## Abstract

The cross joint is one of the standard connection types for urban water supply pipelines. A pipeline containing cross joints was taken as the research object using NaCl as the tracer. The turbulent mixing characteristics of water and pollutants at the cross joints and pollutant migration and diffusion were studied and analysed by numerical analysis and experimental measurement. The primary purpose is to check the comprehensive influence of the six factors of pipe diameter, inlet flow ratio, outlet flow ratio, location of damage point, flow of brine, and density of brine on the turbulent mixing of water and brine in the pipeline. The coefficient of variation is used as an evaluation index to evaluate the mixing in the pipeline, and the effective mixing length *L*_{EML} is used to quantify the uniform mixing position of brine in the pipeline. The results show that the inlet flow ratio, outlet flow ratio, and pipe diameter significantly affect *L*_{EML} in the east outlet of the cross joint. Outlet flow ratio and pipe diameter significantly affect *L*_{EML} in the cross joint's south outlet direction. In addition, the dimensionless relationship equation representing *L*_{EML} is fitted through dimensional analysis.

## HIGHLIGHTS

The main and secondary factors affecting the pollutant diffusion at the cross joint were determined by orthogonal test.

The influence of related factors on the turbulent mixing characteristics at the cross joint was studied by using the control variable method.

The dimensionless equation representing the effective mixing length was fitted to quantify the mixing at the cross joint.

### Graphical Abstract

## NOTATION

*D*pipe diameter

*D*_{i}values in row

*i*of*D*in Table 1*Q*_{1}west inlet flow

*Q*_{2}north inlet flow

*Q*_{3}east outlet flow

*Q*_{4}south outlet flow

*α*inlet flow ratio(

*Q*_{1}/*Q*_{2})*α*_{i}values in row

*i*of*α*in Table 1*β*outlet flow ratio (

*Q*_{3}/*Q*_{4})*β*_{i}values in row

*i*of*β*in Table 1*P*location of damage point

*P*_{i}values in row

*i*of*P*in Table 1*Q*_{s}flow of brine

*Q*_{si}values in row

*i*of*Q*_{s}in Table 1*ρ*_{s}density of brine

*ρ*_{si}values in row

*i*of*ρ*_{s}in Table 1

Case . | D
. | P
. | ρ_{s}
. | α
. | Q_{s}
. | β
. |
---|---|---|---|---|---|---|

(m) . | (m) . | (kg/m^{3})
. | (Q_{1}/Q_{2})
. | (m^{3}/h)
. | (Q_{3}/Q_{4})
. | |

1 | 0.05 | 0.50 | 1,005.00 | 0.50 | 0.08 | 0.43 |

2 | 0.06 | 1.00 | 1,071.00 | 1.00 | 0.12 | 1.00 |

3 | 0.07 | 1.50 | 1,148.00 | 1.50 | 0.16 | 2.33 |

Case . | D
. | P
. | ρ_{s}
. | α
. | Q_{s}
. | β
. |
---|---|---|---|---|---|---|

(m) . | (m) . | (kg/m^{3})
. | (Q_{1}/Q_{2})
. | (m^{3}/h)
. | (Q_{3}/Q_{4})
. | |

1 | 0.05 | 0.50 | 1,005.00 | 0.50 | 0.08 | 0.43 |

2 | 0.06 | 1.00 | 1,071.00 | 1.00 | 0.12 | 1.00 |

3 | 0.07 | 1.50 | 1,148.00 | 1.50 | 0.16 | 2.33 |

## INTRODUCTION

The water supply pipeline system is an essential part of urban infrastructure. The safety, stability, reliability, and efficiency of the water supply system are crucial to the daily work and lives of the people who depend on it (Deng *et al.* 2020; Chowdhury & Akter 2021). Due to the complex interaction of physical, environmental, and operational factors, the channel may break (Chung *et al.* 2004), and it is very easy for the pressure in the pipeline to fluctuate significantly, resulting in low pressure or negative pressure. When there is low or negative pressure in the pipe network, the pipeline below the groundwater level is subject to a pressure under the external water (depending on the height of the groundwater level above the pipeline). Pollutants around the buried pipe will enter the water supply network through a breakage or pipe accessories, and the pollutants will invade the water supply pipe (Li *et al.* 2010; Kakoudakis *et al.* 2018; Barton *et al.* 2019). The diffusion of solutes or pollutants in the water distribution network is mainly controlled by mixing at the tube joint. Different flow rates and concentrations can enter the tube joints to cause fluid mixing (Ho 2008). The cross joint is one of the standard connection types for urban water supply pipelines (Ho *et al.* 2008). For the diffusion of point source pollution, studying the turbulent mixing characteristics in the cross joint water supply pipeline has essential reference significance for locating breakage and identifying and tracking the pollution source, including establishing a water quality model of the water supply network.

The turbulent mixing at the cross node is extremely complex, and the predictions of various models are different from each other. Given the importance of accurately describing such factors as velocity distribution, tracer concentration distribution, and eddy diffusivity, computational fluid dynamics were used, and the results of CFD were compared with experimental data to verify the feasibility of the model. Among several turbulence models in simulation, the *k*-*ε* turbulence model simulation results are the most consistent with the experimental results. The model can reasonably simulate the axial diffusion phenomena in the transition and turbulent regions of the pipe flow and has good convergence (Ekambara & Joshi 2003; Lin & Ferng 2016). In addition, Chen & Long (2019) discussed the diffusion term added to the diffusion effect in the water quality simulation and solved the convection diffusion reaction equations of the multi-component mass transfer reaction model using the Euler–Lagrange splitting operator method. The example shows that the pollutant concentration at the downstream node gradually accumulates with time. The axial diffusion and radial mass-transfer reaction significantly change the microbial concentration in the pipe network.

The diffusion process of point source pollution in pipelines is liquid–liquid mixing. To measure the distribution of solute in the liquid, a relatively simple method is to use dyed solute liquid, and then compare the intensity of colour at each point in the pipeline. Han *et al.* (2014) dyed the solute liquid to show its internal movement, then compared the pixel intensity distribution of each point in the pipeline, and carried out quantitative tracking methods and statistical analysis on the mixing of moving droplets. Ahmad *et al.* (2011) proposed an adaptive neuro-fuzzy inference system method to predict the mixing of pollutants in water flow. A more accurate way is to use light-activated fluorophores and standard laser-induced fluorescence technology and then use particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) to measure the liquid distribution after the combination of the two liquids (Hansen *et al.* 2000).

In reality, there are many factors that affect the mixing process, and each factor has different effects. Yu *et al.* (2014) simulated small cross-joint pipe networks with different pipe diameters. The results show that the tracer and clean water flow rate ratio can affect each pipe joint's outlet concentration. The joint connected by a pipe with a more considerable diameter difference has more thorough mixing than the same type of joint connected by a tube with a minor diameter difference. Van Bloemen Waanders & Webb (2006) simulated the mixing in the cross joint and the double T-joint pipe joint. The simulation results show that, because the two incoming flows interact on the mixing interface at the junction of the joint, that is, the ‘impact interface’, there is an unsteady mixing behaviour at the fluid interface, and the two incoming flows diverge at the junction of the node to the adjacent pipe section, resulting in incomplete mixing. Van Bloemen Waanders *et al.* (2005) simulated the mixing behaviour of the water flow at a cross node with a pipe diameter of two inches (50.8 mm). For high Reynolds, the mixing at the node connection was incomplete, and only 12%–14% of the chemical tracers mixed with other inlet pipes. The analysis of solute dispersion under the action of both longitudinal diffusion and radial dispersion shows that when the flow velocity in the pipeline is low, the radial diffusion process has a great influence on the results. In contrast, the radial diffusion process has little effect on the results (Aris 1956; Piazza *et al.* 2020; Ozdemir *et al.* 2021).

So far, previous scholars have made progress in researching mixing and solute diffusion. The comprehensive effects of factors related to the mixing characteristics have not been studied quantitatively, such as pipe diameter, inlet flow ratio, outlet flow ratio, damage point location, pollutant intrusion flow, and pollutant density, and the specific function expression has not been derived. To quantitatively analyse the comprehensive influence of factors on turbulent mixing in pipes, the effects of relevant factors on turbulent mixing in pipes with cross nodes were studied by numerical simulation and experimental measurement. A dimensionless equation was fitted to analyse the mixing in the tube quantitatively.

## METHODS

### Experimental study

#### Layout of test platform

#### Experimental pipeline and data acquisition

### Definition of evaluation indicators

*Cv*< 0.05 is suitable for mixing, and

*Cv*< 0.01 is entirely mixed. In this study, the coefficient of variation method is selected to evaluate the mixing of each cross-section along the tube axis. The coefficient of variation, also known as the dispersion coefficient, is a normalized measure of the degree of dispersion of the probability distribution, which is defined as the ratio of the standard deviation to the average value:where

*S*is the standard deviation of the salt water mass fraction on the cross-section, calculated from the salt water mass fraction at all points on the cross-section.where is the arithmetic square root of the mass fraction of brine at all points on the cross-section:where

*n*is the number of nodes on the section, and is the mass fraction at node

*i*.

### Orthogonal experimental design

Many factors affect the diffusion of point source pollution, including the type of joint, pipe diameter, flow, pollutant intrusion amount, and kind of pollutant. Therefore, the orthogonal test aims to analyse the influence degree of many factors on the *L*_{EML} (the length from the joint to the uniform mixing location), determine the main influencing factors and secondary influencing factors, and take the main influencing factors as the variables of point source pollution diffusion research.

The basic idea of selecting the factors for research is to screen out the factors affecting the test indicators according to the understanding of professional knowledge, previous research experience, and existing research results. According to the existing conclusions, the research factors selected for this test are pipe diameter and inlet flow ratio, outlet flow ratio, damage location, flow of brine, and density of brine. Three levels are selected for each factor. The level table of factors in the orthogonal test is shown in Table 1.

The design of this test does not consider the interaction between factors. There are three levels and six factors, leaving an empty column as the error column. Therefore, the orthogonal table of L_{18}(3^{7}) is selected, and the number of tests is 18. The experimental scheme after the setting is shown in Table 2.

No . | D
. | P
. | ρ_{s}
. | α
. | Q_{s}
. | β
. | Error column . |
---|---|---|---|---|---|---|---|

(m) . | (m) . | (kg/m^{3})
. | (Q_{1}/Q_{2})
. | (m^{3}/h)
. | (Q_{3}/Q_{4})
. | ||

1 | 0.05 | 0.50 | 1,005.00 | 0.50 | 0.08 | 0.43 | 1.00 |

2 | 0.05 | 0.50 | 1,071.00 | 1.00 | 0.16 | 2.33 | 2.00 |

3 | 0.05 | 1.00 | 1,005.00 | 1.50 | 0.16 | 1.00 | 3.00 |

4 | 0.05 | 1.00 | 1,148.00 | 0.50 | 0.12 | 2.33 | 1.00 |

5 | 0.05 | 1.50 | 1,071.00 | 1.50 | 0.12 | 0.43 | 2.00 |

6 | 0.05 | 1.50 | 1,148.00 | 1.00 | 0.08 | 1.00 | 3.00 |

7 | 0.06 | 0.50 | 1,005.00 | 1.50 | 0.12 | 2.33 | 3.00 |

8 | 0.06 | 0.50 | 1,148.00 | 0.50 | 0.16 | 1.00 | 2.00 |

9 | 0.06 | 1.00 | 1,071.00 | 1.00 | 0.12 | 1.00 | 1.00 |

10 | 0.06 | 1.00 | 1,148.00 | 1.50 | 0.08 | 0.43 | 2.00 |

11 | 0.06 | 1.50 | 1,005.00 | 1.00 | 0.16 | 0.43 | 1.00 |

12 | 0.06 | 1.50 | 1,071.00 | 0.50 | 0.08 | 2.33 | 3.00 |

13 | 0.07 | 0.50 | 1,071.00 | 1.50 | 0.08 | 1.00 | 1.00 |

14 | 0.07 | 0.50 | 1,148.00 | 1.00 | 0.12 | 0.43 | 3.00 |

15 | 0.07 | 1.00 | 1,005.00 | 1.00 | 0.08 | 2.33 | 2.00 |

16 | 0.07 | 1.00 | 1,071.00 | 0.50 | 0.16 | 0.43 | 3.00 |

17 | 0.07 | 1.50 | 1,005.00 | 0.50 | 0.12 | 1.00 | 2.00 |

18 | 0.07 | 1.50 | 1,148.00 | 1.50 | 0.16 | 2.33 | 1.00 |

No . | D
. | P
. | ρ_{s}
. | α
. | Q_{s}
. | β
. | Error column . |
---|---|---|---|---|---|---|---|

(m) . | (m) . | (kg/m^{3})
. | (Q_{1}/Q_{2})
. | (m^{3}/h)
. | (Q_{3}/Q_{4})
. | ||

1 | 0.05 | 0.50 | 1,005.00 | 0.50 | 0.08 | 0.43 | 1.00 |

2 | 0.05 | 0.50 | 1,071.00 | 1.00 | 0.16 | 2.33 | 2.00 |

3 | 0.05 | 1.00 | 1,005.00 | 1.50 | 0.16 | 1.00 | 3.00 |

4 | 0.05 | 1.00 | 1,148.00 | 0.50 | 0.12 | 2.33 | 1.00 |

5 | 0.05 | 1.50 | 1,071.00 | 1.50 | 0.12 | 0.43 | 2.00 |

6 | 0.05 | 1.50 | 1,148.00 | 1.00 | 0.08 | 1.00 | 3.00 |

7 | 0.06 | 0.50 | 1,005.00 | 1.50 | 0.12 | 2.33 | 3.00 |

8 | 0.06 | 0.50 | 1,148.00 | 0.50 | 0.16 | 1.00 | 2.00 |

9 | 0.06 | 1.00 | 1,071.00 | 1.00 | 0.12 | 1.00 | 1.00 |

10 | 0.06 | 1.00 | 1,148.00 | 1.50 | 0.08 | 0.43 | 2.00 |

11 | 0.06 | 1.50 | 1,005.00 | 1.00 | 0.16 | 0.43 | 1.00 |

12 | 0.06 | 1.50 | 1,071.00 | 0.50 | 0.08 | 2.33 | 3.00 |

13 | 0.07 | 0.50 | 1,071.00 | 1.50 | 0.08 | 1.00 | 1.00 |

14 | 0.07 | 0.50 | 1,148.00 | 1.00 | 0.12 | 0.43 | 3.00 |

15 | 0.07 | 1.00 | 1,005.00 | 1.00 | 0.08 | 2.33 | 2.00 |

16 | 0.07 | 1.00 | 1,071.00 | 0.50 | 0.16 | 0.43 | 3.00 |

17 | 0.07 | 1.50 | 1,005.00 | 0.50 | 0.12 | 1.00 | 2.00 |

18 | 0.07 | 1.50 | 1,148.00 | 1.50 | 0.16 | 2.33 | 1.00 |

### Numerical simulation

#### Geometric model

*D*(DN60, DN70, and DN80),

*l*

_{1}= 2.5 m upstream, and

*l*

_{4}= 7 m downstream of the cross joint, mutually perpendicular pipes, and tracer injection pipes. The west inlet flow rate is

*v*

_{1}, the flow rate is

*Q*

_{1}, the north inlet flow rate is

*v*

_{2}, the flow rate is

*Q*

_{2}, and the brine inlet flow rate is

*Q*

_{s}. In the experiment, the conductivity of the brine mixture in the sampling tube is measured, and the mixing of the brine and tap water in the tube is determined according to the conductivity and the concentration of the salt solution. The mixing speed and conductivity in the pipe depend on the flow conditions at the inlet and in the pipe.

#### Governing equation

*et al.*2018; Tang

*et al.*2019), which can simulate the interaction between each component of the mixture or with other phases. Therefore, this model is widely used to solve the problem of molecular exchange between two substances. Its continuity equation is as follows (Zhang

*et al.*2022):where

*k*is the number of species in the model, and , , and represent the volume fraction, density and average velocity of species

*k*, respectively. The momentum equation is:where

*p*is the pressure, is the molecular momentum, is the turbulent stress,

*g*is the gravitational acceleration, and

*F*is the volume force. The energy equation is developed as follows:

*T*represent the sensible enthalpy, effective thermal conductivity, and temperature of species

*k*, respectively. In addition, according to the law of conservation of components, the mass conservation equation of species

*k*can be obtained from Equation (7):where is the diffusion coefficient.

#### Solution settings

The flow state in the pipeline is turbulent, so the diffusion process of brine in water is regarded as a turbulent mixing process. The turbulence model of *k*-*ε* is adopted, and gravity acceleration is applied in the *z*-direction. The fluid materials in the numerical model are water and salt water. The model adopts the component transport model, and the wall is set as the standard nonslip wall condition. This study refers to the simulation methods of some classic fluid dynamics cases. The main parameters in the solver are set as follows: the gradient based on the least squares element is selected; the second-order upwind is used to calculate momentum, pressure, and components. In addition, the turbulence kinetic energy and turbulence dissipation rate is of the first-order upwind. The first-order implicit formula is selected for the transient equation. The initial water mass fraction is set to 1, the convergence standard is 10^{−6}, and the number of iterations is not less than 6,000.

### Mesh independence check

*y*= 0.05 m, the maximum error is 1.358%, the minimum error is 0.0027%, and the average error is 0.2817%. The calculated error is less than 2%. Therefore, the grid size of 5 mm can meet the requirements of simulation accuracy.

### Comparison of numerical and experimental results

#### Relationship between conductivity and concentration

*R*

^{2}of which is 0.9950, as shown in Figure 7.where

*C*is the concentration of a salt solution (g/L), and

*EC*is the conductivity of a salt solution (mS/cm).

#### Conversion between solution concentration and mass fraction

*C*

_{0}and the density is

*ρ*

_{0}. The volume of salt used in this solution is much smaller than that of water, so the change of solution volume caused by dissolution is negligible, and so the density of the salt solution is:Suppose that a unit volume of brine is mixed with water of

*V*

_{0}volume, and the solution concentration of brine and water after mixing is

*C*

_{1}; the calculation formula is:where

*ρ*

_{w}is the density of water. Set up

*ρ*

_{1}as the density after mixing, then the mass fraction expression of brine in the mixed solution is:

*y*= 0.5 m; the minimum error is 1.51%, and the average error is 6.30%. Under condition 20, the maximum error between numerical results and experimental results is 25.37%, at

*y*= 1 m; the minimum error is 2.18%, and the average error is 8.61%. In conclusion, the average error between the simulation results and the experimental results is less than 10%, which verifies that the simulation method in this study is feasible for simulating the mixing process of brine and water.

## RESULTS AND DISCUSSION

### Orthogonal test results and analysis

To inspect the pipe diameter, inlet flow rate, outlet flow rate, damage point, density of brine, and flow of brine effect on the effective mixing distance is significant; in this study, the pipe size in the actual project is used, and the flow size of the municipal water supply pipe is considered. The orthogonal working conditions are simulated and calculated, and the results are derived. The uniform mixing position in the two outlet directions is calculated according to Equation (2), and the calculation results are shown in Table 3.

No . | L_{EML} of the east exit direction (m)
. | L_{EML} of the south exit direction (m)
. | No . | L_{EML} of the east exit direction (m)
. | L_{EML} of the south exit direction (m)
. |
---|---|---|---|---|---|

1 | 0.74 | 4.38 | 10 | 1.36 | 4.06 |

2 | 5.53 | 0.95 | 11 | 0.98 | 3.45 |

3 | 4.20 | 2.01 | 12 | 0.94 | 0.43 |

4 | 0.69 | 0.42 | 13 | 6.95 | 3.71 |

5 | 0.88 | 2.80 | 14 | 1.54 | 4.81 |

6 | 2.45 | 1.72 | 15 | 7.78 | 1.30 |

7 | 7.49 | 1.07 | 16 | 0.87 | 5.31 |

8 | 1.80 | 1.53 | 17 | 2.62 | 2.21 |

9 | 3.19 | 2.68 | 18 | 8.80 | 1.27 |

No . | L_{EML} of the east exit direction (m)
. | L_{EML} of the south exit direction (m)
. | No . | L_{EML} of the east exit direction (m)
. | L_{EML} of the south exit direction (m)
. |
---|---|---|---|---|---|

1 | 0.74 | 4.38 | 10 | 1.36 | 4.06 |

2 | 5.53 | 0.95 | 11 | 0.98 | 3.45 |

3 | 4.20 | 2.01 | 12 | 0.94 | 0.43 |

4 | 0.69 | 0.42 | 13 | 6.95 | 3.71 |

5 | 0.88 | 2.80 | 14 | 1.54 | 4.81 |

6 | 2.45 | 1.72 | 15 | 7.78 | 1.30 |

7 | 7.49 | 1.07 | 16 | 0.87 | 5.31 |

8 | 1.80 | 1.53 | 17 | 2.62 | 2.21 |

9 | 3.19 | 2.68 | 18 | 8.80 | 1.27 |

#### Range analysis

Table 4 is the range analysis table, which shows the influence of each test factor on *L*_{EML} in the east outlet direction. *R* is the range. The greater the range, the greater the influence of test factors on test results. The order of influence is outlet flow ratio > inlet flow ratio > pipe diameter > damage point position > density of brine > flow of brine. According to the value of *R*, the *R* of the three factors of outlet flow ratio, inlet flow ratio, and pipe diameter is greater than the range of the blank column, indicating that these three factors have a significant impact on *L*_{EML} in the east outlet direction. The *R* of the damage point location, density of brine, and flow of brine is less than the range of the blank column, indicating that these three factors have no significant impact on the *L*_{EML} in the east outlet direction.

No . | D (m)
. | I (m)
. | ρ_{s} (kg/m^{3})
. | α
. | Q_{s} (m^{3}·h^{−1})
. | β
. | Empty column . |
---|---|---|---|---|---|---|---|

K_{1} | 14.49 | 24.05 | 23.81 | 7.66 | 20.22 | 6.37 | 5.00 |

K_{2} | 15.76 | 18.09 | 18.36 | 21.47 | 16.41 | 21.21 | 12.00 |

K_{3} | 28.56 | 16.67 | 16.64 | 29.68 | 22.18 | 31.23 | 18.00 |

k_{1} | 2.42 | 4.01 | 3.97 | 1.28 | 3.37 | 1.06 | 0.83 |

k_{2} | 2.63 | 3.02 | 3.06 | 3.58 | 2.74 | 3.54 | 2.00 |

k_{3} | 4.76 | 2.78 | 2.77 | 4.95 | 3.70 | 5.21 | 3.00 |

R | 2.35 | 1.23 | 1.20 | 3.67 | 0.96 | 4.14 | 2.17 |

No . | D (m)
. | I (m)
. | ρ_{s} (kg/m^{3})
. | α
. | Q_{s} (m^{3}·h^{−1})
. | β
. | Empty column . |
---|---|---|---|---|---|---|---|

K_{1} | 14.49 | 24.05 | 23.81 | 7.66 | 20.22 | 6.37 | 5.00 |

K_{2} | 15.76 | 18.09 | 18.36 | 21.47 | 16.41 | 21.21 | 12.00 |

K_{3} | 28.56 | 16.67 | 16.64 | 29.68 | 22.18 | 31.23 | 18.00 |

k_{1} | 2.42 | 4.01 | 3.97 | 1.28 | 3.37 | 1.06 | 0.83 |

k_{2} | 2.63 | 3.02 | 3.06 | 3.58 | 2.74 | 3.54 | 2.00 |

k_{3} | 4.76 | 2.78 | 2.77 | 4.95 | 3.70 | 5.21 | 3.00 |

R | 2.35 | 1.23 | 1.20 | 3.67 | 0.96 | 4.14 | 2.17 |

Table 5 shows that the influence degree of each test factor on *L*_{EML} in the south outlet direction is in the order of outlet flow ratio > pipe diameter > damage point location > density of brine > flow of brine > inlet flow ratio. Comparing the *R*, it is found that the *R* of outlet flow ratio, pipe diameter, and damage point position are greater than those in the blank column. The *R* of density of brine, flow of brine and inlet flow ratio are less than those in the empty column, so the outlet flow ratio, pipe diameter, and damage point position are the main factors. In contrast, the secondary factors are the density of brine, flow of brine, and inlet flow ratio.

No . | D (m)
. | l (m)
. | ρ_{s} (kg/m^{3})
. | α
. | Q_{s} (m^{3}·h^{−1})
. | β
. | Empty column . |
---|---|---|---|---|---|---|---|

K_{1} | 12.28 | 16.45 | 14.42 | 14.28 | 15.6 | 24.81 | 15.91 |

K_{2} | 13.22 | 15.78 | 15.88 | 14.91 | 13.99 | 13.86 | 12.85 |

K_{3} | 18.61 | 11.88 | 13.81 | 14.92 | 14.52 | 5.44 | 15.35 |

k_{1} | 2.05 | 2.74 | 2.40 | 2.38 | 2.60 | 4.14 | 2.65 |

k_{2} | 2.20 | 2.63 | 2.65 | 2.49 | 2.33 | 2.31 | 2.14 |

k_{3} | 3.10 | 1.98 | 2.30 | 2.49 | 2.42 | 0.91 | 2.56 |

R | 1.06 | 0.76 | 0.35 | 0.11 | 0.27 | 3.23 | 0.51 |

No . | D (m)
. | l (m)
. | ρ_{s} (kg/m^{3})
. | α
. | Q_{s} (m^{3}·h^{−1})
. | β
. | Empty column . |
---|---|---|---|---|---|---|---|

K_{1} | 12.28 | 16.45 | 14.42 | 14.28 | 15.6 | 24.81 | 15.91 |

K_{2} | 13.22 | 15.78 | 15.88 | 14.91 | 13.99 | 13.86 | 12.85 |

K_{3} | 18.61 | 11.88 | 13.81 | 14.92 | 14.52 | 5.44 | 15.35 |

k_{1} | 2.05 | 2.74 | 2.40 | 2.38 | 2.60 | 4.14 | 2.65 |

k_{2} | 2.20 | 2.63 | 2.65 | 2.49 | 2.33 | 2.31 | 2.14 |

k_{3} | 3.10 | 1.98 | 2.30 | 2.49 | 2.42 | 0.91 | 2.56 |

R | 1.06 | 0.76 | 0.35 | 0.11 | 0.27 | 3.23 | 0.51 |

*K*_{1}, *K*_{2}, and *K*_{3} are respectively the sum of *L*_{EML} corresponding to three levels in each factor, and *k*_{1}, *k*_{2}, and *k*_{3} are respectively the average values of *L*_{EML} corresponding to three groups in each factor.

*L*

_{EML}, based on the range analysis and calculation results, take the level of each factor as the abscissa, and take the mean value of the

*L*

_{EML}corresponding to different levels as the ordinate, and draw the influence chart of other groups on the adequate mixing, as shown in Figure 9(a) and 9(b).

Figure 9(a) shows the *L*_{EML} in the east outlet direction. The *L*_{EML} is positively related to the pipe diameter, inlet flow ratio, and outlet flow ratio. In addition, the *L*_{EML} is negatively associated with the location of the damage point and density of brine, and not related to the flow of brine. Figure 9(b) shows that the *L*_{EML} in the south outlet direction is positively associated with the pipe diameter, negatively related to the location of the damage point and the outlet flow ratio, but not related to the density of brine, flow of brine, and inlet flow ratio.

### Analysis of influencing factors in the mixing process

In this test, first, orthogonal tests are conducted on equal-diameter cross joints with different pipe diameters. Range analysis is applied to the results of orthogonal tests to determine the primary and secondary factors. Based on the orthogonal experimental results, the factors that significantly impact the effective mixing results are selected for analysis. Next, the pipe diameter, inlet flow ratio, outlet flow ratio, and the location of the damage point are analysed. The method of control variables is used to confirm further the magnitude and trend of the impact of these factors on the effective mixing of uniform length.

#### Influence of flow ratio on mixing effect

To study the influence of the inlet flow ratio and outlet flow ratio on the mixing, transportation, and diffusion of brine in the pipeline, the flow at the inlet and outlet is designed. Different inlet flow ratios and outlet flow ratios are designed. This test is divided into three groups of complete analysis tests according to the pipe diameter size. Each group of tests has 5×5 = 25 working conditions, and the three groups of tests have 75 working conditions in total. See Table 6 for the simulation test conditions and results.

Group . | D (m)
. | α
. | β
. | Total . |
---|---|---|---|---|

1 | 0.05 | 0.50, 0.70, 1.00, 1.50, 2.00 | 0.43, 0.67, 1.00, 1.50, 2.33 | 25.00 |

2 | 0.06 | 0.50, 0.70, 1.00, 1.50, 2.00 | 0.43, 0.67, 1.00, 1.50, 2.33 | 25.00 |

3 | 0.07 | 0.50, 0.70, 1.00, 1.50, 2.00 | 0.43, 0.67, 1.00, 1.50, 2.33 | 25.00 |

Group . | D (m)
. | α
. | β
. | Total . |
---|---|---|---|---|

1 | 0.05 | 0.50, 0.70, 1.00, 1.50, 2.00 | 0.43, 0.67, 1.00, 1.50, 2.33 | 25.00 |

2 | 0.06 | 0.50, 0.70, 1.00, 1.50, 2.00 | 0.43, 0.67, 1.00, 1.50, 2.33 | 25.00 |

3 | 0.07 | 0.50, 0.70, 1.00, 1.50, 2.00 | 0.43, 0.67, 1.00, 1.50, 2.33 | 25.00 |

##### Influence of inlet flow ratio on mixing effect

*L*

_{EML}in the east outlet direction with the inlet flow ratio when the outlet flow ratio is unchanged. It can be seen from Figure 10(a) that when the outlet flow ratio is less than 1, the

*L*

_{EML}in the east outlet direction increases with the increase of the inlet flow ratio, and the growth rate gradually increases; otherwise, when the inlet flow ratio is greater than 1, the growth rate of

*L*

_{EML}gradually decreases as shown in Figure 10(b).

When the difference between the inlet flow ratio and the outlet flow ratio is significant, the influence on the mixing rate is greater. When the inlet flow ratio and outlet flow ratio are both high or low, the impact on the mixing speed is low.

##### Influence of outlet flow ratio on mixing effect

*L*

_{EML}in the east outlet direction with the outlet flow ratio. The

*L*

_{EML}in the east outlet direction increases with the increase of the outlet flow ratio, and the growth rate gradually decreases. The change rule is consistent with the impact of the inlet flow ratio on the

*L*

_{EML}in the east outlet direction. Figure 11(b) describes the changing trend of

*L*

_{EML}in the south outlet direction with the outlet flow ratio. It can be seen that the

*L*

_{EML}in the south outlet direction decreases with the increase of the outlet flow ratio, and the decreasing trend gradually slows down. The change in outlet flow ratio affects the flow distribution and brine distribution of the two outlets. The tracer distribution is shown in Figure 12: the tracer concentration at the two outlets is consistent with the trend of

*L*

_{EML}. In addition, an ‘impact surface’ will be formed at the node, on which molecular exchange can be generated.

#### Effect of pipe diameter on mixing

*L*

_{EML}is shortened.

#### Effect of the location of the damage point on mixing

*L*

_{EML}with the increase of the location distance of the failure point. The farther the damage point is from the node, the more thoroughly the brine will be mixed when it flows through the cross node, so the

*L*

_{EML}of the two outlets will be reduced. When the mixed solution and tap water flow through the cross node, an ‘impact surface’ will be formed at the node, where molecular exchange can occur. One side of the ‘impact surface’ flows into the east outlet direction, while the other side flows into the south outlet direction after 90° deflection. Therefore, the mixing characteristics in the two outlet pipes are different. Therefore, the

*L*

_{EML}in the south outlet direction is smaller than in the east.

### Application formula derivation of the *L*_{EML}

*L*

_{EML}under different pipeline structural parameters and hydraulic conditions, the dimensional analysis method is used to establish the relationship between them and the calculation formula applicable to various conditions (Bijankhan & Ferro 2020). Based on analysing the factors affecting the practical uniform mixing length, the seven physical quantities of

*L*

_{EML}, pipe diameter, inlet flow ratio, outlet flow ratio, distance from damage point position, flow of brine and density of brine, were selected to obtain the size relationship:where

*D*,

*g*, and

*ρ*

_{s}are fundamental physical quantities, Equation (15) can be obtained from the

*π*theorem:

*a*,

*b*,

*c*,

*d*,

*e*, and

*f*are coefficients, and the values of the coefficients in Equation (19) can be obtained by fitting the simulated data:

*L*

_{EML}is the effective mixing length,

*D*is the pipe diameter,

*g*is the gravitational acceleration,

*ρ*

_{w}is the water density,

*α*is the inlet flow ratio (

*Q*

_{1}/

*Q*

_{2}),

*β*is the outlet flow ratio (

*Q*

_{3}/

*Q*

_{4}),

*P*is the location of the damage point,

*Q*is the rate of flow of brine, and

_{s}*ρ*

_{s}is the density of brine. Equation (20) can be simplified into the following form:

The correlation coefficients *R*^{2} of Equations (21) and (22) are 0.8694 and 0.8738, respectively, exceeding 0.8, indicating a strong correlation between the *L*_{EML} and the six variables in the equation. As shown in Figures 15, parameters of each working condition were substituted into Equation (21), and the results of Equation (21) were compared with the numerical simulation results. The results show that the two values are close, and the maximum relative error is 9.56%, the minimum relative error is 0.09%, and the average error is 4.01%. Therefore, it is proved that Equation (21) has high calculation accuracy and can accurately predict the *L*_{EML}.

## CONCLUSION

In this paper, the mixing characteristics of pipe flow with cross joints are investigated using numerical simulation and experimental measurement. It is verified that the simulation method is feasible for simulating the mixing process of salt water and water at the cross node. The effects of pipe diameter, inlet flow ratio, outlet flow ratio, location of damage point, rate of flow of brine, and tracer density on *L*_{EML} at the node-outlet were studied using the orthogonal test and control variable method. In addition, based on the above factors, a dimensionless equation representing *L*_{EML} is fitted by dimensional analysis. The main conclusions are as follows:

- (1)
Inlet flow ratio

*α*, outlet flow ratio*β*, and pipe diameter*D*have significant effects on*L*_{EML}in the east outlet direction of the cross joint, while other factors have no significant effects on it. For the south outlet direction of the cross joint, only the outlet flow ratio*β*and pipe diameter*D*have significant effects on*L*_{EML}. - (2)
The

*L*_{EML}at the east outlet of the cross joint varies with the inlet flow ratio*α*, outlet flow ratio*β*, and pipe diameter*D*, increasing and decreasing with the rise of the density of invasive pollutants and the distance from the damage point to the node.*L*_{EML}in the south outlet direction of the cross joint increases with the increase of pipe diameter*D*and decreases with the rise of outlet flow ratio*β*. - (3)
When the difference between the inlet flow ratio and the outlet flow ratio is significant, the joint mixing rate is higher. When the inlet flow ratio and outlet flow ratio are both high or low, the influence on the mixing speed is low.

## ACKNOWLEDGEMENTS

This research was supported by the National Key Research and Development Program of China (No. 2022YFC3801002), the National Natural Science Foundation of China (No. 51978630), the Program for Innovative Research Team (in Science and Technology) in University of Henan Province (No.23IRTSTHN004), the Key Scientific Research Projects of Colleges and Universities in Henan Province (No. 22A570009), the Open Research Fund of Key Laboratory of Water-Saving Irrigation Engineering of the Ministry of Agriculture and Rural Affairs (MARA) (No. FIRI2021020201), Open Research Fund of MWR Key Laboratory of Lower Yellow River Channel and Estuarine Regulation (No. LYRCER202202), the Fundamental Research and Cultivation of Young Teachers of Zhengzhou University in 2022 (No. JC22550027), the First-Class Special Fund of Yellow River Laboratory (Zhengzhou University) (No. YRL22IR11), Special Scientific Research Project of Yellow River Water Resources Protection Institute (No. KYY-KYZX-2022-01).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.