Charging Efficiency of the Electrical Low Pressure Impactor's Corona Charger: Influence of the Fractal Morphology of Nanoparticle Aggregates and Uncertainty Analysis of Experimental Results

Abstract

The charging efficiency function of the Electrical Low Pressure Impactor (ELPI) corona charger was estimated for spherical, cubical, and fractal particles having an electrical mobility diameter D _b ranging from 0.01 to 0.5 μ m. Uncertainty analysis was performed on the experimental results for comparison with previous results for reference aerosols (spherical and cubical). Significant discrepancies were observed for the reference aerosols relative to Marjamäki et al. (2000, 2002). For fractal particles, charging efficiency follows a power-law function; the coefficients show significant discrepancies with those estimated for the reference aerosols. Consequently, fractal particles present higher charging efficiency than reference particles with a similar electrical mobility diameter. Two approaches are proposed to estimate the equivalent charge diameter of particles denoting fractal morphology. The work of Rogak and Flagan (1992) provides the best agreement between the experimental results and the theoretical representation of the ELPI charging efficiency.

INTRODUCTION

Since its introduction, the Electrical Low Pressure Impactor (ELPI Dekati) has been widely used for measuring particulate emissions. Ahlvik et al. (1998), Moisio (1999), Maricq et al. (2000), and Van Gulijk et al. (2004) recently used the ELPI for combustion aerosol characterization. In contrast with reference aerosols, combustion aerosols denote well-known fractal morphology (Jullien and Botet 1987). Few studies deal with the influence of this fractal shape on the ELPI response. Initial investigations were conducted by Matti Maricq et al. (2000), Virtanen et al. (2002), and Van Gulijk et al. (2004), who compared combustion aerosol size distributions obtained using the ELPI and the Scanning Mobility Particle Sizer (SMPS). As a result of these studies, fractal theory is now used in aerosol metrology and the opportunities are numerous. Nevertheless, few studies deal with the influence of fractal morphology on the charging efficiency of the ELPIs. The available work has established the ELPI's charging efficiency for spherical and cubical particles (Keskinen et al. 1992; Marjamäki et al. 2000, 2002). Marjamäki et al. (2000, 2002) reported no charging efficiency discrepancies between spherical and cubical particles, but more recently, Unger et al. (2004) underlined major discrepancies between spherical and cubical particles for unipolar charging in a corona wire charger.

Regarding fractal particles, numerous theoretical studies are available on bipolar charging (Wen et al. 1984; Rogak and Flagan 1992; Lall et al. 2006) and unipolar charging (Chang 1981; Han and Gentry 1993; Biskos et al. 2004); nevertheless, to our knowledge, the only study dealing with experimental measurement of charging efficiency using a corona charger for fractal particles is the work of Ntziachristos et al. (2004). The authors underlined the significant discrepancy between the charging efficiency of reference aerosols and combustion aerosols for the “filter-stage” configuration of the ELPI corona charger: the results agree with the work of Oh et al. (2004). More recently, Biskos et al. (2005) studied unipolar charging of combustion aerosols in a corona charger and, in contrast to Ntziatchristos et al. (2004), described good agreement between combustion and reference aerosol charging efficiency and with the theoretical approach of Fuchs (1963).

In the present work, two distinct aspects have been considered: uncertainty analysis of the charging efficiency function for reference aerosols and the influence of fractal morphology on the charging efficiency of the ELPI corona charger. The experimental procedure, similar to Marjamäki et al. (2000), was first used for cubical, spheroid, and spherical aerosols, then for fractal particles generated by a GFG-1000 PALAS spark discharge generator (Evans et al. 2003) and by acetylene gas and polymer diffusion flames (Ouf et al. 2008).

ELECTRICAL LOW PRESSURE IMPACTOR AND CORONA CHARGING

A. ELPI Principle

Since the ELPI granulometer has been widely described in previous publications (Keskinen et al. 1992; Moisio 1999; Marjamäki et al. 2000, 2002, 2005; Lettemy 2005), we will provide only a brief description of how it works. The ELPI combines a low-pressure impactor with a corona charger. At the inlet of the ELPI, a corona charger imposes an electrical charging state on the particles composing the aerosol. At the charger's outlet, the remaining ions are trapped by an electrostatic precipitator and the particles are classified according to their aerodynamic diameter using a low pressure cascade impactor. Currents induced by particles collected on impaction stages are measured using electrometers and are converted, with the help of the charging efficiency function, into particle number concentration.

B. Principle, Description, and Characteristics of the Corona Charger

The principle of the ELPI corona charger is described in Figure 2 (Dekati Ltd., 2005). First, an ionization field is generated by applying 5000 V (1 μ A in current) to a wire 100 μ m in diameter. Particles flowing through this ionization field acquire electrical charges by diffusion and field charging. At the outlet of the corona charger, non-fixed ions are electrically precipitated using a trap voltage of 400 V. The averaged characteristics of the corona charger are derived from Moisio (1999):

•	intensity of field charging E: 1.67.105 V/m,
•	ion density in field charging Ni: 1.9.1014 #/m3,
•	field charging volume Vcharging: 7.3.10–5 m3 = 73 cm3,
•	sampling flow rate Q: 1.65.10–4 m3/s = 165 cm3/s,
•	residence time in the charging zone t = Vcharging/Q: 0.44 s,
•	radius of the corona charger and outer radius of the ion trap r = router: 3 cm,
•	ion trap voltage U: 400 V,
•	length of ion trap L: 1.5 cm,
•	inner radius of the ion trap rinner: 2.65 cm

For a given size range, the conversion between measured current I and particle number concentration C is defined as E_ch:

where the charging efficiency Pn is defined as:

THEORETICAL CONSIDERATIONS

We present the different relations used for the analysis and after-treatment (see Appendix II) of our experimental results. The specific design of the corona charger imposes two electrical charging particle phenomena, the first by unipolar ion diffusion and the second by field charging. The first phenomenon results from collisions between unipolar ions generated by the corona charger and particles flowing through this charger. The number of charges that a spherical particle acquires by unipolar diffusion as a function of residence time has been theoretically summarized by Hinds (1999), Baron and Willeke (2000), and more recently Kleefsman and Van Gulijk (2008):

where:

• ϵ₀: vacuum permittivity: 8.85.10^–12 C²/N.m²,

• D_p: particle diameter if particles are assumed spherical (m),

• k: Boltzmann constant: 1.380658.10^–23 J/K,

• T: temperature (K),

• e: charge of an electron: 1.6.10^–19 C,

• : mean velocity of ions: 240 m/s.

The second charging phenomenon encountered in the ELPI is field charging. This phenomenon corresponds to the charging of particles by unipolar ions in a strong electrical field. The number of charges that a particle acquires by field charging was also reported by Hinds (1999):

where:

• ϵ_p: relative permittivity of the particle,

• Z_i: electrical mobility of ions: 1.5.10^–4 m²/V.

The global charging of the corona charger can be approximated as the sum of the previously described charges induced by diffusion and field charging:

Charging efficiency, as described in Equation (2), also integrates particle losses inside the corona charger. Two loss mechanisms were identified, one in the charging zone and the other in the ion trap. For losses in the charging zone, Hinds (1999) reported using a numerical approach based on the Deutsch-Anderson equation for penetration coefficient estimation P_charging(D_p,ϵ_p t):

where:

• λ : mean free path (66.5 nm for air at 293.15 °K and 101.3 kPa),

• μ : dynamic viscosity (1.833.10⁻⁵ Pa.s for air at 293.15 °K and 101.3 kPa).

One must notice that the Deustch-Anderson equation assumes that particles acquire their charge as soon as they enter the corona charger. For losses inside the ion trap, the penetration coefficient P_trap(D_p,ϵ_p t) is estimated assuming a perpendicular field inside a symmetrical, cylindrical tube (Hinds 1999):

The total penetration coefficient P_T(D_p,ϵ_p t) corresponds to the product of these penetration coefficients:

Then the charging efficiency function can thus be defined as the product of the total penetration coefficient P_T(D_p,ϵ_p,t) and the total number of charges acquired by the particle n_T(D_p, ϵ_p, t):

MEASUREMENTS

Aerosol Generation

This section describes the type and shape of the aerosol studied. First, for reference aerosols we used a TSI 3076 Constant Output Atomizer with four different solutions:

•	liquid spherical particles of diethyl sebacate (DES) and diethyl phtalate (DEP),
•	solid cubical particles of sodium chloride (NaCl) and spheroid particles of cesium chloride (CsCl).

Solid particle shapes were investigated using transmission electron microscopy (TEM) and are detailed in Figure 1. The cubical shape of NaCl and the intermediate spheroid appearance of CsCl are in good agreement with the previous results of Wise et al. (2005). Table 1 gives the particle characteristics of aerosols produced with the TSI atomizer.

FIG. 1 TEM micrographs of solid particles generated by PALAS GFG 1000, acetylene, and PMMA fuels, and TSI 3076 Constant Output Atomizer for NaCl and CsCl solutions.

TABLE 1 Physical characteristics of reference aerosol particles

Particles nature	Shape (dynamic shape factor)	Dielectric constant (at considered temperature)	Density (g/cm^3)	Concentration solution
DES	Spherical (1)	4.99 (30°C)	0.91	8 ml per ethanol L
DEP	Spherical (1)	7.63 (20°C)	0.98	10 ml per ethanol L
NaCl	Cubical (1.08)	5.9 (25°C)	2.17	6.3 g per water L
CsCl	Spheroid (1–1.08)	6.34 (19°C)	3.99	10 g per water L

Carbon aggregates were generated using a PALAS GFG 1000 under the following experimental conditions:

•	1.5 bar of argon,
•	spark frequency: 270 Hz.

The aggregates produced were mainly composed of graphitic carbon with an assumed density of 2.2 g/cm³ and a dielectric constant ranging from 12 to 15. The recent work of Evans et al. (2003) and Naumann (2003) describes these particles as fractal nanoparticle aggregates with a fractal dimension near 2 and primary particle diameter of 7.4 nm. Figure 1 shows transmission electron microscopy (TEM) images of these particles.

For the combustion aerosol, two fuels were used in this study: a gas (acetylene, C₂H₂) and a solid polymer (polymethyl methacrylate, PMMA, C₅H₈O₂). A detailed study of the combustion aerosol properties produced of these two fuels is available in the recent work of Ouf et al. (2008). Table 2 provides the global characteristics of soot particles produced during acetylene and PMMA combustion in our experimental conditions.

TABLE 2 Physical characteristics of fractal aggregates studied

Particle nature	Fractal dimension Df	Fractal prefactor kf	Primary particle diameter (nm)	Dielectric constant
Acetylene	1.93*	1.68*	64*	12–15
PMMA	1.76*	2.66*	48*	12–15
GFG 1000	2**	2***	7.4**	12–15

Figure 1 presents TEM micrograph examples of soot particles produced by acetylene and PMMA. Size distributions of the seven different aerosols were established using a scanning mobility particle sizer (DMA TSI model 3081, CPC TSI model 3022A). All the size distributions were fitted with a log-normal distribution:

where a is the amplitude, _b is the modal diameter, and σ_g is the geometric standard deviation. Table 3 summarizes these parameters for the different types of particles. The fitting parameters for the experimental size distributions were used in the after-treatment procedure to determine the multiple charge effect on the experimental results (Appendix II).

TABLE 3 Characteristics of aerosol size distributions

Nature of aerosol	Modal diameter (nm)	Geometric standard deviation σg	Amplitude a (#/cm^3)
NaCl	42.3	1.79	15756
CsCl	41.5	1.95	17239
DES	84.8	1.79	49493
DEP	86.6	1.79	58160
GFG 1000	55.8	1.58	295097
Acetylene	242.4	1.90	23625
PMMA	205.6	1.93	21142

Charging Efficiency Determination

Figure 2 presents the experimental device used to determine the charging efficiency of spherical, spheroid, cubical, and fractal particles respectively produced using the TSI atomizer, the GFG 1000, or the acetylene and PMMA flames (Ouf et al. 2008).

FIG. 2 Experimental device for determining the corona charging efficiency of spherical, spheroid, cubical, and fractal aggregates (ELPI corona charger principle taken from Dekati Ltd., 2005).

Both experimental set-ups used generation devices (atomizer, GFG 1000, or diffusion flame) and dilution devices (dilution tube or FPS 4000). Particles were then classified according to their electrical mobility using a Kr-85 radiation source (TSI 3077) and an electroclassifier (TSI EC 3080) with a long DMA column (TSI 3081) allowing mobility diameter selection from 10 to 1000 nm. At the DMA outlet, the aerosol was diluted with filtered air to reach the ELPI nominal flow rate of 10 LPM. The current induced by the charged particles was then measured simultaneously at the inlet I_inlet and outlet I_outlet of the ELPI corona charger, using a TSI 3068 electrometer. The charging efficiency Pn is thus defined as:

The inlet and outlet currents result from two separate current measurements, I_{inlet measured} and I_{outlet measured,}and the background noise of the electrometer I_bgn. The inlet and outlet currents are thus defined as:

It is experimentally impossible to generate monodisperse soot particles in a representative way. We have chose to use both synthetic (GFG 1000) and realistic carbon aggregates (acetylene and PMMA flames). These generation processes tend to produce aerosols denoting highly polydisperse size distributions. According to this dispersion in terms of size, the multiple charge effect on the selected electrical mobility diameter must be taken into account, and the results presented below are corrected using the method detailed in Appendix II.

RESULTS AND DISCUSSION

Comparison with Previous Results

The ELPI's charging efficiency was first experimentally determined by Keskinen et al. (1992). More recently, it was determined by Marjamäki et al. (2000), using a similar procedure. The generation procedure was similar in both studies and is based on a Condensation-Evaporation Monodisperse Aerosol Generator with an electrical mobility diameter lower than 1 μ m. For Keskinen et al. (1992), the charging efficiency was determined by comparing the number concentration measurement at the corona charger's inlet using a condensation particle counter, with the current measured at the outlet using an electrometer. Marjamäki et al. (2000) modified this procedure and directly compared currents measured at the outlet and the inlet of the corona charger. By using this second approach and only one measuring device (in this case an electrometer), experimental uncertainty is reduced. For particles with a mobility diameter higher than 1 μ m, a Vibrating Orifice Aerosol Generator was used by Marjamäki et al. (2000) and charging efficiency was estimated by comparing number concentration at the corona charger's inlet using an Aerodynamic Particle Sizer (APS) with the current measured at the outlet by an electrometer. Using this experimental procedure, Marjamäki et al. (2000) proposed a general expression of the charging efficiency Pn, for spherical dioctyl sebacate particles and cubical sodium chloride particles:

Experiments similar to Marjamäki et al. (2000) were performed in the present study for spherical particles of DES and DEP, cubical NaCl particles, and spheroid CsCl particles, for a size range representative of combustion aerosols (from 10 to 500 nm). Charging efficiency is described in Figure 3 for “trap on” and “filter stage” configurations; these results are mean values estimated using the statistical approach detailed in Appendix I. No significant discrepancies were observed between cubical, spheroid, and spherical particles.

FIG. 3 Charging efficiency of reference aerosol measured for “trap on” (black dots and curve) and “filter stage” (grey dots and curve) configurations.

We compared our results with the theoretical approach (Equation [11]) introduced in the third part of this study and the previous experimental results of Marjamäki et al. (2000). Overall, our results agree with the theoretical approach, but our results are significantly higher than the theoretical charging efficiency. The charging efficiency determined by Marjamäki et al. (2000) is also in overall agreement with the theoretical model, but in this case experimental charging efficiency appears to be lower than the theoretical approach. An overall discrepancy of 10% for D_b < 0.1 μ m and 25% for D_b < 0.5 μ m can be observed between the present work and the previous results of Marjamäki et al. (2000). Similar to Marjamäki et al. (2000), Figure 3 presents the three size ranges considered for fitting the experimental results: below 40 nm, between 40 and 100 nm, and above 100 nm. One way of explaining the discrepancies observed between our results and those of Marjamäki et al. (2000) is that there might be some discrepancies in the characteristics of ELPI corona charger used in the present study and in the work of Marjamäki et al. (2000). To our knowledge there is no publication dealing with the comparison of several ELPI corona charger's charging efficiency.

A widely used ELPI configuration is the 0V ion-trap configuration. Given the ion trap's cut-off diameter of 25 nm (Marjamäki et al. 2000), this device must be disabled for the “filter stage” configuration which allows decreasing the lowest diameter measured by the ELPI from 30 to 8 nm. Measurements were performed by Marjamäki et al. (2002) and charging efficiency functions were introduced:

We did not develop a theoretical approach for the “filter stage” configuration. Figure 3 only compares our present experimental charging efficiency with that of Marjamäki et al. (2002) for this configuration. These two studies are in good agreement, but a significant discrepancy is observed for electrical mobility diameter values higher than 0.25 μ m.

To present the discrepancies between our experimental results and Marjamäki et al. (2000), we developed discrepancy functions comparing experimental and theoretical charging efficiencies:

Figure 4 shows these discrepancy functions according to electrical mobility diameter for the “trap on” configuration. For the present results, charging efficiency is significantly higher than the theoretical values, while significantly lower charger efficiency was observed by Marjamäki et al. (2000). This discrepancy may be due to the theoretical model's simplicity and its assumptions (average values of the corona charger, constant charge of particles in the corona charger).

FIG. 4 Evolution, as a function of diameter, of discrepancies between experimental results and theoretical charging efficiency for the “trap on” configuration.

Estimation of the Uncertainties Associated with the Charging Efficiency Function

Based on this set of experimental results, we propose new power-law charging efficiency functions. Two distinct approaches were considered; the first is similar to Marjamäki et al. (2000, 2002) and does not integrate the uncertainties associated with each experimental result. The second approach involves a charging efficiency function weighted by the uncertainty associated with each experimental result. The main assumption of a fitting procedure is that the parameter associated with the X-axis is a well-known parameter with the smallest uncertainty. According to Mulholland et al. (1999, 2006) and Ehara et al. (2000), the uncertainty associated with measuring the electrical mobility diameter D_b using a DMA is nearly 5%. This uncertainty is mainly due to uncertainties on DMA voltage, DMA flow-rates and air characteristics at the experimental conditions (Mulholland et al. 2006). Since this uncertainty is small compared to the experimental uncertainty for Pn, we have associated D_b with the X-axis. The main advantage of this approach is that it integrates a complete computation of the uncertainty of the fitting function parameters. More information on determining the power-law fitting uncertainty can be found in Appendix I.C. Power-law functions were considered for the entire size range (D_b in μ m):

Table 4 presents the fitting parameters obtained without and with uncertainty weighting (α, β and α′, β′ respectively) for “trap on” and “filter stage” configurations. Uncertainties associated with these parameters (σ_α′ and σ_β′) are also reported in this table. According to the fitting procedure, the overlap of the fitted functions must be taken into account. Fitted functions are obtained from the experimental results but do not always intercept exactly at the limits of the experimental size range. In our case, charging efficiency is defined as a power-law function of the electrical mobility diameter (see Equation [18]). Consequently, the overlapping diameter is defined by the following equation:

where subscripts 1 and 2 define fitting parameters, and α and β represent the lower and upper overlapping fitted functions. The overlapping diameter D_over is then defined as:

Using this overlapping diameter, one can define the fitting size range in contrast to the experimental size range. The discrepancies between weighted and non-weighted fitting parameters are presented in Table 5. These discrepancies were computed according to:

•	discrepancies between weighted and non-weighted parameters:
•	discrepancies between the non-weighted parameters of Marjamäki et al. (2000, 2002) and the weighted parameters of the present study:

TABLE 4 Fitting parameters for “trap on” and “filter stage” configurations for reference aerosols

Trap on configuration
Experimental range	Fitting range	α	β	Fitting range	α ′	2σα ′	β ′	2σβ ′
Db < 0.04 μm	Db < 0.04 μm	13487	3.2432	Db < 0.04 μm	13493	2.64	3.2431	0.1550
0.04 μm ≤ Db ≤ 0.1 μm	0.04 μm ≤ Db ≤ 0.107 μm	197.52	1.9261	0.04 μm ≤ Db ≤ 0.110 μm	194.99	2.22	1.9213	0.0781
0.1 μm < Db ≤ 0.5 μm	0.107 μm < Db ≤ 0.5 μm	66.23	1.4367	0.110 μm < Db ≤ 0.5 μm	68.64	2.06	1.4475	0.0368
Filter stage configuration
0.025 μm < Db ≤ 0.5 μm	0.025 μm < Db ≤ 0.5 μm	82.12	1.3830	0.025 μm < Db ≤ 0.5 μm	82.26	2.02	1.3810	0.0132

TABLE 5 Discrepancies between weighted and non-weighted fitting parameters for “trap on” and “filter stage” configurations

Fitting range	ηwith-without(%)	γwith-without(%)	ηMarjamäki-Ouf(%)	γMarjamäki-Ouf(%)
Trap on configuration
Db < 0.04 μm	< 0.1	<−0.1	—	—
0.04 μm ≤ Db ≤ 0.1 μm	−1.3	−0.2	−11.8	−0.9
0.1 μm < Db ≤ 0.5 μm	+3.6	+0.8	−23.7	−3.9
Filter stage configuration
0.025 μm < Db ≤ 0.5 μm	+0.2	−0.1	−20.6	−3.3

Note that the discrepancies between weighted and non-weighted fitting parameters are lower than 4% for all size ranges. We can thus assume that the experimental uncertainties have little impact on the procedure for fitting the experimental results. However, there is a large discrepancy between the present fitted functions and those proposed by Marjamäki et al. (2000, 2002). This discrepancy is particularly significant for the α parameter in the size range D_b > 0.1 μ m. In both the “trap on” and “filter stage” configurations, significant discrepancies were observed between the present fitted functions and those of Marjamäki et al. (2000, 2002). These discrepancies are less pronounced for the “filter stage” configuration. For the “trap on” configuration, a significant gap is observed for the particles denoting an electrical mobility diameter larger than 0.1 μ m.

Influence of Particle Shape on Charging Efficiency

The second part of this experimental study deals with the influence of particle shape on the corona charging efficiency. Figure 5 shows the charging efficiency of “trap on” and “filter stage” configuration measured for fractal aerosols generated by the GFG 1000 and during acetylene and PMMA combustion.

FIG. 5 Charging efficiency of fractal particles for “trap on” (black dots and curve) and “filter stage” (grey dots and curve) configurations.

For the “trap on” configuration and for particles with electrical mobility diameter larger than 0.1 μ m, the charging efficiency obtained for fractal aggregates appears to be significantly higher than the charging efficiency of a sphere or cube with the same electrical mobility diameter. This higher charging efficiency (25% for the biggest particles) is in good agreement with the experimental results of Oh et al. (2004), which show that aggregates with low fractal dimension can carry 30% more charge than compact particles. It should be noted that charging efficiency is similar for all fuels which produce soot composed of primary particle with different diameters (7.4 nm for GFG 1000 and 64 nm for the acetylene diffusion flame). For the “filter stage” configuration, the charging efficiency of soot particles is also higher than for compact particles and the present results are in good agreement with the work of Ntziachristos et al. (2004) on diesel particles. Nevertheless, while discrepancies are observed between fractal and compact particles, the charging efficiency of fractal particles can be represented by a power-law function, as it can for spherical or cubical particles. In Table 6, we propose new fitting parameters, α and β, of a power-law function for fractal aggregates. There is a significant discrepancy between the power-law coefficients for fractal particles and the same coefficients for reference aerosols (cf. Table 4).

TABLE 6 Fitting parameters for “trap on” and “filter stage” configurations for fractal aggregates

Experimental range	Fitting range	α	β	Fitting range	α ′	2σα ′	β ′	2σβ ′
Trap on configuration
Db < 0.04 μm	Db < 0.04 μm	36059	3.5449	Db < 0.042 μm	41634	6.04	3.5834	0.5964
0.04 μm ≤ Db ≤ 0.1 μm	0.04 μm ≤ Db ≤ 0.102 μm	309.63	2.0724	0.042 μm ≤ Db ≤ 0.097 μm	253.74	2.56	1.9767	0.1908
0.1 μm < Db ≤ 0.5 μm	0.102 μm < Db ≤ 0.5 μm	89.05	1.5275	0.097 μm < Db ≤ 0.5 μm	96.41	2.17	1.5612	0.1091
Filter stage configuration
0.025 μm < Db ≤ 0.5 μm	0.025 μm < Db ≤ 0.5 μm	104.19	1.4234	0.025 μm < Db ≤ 0.5 μm	152.86	2.09	1.6682	0.0449

Influence of the Fractal Morphology of Soot Particles on Charging Efficiency

Experimental results obtained for fractal aggregates shown that the charging efficiency of these particles is higher than for spherical or cubical particles with the same electrical mobility diameter. Furthermore, in the case of bipolar diffusion charging, Rogak and Flagan (1992) have established a link between equivalent charge diameter D_qe and electrical mobility diameter for fractal aggregates of TiO₂:

The link between bipolar diffusion charging of fractal aggregates and corona charging is not obvious, but in this study we only considered particles with an electrical mobility diameter lower than 0.5 μ m, where the diffusion charging phenomenon is predominant. A more recent approach, introduced by Lall and Friedlander (2006) and based on previous work by Chan and Dahneke (1981), Dahneke (1981), Kasper (1982), and Wen et al. (1984a, b), considers fractal aggregates as a chain-like aggregate of primary particles. The equivalent charge diameter is directly linked to the electrical mobility and primary particle diameters; nevertheless, this approach is limited to aggregates composed of primary particles with a diameter ranging in the free-molecular regime (D _pp ≪ λ, c* = 9.17 from Lall and Friedlander 2006):

These two approaches were used to analyze experimental results for fractal aggregates in the present study. For spherical, spheroid and cubical particles, we assumed that the equivalent charge and electrical mobility diameters were identical. Figure 6 and Figure 7 present charging efficiency as a function of equivalent charge diameter for spherical, spheroid, cubical, and fractal particles in the “trap on” and “filter stage” configurations.

FIG. 6 Charging efficiency of fractal particles for “trap on” configuration as a function of equivalent charge diameter.

FIG. 7 Charging efficiency of fractal particles for “filter stage” configuration as a function of equivalent charge diameter.

Regarding equivalent charge diameter, better agreement between reference aerosols and fractal aggregates was observed with the empirical approach developed by Rogak and Flagan (1992). Furthermore, the Lall and Friedlander approach (2006) does not reduce the discrepancy between the fractal and theoretical charging efficiencies.

CONCLUSIONS

To our knowledge, this study represents the first uncertainty analysis of the ELPI charging efficiency function. The experimental procedure for charging efficiency described by Marjamäki et al. (2000, 2002) was applied to spherical, spheroid, and cubical reference aerosols. Our experimental results underline a significant discrepancy with Marjamäki et al. (2000, 2002) that may be explained by some discrepancies between ELPI corona charger used in the present study and in the work of Marjamäki et al. (2000, 2002). We developed new charging efficiency functions and performed a detailed uncertainty analysis. The general shape of the charging efficiency is similar to previous work and can be approximated with a power-law function:

Uncertainties associated with α and β, respectively, range from 0.02% to 3.0% and from 1.0% to 4.8%. Calculating the uncertainties makes it possible to reliably compare the different laws.

The second part of this study demonstrated the influence of the fractal shape of nanoparticle agglomerates and combustion aerosols on the ELPI charging efficiency. The same experimental procedure to determine the charging law was applied to nanoparticle agglomerates of carbon produced by a PALAS GFG-1000 and to fractal aggregates generated during acetylene and PMMA combustion. A significant discrepancy in charging efficiency was observed for fractal aggregates and reference aerosols with the same electrical mobility diameter. Using the theoretical approach developed by Rogak and Flagan (1992), this discrepancy can be reduced if charging efficiency is considered as a function of the charge equivalent diameter:

The uncertainty analysis could be followed up with an inter-laboratory comparison of corona charging functions to investigate possible discrepancies between different ELPI corona chargers. The aim would be to propose universal charging efficiency functions with computed uncertainties for the ELPI's entire size range for a large variety of shape.

APPENDIX I: UNCERTAINTY ANALYSIS

A. Charging Efficiency Uncertainty

The uncertainty for charging efficiency was determined using the following definition:

Considering the definition of the charging efficiency Pn as a function of I_inlet and I_outlet:

The uncertainties associated with I_inlet and I_outlet can be determined from the definitions of these currents and based on Equation (13). Since the definitions of I_inlet and I_outlet are similar, we only described the uncertainty associated with I_inlet;

The uncertainties for I_inlet and I_outlet are defined as:

The uncertainties associated with I_{inlet measured}, I_{outlet measured} and I_bgn are mainly due to measurement variability on one hand and to the electrometer resolution on the other. The uncertainty for the measurement variability is linked to the standard-deviation σ_I determined from the measured values during consecutive measurements of the charging efficiency for one experimental condition (type of aerosol and electrical mobility diameter). The manufacturer's documentation for the electrometer (TSI 3068) indicates a 1 fA resolution, corresponding to half-width of 0.5 fA and an uncertainty u(R):

The uncertainty associated with one current measurement is defined as:

The uncertainties of I_inlet and I_outlet are thus defined as:

Finally, the charging efficiency uncertainty u(Pn) is defined as:

B. Weighted Mean Value and Associated Uncertainty

From the set of experimental results, one can compute mean values weighted with the uncertainty associated with each value. Considering x_i values obtained in the same experimental conditions and σ_i experimental uncertainties associated with these values, the uncertainty σ and the weighted mean value of this series of n measurements of x are defined as:

C. Weighted Fit Function of Charging Efficiency and Associated Uncertainty

Using the experimental results, fitting functions can be established that take into account the uncertainties associated with the experimental results. The present method is reliable for linear fitting; consequently, we considered the charging efficiency functions in a log-normal scale, then used Equation (18) to introduce the following relationship:

For convenience we introduced new variables in the relationship: y = β x + ln (α) where x = ln(D_b) and y = ln(Pn). Using uncertainty analysis, the slope (β) and origin (ln(α)) of this fit can be defined from the experimental results with their associated uncertainties (Neuilly 1998):

Where is the mean value of x (or ln(D_b)), is the mean value of y (or ln(Pn)), and σ (y) is the experimental uncertainty associated with the considered value of y. For the slope σ (β) and the origin σ (ln(α)) of the fit applied to N_b experimental values of y, the associated uncertainties are defined as:

APPENDIX II: MULTIPLE CHARGES CORRECTION

Due to the aerosol generation procedure and the DMA classification, multiple charged particles may interfere with the measurement of charging efficiency. By definition, electrical mobility Z_p(n,D_bn) depends both on particle diameter D_bn and the number of charges n carried by the particle. Thus, when an electrical mobility Z_p(n,D_bn) is selected using a DMA, the aerosol at the outlet of the DMA is composed of particles with different diameters and charging states. We only included four charging states and the response of the electrometer at the DMA outlet can be defined as:

where I(Z_p(n,D_bn)) is the current induced by the particles denoting an electrical mobility Z_p(n,D_bn).

First, the current induced by particles of electrical mobility Z_p(n,D_bn) at the ELPI corona charger's outlet must be defined. For this configuration, the current is a convolution of the aerosol size distribution N(D_b), the probability that a particle can carry n charges Φ (n,D_b), and the DMA transfer function Ω (Z_p(n,D_bn)):

A log-normal size distribution was considered using mode and standard deviation measured experimentally with an SMPS and detailed in Table 3. The charging probability is estimated using the formulation proposed by Wiedensohler (1988) and the DMA transfer function proposed by Soltzenburg (1988) and, more recently, Hagwood et al. (1999). The Wiedensohler empirical functions were considered reliable by Ji et al. (2004) for describing the charging probability of Kr-85 for flow rates lower than 1 LPM and particles larger than 100 nm for which the multiple charge effect is predominant. For the highest flow rates, i.e., 1.5 LPM for particles smaller than 100 nm, Ji et al. (2004) report significant discrepancies between the Wiedensohler empirical functions and experimental results of bipolar charging efficiency, but the multiple charge effect is not significant in this range and does not impact our experimental results.

In the same way, the current measured at the outlet of the ELPI corona charger can be defined as a convolution of the aerosol size distribution N(D_b), the probability that a particle can carry n charges Φ (n,D_b), the DMA transfer function Ω(Z _p(n,D_bn)), and the charging efficiency of the corona charger Pn(D_b):

Concerning the ELPI charging efficiency, two cases were considered, both based on the theoretical approach described in the third part of this study. The first one assumes that, for reference aerosols, the representative diameter is the electrical mobility diameter and the second one assumes, for fractal aggregates, that the equivalent charge diameter is defined according to D_qe≈ 1.1D_b (Rogak and Flagan 1992).

In order to correct the currents experimentally measured for each electrical mobility diameter at the inlet and outlet of the corona charger, correction factors can be introduced:

Notes

*From Ouf et al. 2008.

***Theoretical hypothesis from Nauman (2003).

**From Nauman 2003.