Page 53 - Advanced Mine Ventilation

36 Advanced Mine Ventilation

solution are generally available in other fields such as heat transfer. However, it is
necessary to understand the “physics” of the mining problems and to give
them correct mathematical representations. Similarly, great care needs to be exer-
cised in the selection of boundary and initial conditions. Above all, experimental
verification of such models and determination of system parameters should be
an essential part of such endeavors. In this chapter, an attempt has been made
to model a few examples of mass transfer in mines by means of partial differ-
ential equations, and solutions have been given for typical initial and boundary
conditions.

3.2 Generalized Mass Transfer Model

Modeling of mass transfer processes in mine airways is based largely on extensive
studies of turbulent dispersions in wind tunnels and the lower atmosphere. Two basic
approaches, namely, the gradient transport theory [2] and the statistical theory [3],
have been used, but the former is considered to be more suitable for underground
mines, as discussed by the author elsewhere [4].
In general, mine airways can be represented by a rectangular parallelepiped, but cy-
lindrical and spherical geometry can be used with advantage for special situations. A
generalized mass transfer model for a rectangular parallelepiped is

vc v vc v vc v vc
¼ ε x þ ε y þ ε z
vt vx vx vy vy vz vz
(3.1)
vc vc vc
uðxÞ vðyÞ wðzÞ lðx; y; z; cÞþ f ðx; y; z; tÞ
vx vy vz
Where c is the time-averaged concentration of pollutants in mine air; x, y, z are the
three coordinate directions; and t is the time variable; ε x , ε y , ε z are turbulent dispersion
coefficients in the x, y, and z directions, respectively; u(x), v(y), and w(z) are compo-
nents of air velocity in the three coordinate directions, respectively; l is a generalized
decay coefficient for the pollutant, which could be a constant or a function of the
concentration and space variables; and f (x, y, z, t) is a source term for the pollutant
mass in the airway.
Eq. (3.1) is a nonhomogeneous partial differential equation. Depending on the
nature of l, ε x , ε y , and ε z , it could be linear or nonlinear. In the latter case, an analytical
solution is generally not possible and a solution has to be obtained with digital
computers using well-known numerical techniques [5]. For linear cases, solutions
are obtained in a manner analogous to that used to solve heat transfer problems [6].
Fortunately, the most important cases of mass transfer in mines are simpler than
Eq. (3.1), and their solutions can often be obtained in closed forms with resultant
ease of numerical computation.
Sources of pollutants in mines can be classified in various ways depending on
the nature of the source (e.g., instantaneous or continuous, moving or stationary),