Page 57 - Advanced Mine Ventilation

40 Advanced Mine Ventilation

2
where a ¼ u þ 4ε x l 4ε x and erfc is the complimentary error function. By proper
substitution in Eq. (3.9), the value of u and consequently the quantity of ventilating air
required to meet a given health standard can readily be obtained.

3.6 Dispersion in a Leaky Roadway

In practice, there are air leaks in most of the airways in mines, and the air velocity is,
therefore, nonuniform. The leakage factor is a function of the pressure gradient,
mechanical design of stoppings, doors, air crossings, etc., and the fracture system
in the strata. In many instances, the nonuniform velocity, u(x), can be approximated
by u o expð jxÞ, where u o is the velocity at the initial cross section, and j is the
leakage factor.
The one-dimensional model for a leaky roadway for the dispersion of a tracer is

2
vc v c vc
¼ ε x u o exp ð jxÞ (3.10)
vt vx 2 vx
The boundary condition at x ¼ 0 depends on the mode of release of the tracer. In
reality, it is a function of time, that is, cj ¼ f (t) / o for large values of t.
x¼o
Assuming that cj x¼N ¼ 0 and c ¼ 0at t ¼ 0, a solution of Eq. (3.10) is obtained
after Grekov [12] as follows:
x Z t f ðt sÞ x 2
cðx; tÞ¼ p ffiffiffiffiffiffiffi 1;5 exp ds
2 pε x o s 4 ε x s
t
Z
j x u o f ðt sÞ
1 e p ffiffiffiffiffiffiffiffiffi 1;5
4 j e x p ε x o s
2 p ffiffiffiffiffiffiffiffiffi 2 p ffiffiffiffiffiffiffi
x j p ε x jx j ε x x j ε x s
exp exp þ s erfc p ffiffiffiffiffiffiffi ds
4 ε x s 2 2 4 2 ε x s 2
(3.11)

In practice, values of j are of the order of 10 3 to 10 4 m 1 and hence u(x) can be
approximated by u o (1 jx). Under some mining conditions, a steady-state situation
may develop and Eq. (3.10) then simplifies to:
2
v c vc
εx u o ð1 jxÞ ¼ 0 (3.12)
vx 2 vx
An approximate solution of Eq. (3.12) for very small values of j is obtained as:

u o ð1 jxÞ u o ð1 jxÞ
cðxÞ¼ A Exp þ B Exp (3.13)
ε x ε x
where A and B are constants and have to be determined from boundary conditions.