Page 190 - Advanced Mine Ventilation

170 Advanced Mine Ventilation

11.4.2 Time-Dependent Model

When a diesel engine is moving in a roadway with a velocity, v, the growth of concen-
tration becomes time-dependent. If the velocity of air is u, the relative velocity would
be (v u) ¼ V r . The mass flow diagram is shown in Fig. 11.3.
Assuming v > u and the plane x ¼ 0 moving with the diesel engine, the equation of
convection diffusion becomes

2
vc vc v c
þ V r ¼ E x (11.9)
vt vx vx 2
where, E x is the coefficient of longitudinal turbulent dispersion.
In Eq. (11.9), the term vc is the rate of growth of concentration in the differential
vt
vc
element, whereas V r vx is the net gain of material due to convective transfer. These
two terms balance the total loss of material owing to turbulent dispersion, which is rep-
2
v c
resented by E x 2 . To solve Eq. (11.9), three conditions:two boundaries, and one initial
vx
are needed.
Boundary Condition 1: This is obtained by assuming that mass is conserved at the

origin, i.e., at x ¼ 0. The total input of exhaust from the engine per unit area is qc i
F
where F is the cross-sectional area. Net loss of material at x ¼ 0 given by the algebraic
sum of convection and diffusive terms. Mathematically,

vc qc i
V r c j x¼0 E x ¼ for t > 0 (11.10)
vx F
x¼0
where, q, volume rate of exhaust emission; c i , concentration of species, i, in the
exhaust.
Boundary Condition 2: It is reasonable to assume that at a point very far from the
engine, the concentration of exhausts would be zero, i.e.,

c ¼ 0as x/N for t > 0 (11.11)

Figure 11.3 Mass flow diagram for time-dependent model.