Page 55 - Advanced Mine Ventilation

38 Advanced Mine Ventilation

mine airways. In these circumstances, a one-dimensional model of dispersion is
considered quite accurate. The concentration is then a function of x and t alone
as given below.

" 2 #
Q ðx utÞ
cðx; tÞ¼ 0:5 exp (3.4)
2Aðpε x tÞ 4ε x t

where A is the cross-sectional area of the roadway. Calculations of the dispersion
coefficients from experimental data can be done very simply as discussed in great
detail by Airey [8] and Klebanov [9].

3.4 Continuous Stationary Point Source

This case is represented by a continuous source in the roadway often obtained in
practice. It is also used for the experimental determination of the radial coefficient
of turbulent dispersion. The mathematical model is obtained from Eq. (3.2) by simply
dropping the vc term, that is,
vt
2
vc 1 v vc v c
u ¼ ε r r þ (3.5)
vx r vr vr vx 2
The boundary conditions for a source at the origin are

1. C / 0as s /N
2 0.5
2
where s ¼ (r þ x )
2. c ¼Nas s / 0
A third condition results from the conservation of mass, that is,

vc
2
4ps ε r ¼ Q at s/0
vs
x
A solution of Eq. (3.5) with the above boundary conditions has been derived by
Roberts [10] as given below:

Q uðs xÞ
c ¼ exp (3.6)
4pε r s 2ε r
Taking logarithms on both sides of Eq. (3.6) and rearranging, the following
relationship is obtained:

Q u
ln ¼ ln ε r þ ðs xÞ (3.7)
4psc 2ε r