Novel Formulae for Deposition Efficiency of Electrically Neutral, Submicron Aerosol Particles in Bipolarly Charged Fibrous Filters Derived Using Brownian Dynamics Approach

Abstract

Brownian dynamics (BD) method was employed to calculate the single fiber deposition efficiencies of electrically neutral, submicron aerosol particles in bipolarly charged and non-charged fibrous filters. It was found that a considerable increase in the deposition efficiency can be achieved by increasing the fiber charge density and that it is accompanied by a noticeable shift of the most penetrating particles size towards smaller particles. The results of the BD simulations obtained for various fiber charge densities and various particle diameters allowed us to derive new formulae relating a gain in the deposition efficiency due to action of the electrostatic force to the dimensionless polarization parameter, N _{σ 0} . These equations are applicable in much wider range of N _{σ 0} than other correlations available in the literature. A simple approximate rule for recalculating the form of the proposed formulae for another filter packing density was also presented. Theoretically predicted values of the deposition efficiency in bipolar electret filters were compared with our own experimental data as well as with the data available in the literature and a satisfactory agreement, better than in the case of other existing correlations, was obtained.

1. INTRODUCTION

The two most effective ways to enhance aerosol particles' separation in the fibrous filters are either to make use of electrostatic interactions between the particles and fibers (Baumgartner and Löffler 1986; Romay et al. 1998) or to utilize filtering media made of nanofibers, Podgórski et al. (2006). The former one allows a higher collection efficiency to be achieved without increasing the pressure drop. When the fibers are charged and aerosol particles are neutral, the particles' deposition occurs not only due to the mechanical deposition mechanisms (such as Brownian diffusion, interception, and inertial impaction), but also due to polarization forces. In such case the electrostatic field around a charged fiber induces a dipole in an uncharged particle, causing it to be attracted towards the fiber. When fibers and aerosol particles are both charged, the particles' deposition is influenced then by a strong Coulombic force; however, this case is not considered in this work. Other electrostatic methods for gas cleaning were reviewed by Jaworek et al. (2007).

The filters that are composed of permanently charged fibers are called “electret” filters. They are used for air cleaning applications that require high efficiency and low pressure drop, including disposable respirators, automotive cabin air filters, indoor air filters, vacuum cleaner bags, and industrial gas cleaning devices (Romay et al. 1998; Wang 2001).

Contrary to the mechanical fibrous filters, the efficiency of the electret ones decreases initially with their loading (Baumgartner and Löffler 1987; Martin and Moyer 2000; Ji et al. 2003). This effect might be related to the neutralization of fiber charges by the charges of opposite polarity that are carried by the aerosol particles collected on the fiber (Baumgartner and Löffler 1987), shielding of the electric field by the particles covering the fibers (Walsh and Stenhouse 1996) or by chemical interactions between fibers and aerosol (Barrett and Rousseau 1998). When an electret filter loading achieves a certain level, a filter begins acting as a mechanical one and its efficiency increases. The pressure drop across the filter increases continuously during its loading.

There are miscellaneous techniques to charge polymer fibers: corona charging (in the case of split-fiber electret material), triboelectrification (in the case of mixed-fiber material or resin-wool material), induction (for electrostatically spun material), and freezing in electric polarization (this method is feasible but it is not used in commercially available filter material), Brown and Smith (1999). Depending on the charge state, among electret fibers the unipolar (that have charge of one sign) and bipolar (that have charges in the form of line-dipole) charged fibers can be distinguished (Brown 1993). Electret fibers can be classified as spun and split types. The former ones are unipolarly charged, whereas the latter fibers carry a high bipolar charge (Baumgartner and Löffler 1986; Cao et al. 2004). Unipolarly charged fibers were investigated, e.g., by Baumgartner and Löffler (1987), who assumed that the mechanisms of particle deposition act independently, and by Oh et al. (2002), who used the Brownian dynamics method to calculate the collection efficiency of charged and uncharged monodisperse aerosols of submicron size. More common are the filters with bipolarly charged fibers; however, there are not many information in the literature concerning the calculation of aerosol particle deposition onto such fibers. The simulations for bipolarly charged fibers of rectangular cross-section were done, e.g., by Cao et al. (2004) and Cheung et al. (2005), while the deposition on a cylindrical fiber with a line dipole charge distribution was investigated theoretically, e.g., by Brown (1981). The experimental data of filtration efficiency of neutral aerosol particles in electret filters were reported, e.g., by Lee et al. (2002), Kim et al. (2005), and Bałazy et al. (2006). In this work for the first time the Brownian dynamics method was used to evaluate the deposition efficiency of neutral, submicron aerosol particles onto cylindrical, bipolarly charged fibers. The simulations were performed for various fiber charge densities, particle and fiber diameters as well as air velocities. Obtained discrete numerical data served as a source to establish new simple correlations that allow for fast estimation of the single fiber efficiency increase due to action of polarization force in bipolar electret filters.

2. COMPUTATIONAL METHOD

2.1. Model Formulation

When a neutral, dielectric aerosol particle is placed in a nonuniform electric field E ^(el), it experiences action of a polarization force F ^(el) that is generally expressed as:

where d _P and ε _P are the particle diameter and dielectric constant, respectively, and ε ₀ is the vacuum permittivity. Following Brown (1981), let us assume that a fibrous filter is made of bipolarly charged fibers, being line-dipoles with the distribution of the surface charge density, σ, along the angle θ (that is measured along the fiber circumference) given by:

In the above equation σ _max is the maximum local surface charge density for the assumed sinusoidal angular distribution of charges over the fiber surface. Note that in such case the algebraic sum of the charges on the fiber is zero, which means that the number of positive and negative charges is the same, but they are separated on a fiber surface. It is more convenient to use the linear density of the charges of one sign on a fiber (charge of each sign per unit fiber length), q _F , which is related to σ _max by the following expression: q _F = d _F σ_max, where d _F is the fiber diameter. The average surface density of charges of one sign, σ _av = 1/π ∫_{− π/2} ^π/2 σ (θ)dθ, equals: σ _av = 2σ _max/π = 2q _F /π d _F . Outside the fiber with such distribution of charges a nonuniform electric field is created. In the system of cylindrical co-ordinates 0rθ fixed in the fiber center, the components of the electric field intensity are as follows:

wherein ε _F denotes the fiber's dielectric constant and r is the distance measured from the center of the fiber. Substituting Equations (3) and (4) into the Equation (1) it follows that in the cylindrical co-ordinate system the polarization force has only one, nonzero component, namely the radial one, F _r ^(el), which is expressed by:

Motion of a sufficiently large aerosol particle (typically a few micrometers in diameter in air at STP conditions) is described by the Newton-type, deterministic, ordinary differential equation. However, the motion of a smaller particle becomes stochastic as a result of momentum random fluctuations and it can be described formally by the Langevin equation. Taking into consideration the drag and resistance force, f(u- v), the polarization force, F ^(el), and the fluctuating Brownian force, F ^(B), the stochastic equation of motion of an aerosol particle can be expressed as:

where m _P denotes a particle mass, u and v are the vectors of gas and particle velocity, respectively, t denotes time, and f is the friction coefficient, f = 3 π μ _g d _P /C _C , wherein μ _g stands for the gas viscosity. The Cunningham slip-correction factor, C _C , can be calculated from the following expression:

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

and λ _g denotes the gas mean free path. The values of the coefficients in Equation (7) were given, e.g., by Allen and Raabe (1985): a _Cc = 1.142, b _Cc = 0.558, d _Cc = 0.999.

Let us define the dimensionless variables (denoted by a bar above a symbol) as follows: = v/U ₀, ū = u/U ₀, = t/τ _P and = r/R _F (where τ _P = m _P /f is the particle relaxation time, U ₀ denotes superficial gas velocity and R _F is the fiber radius). Introducing them into Equation (6), the following dimensionless form of the equation of stochastic motion of an aerosol particle in the vicinity of a bipolarly charged fiber can be obtained:

where N _{σ 0}, which is defined as:

is the dimensionless polarization force parameter. F _Stk that occurs in Equation (9) denotes the magnitude of the Stokesian drag force acting on a particle, which moves in a stationary fluid with the velocity U ₀: F _Stk = fU ₀, whereas î _r and î _θ are the unit vectors of the cylindrical co-ordinate system. The fluctuating Brownian force, F ^(B), when averaged over the time interval τ _P (which is the proper time-scale for the Brownian motion analysis), is a vectorial random variable having for each co-ordinate the normal distribution with zero mean value and the standard deviation equal to: σ _{F ^(B)} = √2k _B Tf ²/m _P . Introducing the dimensionless Brownian force ^(B)defined as: ^(B) = F ^(B)/σ _{F ^(B)} , the last term on the right-hand side of Equation (9) can be represented as: F ^(B)/F _Stk = ^(B)(2 Pe Stk)^{− 1/2}. Consequently, the total single fiber deposition efficiency of an electrically neutral Brownian particle on a bipolarly charged fiber should be a unique function of four dimensionless numbers, E _el (Pe, Stk, N _R , N _{σ 0}), where:

and the polarization force parameter N _{σ 0} is defined by Equation (10). Since for a mechanical filter N _{σ 0} = 0, the single fiber deposition efficiency is in this case function of three dimensionless numbers, E _mech (Pe, Stk, N _R ). The gain in the single fiber efficiency for an electret filter caused by the action of polarization force can be formally calculated as: Δ E _el = E _el −E _mech , where E _el and E _mech are the single fiber efficiencies determined for an electret and a mechanical filter, respectively, when both these filters have the same structural characteristics (fiber diameter, d _F , filter packing density, α, filter thickness, L). This difference Δ E _el would be a unique function of N _{σ 0} itself, if the electrostatic mechanism and mechanical mechanisms of deposition (i.e., Brownian diffusion, inertial impaction, and direct interception) are additive. However, this commonly used additivity rule is rather presumption only than a result of any rigorous theoretical analysis. Its validity will be checked in this article.

2.2. Simulation Technique (Brownian Dynamics—BD)

For a few micrometers-sized aerosol particles, Brownian motion effect becomes negligible and the last term (F ^(B)) on the right-hand side of Equation (6) can be omitted, which then transforms itself into a deterministic equation of motion. For a given initial conditions (initial particle position and velocity), it can be easily integrated numerically, yielding unambiguous trajectory of a particle. The stochastic nature of motion of smaller particles leads to infinite number of possible particle trajectories for the same initial position. Thus, solution of the Langevin equation is not a function, but a probability distribution ϕ _i (Δ v _i , Δ L _i ) that during the time interval Δ t the particle will change its ith component of the velocity by Δ v _i and it will be displaced by a distance Δ L _i in ith direction. Making use of Chandrasekhar's (1943) lemmas on random integrals and assuming that during a sufficiently short time-step Δ t the slowly varying variables (u, F ^(el)) can be assumed to be constant, this probability distribution ϕ _i (Δ v _i , Δ L _i ) can be expressed by the bivariate normal distribution (for each co-ordinate, i = 1,2,3, separately) as follows:

This distribution has the following parameters:

•	expected values of the particle velocity change, Δ v i , and the linear displacement of the particle, Δ L i , over the time-step Δ t:
•	standard deviations, σ vi , σ Li :
•	coefficient of correlation, ρ i :

In the above equations β = f/m _P denotes the inverse of the particle relaxation time.

Computational procedure to determine a stochastic trajectory of a Brownian particle in an electric field may be implemented as follows. For a given initial particle position and its initial velocity components, v _i , at a moment t, we calculate the polarization force, F _i ^(el), and the local fluid velocity, u _i (the well-known Kuwabara (1959) cell model was used in this article to describe the gas flow field around a fiber). Then, one computes the expected values Δ v _i and Δ L _i from the Equations (15)–(16), the standard deviations, σ _vi , σ _Li , from the formulae (17)–(18), and the correlation coefficients, ρ _i , from the Equation (19). Next, we generate two non-correlated random numbers, G _Li and G _vi , both having Gaussian distribution with zero mean and unit variance. Finally, we calculate the change of the particle velocity, Δ v _i , and the particle linear displacement, Δ L _i , during the time-step Δ t from the expressions accounting for deterministic and stochastic motion:

All the steps are repeated independently for each co-ordinate, i = 1,2,3. Having determined the increments Δ v _i and Δ L _i from Equations (20)–(21), a new particle velocity at the moment t+Δ t is obtained as: v _i (t+Δ t) = v _i (t)+Δ v _i , and in the same manner a new particle position is calculated as: L _i (t+Δ t) = L _i (t)+Δ L _i . After completing one time-step of simulations, the next step is performed in the same way.

Using this algorithm the single fiber deposition efficiency was calculated as follows. First, the entire inlet to the Kuwabara cell was divided into subintervals of the same height by N_PTS = 40 points and from each point a small number of N_GEN = 40 particles were released and their random trajectories were traced. At this stage the location of the starting particle at the inlet to the Kubawara cell above which no deposition occurred, y _0cr , was determined. Since these initial calculations were done for a small amount of particles, the height of the control entrance window for the main simulations, y _{0 max}, was taken as two and a half times larger than the estimated value of y _0cr . Then, only the part of the inlet to the Kuwabara cell of the height y _{0 max} was divided into new subintervals of the same height by N_PTS = 40 points. N_GEN = 500 random particle trajectories were calculated for each starting point (located at y _0i = y _{0 max}(i – 1)/(N_PTS–1), i = 1, …, N_PTS). The number of particles deposited on the fiber, N_DEPi, which were released from the point y _0i , was counted and then the deposition probability as a function of the initial particle position was calculated as: P _Di(y _0i) = N_DEPi/N_GEN. Calculations were accepted only if the P _Di was zero for at least three points located most distantly from the stagnation line y = 0; otherwise, the height of the entrance control window was increased to assure that the deposition probability for particles entering the Kuwabara cell above y _{0 max} is really negligible. Having determined P _Di(y _0i), the cell deposition efficiency was calculated by numerical integration as: E _cell = (∫₀ ^RK P _Di(y _0i)dy _0i)/RK = (∫₀ ^{y _0i} P _Di (y _0i)dy _0i)/RK, where RK = R _F /α^1/2 is the radius of the Kuwabara cell. Then, the cell deposition efficiency was recalculated into the commonly used single fiber deposition efficiency, E (which is related to the fiber diameter) as: E = E _cell RK/R _F (due to symmetry only the half-plane y ≥ 0 was considered). Such simulations were repeated N_SMPL = 10 times and averaged efficiencies and standard deviations were determined. The algorithm described above allows one motion and deposition of a solid spherical aerosol particle to be studied, taking simultaneously into consideration a coupling between the Brownian random walk, particle inertia, convection in a moving fluid and the effect of external forces. This method was also used to calculate deposition efficiency of fractal-like aggregates in mechanical fibrous filters by Bałazy and Podgórski (2007) and it was extended by Podgórski (2001) to account for variation of a particle friction and diffusion coefficients near a solid wall as a result of hydrodynamic interactions. Alternatively formulated BD algorithms, e.g., by Ermak and Buckholtz (1980) and by Elimelech et al. (1995), were reviewed and compared by Podgórski (2002).

3. RESULTS AND DISCUSSION

3.1. Determination of Δ E _el (N _{σ 0}) Correlation

Using the Brownian dynamics method, the calculations of the single fiber deposition efficiency were carried out for eight values of the fiber charge density q _F in the range from 0 to 13 nC/m and for eighteen various particle diameters from 0.01 to 1 μ m. If not stated otherwise, the simulations were performed for the following set of parameters: the fiber diameter, d _F = 7.84 μ m, the filter packing density, α = 0.069, and the superficial air velocity U ₀ = 0.129 m/s (they corresponded to the conditions during the experiments performed by Bałazy et al. (2006), who investigated penetration of nanoparticles through a commercial respirator). Calculated single fiber deposition efficiencies as a function of particle diameter for various fiber charge densities are shown in Figure 1. It can be observed that using an electret filter media of a higher electrization allows one to increase the single fiber efficiency even by a few orders of magnitude compared to a mechanical fibrous filter with the same structure. Moreover, significant shift of the most penetrating particle size (MPPS – the size of a particle that corresponds to a minimum of fractional efficiency) towards smaller particles with the increase in the fiber charge density can be noticed. It is clearly visible that the MPPS drops from 0.3 μ m for a neutralized fiber to about 0.02 μ m when the fiber has charge density 5 nC/m. In the case of the highest considered q _F (equal to 13 nC/m) the lowest single fiber efficiency is observed for the particle diameter of 0.01 μ m that was the lowest limit of d _P during the calculations.

FIG. 1 The influence of the fiber charge density, q _F , on the fractional single fiber efficiency calculated for the neutral and bipolarly charged fibers; U ₀ = 0.129 m/s, d _F = 7.84 μ m, and α = 0.069.

Having determined the single fiber efficiencies for an electret filter, E _el , and for a structurally identical, uncharged (mechanical) filter, E _mech , both calculated directly using the standard Brownian dynamics approach, the difference Δ E _el = E _el − E _mech was computed then, which can be interpreted as an increase in the single fiber deposition efficiency attributed to the action of polarization forces. The values of this efficiency difference can be correlated with respect to the polarization force parameter, N _{σ 0}. To interpolate the discrete numerical data of Δ E _el we proposed the following simple fit:

where a, b, c, d are the constants to be determined by fitting Equation (22) to the results of simulations performed using the Brownian dynamics method. This expression was proposed for the entire range of the polarization force parameter N _{σ 0} and it turned out to be extraordinarily useful, allowing one a very precise interpolation of the results of numerical simulations. Figure 2 shows two such examples obtained for fixed fibers' charge densities, q _F = 1 nC/m and q _F = 5 nC/m; in this case variability of N _{σ 0} is the result of different fibers' diameters. Similarly, Figure 3 presents two examples of fitting numerical data to the Equation (22), when the particle diameters were fixed (d _P = 0.1 μ m and d _P = 1 μ m), while N _{σ 0} varied because of the fibers' charge variability. As can be seen, in all cases such interpolation is excellent with the correlation coefficients R² far above 0.99. The best fits to the Equation (22) are shown in the plots' legends in the Figure 2 and Figure 3. The key problem is whether deposition due to action of polarization force and all other mechanical mechanisms are additive or not. An universal relationship Δ E _el (N _{σ 0}) is expected to exist if this is true. Figure 4 shows all the results of numerical simulations obtained for various values of q _F and d _P . Different symbols in this plot represent discrete data for fixed values of the fiber charge density (but for variable particle diameter) and thin dotted lines are the best fits to the Equation (22) for particular values of q _F . As can be seen, all the data Δ E _el (N _{σ 0}) form almost an unique curve, apart from the smallest particles. Specific values of Δ E _el for a fixed q _F start to depart from the common curve Δ E _el (N _{σ 0}) for d _P = 0.02 μ m, when the fiber charge density was the highest (13 nC/m), and for d _P = 0.05 μ m, when the fiber charge density was the lowest (0.1 nC/m) among the considered ones. This means that electrostatic and diffusional mechanisms of deposition are not completely independent for nanoparticles and the coupling between them is stronger when fibers are charged less. Nevertheless, an approximate correlation can be derived to describe the relationship Δ E _el (N _{σ 0}) in a broad range of the parameter N _{σ 0} variability. Taking into account the data obtained with the use of the BD method for all considered cases, we have found the following formula:

FIG. 2 Effect of the polarization force parameter (variable particle diameter) on the single fiber deposition efficiency due to electrostatic forces for q _F = 1 nC/m (left) and q _F = 5 nC/m (right).

FIG. 3 Effect of the polarization force parameter (variable fiber charge density) on the single fiber deposition efficiency due to electrostatic forces for d _P = 0.1 μ m (left) and d _P = 1 μ m (right).

FIG. 4 Collection of all numerical data (points) obtained using the BD method and comparison with individual fits to Equation (22) for fixed q _F values (thin dotted lines) and with global fits (two thick, solid and dashed lines, for various ranges of N _{σ 0} variability).

The thick solid line in Figure 4 is plotted according to Equation (23). On the other hand, excluding from the fitting procedure the data deviating the most from the common relationship Δ E _el (N _{σ 0}), i.e., for N _{σ 0} < 5 × 10^{− 3} and for d _P ≤ 0.3 μ m, another correlation was derived, which is shown in Figure 4 as the thick dashed line:

Note that both these formulae—Equations (23) and (24)—predict almost the same values of Δ E _el for N _{σ 0} > 1. Since the definition of the polarization parameter N _{σ 0}, Equation (10), contains—in addition to q _F and d _P —also the gas velocity U ₀ and the fiber diameter d _F , it could be worth to consider whether variability of these two parameters has an effect on the Δ E _el (N _{σ 0}) relationship. To verify a universality of the proposed form of correlation, Equation (22), the calculations were repeated for other air velocities and fiber diameters. The results obtained for three different air velocities: 0.129, 0.2, and 0.3 m/s, for constant values: q _F = 5 nC/m, d _F = 7.84 μ m, and α = 0.069, and for the considered particle size range (0.01–1 μ m) are shown in Figure 5a. Similarly, the results for three different fiber diameters: 7.84, 15, and 20 μ m, at fixed gas velocity 0.129 m/s, and with all other parameters the same as in the previous case, are collected in Figure 5b. The data shown in these plots indicate that the Equation (22) is quite universal in terms of U ₀ and d _F for a wide range of the polarization force parameter, but for the smallest nanoparticles the BD approach should be preferred to obtain more precise results for each individual case.

FIG. 5 (a) Effect of gas velocity (left) and (b) effect of fiber diameter (right) on the Δ E _el (N _{σ 0}) relationship.

It should be borne in mind that the Equations (23) and (24) were derived using the data obtained for the constant filter packing density α = 0.069. Although the definition of the polarization force parameter N _{σ 0} does not contain α, it is clear that the single fiber deposition efficiency remarkably depends on the filter packing density in both mechanical and electret filters. Thus, to make our correlations more widely applicable, we have proposed approximate generalization of the Equations (23) and (24). The method to recalculate these formulae for other values of α is described in detail in the Appendix. This generalization involves introduction of the Kuwabara number, Ku (called also the hydrodynamic factor), which is related to α as follows:

Hence, our final correlations proposed on the basis of the BD simulations for any values of the Kuwabara number take the following form:

	Equation (23) is extended to:
	Equation (24) transforms to:

Equation (26), derived on the basis of a more scattered data set, was proposed for a slightly higher range of applicability and may be even more accurate when nanoparticles are filtered in weakly charged electret media. For intermediate strength of electrostatic interactions and in the case of particles larger than around 0.3 μ m, Equation (27) may turn out to be better. For stronger electrostatic effects, both of them are expected to be of a comparable accuracy.

3.2. Comparison with Experimental Data and Other Correlations

Bałazy et al. (2006) measured penetration of neutral aerosol particles through a commercial N95 filtering-facepiece respirator that consisted of three polypropylene fibrous layers. Two of them—the inner and the outer—were electrically neutral, while the middle layer was electrically active. Structural characteristics of this respirator reported by the authors are given in Table 1.

TABLE 1 Characteristics of the respirator investigated experimentally by Bałazy et al. (2006)

Layer	Material	Thickness L [mm]	Fiber diameter d F [μ m]	Packing density α [−]	Basis weight ρ SF [g/m^2]
Outer	Polypropylene, neutral	0.31 ± 0.05	39.49 ± 1.80	0.165	46.67
Middle	Polypropylene, electret	1.77 ± 0.12	7.84 ± 2.00	0.069	110.67
Inner	Polypropylene, neutral	1.05 ± 0.16	40.88 ± 2.26	0.200	191.67

Using the BD method described in this article, we determined penetrations of aerosol particles through each layer of the N95 respirator at superficial gas velocity U ₀ = 0.129 m/s, i.e., for the same conditions as during the experiments, taking q _F = 0 for the inner and the outer layers, and for several assumed values of q _F for the middle layer. Then, the overall penetration through the respirator was calculated as the product of penetrations through each layer. As can be observed in Figure 6, a fairly good agreement between results of direct BD simulations and experimental data was achieved when the charge density of the fibers forming the middle layer was taken to be of the order 1.2–1.3 nC/m. Therefore, it seems that the BD approach may be also a method of indirect theoretical estimation of fibers' charge density in electret filters, the property which is difficult to measure in the case of bipolarly charged fibers.

FIG. 6 Validation of the results of direct BD simulations using experimental data of Bałazy et al. (2006) obtained for a commercial respirator.

We have also made an attempt to verify applicability of the proposed correlations—Equations (26) and (27)—using the experimental data taken from literature, and to compare our new formulae for Δ E _el (N _{σ 0}) with the ones proposed by other authors. Brown (1981, 1993) derived the following expression for a fixed filter packing density α = 0.05, assuming that the particle trajectory differs from the fluid streamline only due to the action of the electric field of the fiber (i.e., neglecting particle inertia and Brownian motion):

He suggested that for other packing densities this equation may be generalized as follows:

The same formula was also quoted by Kim et al. (2005), Lee et al. (2002), and Otani et al. (1993). However, Equation (29) is inconsistent with the original Brown's Equation (28) obtained for α = 0.05. As it was explained in the Appendix, the numerical constant is incorrect—instead of 0.54 it should be 0.41. For values of N _{σ 0} smaller than one, Otani et al. (1993) proposed the following expressions [see also Kim et al. (2005); Lee et al. (2002)]:

Experimental data of the single fiber deposition efficiency due to the polarization force as a function of the polarization parameter, Δ E _el (N _{σ 0}), were reported for various electrically charged filters by Lee et al. (2002) and by Kim et al. (2005). Lee et al. (2002) studied a polypropylene, melt-blown, high-performance electret filter having the characteristics: d _F = 2.64 μ m, α = 0.15, L = 0.31 mm, while Kim et al. (2005) investigated a wool filter impregnated with PTFB resin (p-t-butyl-phenol-formaldehyde) with the following parameters: d _F = 13.1 μ m, α = 0.075, L = 5.85 mm. Results of Brown's, Otani's and two our correlations, Equations (29), (31), (26), and (27), respectively, were compared with the experimental data of Kim et al. (2005)—see Figure 7, and with these ones of Lee et al. (2002)—Figure 8. Comparing predictions of our new correlations it may be stated that Equation (26) seems to agree better with Kim's data, while for the Lee's data a better agreement is observed for Equation (27). Nevertheless, in both cases our two newly derived formulae for the single fiber efficiency due to the polarization force describe better available experimental data than other correlations that can be found in the literature, Equations (29), (30), (31). Especially, our new theoretical equations cover in a continuous way a very broad range of values of the polarization force parameter. Contrary to that, each of the Equations (29), (30), (31) was recommended for a narrower scope of N _{σ 0}, and they are, in general, mutually discontinuous at the boundaries (for N _{σ 0} = 10^–2 and N _{σ 0} = 1).

FIG. 7 Comparison of various correlations Δ E _el (N _{σ 0}) with experimental data of Kim et al. (2005) for an electrically active resin wool filter.

FIG. 8 Comparison of various correlations Δ E _el (N _{σ 0}) with experimental data of Lee et al. (2002) for a melt-blown polypropylene electret filter.

4. SUMMARY AND CONCLUSIONS

The main aim of this article was to derive simple, but more general and accurate formulae than available in the literature, to describe an increase in the single fiber deposition efficiency, Δ E _el , due to action of the polarization force, in the case of filtration of neutral submicron aerosol particles through bipolarly charged fibrous filters. For this purpose the Brownian dynamics method was used, which takes simultaneously into account particle convection by moving fluid, random Brownian motion, particle inertia and action of electrostatic forces. Calculations showed that the increase in the fiber charge density results in a significant increase in the deposition efficiency together with the reduction of the most penetrating particle size. Results of extensive numerical simulations allowed us to determine dependence of Δ E _el on the dimensionless polarization parameter, N _{σ 0}, for various fiber charge densities and diameters, particle sizes and gas velocities. A simple interpolating formula Δ E _el (N _{σ 0}) was proposed and it was proven that it works perfectly when variation of N _{σ 0} is related to variation of either fiber charge density or particle diameter. Closer inspection of all numerical results led us to derivation of general correlations Δ E _el (N _{σ 0}), approximately valid for a very broad range of N _{σ 0} values. It was also noted that for the finest aerosol particles and weakly charged fibers there is a coupling between Brownian motion and drift of a particle in an electric field, thus the direct BD simulations are preferred then if one requires more precise results. A good agreement between the results of BD calculations and our own experimental data was obtained. Our newly derived correlations Δ E _el (N _{σ 0}) were successfully validated using experimental data taken from the literature, showing better agreement than other formulae available so far.

APPENDIX: METHOD OF CORRELATION RECALCULATION FOR ANOTHER PACKING DENSITY

Let us assume that we have determined a correlation for Δ E _el for a fixed value of the Kuwabara number, namely Ku₁ = const [that corresponds to a filter packing density α ₁, Ku₁ = Ku(α ₁)], which can be generally expressed as an explicit function of N _{σ 0} as follows:

Brown (1981) suggested that for any other values of Ku(α), the form of correlation Δ E _el (N _{σ 0}, Ku) can be expressed as:

where the function f has the same general form as f ₁ in Equation (A1), but it has a different argument than f ₁ (i.e., N _{σ 0}Ku instead of N _{σ 0}) and numerical constants in f and f ₁ may be different. The new numerical constants in Equation (A2) can be determined from the condition that both Equations (A1) and (A2) must give the same values of Δ E _el for Ku = Ku₁ for any value of N _{σ 0}; thus:

The procedure of finding new numerical constants using Equation (A3) can be explained the best with the use of examples.

Example 1: Correlation in a Form of Power Function

Let us assume that Equation (A1) is a power function of N _{σ 0}, i.e.:

where a ₁, b ₁ are numerical constants found for Ku₁ = const. Hence, for any value of Ku, Equation [A2] should have a general form as follows:

where the new constants a, b valid for any value of Ku may be determined from the condition expressed by Equation (A3), which reads in this case as:

Equation (A6) can be satisfied for any value of N _{σ 0} if, and only if, the new constants a, b are related to a ₁, b ₁ by:

Thus, Equation (A5) can be finally written in the following form:

Let us analyze in detail the case described by Brown (1981) who claimed that the constants a ₁, b ₁ in Equation (A4) are equal to: a ₁ = 0.47, b ₁ = 0.4, for α ₁ = 0.05 (i.e., for Ku₁ = 0.797), if 1 < N _{σ 0} < 100. Thus, Equation (A4) reads as: Δ E _el (N _{σ 0})_{Ku1 = 0.797} = 0.47N _{σ 0} ^0.4. Hence, according to Equation (A7), we have: a = 0.47 × 0.797^{1 − 0.4} = 0.41, and Equation (A5) takes then the form:

This result differs from that one quoted by Brown (1981) who gave incorrect numerical constant 0.54 (cf. Equation [29]) instead of the proper value 0.41 in Equation (A10). It had to be simple computational mistake and the author probably divided a ₁ by Ku₁ ^{1 − b ₁} instead of multiplying those values.

Example 2: Correlation in a Form of Quotient of Power Functions

As it has been shown in this article, Equation (A1) can be conveniently formulated for a wide-range of N _{σ 0} values in the following form:

where numerical constants a ₁, b ₁, c ₁, d ₁ are determined for Ku₁ = Ku(α ₁) = const. Equation (A2) reads in this case as:

Thus, the condition expressed by Equation (A3) implies that:

This equation can be satisfied for any value of N _{σ 0} if, and only if:

Therefore, Equation (A12) may be finally rewritten for any value of Ku as follows:

Equation (A18) was used in this article to recalculate correlations (23)–(24) derived for α = 0.069 for other values of the filter packing density, resulting in Equations (26)–(27).

Note that the above described procedure of recalculation of the correlation established for a fixed value of α for another value of the filter packing density is approximate (in phenomenological sense, not numerically) and direct BD simulations for a considered value of α should be rather preferred if very precise results are required. Notwithstanding, for typical ranges of values of fibrous filters' packing densities, this simple approach can be fairly satisfactory and it may serve as a quick and reasonable estimate.

Acknowledgments

The work was financed from the budget means for science in 2006–2009 as a research project.