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104 4 Properties of Aerosol Particles
4.2.6 Particle Deposition on Surface by Diffusion
As introduced above, most of the fine particles adhere when they impact on a
surface. In this case, the particle concentration in the space where the surface is can
be assumed to be zero and there is a concentration gradient established in the region
near the surface. This results in a continuous diffusion of particles to the surface and
a gradual decrease in particle concentration in the gas.
Consider a plane vertical surface that is immersed in a large space filled with gas
and particles. It can be assumed that the gas velocity near the surface is zero. Then
the rate at which particles are removed from the space, by deposition onto the
surface can be determined following the analysis below. Let x be the horizontal
distance from the surface. Then the particle concentration, n(x, t), in the space at x at
any time t, must satisfy Fick’s second law of diffusion.
2
dn d n nðx; 0Þ¼ n 0 for x [ 0
¼ D p 2 ð4:30Þ
dt dx nð0; tÞ¼ 0 for t [ 0
It is assumed that the initial particle concentration in the entire space is n 0 . The
boundary condition of nð0; tÞ¼ 0 is based on the assumption that all particles will
adhere on the surface once they come into contact. The general solution of this
equation is
Z x 2
n 0 z
exp dz ð4:31Þ
nðx; tÞ¼ p ffiffiffiffiffiffiffiffiffiffi
pD p t 4D p t
0
where z is the dummy variable for integration.
Then the concentration gradient at the surface, dn/dx at x =0, is
3
2
Z x
2
d z
dn n 0 n 0
¼ 4 p ffiffiffiffiffiffiffiffiffiffi expð Þdz 5 ¼ p ffiffiffiffiffiffiffiffiffiffi ð4:32Þ
dx dx pD p t 4D p t pD p t
x¼0
0 x¼0
The rate of deposition of particles onto a unit area of surface at any time, t, is
then described as
r ffiffiffiffiffiffi
dn D p
J ¼ D p j x¼0 ¼ n 0 ð4:33Þ
dx p t
Integrating over time in this equation from 0 to t, the cumulative number of
particles, N(t), deposited per unit area of surface is
