# Pollutant Dispersion Modeling in Natural Streams Using the Transmission Line Matrix Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- Vertical and transversal dispersions are very small;
- The pollutant is completely miscible in water;
- Chemical reactions between the pollutant and its environment are absent; and
- The overall mass of pollutant is maintained during transport.

_{L}is the longitudinal dispersion coefficient (m

^{2}/s), x is the distance (m), and t is the time (s).

## 2. Methods

#### 2.1. Physical Phenomenon

#### 2.1.1. Physical Phenomenon Description

_{0}), the maximum concentration (C

_{max}), the peak time (t

_{p}), and the end time of pollution (t

_{f}).

Station | Nomenclature | x (m) | t_{0}(s) | C_{max}(μg/l) | t_{p}(s) | t_{f}(s) |
---|---|---|---|---|---|---|

A | Llanidloes Meadow | 210 | 60 | 1050 | 300 | 900 |

B | Dol-llys | 1175 | 1320 | 225 | 1560 | 8580 |

C | Morfodion Ford | 2875 | 3480 | 110 | 4140 | 16,560 |

D | Dolwen | 5275 | 6480 | 58 | 8880 | 16,440 |

E | Rickety Bridge | 7775 | 10,440 | 34.5 | 13,440 | 25,980 |

F | Llandinam | 10,275 | 14,220 | 21 | 18,720 | 27,120 |

G | Carnedd | 13,775 | 17,713 | 20 | 23,533 | 33,433 |

^{2}(standard deviation: 2.8 m

^{2}) while below this, it is 13.8 m

^{2}(standard deviation: 4.1 m

^{2}) [12].

#### 2.1.2. Moment Method and Experimental Parameters Determination

_{L}:

_{i}is the distance between station i and dumping point (m).

_{L}and U are essentially the two experimental parameters of the pollutant dispersion.

#### 2.1.3. General Assumptions and Hydraulic Conditions

_{m}is the length of the mixture (m), K

_{2}is the constant depends on the manner of tracer pouring, U is the flow speed (m/s), B is the average width between the dumping and the sampling points (m), and E

_{Y}is the lateral dispersion coefficient (m

^{2}/s).

_{0}is the dumped tracer initial concentration (g/l). Concerning the boundary conditions, the tracer disperses only downstream from the injection point in the flow direction: $\forall x<0,\forall t>0,c\left(x,t\right)=0$ and the tracer concentration is null at x = ℓ, $\text{if}x=\ell ,\text{}\forall t0,c\left(x,t\right)=0$, where ℓ is the test distance (m).

#### 2.2. Modeling Method

#### 2.2.1. Transmission Line Matrix Method

**Figure 1.**Dispersion of an injected pulse. (

**a**) pulse reflection; (

**b**) the first iteration result; and (

**c**) the second iteration result.

#### 2.2.2. TLM Model for Pollutant Dispersion Phenomenon

_{m}(A) at each node. Hence, a TLM model of a pollutant dispersion phenomenon is a perfectly insulated transmission line (conductance G(Ω) null) of length Δx(m), that is electrically represented by two resistors R(Ω), two inductors L(H), and a capacitor C(F) (Figure 2). The TLM node is given on Figure 3, where I(x), V(x), and Z are, respectively, the intensity of current (A), the voltage (Volt), and the characteristic impedance (Ω) of the line. The latter is defined by the relation from Equation (12).

_{d}, C

_{d}, and L

_{d}are, respectively, distributed resistance, capacitance, and inductance over the entire length Δx of the transmission line and are given by the following relationships:

_{m}at the node (n) as indicated by Equation (17):

#### 2.2.3. TLM Initial and Boundary Conditions

#### 2.2.4. TLM parameters Optimization

Input Data | Parameter/Unit | Value |
---|---|---|

Experiment duration | t (s) | 35,000 |

Time step | ∆t (s) | 10 |

Test distance | ℓ (m) | 14,000 |

Distance step | ∆x (m) | 25 |

Characteristic impedance | Z (Ω) | 10 |

Injected total tension | V(1) (Volt) | 1110 |

_{d}, g

_{m}, and initial total tension V

_{0}have been optimized to keep the modeling simple. In TLM technique, the time step ∆t must be less than the time constant (RC) leading to a choice of R

_{d}> 0.8. The conductance is the resistance inverse, so g

_{m}< 1.25. The initial total tension at each node must be inferior to the injected total tension (first node), so V

_{0}(n ≠ 1) < 1110.

_{0}, V

_{max}, k

_{p}, k

_{f}) are compared to the electrical equivalents of experimental characteristic parameters (t

_{0}, C

_{max}, t

_{p}, t

_{f}) (Table 1). The electrical equivalents (k'

_{0}, V'

_{max}, k'

_{p}, k'

_{f}) are deduced from Equation (11) and Equation (29). The optimized TLM parameters (R

_{d}, g

_{m}, and V

_{0}) correspond to the minimal values (∆k

_{0}, ∆V

_{max}, ∆k

_{p}, ∆k

_{f}).

#### 2.3. Statistical Tests for TLM Model Validation

_{tlm}and U), longitudinal dispersion coefficient (D

_{tlm}and D

_{L}), maximum concentration (C

_{maxtlm}and C

_{max}), peak time (t

_{ptlm}and t

_{p}), start time of pollution (t

_{0tlm}and t

_{0}), and end time of pollution (t

_{ftlm}and t

_{f}).

_{div}with R

_{div}>1 (R

_{div}< 1) indicates an over-prediction (under-prediction) of a value; the mean percentage error E(%) representing a convenient model when it is small; the Mean Relative Square Error MRSE indicating a successful model if it is close to 0; the Factor Of EXcedence FOEX(%), signifying an overestimation (underestimation and/or correct estimation) of all values when it is equal to 100% (0%); the factor of two FA

_{2}(%) representing a reliable model if it is close to 100% and finally the scatter diagram indicating a situation of overestimation (underestimation) when the point is above (below) “y = x” line.

## 3. Results and Discussion

#### 3.1. Determination of Experimental Parameters

_{L}is determined using Equation (4). The results are presented in Table 3:

Station | U (m/s) | D_{L} (m^{2}/s) |
---|---|---|

B | 0.71 | 18.76 |

C | 0.68 | 23.39 |

D | 0.45 | 11.47 |

E | 0.55 | 62.62 |

F | 0.50 | 22.97 |

G | 0.72 | −101.48 |

#### 3.2. Determination of TLM Parameters

**Figure 4.**TLM modeling and experimental data of Atkinson and Davis. The total tension (tracer concentration) is traced according to the iteration number k at stations B, C, D, E, and F.

Station | TLM Parameters | Physical Equivalents | ||||
---|---|---|---|---|---|---|

−g_{m} (Ω) | R_{d} (Ω) | V_{0}(Volt) | U_{tlm} (m/s) | D_{tlm}(m^{2}/s) | C_{0tlm} (μg/L) | |

B | 0.03 | 1.33 | 990 | 0.76 | 18.76 | 990 |

C | 0.03 | 0.89 | 863.5 | 0.70 | 28.2 | 863.5 |

D | 0.02 | 1.04 | 690 | 0.59 | 24 | 690 |

E | 0.02 | 0.83 | 522 | 0.576 | 30 | 522 |

F | 0.02 | 0.89 | 370 | 0.56 | 28 | 370 |

_{tlm}, D

_{tlm}, and C

_{0tlm}are, respectively, the flow velocity, the longitudinal dispersion coefficient, and the initial concentration predicted by the TLM model.

#### 3.3. TLM Model Performance

**Table 5.**Statistical tests for flow velocity, longitudinal dispersion coefficient and maximum concentration.

Parameters | Indices | B | C | D | E | F | Average |
---|---|---|---|---|---|---|---|

Flow velocity (U_{tlm}, U) | R_{div} | 1.07 | 1.03 | 1.31 | 1.05 | 1.12 | 1.12 |

E (%) | 7.00 | 2.90 | 31.10 | 4.70 | 12.0 | 11.60 | |

MRSE | 0.02 | ||||||

FOEX (%) | 100 | ||||||

Fa_{2} (%) | 100 | ||||||

Longitudinal dispersion coefficient (D_{tlm}, D_{L}) | R_{div} | 1.00 | 1.21 | 2.09 | 0.48 | 1.22 | 1.20 |

E (%) | 0.00 | 20.60 | 109.20 | 52.10 | 21.90 | 40.80 | |

MRSE | 0.21 | ||||||

FOEX (%) | 60 | ||||||

Fa_{2} (%) | 60 | ||||||

Maximum concentration (C_{maxtlm}, C_{max}) | R_{div} | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

E (%) | 0.00 | 0.00 | 0.10 | 0.00 | 0.00 | 0.00 | |

MRSE | 0.00 | ||||||

FOEX (%) | 0 | ||||||

Fa_{2} (%) | 100 |

_{2}, MRSE, and E indicate that the model representation of the experimental flow velocity is acceptable.

**Figure 5.**Scatter diagrams for (

**a**): flow velocity; (

**b**): longitudinal dispersion coefficient; (

**c**): maximum concentration; (

**d**): peak time; (

**e**): pollution start time; and (

**f**): pollution end time.

_{L}decreases with U. The overestimation of the flow velocity explains the overestimation of the longitudinal dispersion coefficient at this station. For station E, its coefficient is the highest because of an obvious and sudden change in river geometry upstream of the station E at 6.5 km from the injection point. In this region, the channel is much deeper and starts to widen; station E has the biggest section. The longitudinal dispersion coefficient depends on the width and depth of the channel; thus, it is strongly related to the channel geometry. Therefore, the difference between the predicted and observed longitudinal dispersion coefficients at station E is due to our model not taking into account the channel geometry. The values of Fa

_{2}, MRSE, and E indicate that the model representation of the experimental longitudinal dispersion coefficients is fairly acceptable. Moreover, the numerical estimation of this coefficient is very difficult; in the literature [7,8,9,14,17,38,43,44,45,46] for natural rivers, there is a great disparity between the values of longitudinal dispersion coefficients calculated by different methods and empirical formulas. This shows that such a coefficient is difficult to estimate, its experimental determination is the fairest approach because it is based on spatio-temporal monitoring of pollutant concentrations in streams with the real effect of skewness, topography, and water storage areas.

_{2}factor is equal to 100%. All the statistical tests indicate that the model provides a very well representation for the experimental maximum concentrations and their peak times. The principal role of a pollutant model is to correctly give the pollution degree (maximum concentration and its peak time) at a given distance; the hereby-presented original model satisfies this requirement.

_{2}factor is equal to 100%. For the pollution end time, E is equal to 36%, MRSE is 0.32, FOEX is null, and Fa

_{2}factor is equal to 60%. These statistical tests show that the model provides a good estimate of pollution start time but less of an estimate of the pollution end time. It is preferable for a pollution model to predict early tracer cloud passage so one can react as quickly as possible to preserve the environment. The hereby-presented original model fulfills this demand.

#### 3.4. TLM Model Stability

Stations | s | r |
---|---|---|

B | 0.15 | 0.30 |

C | 0.14 | 0.45 |

D | 0.12 | 0.38 |

E | 0.12 | 0.48 |

F | 0.11 | 0.45 |

#### 3.5. Final TLM Model

_{L}/D

_{tlm}, β = U/U

_{tlm}, and γ = C

_{0}/C

_{0tlm}. These factors (Table 7) are different because of the channel geometry and topology variation from a station to another.

Station | α | β | γ |
---|---|---|---|

B | 1 | 0.93 | 1.12 |

C | 0.83 | 0.97 | 1.29 |

D | 0.48 | 0.76 | 1.61 |

E | 2.09 | 0.95 | 2.13 |

F | 0.82 | 0.89 | 3.00 |

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Chapra, S.C. Surface Water-Quality Modeling; McGraw-Hill Series in Water Resources and Environmental Engineering: New York, NY, USA, 1997. [Google Scholar]
- Fischer, H.B.; List, E.J.; Koh, R.C.Y.; Imberger, J.; Brooks, N.H. Mixing in Inland and Coastal Waters; Academics Press: San Diego, CA, USA, 1979. [Google Scholar]
- Graf, W.H. Fluvial Hydraulics: Flow and Transport Processes in Channels of Simple Geometry; Wiley: Hoboken, NJ, USA, 1998. [Google Scholar]
- Runkel, R.L.; Broshears, R.E. One-Dimensional Transport with Inflow and Storage (OTIS)—A Solute Transport Model for Small Streams; University of Colorado: Boulder City, CO, USA, 1991. [Google Scholar]
- Marsalek, J.; Sztruhar, D.; Giulianelli, M.; Urbonas, B. Enhancing Urban Environment by Environmental Upgrading and Restoration; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2004. [Google Scholar]
- Jabbour, D. Etude Expérimentale et Modélisation de la Dispersion en Champ Lointain Suite à un Rejet Accidentel d’un Polluant Miscible Dans un Cours d’eau: Application à la Gestion de Crise. Ph.D. Thesis, University of Provence, Aix-en-Provence, France, January 2006. [Google Scholar]
- Shen, C.; Niu, J.; Anderson, E.J.; Phanikumar, M.S. Estimating longitudinal dispersion in rivers using acoustic doppler current profilers. Adv. Water Resour.
**2010**, 33, 615–623. [Google Scholar] [CrossRef] - Etemad-Shahidi, A.; Taghipour, M. Predicting longitudinal dispersion coefficient in natural streams using M5’ Model Tree. J. Hydraul. Eng.
**2012**, 138, 542–554. [Google Scholar] [CrossRef] - Rodrigues, P.P.G.W.; González, Y.M.; de Sousa, E.P.; Neto, F.D.M. Evaluation of dispersion parameters for river Sao Pedro, Brazil, by the simulated annealing method. Inverse Probl. Sci. Eng.
**2013**, 21, 34–51. [Google Scholar] [CrossRef] - Tealdi, S.; Camporeale, C.; Perucca, E.; Ridolfi, L. Longitudinal dispersion in vegetated rivers with stochastic flows. Adv. Water Resour.
**2010**, 33, 562–571. [Google Scholar] [CrossRef] - De Cogan, D. Transmission Line Matrix (TLM) Techniques for Diffusion Applications; The Gordon and Breach Science Publishers: Amsterdam, The Netherlands, 1998. [Google Scholar]
- Atkinson, T.C.; Davis, P.M. Longitudinal dispersion in natural channels: 1. Experimental results from the river Severn, UK. Hydrol. Earth Syst. Sci.
**2000**, 4, 345–353. [Google Scholar] [CrossRef] - Fischer, B.H. Longitudinal Dispersion in Laboratory and Natural Streams; California Institute of Technology: Pasadena, CA, USA, 1966. [Google Scholar]
- Perucca, E.; Camporeale, C.; Ridolfi, L. Estimation of the dispersion coefficient in rivers with riparian vegetation. Adv. Water Resour.
**2009**, 32, 78–87. [Google Scholar] [CrossRef] - Pathak, S.K.; Pande, P.K.; Kumar, S. Effect of circulation on longitudinal dispersion in open channel. In Proceedings of the 6th Australasian Hydraulics and Fluid Mechanics Conference, Adelaide, Australia, 5–9 December 1977; pp. 597–600.
- Davis, P.M.; Atkinson, T.C.; Wigley, T.M.L. Longitudinal dispersion in natural channels: 2.The roles of shear flow dispersion and dead zones in the river Severn, UK. Hydrol. Earth Syst. Sci.
**2000**, 4, 355–371. [Google Scholar] [CrossRef] - Baek, K.O.; Seo, I.W. Routing procedures for observed dispersion coefficients in two-dimensional river mixing. Adva. Water Resour.
**2010**, 33, 1551–1559. [Google Scholar] [CrossRef] - International Organization for Standardization (ISO). Mesure de Débit Des Liquides Dans Les Canaux Découverts. Longueur de Bon Mélange d’un Traceur; TR 11656-1993; International Organization for Standardization: Geneva, Switzerland, 1993. (In French) [Google Scholar]
- Toledo-Redondo, S.; Salinas, A.; Morente-Molinera, J.A.; Méndez, A.; Fornieles, J.; Portí, J.; Morente, J.A. Parallel 3D-TLM algorithm for simulation of the earth-ionosphere cavity. J. Comput. Phys.
**2013**, 236, 367–379. [Google Scholar] [CrossRef] - Amri, A.; Saidane, A.; Pulko, S. Thermal analysis of a three-dimensional breast model with embedded tumour using the transmission line matrix (TLM) method. Comput. Biol. Med.
**2011**, 41, 76–86. [Google Scholar] [CrossRef] [PubMed] - Feradji, A.; Pulko, S.H.; Saidane, A.; Wilkinson, A.J. Transmission line matrix modelling of self heating in multi-finger 4H-SiC MESFETs. J. Appl. Sci.
**2012**, 12, 32–39. [Google Scholar] [CrossRef] - Rao, R.S.; Subbaiah, P.V.; Rao, B.P. Electromagnetic transient scattering analysis in time-domain comparison of TLM and TDIE methods. J. Comput. Electron.
**2012**, 11, 315–320. [Google Scholar] - Johns, P.B.; Beurle, R.L. Numerical solution of 2-dimensional scattering problems using a transmission line matrix. Proc. IEEE
**1971**, 118, 1203–1208. [Google Scholar] [CrossRef] - Guillaume, G. Application de la Méthode TLM à la Modélisation de la Propagation Acoustique en Milieu Urbain. Ph.D. Thesis, University of Maine, Orono, ME, USA, October 2009. [Google Scholar]
- Le Maguer, S. Développement de Nouvelles Procédures Numériques Pour la Modélisation TLM: Application à la Caractérisation de Circuits Plaqués et de Structures à Symétrie de Révolution en Bande Millimétrique. Ph.D. Thesis, University of Western Brittany, Brest, France, November 1998. [Google Scholar]
- Amri, A. Numerical Analysis of Microwave Sintering of Ceramic Materials Using a 3D-TLM Method. Ph.D. Thesis, University USTO-MB, Oran, Algeria, June 2005. [Google Scholar]
- Aliouat, B.S.; Saidane, A.; Benzohra, M.; Saiter, J.M.; Hamou, A. Dimensional soft tissue thermal injury analysis using transmission line matrix TLM method. Int. J. Numeri. Model. Electron. Netw. Devices Fields
**2008**, 21, 531–549. [Google Scholar] - Christopoulos, C. The Transmission Line Modeling Method TLM; University of Nottingham: Nottingham, UK, 1995. [Google Scholar]
- Johns, P.B. On the relationship between TLM and finite-difference methods for Maxwell’s equations. IEEE Trans. Microw. Theory Tech.
**1987**, 35, 60–61. [Google Scholar] [CrossRef] - Smith, A. Transmission Line Matrix Modeling, Optimization and Application to Adsorption Phenomena. Bachelor’s Thesis, Nottingham University, Nottingham, UK, July 1988. [Google Scholar]
- De Cogan, D.; Henini, M. Transmission line matrix TLM: A novel technique for modeling reaction kinetics. Faraday Trans. 2: Mol. Chem. Phys.
**1987**, 83, 843–855. [Google Scholar] [CrossRef] - Al-Zeben, M.Y.; Saleh, A.H.M.; Al-Omar, M.A. TLM modeling of diffusion, drift and recombination of charge carriers in semiconductors. Int. J. Numer. Model. Electron. Netw. Devices Fields
**1992**, 5, 219–225. [Google Scholar] [CrossRef] - Gui, X.; Gao, G.B.; Morkoç, H. Transmission-line matrix method for solving the multidimensional continuity equation. Int. J. Numer. Model. Electron. Netw. Devices Fields
**1993**, 6, 233–236. [Google Scholar] [CrossRef] - Courant, R.; Friedrichs, K.; Lewy, H. On the partial difference equations of mathematical physics. IBM J. Rese. Dev.
**1967**, 11, 215–234. [Google Scholar] [CrossRef] - Mosca, S.; Graziani, G.; Klug, W.; Bellasio, R.; Bianconi, R. A statistical methodology for the evaluation of long range dispersion models: An application to the ETEX exercise. Atmos. Environ.
**1998**, 32, 4307–4324. [Google Scholar] [CrossRef] - Boybeyi, Z.; Ahmad, N.N.; Bacon, D.P.; Dunn, T.J.; Hall, M.S.; Lee, P.C.S.; Sarma, R.A.; Wait, T.R. Evaluation of the operational multiscale environment model with grid adaptivity against the European tracer experiment. J. Appl. Meteorol.
**2001**, 40, 1541–1558. [Google Scholar] [CrossRef] - Duijm, N.J.; Ott, S.; Nielsen, M. An evaluation validation procedures and test parameters for dense gas dispersion models. J. Loss Prev. Process Ind.
**1996**, 9, 323–338. [Google Scholar] [CrossRef] - Bashitialshaaer, R.; Bengtsson, L.; Larson, M.; Persson, K.M.; Aljaradin, M.; Al-Itawi, H.I. Sinuosity effects on longitudinal dispersion coefficient. Int. J. Sustain. Water Environ. Syst.
**2011**, 2, 77–84. [Google Scholar] - Boxall, J.B.; Guymer, I. Longitudinal mixing in meandering channels: New experimental data set and verification of a predictive technique. Water Res.
**2007**, 41, 341–354. [Google Scholar] [CrossRef] [PubMed] - Zhang, W.; Boufadel, M.C. Pool effects on longitudinal dispersion in streams and rivers. J. Water Resour. Prot.
**2010**, 2, 960–971. [Google Scholar] [CrossRef] - Keylock, C.J.; Constantinescu, G.; Hardy, R.J. The application of computational fluid dynamics to natural river channels: Eddy resolving versus mean flow approaches. Geomorphology
**2012**, 179, 1–20. [Google Scholar] [CrossRef] - Chen, D.; Tang, C. Evaluating secondary flows in the evolution of sine-generated meanders. Geomorphology
**2012**, 163, 37–44. [Google Scholar] [CrossRef] - Kashefipour, S.; Falconer, R. Longitudinal dispersion coefficients in natural channels. Water Res.
**2002**, 36, 1596–1608. [Google Scholar] [CrossRef] - Dongsu, K. Assessment of longitudinal dispersion coefficients using acoustic doppler current profilers in large river. J. Hydro-Environ. Res.
**2012**, 6, 29–39. [Google Scholar] - Piotrowski, A.P.; Rowinski, P.M.; Napiorkowski, J.J. Comparison of evolutionary computation techniques for noise injected neural network training to estimate longitudinal dispersion coefficients in rivers. Expert Syst. Appl.
**2012**, 39, 1354–1361. [Google Scholar] [CrossRef] - Azamathulla, H.M.; Ghani, A.A. Genetic programming for predicting longitudinal dispersion coefficients in streams. Water Resour. Manag.
**2011**, 25, 1537–1544. [Google Scholar] [CrossRef]

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Meddah, S.; Saidane, A.; Hadjel, M.; Hireche, O.
Pollutant Dispersion Modeling in Natural Streams Using the Transmission Line Matrix Method. *Water* **2015**, *7*, 4932-4950.
https://doi.org/10.3390/w7094932

**AMA Style**

Meddah S, Saidane A, Hadjel M, Hireche O.
Pollutant Dispersion Modeling in Natural Streams Using the Transmission Line Matrix Method. *Water*. 2015; 7(9):4932-4950.
https://doi.org/10.3390/w7094932

**Chicago/Turabian Style**

Meddah, Safia, Abdelkader Saidane, Mohamed Hadjel, and Omar Hireche.
2015. "Pollutant Dispersion Modeling in Natural Streams Using the Transmission Line Matrix Method" *Water* 7, no. 9: 4932-4950.
https://doi.org/10.3390/w7094932