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408 13 Nanoaerosol

and change in size and number concentration. Thus, the assumption of Max-
well–Boltzmann distribution of nanoparticle thermal speed might be valid for
diluted cases only. Nonetheless, we have to carry on with the analysis before a
better hypothesis is established.

13.4.3 Critical Thermal Speed

The critical particle speed that enables thermal rebound of aerosol particles is a
function of adhesion energy (E ad ), the coefficient of restitution (e) and particle mass
(m).

r ffiffiffiffiffiffiffiffiffi
2E ad
v cr ¼ ð13:21Þ
me 2
The particle critical velocity, above which particle rebounds from the surface, is
3
calculated using Eq. (13.21) and, with m = ρ p πd p /6, it becomes
v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
12E ad
u
ð13:22Þ

v cr ¼ t
pq d 3 e 2
p p
where E ad is the adhesion energy, which will be elaborated shortly, e is the coef-
ficient of restitution defined as the particle velocity at rebound over the normal
particle velocity at the instant of contact. While intuitively one may assume that
e ≈ 1 for nanoaerosol because of the great rigidity, it is not true. The absolute value
is unknown [16]. The coefficient of restitution is dependent on the material of the
nanoparticles and the filter surface and the impact velocity of the nanoparticles [3].
For the impact velocities close to the critical velocity, the coefficient of restitution is
small, and it leads to small rebound velocities. Molecular dynamics simulation by
Ayesh et al. [3] showed that, for solid nanoparticles,
e 0:6

Unfortunately, the database for coefficient of restitution for nanoparticles is still
not well developed yet.

13.4.4 Adhesion Efficiency

Since the particles are considered as being collected when their thermal velocities are
below v cr , the fractional adhension efficiency can be mathematically described by