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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.

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 [

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.

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.

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 [^{2}

Such independent variables

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 [^{3}) and ^{2}).

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 [^{3}), ^{2}), ^{−1}), and ^{3}s).

The initial conditions of (

An opened-closed reservoir with contaminant removal mechanism zones and monitoring points M1, M2, M3, M4, and M5.

The boundary conditions of the model in an opened-closed reservoir are assumed as follows:

Initial and boundary conditions of hydrodynamic model in opened-closed reservoir.

The initial pollutant concentration in reservoir is ^{3}). The water pollutant is discharged from the open gate into the opened-closed reservoir which are assumed to be ^{3}) is the averaged discharged pollutant concentration along the entrance gate. The opened-closed reservoir drain water at the exit gate by assuming rate of water drain as

The initial and boundary conditions of water pollutant dispersion model in opened-closed reservoir.

The removal terms can model a variety of different phenomena, that is, the removal of pollutant concentration in (^{3}s), where

The term

The hydrodynamic model provides the velocity field and elevation of the water. Then the calculated results will be input into the dispersion model that provides the pollutant concentration results.

We apply the numerical method for solving the single hyperbolic partial differential equations known as the Lax-Wendroff method. The Lax-Wendroff method involves starting to calculate a first half step and then using the result from the half step to calculate the full step [

The first half step defines values of

The values of vector

We can then approximate

Taking the forward time central space technique [^{2}/s), ^{−1}), and ^{3}s).

If

For the left boundary condition, we have

Assume that the western gate is opened and the water elevation along the gate is described as a function ^{3}). The water pollutant is released from the open western gate into the reservoir, which is the averaged pollutant concentration along the entrance gate defined by ^{3}). The reservoir has drained water off through the exit eastern gate with the rate of change of pollutant concentration across the gate defined by ^{3}s). There is a pollutant concentration which is loss from the water surface, the decaying rate ^{−1}).

Removal mechanism locations

Removal mechanism | Northern | Center | Southern |
---|---|---|---|

Zone 1 | (200, 1600) | (200, 1080) | (200, 400) |

Zone 2 | (800, 1600) | (800, 1080) | (800, 400) |

Zone 3 | (1200, 1600) | (1200, 1080) | (1200, 400) |

The water flow velocity in

Approximated water velocity in an opened-closed reservoir.

The approximated pollutant concentration levels in removal mechanism system and nonremoval mechanism system are compared in Figure

Comparison of approximated pollutant concentrations in removal mechanism system and nonremoval mechanism system.

Approximated pollutant concentrations when the first zonal removal mechanism system is activated.

Approximated pollutant concentrations when the second zonal removal mechanism system is activated.

Approximated pollutant concentrations when the third zonal removal mechanism system is activated.

The approximated pollutant concentrations along the northern part,

Comparison of approximated pollutant concentrations along the northern part,

Comparison of approximated pollutant concentrations along the middle part,

Comparison of approximated pollutant concentrations along the southern part,

Assuming that the third zonal removal mechanism systems are activated. The approximated pollutant concentrations over 5 monitoring nodes, when the third zonal removal mechanism systems are activated and the eastern exit gate is opened, are shown in Figure

Monitoring point locations

Monitoring points | M1 | M2 | M3 | M4 | M5 |
---|---|---|---|---|---|

Locations | (1600, 1600) | (1600, 1200) | (1600, 800) | (1600, 400) | (2000, 1000) |

Approximated water flow velocity in

| 0 | 400 | 800 | 1200 | 1600 | 2000 |
---|---|---|---|---|---|---|

800 | 0.0000 | 0.0870 | −0.0356 | 0.0411 | 0.0219 | −0.1939 |

1000 | 0.0000 | −0.0344 | −0.0021 | 0.0103 | −0.0487 | 0.0305 |

1200 | 0.0000 | −0.1260 | 0.0491 | −0.0584 | 0.0283 | 0.1866 |

Approximated water flow velocity in

| 0 | 400 | 800 | 1200 | 1600 | 2000 |
---|---|---|---|---|---|---|

800 | 0.6213 | 0.0256 | −0.1082 | 0.0412 | 0.0323 | 0.1251 |

1000 | −0.2959 | 0.4853 | −0.1423 | −0.0040 | −0.1365 | −0.5382 |

1200 | 0.5779 | −0.0048 | −0.1376 | 0.1127 | 0.0168 | 0.1442 |

Approximated pollutant concentration ^{3}).

| 0 | 400 | 800 | 1200 | 1600 | 2000 |
---|---|---|---|---|---|---|

800 | 5.7854 | 2.9295 | 1.5580 | 0.7936 | 1.0637 | 1.3736 |

1000 | 10.0000 | 4.0758 | 1.6448 | 0.5297 | 1.0873 | 1.3920 |

1200 | 5.8178 | 2.9421 | 1.5798 | 0.9594 | 1.1355 | 1.3860 |

The approximated pollutant concentrations over 5 monitoring nodes, when the eastern exit gate is opened on 0–100 min, 33–100 min, and 66–100 min.

The approximated pollutant concentrations over the eastern exit gate is opened on 0–100 min, 33–100 min, and 66–100 min, when

The overall approximated pollutant concentrations when the eastern exit gate is opened on the period 0–100 min.

The overall approximated pollutant concentrations when the eastern exit gate is opened on the period 33–100 min.

The overall approximated pollutant concentrations when the eastern exit gate is opened on the period 66–100 min.

In simulation 1, the different activated zonal removal mechanism systems are considered. The approximated pollutant concentration levels with the contaminant removal mechanism system are shown in Figure

In simulation 2, the eastern exit gate opening on different periods of time is considered. The suitable third zonal removal mechanism system has been chosen to be activated. If the eastern exit gate is opened or shut on different periods of time such as 0–100 min, 33–100 min, and 66–100 min, the approximated pollutant concentrations over 5 monitoring nodes are shown in Figure

If we consider the approximated pollutant concentrations along the eastern exit gate, when the the eastern exit gate is opened or shut on the difference periods of time as shown in Figure

However, the overall approximated pollutant concentrations, when the eastern exit gate is opened on the different period of times, are shown in Figures

The monitoring point placement for water quality measurement in an opened-closed reservoir can give accurate or inaccurate assessment. However, the proposed numerical simulations have given the overall approximated water quality when the removal mechanism systems are activated in different zones. The proposed techniques give the zonal removal mechanism placement to specialists who want to control the water quality. Moreover, we obtain the overall and specified approximated water quality for each points and time when the exit gate is opened on the different periods of time. The techniques can give a suitable period of time to open the exit gate to specialists who want to get a good agreement water quality by using contaminant removal mechanism.

The authors declare no conflicts of interest.

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.