research article evolution of aerosol particles in the

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 709497, 7 pageshttp://dx.doi.org/10.1155/2013/709497

Research ArticleEvolution of Aerosol Particles in the Rainfall Process viaMethod of Moments

Fangyang Yuan and Fujun Gan

School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China

Correspondence should be addressed to Fangyang Yuan; [email protected]

Received 20 July 2013; Accepted 13 August 2013

Academic Editor: Jianzhong Lin

Copyright © 2013 F. Yuan and F. Gan. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The method of moments is employed to predict the evolution of aerosol particles in the rainfall process. To describe the dynamicproperties of particle size distribution, the population balance equation is converted to moment equations by the method ofmoments and the converted equations are solved numerically. The variations of particle number concentration, geometric meandiameter, and geometric standard deviation are given in the cases that the Brownian diffusion and inertial impaction of particlesdominate, respectively. The effects of raindrop size distribution on particle size distribution are analyzed in nine cases. The resultsshow that the particle number concentration decreases as time goes by, and particles dominated by Brownian diffusion are removedmore significantly. The particle number concentration decreases much more rapidly when particle geometric mean diameter issmaller, and the particle size distribution tends to bemonodisperse. For the same water content, the raindrops with small geometricmean diameters can remove particles with much higher efficiency than those with large geometric mean diameters. Particles in the“Greenfield gap” are relatively difficult to scavenge, and a new method is needed to remove it from the air.

1. Introduction

Our surroundings are filled with aerosol particles which notonly affect the environment such as the air visibility, weather,and climate, but also cause respiratory diseases. Availableresearches show that the respiratory diseases are not onlyrelated to the particle mass concentration but also to theparticle size and number concentration [1]. Therefore, it isnecessary to remove aerosols fromair. In nature, precipitationis one of the most effective approaches to remove aerosolsin air, in which raindrops collide with aerosols and thencollect them. The removing process is affected by externalfactors including aerosol size distribution [2], raindrop sizedistribution (RSD) [3], rainfall intensity [4], and physicaland chemical properties [5] and internal factors includingcollision mechanisms between raindrops and aerosols andcondensation/evaporation of raindrops and aerosols [6].The mechanisms involved in the above processes includeparticle Brownian diffusion, direction interception, inertialimpaction, thermo- and diffusiophoresis forces, and electri-cal forces [2, 7]. Although these forces are coupled, one ormore of them may dominate for various regions of particle

size, drop size, particle density and hydrodynamic tempera-ture, and diffusion fields.

In the rainfall process, the most interesting thing ishow the particle size distribution (PSD) changes as timeprogresses and how PSD is affected by different RSD. Inorder to answer the questions, the population balance equa-tion (PBE) for particles is introduced [8]. The kernel ofPBE is how to express the scavenging coefficient whichrepresents the removing rate of aerosols by raindrops andis a function of collision efficiency of raindrops, terminalvelocity of raindrops, and PSD [4]. In the pioneering work,Slinn [9] obtained a semiempirical formula of collisionefficiency according to the Navier-Stokes equation using thedimensionless analysis coupled with the experimental data,which has been widely used when referring to the below-cloud scavenging process. Chate and Kamra [6] showed thatthe collision efficiency of water drops increases with theincreasing impaction parameter. Chate et al. [5] evaluatedthe scavenging coefficient for aerosols of diameters in therange of 0.02–10 𝜇m with various densities in accordancewith their chemical compositions for heavy rain regime.Theyfound that the inertial impactionmechanism is the dominant

2 Abstract and Applied Analysis

one in removing particles of all sizes for the heavy rainregime, and the scavenging coefficient is highly dependenton relative humidity for hygroscopic particles. Mircea andStefan [4] obtained an exponential expression of the scav-enging coefficient as a function of rainfall intensity andcollision efficiency between raindrops and aerosol particles.In their later work, Mircea et al. [8] got the linear relationsbetween the scavenging coefficient and the rainfall intensityvia numerical analysis. Andronache [10, 11] concluded thatthe below-cloud scavenging (BCS) coefficients of aerosolsby rainfall depend mainly on the aerosol size distributionparameters and on rainfall intensity, decreasing significantlywith aerosol diameter, increasing with rainfall rate, and aver-age raindrop and aerosol electric charge. Later, Andronacheet al. [12] developed a more complicated model to predictthe scavenging coefficient, finding it sensitive to the choiceof representation of mixing processes, raindrop size distribu-tion, phoretic effects in aerosol-raindrop collisions, and clouddroplet activation.

The researchmentioned above was focused on getting therelation between the scavenging coefficient and the externalfactors. Jung et al. [13] expressed the collision efficiency aspolynomial expression of particle diameter and applied themethod of moments (MOM) to get analytical solutions ofPBE. In their following work [3], they employed the collisionefficiency proposed by Slinn [9] and applied the MOM tostudy the evolution of PSDwhen the RSDobeys theMarshall-Palmer (MP) and Krigian-Mazin (KM) distributions, andthey got relatively good results. Besides, Bae et al. [14] devel-oped a good analytical expression for scavenging coefficient.The value of scavenging coefficient can be calculated whenthe initial three key parameters of PSD and rainfall intensityare given. While in the scavenging process, these parametersevolve as time goes by and the initially calculated scavengingcoefficient will be inaccurate due to the evolution of thePSD, even if the rainfall intensity remains the same. In fact,the three parameters of PSD and the scavenging coefficientcouple together. Thus, we suggest that more attention shouldbe paid to the dynamic evolution of PSD before getting thevalue of the scavenging coefficient.

The method of moments has been extensively usedto deal with the PBE when referring to physical/chemicalchanges of particles, including aggregation/breakage [15],condensation/evaporation [16], coagulation [17, 18], depo-sition/removal [19–21], and chemical reaction [22, 23]. Theperiodic moment method (PMM), proposed by Pratsinis[24], can be used to solve the evolution of PSD for simul-taneous nucleation, condensation, and coagulation in theentire particle size spectrum through approximating the sizedistribution by a unimodal log-normal function.

In the present study, the PMM is adopted to deal with thePBE when referring to the wet scavenging process, and thescavenging coefficient is expressed as a polynomial functionof aerosol diameter, raindrop diameter, and raindrop velocity.The evolutions of PSD are simulated numerically and theeffects of RSD on PSD are studied. The variations of particlenumber concentration, geometric mean diameter, and geo-metric standard deviation are givenwhen Brownian diffusionand inertial impaction of particles dominate, respectively.

2. Theory

2.1. Basic Theory of Wet Removal. The governing equationdescribing the time-dependent removal of particles in air bycollision with raindrops [8] is as follows:

−𝜕𝑛 (𝑑𝑝, 𝑡)

𝜕𝑡= Λ (𝑑𝑝) ⋅ 𝑛 (𝑑𝑝, 𝑡) ,

(1)

where 𝑛(𝑑𝑝, 𝑡) is the particle size distribution in air andΛ(𝑑𝑝) is the scavenging coefficient representing the rate ofscavenged particles by raindrop; consider

Λ(𝑑𝑝) = ∫

∞

0

𝐾(𝑑𝑝, 𝐷𝑑) 𝐸 (𝑑𝑝, 𝐷𝑑) 𝑑𝐷𝑑, (2)

where 𝐾(𝑑𝑝, 𝐷𝑑) is the collision kernel (or collection kernel)which describes the probability of collisions between particleswith diameter 𝑑𝑝 and raindrops with diameter 𝐷𝑑 [3, 13];consider

𝐾(𝑑𝑝, 𝐷𝑑) =𝜋𝐷2

𝑑

4𝑈 (𝐷𝑑) 𝑛 (𝐷𝑑) ,

(3)

where 𝐸(𝑑𝑝, 𝐷𝑑) is the collision efficiency which will bediscussed below. 𝑈(𝐷𝑑) is the velocity of a falling raindropwith diameter𝐷𝑑 and can be expressed as follows:

𝑈 (𝐷𝑑) = 30.75𝐷2

𝑑× 106, 𝐷𝑑 < 100 𝜇m,

𝑈 (𝐷𝑑) = 38𝐷2

𝑑× 103, 100 𝜇m < 𝐷𝑑 < 1000 𝜇m,

𝑈 (𝐷𝑑) = 133.046𝐷0.5

𝑑, 𝐷𝑑 > 1000 𝜇m.

(4)

The collision efficiency, 𝐸(𝑑𝑝, 𝐷𝑑), represents the ratio of theactual frequency of collisions to the theoretical frequency andis affected by turbulent diffusion, Brownian diffusion, van derWaals force, thermophoresis, electrostatic adsorption, and soon. Mircea et al. [4, 8] suggested the collision efficiency tobe a constant or a function of rainfall intensity (includinglinear and power law functions). Jung et al. [13] got aset of analytical solutions for polydispersed particles by awet removal process for submicrometer particles. Slinn [9]obtained the following semiempirical formula according tothe Navier-Stokes equation using the dimensionless analysiscoupled with the experimental data:

𝐸 (𝑑𝑝, 𝐷𝑑) =1

Re ⋅Sc[1 + 0.4Re1/2Sc1/3 + 0.16Re1/2Sc1/2]

+ 4𝑑𝑝

𝐷𝑑

[𝜇𝑎

𝜇𝑤

+ (1 + Re1/2)𝑑𝑝

𝐷𝑑

]

+ (St − 𝑆∗

St − 𝑆∗ + 2/3)

3/2

,

(5)

where Re = 𝐷𝑑𝑈(𝐷𝑑)𝜌𝑎/(2 𝜇𝑎) is the Reynolds number basedon the raindrop diameter, Sc = 𝜇𝑎/(𝜌𝑎𝐷diff) is the Schmidtnumber of particles with the diffusion coefficient 𝐷diff =

𝑘𝑏𝑇𝐶𝑐/(3𝜋𝜇𝑎𝑑𝑝), St = 2𝜏𝑈(𝐷𝑑)/𝐷𝑑 is the Stokes number of

Abstract and Applied Analysis 3

particles with relaxation time 𝜏 = 𝜌𝑝𝑑2

𝑝/(18𝜇𝑎), and 𝑆

∗=

[1.2 + ln(1 + Re)/12]/[1 + ln(1 + Re)] is a dimensionlessparameter. Here, 𝜌𝑎 and 𝜇𝑎 are the density and the viscosityof air, respectively, 𝜇𝑤 is the viscosity of a water drop, 𝑘𝑏 isBoltzmann’s constant, 𝑇 is the absolute temperature of air,and 𝐶𝑐 is the Cunningham slip correction factor and can beapproximated as follows [25]:

𝐶𝑐 = 1 + 2.493𝜆

𝑑𝑝

+ 0.84𝜆

𝑑𝑝

exp(−0.435𝑑𝑝

𝜆) ≅ 1 + 3.34

𝜆

𝑑𝑝

,

(6)

where 𝜆 is the molecular mean free path.The terms on the right-hand side (RHS) of (5) represent

the effects of Brownian diffusion, interception, and inertialimpaction, respectively.

2.2. Application of the Method of Moments. According to(2)-(3), the scavenging coefficient, Λ(𝑑𝑝), is related to thecollision kernel between particles and raindrops, collisionefficiency 𝐸(𝑑𝑝, 𝐷𝑑), and RSD 𝑛(𝐷𝑑). The RSD based onparticle diameter can be described with the log-normaldistribution [26], and the particle size distribution can also beapproximated by the log-normal distribution [27] as follows:

𝑛 (𝐷𝑑) =𝑁𝑑

√2𝜋 ln𝜎𝑑𝑔exp[−

ln2 (𝐷𝑑/𝐷𝑑𝑔)

2ln2 (𝜎𝑑𝑔)]1

𝐷𝑑

,

𝑛 (𝑑𝑝) =𝑁𝑝

√2𝜋 ln𝜎𝑝𝑔exp[−

ln2 (𝑑𝑝/𝑑𝑝𝑔)

2ln2 (𝜎𝑝𝑔)]1

𝑑𝑝

,

(7)

where 𝑁𝑑, 𝐷𝑑𝑔, and 𝜎𝑑𝑔 are the number concentration,geometric mean diameter, and geometric standard deviationof raindrop, respectively. Definitions of the 𝑘th moment ofraindrops and particles are 𝜉𝑘 and 𝑚𝑘, respectively [24];consider

𝜉𝑘 = ∫

∞

0

𝐷𝑘

𝑑𝑛 (𝐷𝑑) 𝑑𝐷𝑑 = 𝜉0𝐷

𝑘

𝑑𝑔exp[𝑘

2

2ln2𝜎𝑑𝑔] ,

𝑚𝑘 = ∫

∞

0

𝑑𝑘

𝑝𝑛 (𝑑𝑝) 𝑑𝑑𝑝 = 𝑚0𝑑

𝑘

𝑝𝑔exp[𝑘

2

2ln2𝜎𝑑𝑝] .

(8)

The last term on RHS of (5) is difficult to deal with whenapplying the moment method to (1). Thus, the approximateexpression proposed by Jung et al. [3] is adopted as follows:

(St − 𝑆∗

St − 𝑆∗ + 2/3)

3/2

≅ 1 − 0.9St−0.5. (9)

The final expression of scavenging coefficient and govern-ing moment equation are as follows [3, 14, 28]:

Λ(𝑑𝑝) = 𝛾1𝜉1 (𝑑−1

𝑝+ 𝐴𝑑−2

𝑝) + 𝛾2𝜉7/4 (𝑑

−2/3

𝑝+ 2𝐴𝑑

−5/3

𝑝/3)

+ 𝛾3𝜉7/4 (𝑑−1/2

𝑝+ 𝐴𝑑−3/2

𝑝/2) + 𝛾4𝜉3/2𝑑𝑝 + 𝛾5𝜉1/2𝑑

2

𝑝

+ 𝛾6𝜉5/4𝑑2

𝑝+ 𝛾7𝜉5/2 + 𝛾8𝜉11/4𝑑

−1

𝑝,

(10)

−𝜕𝑚𝑘

𝜕𝑡= 𝛾1𝜉1 (𝑚𝑘−1 + 𝐴𝑚𝑘−2) + 𝛾2𝜉7/4

× [𝑚𝑘−2/3 + 2𝐴𝑚𝑘−5/3/3] + 𝛾3𝜉7/4

× [𝑚𝑘−1/2 + 𝐴𝑚𝑘−3/2/2] + 𝛾4𝜉3/2𝑚𝑘+1

+ 𝛾5𝜉1/2𝑚𝑘+2 + 𝛾6𝜉5/4𝑚𝑘+2

+ 𝛾7𝜉5/2𝑚𝑘 + 𝛾8𝜉11/4𝑚𝑘−1,

(11)

where

𝛾1 =𝑘𝑏𝑇

6𝜇𝑎

, 𝛾2 = (0.4 × 130𝜋

4)(

2 𝜇𝑎

130𝜌𝑎

)

1/2

(𝑘𝑏𝜌𝑎𝑇

3𝜋𝜇2𝑎

)

2/3

,

𝛾3 = (0.16 × 130𝜋

4)(

2𝑘𝑏𝑇

3 × 130𝜋𝜇𝑎

)

1/2

, 𝛾4 =130 𝜋𝜇𝑎

𝜇𝑤

,

𝛾5 = 130𝜋, 𝛾6 = 130𝜋(130𝜌𝑎

2 𝜇𝑎

)

1/2

, 𝛾7 =130𝜋

4,

𝛾8 = −(0.9 × 130𝜋

4)(

18 𝜇𝑎

2 × 130𝜌𝑝

)

1/2

, 𝐴 = 3.34𝜆.

(12)

According to the definition of𝑚𝑘,𝑚0 is the total particlenumber concentration, and (𝜋/6)𝑚3 is the total volume ofparticles. In the following calculation, we solve the first threemoment equations, that is, 𝑘 = 0, 1, and 2. The geometricmean particle diameter, 𝑑𝑝𝑔, and the geometric standarddeviation, 𝜎𝑝𝑔, can be expressed as the function of the firstthree moments as follows:

𝑑𝑝𝑔 =𝑚2

1

𝑚3/2

0𝑚1/2

2

, 𝜎𝑝𝑔 = exp[ln(𝑚2𝑚0

𝑚21

)]

1/2

. (13)

3. Results and Discussions

3.1. Numerical Specifications. The 4th-order Runge-Kuttamethod with fixed time step is employed to solve (11)with 𝑘 = 0, 1, and 2. The evolutions of PSD are simulatednumerically and the effects of RSD on PSD are studied innine cases.The values of the initialization parameter are listedin Table 1. Case 1 and Case 4 with initial geometric standarddeviation 𝜎𝑝𝑔0 = 1.5 are selected to validate the computationcodes. Cases 2–5 are selected to obtain the evolution of PSDfor clarifying the function by different mechanisms (i.e.,Brownian diffusion is dominant for 𝑑𝑝𝑔0 = 1–10 nm; bothBrownian diffusion and interception are dominant for 𝑑𝑝𝑔0= 0.1–0.5𝜇m; and inertial impaction is dominant for 𝑑𝑝𝑔0 =5–8𝜇m). Cases 6–9 are selected to study the effect of RSDon PSD for a given water content defined by (14) [4], andthe corresponding raindrop number concentration, 𝑁𝑑, iscalculated via (15). The computational programs are written


Table 1: The values of the initialization parameter for Cases 1–9.

Case 𝑁𝑑/m−3

𝐷𝑑𝑔/mm 𝜎𝑑𝑔 𝑁𝑝0/m−3

𝑑𝑝𝑔0/𝜇m 𝜎𝑝𝑔0

1 1.0 × 105 0.1 1.2 1.0 × 106 0.001 1.2/1.5/1.82 1.0 × 105 0.1 1.2 1.0 × 106 0.01 1.2/1.5/1.83 1.0 × 105 0.1 1.2 1.0 × 106 0.1 1.2/1.5/1.84 1.0 × 105 0.1 1.2 1.0 × 106 5.0 1.2/1.5/1.85 1.031 × 107 0.2 1.1 1.0 × 106 0.01/0.5/8.0 1.36 9.569 × 104 0.2 1.5 1.0 × 106 0.01/0.5/8.0 1.37 6.601 × 105 0.5 1.1 1.0 × 106 0.01/0.5/8.0 1.38 6.124 × 103 0.5 1.5 1.0 × 106 0.01/0.5/8.0 1.39 6.786 × 104 0.5 1.5 1.0 × 106 0.01/0.5/8.0 1.3

using the C Programming Language and are performed onMicrosoft Visual C++ 6.0 complier. Consider

𝑤 =𝜋

6∫𝐷𝑑

𝜌𝑤𝐷3

𝑑𝑛 (𝐷𝑑) 𝑑𝐷𝑑, (14)

𝑁𝑑 =6𝑤

𝜋𝜌𝑤𝐷3

𝑑𝑔exp (9𝜎2

𝑑𝑔/2)

. (15)

3.2. Validation of Computation Codes. Particles with diam-eter 𝑑𝑝𝑔0 = 1 nm and 0.5 𝜇m are selected to validate thecomputation codes. Figure 1 shows the numerical results ofparticle number concentration based on different collisionefficiencies proposed by Slinn [9] and Jung and Lee [29],respectively. The collision efficiency proposed by Slinn isgiven with a semiempirical formula based on the Navier-Stokes equation using dimensionless analysis combined withexperimental data.While the collision efficiency proposed byJung and Lee, as shown in (16), is valid for 𝑑𝑝𝑔0 < 1𝜇m,

𝐸 (𝑑𝑝, 𝐷𝑑) = 2(√3𝜋

4𝑃𝑒)

2/3

[(1 − 𝛼) (3𝜎 + 4)

𝐽 + 𝜎𝐾] . (16)

3.3. Evolution of Particle Geometric Mean Diameter andGeometric Standard Deviation. Figures 2 and 3 show theevolution of particle geometric mean diameter, 𝑑𝑝𝑔, andgeometric standard deviation, 𝜎𝑝𝑔, in the rainfall process,respectively. From Figure 2 we can see that 𝑑𝑝𝑔 increaseswhen the initial geometric mean diameter 𝑑𝑝𝑔0 is equal to1 nm, 10 nm, and 0.1 𝜇m, and it decreases when 𝑑𝑝𝑔0 is equalto 5 𝜇m. Thus, we can deduce that there exists one kind ofparticle diameter, for which 𝑑𝑝𝑔 never changes in the wholescavenging process. For particles with 𝑑𝑝𝑔0 = 1 nm and 10 nm,𝑑𝑝𝑔 grows much faster when the value of 𝜎𝑝𝑔0 is larger. Andfor particles with 𝑑𝑝𝑔0 = 5 𝜇m, 𝑑𝑝𝑔 decreases much morerapidly when 𝜎𝑝𝑔0 is larger. The region that particle diameterlocates between 0.01 𝜇m and 2 𝜇m is called “Greenfield gap”[30] because the collision efficiency of particles in this regionand raindrops is relatively low. While for particles in the“Greenfield gap” (e.g., 𝑑𝑝𝑔0 = 0.1 𝜇m), 𝑑𝑝𝑔 hardly changes inthe whole process. The evolution of 𝑑𝑝𝑔 in the present studyis qualitatively consistent with the result given by Jung et al.[13].

100

101

102

103

104

105

106

107

10−2

10−1

100

Jung and Lee (1998)

t (s)

Np

/Np0

dpg0 = 1nm

dpg0 = 0.5 𝜇m

Numerical results using the collision efficiency proposed by

Slinn (1983) (this study)

Figure 1: Comparison of particle number concentrations based ondifferent collision efficiencies.

In Figure 3, the geometric standard deviation of particlediameter, 𝜎𝑝𝑔, converges to 1.0 as time goes by in the rainfallprocess for all particle sizes, which means that the particlesize distribution tends to be monodisperse. For particles withthe same value of 𝑑𝑝𝑔0, 𝜎𝑝𝑔 decays much faster when thevalue of 𝜎𝑝𝑔0 is larger. For particles with the same initial valueof 𝜎𝑝𝑔0, 𝜎𝑝𝑔 decreases at an early stage for small particlesdominated by the Brownian diffusion, and at a later stage forlarge particles controlled by the inertial impaction.𝜎𝑝𝑔 hardlychanges for particles in the “Greenfield gap”.

3.4. Effect of Raindrop Size Distribution on Particle SizeDistribution. The effect of raindrop size distribution (RSD)on the particle size distribution (PSD) is studied using thevalues of initialization parameter for Cases 6–9 with thegiven water content (10 g/m3). Figure 4 shows the evolutionof particle number concentration. For Cases 6 and 8 thegeometric standard deviation of raindrop diameter, 𝜎𝑑𝑔, isthe same but the geometric mean raindrop diameter, 𝐷𝑑𝑔, isdifferent. The particle number concentration decreases fasterin Case 6 than in Case 8 as shown in Figure 4. For particles


100

101

102

103

104

105

106

107

t (s)

dpg0 = 1nm

dpg0 = 5𝜇m

dpg0 = 0.1 𝜇m

dpg0 = 10nm

𝜎pg0 = 1.2

𝜎pg0 = 1.5

𝜎pg0 = 1.8

dpg/d

pg0

100

Figure 2: Evolution of particle geometric mean diameter as timeprogresses for Cases 2–5.

0.6

0.7

0.8

0.9

1.0

1.1

100

101

102

103

104

105

106

107

t (s)

𝜎pg

/𝜎pg0

𝜎pg0 = 1.2

𝜎pg0 = 1.5

𝜎pg0 = 1.8

dpg0 = 1nm

dpg0 = 0.1 𝜇m

dpg0 = 5𝜇m

dpg0 = 10nm

Figure 3: Evolution of geometric standard deviation of particlediameter for Cases 2–5.

with 𝑑𝑝𝑔0 = 0.5 𝜇m and 𝑑𝑝𝑔0 = 8 𝜇m in Cases 6 and 8, thesame tendency can be observed, which demonstrates that theRSD in Case 6 can scavenge particles with a much higherefficiency than that in Case 8.

For Cases 8 and 9, the geometric mean raindrop diam-eter, 𝐷𝑑𝑔, is the same but the geometric standard deviationof raindrop diameter, 𝜎𝑑𝑔, is different. It can be seen thatthe RSD in Case 8 can remove particles with a much higherefficiency than that in Case 9 for any kind of particles.Meanwhile, it takes a much longer time to scavenge particleswhen 𝑑𝑝𝑔0 = 0.5 𝜇m in Case 9 than that for any other kind ofparticle diameters in any case.

Figure 5 shows the effect of RSD on 𝑑𝑝𝑔 and 𝜎𝑝𝑔 forparticles with 𝑑𝑝𝑔0 = 10 nm. It can be seen that 𝑑𝑝𝑔 increasesand 𝜎𝑝𝑔 decreases in all cases, which is consistent with theresults given in Figures 2 and 3. It takes the longest time for𝑑𝑝𝑔/𝑑𝑝𝑔0 and 𝜎𝑝𝑔/𝜎𝑝𝑔0 to get the same values in Case 9, andthe shortest time in Case 6.The overall tendency of the effectof RSD on 𝑑𝑝𝑔 and 𝜎𝑝𝑔 for particles with 𝑑𝑝𝑔0 = 0.5 𝜇m and

Case 9Case 8

Case 7Case 6

100

101

102

103

104

105

106

107

t (s)

Np

/Np0

dpg0 = 10nmdpg0 = 0.5 𝜇mdpg0 = 8.0 𝜇m

10−3

10−2

10−1

100

Figure 4: Effect of RSD on particle number concentration forCases 6–9.

Case 6 Case 7 Case 8 Case 9

100

101

102

103

104

105

106

t (s)

dpg0 = 10nm

dpg/d

pg0

𝜎pg

/𝜎pg0

1

2

Figure 5: Effect of RSD on particle geometric mean diameter andgeometric standard deviation (𝑑𝑝𝑔0 = 10 nm).

𝑑𝑝𝑔0 = 8𝜇m, which is not shown in Figure 5, is the same asthat for particles with 𝑑𝑝𝑔0 = 10 nm.

From the above numerical results and analysis, we cansee that, for the same water content, the RSD with small𝐷𝑑𝑔 can remove the particles more easily than with large𝐷𝑑𝑔. The RSD with small 𝜎𝑑𝑔, that is, a narrow distributionof particle diameter, can collect particles more easily thanwith large 𝜎𝑑𝑔. Thus, the RSD in Case 6 is the most efficientin scavenging particles, the RSD in Case 8 takes secondplace, the RSD in Case 7 the third, and the RSD in Case 9is the worst, which is consistent with the experimentalresult [31]. This can be explained in the following way. Theraindrop number concentration, 𝑁𝑑, increases as 𝐷𝑑𝑔 and𝜎𝑑𝑔 decrease with a given water content according to (15),leading to the increase of the total surface area of raindrops,which enhances the probability of capturing small particlesdominated by the Brownian diffusion. The intermediate size


particles controlled by the combined effect of Browniandiffusion and interception depend on the value of 𝑑𝑝/𝐷𝑑. Forthe particles with smaller 𝐷𝑑𝑔 and 𝜎𝑑𝑔, 𝑑𝑝/𝐷𝑑 is larger; thatis, the RSD is more monodisperse, which makes the collisionefficiency become large as shown in the second term on theright-hand side of (5). Thus, the RSD with smaller 𝐷𝑑𝑔 and𝜎𝑑𝑔 can scavenge intermediate size particles with a muchhigher efficiency. While for particles with a relatively largesize, the scavenging coefficient becomes smaller with larger𝐷𝑑𝑔 and 𝜎𝑑𝑔.

4. Conclusions

The removal of aerosol particles in the rainfall process isstudied with the periodic moment method. Both the aerosolparticle size distribution and raindrop size distribution areassumed to be log-normal. The effects on the collisionefficiency of the mechanisms of Brownian diffusion, inter-ception, and inertial impaction have been investigated. Thebinomial formulas are reexpanded to get a more accurateexpression of scavenging coefficient. The first three momentsof particle size distribution are simulated and the effects ofraindrop size distribution on particle size distribution arestudied in nine cases.

The results show that the particle number concentrationdecreases as time goes by. Particles dominated by Browniandiffusion are removed more easily. The particles in the“Greenfield gap” are the most difficult to be removed. Theparticle number concentration decreases much more rapidlywhen particle geometric mean diameter is smaller. In thescavenging process, the particle geometric mean diameterincreases when 𝑑𝑝𝑔0 < 10 nm, and decreases when 𝑑𝑝𝑔0 ≥1 𝜇m, but changes a little when 10 nm < 𝑑𝑝𝑔0 < 1 𝜇m.The geo-metric standard deviation of particle diameter converges to1.0 as time progresses for any kind of particles, which meansthat the particle size distribution tends to be monodisperse,and it decays much faster for particles with large 𝜎𝑝𝑔0.

For the same water content, the raindrop size distributionwith small 𝐷𝑑𝑔 can remove particles with a much higherefficiency than that with large 𝐷𝑑𝑔, and the raindrop sizedistribution with small 𝜎𝑑𝑔 can collect particles more easilythan with large 𝜎𝑑𝑔. Particles in the “Greenfield gap” arerelatively difficult to scavenge, and a new method is neededto remove it from the air.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This work was supported by the Major Program of the Na-tional Natural Science Foundation of China (11132008).

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