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108 4 Properties of Aerosol Particles

Equation (4.37) becomes,

N 0
NtðÞ ¼ ð4:39Þ
1 þ t=s
Since N ¼ N 0 =2 when t ¼ s, s is also referred to as the half-value time, which is
the time it takes for the particle number concentration to drop to half of its initial
value.

Example 4.6: Half value time of monodisperse aerosol
Calculate the half-value time of spherical monodisperse aerosol with an initial
3
3
concentration of 10 particles/cm and an initial particle diameter of d p ¼ 1 nm.
Assume standard condition and only Brownian coagulation is considered.
Solution
The Knudsen number of 1 nm particle is

Kn 1nmÞ ¼ 2 0:066=0:001 ¼ 132 [ 100
ð
It is within the free molecule region. Therefore,

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s ffiffiffiffiffiffiffiffiffi
kT ð 1:38 10 23 Þ 298 7
K ¼ 9:8 ¼ 9:8 9 ¼ 6:28 10
ð
p
q d p 1;000 1 10 Þ
Then the half-value time is
2 2
s ¼ ¼ ¼ 0:003 s
KN 0 6:28 10 7 10 3 1 3 10 6 cm 3
cm m 3

As seen from the result, particles at 1 nm do not have a great life time. While this
is practically inaccurate, it does qualitatively show that aerosol particle diameter is a
dynamic parameter.

4.4.2 Polydisperse Coagulation

Coagulation coefficients for different mechanisms have been summarized by Geng
et al. [10]. Brownian motion is the dominating mechanisms for the coagulation of
very fine particle under normal atmospheric condition; for this reason, our analysis
is focused on Brownian coagulation.
One of the widely used Brownian coagulation coefficients is the Fuchs (1964)
equation for binary collision.