Page 126 - Advanced Mine Ventilation

Characteristics of Respirable Coal Dust Particles 107

The mathematical equations to predict v S varies depending on the size of the particle, r,
in relation to its mean force path, [. The following three cases will be considered:

Case 1. r[[
Case 2. r [
Case 3. r z [
In all cases, the derivation of mathematical equation is based on the following
assumptions:
1. Incompressibility of the medium (air).
2. Infinite extent of the medium (no boundary influence).
3. Very small rate of movement.
4. Constant rate of movement.
5. Rigidity of particle (always true for solid particles).
6. Absence of slipping at its surface.
For a detailed discussion of the above factors, a reference can be made to Fuchs [2].

8.1.1 Derivation of Settling Velocity When r [ [
For a force balance, the rate of change of momentum, M v equals force due to gravity,
0
Fg, less resistance due to medium, Fm. Let the particle radius be r, particle density, r,
0
and air density equals r air .M is the apparent weight of particle and equals
4 3
air
3 pr ðr r Þ
Fm ¼ 6p rVh (8.2)

Hence,

0
M dv
¼ Fg Fm
dt
(8.3)
0
M dv
0
or ¼ M g 6p rV h
dt
0
Substituting for M and transposing, dv ¼ g v .
dt 2
2
r ðr r air Þ h
9
2
2 r ðr r air Þ
Let us call T ¼ ; it is known as the relaxation time of the particle.
9 h
Hence,
dv v
¼ g (8.4)
dt T
The constant settling velocity of the particle is called V ¼ V S at t ¼ T. Thus in Eq.
(8.4), dv ¼ 0att ¼ T.
dt