Minimum Collection Efficiency Particle Diameter during Precipitation as a Function of Rain Intensity

This study found that the minimum collection efficiency particle diameter during precipitation is a function of rain intensity. The raindrop size distribution was parameterized with a log normal size distribution as a function of rain intensity, and the minimum collection efficiency and minimum collection efficiency diameter were obtained analytically. The results show that, during precipitation, both the minimum scavenging coefficient diameter and the minimum scavenging coefficient increase along with rain intensity. A comparison of the minimum collection efficiency diameters for various falling velocities reveals that there is not much of a difference in the diameters. Both the numerical and analytical results of this study agree well, without much loss of accuracy.


INTRODUCTION
Wet deposition of aerosol particles is an important removal mechanism that serves to maintain a balance between atmospheric sources and sinks (Maria and Russell, 2005;Chen et al., 2011).The scavenging of atmospheric aerosols by the capture of raindrops, called below-cloud scavenging, is a wet deposition process (Tiwari et al., 2012).Wet deposition played an important role in a simulation of fate and transport, removing as much as 70% of the released aerosol (Loosmore and Cederwall, 2004).
The characteristics of rainfall can be determined by many factors, such as the size and speed of the falling raindrops, and the variability in these different terms under various weather conditions (Depuydt, 2012).Abel and Boutle (2012) analyzed observations of the size distribution of raindrops from a variety of cloud types and precipitation rates.
Among the various concepts and theories in explaining aerosol wet deposition or scavenging process, the prediction of the most penetrating particle size is crucially important.
Scavenging gap or minimum collection efficiency diameter in scavenging process is the size where aerosols are not effectively removed by rain droplets.This is the size where the penetration of particles through the rain is maximized, and the collection efficiency minimized.Usually, the minimum collection efficiency diameter can be numerically calculated by differentiation of the collection efficiency.
The minimum collection efficiency diameter and corresponding minimum collection efficiency depend on the raindrop size distribution, falling velocity and aerosol removal mechanisms.Thus, estimating the minimum collection efficiency diameter and corresponding minimum collection efficiency can contribute the understanding of the characteristics in raindrop size and aerosol relation as well as atmospheric conditions.
For example, in predicting the harmful impact of wet deposition such as Chernobyl or volcanic eruption event, it is important to estimate the size resolved deposition characteristics during the deposition process.In that case, increasing minimum collection efficiency means that the amount of wet deposited particles increases.On the while, minimum collection efficiency diameter means the diameter at which the particles are mostly suspended without removing after the wet deposition.Because the particle size is very critical in human health, it is really important to properly simulate the minimum collection efficiency diameter and corresponding minimum correction efficiency.
Basically, the main process of aerosol scavenging is similar with aerosol filtration theory and scavenging gap can be explained in terms of the most penetrating particle size.
Lee (1981) obtained the most penetrating particle size for granular filtration when diffusion, interception and gravitation are considered.Jung and Lee (2007) derived the most penetrating particle size for multiple fluid collectors and showed the similar tendency with the solid collector of Lee (1981).The concept of most penetrating particle size can also be used for granular media filtration processes as critical suspended particle size in water (Qi, 1998;Jung and Lee, 2006).Jung et al. (2011) obtained the approximate analytical solution for the scavenging gap particle size during precipitation.However, the parameters of the raindrop size distribution used in Jung et al. (2011) were not sensitive to the change of rain type or rain intensity, which had limits with regard to its practical application.In general, precipitation is characterized based on measurements of the rain intensity, because it is difficult to measure raindrop size distribution.For the purpose of easy manipulation, the scavenging coefficients can be usefully used when they are parameterized as a function of rain intensity.Andronache (2003) parameterized the calculated scavenging coefficients as a power law function of rain intensity.Mircea et al. (2000) and Bae et al. (2006) also approximated the calculated scavenging coefficients as a function of rain intensity.
In this study, we adopted parameterized raindrop size distribution with log normal size distribution as a function of rain intensity.The minimum collection efficiency and the minimum collection efficiency diameter were then obtained analytically, and their variation according to rain intensity was verified.The obtained results were compared with numerical results.

MINIMUM COLLECTION EFFICIENCY DIAMETER
The scavenging efficiency represents a first-order approximation of particle transfer rate into raindrops (Mircea et al., 2000).The governing equation of the wet removal process of particles (scavenging efficiency) can be expressed as follows: ( , ) ( ) ( , ) where d p is the diameter of the collected particle, Λ(d p ) is the scavenging coefficient, and n(d p ,t) is the number size distribution of particles.Λ(d p ) can be defined as (Seinfeld and Pandis, 1998): where, ( ) Re 2 , and where, Re is the Reynolds number of a raindrop based on its radius, Sc is the Schmidt number of a particle, ρ a is the density of a particle in g/cm 3 , μ a and μ w are the viscosity of air and a water in g/cm/s, respectively, D diff is the vapor diffusivity in cm 2 /s, τ is the relaxation time, k b is Boltzmann's constant in g cm 2 /K/s 2 , λ is the mean free path in cm, Cc is the Cunnungham slip correction factor, and T is the temperature in K.
The collection efficiency due to diffusion increases as the particle diameter decreases.At the same time, the collection efficiency due to interception and impaction increases as the particle diameter increases.
One can define the critical particle diameter as a diameter, from which the impaction mechanism become important.
The critical diameter is a particle diameter at where The subsequent critical diameter can be expressed as follows from Eq. ( 4): It has been shown that the critical diameter is more than several microns in diameter, which means the impaction mechanism does not influence the determination of the minimum collection efficiency diameter (Jung et al., 2011).This means that only diffusion and interception need to be considered to obtain the minimum-collection-efficiency particle diameter.
Therefore, the most penetrating particle size exists between the two dominant mechanisms.Usually, the minimum efficiency particle size, which is known as the scavenging gap, ranges from 0.1 to several microns in diameter.
From Eqs. ( 2) and ( 3), the scavenging coefficient is given as follows (Jung et al., 2011): where If the drop size distribution is log-normal, the polydisperse drop size distribution can be expressed as a moment formula (Jung et al., 2002): where M k is the k-th moment expression for a polydisperse drop size distribution given by.
where M 0 (= N d ) is the total raindrop number concentration and D Dg is the geometric mean drop diameter, and σ Dg is the geometric standard deviation of drops.A detailed description of the moment formula can be found in the literature (Jung et al., 2002;Bae et al., 2006).
In this study, we did not consider the Cunningham slip correction factor.Normally, the Cunningham slip correction factor becomes important with particle diameters less than 1 μm.This simplification is expected to degrade, to some degree, the accuracy of the analytical results (Lee and Liu, 1980).The degradation of the accuracy in estimating minimum collection efficiency diameter depends on aerosol diameter.According to Lee and Schmidt (1981), the discrepancy is about 5% for a particle size of 0.4 μm for solid collector, which can be negligible.However, it is necessary to evaluate the accuracy through the more quantitative analysis for scavenging drop.
It is difficult to measure raindrop size distribution directly.Rain intensity, however, can be easily measured.Thus, various scavenging coefficients, expressed as constants or functions of precipitation intensity, are still used in regional or mesoscale Lagrangian and Eulerian models to describe precipitation and the wet removal of pollutants (Seinfeld and Pandis 1998;Bae et al., 2006).
Several measurement results have shown that a lognormal function can also correspond well with raindrop size distributions (Feingold and Levin, 1986).The lognormal distribution of raindrops is given by where N d is the total raindrop number concentration in cm -3 , D dg is the geometric mean diameter of the raindrops in cm, D d is the raindrop diameter in cm, and σ dg the geometric standard deviation of raindrops.
In this study, we applied the parameterized raindrop size distribution obtained by Feingold and Levin (1986) and Cerro et al. (1997).They approximated the raindrop size distribution with a lognormal function dependent on the rain intensity, which is shown in Table 1.Based on the raindrop size distribution parameterization, the moment formula, as a function of rain intensity, can be expressed as follows: for the Feingold and Levin (1986) Cerro et al. (1997)

distribution
The raindrop size distributions obtained by Feingold and Levin (1986) and Cerro et al. (1997) show that the number concentration and geometric mean diameter increase, and the geometric standard deviation decreases as rain intensity increases.The number concentration and the geometric standard deviation obtained by Feingold and Levin (1986) are lower than those obtained by Cerro et al. (1997).
One of the important factors affecting the scavenging of aerosol through rain is the falling velocity.The general mathematical expression for the falling velocity based on the drop diameter is the power-law type function of the drop diameter (Jung et al., 2002): where α and β are constants.
Table 2 shows a falling velocity as a function of drop diameter.Three formulas for the falling velocity were listed.The two parameters for each formula are shown in Table 2.The falling velocity increases as the drop diameter increases.The falling velocity obtained by Atlas and Ulbrich (1977) is larger than those obtained by Kessler (1969) and Depuydt (2012), especially for large drop diameter.
This study compared the minimum scavenging efficiency and the minimum scavenging coefficient diameter based on the previously studied falling velocity (Table 2).
From Eqs. ( 8) and (A4), namely the resulting minimum scavenging coefficient, the following can be obtained: The detailed procedure for obtaining analytic expression for minimum scavenging coefficient and minimum scavenging coefficient diameter can be described in Appendix.
It should be noted that in this study, the moment is a function of rain intensity only.

RESULTS
Fig. 1 shows a comparison of the minimum scavenging coefficient diameter as a function of rain intensity with different rain types.The numerical and analytic results are compared.The falling velocity is based on Kessler's formula.From Eq. (A4), the minimum scavenging coefficient diameter can be obtained numerically by using root-finding algorithm such as the Newton's method, Jenkins-Traub method, Laguerre's method, or Durand-Kerner method.In this study, we applied Newton's method.
As Fig. 1 shows, the minimum scavenging coefficient diameter increases with the rain intensity.Fig. 1 also shows good agreement between the numerical and analytical results.The minimum scavenging coefficient diameter for the raindrop size distribution obtained by Feingold and Levin (1986) is shown to be higher than for that obtained by Cerro et al. (1977).However, the difference between the numerical and analytic results is small.
The minimum collection efficiency diameter is less than 1 μm in all of the simulation conditions considered.The critical diameter (i.e. the diameter at which impaction is  (Jung et al., 2011), and the impaction mechanism is neglected in this study.Fig. 2 shows a comparison of the minimum scavenging coefficient as a function of rain intensity.
As shown in the figure, the analytic solutions agree very well with the numerical results.The minimum scavenging coefficient increases with rain intensity.
The minimum scavenging coefficient for the raindrop size distribution obtained by Feingold and Levin (1986) is shown to be lower than for that obtained by Cerro et al. (1977), and the difference increases with rain intensity.As Fig. 2 shows, both studies from Feingold and Levin (1986) and Cerro et al. (1997) show different size distribution and parameters.However, the three parameters for raindrop size distribution show similar tendency and subsequent minimum collection efficiency show a similar value.The minimum collection efficiency in Cerro distribution is larger than Feingold and Levin's distribution at high rain intensity.Because those raindrop size distributions are based on measurement data and it is difficult to state that which distribution is better for simulation study.However, one can state that the exact raindrop measurement and parameterization of size distribution can improve the accuracy of the simulation results.
Fig. 3 shows the minimum scavenging coefficient diameter and the minimum scavenging coefficient as a function of rain intensity with different falling velocities.The raindrop size distribution obtained by Feingold and Levin was estimated using the falling velocity obtained by Kessler (1969), Atlas andUlbrich (1977), andDepuydt (2012).As shown in Fig. 3, the minimum scavenging coefficient diameters mainly exist between 0.4 and 0.5 μm along the rain intensity of 0.1-100 mm/hr.The comparison of the minimum collection efficiency diameter among different types of falling velocity shows that the increasing rate of minimum collection efficiency particle size is larger for the falling velocity formula from Kessler (1969) and Depuidt (2012) than that of Atlas and Ulbrich (1977).Although there are some differences, the minimum scavenging coefficient diameters are comparable.It is shown that the minimum scavenging coefficient does not vary much with different falling velocities.analytic minimum scavenging coefficient diameter as a function of rain intensity.As is shown, the error decreases as the rain intensity increases.Through a rain intensity range of 0.1-100 mm/hr, the error of the analytic solution is within 3% compared with numerical results.This study assumed the raindrop size distribution as a log-normal function.Although gamma distribution is widely used representing raindrop size distribution, log-normal size distribution can provide informations for polydispersed size distribution such as total number concentration, geometric mean diameter and geometric standard deviation.The widely used gamma type rain drop functions can be applied using this methodology.However, detailed integration may depend on the gamma type raindrop size distribution.It should be noted that Feingold and Levin's or Cerro's raindrop size distributions are one of the examples which representing rain drop size distribution and can be replaced by other raindrop size distributions if they represent raindrop size distribution better.

CONCLUSIONS
In this study, we obtained an analytic solution of the minimum collection efficiency and the corresponding minimum collection efficiency diameter as a function of rain intensity.The harmonic type of approximation from Jung et Rain Intensity (mm/hr) In this study, we took into account diffusion and interception when deriving the minimum collection efficiency diameter.Various mechanisms, such as duffusiophoresis, thermophoresis, and electric charging effects, were not considered.These effects should be considered in future studies.In this study, only two types of lognormal raindrop size distributions were tested.In the future, more measurement based raindrop size distributions should also be considered.
The raindrop size distributions and falling velocities are based on the experimental results and it is difficult to state the better expressions in current study.However, the exact falling velocity can derive the exact minimum scavenging coefficient and minimum scavenging coefficient diameter.

APPENDIX
This Appendix shows the procedure of analytical expression for minimum scavenging coefficient particle size.
From the above mentioned parameterization, the governing equation for the minimum scavenging coefficient particle size can be expressed as in Eq. (A1) by differentiating the scavenging coefficient over the particle size.
The terms in Eq. (A1) refer to diffusion and interception mechanisms.The collection efficiency due to diffusion increases as the particle diameter decreases.At the same time, the collection efficiency due to interception increases as the particle diameter increases.
Thus, one can estimate the leading terms for each dominant particle or collector size region.If two dominant terms in Eq. (A1) are selected, the resulting approximated equations can be generated as follows (Jung andLee, 2006, 2007;Jung et al., 2011): The subsequent solutions from Eq. (A2) can be expressed as follows: It should be noted that M k is the function of rain intensity from Eq. ( 11).Subsequently, the minimum scavenging coefficient particle size can be expressed as in Eq. (A4) by applying the harmonic mean type approximation (Jung et al., 2011)

Fig. 4 Fig. 1 .Fig. 2 .
Fig. 1.Comparison of the minimum scavenging coefficient diameter as a function of rain intensity with different rain type (numerical vs. analytic).

Table 1 .
Expressions for the parameters of lognormal raindrop size distribution as a function of rain intensity (R).