Page 188 - Advanced Mine Ventilation

168 Advanced Mine Ventilation

Figure 11.2 Diffusion from a point source in a uniform velocity stream.

The following differential equation is obtained by taking a mass balance over the
cylindrical ring in Fig. 11.2.

2
vc 1 v vc v c
u ¼ E r r þ (11.1)
vx r vr vr vx 2
where, u ¼ average air velocity, E r ¼ the coefficient of transverse turbulent dispersion,
2 2 1/2
r ¼ (Y þ Z ) , c ¼ concentration of pollutant in the general body of air.
In Eq. (11.1), u vc represents the dispersion of material by the velocity of air, u, i.e.,
vx
2
v c
it is the convective term. E r 2 gives turbulent dispersion of material in the x direction,
vx

whereas E r v r vc is an analogous term for dispersion in the radial direction. Eq.
r vr vr
(11.1) is obtained by simply equating the input and output.
To solve Eq. (11.1), two boundary conditions are needed, which are obtained by
considerations of the physical situations.
Boundary Condition 1: It is reasonable to assume that at an infinitely distant point,
the concentration of diesel exhaust would be zero, i.e.,

c ¼ 0ats ¼ N (11.2)

2
2 1/2
2
where, s ¼ (x þ y þ z ) .
Boundary Condition 2: It can also be assumed that at any cross section in the
roadway, the concentration would be the highest at the center, i.e.,
vc
¼ 0at r ¼ 0 for any x (11.3)
vr