8.4 Equations of Particle Motion
To design particle collection devices and to predict their performance, engineers must be able to
predict the trajectories of particles passing through the devices. The objective of this section is to establish
equations that predict the trajectories of particles. As an aerosol particle moves through air, it disturbs the
air, changing the velocity and pressure fields of the air flow. In addition, particles collide with each other
and are influenced by each others’ wakes. Exact analysis of particles moving in air is therefore nearly
impossible, even for simple air flows, since the equations for the air flow and for each particle’s trajectory
are coupled. Fortunately, some simplifications are possible, which make the problem more tractable. Two
major assumptions are made here:
1.
Particles move independently of each other.
2.
Particles do not influence the flow field of the carrier gas.
The first assumption is valid if two conditions are met: (a) particle collisions are infrequent and
inconsequential, and (b) particles are not significantly affected when they pass through each others’ wakes.
A useful rule of thumb that quantifies these conditions for a monodisperse aerosol is that the average
distance between particles is at least 10 times the particle diameter. Assuming that 8 particles are located at
the corners of a cube of dimension L, one finds that L/D
p
> 10 when
()
p
3
number
p
4
8
1000
3c
c
D
πρ
=>
(8-51)
where c is the mass concentration of particles,
ρ
p
is the particle density, and c
number
= n
t
/V is the
particle
number concentration
, defined as the total number of particles per unit volume of gas. Table 8.3
illustrates these upper limits and indicates that the particle number concentration has to be exceedingly
large for the particles to influence each other. For water droplets (
ρ
p
= 1,000 kg/m
3
), application of Eq. (8-
51) leads to a particle mass concentration c = 4.2 kg/m
3
, corresponding to the upper limit of 1,000 in Eq.
(8-51). For most problems in indoor air pollution, particle concentrations of water and particulate
pollutants are
hundreds
of times smaller than 4.2 kg/m
3
. Consequently the first assumption above is clearly
valid, and one can calculate the trajectory of one individual particle at a time.
The second assumption is more difficult to quantify. Large objects moving through air generate
air flows due to displacement effects and the effects of aerodynamic wakes. For example, a pedestrian
walking along the side of a road feels the “wind” created by a passing vehicle. The same effect can be
experienced when a person walks past another person in otherwise quiescent air. Automobiles and people,
however, are very large objects; particles usually associated with indoor air pollution are many orders of
magnitude
smaller.
As
shown
below,
most
particles
of
interest
to
indoor
Table 8.3
Particle number concentrations beyond which particles influence each other.