traditional DPFs rely on, and filtration efficiency will be sensitive to the microstructure of the filter. As a result, the filtration efficiency would drop significantly without “soot cake”, and conventional particle filters would be expected to fail for filtering smaller particulates from advanced combustion engines. Filtration can be defined as the process of separating dispersed particles from a dispersing fluid by means of a porous medium.4 Theoretically, the characteristics of filtration are dependent on all aspects of the dispersed particles, the dispersing fluid, and the porous medium. The classical filtration theory uses the “unit-cell” approach.1,2 The “unit-cell” approach is based on the flow field solution derived by Kuwabara5 and Happel6 for low speed laminar flow in a randomly packed bed. By solving a transport equation of particulates (a convectiondiffusion equation) numerically and conducting analysis of the trajectory of particles, single collector efficiency of different © 2015 American Chemical Society

individual collection mechanisms can be theoretically derived. The analysis treats the particles as if they are diffusion “points” since the particles are small compared to the collectors. Friedlander7 calculated the diffusion efficiency by solving the boundary layer form of the steady-state convective diffusion equation considering the particle capture process occurs near the surface of the collector. By applying Friedlander’s analysis to packed beds, Lee et al.8 derived a single collector efficiency due to diffusion and interception based on Kuwabara’s flow.5 This “unit-cell” filtration model has been widely used in DPF modeling.1,2,9 Some correlations for predicting single collector efficiency were proposed later. Rajagopalan and Tien10 developed a correlation for single collector efficiency by using the “unit-cell” approach based on Happel’s flow solution.6 Tufenkji and Elimelech11 then developed a correlation for a single collector efficiency, which considers van der Waals forces and hydrodynamic interactions. These researchers attempted to obtain theoretical formulas that express particle capture without firm commitment to a particular flow field. Therefore, the appropriate criterion to choose filtration models is the agreement of filtration efficiency between model and experiments. Recently, Long et al.12 developed a correlation for the clean bed filter coefficient for Brownian particles (50 nm < dp < 300 nm) based on Lattice Boltzmann (LB) simulations in Received: Revised: Accepted: Published: 4963

January 19, 2015 March 29, 2015 March 30, 2015 March 30, 2015 DOI: 10.1021/acs.est.5b00329 Environ. Sci. Technol. 2015, 49, 4963−4970

Article

Environmental Science & Technology random packing of spheres of uniform diameter. A porosity dependent term based on the Carman−Kozeny relation4 was proposed in the diffusion efficiency from numerical breakthrough simulations of flow-through filters with porosities around 0.3−0.42. However, additional investigations may be needed to test the model over a wide range of porosities. Filtration efficiency correlations, based on the mean pore size or collector size, can be classified as “mean” models. These correlations use a representative mean pore size and a mean porosity to model the porous medium. The porous medium is assumed to be homogeneous. The flow field and the capture efficiency are dependent on the mean properties of the porous medium. However, in most realistic filters, this assumption does not hold. The porous medium has heterogeneous microstructure, which spans a wide range of scales.13 The microstructure of the porous wall plays a significant role in filtering finer particulates from advanced combustion sources.14,15 In this paper, a PDF-based heterogeneous multiscale filtration (HMF) model is presented to predict filtration efficiency of clean particulate filters. A new methodology based on statistical theory is developed. A PDF-based pore size distribution is introduced to represent the heterogeneous filter wall. Results are compared to experimental data from a range of laboratory scales of filter samples.

Figure 1. Pore size distributions of 12 samples of a cordierite DPF measured by mercury porosimeter.

data) and a variance of 60 μm. The variance of 60 μm is assumed from the best fitting of the experimental pore size distribution. From statistical theory, the integral of the PDF over the normalized pore size space gives unity, which is shown in eq 1.

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∫ pdfd d(doi) = 1

MODEL DESCRIPTION Pore Size Distribution. The porous wall of a filter typically has a heterogeneous microstructure. Physically, the heterogeneity of a porous filter can be described by heterogeneous porosity, pore size, shape factor of pores, pore connectivity, and so on. Since the size of the pore is relatively easy to measure compared to other heterogeneity properties and the pore size plays a significant role in filtration efficiency modeling,16 the heterogeneous pore size distribution is investigated and introduced in the model. One of the most common techniques for porous material characterization is mercury intrusion porosimetry measurements. Pore size distributions from porosimetry measurements were found to be able to represent heterogeneity and multiscale characteristics of the porous filter wall.16 Porosimetry data of common DPF samples were reported by Wirojsakunchai et al.17 and Merkel et al.18 Some studies found that the pore structure of the DPF based on porosimetry data analysis had significant effects on the filter performance.19−21 From reported porosimetry data,17,18,21 most of the pore size distributions follow a narrow or wide log-normal distribution. Figure 1 shows the experimental porosimetry data of 12 samples of a cordierite DPF by using a porosimeter.22 First, the pore size space covers a wide range of scales from 0.01 to 1000 μm. Second, the 12 pore size distributions are very consistent with each other. Also, some statistics of the experimental pore size distribution data are analyzed. The mean porosity of the filter samples is 0.48, and the experimental statistical mean pore size is around 114.7 μm. However, a most likely pore size corresponding to the peak of the pore size distribution is around 18 μm. This most likely pore size was found to be the most critical pore size that governs the filtration performance and was commonly used in classic mean filtration models as the mean pore size.1,2,9,23 The averaged pore size distribution of 12 samples is fitted by a log-normal distribution function with a mean pore size of 18 μm (the same as most likely pore size from the experimental

(1)

oi

The mean pore size of the distribution can be calculated as well from eq 2.

∫ pdfd doid(doi) = dom

(2)

oi

In order to use the solution of Kuwabara flow in a porous medium, the pore space has to be transformed into a collector space. However, there is no straightforward way to derive an accurate correlation between the pore diameter and collector diameter since the pore size space in the porous filter is a complex network system. By assuming the pore diameter is four times the ratio of the total pore volume to the total pore surface area,4 the diameter of the collectors can be estimated from eq 3, where ϵ is the porosity of the filter. This assumption was widely used in filtration modeling of porous filters.1,2,4,9 As a result, a PDF of collector size can be calculated according to “change of variables” in eq 4. The PDF of collector size is usually used in the calculation of collection efficiency, which will be discussed later. 5

dci =

3(1 − ϵ) doi 2ϵ

pdf d = ci

2ϵ pdf 3(1 − ϵ) doi

(3)

(4)

Resolved Flow Velocity. An analytical solution of the flow field in a complex porous medium is impossible. Even though the flow is laminar in most operating conditions of particulate filters, the flow velocity inside the porous medium is highly nonlinear.24,25 Furthermore, it is difficult to use numerical simulations from 3D CFD or Lattice Boltzmann method (LBM) analysis to help to derive a correlation or solution of flow velocity inside the porous medium that can be directly used for filtration calculation in classic filtration models. Therefore, an analytical solution of Stokes flow in a system of randomly packed spheres derived by Kuwabara5 is used. 4964

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Environmental Science & Technology In a creeping flow regime, the drag force exerted by the flow on an isolated sphere can be calculated by FD = 3πμdcU̅ , where U̅ is the mean superficial velocity far away from the sphere with a diameter of dc. Practically, the mean superficial velocity can be calculated from pressure drop with known permeability. In this paper, the mean superficial velocity is controlled and specified from fundamental filtration experiments. It is assumed to be a constant for clean filters during the early stage of filtration. In the case of Stokes flow around a system of spheres with a packing density of (1− ϵ), the drag exerted by the flow on a representative sphere in a system of spheres can be calculated by eq 5. FD =

3πμdcU̅ K (ϵ)

engine exhaust conditions are diffusion, direct interception, and inertial impaction mechanisms.1 A single collector efficiency due to diffusion and interception from Lee et al.8 was widely and successfully applied in DPF filtration modeling.1,2,9,26 The objective of this paper is not to derive a new correlation for single collector efficiency but to develop a new methodology for filtration efficiency calculation. Therefore, the single collector efficiency based on a convection−diffusion equation from Lee et al.8 is used as the basis for calculating filtration efficiency. a. Diffusion. Single collector efficiency due to diffusion can be calculated from a flow field analysis and by solving the convective diffusion equations near the boundary layer of a representative sphere. Analytical derivation of the diffusion efficiency of a single sphere can be found in Lee et al.8

(5)

Here K(ϵ) = 1 − /5(1 − ϵ) + (1 − ϵ) − /5(1 − ϵ) is a function of the porosity, which is a hydrodynamic factor of the system of spheres derived from flow field solution from Kuwabara.5 At a steady state, assuming the pressure drop across the porous wall with thickness h is a simple summation of all the drag force around each sphere, a linear momentum balance on the fluid in the control volume gives 9

1/3

1

AΔP = NsFD

2

1/3 ⎛ 3π ⎞2/3⎛ ϵ ⎞ −2/3 E D, i = 2⎜ ⎟ ⎜ ⎟ Pei ⎝ 4 ⎠ ⎝ K (ϵ) ⎠

The Peclet number in eq 12 is defined as

Pei =

(6)

Ah(1 − ϵ) 1 πdc3 6

(7)

From eq 5 to eq 7, the mean superficial velocity is U̅ =

K (ϵ) ΔP 2 dc 18μ(1 − ϵ) h

However, a realistic porous filter is made spheres with different diameters. Similarly, velocity across the system of monodispersed diameter of dci at the same pressure drop and given by Ui =

K (ϵ) ΔP 2 dci 18μ(1 − ϵ) h

(8)

E R, i

of a cluster of the superficial spheres with a porosity can be

∫ pdfd Ui d(dci) ci

NR, i 2 3(1 − ϵ) = 2K (ϵ) (1 + NR, i)2

(9)

NR, i =

(14)

dp lR, i

(15)

c. Inertial Impaction. At high Stokes numbers, the controlling collection mechanism is inertial impaction. A single collector efficiency due to inertial deposition of particles can be described by a dimensionless Stokes number. In the applications of particulate filtration from engine exhaust, the Stokes number is very small and the inertial impaction makes a limited contribution to the total collection efficiency. However, the inertial impaction is still considered in this model. In order to take account the influence of the Reynolds number on the inertial efficiency, an effective Stokes number was proposed by D’Ottavio and Gorden,28 Gal et al.,29 and Otani et al.30 The effective Stokes number is calculated as the product of the Stokes number Stki = (ρpdp2Ui)/(18μdci) and a hydrodynamic term involving the Reynolds number and porosity. From Ergun’s equation, the hydrodynamic term was modeled as f(Rei,ϵ) = 1 + 1.75Rei(ϵ/150(1 − ϵ)), where Rei = ρgUdci/μ. The inertial efficiency correlation in eq 16 from Otani et al.30 is used in this paper.

(10)

According to eqs 9 and 10, the velocity for each individual sphere with a diameter dci can be calculated as ⎛ dci ⎞2 Ui = ⎜ ⎟ U̅ ⎝ ldc ⎠

(13)

Here, NR,i in eq 14 is a nondimensional parameter to characterize the direct interception efficiency. A length scale of lR,i is defined to represent the critical length for direct interception, which will be discussed later.

As discussed earlier, the size distribution of spheres can be described by a log-normal probability density function. The mean velocity can be related to the individual velocity from eq 9.

U̅ =

Ud i ci Ddiff

Here, D diff = ((kB T)/(3πμdp))f(K n) is the diffusion coefficient of a particle with a diameter of dp. f(Kn) = 1 + Kn[1.257 + 0.4 exp(−1.1/Kn)] is the slip correction factor where Kn = 2λ/dp is the Knudsen number. b. Direct Interception. As particle size becomes larger, its effect known as direct interception may be taken into consideration for particle collection. A single collector efficiency due to direct interception was derived from the solution of creeping flow around a system of spheres by Lee et al.27

where the number of the sphere in the system can be estimated by eq 7. Ns =

(12)

(11)

where ldc is a length scale that represents the overall filter collector size and is defined as ldc = [∫ pdfdcidci2d(dci)]1/2. Single Collector Efficiency. There are different collection mechanisms involved in filtration process: diffusion (Brownian), direct interception, inertia, gravity, electrostatic, and van der Waals.4 The main mechanisms that dominating filtration performance of diesel/gasoline particulate filters at typical 4965

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Environmental Science & Technology E I, i =

Stk eff, i 3 (0.014 + Stk eff )3

Stk eff, i = f (Rei , ϵ)Stk i

(16) (17)

Total Filtration Efficiency. In the filtration process, particles may be subjected to a simultaneous effect of all deposition mechanisms. One of the basic but most difficult problems in filtration theory is to find the total collection coefficient. In general, this problem has not been exactly solved owing to mathematical difficulties.31 In this paper, a common approach (the “independence rule”), which assumes that all mechanisms act independently, is used to calculate the total collection efficiency. On the basis of the “independence rule”, the overall single collector penetration Pt,i can be calculated as a product of all penetrations from different deposition mechanisms. As a result, the total filter efficiency can be calculated as Et,i = 1 − Pt,i = 1 − (1 − ED,i)(1 − ER,i)(1 − EI,i). The filtration efficiency of a filter consisting of collectors with a specific diameter dci can be calculated from eq 18. In eq 18, h is the thickness and ϵ is the mean porosity of the filter. ⎡ 3(1 − ϵ)h Et, i(dci , d p) ⎤ ⎥ η(dci , d p) = 1 − exp⎢ − ⎢⎣ ⎥⎦ 2ϵ dci

Figure 2. Pore size PDF (red dashed line with left and top axis) and contour plot of PDF weighted filtration efficiencies from each single pore size at different particulate mobility diameters.

probability density of smaller pores. Figure 2 shows that there are two high efficiency zones in the pore size−particulate size space. As the mobility diameter of the particulates increases, the interception filtration mechanism starts to take effect. The most effective pores (diameters between 10 to 30 μm) that contribute to high filtration efficiency are consistent to the pores that have high probabilities in the pore size PDF. Essentially, this contour plot represents how the PDF of pore size impacts the total filtration efficiency in the model. Sensitivity of Pore Size Variance. The variance of a pore size distribution determines how the pore size varies from the mean pore size. Physically, a pore size distribution with a small variance means the porous medium has more homogeneous microstructure or monodispersed pores. In contrast, a pore size distribution with a large variance means the porous medium has more heterogeneous microstructure or poly dispersed pores. To investigate how the pore size distributions change the filtration efficiency in the HMF model, a few pore size distributions are created with the same mean pore size of 20 μm and different variances. As expected, as pore size variance increases, filtration efficiency tends to decrease over the whole range of particulate sizes (see Figure 3). Large pores are less effective for collecting particulates compared to small pores. As the variance reaches a very small value, the filtration efficiency of the HMF model is

(18)

The total filtration efficiency of a filter is calculated as the sum of the contributions from all the collectors with different sizes. By summing the fitted PDF weighted contributions from all of the collectors, the total filtration efficiency can be calculated. From eq 19, it can be seen that the total filtration efficiency is dependent on the size of particulates. In other words, a filter selectively filters the particulates with a particular size. η(d p) =

∫ Uiη(dci , d p)pdfdcid(dci) ∫ Ui pdfdcid(dci)

(19)

By substituting Ui from eq 11 into eq 19, the final filtration efficiency can be written as η(d p) =

∫ η(dci , d p)dci 2 pdfdcid(dci) ldc 2

(20)

Effects of Pore Size PDF on Filtration Efficiency. To investigate the interaction between the pore size PDF and filtration efficiency, a filtration study is conducted on a typical particulate filter sample. The filter sample has a mean porosity of 0.5 and a thickness of 12 mil (1 mil = 0.001 in.), which is very comparable to most commercial DPFs or GPFs. The microstructure of the filter is represented by a pore size distribution with a mean pore size of 20 μm and a variance of 60 μm. The size of particulates for this study scales from 20 to 1000 nm, which covers the typical particulates from diesel and gasoline combustion engines. Each pore at a specific size corresponding to a specific collector makes a contribution to total filtration efficiency. A contour plot of the PDF weighted filtration efficiency from each pore size as well as the pore size PDF is shown in Figure 2. The PDF-weighted filtration efficiency from a specific pore depends on the probability density of the pore and the diameter of the pore. Generally, smaller pores have higher filtration efficiencies due to higher diffusivities. However, contributions from smaller pores to total filtration efficiency can be low due to lower

Figure 3. Comparison of particulate size-dependent filtration efficiencies between mean filtration model and HMF models with different pore size distributions (ν is the variance). 4966

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Environmental Science & Technology

velocities (1, 2, 3, and 4 cm/s). Since there is no pore size distribution measurement for this filter, a pore size variance of 2 μm is assumed. The mean pore size of 10.8 μm from experimental data and the assumed pore size variance are used to create a pore size PDF. In Figure 5 the HMF model gives good predictions of filtration efficiency at four different face velocities, especially for

getting close to the mean model, which solely uses the mean pore size for filtration efficiency calculations. In other words, the HMF model recovers the mean model when the variance is small. Individual collection efficiency due to different collection mechanisms may be controlled by different critical length scales. To demonstrate the effect of length scales on filtration efficiency, the length scale in the interception efficiency is modeled as a product of a constant f and the diffusion length scale dci (see eq 21). As shown in Figure 4, changing

Figure 5. Comparison of the filtration efficiencies between the HMF model and experiment at four different face velocities (symbol, experimental data; solid line, model).

small particles in the diffusion-dominated regime. As face velocity increases, the filtration efficiency decreases dramatically for relatively small particles (in diffusion dominated regime) due to decrease of the residence time of particles in the filter. The dependence of face velocity on filtration efficiency has been captured by the model. Moreover, the maximum penetrating particle size (the particle size with minimum filtration efficiency) has been correctly predicted by the HMF model. The decrease in filtration efficiency with increased velocity should diminish upon entering the direct interception regime, where the efficiency depends only on the particle size and not the velocity. This tendency is evident in Figure 5. There are some discrepancies between the HMF model and data for very large particles (>500 nm). These discrepancies could be due to the differences between the experimental pore size distribution and modeled pore size PDF as well as the interception length scale modeled in the HMF model. The experimental pore size distribution is generally asymmetric. When the log-normal pore size PDF is applied, relatively large scales from the experimental pore size distribution are missed. In other words, there are more relatively small pores scales representing the microstructure of the filter. The interception efficiency is higher with smaller pore size, which can be seen from eq 14 and Figure 4. That is one of the reasons why the filtration efficiency at the interception regime is overpredicted compared to the data, even though the HMF model showed good agreement with experimental data at diffusion regime. On the other hand, the interception efficiency is dominantly controlled by the interception length scale as illustrated in Figure 4. To establish the relationship between the experimental pore size distribution and the critical length scales of interception is a great challenge. Cordierite-Bonded SiC DPF. Filtration experimental data from Ohara et al.34 are used to further investigate the effects of filter microstructure (in terms of pore size distribution) on filtration efficiency. Three cordierite-bonded SiC DPF samples

Figure 4. Effect of interception length scale on filtration efficiency.

interception length scale has almost no effects on the filtration efficiency in the diffusion regime (small size particles). The interception efficiency increases as f decreases as expected. Also, note that the most penetrating particle size shifts to the left. From a sensitivity study, the order of magnitude of the constant f is close to 1. Its value changes from 0.5 to 1.2 for most of the filters investigated in this paper. lRi = fdci = f

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3(1 − ϵ) doi 2ϵ

(21)

RESULTS AND DISCUSSION There are few high quality experimental filtration data of particulates from combustion engines. First, particulates from combustion engine have complicated morphology and chemical composition and are difficult to characterize.32 On the other hand, it is very difficult to get solid, reproducible filtration data. Particle filtration is highly dependent on filter preparation, particle sampling, and measurement approaches. In this paper, several fundamental filtration experiments in the literature are selected for model validation.33−35 Sintered Granular Ceramic Filter. The first filter used for model validation is a sintered granular ceramic filter, with a mean pore size of 10.8 μm and a mean porosity of 45%.33 Filtration efficiency equations are derived based on the assumption that a porous medium consists of spherical grains. Therefore, filtration experiments from granular ceramic filters are very appropriate to test the filtration model. The filter has a mean pore size of 10.8 μm and a thickness of 1.65 mm, which is much higher than the wall thickness of typical DPFs or GPFs. The particles in the filtration study are dioctyl phthalate (DOP 1%) aerosol particles with a diameter range from 100 to 700 nm. The filter was operated at four face 4967

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Environmental Science & Technology with different mean pore sizes of 9, 12, and 17 μm are selected from the filtration experiments conducted by Ohara et al.34 The round disk filter samples all have the same diameter of 25 mm and thickness of 0.305 mm (12 mil). The filter wall thickness is very close to typical DPF filter wall thickness. During the filtration tests, ZnCl2 particles with a diameter in the range of 20−500 nm were used instead of particulates from diesel engine exhaust. The comparisons of filtration efficiencies of the three filter samples between the model and experimental data at a face velocity of 2.5 cm/s are shown in Figure 6. Three pore size

Table 1. Specification of a Single Channel Filter Sample from a Corning Duratrap AC Filter filter sample name

Corning Duratrap

filter material mean pore size (μm) pore size variance (μm) cell density (cpsi) thickness (mil) filter length (cm) mean porosity (−) superficial face velocity (cm/s) cas temperature (°C) particle size (nm)

cordierite 18.8 43 200 12 5.0 0.5435 0.56−1.8 20 40−400

Figure 7. Pore size PDF and comparison of the filtration efficiency for the Duratrap filter at four flow rates (symbol, experimental data; solid line, model).

Figure 6. Filtration efficiency comparisons of three filter samples at face velocity of 2.5 cm/s (black symbol, experimental data; solid line, model).

distributions with different mean pore sizes (9, 12, and 17 μm) and variances (7, 15, and 30 μm) corresponding to the three filter samples are shown in Figure 6. Even through the experimental data were poorly resolved over the particle size domain, the model captures the particle size dependent filtration efficiency for all filter samples. Both the data and model indicate that the smaller the mean pore size, the greater the filtration efficiency. The effects of the pore size distribution on filtration efficiency are correctly captured by the model. Single Channel Filter Sample. The HMF model is further tested using filtration data recently collected on commercial particulate filters at laboratory scales. The fundamental filtration experiments were conducted at Pacific Northwest National Laboratory (PNNL) for an uncoated, single channel filter sample.33 The single channel was prepared by extracting a channel from a larger DPF substrate. As shown in Table 1, the mean pore size of the filter is 18.8 μm, a variance of pore size 43 μm is assumed to generate a PDF of pore size distribution in the HMF model. Ammonium sulfate particles which have a spherical shape were used as filtration particles. The comparison of the filtration efficiency between the model and experimental data at four flow rates are shown in Figure 7. The filtration efficiency drops significantly as particle size increases. The HMF model correctly predicts the filtration efficiency at small and large particle sizes. The most penetrating particle size of this filter is around 200 to 300 nm, which is captured by the model as well. Development of a Pore Size PDF. To apply the HMF model, a pore size PDF representing the microstructure of a porous wall is necessary to be developed. This is because applying the experimental pore size distribution directly to the

HMF model does not work. The experimental statistical mean pore size is larger than the most likely pore size. Filtration efficiency is sensitive to the mean pore size and decreases significantly as the mean pore size increases. Directly applying the experimental pore size distribution will result in significantly lower filtration efficiencies compared to the experimental data. When the experimental pore size distribution data is available, a log-normal distribution is fitted to the data. One of the examples pore size PDF fitting is shown in the Supporting Information. The value of the mean and variance from the fitting are taken as the mean and variance of the pore size PDF in the model. When the experimental pore size distribution data is not available, the mean pore size, which is essentially an intrinsic physical property of a porous filter, can be obtained from a filter supplier. The variance of the pore size distribution can be tuned as a model constant to calibrate the model. Application of the HMF Model. There are a couple of advantages for the HMF model against the classical mean models. First of all, the heterogeneous microstructure of a filter in terms of pore size distribution has been resolved in the HMF model but unfortunately neglected in the mean models. Clearly, the heterogeneous microstructure has significant effect on the filtration efficiency. Second, the PDF-based statistical approach used in the HMF model overcomes the limitations of classic mean filtration models which rely on tuning of the mean collector size. Furthermore, the HMF model delivers good prediction of filtration efficiencies at various operating conditions, especially at diffusion controlled regime. The effects of the microstructure of filters on filtration efficiency as well as 4968

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Environmental Science & Technology Symbols

the most penetrating particle size can be correctly captured by the HMF model. The PDF based pore size distribution methodology presented in this paper is a general approach. The HMF model can be applied in different ways to simulate particulate filtration process. First, additional filtration mechanisms or more accurate single collector filtration efficiencies (eqs 12, 14, and 16) can be integrated into the HMF model. Second, the HMF model can be implemented as a subroutine into a multidimensional simulation code to calculate the local contribution of collection efficiency produced by local heterogeneity in a full scale DPF or GPF. A full scale filter model can then be used as a tool to design and optimize advanced filters for filtering particulates from various combustion sources. In addition, this PDF based statistical approach can be potentially used to model different types of heterogeneities of a filter. For example, a catalyst wash coat layer above the porous filter surface can be modeled separately by a different pore size PDF. In dynamic filtration,36 the filter will not be clean but loaded with soot as particulates accumulate in a filter with time. The evolution of the pore size PDF and dynamic filtration efficiency with time will be interesting. Besides filtration efficiency, pressure drop is another important item to evaluate filtration performance. The ultimate objective of this work is to develop a full-scale filtration model that is capable of predicting dynamic filtration efficiency and pressure drop. In our future work, the HMF model will be integrated into a full-scale 1-D GPF model to study the dynamic filtration characteristics.

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pdfdoi doi dom v dci pdfdci ϵ FD dc μ U̅ K(ϵ) A ΔP Ns h Ui ldc Pei Ddiff f(Kn) Kn dp ED,i ER,i NR,i lR,i EI,i Stki ρp ρg f(Rei,ϵ) Stkeff,i Rei η(dci,dp)

ASSOCIATED CONTENT

S Supporting Information *

Further model validations on mini-DPF samples of NGK 558 and NGK 650. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

η(dp) Pt,i Et,i

Corresponding Author

*E-mail: [email protected].

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Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank General Motors Research and Development for their continued funding and support of this research through the GM−UW collaborative research laboratory (CRL) program. We also thank Mark Stewart from PNNL for providing filtration experimental data to support the modeling work.

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REFERENCES

(1) Konstandopoulos, A. G.; Johnson, J. H. Wall-flow diesel particulate filters-Their pressure drop and collection efficiency. SAE Technical Paper 890405, 1989. (2) Konstandopoulos, A. G.; Kostoglou, M.; Skaperdas, E.; Papaioannou, E.; Zarval, D.; Kladopoulou, E. Fundamental studies of diesel particulate filters: transient loading, regeneration and aging. SAE Technical Paper 2000-01-1016, 2000. (3) Barone, T. L.; Storey, J. M. E.; Youngquist, A. D.; Szybist, J. P. An analysis of direct-injection spark-ignition (DISI) soot morphology. Atmos. Environ. 2012, 49, 268−274. (4) Matteson, M. J.; Orr, C. Filtration: Principles and Practices, 2nd ed.; Marcel Dekker: New York, 1987. (5) Kuwabara, S. The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds number. J. Phys. Soc. Jpn. 1959, 14, 527−532. (6) Happel, J. Viscous flow in multiparticle systems: Slow motion of fluids relative to beds of spherical particles. AIChE J. 1958, 4, 197− 201. (7) Friedlander, S. K. Mass and heat transfer to single spheres and cylinders at low Reynolds numbers. AIChE J. 1957, 3, 43−48. (8) Lee, K. W.; Gieseke, J. A. Collection of aerosol particles by packed beds. Environ. Sci. Technol. 1979, 13, 466−470.

NOMENCLATURE

Abbreviations

HMF PFI SIDI EFA PDF DPF GPF PNNL SEM LBM

probability of pore size pore size diameter mean pore size pore size variance collector diameter PDF of a system of collectors porosity drag force diameter of collector or sphere gas viscosity mean superficial face velocity hydrodynamic factor cross-section area of a filter pressure drop across a filter number of spheres thickness of filter wall superficial face velocity of individual collector length scale of the filter collector Peclet number particulate diffusion coefficient slip correction factor particle Knudsen number mobility particulate diameter single collect diffusion efficiency single collect interception efficiency ratio of particle size to collector size length scale for particle interception single collect inertial efficiency Stoke number particulate density gas density hydrodynamic term effective Stokes number Reynolds number filter filtration efficiency of a collector with diameter of dci at particulate diameter of dp size-dependent filtration efficiency total penetration Pt,i = 1 − Et,i total single collect efficiency

heterogeneous multiscale filtration port fuel injection spark ignition direct injection exhaust filtration analysis probability density function diesel particulate filter gasoline particulate filter Pacific Northwest National Laboratory scanning election microscopy lattice Boltzmann method 4969

DOI: 10.1021/acs.est.5b00329 Environ. Sci. Technol. 2015, 49, 4963−4970

Article

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DOI: 10.1021/acs.est.5b00329 Environ. Sci. Technol. 2015, 49, 4963−4970