Page 132 - Advanced Mine Ventilation

Characteristics of Respirable Coal Dust Particles 113

8.4.1 Normal Distribution

This is found to be most applicable to only particles created by chemical processes,
such as condensation and precipitation. It usually does not represent well the size dis-
tribution of solid particles.
Mathematically,
⎛ ( 2 ⎞
1 ⎜ − ) xx ⎟
y= ∙ exp − (8.18)
σ n 2π ⎜ ⎜ 2 σ 2 ⎟ ⎟
⎝ n ⎠
where y is the probability density; x is the particle size; x is the arithmetic mean;
particle size s n is the number standard deviation.
The number of particles with a diameter between x 1 and x 2 is given by:
⎛ ( 2 ⎞
Σ n x 2 ⎜ − ) xx ⎟
n= i 1 x ∫ exp − dx (8.19)
i
σ 2π ⎜ ⎜ 2 σ 2 ⎟ ⎟
n ⎝ n ⎠

The normal distribution has very limited application for particles created by broken
solid materials.
The number distribution of Eq. (8.19) can be easily converted to a mass (weight)
distribution as shown in Eq. (8.20).
⎛ ( 2 ⎞
ρ S Σ n x 3 X ⎜ − ) xx ⎟
w= V i i X ∫ exp − dx
i ⎜ 2 ⎟ (8.20)
σ w2π ⎜ 2σ w ⎟
⎝ ⎠

where r,S V and s w are density, volume shape factor, and weight standard deviation,
respectively, and w i is the weight fraction between size x 1 and x 2 .
8.4.2 The Log Normal Distribution

This distribution appears to fit many fine solid particles created by comminution (mill-
ing, grinding, crushing). Pulverized silica, clay, granite, limestone, and quartz yield
size distributions that satisfactorily fit the log normal distribution [10].
Mathematically, it is obtained by replacing x in Eq. (8.18) by log x as shown in Eq.
(8.21).
⎡ log x )log x − 2 ⎤
y= exp ⎢ ( 1 g ⎥ − (8.21)
log σ g 2π ⎢ ⎣ 2log σ g ⎥ ⎦
2

where x g is the geometric mean size; s g is the geometric standard deviation.