Moment method for aerosol deposition

J Aerosol Sci. Vol. 31, Suppl. 1, pp. $845-$846, 2000

Pergamon www.elsevier.com/locate/jaerosci

P o s t e r Session II. Aerosol d y n a m i c s a n d t r a n s p o r t MOMENT METHOD FOR AEROSOL DEPOSITION

S.H. PARK and K.W. LEE Department of Environmental Science and Engineering, Kwangju Institute of Science and Technology, 10ryong-dong, Puk-gu, Kwangju 500-712, Korea. Keywords: deposition, moment method. INTRODUCTION Particle size distribution of an aerosol suspended in closed vessels changes due to several mechanisms such as coagulation, condensation, and deposition. The deposition on the walls of a stirred vessel is of importance m many chemical processes including aerosol reactor and nuclear reactor safety analyses. Several theoretical studies have investigated the deposition rate of monodispersed suspended particles m a vessel (Comer and Pendlebury, 195 l; Fuchs, 1964; Crump and Seinfeld, 1981). For the size distribution change of polydispersed aerosol particles, however, these have yet to be solved. In this study, we develop the moment method for the deposition problem by modifying the deposition coefficient suggested by Cramp and Semfeld (1981) which has been validated by several experiments. DEPOSITION COEFFICIENT The change of the size distribution of particles by simultaneous Brownian and turbulent diffusion coupled with gravitational sedimentation is represented by the following equation (Fuchs, 1964):

~n(v,t) _

fl(v).n(v,t ) (1) /gt where n(v, t) is the particle size distribution function at time t and/~v) is the deposition coefficient of particles with volume v. Crump and Seinfeld (1981) developed a general theory for the deposition coefficient that accounts for the Brownian and turbulent diffusion and gravitational sedimentation. According to the theory of Crump and Seinfeld (1981), the deposition coefficient for a vessel of arbitrary shape is represented by the following equation: u t n(x)-kdA(x) fl(v)=l ! [ lru, n(x)'k 1. (2) exp~ . . . . . ~-l

[msm(Tr/m)qk.D'-' J

where 27 denotes the surface of the reactor, V is volume of the reactor, ut is the terminal particle settling velocity, n(x) is the unit outward vector normal to the surface, k is the unit vector m the vertical dirge.ion, dA(x) is the differential area elment on 27, D is the Brownian diffusion co~ciem, k, is the coefficient of

the eddy diffusivity which can be evaluated from the turbulent energy dissipation rate (Comer and Pendlebury, 1951), and m is a constant known to lie between 2 and 3. In Equation (2), D and ut depend on the particle size as follows:

$845

$846

Abstracts of'the 2000 European Aerosol Confcrcncc

D = kfl'C

(3)

6zktr

ut -

2r 2pgC (4)

9/J

where ks is the Boltzmann constant, T is the absolute temperature,/J is the gas viscosity, r is the particle radius, p is the particle density, g is the gravity constant, and C is the slip correction factor which is represented as (Fuehs, 1964):

C= l+A--~l.246+O.418¢xp( -0"867r )1 rL where 2. is the moan free path length of the gas molecules.

\

A

(5)

)J

Using appropriate approximations, Equation (2) is rewrit/en for a reactor with a vertical sidewall as follows: fl(V) = A.v -2(rn-1)/(3m)+B.v 2/3 (6) 2(ra l)

m

I/

r~\

( 3 ~2/3(2pg ~ ) ' B = 1 4 z ) 19/d-/) ' S i s t h e t ° t a l s u r f a c e

_ ( 4 ~ r ~ g - ~ (1.664AkzT1 m [ msin(x/m)Sgk e ]

where A - ~ - ~ - )

l

~

)

~

~

area of the reactor, and H is the height of the reactor. Equation (6) has an appropriate functional form for the moment method. MOMENT EQUATIONS The log-normal size distribution function for particles whose volume is v is written as

n(v,t)dv=

N(t) F-in2{v/vg(t)}] , . 3x[~ lna(t ) exP[ 1-ff~ - ~ ( t ) -J dOnv )

(7)

where N(t) is the total number concentration of particles, vg(t) is the geometric number mean particle volume, and o(0 is the geom~m'ie standard deviation based on the particle radius. Using the moment method, Equation (1) is converted into the following throe equations:

dM°

(~ AA(20m2-13m+2)/9m2.~f(-16mZ+20m-a)/gmZJL~(5m2-7m+2)/9m24-J~ ~ ~'"'"0 ""1 -'2 -u""0

dt dM1

dt -

riM2

(,d PtA(5m2-7m+2)/gm2/~Ar(Sm2+gm

r'""o

""1

4)/9m2/IAt( m2

""2

m+2)/9m2-t-R ]IA -u""o

2/91~lfg/9]L4 1/9

""1 ""2

I,

(8)

1/9A//5/9/L45/9)

""1 ""2 /,

(9)

(A J~.4( m2-m+2)/9m21~(gm2-4m4)/9m21L4(2m2+Sm+2)/9m2 n . . 5 / 9 1 . 16/9..20/9~ +~'Mo m~ M2 l" (10) ~-'---o ""l ""2 The governing equation has been converted into a set of three first-order ordinary differential equations. Equations (8) through (10) can be solved for an appropriate value of m using any standard numerical package for solving first-order ordinary differential equations. ARer Mo, M1, and M2 are solved from Equations (8) through (10), vg and trcan be computed using Equations (1 l) and (12). Subsequ~tly, the size distribution for any time can be constructed using Equation (7).

dt

-

m~

v g - M3o/2M~/2

(11)

In2cr = T11n(MoM2 I

9

(12)

REFERENCES Comer, J. and Pendlebury, E. D. (1951) The coagulation and deposition of a stirredaerosol. Proc. Phys. Soc. B64,

645-654. Crump, J. G. and Seinfeld, J. H. (1981) Turbulent deposition and gravitational sedimentation of an aerosol in a vessel of arbitrary shape. J. Aerosol Sci., 12, 405-415. Fuchs, N.A. (1964) The Mechanics of Aerosols, Pergamon Press, New York.