Page 263 - Advanced Mine Ventilation

240 Advanced Mine Ventilation

Thakur [1] has shown mathematically that

1
M t 6 D 2 1
¼ p ffiffiffi 2 t 2 (14.16)
M N p a
where M t is the volume of gas diffused in time t; M N is the Langmuir volume, V m ,
discussed earlier, and t is the time.
1
A plot of M against t / 2 will yield a straight line. The gradient of the straight line is
M N
1
2

6 D
p 2 .
p a
equal to ffiffiffi
The time taken for a piece of coal to desorb (1 e 1/e) or 63.21% of gas is called its
“sorption time” or s. This expresses the rate of desorption in mining parlance better
2
than the absolute value of D or (D/a ).
Modifying Eq. (14.16), we can write
1
M = ( 1− 1 ) = 6 ⎛ ⎞ D 2 τ 1 2 (14.17)
M ∞ e π ⎝ ⎜ a ⎠ 2 ⎟
Rearranging and solving for s, we get

3:49 10 2
s ¼ 2 (14.18)
ðD=a Þ
Another way to determine s is to solve the equation.

n
M t
¼ 1 exp (14.19)
M N s
To illustrate and compare Eqs. (14.17) and (14.19), a typical gas desorption curve
shown in Fig. 14.9 was plotted both ways and results were compared.
The plot of curve in Fig. 14.9 using Eq. (14.17) results in a straight line with an
equation:

M 1
¼ 0.000436 s 2 (14.20)
M N
2
This gives (D/a ) equal to 1.6589 10 8 1 and s ¼ 24.35 days.
s
Eq. (14.19) was rewritten as
0 1
1
B C
ln ln B C ¼ nlnt nln s (14.21)
@ M t A
1
M N