# Validation of a Computational Fluid Dynamics Model for a Novel Residence Time Distribution Analysis in Mixing at Cross-Junctions

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{4}) as a tracer. Residence Time Distribution (RTD) curves are good features and a novel proposal on mixing cross-junctions, in order to calibrate simulation processes, with the help of stimulus-response techniques [22,23,24]. There were generated turbulent regime scenarios in Reynolds numbers from 43,343 to 155,793. Therefore, different operating conditions that are close to real WDN were performed. To analyse the tracer distribution in the outlets of the cross-junction, the curves of the pulses are registered, visualizing by RTD curves the approximation of the Computational Fluid Dynamics (CFD) simulations. According to the cross-junction mixing models that are reviewed in the literature, this is the first time that the hydraulic pipe flow has been simulated with success using RTD analysis, which was addressed by solving the RANS equations with the k-epsilon turbulence model coupled to the diffusion-convection equation.

#### Complete Mixing Model

_{k}, set of links with flow into k; L

_{j}, length of link; Q

_{j}, flow (volume/time) in link j; Q

_{k,ext}, external source flow entering the network at node k; and, C

_{k},

_{ext}, concentration of the external source flow entering at node k.

## 2. Materials and Methods

#### 2.1. Experimental Model

^{®}, model MP-0400 (Figure 2b), both with datalogger capabilities. For measures at the outlets, the outflowing water is conducted up to four flowmeters, two for each outlet. For the East outlet, the flowmeters are of electromagnetic type, of Badger Meter Inc. brand, model M2000 (Figure 2c). For the South outlet, the water is conducted to two propeller volumetric flow meters (Figure 2d).

^{®}brand model TDS-2004C (Figure 2e) was available to the pressure registration on the four boundaries of the cross-junction. In this case, the oscilloscope visually represents the electrical signals that are captured by pressure transducers. These transducers are of WIKA

^{®}brand model S-10, comprise a measuring range of 0–10 V of electrical signal intensity corresponding to the range −1 to 9 bar of pressure.

_{4}·5H

_{2}O) of fine grade, Fermont

^{®}brand. The 0.25 mol/L solutions are prepared, which return an effective response to the concentration measurement. The tracer injection is done by pumping, with a monophasic equipment, Siemens brand, 1HP power and rotor speed of 3515–3535 rpm. The tracer is stored in a volumetric test tube of four liters size, and then, in each test, the tracer is pumped 150 mL approximately. The injection point is located at 3.50 m before the cross-junction (Figure 3). The pump equipment allows for overcoming the pressures of water flow in the network and the controlled intrusion of tracer that is injected through the opening and closing ball valve of a 0.5 inch.

#### 2.2. Formulation of Numerical Simulation

#### 2.2.1. Turbulent Flow

_{T}, according to the standard k-ε turbulence model; Equations (4) to (6).

_{k}is the energy production term (${P}_{k}={\mu}_{T}[\nabla u:(\nabla u+{(\nabla u)}^{T}]$), and C

_{µ}(0.09), C

_{e1}(1.44), C

_{e2}(1.92), σ

_{k}(1), and σ

_{ε}(1.3) are dimensionless constant values that are obtained by data fitting over a wide range of turbulent flows [38]. K. Hanjalik [41] used the k-ε model over other types of turbulence models for simulation in pressure pipes with high Reynolds numbers, with the same empirical constants. Furthermore, RANS k-ε models have proposed good results also in mixing and concentration analysis, when these models have been validated with measurements, as in [42,43].

#### 2.2.2. Boundary Conditions

^{+}is the normalized velocity component in the logarithmic layer of the wall, κ is the Von Karman constant (0.4187), E is a constant that depends on the roughness of the wall, and y

^{+}is the dimensionless distance from the wall.

- Inlet velocity at N;
- Inlet velocity at W;
- Pressure at E outlet; and,
- Pressure at S outlet.

_{0}and ε

_{0}must be defined, which are obtained from the Equations (8) and (9).

_{T}(dimensionless) varies between 0.05 and 0.10. For the turbulent scale length, L

_{T}(long unit) can be obtained according to the pipe radius r, obtaining 7% of it. L

_{T}= 0.07r [39]. Therefore, the values were fixed on I

_{T}= 0.05 and L

_{T}= 0.07 × 0.0508 m = 0.003556 m.

#### 2.2.3. Residence Time Distribution

_{T}. This term can be evaluated considering the Schmidt Number (turbulent), Sc

_{T}, for which, the equation proposed by Kays-Crawford [41] was used, Equations (11) and (12).

_{T∞}is the value of Sc

_{T}at a distance far from the wall, and it is fixed with a value Sc

_{T∞}= 0.85. It is considered to be a perfect mix at the tracer input boundary due to the distance with which it is injected.

## 3. Results

#### 3.1. Mesh Selection and Sensibility Analysis

^{+}there is an adequate representation of the velocity profile.

^{+}values in a range of (11.06 to 12.00) have important effects to turbulence production near the wall. As shown in the graph of Figure 6, the u velocity with respect to the mesh degree (by number of elements) stops varying from the FINE mesh degree. Therefore, this mesh degree was selected for the CFD model, which guarantees that the mesh tetrahedra established in the contours maintains a minimum y

^{+}value of 11.1 (those were verified plotting values of y

^{+}in a mesh graphic). The wall roughness was assumed to have a negligible effect. The solver that was employed was iterative, a generalized minimal residual, and a relative tolerance of accuracy of the CFD simulations considered a convergence criterion below 1 × 10

^{−5}. The typical solution around these meshes elements was unchanged.

#### 3.2. CFD Simulation

#### 3.3. Residence Time Distribution Curves

#### 3.4. Variation of Sc_{T} Coefficient

_{T}coefficient is modified. The turbulent diffusivity was simulated in different values, using assignments to the Schmidt Turbulent Sc

_{T}number. The range of values for this parameter is 0.61, 0.71, 0.81, and the value that was obtained by the Kays-Crawford model [41], which amounts to 0.5666.

_{T}coefficients, since the parameterization of the curves reflects practically homogeneous results. This conclusion is verified, if a RMSE for each case is obtained again (Table 5).

_{T}was based on the model of Kays-Crawford [41], and it was the most appropriate in most cases. The values of Sc

_{T}have a very low impact in scale for the simulated model. In Figure 12, it is observed that the diffusion increases in the zones of greater recirculation, and there is less speed in the model. This should have a more significant influence, according to the bibliography. For the moment, it can be determined that the speed and pressure range far exceed the studied cases, and this tells us that at high levels of speed and pressure, the turbulent diffusivity does not have a significant impact on the transport of solute.

#### 3.5. Incomplete Mixing Simulations

_{N}/C

_{W}); in the same way, Outlet relation (C

_{E}/C

_{S}). The diverse scenarios implemented were made by the combination of concentration at the inlets, in intervals of 0.25 mol/L, in a range Inlet relation = [0.00–2.00]. In the case of Inlet relation = 0.00, it means that C

_{N}= 0.00 mol/L and C

_{W}= 5 mol/L. At the same way, for an Inlet relation = 1.50, it means that C

_{N}= 7.5mol/L and C

_{W}= 5mol/L. The scenarios are represented in Figure 13, with the results for the outlet relation after the cross-junction mixing.

_{N}/C

_{W}= 1) might not be a complete mixing. That is, since the result contains the same concentration at the inlets, it does not mean that a total mixture of the two incoming flows has been made. The complete mixing model assumes that “Outlet relation” must be equal to one in all cases, by the reason that the estimation assigns the same concentration at the two outlets. No one scenario has the behavior like a complete mixing.

## 4. Discussion and Conclusions

_{T}solved many questions about the role of turbulent diffusion and the impact that it generates in these operating conditions. The RTD curves showed a minimal change between them. The change is almost inappreciable graphically. Therefore, more checks were obtained using the RMSE error for each curve, and the variation is seen up to the fourth decimal place of precision of the RMSE. This allows for concluding that convection is the main transport in diluted species, and that the diffusion does not affect much in the simulation of the tracer course through the cross-junction. Even with this, it was found that the model that was proposed by Kays-Crawford [41] was the adequate parameter when presenting the minimum RMSE values with respect to the turbulent Schmidt variation.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Ahn, J.C.; Lee, S.W.; Choi, K.Y.; Koo, J.Y. Application of EPANET for the determination of chlorine dose and prediction of THMs in a water distribution system. Sustain. Environ. Res.
**2012**, 22, 31–38. [Google Scholar] - Shanks, C.M.; Sérodes, J.B.; Rodriguez, M.J. Spatio-temporal variability of non-regulated disinfection by-products within a drinking water distribution network. Water Res.
**2013**, 47, 3231–3243. [Google Scholar] [CrossRef] [PubMed] - Vasconcelos, J.J.; Rossman, L.A.; Grayman, W.M.; Boulos, P.F.; Clark, R.M. Kinetics of chlorine decay. Am. Water Works Assoc.
**1997**, 89, 54. [Google Scholar] [CrossRef] - Ozdemir, O.N.; Ucak, A. Simulation of chlorine decay in drinking-water distribution systems. J. Environ. Eng.
**2002**, 128, 31–39. [Google Scholar] [CrossRef] - Geldreich, E.E. Microbial Quality of Water Supply in Distribution Systems; CRC Press LLC: Boca Raton, FL, USA, 1996; ISBN 1-56670-194-5. [Google Scholar]
- Knobelsdorf, M.J.; Mujeriego, S.R. Crecimiento bacteriano en las redes de distribución de agua potable: Una revisión bibliográfica. Ing. del Agua
**1997**, 4, 17–28. [Google Scholar] [CrossRef] - Alcocer-Yamanaka, V.H.; Tzatchkov, V.G.; Arreguín-Cortés, F.I. Modelo de calidad del agua en redes de distribución. Ing. Hidraul. Mex.
**2004**, 19, 77–88. [Google Scholar] - Rodríguez, M.J.; Rodríguez, G.; Serodes, J.; Sadiq, R. Subproductos de la desinfección del agua potable: Formación, aspectos sanitarios y reglamentación. Interciencia
**2007**, 32, 749–756. [Google Scholar] - Wang, W.; Ye, B.; Yang, L.; Li, Y.; Wang, Y. Risk assessment on disinfection by-products of drinking water of different water sources and disinfection processes. Environ. Int.
**2007**, 33, 219–225. [Google Scholar] [CrossRef] [PubMed] - Castro, P.; Neves, M. Chlorine decay in water distribution systems case study-lousada network. Electron. J. Environ. Agric. Food Chem.
**2003**, 2, 261–266. [Google Scholar] - Parks, S.L.I.; VanBriesen, J.M. Booster disinfection for response to contamination in a drinking water distribution system. J. Water Res. Plan. Manag.
**2009**, 135, 502–511. [Google Scholar] [CrossRef] - Tabesh, M.; Azadi, B.; Rouzbahani, A. Optimization of chlorine injection dosage in water distribution networks using a genetic algorithm. J. Water Wastewater
**2011**, 22, 2–11. [Google Scholar] - Kansal, M.L.; Dorji, T.; Chandniha, S.K.; Tyagi, A. Identification of optimal monitoring locations to detect accidental contaminations. In Proceedings of the World Environmental and Water Resources Congress 2012: Crossing Boundaries, Albuquerque, NM, USA, 20–24 May 2012; pp. 758–776. [Google Scholar]
- Hernández, D.; Rodríguez, J.M.; Galván, X.D.; Medel, J.O.; Magaña, M.R.J. Optimal use of chlorine in water distribution networks based on specific locations of booster chlorination: Analyzing conditions in Mexico. Water Sci. Technol. Water Supply
**2016**, 16, 493–505. [Google Scholar] [CrossRef] - Weickgenannt, M.; Kapelan, Z.; Blokker, M.; Savic, D.A. Risk-based sensor placement for contaminant detection in water distribution systems. J. Water Res. Plan. Manag.
**2010**, 136, 629–636. [Google Scholar] [CrossRef] - Rathi, S.; Gupta, R. Monitoring stations in water distribution systems to detect contamination events. ISH J. Hydraul. Eng.
**2014**, 20, 142–150. [Google Scholar] [CrossRef] - Seth, A.; Klise, K.A.; Siirola, J.D.; Haxton, T.; Laird, C.D. Testing contamination source identification methods for water distribution networks. J. Water Res. Plan. Manag.
**2016**, 142. [Google Scholar] [CrossRef] - Xuesong, Y.; Jie, S.; Chengyu, H. Research on contaminant sources identification of uncertainty water demand using genetic algorithm. Cluster Comput.
**2017**, 20, 1007–1016. [Google Scholar] [CrossRef] - Montalvo, I.; Gutiérrez, J. Water quality sensor placement with a multiobjective approach. In Proceedings of the Congress on Numerical Methods in Engineering, Valencia, Spain, 3–5 July 2017. [Google Scholar]
- Rathi, S.; Gupta, R. Optimal sensor locations for contamination detection in pressure-deficient water distribution networks using genetic algorithm. Urban Water J.
**2017**, 14, 160–172. [Google Scholar] [CrossRef] - Rossman, L.A. EPANET 2: User’s Manual; EPA/600/R-00/057; U.S. Environmental Protection Agency: Washington, DC, USA, 2000.
- Sandoval, M.; Rosalba, F.; Walsh, F.C.; Nava, J.L.; Ponce de León, C. Computational fluid dynamics simulations of single-phase flow in a filter-press flow reactor having a stack of three cells. Electrochim. Acta
**2016**, 216, 490–498. [Google Scholar] [CrossRef] [Green Version] - Castañeda, L.F.; Antaño, R.; Rivera, F.F.; Nava, J.L. Computational fluid dynamic simulations of single-phase flow in a spacer-filled channel of a filter-press electrolyzer. Int. J. Electrochem. Sci.
**2017**, 12, 7351–7364. [Google Scholar] [CrossRef] - Rodríguez, I.L.; Valdés, J.A.; Alfonso, E.; Estévez, R.D. Dopico. Uso de técnicas estímulo—Respuesta para simular diagnósticos en esófago humano. In Proceedings of the Congreso Latinoamericano de Ingeniería Biomédica, La Habana, Cuba, 23–25 May 2001. [Google Scholar]
- Van Bloemen Waanders, B.; Hammond, G.; Shadid, J.; Collis, S.; Murray, R. A comparison of Navier-Stokes and network models to predict chemical transport in municipal water distribution systems. In Proceedings of the World Water and Environmental Resources Congress, Anchorage, AK, USA, 15–19 May 2005; pp. 1–10. [Google Scholar]
- Webb, S.W. High-fidelity simulation of the influence of local geometry on mixing in crosses in water distribution systems. In Proceedings of the ASCE World Water & Environmental Resources Congress, Tampa, FL, USA, 15–19 May 2007. [Google Scholar]
- Ho, C.K.; Khalsa, S.S. EPANET-BAM: Water quality modeling with incomplete mixing in pipe junctions. In Proceedings of the Water Distribution Systems Analysis 2008 Conference, Kruger National Park, South Africa, 17–20 August 2008; pp. 1–11. [Google Scholar]
- Song, I.; Romero-Gomez, P.; Choi, C.Y. Experimental verification of incomplete solute mixing in a pressurized pipe network with multiple cross-junctions. J. Hydraul. Eng.
**2009**, 135, 1005–1011. [Google Scholar] [CrossRef] - Liu, H.; Yuan, Y.; Zhao, M.; Zheng, X.; Lu, J.; Zhao, H. Study of Mixing at Cross-junction in Water Distribution Systems Based on Computational Fluid Dynamics. In Proceedings of the International Conference on Pipelines and Trenchless Technology, Beijing, China, 26–29 October 2011; pp. 552–561. [Google Scholar]
- Romero-Gomez, P.; Lansey, K.E.; Choi, C.Y. Impact of an incomplete solute mixing model on sensor network design. J. Hydroinform.
**2011**, 13, 642–651. [Google Scholar] [CrossRef] [Green Version] - Yu, T.C.; Shao, Y.; Shen, C. Mixing at cross joints with different pipe sizes in water distribution systems. J. Water Res. Plan. Manag.
**2014**, 140, 658–665. [Google Scholar] [CrossRef] - Shao, Y.; Yang, Y.J.; Jiang, L.; Yu, T.; Shen, C. Experimental testing and modeling analysis of solute mixing at water distribution pipe junctions. Water Res.
**2014**, 56, 133–147. [Google Scholar] [CrossRef] [PubMed] - Mompremier, R.; Pelletier, G.; Mariles, Ó.A.F.; Ghebremichael, K. Impact of incomplete mixing in the prediction of chlorine residuals in municipal water distribution systems. J. Water Supply Res. Technol.
**2015**, 64, 904–914. [Google Scholar] [CrossRef] - Mckenna, S.A.; O’hern, T.; Hartenberger, J. Detailed Investigation of Solute Mixing in Pipe Joints Through High Speed Photography. Water Distrib. Syst. Anal.
**2008**, 1–12. [Google Scholar] [CrossRef] - Ho, C.K.; O’Rear, L., Jr. Evaluation of solute mixing in water distribution pipe junctions. Am. Water Works Assoc. J.
**2009**, 101, 116. [Google Scholar] [CrossRef] - Choi, C.Y.; Shen, J.Y.; Austin, R.G. Development of a comprehensive solute mixing model (AZRED) for double-tee, cross, and wye junctions. Water Distrib. Syst. Anal.
**2008**, 1–10. [Google Scholar] [CrossRef] - Gualtieri, C.; Jiménez, P.L.; Rodríguez, J.M. A comparison among turbulence modelling approaches in the simulation of a square dead zone. In Proceedings of the 19th Canadian Hydrotechnical Conference, Vancouver, BC, Canada, 9–14 August 2009. [Google Scholar]
- Versteeg, H.K.; Malalasekera, W. An Introduction to Computational Fluid Dynamics: The Finite Volume Method; Pearson Education: New York, NY, USA, 2007. [Google Scholar]
- Rosales, M.; Pérez, T.; Nava, J.L. Computational fluid dynamic simulations of turbulent flow in a rotating cylinder electrode reactor in continuous mode of operation. Electrochim. Acta
**2016**, 194, 338–345. [Google Scholar] [CrossRef] - Hanjalic, K. Two-Dimensional Asymmetric Turbulent Flow in Ducts. Ph.D. Thesis, University of London, London, UK, 1970. [Google Scholar]
- Kays, W.M.; Crawford, M.E.; Weigand, B. Convective Heat and Mass Transfer; McGraw-Hill Education: New York, NY, USA, 2005. [Google Scholar]
- Martinez-Solano, F.J.; Iglesias-Rey, P.L.; Gualtieri, C.; López Jiménez, P.A. Modelling flow and concentration field in rectangular water tanks. In Proceedings of the International Environmental Modelling and Software Society (iEMSs), Ottawa, ON, Canada, 5–8 July 2010. [Google Scholar]
- Moncho-Esteve, I.J.; Palau-Salvador, G.; López Jiménez, P.A. Numerical simulation of the hydrodynamics and turbulent mixing process in a drinking water storage tank. J. Hydraul. Res.
**2005**, 2, 207–217. [Google Scholar] [CrossRef] - Georgescu, A.M.; Georgescu, S.C.; Bernad, S.; Coşoiu, C.I. COMSOL Multiphysics versus Fluent: 2D numerical simulation of the stationary flow around a blade of the Achard turbine. In Proceedings of the 3rd Workshop on Vortex Dominated Flows, Timisoara, Romania, 1–2 June 2007. [Google Scholar]
- COMSOL. CFD Module User’s Guide; Version 4.3bCOMSOL Multiphysics; COMSOL, Inc.: Burlington, MA, USA, 2013. [Google Scholar]
- Moukalled, F.; Mangani, L.; Darwish, M. The Finite Volume Method in Computational Fluid Dynamics; Springer: Berlin, Germany, 2016; Volume 113. [Google Scholar]

**Figure 2.**Equipment to measure flow (electromagnetic: (

**a**,

**c**), propeller volumetric: (

**b**,

**d**)) and pressure around the cross-junction (oscilloscope and pressure transducer: (

**e**)).

**Figure 12.**Sc

_{T}and eddy diffusivity plots based on the Kays-Crawford [41] model (Equation (12)). High values appear in the turbulent flow zones.

Material | Solid Copper |
---|---|

Electrode diameter | 4.4 mm |

Length of electrodes | 9.5 cm (covers more than 90% of the pipe diameter) |

Electrodes separation | 4.6 mm |

Insulation material | resin |

Front | V1 | V2 | V3 | V4 | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

V (m/s) | P (bar) | Re | Q (l/s) | V (m/s) | P (bar) | Re | Q (l/s) | V (m/s) | P (bar) | Re | Q (l/s) | V (m/s) | P (bar) | Re | Q (l/s) | ||

Inlets | N | 1.112 | 1.88 | 112,928 | 9.01 | 1.116 | 1.60 | 113,396 | 9.05 | 0.808 | 1.57 | 82,123 | 6.55 | 0.672 | 1.96 | 68,295 | 5.45 |

W | 1.059 | 1.88 | 107,625 | 8.59 | 1.269 | 1.60 | 128,971 | 10.29 | 1.533 | 1.56 | 155,793 | 12.43 | 0.989 | 1.96 | 100,503 | 8.02 | |

Outlets | E | 1.005 | 1.88 | 102,067 | 8.14 | 1.062 | 1.56 | 107,899 | 8.61 | 1.039 | 1.56 | 105,654 | 8.43 | 1.246 | 1.96 | 126,624 | 10.10 |

S | 1.167 | 1.88 | 118,547 | 9.46 | 1.324 | 1.56 | 134,468 | 10.73 | 1.302 | 1.56 | 132,273 | 10.55 | 0.427 | 1.96 | 43,343 | 3.46 |

V1 | V2 | V3 | V4 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Exp | CFD | % Error | Exp | CFD | % Error | Exp | CFD | % Error | Exp | CFD | % Error | ||

P (bar) | N | 1.88 | 1.893 | 0.670% | 1.60 | 1.577 | 1.437% | 1.57 | 1.578 | 0.502% | 1.96 | 1.971 | 0.575% |

W | 1.88 | 1.893 | 0.677% | 1.60 | 1.576 | 1.478% | 1.56 | 1.575 | 0.982% | 1.96 | 1.971 | 0.557% | |

V (m/s) | E | 1.005 | 0.987 | 1.720% | 1.062 | 1.060 | 0.211% | 1.040 | 1.044 | 0.381% | 1.246 | 1.237 | 0.744% |

S | 1.167 | 1.180 | 1.130% | 1.324 | 1.322 | 0.133% | 1.302 | 1.294 | 0.593% | 0.427 | 0.422 | 1.192% |

**Table 4.**Root Mean Square Error (RMSE) values for the four scenarios to evaluate variances in the RTD curves.

RMSE | V1 | V2 | V3 | V4 |
---|---|---|---|---|

E outlet | 0.0215 | 0.0222 | 0.0093 | 0.0509 |

S outlet | 0.0072 | 0.0109 | 0.0181 | 0.0140 |

RMSE | Sc_{T} (adim) | |||
---|---|---|---|---|

0.61 | 0.71 | 0.81 | K-C (0.566) | |

V1 E | 0.0215 | 0.0216 | 0.0216 | 0.0215 |

V1 S | 0.0073 | 0.0074 | 0.0076 | 0.0072 |

V2 E | 0.0223 | 0.0226 | 0.02282 | 0.0220 |

V2 S | 0.0109 | 0.0107 | 0.0106 | 0.0106 |

V3 E | 0.0090 | 0.0085 | 0.0081 | 0.0093 |

V3 S | 0.0182 | 0.0184 | 0.0186 | 0.0181 |

V4 E | 0.0509 | 0.0509 | 0.0509 | 0.0509 |

V4 S | 0.0143 | 0.0147 | 0.0151 | 0.0140 |

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## Share and Cite

**MDPI and ACS Style**

Hernández-Cervantes, D.; Delgado-Galván, X.; Nava, J.L.; López-Jiménez, P.A.; Rosales, M.; Mora Rodríguez, J.
Validation of a Computational Fluid Dynamics Model for a Novel Residence Time Distribution Analysis in Mixing at Cross-Junctions. *Water* **2018**, *10*, 733.
https://doi.org/10.3390/w10060733

**AMA Style**

Hernández-Cervantes D, Delgado-Galván X, Nava JL, López-Jiménez PA, Rosales M, Mora Rodríguez J.
Validation of a Computational Fluid Dynamics Model for a Novel Residence Time Distribution Analysis in Mixing at Cross-Junctions. *Water*. 2018; 10(6):733.
https://doi.org/10.3390/w10060733

**Chicago/Turabian Style**

Hernández-Cervantes, Daniel, Xitlali Delgado-Galván, José L. Nava, P. Amparo López-Jiménez, Mario Rosales, and Jesús Mora Rodríguez.
2018. "Validation of a Computational Fluid Dynamics Model for a Novel Residence Time Distribution Analysis in Mixing at Cross-Junctions" *Water* 10, no. 6: 733.
https://doi.org/10.3390/w10060733