Encyclopedia of Environmental Science and Engineering, Volume I and II

22 AEROSOLS

Brownian Coagulation

Coagulation of aerosols causes a continuous change in

number concentration and size distribution of an aerosol with

the total particle volume remaining constant. Coagulation

can be classified according to the type of force that causes

collision. Brownian coagulation (thermal coagulation) is a

fundamental mechanism that is present whenever particles

are present in a background gas.

In the special case of the initial stage of coagulation of a

monodisperse aerosol having uniform diameter D p , the par-

ticle number concentration N decreases according to

ddNt KN

KKDDpp

⁄ 



050 2

0

.

(),

(21)

where K ( D p , D p ) is the coagulation coefficient between par-

ticles of diameters D p and D p.

When the coagulation coefficient is not a function of

time, the decrease in particle number concentration from N 0

to N can be obtained from the integration of Eq. (21) over a

time period from 0 to t,

N  N 0 /(1 
 0.5 K 0 N 0 t ). (22)

The particle number concentration reduces to one-half its ini-

tial value at the time 2( K 0 N 0 )^ ^1. This time can be considered

as a characteristic time for coagulation.

In the case of coagulation of a polydisperse aerosol, the

basic equation that describes the time-dependent change in

the particle size distribution n ( v, t ), is

 



nvt

t

Kvvvnvtnvvtdv

nvt K vv

, v

,

,,

( )

( ) ( ) ( )

( ) ( )

∫

1

2 0

′ ′ ′, ′, ′

00 ′

∞

∫ nv tdv( ′, ′)

(23)

The first term on the right-hand side represents the rate of

formation of particles of volume v due to coagulation, and

the second term that rate of loss of particles of volume v by

coagulation with all other particles.

The Brownian coagulation coefficient is a function of

the Knudsen number Kn  2 l / D p , where l is the mean free

path of the background gas. Figure 6 shows the values of

the Brownian coagulation coefficient of mono-disperse par-

ticles, 0.5 K ( D p , D p ), as a function of particle diameter in

10 –2 10 –1 100 101

10 –2

10 –1

100

101

102

103

Dp (mm)

Equilibrium charge

distribution by

bipolar ions

Diffusion charging by

unipolar ions

NSt=10^13 s/m^3

NS : ion number concentration

1 : charging time

Field charging by

unipolar ions

E = 3 

105 V/m

NSt = 10^13 s/m

n^

n^

n(n

)

n(n

)

n



=

=^ –

n^

/=^ –

FIGURE 4 The average number of charges on particles by both

field and diffusion charging.

FIGURE 5 Equilibrium charge distribution through bipolar ion

charging. The height of each section corresponds to the number

concentration of particles containing the indicated charge..

0.02 0.04 0.1 0.2 0.5 12

Dp (mm)

Particle number concentration

np 2

np 3

 4

 4

np

np

np

+2

+1

+3

–3

–2

–1

–1

+1

+1

–1

+2

–2

0

0

+3

–3

+1

–1

+2

–2

 (^01000)
 4
 3
 2
 1
Particle size distribution
Charge distribution
 5
FIGURE 6 Brownian coagulation coefficient for coagulation of
equal-sized particles in air at standard conditions as a function of
particle density.
0.001 0.01 0.1 1.0
Dp (mm)
10 –10
10 –9
10.0
5.0
2.5
1.0
p 0.5
=^ 0.25
20 10 5 4 3 2 1 0.5 .4 .3 .2 0.1
Knudsen number Kn
0.5K
(DB
, Dp
) (cmp
3 /
s)
ρ
C001_002_r03.indd 22C001_002_r03.indd 22 11/18/2005 10:09:11 AM11/18/2005 10:09:11 AM