Page 54 - Advanced Mine Ventilation

Turbulent Dispersion of Pollutants in Mine Airways 37

the nature of the pollutant (e.g., gaseous or particulate), and the geometry of the source
(e.g., point, linear, or areal). Different combinations of these occur in practice, and
some of these will be studied here.

3.3 Instantaneous Stationary Point Source

This is a very common situation in the experimental determination of the longitu-
dinal coefficient of turbulent dispersion. A known quantity, Q, of a tracer such as
sulfur hexafluoride, SF 6 , is released instantaneously at the origin and its concentra-
tion measured downstream at several points. A mathematical relation between
concentration and space and time variables is needed to determine the coefficient
of turbulent dispersion. Eq. (3.1) is modified in this case using the following
assumptions:

1. There is no sink or additional source for the tracer in the roadway.
2. Coefficients of turbulent dispersion are constants and homogeneous.
3. Velocity components in the y and z coordinates are negligible and that in the x coordinate is a
constant.
Using polar coordination, the mathematical model for this case is given by:

2
vc 1 v vc v c vc
¼ ε r r þ u (3.2)
vt r vr vr vx 2 vx
Typical boundary and initial conditions are as follows:
1. ! y cdv ¼ Q for t > 0, that is, the tracer mass is conserved

¼ 0; at r ¼ a
vc
2.
vr
where a is the radius of the roadway. This assumes that there is no material transfer at
walls of the airway.
3. c ¼ 0at t ¼ 0 for all x except at the origin
According to Seager and Fitzpatrick [7], a complete solution of Eq. (3.2) with the
above boundary and initial condition is

" N 2 #
Q 1 X Joðb RÞ εrb t
m
m
cðx; r; tÞ¼ 0:5 1 þ exp $exp
2
pa ð4pε r tÞ ½Joðb ފ 2 a 2
m¼1 m
(3.3)
2
!
ðx utÞ

4ε r t
where R ¼ r/a, Jo is the Bessel function of order zero, and b m is the m-th zero of J 1 (R).
In practice, however, the tracer gets mixed with mine air very intimately
across any given cross section because of obstructions in and the roughness of