Page 191 - Advanced Mine Ventilation

Diesel Exhaust Control 171

Initial Condition 1: Just before the engine starts working, concentration of diesel
exhaust in the mine air is zero, i.e.,

c ¼ 0at t ¼ 0 for all x > 0 (11.12)

The solution of Eq. (11.9) with the above boundary and initial condition is derived
by Thakur [8] and is given below:
⎡ xV t ⎛ − V x Vr ⎞ 2t V x ⎤
⎢ erfc r − ⎜ 1+ r + ⎟ e r erfc ⎥
qc ⎢ 2E t ⎝ E x E x ⎠ E x ⎥
x
c= i ⎢ 2 ⎥ (11.13)
2FV 1/2 ⎛
r ⎢ x+V t + 2V ⎛ t ⎞ exp − x V t ⎞ − r ⎥
r
⎢ 2E t r ⎜ E π ⎜ ⎟ ⎜ 2 E t ⎟ ⎟ ⎥
⎣ x ⎝ x ⎠ ⎝ x ⎠ ⎦
Concentration, c, assumes its maximum value at x ¼ 0 and is readily obtained from
Eq. (11.13).
⎡ Vt ⎛ − V t ⎞ 2 ⎤
⎢ erfc r − ⎜ 1+ r ⎟ erfc ⎥
qc i ⎢ 2E t ⎝ x E x ⎠ ⎥
c = ⎢ ⎥ (11.14)
x=0
2FV r ⎢ Vt ⎛ t ⎞ 1/2 ⎛ V t ⎞ r 2 ⎥
r
⎢ ⎣ 2E t + 2V r ⎜ ⎝ E π ⎟ exp − ⎜ ⎝ 4 E x ⎠ ⎟ ⎥ ⎦
x ⎠
x
Now, if we let t /N, the limiting value of the maximum concentration at x ¼ 0is

qc i qc i
C L ¼ ð1 þ erfðNÞÞ ¼ (11.15)
2FV r FV r
This is the same as the effective ventilation formula of Holtz and Dalzell [10].
Mathematically, the maximum concentration growth around the engine for any finite
travel time will, of course, be less than that predicted by the effective ventilation for-
mula. However, for large travel times the concentration around the engine is likely to
reach the limiting value.
To evaluate Eq. (11.13), q and c i are available from engine test data. E x can be
calculated from friction factors. For noncircular roadways, the radius, a, is substituted
by the hydraulic radius defined as R h ¼ (2[Area]/[Perimeter]). To determine the con-
centration growth around the engine with time, several evaluations of Eq. (11.13) are
needed for different x and t. A computer program was written to solve Eq. (11.13) and
is available in Thakur’s thesis [8].
A special case of Eq. (11.13) is when the diesel engine travels in the same direction
as the air current with the same velocity. This is the worst possible situation. In this
q vc
condition, the only convective term results from exhaust emissions, i.e., 2F vx instead
vc
of V r vx in Eq. (11.9). All other terms have the same meaning as before.