Page 196 - Advanced Mine Ventilation

176 Advanced Mine Ventilation

Eq. (11.23) is a linear differential equation with variable coefficients. A solution is
obtained by reducing it to the canonical form, i.e., by substituting:

1 1 U m
cðxÞ¼ zðxÞ$exp $ x (11.24)
2 AvDt
where, z(x) satisfies the differential equation.

00
z f ðxÞz ¼ 0 (11.25)
the solution of Eq. (11.25) is

Z ¼ B 1 expðK 1 xÞþ B 2 expð K 2 xÞ (11.26)

The particular solution B 2 exp ( K 2 x) has no physical meaning in this problem
because the concentration, c, is an increasing function of x. To solve the equation
Z ¼ B 1 exp (K 1 x), the following boundary condition is used:

0:14 c i q
cj ¼ c ¼ (11.27)
x¼0
B½1 U L ŠU L
2
where, B is a constant and q* ¼ (q/pa v). The derivation of this condition is shown by
Thakur [8].
Using this boundary condition and superposing the concentrations due to the move-
ment of diesel engine parallel and opposite to air current, the solution for Eq. (11.23) is

( ( )1)
2 2 2 3
c ip q p 1 1 U m 1 1 U m U L
exp þ x
6 2 AvDt 4 AvDt 8aAvDt 7
½1 U L ŠU L
6 7
7
0:14 6
c ¼
6 7
( (
B 6 )1) 7
2 2
4 5
c iq q q 1 1 þ U m 1 1 þ U m U L
þ exp þ x
½1 þ U L Š U L 2 AvDt 4 AvDt 8aAvDt
(11.28)
where the subscripts p and q refer to parallel and opposite movements of diesel
engines.
Eq. (11.28) has been derived with a coefficient of absorption mainly to make the
approach general. In practice, the absorption of diesel exhausts is not significant and
can be discarded. Eq. (11.28) now becomes

0:14 c ip q p 1 U m c iq q q 1 þ U m
c ¼ exp x þ exp x (11.29)
B ð1 U L ÞU L AvDt ð1 þ U L ÞU L AvDt