COMPUTER SIMULATION OF DEAD-END MINE WORKING VENTILATION

Dep. «Fluid Dynamics, Energy and Mass Transfer», Oles Honchar Dnipro National University, Kazakova St., 18, Dnipro, Ukraine, 49000, tel. +38 (056) 374 98 22, e-mail water.supply.treatment@gmail.com, ORCID 0000-0003-2399-3124 Dep. «Heat and Gas Supply, Drainage and Ventilation», Kryvyi Rih National University, Vitaliia Matusevycha St., 11, Kryvyi Rih, Ukraine, 50027, tel. +38 (056) 409 06 06, e-mail pasha_129@ukr.net, ORCID 0000-0002-0793-9593 Dep. «Hydraulics and Water Supply», Dnipro National University of Railway Transport named after Academician V. Lazaryan, Lazaryana St., 2, Dnipro, Ukraine, 49010, tel. +38 (056) 273 15 09, e-mail water.supply.treatment@gmail.com, ORCID 0000-0001-9257-763X Dep. «Track and Track Facilities», Dnipro National University named after Academician V. Lazaryan, Lazaryana St., 2, Dnipro, Ukraine, 49010, tel. +38 (063) 802 21 80, e-mail irina_bondarenko@ua.fm, ORCID 0000-0003-4717-3032 Dep. «Hydraulics and Water Supply», Dnipro National University of Railway Transport named after Academician V. Lazaryan, Lazaryana St., 2, Dnipro, Ukraine, 49010, tel. +38 (056) 273 15 09, e-mail water.supply.treatment@gmail.com, ORCID 0000-0003-3057-9204 Dep. «Physics», Ukrainian State University of Chemical Technology, Haharina Av., 8, Dnipro, Ukraine, 49000, tel. +38 (056) 753 56 38, e-mail physics@udhtu.edu.ua, ORCID 0000-0002-9893-3479


Introduction
The development of the mining industry is based on increasing the intensity of work in the mine workings, which leads to an increase in dust and gas emission [2,6,9,11]. This problem is connected with risk assessment [10,12]. This places high demands on the efficiency of the ventilation system. The increase in the volume of cleaning space and the intensity of mining operations causes an increase in the volume of air, and this requires rational use of the supplied air. In this connection, it is necessary to know how the impurity concentration changes in the process of airing the mine working. Airing of underground workings is one of the most urgent problems of aerology in the mining industry. As part of this problem, the task of developing forecasting methods for calculating the ventilation time of underground workings should be highlighted [1,2].
In recent years numerical models, computational fluid dynamics models are widely used by engineers to solve different problems [3][4][5][6][7]. Numerical simulation of the aerodynamics of air flow in under-26 ground workings helps to optimize the process of efficient air circulation and the removal of pollution.
Unfortunately, the development of numerical models for solving problems of ventilation of underground workings is carried out in Ukraine is not as active as abroad. The current approaches in Ukraine to calculate the parameters of the ventilation workings are based either on theoretical assumptions that require experiments to determine empirical coefficients, or use the values of average velocity over the mine working cross section and constant coefficients of turbulent diffusion throughout the volume. It is not possible to determine the concentration fields of pollutants at any given time, and thus control the process of ventilation.

Purpose
The aim of the work is to develop computing numerical models to assess dumps influence on air pollution and to solve the problem of choosing of rational dump location.

Methodology
An underground dead end mine working of a given size is considered, in which the air environment is polluted with fine dust of a known concentration. The space ventilation is carried out by supplying clean air through the discharge duct. The task is to develop a mathematical model for the operational calculation of the process of ventilation of the dead end mine working using suction of polluted air.
Modeling equations. To compute the velocity field in the underground dead end mine working when the suction of polluted air takes place, the potential flow model is used. In this case, the governing equation has the following form [1,6,8]: where Рis velocity potential, x, yare Cortesian coordinates, m. When applying this equation, it is assumed that the Y axis is directed vertically upwards.
To solve the equation (1) the following boundary conditions is used [4]: 1) on the walls of the dead end mine working, as well as on other solid surfaces located inside it, a boundary condition is set: P/n = 0, where nis unit vector of external normal to solid wall; 2) at the inlet boundary the boundary condition is P/n = Vn, where Vnis known airflow velocity, m·s -1 ; 3) at the outlet boundary the boundary condition is: P = P0 +const, P0is an arbitrary number (Dirichlet condition).
To simulate the dispersion of dust in the underground mine, the mass transfer equation is used [3,8]: where Сis pollutant concentration in the air, g·m -3 ; u, vare airflow velocity components in the mine working, m·s -1 ; =(x, y)are turbulent diffusion coefficients, m 2 ·s -1 ; (xi, yi)are emission source coordinates, m; Qiis pollutant emission rate at the point (xi, yi), g·s -1 ; (x  xi)(y  yi)are Dirac's delta function, which is used to simulate the entry of a pollutant into the mine working.
Boundary conditions for (2) are discussed in [3,4]. The initial condition is C=C0, for t = 0. Here, C0 is known concentration of pollutant in dead end mine working.
Numerical model. Numerical integration of modeling equations is carried out using a rectangular difference grid.
Before a numerical solution of equation (2), it was splitted, at the differential level, as follows [3]: The construction of a numerical model is carried out by applying the following procedure. Convective derivatives are represented as [4]: The approximation of these derivatives is carried out according to the formulas [4]: The time derivative is approximated as follows: To approximate the second derivatives, the following formulas are used [4]: The two-dimensional dust transport equation is written in a difference form [4]: In this equation, a symbol ij means either «1» or «0», depending on whether or not there is a source of dust emission in the differential cell «ij». Value Qij is calculated as: where Qkthe intensity of the emission of the k-th point source of dust emission, which is located in the difference cell «ij». The splitting of difference equation (4) is carried out as following [4]: 1) on the first step of splitting k = 1/4 the difference equation has the appearance: 4) on the fourth step of splitting k = n + 1; c = n + 3/4 the difference equation has the appearance: From equations (5) -(8) unknown value of the dust concentration on the highest time layer is calculated by explicit formula of running calculation [4]. The initial condition for these equations is written as: To construct the boundary condition on the solid walls of the form C/n = 0 fictitious cells are used.
In conclusion, we note that the applied difference schemes have an important advantage  the calculation of the unknown value of the dust concentration in the working area is carried out using explicit formulas, which ensures a simple software implementation of these formulas.
For the numerical integration of this equation, the method of total approximation is applied. To consider the two-dimensional equation for the velocity potential, first of all, this equation in evolutionary form is written [11]: where tfictitious time (dimensionless).
It is known, that when t   the solution of this equation will approach to the Laplace equation solution for the velocity potential. In the numerical solution of equation (9), it is necessary to specify the potential field at t = 0. For example, before starting the calculation, you can take P = 0 in the entire computational domain for t = 0.
The solution of equation (9) is carried out on a rectangular grid, the function P is determined in the center of the difference cells. The solution of this equation is split into two steps. Difference equations at each step of the splitting are written as:    At each splitting step, the unknown value of the velocity potential is determined by the explicit running calculation formula. The calculation is terminated when the following condition is fulfilled: where εis small number (e.g., ε =0.001); niteration number. We also used Libman method for numerical integration of equation (1). In this case the calculation formula is as follows The components of the air velocity vector are calculated on the sides of the difference cells (control volumes), which makes it possible to construct a conservative difference scheme for the dust transport equation (2).
To code the difference formulae of the developed numerical model Fortran language has been used.

Findings
The developed numerical model was used to simulate the air cleaning in the dead-end mine working. To clean the air suction of the polluted air takes place. The suction opening is situated as it is shown in Figure 1.
To make all parameters dimensionless, we have chosen the following scales: 1) vw is the air velocity at the left boundary, vw=2m/s; 2) Lm, m is the length of the dead-end mine working; 3) C0, μg·m-3 is the initial dust concentration in the dead-end mine working for t=0.
The dimensionless parameters are calculated as following: 1) t=tp·vw/Lm, where tp is time, s; 2) C=Cp/C0, where Cp is dust concentration, μg·m-3; 3) L=Lp/Lm, where Lp is length, m; 4) v=vp/vw, vp is local air velocity, m/s. At the initial moment of time throughout the mine working, a uniform impurity concentration is set С=1 (in dimensionless form). Air supply for ventilation is carried out through the duct (Fig. 1). The length of the dead-end mine working is L=1 (dimensionless), the width is W=0.3 (dimensionless). The length of the computational domain Lx=2.5, the width of the computational domain is Ly=1.5.
The computational experiment was carried out in two stages. At the first stage, the calculation of dead end mine working ventilation was carried out without the suction system.
In Figures 2 -4 the change of the polluted zone in the dead end mine working for different time is shown. Time is dimentionless. In Figures 2-4, 6-7 the arrow indicates the wind flow direction. In Figure 5 the maximum dust concentration change in the mine working during time is shown. From Figure 5 it is seen that the process of concentration decrease takes part very slowly. It depends upon the aerodynamics process of ventilation: in this case the local speed in the dead-end mine working is very small. This results in small intensity of dust evacuation from the dead-end mine working. So, to increase the intensity of dust evacuation from dead-end mine working it is necessary to imply "external" impact. For example, we can use suction of dust from dead-end mine working.
It is well known, that suction application can be efficient if the suction opening is situated properly in the dead-end mine. The process of dust suction strictly depends on the length from the walls of the dead-end mine, coal heaps to the suction opening. Obtaining the proper position of the suction opening is possible by numerical experiment. In Figure 8, for this scenario, the maximum dust concentration change in mine working during time is shown. In conclusion, it should be noted that the calculation took 10 seconds of computer time.

Originality and practical value
Fluid dynamics numerical models were developed to predict the efficiency of dead-end mine working ventilation. For ventilation the suction system was used. The models are based on equation of pollutant dispersion and equation for potential flow. Difference schemes were used for numerical integration. The developed models can be used on the stage the ventilation system development.

Conclusions
An effective numerical model was developed for calculating the process of ventilation of deadend mine workings. The calculation of the aerodynamics of air flow is based on the model of potential flow. The process of dispersion of impurities is modeled on the basis of the mass transfer equation. Practical implementation of the model requires small amount of computer time. The model makes it possible to improve the quality of engineering calculations. Further development of this direction is associated with the creation of a threedimensional numerical model of the process of ventilation of underground workings.