International Journal of Research in Engineering, Technology and Science, Volume VII, Special Issue, Feb 2017 www.ijrets.com, [email protected], ISSN 2454-1915 MATHEMATICAL MODELING ON CONCENTRATION LEVELS FOR MAXIMUM DISPOSAL OF COAL WASTE IN DAMODAR RIVER Dr. Shafique Ahmad1, Dr. Narayan Mahto2 and Dr. P. Mahto3 1Department of Mathematics, B.D.A. College, Pichhri, Bokaro 2Department of Mathematics, R.S.P. College, Jharia 3Department of Mathematics, R.S. More College , Dhanbad ABSTRACT: The quantity and quality of water now a day a very serious problem in many region especially in industrial and mining area of India. Damodar river basin is repository of 45% of India coal reserve. Exploitation of coal & related industries in the area has exerted a great impact on the water pollution. In this paper an attempt has been made to develop a mathematical model to obtain concentration levels of Damodar river under different dumping policies so as to select the policy which disposes maximum waste without exceeding acceptable limits. An analytical solution is obtained to study the concentration levels of pollution for different time histories in Damodar river. The model can be useful in guiding engineering and management decision concerned with the efficient utilization of Damodar river water & protect their quality. Keywords: Damodar River; water pollution; coal mining activities; time history; concentration level;environment; coal waste; Dhanbad. [1] INTRODUCTION Damodar basin extends from 23˚28’32” N. Latitudes to 24˚31’30”N. Latitude and 84˚41’ 00”E. Longitude to 86 51’45” E. Longitude. The basin starts from high hills of Palamu in Jharkhand to Hooghly in West Bengal and stretches over nearly 18,100 Sq.Km. area. Today, many stretches of the Damodar and its tributaries resemble large drains carrying black, highly turbid water. The Total Suspended Solid (TSS) count at most places along the upper and middle sterthes of the river is 40-50 times higher than the permissible limit. The crux of the problem is extensive mining in region mine rejects, minerals and toxic effluent are regularly thrown into the river and its tributaries. Damodar basin has been blessed by the nature in the form of huge reserve of different types of minerals. In fact except the southern part of the basin in Burdwan district of West Bengal, rest part of this has got mineral deposit in abundance. The most important mineral is of course coal. The basin has got some important coalfields like Raniganj coalfield, Jharia coalfield, Giridih coalfield, East and West Bokaro coalfield including Ramgarh and East and West Karnpura coalfield. These coalfields stretch in Burdwan, Corresponding author : [email protected] IJRETS : International Journal of Research in Engineering, Technology and Science, Volume IV, Issue I & II, Dec.14 Bankura and Birbhoom of West Bengal and Dhanbad, Giridih ,Hazaribagh, Palamu and Bokaro districts of Jharkhand. Jharia coalfield is about 40 Km. wide with a total Bokaro coalfield lie nearly 32 kms. West of Jharia coalfield. Some best Quality of coke is produced in this coalfield. Giridih coalfield covers and be nearly 28.5sq. kms., which has got good quality of steam grade coking coal. South and North Karnpura lying west of Bokaro coalfield have an area o 14.25 sq Kms. Ramgarh coalfield extends over an area of 98 sq. Km. The Raniganj coalfield is about 1100 sq.Km.in area. Apart from coal, the basin is also richly endowed with good quality of fire – clay which is used for making refractory’s on the north western flank of the basin lies the famous mica coalfield of Koderma [8].There are several companies which perform coal mining activities and is located on the banks of Damodar river .In the past it has disposed of its waste by indiscriminately dumping it into the river,causing high levels of pollution. As a consequence the local authorities passed new legislation with very high fines if pollution in the river exceeds certain specified concentration limits. The problem in this paper is to frame a policy for discharging the waste so as to ensure that the concentration levels never exceed the specified limits. Mirbagheri et al developed mathematical model of water quality in river systems of Jajrood river in Tehran- Iran [1]. The water quality of Danube River and its analysis through mathematical model was studied by Valerian et al [2]. Agusto et al studied the numerical treatment of the Mathematical models for water pollution [3]. Shukla et al developed Mathematical models and studied its analysis of the depletion of dissolved oxygen in eutrophied water bodies affected by organic pollutants [6]. The remedy through a new technique for water pollution in river was well studied by Pimpunchat et al [7]. Benedini[9] studied Water Quality model for River. R.K.Tiwari et al[10,11]studied environmental pollution from coal mining activities in Damodar river basin in Dhanbad area. Burley [12]developed some Mathematical models in biology and medicine. In order to solve the problem of how to keep concentration level of pollutants below prescribed level of pollutants levels, companies need to obtain changes to concentration levels in the river over time under different dumping policies for chemical waste [4, 5]. A policy for dumping is defined by time history of dumping .We assumes that the dumping is carried out at discrete time instants. The effect of dumping is increase the concentration level suddenly at the moment it occurs. With the passage of time the chemical waste diffuses into the water and also gets transported downstream. Depending on the waste, it can either go through a process of transformation so that it ceases to be an undesirable waste or relation its chemical structure or continue to be an undesirable chemical waste. [2] ASSUMPTION We treat river as a single object with the waste dumped being an input variable (from the system environment) and the water discharge to the sea being an output variable (to the system environment). In the simplest characterization the river is viewed as a one-dimension al channel of constant cross- section A, with x = 0 corresponding to the location where the waste is dumped into the river. Let L1 &L2 represents the length of the river upstream and downstream from the point where the waste is dumped. As a result ,we have , - L1< x < L2 with distances measured downstream being positive &those measured upstream being negative. Dr. Shafique Ahmad, Dr. Narayan Mahto and Dr. P. Mahto MATHEMATICAL MODELING ON CONCENTRATION LEVELS FOR MAXIMUM DISPOSAL OF COAL WASTE IN DAMODAR RIVER Let T (t, x) be the concentration level of chemical waste at location x from the point of dumping at time t. The variable changes continuously with t & x. Let v be the velocity of flow which we treat as a constant. Since the river receive clean water from an external source the concentration is zero at x = - L1hence T(- L1, t) = 0.We assume that the sea to which the river discharge is very then the concentration level there is also negligible. HenceT (L2 ,t) = 0.The diffusion & transportation of the waste is predictable & governed by the law of fluid mechanics. We describe this model as an open deterministic dynamic system. The main objective of this paper is to obtain time histories of concentration levels under different dumping policies so as to select the policy which disposes maximum waste without exceeding acceptable limits. In the above characterization, we assume the river as a one-dimensional channel with cross-section and ignore the lateral diffusion of the chemical waste across the width. Since the waste is dumped at a point, rather than along the whole cross-section, the above characterization is over-simplification. A more redefined characterization includes the spatial variations. For example, we can treat the depth as being nearly constant and we represent this by a parameter h and the width of the river is changing with x. Let w(x) represent the width of the river at a distance x from the point of dumping along the river, then the cross-sectional area A(x) changes with x. The concentration level is given by a variable T (t, x, y, z), the concentration at a point whose coordinates are given by (x, y, z) as depicted in figure 1 at time t. The limits for y and z are given by 0 < z< h and 0 < y < w(x). Since the flow volume must be constant for all x, the velocity of flow U(x), is a function of x. x = 0 W(x) x y z T (t, x, y, z) Figure 1: Coordinate frame for river pollution IJRETS : International Journal of Research in Engineering, Technology and Science, Volume IV, Issue I & II, Dec.14 [3] MATHEMATICAL MODEL FORMULATION: [3.1] MODEL 1: Let T(x, t) represent the concentration level of pollution at a distance x from the point of discharge at time t. From the system characterization, we have, - L1 ≤ x ≤ L2 and 0 ≤ t < ∞ (1) The Concentration level changes due to diffusion & transportation. Let D be the coefficient of diffusion & Let U be the velocity of the river which is assumed to be constant. The diffusion is governed by Fick’s Law of diffusion current vector J is given by J = - D (2) From the equation of continuity, for the diffusing substance we have, (3) This is obtained by considering a small volume centered at x at time t & evaluating the net inflow & out flow of pollution in a small time . The net change in the small volume can be related to the net inflow & net outflow and in the limit , we have the above equation.