# Computational Modeling of Pollution Transmission in Rivers

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## Abstract

Modeling of river pollution contributes to better management of water quality and this will lead to the improvement of human health. The advection dispersion equation (ADE) is the government equation on pollutant transmission in the river. Modeling the pollution transmission includes numerical solution of the ADE and estimating the longitudinal dispersion coefficient (LDC). In this paper, a novel approach is proposed for numerical modeling of the pollution transmission in rivers. It is related to use both finite volume method as numerical method and artificial neural network (ANN) as soft computing technique together in simulation. In this approach, the result of the ANN for predicting the LDC was considered as input parameter for the numerical solution of the ADE. To validate the model performance in real engineering problems, the pollutant transmission in Severn River has been simulated. Comparison of the final model results with measured data of the Severn River showed that the model has good performance. Predicting the LDC by ANN model significantly improved the accuracy of computer simulation of the pollution transmission in river.

### Keywords

Pollution transmission Advection dispersion equation (ADE) Multilayer perceptron neural network (MLP) Finite volume method (FVM)## Introduction

The study of rivers’ water quality is extremely important. This issue is more important when the rivers are one of the main sources of water supply for drinking, agriculture, and industry. Unfortunately, river pollution has become one of the most important problems in the environment (Benedini and Tsakiris 2013). When a source of pollution is transfused into the river, due to molecular motion, turbulence, and non-uniform velocity in cross section of flow, it quickly spreads and covers all around the cross section and moves along the river with the flow (Holzbecher 2012; Chanson 2004). Defining the mechanism of pollutant transmission in various types of rivers' geometry helps reduce the effects of water pollution on public health in human societies. The study of the mechanism of pollutant transmission in the rivers has become a major part of the knowledge of environmental engineering (Riahi-Madvar et al. 2009). The governing equation of pollutant transmission in river is advection dispersion equation (ADE). This equation is a partial differential equation, named Convection Equation in general (Aleksander and Morton 1995; Chau 2010; Portela and Neves 1994). Computer simulation of pollution transmission in rivers needs to solve the ADE by analytical or numerical approaches. The ADE has analytical solution under simple boundary and initial conditions but when the flow geometry and hydraulic conditions become more complex such as practical engineering problems, the analytical solutions are not applicable. Therefore, to solve this equation, several numerical methods have been proposed (Kumar et al. 2009; Buske et al. 2011; Chanson 2004; Holzbecher 2012; Zoppou and Knight 1997). In this regard, using the finite difference method (FDM), finite volume method (FVM), and finite element method (FEM), etc. can be stated. Numerical modeling of pollution transmission in rivers in addition to numerical solution of the ADE included the LDC estimation. Fortunately, recently suitable numerical methods have provided. The major part of studies on the river water quality has focused on the measurement, calculating and estimation of the LDC (Kashefipour and Falconer 2002; Deng et al. 2001, 2002). Nowadays, by advancing the soft computing techniques in water engineering, researchers try to use these methods to predict LDC (Najafzadeh and Sattar 2015). Based on the scientific reports, the precision all of these methods were better than the empirical formulas. In the field of soft computing, using the multilayer perceptron neural network (MLP), adaptive neuro-fuzzy inference system (ANFIS), support vector machine (SVM), neuro-fuzzy GMDH, and genetic programming (GP) has been reported (Riahi-Madvar et al. 2009; Tayfur and Singh 2005; Toprak and Cigizoglu 2008; Azamathulla and Ghani 2011; Sahay 2011; Najafzadeh and Azamathulla 2013; Parsaie et al. 2015). The conclusions derived from the review of past researches suggest that studying the river water quality has been individually conducted by studying the numerical solution of the ADE and LDC measurement or prediction. In this paper, by getting the idea of research which was conducted by Parsaie and Haghiabi (2015), a novel approach is proposed for computer simulation of engineering phenomena. In this paper, both of the FVM and MLP model is used together in computer simulation.

## Materials and methods

*C*is the concentration,

*u*is the mean flow velocity, and

*x*is the distance from the place of perfect mixing in the cross section of flow. The ADE includes two different parts advection and dispersion. The pure advection is given in Eq. (2) and the pure dispersion term is given in Eq. (3).

Here F is the flow flux and D_{L} is the longitudinal dispersion coefficient. The results of advection and dispersion term will be combined by the time splitting technique. To calculate the LDC, several ways as empirical formulas and artificial intelligent techniques have been proposed and development of these is based on dimensionless parameters that are derived using the Buckingham theory, which will be explained in the next section.

## Longitudinal dispersion coefficient

Empirical equations for estimating the longitudinal dispersion coefficient

Row | Author | Equation |
---|---|---|

1 | Elder (1959) | \(D_{\text{L}} = 5.93\;hu_{*}\) |

2 | McQuivey and Keefer (1974) | \(D_{\text{L}} = 0.58\left( {\frac{h}{{u_{*} }}} \right)uw\) |

3 | Fischer (1966) | \(D_{\text{L}} = 0.011\frac{{u^{2} w^{2} }}{{hu_{*} }}\) |

4 | Li et al. (1998) | \(D_{\text{L}} = 0.55\frac{{wu_{*} }}{{h^{2} }}\) |

5 | Liu (1977) | \(D_{\text{L}} = 0.18\left( {\frac{u}{{u_{*} }}} \right)^{1.5} \left( \frac{w}{h} \right)^{2} hu_{*}\) |

6 | Iwasa and Aya (1991) | \(D_{\text{L}} = 2\left( \frac{w}{h} \right)^{1.5} hu_{*}\) |

7 | Seo and Cheong (1998) | \(D_{\text{L}} = 5.92\left( {\frac{u}{{u_{*} }}} \right)^{1.43} \left( \frac{w}{h} \right)^{0.62} hu_{*}\) |

8 | Koussis and Rodriguez-Mirasol (1998) | \(D_{L} = 0.6\left( \frac{w}{h} \right)^{2} hu_{*}\) |

9 | Li et al. (1998) | \(D_{\text{L}} = 5.92\left( {\frac{u}{{u_{*} }}} \right)^{1.2} \left( \frac{w}{h} \right)^{1.3} hu_{*}\) |

10 | Kashefipour and Falconer (2002) | \(D_{\text{L}} = 2\left( {\frac{u}{{u_{*} }}} \right)^{0.96} \left( \frac{w}{h} \right)^{1.25} hu_{*}\) |

11 | Tavakollizadeh and Kashefipur (2007) | \(D_{\text{L}} = 7.428 + 1.775\left( {\frac{u}{{u_{*} }}} \right)^{1.752} \left( \frac{w}{h} \right)^{0.62} hu\) |

12 | Sahay and Dutta (2009) | \(D_{\text{L}} = 10.612\left( {\frac{u}{{u_{*} }}} \right)hu\) |

Range of collected data related to the LDC

| | | | \(D_{\text{L}} \;({\text{m}}^{2} /{\text{s}})\) | |
---|---|---|---|---|---|

Min | 11.9 | 0.2 | 0.0 | 0.0 | 1.9 |

Max | 711.2 | 19.9 | 1.7 | 0.6 | 1486.5 |

Avg | 73.2 | 1.5 | 0.5 | 0.1 | 115.3 |

Std dev | 106.9 | 2.3 | 0.4 | 0.1 | 218.7 |

## Artificial neural network (ANN)

## Results and discussion

## Model validation

Universal coordinates of sampling stations

Site | UK (grid reference) | Distance (m) |
---|---|---|

Injection | SN 9549 8479 | 0 |

A | SN 9570 8488 | 210 |

B | SN 9621 8561 | 1175 |

C | SN 9748 8558 | 2875 |

D | SN 9969 8518 | 5275 |

E | SO 0160 8677 | 7775 |

F | SO 0252 8858 | 10275 |

G | SO 0220 9090 | 13775 |

Result of LDC calculation by routing method for Severn River

River | Station | Δ | Δ | | |
---|---|---|---|---|---|

Severn River | A | 4 | 2 | 41.5 | |

B | 4 | 2 | 26.5 | ||

C | 4 | 2 | 12.5 | ||

D | 4 | 2 | 26.5 | ||

E | 4 | 2 | 37.5 | ||

F | 4 | 2 | 29.5 | ||

G | 4 | 2 |

Calculating LDC by empirical formulas and MLP model

Model | St. (A–B) | St. (B–C) | St. (C–D) | St. (D–E) | St. (E–F) | St. (F–G) | |
---|---|---|---|---|---|---|---|

\(D_{{{\text{L}} - {\text{R}}}} ({\text{m}}^{2} /{\text{s)}}\) | 41.5 | 26.5 | 12.5 | 38.5 | 37.5 | 29.5 | |

Eq. (1) | 0.11 | 0.12 | 0.14 | 0.14 | 0.12 | 0.15 | 0.21 |

Eq. (2) | 1069.3 | 905.96 | 1071.9 | 1069 | 957 | 1166.76 | 0.00 |

Eq. (3) | 150.1 | 132.5 | 124.8 | 116.4 | 147.9 | 117.7 | 0.17 |

Eq (4) | 2.81 | 2.7 | 2.38 | 2.43 | 3.06 | 2.35 | 0.11 |

Eq. (5) | 12.92 | 34.6 | 34.8 | 35.9 | 43.1 | 37.8 | 0.7 |

Eq. (6) | 74.03 | 13.8 | 14.7 | 15.3 | 16.8 | 16.5 | 0.38 |

Eq. (7) | 74.4 | 69.07 | 72.53 | 66.24 | 67.35 | 69.7 | 0.01 |

Eq. (8) | 27.1 | 29.04 | 29.8 | 31.8 | 37.5 | 34 | 0.02 |

Eq. (9) | 18.26 | 17.42 | 17.73 | 16.93 | 18.85 | 17.71 | 0.05 |

Eq. (10) | 3.49 | 3.39 | 3.77 | 3.45 | 3.15 | 3.71 | 0.4 |

Eq. (11) | 586.1 | 490.8 | 490.5 | 416.2 | 441.1 | 424.4 | 0.00 |

Eq. (12) | 56.13 | 53.29 | 55.25 | 52 | 55 | 54.47 | 0.01 |

ANN (MLP) | 38.2 | 29.8 | 14.5 | 49.7 | 40.4 | 32.6 | 0.83 |

## Conclusion

Rivers are one of the main sources of water supply for drinking, agricultural, and industrial usage. Therefore, controlling the quality of them is important, since the water quality of the rivers is directly related to human and environment health. Unfortunately, sometimes it seems that river has been considered as a place for injection of sewages. One method to manage the water quality is mathematical modeling of pollution transmission in the river. In mathematical modeling, governing partial differential equations is solved by suitable and powerful methods. In some governing equations, there are coefficients that researchers have measured and calculated, and they have also proposed some empirical formulas to calculate them. Recently, the soft computing techniques were used as powerful tools to predict these coefficients by researchers. It seems that, for developing software or commercial computer models, in addition to using a suitable numerical method, the coefficients could be predicted using soft computing methods. This approach leads to the increase the performance of mathematical modeling of phenomena, especially when the coefficients are very sensitive and the range variation of them is much more. In other words, these coefficients may be probability. In this paper, for numerical solution of ADE, the finite volume has been used and to predict the longitudinal dispersion coefficient, the MLP model was prepared. It is shown that the results are suitable, when the AI models have been used as a powerful tool to predict the LDC.

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