137 THERMAL SCIENCE, Year 1, Vol. 16, No. 5, pp. 137-1376 EVOLUTION OF PARTICLE SIZE DISTRIBUTION IN AIR IN THE RAINFALL PROCESS VIA THE MOMENT METHOD by Fu-Jun GAN a an Jian-Zhong LIN a,b * a Department of Mechanics, Zhejiang University, Hangzhou, China b China Jiliang University, Hangzhou, China Short paper DOI: 1.98/TSCI1537G Population balance equation is converte to three moment equations to escribe the ynamical behavior of particle size istribution in air in the rainfall. The scavenging coefficient is expresse as a polynomial function of the particle iameter, the rainrop iameter an the rainrop velocity. The evolutions of particle size istribution are simulate numerically an the effects of the rainrop size istribution on particle size istribution are stuie. The results show that the rainrops with smaller geometric mean iameter an geometric stanar eviation of size remove particles much more efficiently. The particles which fall in the greenfiel gap are the most ifficult to be scavenge from the air. Key wors: particles in air in the rainfall process, scavenging coefficient, istribution of particle, numerical simulation Introuction Our surrounings are fille with aerosol particles. The respiratory iseases are not only relate to the particle mass concentration but also the particle size an number concentration. Therefore, it is necessary to remove aerosols from the air. Rainfall process is one of the most effective approaches to remove aerosols, in which rainrops collie with aerosols an then collect them. The removing process is affecte by external factors, an in the rainfall process, the most interesting thing is how the particle size istribution (PSD) changes as time progresses an how PSD is affecte by ifferent rainrop size istribution (RSD). In orer to answer the questions, the population balance equation (PBE) for particles is introuce. Slinn [1] obtaine a semi-empirical formula of collision efficiency which has been wiely use. Anronache [, 3] conclue that the below-clou scavenging (BCS) coefficients of aerosols by rainfall epens mainly on the aerosol size istribution parameters an on rainfall intensity. Later, Anronache et al. [4] evelope a more complicate moel to preict the scavenging coefficient. The research mentione above was focuse on getting the relation between the scavenging coefficient an the external factors. In the preset stuy the PBE is solve with referring to the wet scavenging process, an the scavenging coefficient is expresse as a polynomial function. * Corresponing author; e-mail: mecjzlin@public.zju.eu.cn
THERMAL SCIENCE, Year 1, Vol. 16, No. 5, pp. 137-1376 1373 Theory [5]: The equation escribing the removal of particles in air by collision with rainrops is n (, ) p t π D U D n D E D D n t p p t 4 = ( ) ( ) (, ) (, ) (1) where n( p, t) is the particle size istribution in air, U(D ) the velocity of a falling rainrop with iameter D an can be expresse as follows: 6 U( D) = 3.75D 1 D < 1 μm 3 U( D) = 38D 1 1 μm< D < 1 μm.5 U( D) = 133.46 D D > 1 μm () The collision efficiency is given as [1]: 1 3 E ( p, D ) = [1 + Re Sc +.16 Re Sc] + Re Sc * p μ a p St S +4 + ( 1+ Re) + D * μw D St S + 3 3 (3) where Re = D U(D )ρ a /(μ a ) is the Reynols number, Sc = μ a /(ρ a D iff ) the Schmit number with the iffusion coefficient D iff = b TC c /(3πμ a p ), St = τu(d )/D the Stoes number of particles with relaxation time τ = ρ p p /(18μ a ), an S * = [1. + ln (1 + Re)/1]/[1 + ln (1 + + Re)] a imensionless parameter. Here, μ w is the viscosity of a water rop, b the Boltzmann s constant, T the absolute temperature of air, an C c the Cunningham slip correction factor an can be approximate as follows [6]: C c λ λ p λ = 1+.493 + 4 exp 35 1+ 3.34 p p λ p (4) where λ is the molecular mean free path. The terms on the right-han sie of eq. (3) represent the effects of Brownian iffusion, interception an inertial impaction, respectively. In eq. (1) the particle size istribution can also be approximate by : N ln ( D/ D ) g 1 nd ( ) = exp π lnσg ln ( σg ) D N p ln ( p/ ) g 1 n ( p ) = exp π lnσ pg ln ( σ pg ) p (5)
1374 THERMAL SCIENCE, Year 1, Vol. 16, No. 5, pp. 137-1376 where N, D g, an σ g are the number concentration, geometric mean iameter an geometric stanar eviation of rainrop, respectively. The -th moment of rainrops an particles are : DnD ( )D Dg exp ln g ξ = = ξ σ p ( p ) p pg exp ln σ p m = n = m (6) [7]: The final expression of scavenging coefficient an governing moment equation are 53 3 A 1 3 p A 1 p Λ( p) = γξ 1 1( p + Ap ) + γξ7 4 p + + γξ 3 7 4 p + + 3 1 4 3 p 5 1 p 6 5 4p 7 5 8 11 4p + γξ + γξ + γξ + γξ + γξ (7) m Am 53 Am 3 = γξ 1 1( m 1+ Am ) + γξ7 4 m 3+ + γξ 3 7 4 m 1 + + t 3 + γ ξ m + γ ξ m + γ ξ m + γ ξ m + γ ξ m (8) 4 3 + 1 5 1 + 6 5 4 + 7 5 8 11 4 1 1 b 13π μa bρa 1 γ 6μa 4 13ρa 3πμa T T γ =, =, 1.16 13π T b 13πμa 3 4 5 4 3 13πμa μw γ =, γ =, γ = 13π, 1 ρa 13π 6 7 8 μa 13.9 13π 18μ a γ =13π, γ =, γ =, A = 3.34λ 4 4 13ρp Accoring to the efinition of m, m is the total particle number concentration, an (π/6) m 3 is the total volume of particles. We solve the first three moment equations an to get the geometric mean particle iameter pg, an the geometric stanar eviation σ pg. Results an iscussions The Runge-Kutta metho is employe to solve eq. (8) with =, 1, an. The evolutions of PSD are simulate numerically an the effects of RSD on PSD are stuie in nine cases. The values of the initialization parameter are liste in tab. 1. Figures 1 an show the evolution of particle number concentration an instantaneous PSD. For case 6 an 8 the geometric stanar eviation of rainrop iameter, σ g, is same but the geometric mean rainrop iameter, D g, is ifferent. 3 1
THERMAL SCIENCE, Year 1, Vol. 16, No. 5, pp. 137-1376 1375 1 N p /N p Case 6 Case 7 1 1 Case 8 Case 9 1 pg = 1 nm pg =.5 μm pg = 8. μm 1 3 1 1 1 1 1 3 1 4 1 5 1 6 1 t [s] 7 Figure 1. Effect of RSD on particle number concentration for cases 6-9 n( p ) p /N p 1.. Case 6 t =1 min t = min t =3 min t =3 min t = h t =3 h t =1 min t = min t =3 min 1.. Case 7 t = h t = h t =5 h t =1 h t = h t =4 h t = h 1 3 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 n( p ) p /N p 1.. Case 8 t = min t = min t =3 min t = h t = min t = min 1.. Case 9 t = h t =3 h t =1 h t = h t =4 h t =8 h t =16 h t = h t =3 h 1 3 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 p [μm] p [μm] Figure. Evolution of PSD uner the effect of RSD as time progresses for cases 6-9 As shown in fig. the particles with pg = 1 nm have been almost remove within 1 minutes in case 6, while it nees about 3 minutes in case 8. The particle number concentration ecreases faster in case 6 than that in case 8. For particles with pg =.5 μm an pg = 8 μm in cases 6 an 8, the same tenency can be observe, which emonstrates that the RSD in case 6 can scavenge particles with much higher efficiency than that in case 8.
1376 THERMAL SCIENCE, Year 1, Vol. 16, No. 5, pp. 137-1376 Table 1. The values of the initialization parameter for cases 1-9 Case N m 3 D g mm 1 σ g N p m 3 pg μm 1 σ pg 1 1 5.1 1. 1 6 1 1./1.5/1.8 1 5.1 1. 1 6 1 1./1.5/1.8 3 1 5.1 1. 1 6.1 1./1.5/1.8 4 1 5.1 1. 1 6 5. 1./1.5/1.8 5 31 1 7. 1.1 1 6 1/.5/8. 1.3 6 9.569 1 4. 1.5 1 6 1/.5/8. 1.3 7 6.61 1 5.5 1.1 1 6 1/.5/8. 1.3 8 6.14 1 3.5 1.5 1 6 1/.5/8. 1.3 9 6.786 1 4.5 1.5 1 6 1/.5/8. 1.3 Conclusions Particles ominate by Brownian iffusion are remove more easily. The particle number concentration ecreases much more rapily when pg is smaller. In the scavenging process, the particle geometric mean iameter increases when pg < nm, an ecreases when pg 1 μm, but changes a little when nm < pg < 1 μm. The PSD tens to be monoisperse, an ecays much faster for large σ pg. The RSD with small D g can remove particles with much higher efficiency than that with large D g, an the RSD with small σ g can collect particles more easily than that with large σ g. Acnowlegments This wor was supporte by National Natural Science Founation of China (11138). References [1] Slinn, W. G. N., Precipitation Scavenging, in: Atmospheric Sciences an Power Prouction (E. D. Raerson), Division of Biomeical Environmental Research, US Department of Energy, Washington D. C., 1983, pp. 386-39 [] Anronache, C., Estimate Variability of Below-Clou Aerosol Removal by Rainfall for Observe Aerosol Size Distributions, Atmospheric Chemistry an Physics, 3 (3), 6, pp. 131-143 [3] Anronache, C., Diffusion an Electric Charge Contributions to Below-Clou Wet Removal of Atmospheric Ultra-Fine Aerosol Particles, Journal of Aerosol Science, 35 (4), 1, pp. 1467-148 [4] Anronache, C., et al., Scavenging of Ultrafine Particles by Rainfall at a Boreal Site: Observations an Moel Estimations, Atmospheric Chemistry an Physics, 6 (6), 1, pp. 4739-4754 [5] Mircea, M., Stefan, S., Fuzzi, S., Precipitation Scavenging Coefficient: Influence of Measure Aerosol an Rainrop Size Distribution, Atmospheric Environment, 34 (), 9, pp. 5169-5174 [6] Lee, K. W., Liu, B. Y., Theoretical Stuy of Aerosol Filtration by Fibrous Filters, Aerosol Science an Technology, 1 (198), pp. 147-161 [7] Bae, S. Y., Jung, C. H., Kim, Y. P., Derivation an Verification of an Aerosol Dynamics Expression for the Below-Clou Scavenging Process Using the Moment Metho, Journal of Aerosol Science, 41 (1), 3, pp. 66-8 Paper submitte: August 1, 1 Paper revise: September 8, 1 Paper accepte: September 1, 1