ELPI Response and Data Reduction I: Response Functions

Response functions of the ELPI impactor with normal and porous substrates are presented together with fit functions to describe particle collection. In addition to primary collection efficiency, fits for the secondary collection mechanisms, diffusion, image charge force, and space charge field are presented. Charging efficiency for different configurations is also presented. Presented response functions can be used in data reduction and inversion of ELPI data.

INTRODUCTION

Electrical Low Pressure impactor (ELPI; Keskinen et al. 1992) combines electrical detection with a conventional cascade impactor to acquire real-time aerosol measurement. Since the introduction of the ELPI, it has been used in a wide variety of applications, including radioactive aerosols, pharmaceutical applications (e.g., Crampton et al. 2004; Glover and Chan 2004), combustion studies (e.g., Kymäläinen et al. 1996; Latva-Somppi et al. 1998; Hayes et al. 2004), ambient air quality studies, and atmospheric aerosol studies (e.g., Shi et al. 1999; Temesi et al. 2001). Because of the real-time measurement capability of the ELPI, it has seen wide use in automotive emission measurements, both in the industry and in universities and institutes (e.g., Ahlvik et al. 1998; Maricq et al. 1999; Maricq et al. 2000; Ntziachristos and Samaras 2000; van Gulijk et al. 2003).

In ELPI, particles are first charged in a unipolar (diffusion) charger and then classified to size fractions by a cascade impactor. Particle detection is based on the particle charge by measuring the currents generated by collected particles. Therefore, the response functions of the ELPI consist of the efficiency of the charger and collection efficiency of the impactor. There are four main ways to use the ELPI. The impactor can be assembled with or without the electrical filter stage, which is associated with different charger operation parameters. In addition, the impactor stages may have smooth or porous collection plates. The response functions for each case are different.

The charger efficiency functions without the filter stage were presented by Marjamäki et al. (2000) and with the filter stage by Marjamäki et al. (2002). The primary collection efficiency curves of the ELPI-impactor with normal substrates have been presented by Marjamäki et al. (2000). In addition to impaction, there are secondary collection mechanisms in the impactor that are responsible for the “tails” of the collection efficiency curves. These mechanisms have been studied by Virtanen et al. (2001). Marjamäki and Keskinen (2004a, b) studied the effect of collection substrate porosity and roughness on the collection efficiency characteristics of the low cut-point impactor stages of ELPI. They noted that the difference caused by porous plates is so large that the impactor must be re-calibrated. In this paper we present calibration results of the collection characteristics of the ELPI impactor with porous collection plates.

The main aim of this study is to present the response functions of the ELPI in a form that is suitable for data reduction and inversion. For this purpose we review the previous results and present new results for the porous substrates. We present the response functions for the different ELPI configurations in a functional form. Detailed numerical values of the functions are needed when applying inversion algorithms to obtain size distribution.

Impactor Collection Characteristics with Porous Plates

Impactors are characterized by their collection efficiency. Collection efficiency function for each stage i, E _i , can be described as a continuous function of particle size describing the fraction of particles collected by the stage. We use the data measured for earlier papers (Marjamäki and Keskinen 2004a, b) to construct the low-concentration collection efficiency curves for the six lowest impactor stages. The higher cut-point stages were calibrated with the methodology presented by Marjamäki et al. (2000) and Virtanen et al. (2001).

Primary Particle Collection Efficiency

In the ELPI there are 13 impactor stages. Stages are numbered so that the stage with the lowest cut-point is stage 1 and the stage with the highest cut-point is stage 13. Stage 13 is used as a pre-cut and no signal is measured from it, it only acts as upper limit for the impactor. If the electrical filter stage (Marjamäki et al. 2002) is used, stage 12 is removed from the impactor and size range of stage 11 is increased. The primary collection takes place on the impactor plates below the nozzles. The measured collection efficiency curves for stages 1–12 are shown in Figure 1a. Figure 1b compares collection efficiency curves of individual stages with smooth and porous collection plates. As discussed by Marjamäki and Keskinen (2004a, b) the collection efficiency curves are less steep and have a lower cut-point when porous plates are used. However, the difference gets smaller for stages with higher cut point. For stages 8 and above, the difference in cut point is negligible.

Figure 1 (a) Measured and fitted functions for the collection efficiencies of the ELPI-impactor with 40 μ m porous substrates. Measured values shown with circles, and lines represent the fit functions. (b) Collection efficiencies of stages 2, 4, and 7 with 40 μm porous substrates and normal substrates.

Secondary Particle Collection Efficiency

There are several mechanisms that can contribute to the deposition of particles smaller than the cut size of the impactor. Collection caused by these mechanisms takes place not only on the collection plate but also on the stage hull. At low aerosol concentrations the dominant deposition mechanisms are diffusion and deposition caused by the electrical image force. In most cases diffusion is more relevant of these two, image charge deposition is significant to particles larger than 0.2 μ m. Experimental results for the combined effect of diffusion and image charge deposition can be seen in Figure 2. Figure 2a presents the results for smooth collection plates and Figure 2b presents the results for the porous collection plates. The porous plates cause a much higher level of secondary particle collection efficiency. This is partly explained by changes in the flow characteristics near the collection plate. However, there is also an effect caused by the fact that part of the flow penetrates the substrate and gets filtered by the porous substate material it passes (Huang et al. 2001; Marjamäki and Keskinen 2004a, b).

Figure 2 Measured secondary particle deposition caused by diffusion and image charge. Smooth plates on the left (a) and porous ones on the right (b).

Construction of the Response Functions

Impactor Collection Efficiency Functions

We use a simple fit function for the primary collection efficiency already described by Winklmayr et al. (1990) and Dzubay and Hasan (1990). This was also used by Dong et al. (2004) to produce fit functions for the primary collection efficiency E ^p , obtained by fitting curves to the graphs published by Marjamäki et al. (2000).

where D _p,50 is the cutpoint diameter and s is a parameter affecting the steepness of the collection efficiency curve. For the electrical filter stage collection efficiency is assumed to be 1. The same functional form is used both for smooth and porous collection plates (Figure 1).

Fit curves and measured efficiencies are different at low efficiencies because the fits for the primary collection do not include the effects of secondary collection mechanisms: diffusion, image charge, and space charge. The collection efficiency E ^D caused by diffusion depends on the diffusion coefficient of the particles and the geometry and flow conditions of the impactor stage in question. Deposition takes place both on the collection plate and the rest of the stage. The flow geometry is therefore a complex mix of planar, tubular, and annular components. For simplicity, a fit based on tube flow geometry was used for the diffusive part:

D (m²/Vs) is the diffusion coefficient of the particles, L (m) is the effective length of the stage in question, Q (m³/s) is the flow rate, Sh is Sherwood number for laminar flow (see e.g., Sampling and transport of aerosols by J.E. Brockmann in Willeke and Baron 1993).

For image charge the deposition velocity is proportional to the product of the square of the charge and mechanical mobility of the particle. With these assumptions collection efficiency, E ^IM , can be estimated to be:

where c is an arbitrary constant describing the intensity of the image charge deposition and q is the number of the elementary charges in the particle (can be estimated as max[1; 74.714*D _p ^1.1099], see Virtanen et al. 2001).

When the measured concentration is high, the repulsive electric field generated by the charged particles becomes significant. Particle properties affecting space charge deposition are particle concentration, particle charge, and electrical mobility of the particles. Space charge collection efficiency, E _i ^SC , for stage i can be estimated using the following equation (Virtanen et al. 2001).

where Z is the electrical mobility of the particles, I _j is the measured current on stage j, t is the characteristic residence time of the stage, Q is the flow rate of the impactor and ϵ₀ is the permittivity of vacuum. The current and the characteristic time are the corresponding values of the stage in question, but electrical mobility is evaluated differently for each particle size. Characteristic time is dependent on the actual volume of the stage, but it also depends on the flow profile, that is, how much mixing occurs inside of the stage.

When using porous collection plates, secondary collection is enhanced by change of flow conditions near the plate surface, increasing both diffusion and image charge collection efficiency terms. In addition, part of the jet flow penetrates the porous plate. Particles within that part of the flow get filtered by the porous plate material, with varying efficiency. In practice it is rather difficult to estimate the effect of this mechanism or to separate it from the other collection efficiency mechanisms. Therefore the increased collection efficiency of porous plates is treated as if caused by increased diffusion and image charge deposition.

To obtain overall collection efficiency values, the collection processes described above are assumed to be independent and penetration of particles the product of the penetration values of the different processes. Therefore, overall collection efficiency E _i for stage i can be calculated from the primary collection efficiency (Equation 1), diffusion and image charge deposition (Equations 2 and 3), and space charge deposition (Equation 4)

The parameter values of Equations 1, 2, 3, 4 were fitted to produce best agreement of Equation 5 and the measured collection efficiency values. Figure 3 shows the contribution of primary collection, diffusion, and image charge deposition to the overall collection efficiency of stage 9. There is an increase in deposition for large particles caused by the image charge part of the fit, but in practice it is always lost in the steep rise of the collection curve by impaction. Values for stages 1–4 have to be approximated, because the tail caused by secondary mechanisms cannot be separated from the primary collection efficiency. However the fit using the approximations fits the measured collection efficiencies well.

Figure 3 Measured efficiencies (marked with circles) and fitted functions (marked with lines) for stage 9: (a) Whole collection efficiency curve, fit labeled as total is calculated with Equation 5. (b) same collection efficiency with different y-axis and additional curves showing the contribution of primary, diffusion and image charge depositions (calculated with the Equations 1, 2, and 3 with the constants from Table 1). Cut point of the stage shown in both figures with dotted vertical line. (Adapted from Marjamäki 2003).

The fitted values for steepness (s) effective length (L), image charge deposition intensity (c), and residence time (t) for each stage are listed in Table 1. Using the constants from Table 1, deposition for each stage as a function of particle size can be calculated. Values can be adapted for other ELPI-impactor units with differing cutpoints by changing the corresponding D _p,50 value. In the case of porous collection plates we have given a constant C _p that can be used to estimate the cutpoint if calibration data is unavailable:

Table 1 Fitted values used in Equation 1 for the primary collection efficiency curve. Values for normal and 40 μ m porous substrates

	Primary collection				Secondary collection
	Normal		Porous		Normal		Porous
Stage	D p,50	s	C p	s	L, m	c	L, m	c	Residence time, s
1	0.029	3.4130	0.74	1.8365	2.44*	5.00*	4.1*	25*	0.017
2	0.054	4.2936	0.69	1.6728	2.44*	5.00*	4.1*	25*	0.030
3	0.090	2.9415	0.60	1.3924	2.44*	5.00*	4.1*	25*	0.053
4	0.15	3.1026	0.74	1.5889	2.44*	5.00*	4.1*	25*	0.069
5	0.26	3.5774	0.80	1.7033	2.44	5.00	10.72	5.16	0.076
6	0.38	9.2669	0.85	2.4652	3.60	2.54	8.37	3.85	0.078
7	0.62	5.8714	0.97	3.0448	2.92	1.96	6.40	3.22	0.09
8	0.92	8.7742	1	3.3844	1.84	2.18	5.54	1.96	0.1
9	1.6	4.8827	1	3.1656	1.84	1.15	3.81	1.71	0.1
10	2.4	5.5918	1	3.1002	1.83	0.53	3.00	1.57	0.11
11	4.0	4.5316	1	3.1244	1.88	0.58	3.19	0.67	0.12
12	6.5	4.4947	1	3.1881	2.53	0.26	3.19	0.67	0.14
13	10.0	3.8548
D p,50 = cut-point in micrometers, s = steepness, L = effective length, c = image charge intensity values, and C p = constant used to estimate the cut-point with porous substrates. Values marked with asterisk are approximations because of the data limitations. Stage 12 is removed when the filter stage is used.

Charging Efficiency

The ELPI is based on the electrical measurement of the collected particles. This means that, in addition to the collection efficiency of the impactor, efficiency of the charger has to be known. ELPI charger has two operation modes, one with the trap voltage (400 V) on and one without the trap voltage. Usually trap voltage is switched off when using the electrical filter stage to allow the passing of smallest particles into the impactor.

The efficiency of the charger is needed to convert the measured currents to number concentrations. Number concentration, C, can be calculated from the currents with the following equation

where I is the measured current, e is the elementary charge, Q is the flow rate. P _ch is the particle penetration through the charger and n is the average number of elementary charges per particle. Both the particle penetration through the charger and average number of charges per particle are functions of the particle mobility diameter. Charging efficiency was determined experimentally and following fit for the dimensionless product P _ch n was presented by Marjamäki et al. (2000) for the case of 400 V trap voltage:

where particle size is given in micrometers. When the trap is not used to capture the ions, the penetration of small particles increases and P _ch n takes the form (Marjamäki et al. 2002).

Response Functions

Impactor Kernel Functions

Impactor response functions are usually labeled as kernel functions. In this paper we use kernel functions to describe the impactor response functions. Response functions are used to describe the overall response functions that include the response functions of both the charger and the impactor.

Kernel functions inform how particles are distributed among each impactor channel. The sum of kernel functions at each particle size is one. Kernel functions, k _i , are functions of the particle size and can be calculated from the collection efficiency functions by:

where E _i is the collection efficiency of the stage i. Numbering is such that the stage with lowest cut size is stage 1 and stage with highest cut size is stage N. When the filter stage is used, stage 12 is removed and filter stage is labeled as stage 0. Efficiency of the filter stage is assumed to be 1. Here it should be noted that the primary collection efficiency is a function of particle aerodynamic diameter but space charge-, diffusion-, and image charge deposition are functions of mobility diameter. This means that the kernel functions and the response of the impactor are dependent on the effective density of particles.

Kernel functions of the ELPI impactor for the different configurations are shown in Figure 4. Results are calculated assuming unit density particles. Deposition caused by the space charge is evaluated to be zero, that is, aerosol concentration is assumed to be low. Compared to normal smooth substrates, kernel functions with porous substrates are wider, lower, and have more overlap between stages. With the filter stage, kernel function of stage 11 is wider because stage 12 has been removed. The kernel function of the filter stage has similar shape to impactor kernels, because secondary collection in impactor stages increases with decreasing particle size. Since secondary collection is a function of the mobility size of the particles, the kernel of filter stage has a strong dependence on the effective density of particles. This can cause problems especially when measuring particles of unknown composition and density. In the case of normal substrates, filter stage's kernel is wide. As described in Marjamäki et al. (2002), this is problematic when using cut diameter concept for converting the measured currents to number concentrations.

Figure 4 ELPI-impactor kernel functions, k _i , of the in different configurations. Functions are calculated according to Equation 10 from the collection efficiencies.

ELPI Response Functions for Moment Distributions

Size distribution functions can be presented as a moment distribution linked to the number size distribution f(D _p ) by

where α and β are constants. For example, β = 3 for volume and mass distributions. In addition, the sensitivity functions of many instruments can be described as moment functions. In the ELPI, the charge carried by the particles is distributed as

in which E _ch(D _p ) is the charging efficiency. The last part is an approximation, the true exponent varying as a function of the particle size. The approximation is made so that we can discuss the different aerosol distributions measured by the ELPI in terms of their moments.

The ideal response of each ELPI channel is calculated by integrating the product of the charger efficiency E _ch(D _p ), the impactor kernel function k _i (D _p ), and the number size distribution f(D _p ).

Using Equations 11 and 12, this can be written for any moment distribution β as:

This means that for moment β, ELPI overall kernel function K ^β(D _p ) = k(D _p )D _p ^{1.5 − β}.

The type of integral equation in Equations 13 and 14 is called the Fredholm integral equation of the first kind. Solving this equation for size distribution in not unambiguous, and the procedure is called inversion. The inversion of ELPI data can be made to return different moment distributions. From Equation 12 it is obvious that inversion into moment β = 1.5 is similar to inversion of gravimetric impactor data. The two most widely used distributions are the number distribution (i.e., zeroth moment) and the mass distribution, the third moment. Response functions for all of these cases are shown in Figure 5.

Figure 5 Response functions K ^β of the ELPI with filter stage and normal collection substrates for different moments: (a) 0-moment (number), (b) 1.5-moment, and (c) 3-moment (volume and mass). Response functions are the combination of the charger response and impactor kernels. Note that the y-axis is logarithmic and in arbitrary units.

Response functions for moment 1.5 are most even and therefore it would be most natural to use them in inversion (i.e., 1.5th moment distribution, β = 1.5). For the zeroth-moment (number size distribution) and 3rd-moment (mass and volume distribution) there are large differences in the magnitude of the response for different stages. For the 3rd-moment there is also duality in the response functions, that is, high values for both small and large particles, that can make the inversion difficult.

DISCUSSION

Impactor kernels are very similar to the ones reported by other authors for multijet cascade impactors. However the data reduction of ELPI differs significantly from the cascade impactor data reduction because the charging efficiency has to be considered. This gives the response functions a unique shape and makes the data reduction more challenging.

The most important phenomenon not included in this study is probably the particle bounce. As in all impactor measurements, this might cause a substantial error. The bounce can be reduced by using greased impaction plates, which is the standard procedure. In cases where the particle bounce is a particularly severe problem, the porous collection plates need to be used and therefore their response functions are also given.

Another ELPI-related phenomenon not included in this study is the induced currents caused by the rapid changes in aerosol concentration (Marjamäki et al. 1995). If the aerosol concentration—rapidly changes inside the impactor stage this induces a current that is proportional to the time derivative of the charge concentration. When the concentration increases it induces a positive current and when it decreases it induces a negative current. Total time integral of these induced currents is zero but they can be a significant source of uncertainty when studying rapidly changing concentrations in real time.

The particle charging, diffusion and the image charge deposition are parameters that depend on the particle mobility diameter. However, the primary collection efficiency depends on the aerodynamic diameter of the particle. Therefore to get exact results particle effective density has to be known or at least estimated. Fits for the response functions are semi-empirical and they are not supposed to be exactly correct, but they can be used to generate ELPI response functions for aerosols having variable effective densities.

SUMMARY

Response functions for the charger and impactor in different configurations have been presented. The overall response function of the ELPI is the product of the impactor kernels and the charger efficiency. Presented collection efficiency functions for the ELPI-impactor include the primary particle collection caused by impaction and secondary collection caused by diffusion and image charge. Charging efficiency is given in two different trap voltages. Presented equations allow construction of the ELPI response functions for different particle densities. These response functions can be used to develop algorithms for the inversion of ELPI data. A study of the inversion of the ELPI data and an example of an inversion algorithm for the ELPI is presented in the second part of this paper.