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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 LEML 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 LEML in the east outlet of the cross joint. Outlet flow ratio and pipe diameter significantly affect LEML in the cross joint's south outlet direction. In addition, the dimensionless relationship equation representing LEML is fitted through dimensional analysis.

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  • 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

Graphical Abstract
Graphical Abstract
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Graphical Abstract
Graphical Abstract
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cross joint, dimensional analysis, migration and diffusion, numerical analysis, orthogonal test, turbulent mixing
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D

pipe diameter

Di

values in row i of D in Table 1 

Q1

west inlet flow

Q2

north inlet flow

Q3

east outlet flow

Q4

south outlet flow

α

inlet flow ratio(Q1/Q2)

αi

values in row i of α in Table 1 

β

outlet flow ratio (Q3/Q4)

βi

values in row i of β in Table 1 

P

location of damage point

Pi

values in row i of P in Table 1 

Qs

flow of brine

Qsi

values in row i of Qs in Table 1 

ρs

density of brine

ρsi

values in row i of ρs in Table 1 

Table 1

Factors and levels of orthogonal test

CaseDPρsαQsβ
(m)(m)(kg/m3)(Q1/Q2)(m3/h)(Q3/Q4)
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 
CaseDPρsαQsβ
(m)(m)(kg/m3)(Q1/Q2)(m3/h)(Q3/Q4)
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 
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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.

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Experimental study

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Layout of test platform

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To study the migration, diffusion, and mixing of point source pollutants near the cross joint, a cross joint pipeline test system was designed and built in Zhengzhou University. As shown in Figures 1 and 2, the experimental device mainly consists of a clean water supply system, tracer injection system, flow control system, and test pipeline. Due to the complex topological structure of the water supply network, many types of joints, and the impact between joints, this study only considers the case of cross joints, and the migration, diffusion, and mixing of point source pollution at this joint. Considering tracers' safety, non-toxicity, and accessibility, sodium chloride is selected as the tracer to simulate the point source diffusion behaviour of water-soluble pollutants in the pipeline.
Figure 1
Experimental layout: (a) electromagnetic flowmeter; (b) tracer liquid storage tank; (c) liquid injector; (d) north inlet float flowmeter; (e) ball valve; (f) outlet float flowmeter.
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Experimental layout: (a) electromagnetic flowmeter; (b) tracer liquid storage tank; (c) liquid injector; (d) north inlet float flowmeter; (e) ball valve; (f) outlet float flowmeter.

Figure 1
Experimental layout: (a) electromagnetic flowmeter; (b) tracer liquid storage tank; (c) liquid injector; (d) north inlet float flowmeter; (e) ball valve; (f) outlet float flowmeter.
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Experimental layout: (a) electromagnetic flowmeter; (b) tracer liquid storage tank; (c) liquid injector; (d) north inlet float flowmeter; (e) ball valve; (f) outlet float flowmeter.

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Figure 2
Top view at cross joint: (g) pipeline damage point; (h) submersible pump; (i) sampling pipe section; (k) west entrance; (l) north entrance; (m) south exit; (n) east exit.
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Top view at cross joint: (g) pipeline damage point; (h) submersible pump; (i) sampling pipe section; (k) west entrance; (l) north entrance; (m) south exit; (n) east exit.

Figure 2
Top view at cross joint: (g) pipeline damage point; (h) submersible pump; (i) sampling pipe section; (k) west entrance; (l) north entrance; (m) south exit; (n) east exit.
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Top view at cross joint: (g) pipeline damage point; (h) submersible pump; (i) sampling pipe section; (k) west entrance; (l) north entrance; (m) south exit; (n) east exit.

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Experimental pipeline and data acquisition

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The experimental pipeline includes polyethylene and PVC pipe, and the total pipeline length is 20 m. The sizes of tubes are DN60, DN70, and DN80. The downstream part of the cross joint is the sampling pipe, as shown in Figure 3. To obtain the tracer concentration at each section of the pipe under each working condition, a series of measuring points is set in the sampling pipe section. A group of measuring points is assigned within the first 5 m range, and then a group of measuring points is assessed. Each measuring point group is located on the upper and lower wall of the sampling pipe as shown in Figure 3(b). Each measuring point comprises an L-shaped connector, sampling tube, and bottle. The mixed liquid at this position is obtained from each measuring point. The conductivity of the mixed fluid is measured with a conductivity meter, and the concentration of the salt solution is obtained according to the relationship between the conductivity and the concentration of the salt solution.
Figure 3
Schematic diagram of sampling pipeline: (a) layout of measuring points; (b) section at measuring points; (c) experimental layout of measuring points.
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Schematic diagram of sampling pipeline: (a) layout of measuring points; (b) section at measuring points; (c) experimental layout of measuring points.

Figure 3
Schematic diagram of sampling pipeline: (a) layout of measuring points; (b) section at measuring points; (c) experimental layout of measuring points.
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Schematic diagram of sampling pipeline: (a) layout of measuring points; (b) section at measuring points; (c) experimental layout of measuring points.

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Definition of evaluation indicators

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The evaluation index is to quantitatively study the diffusion of point source pollution and provide a theoretical basis for studying the main influencing parameters affecting the diffusion of point source pollution. Currently, the mature method is to judge the mixing effect according to the coefficient of variation. Generally, it is considered that 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:
(1)
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.
(2)
where is the arithmetic square root of the mass fraction of brine at all points on the cross-section:
(3)
where n is the number of nodes on the section, and is the mass fraction at node i.

Orthogonal experimental design

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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 LEML (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 L18(37) is selected, and the number of tests is 18. The experimental scheme after the setting is shown in Table 2.

Table 2

Orthogonal experimental conditions

NoDPρsαQsβError column
(m)(m)(kg/m3)(Q1/Q2)(m3/h)(Q3/Q4)
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 
NoDPρsαQsβError column
(m)(m)(kg/m3)(Q1/Q2)(m3/h)(Q3/Q4)
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 
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Numerical simulation

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Geometric model

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Figure 4 shows the physical model structure of this study. The simulation domain includes equal-diameter cross pipes with diameters of D (DN60, DN70, and DN80), l1 = 2.5 m upstream, and l4 = 7 m downstream of the cross joint, mutually perpendicular pipes, and tracer injection pipes. The west inlet flow rate is v1, the flow rate is Q1, the north inlet flow rate is v2, the flow rate is Q2, and the brine inlet flow rate is Qs. 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.
Figure 4
Physical model: (a) 3D view and (b) side view.
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Physical model: (a) 3D view and (b) side view.

Figure 4
Physical model: (a) 3D view and (b) side view.
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Physical model: (a) 3D view and (b) side view.

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Governing equation

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The pipeline is a mixing process of brine and water. Since the two materials in the mixing process can be mutually soluble in any proportion, a component transport model is adopted (Mohammed 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):
(4)
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:
(5)
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:
(6)
where ⁠, and 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):
(7)
where is the diffusion coefficient.

Solution settings

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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

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To discretize the solution domain, the mesh is used to generate unstructured meshes of tetrahedral elements. Figure 5 shows an example of the mesh structure of the computational domain.
Figure 5
Grid division diagram: (a) expansion layer division; (b) cross joint grid division; (c) damage point grid division.
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Grid division diagram: (a) expansion layer division; (b) cross joint grid division; (c) damage point grid division.

Figure 5
Grid division diagram: (a) expansion layer division; (b) cross joint grid division; (c) damage point grid division.
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Grid division diagram: (a) expansion layer division; (b) cross joint grid division; (c) damage point grid division.

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In this study, to eliminate the influence of grid size on the calculation results, different grid sizes (3 mm, 4 mm, and 5 mm) were simulated, and the corresponding number of grids was 5,772,442, 3,241,392, and 2,133,249. Using the same boundary conditions, by comparing the brine mass fraction at the centreline of the pipe, as shown in Figure 6, the maximum error is at 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.
Figure 6
Mass fraction of brine in the pipeline section along the north outlet.
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Mass fraction of brine in the pipeline section along the north outlet.

Figure 6
Mass fraction of brine in the pipeline section along the north outlet.
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Mass fraction of brine in the pipeline section along the north outlet.

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Comparison of numerical and experimental results

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Relationship between conductivity and concentration

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Under certain conditions, the conductivity of the salt solution is linear with the concentration. To establish the relationship between conductivity and concentration, a series of salt solutions with different concentrations are prepared, and their conductivity is measured respectively to fit a higher correlation equation, R2 of which is 0.9950, as shown in Figure 7.
(8)
where C is the concentration of a salt solution (g/L), and EC is the conductivity of a salt solution (mS/cm).
Figure 7
Relationship between conductivity and concentration of the salt solution.
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Relationship between conductivity and concentration of the salt solution.

Figure 7
Relationship between conductivity and concentration of the salt solution.
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Relationship between conductivity and concentration of the salt solution.

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Conversion between solution concentration and mass fraction

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Assume that the concentration of brine used is C0 and the density is ρ0. 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:
(9)
Suppose that a unit volume of brine is mixed with water of V0 volume, and the solution concentration of brine and water after mixing is C1; the calculation formula is:
(10)
where ρw is the density of water. Set up ρ1 as the density after mixing, then the mass fraction expression of brine in the mixed solution is:
(11)
According to Equation (9), the volume of water in the hybrid solution is:
(12)
Substitute Equation (12) into Equation (11) to obtain the mass fraction of brine in the mixed solution:
(13)
In this test, the brine is transported and diffused radially and axially in the pipeline. In the actual experiment, the conductivity of the mixed solution can be measured. The calculated data in the simulation results is the mass fraction of brine species in the total species. It is necessary to convert the concentration into a mass fraction to compare the simulation results with the experimental results. The relationship between conductivity and concentration obtains the concentration of the mixed solution, and the solute concentration is converted into a mass fraction by Equation (13). Based on working-conditions 9, 14, and 20, the numerical simulation mass fraction results are compared with the experimental results. As shown in Figure 8, each colour curve represents the mass fraction of brine at the upper and lower pipe walls in a working-condition, the curve represents the numerical simulation results, and the point data represents the sampling results of the upper and lower wall surfaces of the test. Under the three working-conditions, the changing trend of the salt water mass fraction at the upper and lower walls of the simulation results along the mixing distance is almost the same as that of the experimental results. Under working-condition 9, the maximum error between the simulation results and the experimental results is 23.16%, at 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.
Figure 8
Comparison of experimental and simulation results under working-conditions 9, 14 and 20.
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Comparison of experimental and simulation results under working-conditions 9, 14 and 20.

Figure 8
Comparison of experimental and simulation results under working-conditions 9, 14 and 20.
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Comparison of experimental and simulation results under working-conditions 9, 14 and 20.

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Orthogonal test results and analysis

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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.

Table 3

Results of orthogonal test

NoLEML of the east exit direction (m)LEML of the south exit direction (m)NoLEML of the east exit direction (m)LEML 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 
NoLEML of the east exit direction (m)LEML of the south exit direction (m)NoLEML of the east exit direction (m)LEML 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 
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Range analysis

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Table 4 is the range analysis table, which shows the influence of each test factor on LEML 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 LEML 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 LEML in the east outlet direction.

Table 4

Range analysis of LEML in the east exit direction

NoD (m)I (m)ρs (kg/m3)αQs (m3·h−1)βEmpty column
K1 14.49 24.05 23.81 7.66 20.22 6.37 5.00 
K2 15.76 18.09 18.36 21.47 16.41 21.21 12.00 
K3 28.56 16.67 16.64 29.68 22.18 31.23 18.00 
k1 2.42 4.01 3.97 1.28 3.37 1.06 0.83 
k2 2.63 3.02 3.06 3.58 2.74 3.54 2.00 
k3 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 
NoD (m)I (m)ρs (kg/m3)αQs (m3·h−1)βEmpty column
K1 14.49 24.05 23.81 7.66 20.22 6.37 5.00 
K2 15.76 18.09 18.36 21.47 16.41 21.21 12.00 
K3 28.56 16.67 16.64 29.68 22.18 31.23 18.00 
k1 2.42 4.01 3.97 1.28 3.37 1.06 0.83 
k2 2.63 3.02 3.06 3.58 2.74 3.54 2.00 
k3 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 
View Large

Table 5 shows that the influence degree of each test factor on LEML 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.

Table 5

Range analysis of LEML in the south exit direction

NoD (m)l (m)ρs (kg/m3)αQs (m3·h−1)βEmpty column
K1 12.28 16.45 14.42 14.28 15.6 24.81 15.91 
K2 13.22 15.78 15.88 14.91 13.99 13.86 12.85 
K3 18.61 11.88 13.81 14.92 14.52 5.44 15.35 
k1 2.05 2.74 2.40 2.38 2.60 4.14 2.65 
k2 2.20 2.63 2.65 2.49 2.33 2.31 2.14 
k3 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 
NoD (m)l (m)ρs (kg/m3)αQs (m3·h−1)βEmpty column
K1 12.28 16.45 14.42 14.28 15.6 24.81 15.91 
K2 13.22 15.78 15.88 14.91 13.99 13.86 12.85 
K3 18.61 11.88 13.81 14.92 14.52 5.44 15.35 
k1 2.05 2.74 2.40 2.38 2.60 4.14 2.65 
k2 2.20 2.63 2.65 2.49 2.33 2.31 2.14 
k3 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 
View Large

K1, K2, and K3 are respectively the sum of LEML corresponding to three levels in each factor, and k1, k2, and k3 are respectively the average values of LEML corresponding to three groups in each factor.

To clearly and intuitively analyse the influence of various factors on the LEML, based on the range analysis and calculation results, take the level of each factor as the abscissa, and take the mean value of the LEML 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
The change curve of LEML with the type and value of factors: (a) LEML in the east exit direction; (b) LEML in the south exit direction.
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The change curve of LEML with the type and value of factors: (a) LEML in the east exit direction; (b) LEML in the south exit direction.

Figure 9
The change curve of LEML with the type and value of factors: (a) LEML in the east exit direction; (b) LEML in the south exit direction.
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The change curve of LEML with the type and value of factors: (a) LEML in the east exit direction; (b) LEML in the south exit direction.

Close modal

Figure 9(a) shows the LEML in the east outlet direction. The LEML is positively related to the pipe diameter, inlet flow ratio, and outlet flow ratio. In addition, the LEML 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 LEML 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

Listen

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

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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.

Table 6

Summary of test conditions affected by flow ratio

GroupD (m)αβTotal
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 
GroupD (m)αβTotal
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 
View Large

Influence of inlet flow ratio on mixing effect
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Figure 10(a) and 10(b) describe the changing trend of the LEML 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 LEML 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 LEML gradually decreases as shown in Figure 10(b).
Figure 10
Impact of inlet flow ratio on LEML in the east outlet direction: (a) β< 1; (b) β > 1.
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Impact of inlet flow ratio on LEML in the east outlet direction: (a) β< 1; (b) β > 1.

Figure 10
Impact of inlet flow ratio on LEML in the east outlet direction: (a) β< 1; (b) β > 1.
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Impact of inlet flow ratio on LEML in the east outlet direction: (a) β< 1; (b) β > 1.

Close modal

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
Listen
Figure 11(a) describes the changing trend of LEML in the east outlet direction with the outlet flow ratio. The LEML 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 LEML in the east outlet direction. Figure 11(b) describes the changing trend of LEML in the south outlet direction with the outlet flow ratio. It can be seen that the LEML 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 LEML. In addition, an ‘impact surface’ will be formed at the node, on which molecular exchange can be generated.
Figure 11
Impact of outlet flow ratio on LEML: (a) LEML in the east exit direction; (b) LEML in the south exit direction.
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Impact of outlet flow ratio on LEML: (a) LEML in the east exit direction; (b) LEML in the south exit direction.

Figure 11
Impact of outlet flow ratio on LEML: (a) LEML in the east exit direction; (b) LEML in the south exit direction.
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Impact of outlet flow ratio on LEML: (a) LEML in the east exit direction; (b) LEML in the south exit direction.

Close modal
Figure 12
Tracer distribution: (a) β = 3/7; (b) β = 1; (c) β = 7/3.
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Tracer distribution: (a) β = 3/7; (b) β = 1; (c) β = 7/3.

Figure 12
Tracer distribution: (a) β = 3/7; (b) β = 1; (c) β = 7/3.
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Tracer distribution: (a) β = 3/7; (b) β = 1; (c) β = 7/3.

Close modal

Effect of pipe diameter on mixing

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In the mixing process, the pollution and water are diffused along the axial and radial directions, respectively. The flow velocity accelerates the brine's radial diffusion while prolonging the brine's axial transport distance. As shown in Figure 13, the smaller the pipe diameter is, the greater the slope of the brine distribution variation coefficient in the pipe section is, and the faster the uniform mixing condition is achieved. When other conditions remain unchanged, the smaller the pipe diameter, the shorter the effective mixing uniform distance, because under the same flow rate, the smaller pipe diameter has a higher velocity, which makes the turbulence intensity of the water-flow stronger, and accelerates the radial mixing rate of the brine, so the LEML is shortened.
Figure 13
Variation trend of variation coefficient of brine distribution section along the pipeline axis under different pipe diameters.
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Variation trend of variation coefficient of brine distribution section along the pipeline axis under different pipe diameters.

Figure 13
Variation trend of variation coefficient of brine distribution section along the pipeline axis under different pipe diameters.
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Variation trend of variation coefficient of brine distribution section along the pipeline axis under different pipe diameters.

Close modal

Effect of the location of the damage point on mixing

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Figure 14 shows, under other conditions, the LEML 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 LEML 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 LEML in the south outlet direction is smaller than in the east.
Figure 14
Effect of the distance of breakage point in front of the node on effective mixed length.
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Effect of the distance of breakage point in front of the node on effective mixed length.

Figure 14
Effect of the distance of breakage point in front of the node on effective mixed length.
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Effect of the distance of breakage point in front of the node on effective mixed length.

Close modal
Figure 15
Comparison between Equation (21) and numerical simulation result LEML in the east exit direction.
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Comparison between Equation (21) and numerical simulation result LEML in the east exit direction.

Figure 15
Comparison between Equation (21) and numerical simulation result LEML in the east exit direction.
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Comparison between Equation (21) and numerical simulation result LEML in the east exit direction.

Close modal

Application formula derivation of the LEML

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To predict the LEML 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 LEML, 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:
(14)
where D, g, and ρs are fundamental physical quantities, Equation (15) can be obtained from the π theorem:
(15)
Through the π theorem, the following equation can be obtained:
(16)
Further, Equation (17) can be obtained from Equation (15) and Equation (16) as follows:
(17)
Equation (17) can be transformed into Equation (18):
(18)
In this, 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:
(19)
Through transformation, the above equation can be transformed into Equation (20):
(20)
where LEML is the effective mixing length, D is the pipe diameter, g is the gravitational acceleration, ρw is the water density, α is the inlet flow ratio (Q1/Q2), β is the outlet flow ratio (Q3/Q4), P is the location of the damage point, Qs is the rate of flow of brine, and ρs is the density of brine. Equation (20) can be simplified into the following form:
(21)
Similarly, the LEML formula of the south exit direction is fitted as follows:
(22)

The correlation coefficients R2 of Equations (21) and (22) are 0.8694 and 0.8738, respectively, exceeding 0.8, indicating a strong correlation between the LEML 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 LEML.

Listen

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 LEML 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 LEML 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 LEML 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 LEML.

  • (2)

    The LEML 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. LEML 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.

Listen

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).

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

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The authors declare there is no conflict.

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© 2023 The Authors
This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY-NC-ND 4.0), which permits copying and redistribution for non-commercial purposes with no derivatives, provided the original work is properly cited (http://creativecommons.org/licenses/by-nc-nd/4.0/)