Research Article Evolution of Aerosol Particles in the ...
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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 Β© 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 theβGreenfield gapβ 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 0.02β10 πm 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 [19β21], 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:
βππ (ππ, π‘)
ππ‘= Ξ (ππ) β π (ππ, π‘) ,
(1)
where π(ππ, π‘) is the particle size distribution in air andΞ(ππ) is the scavenging coefficient representing the rate ofscavenged particles by raindrop; consider
Ξ(ππ) = β«
β
0
πΎ(ππ, π·π) πΈ (ππ, π·π) ππ·π, (2)
where πΎ(ππ, π·π) is the collision kernel (or collection kernel)which describes the probability of collisions between particleswith diameter ππ and raindrops with diameter π·π [3, 13];consider
πΎ(ππ, π·π) =ππ·2
π
4π (π·π) π (π·π) ,
(3)
where πΈ(ππ, π·π) is the collision efficiency which will bediscussed below. π(π·π) is the velocity of a falling raindropwith diameterπ·π and can be expressed as follows:
π (π·π) = 30.75π·2
πΓ 106, π·π < 100 πm,
π (π·π) = 38π·2
πΓ 103, 100 πm < π·π < 1000 πm,
π (π·π) = 133.046π·0.5
π, π·π > 1000 πm.
(4)
The collision efficiency, πΈ(ππ, π·π), 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:
πΈ (ππ, π·π) =1
Re β Sc[1 + 0.4Re1/2Sc1/3 + 0.16Re1/2Sc1/2]
+ 4ππ
π·π
[ππ
ππ€
+ (1 + Re1/2)ππ
π·π
]
+ (St β πβ
St β πβ + 2/3)
3/2
,
(5)
where Re = π·ππ(π·π)ππ/(2 ππ) is the Reynolds number basedon the raindrop diameter, Sc = ππ/(πππ·diff) is the Schmidtnumber of particles with the diffusion coefficient π·diff =
ππππΆπ/(3πππππ), St = 2ππ(π·π)/π·π is the Stokes number of
Abstract and Applied Analysis 3
particles with relaxation time π = πππ2
π/(18ππ), and π
β=
[1.2 + ln(1 + Re)/12]/[1 + ln(1 + Re)] is a dimensionlessparameter. Here, ππ and ππ are the density and the viscosityof air, respectively, ππ€ is the viscosity of a water drop, ππ isBoltzmannβs constant, π is the absolute temperature of air,and πΆπ is the Cunningham slip correction factor and can beapproximated as follows [25]:
πΆπ = 1 + 2.493π
ππ
+ 0.84π
ππ
exp(β0.435ππ
π) β 1 + 3.34
π
ππ
,
(6)
where π 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, Ξ(ππ), is related to thecollision kernel between particles and raindrops, collisionefficiency πΈ(ππ, π·π), and RSD π(π·π). 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:
π (π·π) =ππ
β2π lnπππexp[β
ln2 (π·π/π·ππ)
2ln2 (πππ)]1
π·π
,
π (ππ) =ππ
β2π lnπππexp[β
ln2 (ππ/πππ)
2ln2 (πππ)]1
ππ
,
(7)
where ππ, π·ππ, and πππ are the number concentration,geometric mean diameter, and geometric standard deviationof raindrop, respectively. Definitions of the πth moment ofraindrops and particles are ππ and ππ, respectively [24];consider
ππ = β«
β
0
π·π
ππ (π·π) ππ·π = π0π·
π
ππexp[π
2
2ln2πππ] ,
ππ = β«
β
0
ππ
ππ (ππ) πππ = π0π
π
ππexp[π
2
2ln2πππ] .
(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 β πβ
St β πβ + 2/3)
3/2
β 1 β 0.9Stβ0.5. (9)
The final expression of scavenging coefficient and govern-ing moment equation are as follows [3, 14, 28]:
Ξ(ππ) = πΎ1π1 (πβ1
π+ π΄πβ2
π) + πΎ2π7/4 (π
β2/3
π+ 2π΄π
β5/3
π/3)
+ πΎ3π7/4 (πβ1/2
π+ π΄πβ3/2
π/2) + πΎ4π3/2ππ + πΎ5π1/2π
2
π
+ πΎ6π5/4π2
π+ πΎ7π5/2 + πΎ8π11/4π
β1
π,
(10)
βπππ
ππ‘= πΎ1π1 (ππβ1 + π΄ππβ2) + πΎ2π7/4
Γ [ππβ2/3 + 2π΄ππβ5/3/3] + πΎ3π7/4
Γ [ππβ1/2 + π΄ππβ3/2/2] + πΎ4π3/2ππ+1
+ πΎ5π1/2ππ+2 + πΎ6π5/4ππ+2
+ πΎ7π5/2ππ + πΎ8π11/4ππβ1,
(11)
where
πΎ1 =πππ
6ππ
, πΎ2 = (0.4 Γ 130π
4)(
2 ππ
130ππ
)
1/2
(πππππ
3ππ2π
)
2/3
,
πΎ3 = (0.16 Γ 130π
4)(
2πππ
3 Γ 130πππ
)
1/2
, πΎ4 =130 πππ
ππ€
,
πΎ5 = 130π, πΎ6 = 130π(130ππ
2 ππ
)
1/2
, πΎ7 =130π
4,
πΎ8 = β(0.9 Γ 130π
4)(
18 ππ
2 Γ 130ππ
)
1/2
, π΄ = 3.34π.
(12)
According to the definition ofππ,π0 is the total particlenumber concentration, and (π/6)π3 is the total volume ofparticles. In the following calculation, we solve the first threemoment equations, that is, π = 0, 1, and 2. The geometricmean particle diameter, πππ, and the geometric standarddeviation, πππ, can be expressed as the function of the firstthree moments as follows:
πππ =π2
1
π3/2
0π1/2
2
, πππ = exp[ln(π2π0
π21
)]
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 π = 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 πππ0 = 1.5 are selected to validate the computationcodes. Cases 2β5 are selected to obtain the evolution of PSDfor clarifying the function by different mechanisms (i.e.,Brownian diffusion is dominant for πππ0 = 1β10 nm; bothBrownian diffusion and interception are dominant for πππ0= 0.1β0.5πm; and inertial impaction is dominant for πππ0 =5β8πm). Cases 6β9 are selected to study the effect of RSDon PSD for a given water content defined by (14) [4], andthe corresponding raindrop number concentration, ππ, iscalculated via (15). The computational programs are written
4 Abstract and Applied Analysis
Table 1: The values of the initialization parameter for Cases 1β9.
Case ππ/mβ3
π·ππ/mm πππ ππ0/mβ3
πππ0/πm πππ0
1 1.0 Γ 105 0.1 1.2 1.0 Γ 106 0.001 1.2/1.5/1.82 1.0 Γ 105 0.1 1.2 1.0 Γ 106 0.01 1.2/1.5/1.83 1.0 Γ 105 0.1 1.2 1.0 Γ 106 0.1 1.2/1.5/1.84 1.0 Γ 105 0.1 1.2 1.0 Γ 106 5.0 1.2/1.5/1.85 1.031 Γ 107 0.2 1.1 1.0 Γ 106 0.01/0.5/8.0 1.36 9.569 Γ 104 0.2 1.5 1.0 Γ 106 0.01/0.5/8.0 1.37 6.601 Γ 105 0.5 1.1 1.0 Γ 106 0.01/0.5/8.0 1.38 6.124 Γ 103 0.5 1.5 1.0 Γ 106 0.01/0.5/8.0 1.39 6.786 Γ 104 0.5 1.5 1.0 Γ 106 0.01/0.5/8.0 1.3
using the C Programming Language and are performed onMicrosoft Visual C++ 6.0 complier. Consider
π€ =π
6β«π·π
ππ€π·3
ππ (π·π) ππ·π, (14)
ππ =6π€
πππ€π·3
ππexp (9π2
ππ/2)
. (15)
3.2. Validation of Computation Codes. Particles with diam-eter πππ0 = 1 nm and 0.5 πm 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 πππ0 < 1πm,
πΈ (ππ, π·π) = 2(β3π
4ππ)
2/3
[(1 β πΌ) (3π + 4)
π½ + ππΎ] . (16)
3.3. Evolution of Particle Geometric Mean Diameter andGeometric Standard Deviation. Figures 2 and 3 show theevolution of particle geometric mean diameter, πππ, andgeometric standard deviation, πππ, in the rainfall process,respectively. From Figure 2 we can see that πππ increaseswhen the initial geometric mean diameter πππ0 is equal to1 nm, 10 nm, and 0.1 πm, and it decreases when πππ0 is equalto 5 πm. Thus, we can deduce that there exists one kind ofparticle diameter, for which πππ never changes in the wholescavenging process. For particles with πππ0 = 1 nm and 10 nm,πππ grows much faster when the value of πππ0 is larger. Andfor particles with πππ0 = 5 πm, πππ decreases much morerapidly when πππ0 is larger. The region that particle diameterlocates between 0.01 πm and 2 πm is called βGreenfield gapβ[30] because the collision efficiency of particles in this regionand raindrops is relatively low. While for particles in theβGreenfield gapβ (e.g., πππ0 = 0.1 πm), πππ hardly changes inthe whole process. The evolution of πππ in the present studyis qualitatively consistent with the result given by Jung et al.[13].
100
101
102
103
104
105
106
107
10β2
10β1
100
Jung and Lee (1998)
t (s)
Np
/Np0
dpg0 = 1nm
dpg0 = 0.5 πm
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, πππ, 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 πππ0, πππ decays much faster when thevalue of πππ0 is larger. For particles with the same initial valueof πππ0, πππ decreases at an early stage for small particlesdominated by the Brownian diffusion, and at a later stage forlarge particles controlled by the inertial impaction.πππ hardlychanges for particles in the βGreenfield gapβ.
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 6β9 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, πππ, isthe same but the geometric mean raindrop diameter, π·ππ, isdifferent. The particle number concentration decreases fasterin Case 6 than in Case 8 as shown in Figure 4. For particles
Abstract and Applied Analysis 5
100
101
102
103
104
105
106
107
t (s)
dpg0 = 1nm
dpg0 = 5πm
dpg0 = 0.1 πm
dpg0 = 10nm
πpg0 = 1.2
πpg0 = 1.5
πpg0 = 1.8
dpg/d
pg0
100
Figure 2: Evolution of particle geometric mean diameter as timeprogresses for Cases 2β5.
0.6
0.7
0.8
0.9
1.0
1.1
100
101
102
103
104
105
106
107
t (s)
πpg
/πpg0
πpg0 = 1.2
πpg0 = 1.5
πpg0 = 1.8
dpg0 = 1nm
dpg0 = 0.1 πm
dpg0 = 5πm
dpg0 = 10nm
Figure 3: Evolution of geometric standard deviation of particlediameter for Cases 2β5.
with πππ0 = 0.5 πm and πππ0 = 8 πm 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, π·ππ, is the same but the geometric standard deviationof raindrop diameter, πππ, 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 πππ0 = 0.5 πm in Case 9 than that for any other kind ofparticle diameters in any case.
Figure 5 shows the effect of RSD on πππ and πππ forparticles with πππ0 = 10 nm. It can be seen that πππ increasesand πππ decreases in all cases, which is consistent with theresults given in Figures 2 and 3. It takes the longest time forπππ/πππ0 and πππ/πππ0 to get the same values in Case 9, andthe shortest time in Case 6.The overall tendency of the effectof RSD on πππ and πππ for particles with πππ0 = 0.5 πm and
Case 9Case 8
Case 7Case 6
100
101
102
103
104
105
106
107
t (s)
Np
/Np0
dpg0 = 10nmdpg0 = 0.5 πmdpg0 = 8.0 πm
10β3
10β2
10β1
100
Figure 4: Effect of RSD on particle number concentration forCases 6β9.
Case 6 Case 7 Case 8 Case 9
100
101
102
103
104
105
106
t (s)
dpg0 = 10nm
dpg/d
pg0
πpg
/πpg0
1
2
Figure 5: Effect of RSD on particle geometric mean diameter andgeometric standard deviation (πππ0 = 10 nm).
πππ0 = 8πm, which is not shown in Figure 5, is the same asthat for particles with πππ0 = 10 nm.
From the above numerical results and analysis, we cansee that, for the same water content, the RSD with smallπ·ππ can remove the particles more easily than with largeπ·ππ. The RSD with small πππ, that is, a narrow distributionof particle diameter, can collect particles more easily thanwith large πππ. 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, ππ, increases as π·ππ andπππ 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
6 Abstract and Applied Analysis
particles controlled by the combined effect of Browniandiffusion and interception depend on the value of ππ/π·π. Forthe particles with smaller π·ππ and πππ, ππ/π·π 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 π·ππ andπππ can scavenge intermediate size particles with a muchhigher efficiency. While for particles with a relatively largesize, the scavenging coefficient becomes smaller with largerπ·ππ and πππ.
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 theβGreenfield gapβ 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 πππ0 < 10 nm, and decreases when πππ0 β₯1 πm, but changes a little when 10 nm < πππ0 < 1 πm.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 πππ0.
For the same water content, the raindrop size distributionwith small π·ππ can remove particles with a much higherefficiency than that with large π·ππ, and the raindrop sizedistribution with small πππ can collect particles more easilythan with large πππ. Particles in the βGreenfield gapβ 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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