Page 287 - Advanced Mine Ventilation

264 Advanced Mine Ventilation

15.6.2 Mathematical Derivation of Limiting Methane
Concentration at the Tailgate
Assuming that q is the net methane emission ([emission from solid coal þ emissions
from broken coal] minus methane lost with leak-off air) in a differential element, dx,
the mathematical equation for methane concentration is (refer to Chapter 3) given by
3
Equation (15.7). u is the ventilation rate, ft /min.
2
v c vc
þ u q ¼ 0 (15.7)
vx 2 vx
The boundary conditions are as follows:

vc
¼ 0at x ¼ 0 ðmethane concentration is constantÞ
vx

vc
ε x þ ucj x¼L ¼ qL ðfrom mass conservationÞ
vx
x¼L
The solution of Eq. (15.7) is as follows:

q h ε x i
cðxÞ¼ x 1 e ux=ε x (15.8)
u u
At x ¼ L,

q h ε x i
C L ¼ L 1 e uL=ε x (15.9)
u u
If L is large, we can discard e uL=ε x term as zero, and

qL ε x
C L ¼ L (15.10)
u u
Thus, the limiting value of C L is qL .
u
An example:
3
Assume L ¼ 1000 ft, q ¼ 0.3 ft /ft-min
C L ¼ 0.8% (or 0.008).
0:3 1;000
Hence, 0.008 ¼
u
or u ¼ 37,500 CFM.
A check of gas layering index should be made to make sure it exceeds 5.00 (refer to
Chapter 4).