Page 189 - Advanced Mine Ventilation

Diesel Exhaust Control 169

Finally, to relate the solution to the rate of emission of exhaust, one can consider
conservation of mass at the source, i.e., at the origin. Mathematically, it is expressed
as

vc

2
4p s E r ¼ qc i at s/0 (11.4)
vs
x
where, q, volume rate of engine exhaust discharge; c i , concentration of pollutants in the
engine exhaust.
The solution of Eq. (11.1) with the special boundary conditions has been derived
elsewhere [9] and is given below:

qc i uðs xÞ
C ¼ exp (11.5)
4pE r s 2E r
By inspection, it can be easily seen that the concentration, c, is the maximum at
x ¼ 0, i.e., in the plane of the source itself. Hence, concentration at the diesel engine
is given by

qc i ur
Cj ¼ exp (11.6)
x¼0
4pE r r 2E r
Eq. (11.6) can be further simplified if it is assumed that the concentration is
measured in a manner to obtain average concentration within a radius, a, from the en-
gine. Designating this concentration as C L , one obtains

R 1
a ur=2E r
qc i 0 r e $rdr
C L ¼ R a (11.7)
4 p E r rdr
0
On integration, Eq. (11.7) reduces to

qc i ua
C L ¼ 1 e (11.8)
pua 2 2E r
For any finite value of ua , the expression e ua is greater than zero. Hence, C L is
2E r 2E r
always smaller than qc i 2 , which is the concentration predicted by the static dilution for-
pua
mula. Derivation of E r will be discussed later in this chapter. E r in coal mines is typi-
2
cally higher than smooth pipes, at approximately 0.8 m /s. Eq. (11.8) yields dilution air
quantities that is 50%e100% of static dilution air quantities [8].