Page 125 - Advanced Mine Ventilation

106 Advanced Mine Ventilation

where dQ is the amount passing through an area dy$dz in the direction of x in time, dt,
creating an increase of volume concentration by dc .
dx
6. Displacement by Brownian motion: Fine dust particles have a random Brownian motion too.
The finer the size of a particle, the greater the Brownian displacement. Thus, below a certain
size (typically around 0.5 mm in diameter) the dust particle may not settle and that fraction is
called “nonsettling fraction” of respirable dust [1].
7. Relaxation time, (T): Each particle has a characteristic “relaxation time”, T, that has a unit of
time (s). The settling velocity, v S , ¼ T$g, where “g” is the acceleration due to gravity. It will
be mathematically derived later in the text.
8. Mean free path: It is the average distance travelled by a moving particle between successive
impacts (collisions) that modify its direction or energy or other particle properties.
d$v
9. Particle Reynolds number: It is defined as R, where R ¼ , where d is the particle diameter,
s
v is the settling velocity, and s is the kinematic viscosity of air.
These characteristics of particles ranging from 10 3 to 10 mm are shown in
Table 8.1 [2].
It is to be noted that mean free path, [ does not change significantly with the size of
the particle in this range.

8.1 Settling Velocity of Small Particles Due to Gravity
(Stoke’s Formula)

When a small dust particle is floated in a medium such as air, it is pulled down by grav-
ity, Fg, and the motion is resisted by the viscous force of the medium, Fm. The settling
velocity, v S , is the final constant rate at which the particle is pulled down by gravity.

Table 8.1 Characteristics of Fine Particles

Mean
Settling Brownian
Velocity, X S
Radius, Diffusivity, Relaxation Mean Free Movement,
2 L1
r(mm) D (cm s ) Time, T (s) Path, [ (cm) (cm/s) X B (cm/s)
10 1.38 10 8 1.23 10 3 6.08 10 6 1.21 1.23 10 4
5 2.38 10 8 3.08 10 4 4.32 10 6 3.02 10 1 1.74 10 4
1 1.27 10 7 1.31 10 5 2.06 10 6 1.28 10 2 4.02 10 4
0.5 2.74 10 7 3.54 10 6 1.53 10 6 3.47 10 3 5.90 10 4
0.2 8.32 10 7 6.87 10 7 1.21 10 6 6.73 10 4 1.03 10 3
0.1 2.21 10 6 2.28 10 7 1.13 10 6 2.24 10 4 1.68 10 3
0.01 1.35 10 4 1.40 10 8 2.20 10 6 1.37 10 5 1.31 10 2
0.001 1.28 10 2 1.33 10 9 6.59 10 6 1.31 10 6 1.28 10 1

Adapted from Fuchs NA. The mechanics of aerosols. Pergamon Press; 1964. p. 408.