A Comparative Study Of Several Explicit Finite Difference Schemes For The Solution Of Mass Transfer Problems In Mine Ventilation

In order to be able to design an adequate mine ventilation system it is necessary to understand how pollutant masses disperse into turbulent ventilation air streams. The mixing of gases and dusts with ventilating air, and dispersion of these pollutants in mine airways, are important flow or transfer problems. For the solution of such problems, the finite difference technique is often used. In this paper, numerical approximation schemes for the solution of the mass transfer problem have been developed using several explicit finite difference schemes. Results from these schemes have been compared with the analytical solution for spatial and time increments using the concept of Root Mean Square error. The comparison of various schemes reveals that the explicit finite difference schemes analyzed here give more accurate results at lower velocities, though the results are dependent on the spatial and time steps considered. At higher velocities the schemes tends to oscillate about the analytical solution. INTRODUCTION In mining, most of the mine air pollutants, such as respirable dust, methane, blasting fumes and diesel exhaust, are produced at the mine face and in haulageways. Injection of any of these pollutants in the mine ventilating air causes disturbances of the equilibrium state of the pollutant concentration of the ventilating air. The mixing of these gases and dusts with the ventilating air are typical examples of mass transfer processes. Pollutants are diluted by the stream of ventilating air flow along airways and discharged into the outside atmosphere of the mine through a return shaft. The dispersions of these pollutants in mine airways are important flow or transfer problems. Knowledge about pollutant sources, emission and transport of these pollutants are essential for ventilation planning and design to ensure good engineering control over air quality, quantity, and environmental conditions. Modeling of mass transfer phenomena in mine ventilation is largely based on extensive studies of turbulent dispersion in wind tunnels and the lower atmosphere of the earth. The general approach in the formulation of models lies in the recognition that these mass transfer phenomena can be conceptualized as a flow problem that can be solved by the general application of Taylor's theory (1954) of turbulent flow and diffusion with modification appropriate static and dynamic parameters. If the face zone is initially contaminated by a pollutant over a long section (e.g. instantaneous exposure of large gas accumulation, rapid methane release etc.), the process of decontaminating the mine workings can be regarded as displacement and diffusion of the tailing edge of the cloud of contaminates in the form of a semi-infinite gas body, for which a one-dimensional mass transfer equation is represented by: [at + u • ax = Ex ex2 + f (x,t)(1.1)] In Eq. (1.1), the term [~~] is the rate of growth of the concentration in the differential element, while a [(5xx )] is the net gain of material due to convective transfer. The two terms balance the total loss of material due to turbulent dispersion that is represented by [Ex . 21] and f(x,t), the source term for the pollutant in the roadway. Several cases of mass transfer that occur in mining can be identified and modelled using the conservative volume element method. The governing equation, along with the associated boundary and initial conditions, source terms and their solutions are detailed elsewhere (Bandopadhyay and Ramani, 1988, Devala, 1989). Only one mass transfer situation is presented (Figure 1) here for the present analysis. Mixed Evolution of Pollutant (multiple pulses) Typical examples of these situations are emission of pollutants from multiple diesel vehicles in the face area, and gas emission from standing pillars as well as from broken coal seam during coal cutting with a continuous miner. The mathematical statement of the problem along with the initial and boundary conditions are given below, Governing Equation: [do+u- =E d2c(1.2) ataxx dx2] Initial conditions: