Penetration of Monodisperse, Singly Charged Nanoparticles through Polydisperse Fibrous Filters

Abstract

The article presents experimental results and theoretical analysis of aerosol nanoparticle penetration through fibrous filters with a broad fiber diameter distribution. Four fibrous filters were produced using the melt-blown technique. The analysis of the filters’ SEM images indicated that they had log-normal fiber diameter distribution. Five kinds of proteins and two types of silica particles were generated by electrospraying and were then classified using a Parallel Differential Mobility Analyzer to obtain well-defined, monodisperse, singly charged challenge aerosols with diameters ranging from 6.3 to 27.2 nm. Particle penetration through the filters was determined using a water-based CPC. Experimental results were compared first with predictions derived from the classical theory of aerosol filtration. It is demonstrated that it is inappropriate to apply it to the arithmetic mean fiber diameter, as this results in turn in a huge underestimation of nanoparticle penetration. A better, but still unsatisfactory agreement is observed when that theory was used together with the pressure drop equivalent fiber diameter or when the Kirsch model of nonuniform fibrous media was applied. We show that the classical theory applied to any fixed fiber diameter predicts a stronger dependence of nanoparticle penetration on the Peclet number as compared to experimental data. All these observations were successfully explained by using our original partially segregated flow model that accounts for the filter fiber diameter distribution. It was found that the parameter of aerosol segregation intensity inside inhomogeneous filters increases with the increase in particle size, when the convective transport becomes more pronounced in comparison to the diffusive one.

1. INTRODUCTION

Particle filtration is a very efficient, cost-effective, and widely applied method of gas cleaning, hence many filtration studies have been carried out to determine the fate of particles in filter media. For the most part, however, the research has focused on micrometer sized particles. There is an ever increasing need to remove nanoparticles from the gas and the question of how well nanoparticles can be captured by a filter is not yet completely answered. The extension of filter performance studies to nanometer sized particles still needs attention and the experimental basis is incomplete. Due to the complexity of the factual filtration process, its theoretical description is always an approximation, and therefore experimental studies of filter behavior allow on the one hand the judgment of the actual filter performance, while on the other hand the verification of theoretical approaches, which if successfully confirmed provide an insight into the nanoparticle filtration mechanisms. However, the number of contributions investigating the filtration of nanoparticles is still limited and the results are not yet completely satisfactory. Penetration measurements of NaCl particles below 20 nm through metal filters and metal or plastic meshes (Heim et al. 2005) indicate rather good agreement with the classical single-fiber filtration theory in the case of meshes composed of nearly identical collectors. However, for stainless steel filters with slightly polydisperse fiber sizes, the authors measured lower filtration efficiency than predicted by the classical theory. Observed deviations are believed to be due to filter media inhomogeneities. Kim et al. (2006) investigated the influence of relative humidity on NaCl-nanoparticle filtration using fiberglass filters. It was found that the filtration efficiency for particles with diameters above 3 nm appears to be independent of relative humidity. The authors believe that there was an indication of particle thermal rebound for smaller particles; however, they did not compare their data with any filtration models. Investigation with silver nanoparticles in charge equilibrium with sizes from 3 to 20 nm in diameter (Kim et al. 2007) show however no indication of the thermal rebound for the filter used, yet measurements down to 3 nm were only possible for one filter due to counting efficiency issues. Studies of nanoparticle filtration with electrospun polymer fibers with a narrow size distribution in the particle size range above 10 nm in diameter (Yun et al. 2007) did not indicate any substantial differences for charged and uncharged particles and show a good agreement with the classical filtration theory (Kirsh et al. 1975). However, the same authors reported much higher experimentally measured penetration of nanoparticles when compared to the classical theory in the case of a glass HEPA filter with a broad fiber diameter distribution. Similarly, Wang et al. (2007) measured higher penetration of nanoparticles than was expected from the classical theory for four inhomogeneous fiberglass filters. The authors noted that a better agreement between experimental data and model calculations could be obtained when the single fiber efficiency for Brownian diffusion was assumed to be proportional to the Peclet number to the power between −0.46 and −0.41, instead of the theoretical value −2/3. A mathematical model which explained these experimental findings was proposed by Podgórski (2009). Experimental investigation into nanoparticle filtration through nanofibrous media include works by Podgórski et al. (2006), Wang et al. (2008), and Leung et al. (2009). Deposition of neutral, singly, doubly, and triply charged nanoparticles with diameters in the range 25–65 nm on metal wire screens was measured by Alonso et al. (2007), and Heim et al. (2010) measured deposition of singly charged particles with diameters 1.2–8 nm on metal wire grids. Consequently, it is evident why experimental investigations of nanoparticle filtration are still limited. It is primarily because of the difficulties linked to generation of high concentrations of well-defined nanoparticles with respect to their size in order to achieve reliable counting statistics. Furthermore, problems are also linked to a reliable measurement of nanoparticles, especially in the diameter range below 10 nm.

In this work we address the nanoparticle filtration in polydisperse, polypropylene fibrous filter media made using melt-blown technology. This technique is also capable of producing nanofibrous filters (Gradoń et al. 2005; Podgórski et al. 2006). The challenge aerosols were monodisperse, electrostatically size-classified globular particles (proteins, silica particles) from 3.2–27.2 nm. They were aerosolized from appropriate suspensions by means of the electrospray (EAG Mod. 3480, TSI, Inc.). The sizing and delivery of challenge particles was carried out using the Parallel Differential Mobility Analyzer—PDMA (Laschober et al. 2007), a technique which simultaneously allows a measurement of the size distribution of the particles in question and also in parallel an extraction of a narrow, specific size fraction, which was then used in the filter penetration studies. Experimentally determined penetrations of aerosol nanoparticles through four polydisperse filters were compared with the classical filtration theory used for the arithmetic mean fiber diameter and an equivalent fiber diameter obtained from the pressure drop measurements. Then, the theoretical description was extended to account for complete fiber diameter distribution by formulating a partially segregated flow model, which was successfully validated.

2. THEORY OF AEROSOL NANOPARTICLES FILTRATION IN FIBROUS FILTERS

Classical single-fiber theory of aerosol filtration in fibrous media, developed originally for monodisperse filters made of identical fibers, is outlined first in this section, and then it is generalized to account for a distribution of fibers’ diameters, which is an inherent feature of most real fibrous filters. For the consistency of analysis, we selected only theoretically justified correlations pertinent to nanoparticle filtration in order to calculate the single fiber efficiencies for various deposition mechanisms, all of them being based on the most commonly used Kuwabara (1959) cell model. Although the models proposed in Section 2.2 for polydisperse filters are entirely general and they can be applied to particles of arbitrary sizes if appropriate deposition mechanisms are accounted for, further discussion will be focused on filtration mechanisms that are relevant to nanoparticles which have been studied experimentally in this work.

2.1. Classical Description for Homogeneous Filter Media

According to the classical depth filtration theory of aerosols in fibrous filters, the penetration, P, of particles through a filter with a thickness L and a packing density (solidity) α is given by:

wherein E is the total single fiber efficiency, which depends, among other factors, on the fiber diameter, d_F . This total single fiber efficiency may be calculated, in the first approximation, by totaling the individual single fiber efficiencies, E_m , which correspond to particular mechanisms of deposition discussed below: .

The diffusional mechanism of deposition, resulting from a random Brownian motion of particles, is obviously the prevalent one in the case of nanoaerosols. Notwithstanding the existence of numerous specific formulae found in the literature, the general consensus is that the single fiber deposition efficiency due to Brownian diffusion, E_D , is proportional to the Peclet number, Pe, raised to the minus two-third power: E_D ∼ Pe^−2/3. The Peclet number is defined as:

wherein is the superficial (face) mean gas velocity and D is the coefficient of Brownian diffusion of an aerosol particle, which is given by:

In Equation (3) k_B denotes the Boltzmann constant, T the absolute temperature, and f is the Stokesian friction coefficient of a particle with diameter d_P , expressed as:

wherein μ is the gas viscosity, and the Cunningham slip correction factor, C_C , can be calculated as follows (Allen and Raabe 1985):

The Knudsen number for a particle, Kn _P , is defined as:

and λ _g denotes the gas mean free path.

Using the Kuwabara flow field model around a fiber, Lee and Liu (1982) showed theoretically that E_D =C_d [(1−α)/Ku]^1/3Pe^−2/3, and they quoted the value of the numerical constant C_d = 2.6. The symbol Ku denotes the Kuwabara number, also known as the hydrodynamic factor:

On the other hand, a slightly different value of C_d = 2.9 was reported by Natanson (1957) for the Lamb flow, by Spielman and Goren (1968) for the Brinkman flow and by Stechkina and Fuchs (1966) for the simplified Kuwabara flow, but neither of those three works contained the term (1−α)^1/3 (which is practically equal to one for typical modern highly porous filters). In the case of filters examined in this work—see Section 3.2.- - this term varies between 0.980 for the filter N1 and 0.994 for the filter N6. The above cited authors used the Eulerian approach to determine deposition efficiency due to diffusion only, i.e., their analyses were based on solving the convective diffusion equation. Another possible method to study theoretically diffusional filtration of nanoparticles is the Brownian dynamics technique (Lagrangian approach) based on stochastic simulations of aerosol particles’ trajectories. Results obtained in such a way by Podgórski (2002) and by Bałazy and Podgórski (2007) agreed—as for the functional structure—with the form of expression for E_D derived by Lee and Liu (1982), i.e., with the term (1−α)^1/3 retained, but at the same time the use of the numerical constant C_d = 2.9 instead of 2.6, as suggested by other researches, yielded a somewhat better agreement. Thus, here we will use the following theoretical formula for the single fiber deposition efficiency due to Brownian diffusion:

Lee and Liu (1982) also derived a theoretical expression for the single fiber efficiency related to the direct interception, E_R , that accounts for a non-zero particle size (a “point-particle” model is commonly assumed when formulating boundary conditions at the fiber surface in classical diffusion theory); their compact result reads as:

wherein

is the dimensionless interception parameter.

Other mechanical means of filtration, like inertial impaction or sedimentation, are utterly negligible in the case of aerosol nanoparticles, but electrostatic mechanisms might play some role, if present. We can distinguish four principal types of electric forces that could increase filtration efficiency of aerosol particles in fibrous filters: (1) Coulombic attraction between oppositely charged fibers and particles; (2) polarization force in the case of charged fibers and neutral particles; (3) image force induced by charged particles in a neutral filter; (4) dielectrophoretic force arising when a neutral filter and a neutral aerosol particle are placed in an external electric field. Since in current experiments neutral filters and singly charged particles were used, and no external electric field was applied through the filters, only the third kind of the abovementioned interactions will be considered later on in this article. Brown (1993) reported the following theoretical expression for the single fiber efficiency due to action of image force, derived for the Kuwabara flow field model:

wherein

denotes the dimensionless parameter describing particle capture by image forces. Other symbols have the following meaning: q_P —the particle electric charge, ϵ _F —the fiber dielectric constant (ϵ _F = 2.25 was taken for polypropylene fibers), and ϵ ₀—vacuum permittivity. Brown (1993) also noted that some authors reported other values—than theoretically justified factor 2—of the numerical constant in Equation (11), for example, 1.5 or 2.3, obtained by fitting procedures.

Let us compare the relative importance of the three above mentioned mechanisms of deposition, i.e., Brownian diffusion, direct interception and image force action, in the case of nanoparticle filtration under conditions used in our experiments (Section 3.3. Figure 1a shows an example of the single fiber efficiencies—E_D , E _R , and E_IF —calculated from Equations (8), (9), and (11), respectively, for particle size range 3–100 nm for a selected experimentally examined filter (filter N6 described in Section 3.2. at the gas velocity U ₀= 5.88 cm/s used in the experiments. These calculations were performed for the arithmetic mean fiber diameter, which was equal to 1.91 μm, and assuming that all particles carry single elementary charge (q_P = 1.622·10⁻¹⁹ C), as explained in Section 3.3. It can be observed that both E_D and E_IF decrease with an increasing particle size (E_D faster than E_IF in the considered particle size range, primarily because of a rapid fall of the diffusion coefficient with a rise of a nanoparticle's size), while E_R increases then. Notwithstanding this, the direct interception effect is virtually negligible even for particles with a diameter of 100 nm. Similarly, influence of the image force on the deposition efficiency is of secondary importance when compared to diffusional deposition of nanoparticles, although it is much more noticeable than direct interception. Confining analysis to the particle sizes used in our experiments (6.3–27.2 nm) we can conclude that the Brownian diffusion is a completely predominating filtration mechanism, contributing to 96–99% of the overall single fiber efficiency, depending on the particle diameter, see Figure 1b. Thus, one can anticipate that our experimental results should be primarily governed by the laws of Brownian deposition. In particular, at first glance one might expect a −2/3-power dependence of the single fiber efficiency on the Peclet number, but that would be a premature supposition in the case of polydisperse filters, as will be explained afterwards.

FIG. 1 An example of estimations based on the classical theory applied to the arithmetic mean fiber diameter for one of the filters (N6) used in experiments. (a) Effect of particle diameter on the single fiber deposition efficiencies of nanoparticles for various deposition mechanisms: Brownian diffusion (E_D ), action of image force for a singly-charged particle and a neutral fiber (E_IF ), and direct interception (E_R ); (b) relative contribution of Brownian diffusion to the overall single fiber deposition efficiency for the range of particle sizes studied experimentally.

2.2. Existing Methods of Theoretical Description Extension for Polydisperse Fibrous Media

The classical theory of aerosol filtration described in Section 2.1 is very popular because of its simplicity. It may be straightforwardly applied to an idealized case of homogeneous, monodisperse filters consisting of all identical fibers. However, it becomes troublesome in most real instances, when a filter is polydisperse and it is made of fibers with a wide variety of diameters. In such cases Equation (1) is usually utilized by applying a certain equivalent fiber diameter, e.g., the arithmetic mean diameter, d_Fa , determined by image analysis or the equivalent diameter obtained from the pressure drop measurements, d _FΔp . Yet another, somewhat more sophisticated method was proposed by Kirsch et al. (1975) and by Kirsch and Stechkina (1978). This approach is still based on the arithmetic mean fiber diameter, but it introduces two corrections: one for the effective filter packing density (which is related to the dimensionless variance of the fiber diameter distribution), and another for the effective single fiber efficiency in inhomogeneous media (which is determined by comparing experimental and theoretical drag on the fiber). This method is outlined in Appendix B and it was called “the original Kirsch model,” when the fan model filter is used, or “the modified Kirsch model,” if the Kuwabara model filter is utilized.

2.3. Partially Segregated Flow Model for Polydisperse Fibrous Filters

Any simplified approach, based on an equivalent fiber diameter, is not well grounded and it may result in serious errors. Instead, the filter fiber diameter distribution should be taken into account. Moreover, aerosol penetration through a polydisperse filter depends on the relative arrangement of various fibers. We can consider two limiting cases. When fibers with different sizes are distributed evenly in a filter space, and flow in a filter is well mixed at the mesoscale, the term that appears in square brackets of Equation (1) (that can be interpreted as the unit layer efficiency) may be simply averaged using the fiber diameter distribution function g(d_F ). Such an approach is based on the assumption that superficial gas velocity is the same for all Kuwabara cells containing various fibers and has been named the Perfectly Mixed Flow Model (PMFM), (Podgórski and Jackiewicz 2009). Thus, aerosol penetration according to PMFM can be calculated as:

where summation index m goes over all deposition mechanisms that are taken into account and d _Fmin and d _Fmax are the minimal and maximal observed fiber diameters. PMFM represents an estimation of the lower limit of aerosol penetration through a polydisperse filter. In the second extreme, one can consider a situation in which all fibers with different diameters are completely separated from each other. Assuming that the pressure drop per unit filter thickness is the same for all Kuwabara cells with various fibers, we can formulate a Fully Segregated Flow Model (FSFM) and calculate penetration as, see Podgórski (2009) for details of derivation:

P(d_F ) in Equation (14) denotes penetration calculated using Equation (1) as for a monodisperse filter with fibers of diameter d_F , and the single fiber deposition efficiencies E_m (d_F ) for particular fibers’ diameters should be computed from the appropriate correlations reviewed in Section 2.1 for individual superficial gas velocities U ₀(d_F ), corresponding to various Kuwabara cells with different fibers, which are given by Podgórski (2009):

wherein denotes the mean superficial gas velocity for the entire filter. FSFM is anticipated to be the estimate of the upper limit of aerosol penetration through a polydisperse filter. The real situation lies somewhere in between PMFM and FSFM, thus, we finally postulate a general Partially Segregated Flow Model (PSFM), for which the penetration is calculated as a linear combination of penetrations computed from the two limiting models—Equations (13) and (14):

wherein we introduced a dimensionless phenomenological parameter s that can be named the segregation intensity. It can vary between zero (for perfect mixing) and one (for full segregation) and it should be determined experimentally.

The models described in this section were formulated in a general manner, for any arbitrary fiber size distribution, g(d_F ). Our experience (Section 3.2. as well as numerous data available in the literature indicate that for the majority of real fibrous filters produced using various techniques, the filter's fiber diameter distribution, g(d_F ), can be described by the log-normal one, whose normalized density is given by:

wherein d_Fg is the geometric mean fiber diameter and σ _gdF is the geometric standard deviation. Equations (13)– (15) have the following, working form, which will be used in line with Equation (16) and correlations for the single fiber efficiencies reviewed in Section 2.1 to interpret experimental data:

where

When the Brownian diffusion is the only important mechanism of deposition, the limiting models may be solved to obtain explicit formulae in the case of log-normal fiber diameter distribution—PMFM has an exact analytical solution, and an approximate one was found for FSFM, see Appendix A. Nevertheless, complete numerical solutions of Equations (18)– (19), not limited exclusively to the diffusive deposition, will be used in Section 4 to interpret our experimental data.

3. MATERIALS AND METHODS

This section describes a method of fibrous filter production, their structural characteristics and techniques used to determine penetration of aerosol nanoparticles through the filters.

3.1. Production of Fibrous Filter Media

The melt-blown technique was used for the production of fibrous filter media. The scheme of the process including some details of the die construction and fiber formation in a single nozzle of the die are shown in Figures 2a– c, respectively. Granulated polymer placed in the container 2 is taken by the screw of the extruder and pressed into the die 3. During transportation the polymer is heated from the outside by an electronic heater 1 in three steps. The flow rate of melted and homogenized polymer is determined by the rate of the screw rotation, controlled via an inverter through an electronic motor with the gear system. The power transmission system is protected from the heating zone by the cooling system. The melted polymer is extruded through the row of die nozzles, Figure 1b. The pressure and temperature of the polymer in the die are measured with the sensing elements 6. In the die structure the polymer nozzles are surrounded by the air nozzles. The stream of hot air from compressor 7 flowing along melted polymer filaments extends them to the desired diameter. The solidified fibers 4 are collected on the mandrel of the fiber receiver 5. This mandrel rotates and moves to-and-from to form a proper filter structure defined by the fiber diameter and the local packing density of fibers. The rates of the mandrel movements and its distance from the die determine these parameters. The key factor of the filter performance is the fiber diameter. The estimation of the process parameters for formation of fibers of assumed diameter was accomplished though the modeling of the phenomenon in a single nozzle of the die, Figure 2c. The basic information describing the formation of a thin fiber are derived, Gradoń et al. (2005), under the following assumption; a fiber has a cylindrical shape, a polymer has the same density for melted and solid phases, an axial velocity within a melted fiber is uniform in the fiber's cross-section, melted polymer is a Newtonian liquid with Arrhenius type of dependence of viscosity on temperature, the heat capacity of the polymer is constant, the heat conduction in the axial direction is negligible and finally the temperature distribution is uniform in the fiber cross-section. Taking into account constancy of the polymer volumetric flow rate at a given pressure and temperature in the die and the flow rate and temperature of the air in the nozzle of the die, the process was modeled. Balancing the momentum and using the stress-strain relation and energy balance, we were able to estimate the influence of process parameters on the diameter of the solid fiber R_f produced from the die nozzle of the diameter R_o . The particular filter media characterized below were made of the polypropylene (Borealis). The temperature of the polymer in the die was selected from the range (260–280°C) and pressure in the die was from the range (3–5 atm). The relative velocity of the air to the polymer velocity at the nozzle was equal 52 m/s for all cases. The temperature of the air was equal 360°C.

FIG. 2 System of fibrous filter production (a), the die structure (b), and single fiber formation (c).

3.2. Structural Characteristics of Investigated Filters

Four fibrous filters were used to perform the experiments within the framework of this study. All of them were produced utilizing the melt-blown technology described in previous section. Their structures were fully characterized and the results are presented in Table 1. The following parameters of the filter media were determined: filter thickness, L, packing density, α, basis weight, q_s . Filter thicknesses were measured in a systematic way using a caliper with a resolution 0.01 mm. The values of both latter parameters (α and q_s ) were evaluated by the gravimetric method using an electronic scales with a resolution 0.1 mg. All these measurements (thickness and mass of a filter sample) were repeated eight times for each filter for its various samples cut out from different parts of a flat mat. Standard errors of filters thickness determination ranged from 0.08 mm for filter N6 to 0.14 mm for filters N1 and F3, whilst the standard errors of a filter sample mass measurement varied between 0.4 mg for filter N1 and 0.8 mg for filter F3. This resulted in the estimated relative standard errors of the filter thickness measurement between 8% (filter F1) and 14% (filter N1), and they ranged from 7% (filter F1) to 15% (filter N1) in the case of the filter packing density determination. Knowing the mass of a filter sample, m_F , the surface density was calculated as the ratio of the mass to the filter area, F (which was equal to 16 cm²). Having determined the filter porosity, ϵ, according to the following formula:

we could compute the filter solidity as: α = 1 – ϵ. All used filters were made of polypropylene, whose density, ρ, is 900 kg/m³.

TABLE 1 Summary of filters’ structural parameters

Filter	L[mm]	α [−]	qs [kg/m^2]	dFg [μm]	σ gdF [-]	dFa [μm]	dFmin [μm]	dFmax [μm]
N1	1.00	0.0582	0.0643	4.88	2.24	5.33	2.06	17.64
N6	0.78	0.0171	0.0120	1.66	1.52	1.91	0.87	5.83
F1	1.70	0.0388	0.0593	1.97	1.58	2.55	0.72	11.85
F3	1.58	0.0419	0.0595	2.61	1.68	3.15	0.97	9.98

The filters used in this study were found to be structurally nonuniform, composed of fibers with various diameters. The arithmetic mean fiber diameters, d_Fa , and the observed range of fiber sizes (minimal, d _Fmin, and maximal fiber diameter, d _Fmax) were determined analyzing SEM images obtained under the electron microscope. Figure 3 depicts the photographs of the filters N1 and N6 taken at the magnification of 400×, and filters F1 and F3, when the magnification was 500×. Determined distributions of the fiber diameters are shown for all tested filters in Figure 4. It was found that all of them can be satisfactory described by the log-normal distribution. Parameters of this distribution (the geometric mean fiber diameter, d_Fg , and the geometric standard deviation, σ _gdF ) can be easily and precisely determined by fitting the discrete experimental data of fibers’ diameters to the normalized cumulative log-normal function. Such a procedure allows one to find these parameters with a high accuracy, even for few measured fiber diameters.

FIG. 3 SEM images of the tested filters: (a) N1, (b) N6, (c) F1, and (d) F3.

FIG. 4 Normalized cumulative log-normal distributions of fibers’ diameters for investigated filters.

As can be seen in Table 1, the filter N6 was the thinnest and the filter F1 the thickest amongst the analyzed filters. The most densely packed was filter N1. Consequently, the basis weight of the #N1 was lower than that for the other filters. The filter N1 was composed of fibers with diameters in the range 2.06–17.64 μm, while all others contained fibers with smaller diameters. The size distributions of fibers for all tested filters were broad. The filter N1 had the largest mean fiber diameter, while the mean fiber diameter of the filter N6 was the lowest one.

3.3. Nanoparticle Aerosol Generation and Measurement

Various proteins and silica nanoparticles with diameters ranging from 6.3 to 27.2 nm were used in this work. In order to obtain well-defined nanoaerosols for filter performance investigation, an arrangement containing an electrospray particle generator with a subsequent electrostatic size classification was used as described previously (Allmaier et al. 2008). Suspensions holding a given type of nanoparticle were prepared in 20 mM aqueous ammonium acetate buffer solutions. Stable output of challenge aerosols was obtained in case of proteins using concentrations between 20 and 100 μg/mL. For silica particle suspensions the original stock solutions were diluted in a ratio of 1:1000. All particle types (Sigma Aldrich, Inc.) used in this work are summarized in Table 2.

TABLE 2 Materials used for filter penetration measurements (all from Sigma Aldrich)

	Particle type	dP [nm]
Proteins:	Bovine Serum Albumin	6.3
	Enolase	7.2
	Gamma-Globulin	8.2
	Ferritin	12.8
	Thyroglobulin	13.3
Silica Particle:	HS-40	18.2
	AS-40	27.2

A schematic diagram of the system used in this study is shown in Figure 5. An electrospray aerosol generator (EAG, Model 3480, TSI Inc., Minneapolis, MN, USA) was used to produce the required particles. A fused silica capillary of 40 μm was immersed in the suspension and a pressure of about 30 kPa was used to push the sample through the capillary into the electrospray chamber. Within the chamber a 2 L/min sheath flow of dry, particle-free air was used together with 0.3 L/min of CO₂ (Air Liquide, Quality N45) to prevent electrostatic discharge and corona formation at the tip of the capillary. Monodisperse droplets with sizes of the order of 150 nm can be obtained by operating the electrospray in the “cone-jet-mode” (Cloupeau and Prunet-Foch 1989; Chen and Pui 1997; Kaufman 1998). For a stable cone-jet operation a positive voltage of 3 kV was applied to the capillary and confirmed by the visual inspection. Within the electrospray chamber a Po-210 source radioactive source was used to control particle loss due to space charge and to prevent evaporating droplets from reaching the Rayleigh limit (Marginean et al. 2006). Due to these processes particles remaining after the droplet evaporation have a steady-state equilibrium charge distribution (Fuchs 1963).

FIG. 5 Experimental arrangement.

The sizing and preparation of challenge particles were carried out using a Parallel Differential Mobility Analyzer (PDMA). The PDMA is an advanced DMA system (Austrian patent no. 502207A1, 2007), which can be used to simultaneously monitor the size distribution of the nanoaerosol particles in question and to select one specific size fraction (Allmaier et al. 2008). The PDMA consists of two in-house built nano-DMAs working in parallel (Figure 5). Both nano-DMAs are identical in construction and run under the identical hydrodynamic conditions. They were operated at a 20 L/min sheath air flow and 1 L/min aerosol flow. The analyzer DMA—nDMA1—is combined with an electrical aerosol detection device working on the Faraday cup principle. The nDMA1 is used for scanning the generated aerosol over the whole size spectrum determining the particle size distribution obtained from the EAG. The classifier DMA—nDMA2—is operated at one specific voltage setting corresponding to a particular particle mobility size class and thus separating one size fraction from the measured distribution.

The quality and the size verification of the thus obtained challenge aerosol from the nDMA2 were also evaluated by a separate DMA combined with a condensation particle counter (CPC, model 3022A, TSI, Inc.). Results are shown in Figure 6. The measured data show well-defined nanoparticle size distributions with the width at the half height corresponding to the resolution of the applied measuring system (Wang and Flagan 1990). The measured distributions show that aerosol particles exiting from the PDMA are monodisperse and mainly singly charged. Concentrations of doubly charged particles are in this case negligible compared to the singly charged particle fraction. Hence, the assumption that monodisperse, singly charged nanoparticles are used for this filter study is justified.

FIG. 6 Size spectra of Enolase, Ferritin and AS-40 silica particles exiting the classifier DMA – nDMA2.

The filter penetration measurement system is similar to an arrangement described previously (Heim et al. 2005). The key aim was to obtain analogous experimental conditions providing a possibility to compare results. Two identical Sartorius^TM aluminum filter holders were connected in parallel via a 3-way magnetic valve. One filter holder was used as a dummy unit. The second one was furnished with the investigated filter media. The filtration area for all filter samples was equal to 284 mm². The flow rate of 1 L/min resulted in the gas face velocity of 5.88 cm/s. The upstream and downstream particle concentrations were determined using water CPC (WCPC Mod. 3785, TSI, Inc., MN, USA). This device was operated in the single particle counting mode for both up and downstream concentration measurements for all particles used here. Before measurements of filter penetration, a leakage test was performed by applying zero voltage to the classifier DMA—nDMA2—to ensure that no particles leaking from the system or possible residual particles in the sampling tubes were disturbing the penetration measurements (Kim et al. 2006). The measuring times were chosen between 120 s and 3600 s depending on particle concentrations in order to achieve statistically satisfactory conditions. Alternating measurements for up and downstream concentrations were performed 3 times for low particle concentration that resulted in measuring times of more than 1800 s. For high concentrations and therefore shorter measuring times, 5 reruns were typically performed to obtain reliable statistics. The filter penetration measurements for all particle sizes and filter media were followed again by a leakage test. The measurement of particle concentrations by the WCPC was done using the totalizer mode for particle concentrations lower than 2000 #/cm⁻³ with an output of particle number/time. For penetration calculations this particle number/time value was calculated to a mean number concentration. For errors of the number concentrations, the ranges of uncertainties of air flow and particle counting of the WCPC specified by the manufacturer were taken into account. For concentrations higher than 2000 #/cm⁻³ the software monitored one value of number concentration every 6 s. Thus, the mean number concentrations were used for penetration calculations and the standard deviation for errors calculations. The errors of penetrations were calculated by Gaussian error propagation using the deviations of the number concentrations. Typically taking into account all experimental uncertainties, measurement errors were typically of the order of 10% or less.

4. RESULTS AND MODEL VALIDATION

Source data of experimentally determined penetrations, P, for seven kinds of well-defined nanoparticles through four precisely described fibrous filters are collected in Table 3 together with the estimated values of the standard deviations (SD). These data were compared with results of theoretical calculations performed using two methods, described in Section 2. namely: (1) applying the classical theory (Section 2.1. to the arithmetic mean fiber diameter, d_Fa , and (2) utilizing the PSFM outlined in Section 2.2. The former approach is self-evident and requires no further elaboration. The approach is simply the single fiber efficiencies—E_D , E_R , and E_IF —are computed for d_Fa from Equations (8), (9), and (11), then the total single fiber efficiency is calculated additively as: E=E_D +E_R +E_IF , and finally this value of E is inserted into Equation (1), which is used in line with d_Fa in place of d_F in its denominator in order to calculate penetration, P, of aerosol particles of a given size. The results obtained in such a way will be discussed afterwards, jointly with those obtained from the latter approach based on the PSFM, which was utilized as follows. For each ith particle diameter an individual value of the segregation intensity, s_i , for the Partially Segregated Flow Model (PSFM) described in the Section 2.2. for which the PSFM matches the experimental data for this particular particle diameter exactly, was calculated as follows:

wherein P _exp,i is experimentally measured penetration for ith particle diameter, whereas and are penetrations calculated for this particle size using the Perfectly Mixed Flow Model, Equation (18), and Fully Segregated Flow Model, Equation (19), respectively, the latter one used in line with Equation (20). Results of such calculations are presented in Table 4, where the segregation intensity is also expressed as a function of the Peclet number related to a given particle diameter and based on the arithmetic mean fiber diameter for each particular filter.

TABLE 3 Summary of experimentally measured penetrations

	Penetration, P±SD [-]
Particle diameter, dP [nm]	Filter N1	Filter N6	Filter F1	Filter F3
6.3	6.1·10^−4 ± 9·10^−5	1.1·10^−3 ± 1·10^−4	1.7·10^−6 ± 2·10^−7	2.0·10^−6 ± 3·10^−7
7.2	1.6·10^−3 ± 2·10^−4	2.1·10^−3 ± 3·10^−4	1.1·10^−6 ± 2·10^−7	2.8·10^−6 ± 5·10^−7
8.2	4.4·10^−3 ± 6·10^−4	4.5·10^−3 ± 6·10^−4	8.3·10^−6 ± 1·10^−6	7.3·10^−6 ± 7·10^−7
12.8	3.4·10^−2 ± 5·10^−3	3.3·10^−2 ± 3·10^−3	2.5·10^−4 ± 4·10^−5	8.0·10^−4 ± 1·10^−4
13.3	4.6·10^−2 ± 4·10^−3	4.2·10^−2 ± 6·10^−3	7.2·10^−4 ± 7·10^−6	1.2·10^−3 ± 2·10^−4
18.2	1.9·10^−1 ± 2·10^−2	9.4·10^−2 ± 1·10^−2	6.4·10^−3 ± 9·10^−4	8.4·10^−3 ± 9·10^−4
27.2	3.8·10^−1 ± 5·10^−2	2.0·10^−1 ± 2·10^−2	4.8·10^−2 ± 7·10^−3	6.7·10^−2 ± 1·10^−2

TABLE 4 Values of the segregation intensity, s, as a function of the Peclet number, Pe, determined from experimental data using the PSFM

	Filter N1		Filter N6		Filter F1		Filter F3
dP [nm]	Pe	s	Pe	s	Pe	s	Pe	s
6.3	2.34	0.00805	0.837	0.0690	1.12	0.000594	1.38	0.000559
7.2	3.04	0.0161	1.09	0.0958	1.46	0.000341	1.80	0.000459
8.2	3.94	0.0351	1.41	0.152	1.89	0.00140	2.33	0.000759
12.8	9.51	0.149	3.41	0.455	4.55	0.0151	5.62	0.0240
13.3	10.3	0.193	3.67	0.541	4.91	0.0401	6.06	0.0330
18.2	19.0	0.582	6.81	0.741	9.09	0.193	11.2	0.125
27.2	41.6	0.861	14.9	0.973	19.9	0.755	24.6	0.536

Analyzing data given in Table 4, one finds that s is not a constant in the case of nanoparticles, and that its value varies by two orders of magnitude in the considered particle size range. So far, we had been very successful in using the PSFM to describe filtration of aerosol particles with diameters above 200 nm in polydisperse fibrous filters, assuming a constant s-value for a given filter and for a specific gas velocity (Jackiewicz and Podgórski 2009), but now, for nanoparticles, this approach failed. Nevertheless, the s-values determined in such a way were found to be in the expected range between zero and one. We believe that this finding can be explained as follows: for larger particles, examined by us previously (>200 nm; up to 10 μm), the effect of Brownian mixing becomes weaker and weaker with the increase of the particle diameter, hence, segregation of aerosol particles within a porous space of inhomogeneous filter structures is mainly related to segregation of the air flow itself (i.e., this phenomenon is mainly of a convective nature—carrying particles by a non-uniformly flowing gas), thus, s may be indeed expected to be independent of the particle size, and it should depend only on the filter structure and on the air flow rate, but not on the particle size. However, in the case of nanoparticles, we have two competing transport phenomena for aerosol particles: Firstly, convection by fluid, resulting in (longitudinal—i.e., along the main direction of gas flow through a filter) segregation of the air flow (and, thereby, to some extent—in segregation of aerosol particles) within an inhomogeneous filter structure. Secondly, by Brownian diffusion that causes lateral, i.e., transversely to that direction (n.b.: not only lateral—also longitudinal, but from the point of view of particles dispersion; the former, which causes a “smoothening” of a local profile of particles’ concentration on the microscale, is more important) mixing of nanoparticles, which finally should reduce the effect of convective segregation. Thereby, one should not be surprised that svaries with the nanoparticles’ size. Our experimental data confirm this expected trend—an increase of s with an increase of particle diameter (or, equivalently, the Peclet number). Thus, the next step in our analysis was an attempt to find a simple correlation that could describe variation of s. Since the relative importance of the Brownian motion (when it is compared to the convective transport) is described by the Peclet (Pe) number, we tried to find a simple approximating formula for a function s= f(Pe). By definition, s may only change between zero and one, so we assumed that this relationship could be described by a sigmoid-shaped function with two asymptotes: s→0 for Pe→0, and s→1 for Pe→∞. Such properties has, for example, log-normal distribution, defined with respect to the s= f(Pe) relationship, by the following cumulative distribution:

This distribution has two fitting parameters: the geometric mean value of the Peclet number, Pe _g , and the geometric standard deviation, σ _gPe , and the symbol erf(…) denotes the error function. Another possible fitting formula with desired asymptotic properties may be based on the log-logistic distribution with the cumulative distribution defined by:

wherein Pe _m is the median of this distribution (i.e., such a value of the Peclet number, for which the segregation intensity equals 0.5), and β is the shape parameter. It was found that both proposed fitting functions enabled one to very accurately interpolate experimental values of the segregation intensity with respect to the Peclet number for all investigated filters. This is illustrated in Figure 7, and determined parameters of both distributions defined by Equations (23) and (24) are listed in Table 5. The log-logistic approximation turned out to be slightly better (the mean—for all filters—value of the coefficient of determination was found for this kind of fit R²= 0.9969, meaning nearly perfect approximation, whilst it was R²= 0.9869 on average for the log-normal approximation). Thus, the interpolating formulae in the form of Equation (24) will be used in further analysis in order to describe dependence of the segregation intensity on the Peclet number. It may be also worthwhile mentioning that values of Pe _g in Equation (23) and Pe _m in Equation (24) were found to be very close.

FIG. 7 Log-logistic (solid lines) and log-normal (dotted lines) approximations of the segregation intensity, s, versus the Peclet number, Pe, based on the arithmetic mean fiber diameter for four examined filters.

TABLE 5 Parameters of the fitting functions s(Pe) and the coefficients of determination, R²

	Log-normal fit, Equation (23)			Log-logistic fit, Equation (24)
Filter	Pe g	σ gPe	R^2	Pe m	β	R^2
N1	18.36	2.26	0.9841	17.75	2.40	0.9941
N6	3.57	2.55	0.9955	3.58	1.87	0.9940
F1	14.30	1.98	0.9788	14.08	3.26	0.9996
F3	24.17	2.20	0.9890	23.14	2.85	0.9998

Having these correlations s= f(Pe) determined, we then compared (Figure 8) experimental data of penetration (points in these plots) with predictions of the PSFM (solid lines) used in line with these interpolating formulae defined by Equation (23), and also with the classical theory of aerosol filtration in monodisperse filters applied to the mean fiber diameter (dotted lines in Figure 8). It may be concluded that PSFM used together with the log-logistic approximation of the segregation intensity versus the Peclet number is capable of very precisely describing penetration of aerosol nanoparticles through all analyzed polydisperse fibrous filters, while the classical theory applied to the arithmetic mean fiber diameter tremendously underestimates nanoparticle penetration by several orders of magnitude.

FIG. 8 Comparison of experimental data of penetration (points) with PSFM used in line with the log-logistic approximation s (Pe)—solid lines, and with the classical theory applied to the arithmetic mean fiber diameter—dotted lines.

Since, as it was shown above, the classical theory entirely fails in the case of nanoparticle filtration when it is used together with the arithmetic mean fiber diameter, we also analyzed whether it could result in more reliable predictions if applied to any other equivalent (or surrogate) fiber diameter with a constant, specific value for the polydisperse filter in question. Figure 9a shows a representative example of such considerations for the filter N6. The dashed line represents the penetration calculated from the classical theory for a fitted fiber diameter d_Ffit = 4.37 μm, for which this classical theory matches experimental data for the smallest examined particles, but, unfortunately, it significantly overestimates experimental values of penetration for all larger particles. On the other hand, the dotted line in Figure 9a was obtained in the same manner but for a fitted fiber diameter d_Ffit = 3.37 μm, when the classical theory agrees with measurement results for largest investigated particles. However, it then considerably underestimates experimental data for all smaller particles. Thus, we can conclude that it is impossible to select any unique value of a fitted fiber diameter in order to obtain a good agreement between the classical theory and experiments for the whole range of particle diameters. We can observe in Figure 9a that the penetration curves resulting from the classical theory are steeper than the experimental curve. It is even more clear when one analyzes the effect of the Peclet number on the single fiber efficiency in a log-log plot, see Figure 9b. Experimental values of the single fiber efficiency (points in Figure 9b), E _exp, were calculated by inverting Equation (1) applied to the arithmetic mean fiber diameter, d_Fa , and using experimental data of penetration, P _exp:

FIG. 9 Analysis of results for the filter N6: (a) an attempt of application of the classical theory for two fitted fiber diameters: d_Ffit = 3.37 μm (dotted line) and d_Ffit = 4.37 μm (dashed line), and for the pressure drop equivalent diameter d _FΔp = 3.64 μm (solid line); (b) interpolation of experimental values of the single fiber efficiency by the power function of the Peclet number: solid line—free exponent of the Peclet number, dotted line—fixed exponent of the Peclet number (=−2/3); (c) comparison of experimental data with the “original” and “modified” Kirsch model; (d) relative errors (indicated by symbols) of calculation of –ln P for various models.

It is evident from Figure 9b that the slope of the curve ln P_exp versus ln Pe considerably differs from what is expected from the classical theory value −2/3 (dotted line) and it was found to be −0.516 (solid line in Figure 9b). This is qualitatively consistent with the Wang et al. (2007) data, who also reported higher than −2/3 values of the Peclet number exponent determined from experimental data of nanoparticle penetration through four various polydisperse fibrous filters (the authors, using the fitting procedure, obtained values of this exponent in the range of −0.46 to −0.41, depending on the filter). Next, we analyzed whether the pressure drop equivalent fiber diameter, d _FΔp , can serve as a better alternative than the arithmetic mean fiber diameter in order to predict penetration of nanoparticles on the basis of the classical theory. The value of d _FΔp was found to be 3.64 μm for filter N6 using experimental data of the pressure drop across the filter Δp and applying theoretical Pich (1971) equation:

wherein Kn_F =2λ _g /d_F denotes the Knudsen number for a fiber. It may be noted that the pressure drop calculated theoretically from Equation (26) with the arithmetic mean fiber diameter d_Fa used in its denominator in place of d_F was several times higher than the pressure drop measured experimentally for all investigated filters (8.7 times for N1, 3.6 times for N6, 5.7 times for F1, and 4.4 times for F3). In other words, the pressure drop equivalent diameters obtained from experimental values of Δp with the use of Equation (26) were always greater than values of d_Fa obtained from the image analysis (3.0 times for N1, 1.9 times for N6, 2.4 times for F1, and 2.1 times for F3). As seen in Figure 9a, it is possible to obtain much better agreement between the classical theory and experiment when d _FΔp is used instead of d_Fa , especially for the two largest analyzed particles. Nevertheless, such a theoretical curve is still much steeper than the pattern of experimental results, yielding underestimation of penetration by an order of magnitude for the finest considered particles. Finally, we compared experimental data of penetration with predictions of two variants of the Kirsch model described in Appendix B. It can be observed in Figure 9c that the “original Kirsch model” (based on single fiber efficiencies for the fan model filter) always underestimates experimental results of nanoparticle penetration, more noticeably for smaller particles. On the other hand, the “modified Kirsch model” (that utilizes formulae for the single fiber efficiencies derived for the Kuwabara flow model) works much better for finest particles, but it more and more gradually overestimates penetration of the bigger ones. Since both variants of the Kirsch model are based on the arithmetic mean fiber diameter, they still retain a −2/3-power dependence of the diffusional single fiber efficiency on the Peclet number, resulting in a steeper shape of the fractional penetration curve, compared to what was measured experimentally.

In order to evaluate quantitatively effectiveness of various methods of theoretical calculation of aerosol nanoparticle penetration, we computed the relative errors, RE _i , of the minus logarithm of penetration predicted by them at ith experimental points:

wherein P _exp,i denotes value of the penetration measured experimentally for ith particle diameter, whereas P _model,i is the penetration calculated theoretically for this particle size from a specific model (PSFM, classical theory for the pressure drop equivalent diameter, “original” and “modified” Kirsch model). We decided not to compare directly theoretical and experimental values of penetration, since these ones may vary by several orders of magnitude for nanoparticles, but the values of −lnP (which are proportional to the single fiber efficiency and they vary in much narrower range). It turned out that for the filter N6 such computed values of RE _i ranged from −12.2% to +35.8% (depending on the particle size), when the classical theory was used in line with the pressure drop equivalent fiber diameter, from +6.0% to +80.3% for the “original” Kirsch model, from −27.7% to +11.3% for the “modified” Kirsch model, and from −1.5% to +2.5% for our PSFM. Distribution of these relative errors is shown in Figure 9d for the filter N6. Finally, we show the root mean square relative error, RMSRE, computed for each model as follows:

wherein N is number of experimental points (N= 7 in our case). The “original” Kirsch model turned out to be the worst approach from this criterion viewpoint, characterizing itself by RMSRE = 42.9% for the filter N6. Over twofold smaller error (RMSRE = 19.9%) was found for this filter, when the classical theory was applied for the pressure drop equivalent diameter. Comparable, but somewhat better effectiveness of experimental data reproduction was noted for the “modified” Kirsch model (RMSRE = 14.6%). Nevertheless, none of these three abovementioned approaches can compare with the precision of description obtained from our PSFM, for which RMSRE was as low as only 1.53% in the case of the filter N6.

5. CONCLUSIONS

Experimental results of singly charged, monodisperse aerosol nanoparticle penetration through four polydisperse fibrous filters were presented. Theoretical analysis indicates that the Brownian diffusion is the entirely predominant mechanism of deposition under selected experimental conditions. It was demonstrated that the classical theory of aerosol filtration drastically underestimates nanoparticle penetration when it is used in line with the arithmetic mean fiber diameter in the case of polydisperse filters. Significantly better agreement was obtained when the pressure drop equivalent diameter was used, but serious discrepancy was still observed for the smallest particles. The “original” Kirsch model of aerosol filtration in polydisperse fibrous filters turned out to be unsatisfactory in the case of nanoparticles, and its “modified” version (obtained by replacing formulae for the single fiber efficiencies for the fan model filter with the corresponding ones derived for the Kuwabara flow model) was found to be a better, but still not a perfect method of approach. It was also shown that it is impossible to select any constant fitted fiber diameter that would enable one to interpolate experimental data precisely for the whole span of investigated particle sizes. Contrary to that, the partially segregated flow model, which accounts for the fiber diameter distribution, was successfully applied. It was found that for the analyzed filters, which had the log-normal fiber size distributions, the segregation intensity increased with the increase in the particle diameter and it was possible to interpolate the segregation intensity versus the Peclet number very precisely by means of either the log-logistic or the log-normal function. When the former interpolation was used together with the partially segregated flow model, a perfect agreement with experimentally measured penetration was obtained for all analyzed particles and filters.

6. APPENDIX A: APPROXIMATE EXPLICIT SOLUTIONS OF LIMITING MODELS (PMFM AND FSFM) FOR DEPOSITION DUE TO BROWNIAN DIFFUSION ONLY IN POLYDISPERSE FILTERS WITH LOG-NORMAL FIBER DIAMETER DISTRIBUTION

Let us consider the case when the Brownian diffusion is the only significant mechanism of deposition and the single fiber efficiency is given by Equation (8). Moreover, let us assume that a polydisperse filter in question has log-normal fiber diameter distribution defined by Equation (17). Then, aerosol penetration according to PMFM, Equation (18), can be expressed as:

We assume that the integral over the finite interval [d _Fmin; d _Fmax] in Equation (18) may be realistically approximated by that over infinite interval [0; ∞] in the case of highly polydisperse filters. E_D (d_Fg ) in Equation (A1) denotes the single fiber deposition efficiency for Brownian diffusion calculated as for a monodisperse filter composed of fibers with diameters equal to the geometric mean fiber diameter, d_Fg , of the considered polydisperse filter:

and Pe _g is the Peclet number based on d_Fg :

Equation (A1) has an exact simple analytical solution:

wherein P_g denotes penetration calculated for the geometric mean fiber diameter d_Fg in the case of diffusional deposition only:

with E_D (d_Fg ) given by Equation (A2).

For the second limiting model, FSFM, penetration of nanoparticles (being deposited by Brownian diffusion only) through a polydisperse filter with log-normal fiber diameter distribution, which is described in general case by Equations (19)– (20), can be written as (cf., Podgórski 2009):

Equation (A6) has no explicit analytical representation. A simple, approximate numerical solution was found in the form:

wherein P_g is defined by Equation (A5), and parameters a and b in Equation (A7) depend solely on σ _gdF . Results of numerical integration of Equation (A6) allowed us to propose the following approximations:

which have the expected limiting properties for monodisperse filters (i.e., when σ _gdF = 1), namely a≈b≈ 0.

Equations (A4) and (A7) can be used for a quick estimation of limiting values of nanoparticle penetration (according to PMFM and FSFM models) through polydisperse fibrous filters with log-normal fiber diameter distribution when the Brownian diffusion is the only significant mechanism of deposition.

7. APPENDIX B: AN OUTLINE OF THE KIRSCH MODEL OF AEROSOL FILTRATION IN POLYDISPERSE, INHOMOGENEOUS FILTER MEDIA

The Kirsch model (cf., Kirsch et al. 1975; Kirsch and Stechkina 1978) of aerosol filtration in non-uniform, polydisperse fibrous media is based on the comparison of the resistance and efficiency of a real filter with those for the fan model filter with the same parameters taken as a reference for an uniform filter. It may be recapitulated by the following equation for aerosol penetration, which is based on the arithmetic mean fiber diameter, d_Fa :

An effective filter packing density, α _eff , is used everywhere in calculations instead of the real packing density, α, where:

and SCV denotes the squared coefficient of variation (or dimensionless variance) of the fiber diameter distribution, defined as:

σ _dF in Equation [B3] is the standard deviation of the fiber diameter distribution (not to be confused with the geometric standard deviation, σ _gdF ), and d_Frms is the root mean square fiber diameter given by:

Thus, the Equation (B2) may be rewritten as:

and reduces in the case of log-normal fiber diameter distribution to:

The effective single fiber efficiency in Equation (B1), E_eff , is obtained by dividing the single fiber efficiency, E(d_Fa , α _eff ), calculated for the arithmetic mean fiber diameter, d_Fa , and for the effective filter packing density, α _eff , by the filter structure nonuniformity factor ϵ ₀:

The value of ϵ ₀ for a polydisperse fibrous filter is defined in the Kirsch model as the ratio of the dimensionless drag acting on unit fiber length for the fan model filter, F^f ₀, to that one for the filter in question, F ₀, and the subscript “0” denotes that both these quantities are determined for non-slip conditions (i.e., when the Knudsen number based on the arithmetic mean fiber diameter, Kn_Fa =2λ _g /d_Fa , tends to zero):

The value of the dimensionless drag on the fiber for a considered polydisperse filter under non-slip conditions, F ₀, is related in the Kirsch model to its value F obtained with an allowance for the effect of gas slip by the following formula:

On the other hand, the value of F may be obtained from the experimentally measured pressure drop, Δp_exp , by means of the equation:

Combining Equations (B8)– (B10), one obtains the following quadratic algebraic equation allowing to calculate ϵ ₀:

which root equals:

To shorten notation, we denoted by A in Equations (B11)– (B12) the term:

The value of F^f ₀ for the fan model filter is given by (cf., Kirsch et al. 1975; Kirsch and Stechkina 1978):

Inserting Equations (B10), (B13), and (B14) into Equation (B12) we can determine the sought value of ϵ ₀. To calculate contribution to the single fiber deposition efficiency due to Brownian diffusion, E_D (d_Fa , α _eff ), and due to direct interception, E_R (d_Fa , α _eff ), the following formulae were proposed in the Kirsch model:

wherein K^f ₁ is the hydrodynamic factor for the fan model filter:

whilst Pe _a and N_Ra denote the Peclet number and the interception parameter, respectively, both based on the arithmetic mean fiber diameter d_Fa :

No formula describing the single fiber efficiency due to action of image force is yet available for the fan filter model, thus, we adapted Equations (11)– (12) derived for the Kuwabara model by means of using d_Fa and α _eff in place of d_F and α:

with

and

It should be noted, that in the case of a negligible effect of gas slip on a filter fiber (Kn _Fa ≪ 1), as it was under experimental conditions in this work, Equations (B12)– (B13) reduce to simple formula:

In other words, ϵ ₀ can be easily calculated then as a ratio of the pressure drop for an assumed model of an uniform filter structure calculated theoretically with the gas slip neglected, Δp _th,0, to the pressure drop across investigated filter measured experimentally, Δp_exp :

Using the Kuwabara model without the slip to calculate Δp _th,0 (i.e., Equation (26) with Kn _F →0 and for Ku _eff defined by Equation (B22)), one obtains:

Such an approach to calculate ϵ ₀ will be referred to in this work as the “modified Kirsch model,” which is based on the Kuwabara flow model (instead of the fan model), taken as a reference. Hence, we will use for consistency in this modified Kirsch model the following formula to calculate contribution to the single fiber efficiency due to Brownian diffusion:

which is obtained by replacing d_F , α and Ku in Equation (8) with d_Fa , α _eff , and Ku _eff , respectively. Similarly, Equation (9) for deposition due to direct interception is rewritten for the modified Kirsch model as:

with N_Ra defined by Equation (B19). Equation (B20) (to calculate the single fiber efficiency for action of image force) completes description of the modified Kirsch model.