International Journal of Differential Equations

Volume 2018, Article ID 1343541, 12 pages

https://doi.org/10.1155/2018/1343541

## Numerical Simulations of Water Quality Measurement Model in an Opened-Closed Reservoir with Contaminant Removal Mechanism

^{1}Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand^{2}Centre of Excellence in Mathematics Commission on Higher Education (CHE), Si Ayutthaya Road, Bangkok 10400, Thailand

Correspondence should be addressed to Kaboon Thongtha; moc.liamg@t.noobak

Received 2 January 2018; Revised 7 March 2018; Accepted 13 March 2018; Published 2 May 2018

Academic Editor: Julian Poulter

Copyright © 2018 Kaboon Thongtha and Jaipong Kasemsuwan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The mathematical simulation of water contaminant measurement is often used to assess the water quality. The monitoring point placement for water quality measurement in an opened-closed reservoir can give accurate or inaccurate assessment. In this research, the mathematical model of the approximated water quality in an opened-closed reservoir with removal mechanism system is proposed. The water quality model consists of the hydrodynamic model and the dispersion model. The hydrodynamic model is used to describe the water current in the opened-closed reservoir. The transient advection-diffusion equation with removal mechanism provides the water pollutant concentration. The water velocity from the hydrodynamic model is plugged into the dispersion model. The finite difference techniques are used to approximate the solution of the water quality model. The proposed numerical simulations give a suitable area of zonal removal mechanism placement. The proposed simulations also give the overall and specified approximated water quality for each point and time when the exit gate is opened on the different periods of time. In addition, the proposed techniques can give a suitable period of time to open the exit gate to achieve a good agreement water quality by using contaminant removal mechanism.

#### 1. Introduction

Field measurement and mathematical simulation are methods to detect the amount of the level of pollutants in water area. In water quality modeling for reservoir, the general governing equations used are the hydrodynamic model and the dispersion model. The two-dimensional shallow water equation and the advection-diffusion-reaction equation govern the first and the second models, respectively.

The several numerical techniques for solving such models were available. In [1–3], they used the hydrodynamic model and the dispersion model with the finite element method to approximate the velocity of the water current in bay, estuaries, and open reservoir, respectively. In [4], the finite element method was used for solving the water pollution levels to the optimal control of the water treatment plants to achieve minimum cost. In [5], the method of the characteristic technique combined with Galerkin finite element method is used to solve the shallow water mass transport problems. In [6, 7], the numerical techniques are used to solve the nonuniform flow of stream water quality model with the advection-dispersion-reaction equations. In [8], the Crank-Nicolson method is used to solve the hydrodynamic model and the backward time central space (BTCS) for the dispersion model. In [9], the approximated solutions of the hydrodynamic model and advection-diffusion-reaction equation in a uniform reservoir are proposed. In [10], a nondimensional form of the hydrodynamic model with variable coefficients using Lax-Wendroff method is presented. In [11], the Lax-Wendroff finite difference method is also proposed to approximate the water elevation and water flow velocity with a rectangular domain. In [12], the mathematical models and numerical methods for approximating water current and pollutant concentration level in Rama-nine reservoirs are presented. In [13], the Lax-Wendroff method for solving the dimensional form of shallow water equation in spherical model with Matlab program is proposed. In [14], an analytical solution to a hydrodynamic model in an open uniform reservoir with the specified tidal wave functions is proposed.

In this research, the mathematical models for water quality measurement which consist of the hydrodynamic model and the dispersion model, used to simulate water quality in a water flow systems, were considered. The first is a hydrodynamic model that provides the water current and the elevation of water in an opened-closed reservoir. The second is a dispersion model that gives the concentration of pollutant in an opened-closed reservoir with the contaminant removal mechanism. For numerical techniques, we used the Lax-Wendroff method to the system of the hydrodynamic model and the forward in time central in space (FTCS) to the dispersion model. The results from the shallow water equation of the hydrodynamic model are the water flow velocity which are input data for advection-diffusion-reaction equation which provides the level of pollutant concentration field. Averaging the equation over the depth with anisotropic bottom topography and discarding the term due to the Coriolis force, surface wind effect, and external forces, it follows that the two-dimensional shallow water and advection-diffusion-reaction equations are applicable.

#### 2. Water Quality Measurement in Opened-Closed Reservoir with Removal Mechanism Model

The mathematical models for water quality measurement in opened-closed reservoir with removal mechanism are described. They are used to simulate time-varying pollutant levels caused by waste water discharges from external source into an opened-closed reservoir with removal mechanism and drain water at the exit gate. The first model is a hydrodynamic model that determined the velocity and elevation of the water at any locations in the reservoir with anisotropic bottom topography, while the second model is a pollutant dispersion with removal mechanism model that determined the pollutant level at any points in the reservoir.

##### 2.1. Hydrodynamic Model

The two-dimensional unsteady water current into and out of the reservoir can be determined by using the system of shallow water equations as the conservation of mass and conservation of momentum. It is taken into account that the equations of the system of shallow water can be derived from depth-averaging Navier-Stokes equations in the vertical direction, neglecting the diffusion of momentum due to turbulence and discarding the terms expressing the effects of friction, surface wind, Coriolis factor, and shearing stresses. The continuity and momentum equation governs the hydrodynamic behavior of the reservoir [15]. The well known two-dimensional shallow water equations are [1, 2, 13, 16] where is the longitudinal distance along the reservoir (m), is the transverse distance along the reservoir (m), is the time (s), is the depth measured from the mean water level to the reservoir bed (m), is the elevation of water surface from the mean water level in reservoir (m), is the elevation of water surface measured from the mean water level to the bed of the reservoir, is velocity in -direction (m/s), is velocity in -direction (m/s), and is gravitational constant m/s^{2}, for all .

Such independent variables , and make up the special dimensions and time. The dependent variables are the depth with respect to the surface and the two-dimensional velocities and . The partial derivatives taken with respect to the same term are grouped into vectors and rewritten as a single hyperbolic partial differential [13, 15],where

##### 2.2. Dispersion Model

###### 2.2.1. Water Pollutant Dispersion Model

Applying the distributed pollutant process, including the transportation and diffusion, we have the mass transfer equation. There is a representation simplified by averaging the equation over the depth, generating the advection-diffusion equation as follows [4]:where is the depth averaged water pollutant concentration at the point and at the time (kg/m^{3}) and is the pollutant dispersion coefficient (m/s^{2}).

###### 2.2.2. Water Pollutant Dispersion with Removal Mechanism Model

The mechanisms of pollutant removal are introduced by decaying chemical reaction and absorptive reduction. A representation is modified by generating the advection-diffusion-reaction with sink term [2],where is the depth averaged water pollutant concentration at the point and at the time (kg/m^{3}), is the pollutant dispersion coefficient (m/s^{2}), is the water pollutant decaying rate (s^{−1}), and is the decreasing rate of water pollutant concentration due to a water pollutant sink (kg/m^{3}s).

##### 2.3. Initial and Boundary Conditions of Hydrodynamic Model in Opened-Closed Reservoir

The initial conditions of (2) are assumed to be motionless, An opened-closed reservoir supplied by the outer water wave is going to flow into the reservoir such that the water is drained as shown in Figure 1. The elevation of water on the opened gate is assumed to be a wave maker function, say . The elevation of water on the exit gates is assumed to be , where is the rate of change between inner and outer water level through the eastern exit gate.