# Numerical and Experimental Study of Turbulent Mixing Characteristics in a T-Junction System

^{1}

^{2}

^{3}

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## Abstract

**:**

_{EML}) was defined for quantitative analysis. The numerical results were found to be in good agreement with the experimental results. The results reveal that the values of L

_{EML}rise as Q or φ increase and decrease with the increase of δ, where the influence of φ is much greater than Q and δ, and there is no obvious regularity between L

_{EML}and θ. By dimensional analysis and multivariate nonlinear regression analysis, a dimensionless relationship equation in harmony with the dimensional analysis was fitted, and a simplified equation with the average error of 4.01% was obtained on the basis of correlation analysis.

## 1. Introduction

## 2. CFD Simulation Methodology

#### 2.1. Mathematical Model

#### 2.2. Computational Grid

#### 2.3. Boundary Conditions and CFD Procedures

^{3}and a viscosity value of 0.089008 mPa·s and the salt solution with a density of 1007.05 kg/m

^{3}and a viscosity value of 0.4 mPa·s, were used.

## 3. Experiment and Model Validation

#### 3.1. Experimental Details

_{e}is more than 2300. In turbulent conditions, the particles are mixed with each other, the motion is disordered, and it has energy dissipation capability and diffusivity, so they have remarkable influence on the mixing of solutions in the pipes.

_{e}values of some typical working conditions are listed in Table 2, which indicated the flow in the test main pipe was turbulent. R

_{e}in the main pipe rose with the increase of the flow rate and decreased with the increase of the main pipe diameter under the same other conditions. In addition, the numbers of R

_{e}in the table were large indicating obvious turbulent characteristics.

#### 3.2. Model Validation

#### 3.2.1. The Relationship between the Volume Fraction and Electrical Conductivity

#### 3.2.2. Comparison between Experimental Data and Simulation Results

## 4. Results and Discussion

_{EML}’ for short which was determined [23].

#### 4.1. Influence Factors in the Mixing Process

#### 4.1.1. The Effect of Pipe Diameter on Mixing Effect

^{3}/h), and incident angles, and it can be observed that L

_{EML}increases obviously with the increase of the pipe diameter. Firstly, the area of pipe sections is reduced with the significant reduction in the pipe diameter, the salt solution diffuses more easily, and mixing time decreases significantly which leads to a shorter observed L

_{EML}. Meanwhile, increased water inlet area reduces the average flow velocity by the reason of constant cross-flow flux. The Reynolds number of a fluid with a lower velocity is a smaller result in the indistinctive characteristics of turbulent flow and the L

_{EML}increased when the impact dispersion fluidity became weak.

#### 4.1.2. The Effects of Cross-Flow Flux and Mixing Ratio on Mixing Effect

_{EML}. It can be observed that the L

_{EML}augments with the increase of cross-flow flux when other conditions remain the same, and increases when the mixing ratio is decreased. However, the effects of cross-flow flux and mixing ratio are minimal. Similar to the law of L

_{EML}with the pipe diameter, the salt solution velocity increases when raising the mixing ratio, which results in the augment of the diffusion rate of salt solution and the enhanced salt solution mixing effect to decrease the L

_{EML}.

_{EML}still increased.

#### 4.1.3. The Effect of Incident Angle of T-Junction on Mixing Effect

_{EML}which vary with the incident angles of T-junctions differ under different main pipe diameters.

_{EML}with the incident angles of T-junctions differ under different main pipe diameters. In the next section, the dimensional analysis (D-A) was adopted to further research the influences of the four variables on L

_{EML}.

#### 4.2. Dimensional Analysis of Effective Mixing Length

_{EML}in the structural formula, dimensional analysis was used to fit the computational expressions of L

_{EML}. Based on a thorough understanding of the research, the main physical quantities affecting the physical process must be correctly determined, to ensure the correct application of dimensional analysis. In this section, a total of eight physical quantities were considered, including effective mixing length (L

_{EML}), main pipe diameter (φ) and cross-flow flux (Q), water density (ρ), mixing ratio ($\delta $), gravitational acceleration (g), incident angle (θ), and average pressure of the mixture (p).

_{EML}is effective mixing length, φ represents main pipe diameter, Q is cross-flow flux, g is gravitational acceleration, p denotes average pressure of the mixture, ρ is water density, δ signifies mixing ratio, and θ is incident angle.

_{EML}is less affected by δ and θ which can be ignored, and affected by Q, φ, p, ρ, g dramatically. In order to simplify the calculation process, δ

^{0.00491}and θ

^{−0.00297}were assigned a value of 1, and the coefficients were simplified simultaneously. The simplified equation is shown below.

## 5. Conclusions

_{EML}were studied by the controlling variable method. The results show that the value of L

_{EML}increases as Q or φ rise and decreases with the augment of δ, of which φ has a much greater magnitude than Q and δ. Meanwhile, θ has no clear influence on L

_{EML}. Furthermore, dimensional analysis was employed to study these influences, and a dimensionless relationship equation in the harmony of dimension was obtained. Moreover, a simplified equation with the average error of 4.01% was derived, proving the applicability of it. This study provides a valuable reference to the turbulent mixing law in a T-junction system, and has broad engineering significance for tracking the origin of sewage in pipelines.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Typical mesh distribution in simulation of T-junction with φ = 51.4 mm, θ = 30°. (

**a**) Side view, (

**b**) cross-sectional view, and (

**c**) 3-D view.

**Figure 5.**Relationship among electrical conductivity (EC), volume fraction (VOF), and concentration (C) of salt solution.

**Figure 8.**Variation tendency of L

_{EML}under three kinds of main pipe diameters: φ = 51.4 mm; φ = 61.4 mm; φ = 73.6 mm.

**Figure 10.**The average cross section velocities of mixture, water, and salt solution in the selected region.

**Figure 11.**Contours of the distribution of salt solution (

**a**) and water velocity (

**b**) in the selected region.

No. | δ | Q (m^{3}/h) | φ (mm) | θ (°) | No. | δ | Q (m^{3}/h) | φ (mm) | θ (°) |
---|---|---|---|---|---|---|---|---|---|

1 | 30 | 31 | 30 | ||||||

2 | 60 | 32 | 60 | ||||||

3 | 51.4 | 90 | 33 | 51.4 | 90 | ||||

4 | 120 | 34 | 120 | ||||||

5 | 150 | 35 | 150 | ||||||

6 | 30 | 36 | 30 | ||||||

7 | 60 | 37 | 60 | ||||||

8 | 8 | 61.4 | 90 | 38 | 8 | 61.4 | 90 | ||

9 | 120 | 39 | 120 | ||||||

10 | 150 | 40 | 150 | ||||||

11 | 30 | 41 | 30 | ||||||

12 | 60 | 42 | 60 | ||||||

13 | 73.6 | 90 | 43 | 73.6 | 90 | ||||

14 | 120 | 44 | 120 | ||||||

15 | 1% | 150 | 45 | 0.5% | 150 | ||||

16 | 30 | 46 | 30 | ||||||

17 | 60 | 47 | 60 | ||||||

18 | 51.4 | 90 | 48 | 51.4 | 90 | ||||

19 | 120 | 49 | 120 | ||||||

20 | 150 | 50 | 150 | ||||||

21 | 30 | 51 | 30 | ||||||

22 | 60 | 52 | 60 | ||||||

23 | 12 | 61.4 | 90 | 53 | 12 | 61.4 | 90 | ||

24 | 120 | 54 | 120 | ||||||

25 | 150 | 55 | 150 | ||||||

26 | 30 | 56 | 30 | ||||||

27 | 60 | 57 | 60 | ||||||

28 | 73.6 | 90 | 58 | 73.6 | 90 | ||||

29 | 120 | 59 | 120 | ||||||

30 | 150 | 60 | 150 |

No | R_{e} | No | R_{e} | No. | R_{e} |
---|---|---|---|---|---|

1 | 616,940 | 6 | 516,461 | 11 | 430,852 |

16 | 925,409 | 21 | 774,691 | 26 | 646,278 |

Diameter (mm) | Incident Angle | L_{EML} of Group A (m) | L_{EML} of Group B (m) | L_{EML} of Group C (m) | L_{EML} of Group D (m) |
---|---|---|---|---|---|

30° | 1.8124 | 1.8150 | 1.8161 | 1.8164 | |

60° | 1.9167 | 1.9168 | 1.9172 | 1.9220 | |

51.4 | 90° | 1.8496 | 1.8520 | 1.8599 | 1.8610 |

120° | 1.7747 | 1.7753 | 1.7796 | 1.7800 | |

150° | 2.0830 | 2.0879 | 2.1070 | 2.1090 | |

30° | 2.6286 | 2.6345 | 2.6347 | 2.6353 | |

60° | 2.7339 | 2.7450 | 2.7520 | 2.7549 | |

61.4 | 90° | 2.6790 | 2.6830 | 2.6850 | 2.6882 |

120° | 2.7638 | 2.7754 | 2.7769 | 2.7804 | |

150° | 2.5650 | 2.5752 | 2.5970 | 2.5990 | |

30° | 3.8650 | 3.8710 | 3.8740 | 3.8790 | |

60° | 3.5270 | 3.5402 | 3.5600 | 3.5632 | |

73.6 | 90° | 3.7007 | 3.7208 | 3.7240 | 3.7439 |

120° | 3.6640 | 3.6808 | 3.6868 | 3.7010 | |

150° | 3.5600 | 3.5680 | 3.6092 | 3.6120 |

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**MDPI and ACS Style**

Sun, B.; Liu, Q.; Fang, H.; Zhang, C.; Lu, Y.; Zhu, S.
Numerical and Experimental Study of Turbulent Mixing Characteristics in a T-Junction System. *Appl. Sci.* **2020**, *10*, 3899.
https://doi.org/10.3390/app10113899

**AMA Style**

Sun B, Liu Q, Fang H, Zhang C, Lu Y, Zhu S.
Numerical and Experimental Study of Turbulent Mixing Characteristics in a T-Junction System. *Applied Sciences*. 2020; 10(11):3899.
https://doi.org/10.3390/app10113899

**Chicago/Turabian Style**

Sun, Bin, Quan Liu, Hongyuan Fang, Chao Zhang, Yuanbo Lu, and Shun Zhu.
2020. "Numerical and Experimental Study of Turbulent Mixing Characteristics in a T-Junction System" *Applied Sciences* 10, no. 11: 3899.
https://doi.org/10.3390/app10113899