Recent studies have focused on mixing behavior at cross junctions, and incomplete mixing at cross junctions in water distribution systems was verified. Nevertheless, the research results on mixing at other junction configurations, such as double-Tee junctions, were insufficient. Double-Tee junctions can potentially be misrepresented as cross junctions because of network skeletonization. Hence, the diffusion and dilution of the contaminants at junctions were largely underestimated. To examine the mixing phenomenon and collect accurate mixing data at the double-Tee junction, a series of laboratory experiments was carried out with various Reynolds number ratios at the inlets and outlets combined with different dimensionless connecting pipe lengths (L/D). Results showed that the dimensionless connecting pipe length served an important function in mixing at double-Tee junctions. The cross junction was the special case of the double-Tee joint when L/D=0. The complete mixing state occurred when L/D→∞. The mixing degree of the double-Tee junction was between the cross junction and the complete mixing state. A conceptual model that described the mixing behavior at double-Tee junctions was developed. The model included the use of the dimensionless parameter φ, which defined the degree of departure from complete mixing.

INTRODUCTION

The potential occurrence of accidental or malevolent contamination events that threaten the safety of water distribution systems (WDS) is poorly understood (Song et al. 2009). Understanding how contaminants move through WDS is vital to the optimization of a sensor network and to the development of a corresponding monitoring method when the contamination event occurs (Romero-Gomez et al. 2011). The EPANET (Rossman 2000) is widely used for modeling hydraulic and water quality behavior in WDS. However, the EPANET quality model assumes that mixing at all intersections, such as in cross and Tee junctions, is instantaneous and complete. Fowler & Jones (1991) previously noted that a ‘perfect’ mixing assumption would cause possible errors. Ashgriz et al. (2001) identified incomplete mixing behavior at a junction with low incoming velocities, given that the two incoming flows reflected off each other.

Recent studies reported that incomplete mixing at cross junctions was a common phenomenon (Van Bloemen Waanders et al. 2005; Orear et al. 2005; Romero-Gomez et al. 2006; Ho et al. 2006). Van Bloemen Waanders et al. (2005) performed a cross junction mixing experiment, where the Reynolds numbers at all four pipe legs are found to be similar (approximately 44,000). The experimental results show that 86–88% of a NaCl tracer is discharged from the north outlet, which is adjacent to the tracer inlet.

To further recognize the mixed features of the junctions, comprehensive computational and experimental investigations were conducted by several authors, including Romero-Gomez et al. (2008), Austin et al. (2008), and Choi et al. (2008). McKenna et al. (2007) found that the Reynolds number ratio of inlets greatly affected the mixing result at junctions. Austin et al. (2008) found that complete mixing potentially creates considerable errors of contaminant concentrations at each outlet branch because of the bifurcation of the incoming flows. Ho (2008) described a bulk-advective mixing (BAM) model which ignores the impinging interface instability and tracer diffusion of the flows at the junction. Ho & O'Rear (2009) also introduced a bulk-advective mixing wrap (BAM-WRAP) model that extends to pipes with unequal sizes. The factors that affect cross mixing were studied by Yu et al. (2014), who concluded that the Reynolds number between two inlets is the most important factor, followed by the pipe diameter ratio.

The computational fluid dynamics (CFD) approach is applied to explore the mixing mechanism at the double-Tee junctions. Webb et al. (2006, 2007) have investigated the mixing phenomenon at double-Tee junctions with a 2.5 diameter connecting length between two T-fittings, and simulation results for the dimensionless concentration of two outlets were 0.59 and 0.41, respectively. These values differ considerably from those obtained in this paper.

The actual municipal water supply system inevitably contains many cross and Tee junctions (Song et al. 2009). However, because comprehensive and applicable mixing models of multi-in and multi-out junctions are not yet established, the complete mixing assumption is still used at these junctions. Compared to that at cross junctions, the solute mixing at double-Tee junctions is insufficiently studied at the time of this research. Thus, we aimed to investigate the mixing behavior at double-Tee junctions through experiments using equal pipe sizes under variable flow conditions. We also aimed to develop a conceptual model for predicting post-mixing solute concentrations at outlets.

METHODOLOGY: MODEL AND EXPERIMENTS

Dimensionless mixing parameters

According to Plesniak & Cusano (2005), complete mixing happens when (L = pipe length; D = pipe diameters), which means that the downstream length is more than 10 times that of the pipe diameters for single-Tee junctions. Given that the connecting pipe length between two T-fittings, the branches are located at different sides along the main pipe, the diameters of the four legs are the same, and a typical double-Tee hydraulic configuration is generated. The geographic notations (W, S, E, and N) are presented in Figure 1, and such notations indicate the flow directions of the west inlet, south inlet, east outlet, and north outlet, respectively. Shao et al. (2014) have studied the flow directions of the other two situations, such as the inflows from the south (north) and north (west) inlets, which flow out to the west (east) and east (south) outlets in the configuration. The dimensionless concentration was introduced to show the mixing degree (Romero Gomez et al. 2006) and was defined as 
formula
1
where is the dimensionless concentration of the east or north outlets. and are the concentrations of the south and west inlets, respectively. C = concentrations at each outlet. The mass fraction (MF) is defined to describe the solute at the east or north outlets, as follows: 
formula
2
where MF is the mass fraction of NaCl at the two outlets; , , , and are the dimensionless concentrations of the west and south inlets, and north and east outlets, respectively. Q is the flow rate at the inlets and outlets. Based on Equation (1), and . Thus, Equation (3) can be obtained as follows: 
formula
3
Figure 1

Sketch of the experimental system, geographic notations, and photograph of the double-Tee junction.

Figure 1

Sketch of the experimental system, geographic notations, and photograph of the double-Tee junction.

Experimental method

To better understand the mixing phenomenon at double-Tee junctions, four different L/D values were considered: . Yu et al. (2014) confirmed that the inlet Reynolds number ratio () and outlet Reynolds number ratio () greatly affect the mixing. Thus, and ReN/E were also investigated. The Re of the four pipes (ReS, ReW, ReE, and ReN) were controlled and were in the range 12,700–50,000 in the experiments. A data matrix was collected by investigating the seven different and ReN/E values, including 0.25, 0.5, 0.67, 1.0, 1.5, 2.0, and 4.0 for the four cases with different L/D values. In total, 196 (7 × 7 × 4) data points were collected. In addition, every experiment was repeated three times to obtain the average results, thereby resulting in 588 experimental runs.

The experimental setup included water supply and pipe systems (Figure 1). The system could adjust the water head from 6.0 to 7.5 m by supplying stable pressure. The pipe system comprises two pressure sensors, four flow sensors, four sampling points, a double-Tee joint, and four diaphragm valves. The conductivity meter was calibrated against a conductivity standard (NaCl, 1,000 μS/cm, 25 °C) before use. The inlet and outlet pipes of the double-Tee junctions are 60 and 100 pipe diameters, respectively. Food grade NaCl was used as the tracer, and the conductivity of clean water ranged from 253 to 330 μS/cm. The NaCl mixed with clean water in Tank 2 (60 g of NaCl dissolved in 100 L of water) produced a final conductivity of 1,550 ± 100 μS/cm.

RESULTS AND DISCUSSION

Experimental error controls

A series of measures were taken to ensure the accuracy of the experimental data. The high-level tanks in this experiment provided stable pressure. The pressure (P) on a typical experimental run was kPa with a minor fluctuation of approximately around the average pressure at the west inlet. During the experimental runs, the flow rates and pressure were monitored constantly. Water samples were collected, and the data were recorded only after the experimental system had been running for 3 min. The percent mass fraction error (PMFE), which was used to check the experimental accuracy, was defined by Mckenna et al. (2007) as follows: 
formula
4
Each experimental case was independently repeated three times by turning off the power of the experimental system. Through the improvements, the average PMFE of this experiment was 1.2%. The experimental results with L/D = 2.5 are listed in Table 1.
Table 1

List of experimental data, including the experimental conditions, and the results for the dimensionless connecting pipe length of L = 2.5D

ReW/SReN/EC*NMFNMass balance error (%)Flow balance error (%)PMFE
0.25 0.25 0.38 0.38 0.2 0.2 −0.1 
0.24 0.57 0.38 0.72 1.6 0.1 3.1 
0.25 0.67 0.39 0.79 1.5 0.1 2.9 
0.24 1.00 0.35 0.89 1.7 0.3 3.1 
0.25 1.49 0.31 0.95 1.7 0.6 2.8 
0.25 2.00 0.29 0.97 1.5 0.7 2.3 
0.25 3.98 0.25 1.00 1.8 1.0 2.7 
0.50 0.25 0.65 0.38 0.4 0.1 0.5 
0.50 0.50 0.61 0.62 0.2 0.2 0.2 
0.50 0.67 0.60 0.72 0.8 0.0 1.3 
0.50 1.00 0.55 0.83 1.9 0.1 3.0 
0.50 1.51 0.50 0.91 0.5 0.2 0.6 
0.50 1.99 0.47 0.94 0.7 0.5 0.8 
0.50 3.94 0.41 0.99 1.3 0.7 1.6 
0.67 0.25 0.71 0.36 0.4 0.0 −0.6 
0.67 0.50 0.70 0.57 1.5 0.3 2.4 
0.66 0.68 0.66 0.67 0.5 0.2 0.8 
0.67 1.00 0.64 0.80 0.3 0.1 0.5 
0.67 1.52 0.59 0.88 0.9 0.3 1.2 
0.67 2.02 0.55 0.92 1.0 0.5 1.3 
0.66 4.00 0.48 0.97 1.1 0.8 1.3 
1.00 0.25 0.71 0.28 0.4 0.5 −0.3 
0.99 0.50 0.76 0.51 0.3 0.4 0.6 
1.00 0.68 0.76 0.61 1.0 0.2 1.4 
1.00 0.99 0.75 0.75 0.4 0.1 0.5 
0.99 1.49 0.69 0.82 0.3 0.1 0.3 
1.00 2.02 0.67 0.90 0.3 0.4 0.3 
1.00 4.00 0.59 0.96 0.7 0.5 0.8 
1.49 0.25 0.72 0.24 0.2 0.9 0.0 
1.51 0.50 0.78 0.43 3.6 0.4 4.9 
1.50 0.67 0.76 0.51 0.0 0.4 0.2 
1.50 1.00 0.75 0.63 0.8 0.1 −1.0 
1.49 1.50 0.73 0.74 1.2 0.1 −1.6 
1.50 1.51 0.77 0.77 1.7 0.0 2.3 
1.50 4.04 0.68 0.91 0.6 0.4 0.7 
1.99 0.25 0.72 0.22 1.0 1.2 −1.1 
2.00 0.50 0.76 0.38 0.1 0.7 0.1 
2.00 0.67 0.74 0.44 2.0 0.7 −2.4 
1.99 1.00 0.76 0.57 0.3 0.6 −0.2 
1.98 1.51 0.76 0.69 0.2 0.2 −0.2 
1.98 2.00 0.76 0.77 1.4 0.1 1.8 
1.99 4.01 0.73 0.88 0.6 0.5 0.7 
4.00 0.25 0.74 0.18 1.1 1.2 −1.2 
4.00 0.50 0.76 0.31 0.4 1.1 −0.2 
3.99 0.66 0.76 0.38 0.6 0.9 0.9 
4.03 1.00 0.79 0.49 0.6 0.7 0.9 
4.05 1.49 0.80 0.59 0.5 0.3 −0.6 
3.98 2.02 0.80 0.67 0.1 0.2 0.2 
3.99 3.97 0.79 0.79 0.6 0.1 −0.8 
ReW/SReN/EC*NMFNMass balance error (%)Flow balance error (%)PMFE
0.25 0.25 0.38 0.38 0.2 0.2 −0.1 
0.24 0.57 0.38 0.72 1.6 0.1 3.1 
0.25 0.67 0.39 0.79 1.5 0.1 2.9 
0.24 1.00 0.35 0.89 1.7 0.3 3.1 
0.25 1.49 0.31 0.95 1.7 0.6 2.8 
0.25 2.00 0.29 0.97 1.5 0.7 2.3 
0.25 3.98 0.25 1.00 1.8 1.0 2.7 
0.50 0.25 0.65 0.38 0.4 0.1 0.5 
0.50 0.50 0.61 0.62 0.2 0.2 0.2 
0.50 0.67 0.60 0.72 0.8 0.0 1.3 
0.50 1.00 0.55 0.83 1.9 0.1 3.0 
0.50 1.51 0.50 0.91 0.5 0.2 0.6 
0.50 1.99 0.47 0.94 0.7 0.5 0.8 
0.50 3.94 0.41 0.99 1.3 0.7 1.6 
0.67 0.25 0.71 0.36 0.4 0.0 −0.6 
0.67 0.50 0.70 0.57 1.5 0.3 2.4 
0.66 0.68 0.66 0.67 0.5 0.2 0.8 
0.67 1.00 0.64 0.80 0.3 0.1 0.5 
0.67 1.52 0.59 0.88 0.9 0.3 1.2 
0.67 2.02 0.55 0.92 1.0 0.5 1.3 
0.66 4.00 0.48 0.97 1.1 0.8 1.3 
1.00 0.25 0.71 0.28 0.4 0.5 −0.3 
0.99 0.50 0.76 0.51 0.3 0.4 0.6 
1.00 0.68 0.76 0.61 1.0 0.2 1.4 
1.00 0.99 0.75 0.75 0.4 0.1 0.5 
0.99 1.49 0.69 0.82 0.3 0.1 0.3 
1.00 2.02 0.67 0.90 0.3 0.4 0.3 
1.00 4.00 0.59 0.96 0.7 0.5 0.8 
1.49 0.25 0.72 0.24 0.2 0.9 0.0 
1.51 0.50 0.78 0.43 3.6 0.4 4.9 
1.50 0.67 0.76 0.51 0.0 0.4 0.2 
1.50 1.00 0.75 0.63 0.8 0.1 −1.0 
1.49 1.50 0.73 0.74 1.2 0.1 −1.6 
1.50 1.51 0.77 0.77 1.7 0.0 2.3 
1.50 4.04 0.68 0.91 0.6 0.4 0.7 
1.99 0.25 0.72 0.22 1.0 1.2 −1.1 
2.00 0.50 0.76 0.38 0.1 0.7 0.1 
2.00 0.67 0.74 0.44 2.0 0.7 −2.4 
1.99 1.00 0.76 0.57 0.3 0.6 −0.2 
1.98 1.51 0.76 0.69 0.2 0.2 −0.2 
1.98 2.00 0.76 0.77 1.4 0.1 1.8 
1.99 4.01 0.73 0.88 0.6 0.5 0.7 
4.00 0.25 0.74 0.18 1.1 1.2 −1.2 
4.00 0.50 0.76 0.31 0.4 1.1 −0.2 
3.99 0.66 0.76 0.38 0.6 0.9 0.9 
4.03 1.00 0.79 0.49 0.6 0.7 0.9 
4.05 1.49 0.80 0.59 0.5 0.3 −0.6 
3.98 2.02 0.80 0.67 0.1 0.2 0.2 
3.99 3.97 0.79 0.79 0.6 0.1 −0.8 

Note: mass balance error (%)(%); flow balance error (%).

Influence of the inlet flows

Figures 2(a) and 2(b) show the normalized concentration and MFN of the double-Tee junctions with different L/D values. All experimental data were located in the region comprising cross mixing (upper boundary) and complete mixing results (lower boundary). On one hand, the junction mixing with a larger L/D was closer to complete mixing and was farther from cross mixing compared with the junction with a smaller L/D. On the other hand, a larger resulted in more complete mixing, when, because higher velocity in the west inlet produced stronger collision and diffusion at the interface of the two inlet flows. Considering the two factors mentioned above, the mixing of and was regarded as complete. Bulk mixing is preferred, especially when . The general tendencies of MFN for all the double-Tee junctions decreased with increasing Reynolds number ratio of inlets . Nearly all experimental data coincided with the complete mixing results when , indicating that complete mixing took place.

Figure 2

Normalized north outlet concentration and mass fraction with different L/D values and results of mixing at the cross junctions and the complete mixing model (a) and (b) ReW ≠ ReS and ReN = ReE; (c) and (d) ReW = ReS and ReN ≠ ReE.

Figure 2

Normalized north outlet concentration and mass fraction with different L/D values and results of mixing at the cross junctions and the complete mixing model (a) and (b) ReW ≠ ReS and ReN = ReE; (c) and (d) ReW = ReS and ReN ≠ ReE.

Based on Figures 2(a) and 2(b), when the velocity in the main pipes (south inlet to the north outlet) was much larger than that in the branch pipes, the west inlet flow was pushed forward by the south inlet flow and was reflected into the adjacent north outlet; thus, the mixing phenomenon at this condition is closer to cross mixing than complete mixing. However, when the velocity in the branch pipes was higher than that in the main pipes, the west inlet flow penetrated through the low-velocity flow and caused a more complete mixing.

Influence of the outlet flows

According to Figure 2(c), with was much closer to cross mixing than complete mixing, and mixing with was located in the middle of the two boundary mixing models. Nevertheless, mixing with and 10.0 was preferred over complete mixing. Figure 2(d) shows the curves of the different values followed similar shapes. The played an important role in the mixing when , as it showed that the slopes of MFN decreased with an increasing. The relative location of either or MFN between cross mixing and complete mixing remained almost unchanged, especially when . Compared with Figures 2(a) and 2(b), the influence of on junction mixing was less than that of.

Influence of the connecting pipe length

Figure 3 shows the end with different values and various flow rates at the inlets and outlets. When , the mixing pattern at the double-Tee junctions approached that of cross junctions. However, when , the complete mixing assumption could be true, and the dimensionless concentrations at the two outlets were equal to 0.5. Both general tendencies can be observed in the experimental data (Figure 3). A typical experimental condition with identical Reynolds numbers at the four inlets and outlets and combined with various Re at the outlets was conducted, as shown in Figure 3.

Figure 3

Dimensionless north outlet concentrations from the experimental results and water quality model (EPANET) based on the complete mixing outcomes at different L/D values when ReW = ReS.

Figure 3

Dimensionless north outlet concentrations from the experimental results and water quality model (EPANET) based on the complete mixing outcomes at different L/D values when ReW = ReS.

The mixing at the cross junctions and double-Tee junctions showed significant differences due to the fact that the entering flows had sufficient time and space to interact and diffuse in the connecting segment of the double-Tee configuration. The double-Tee configuration can be mistakenly simplified into cross configuration when designing and analyzing WDS, such as in EPANET-BAM (Khalsa & Ho 2007), the contaminants in one branch flow may be underestimated because of incomplete mixing at the simplified cross junction, whereas the other branch flow may be overestimated.

CONCEPTUAL MIXING MODELS

The bulk-mixing model assumes that mixing depends on bulk flow interactions, and the mixing caused by the instabilities and diffusion at the interface is disregarded. A scaling parameter s was used to combine the complete and bulk mixing to show the actual mixing in the cross junctions (Ho 2008) 
formula
5
where is the actual solute concentration in the water leaving the junctions, and are the concentrations based on complete mixing and bulk mixing, respectively.
This bulk-mixing model was integrated into EPANET and only considered in mixing at cross junctions (Ho 2008), whereas the double-Tee junctions were neglected in water quality modeling and analysis. The mixing in the double-Tee junctions was between cross junction and complete mixing (Shao et al. 2014). Thus, Equation (5), which describes the mixing at cross junctions, can be inferred to double-Tee junctions, as follows: 
formula
6
whereis the concentration that leaves the junctions, is the concentration in the mixing at cross junctions with the given hydraulic conditions, and is a dimensionless parameter that defines the departure degree from complete mixing . The value ofindicated the impact of . For and the mixing at the double-Tee junction followed the cross mixing, whereas for and , complete mixing occurred. By combining Equations (5) and (6), a new variable is defined as , and Equation (7) can be obtained 
formula
7
Equation (7) is similar to Equation (5), but the mixing at double-Tee junctions is more complete than that at cross junctions; thus, . When L is sufficiently long, then and . Hence, Equation (7) can be used to describe the mixing behavior at both cross junctions and double-Tee junctions.
Based on Equations (5)–(7), a general conceptual model with parameter that describes the mixing at junctions is developed as follows: 
formula
8
where should be a nonlinear function of . L = the connecting pipe length ; the equivalent pipe diameter for the unequal pipe size junction; = the ratio of the pipe diameter. Shao et al. (2014) provide an analytical solution of double-Tee junctions by defining the dimensionless parameter that denotes the departure of the mixing state from complete mixing . However, this solution only considers equal pipes and the momentum ratio of two perpendicular directions. Therefore, a multi-factor assessment of double-Tee junctions that includes various flows, L/D, and pipe diameter ratios should be studied in the future.

Study of parameter φ

Along with the combination of bulk mixing and complete mixing, the experimental parameters φ of the north outlet are listed in Figure 4(a). Most of the φ values in this experiment were less than 0.5, indicating that the mixing at double-Tee junctions was far from bulk mixing (φ = 1) and was close to complete mixing. The value of (when and 5.0) was less than 0 when (Figure 4(a)). Possibly, the west flow penetrated the south flow and was reflected back by the pipe wall and yielded eddies at the downstream node, thereby increasing the number of NaCl tracers traveling into the east outlet. This abnormal phenomenon requires further research through CFD models.

Figure 4

Experimental results of parameter φ with different dimensionless connecting pipe lengths (a) ReN = ReE, ReW ≠ ReS and (b) ReW = ReS, ReN ≠ ReE.

Figure 4

Experimental results of parameter φ with different dimensionless connecting pipe lengths (a) ReN = ReE, ReW ≠ ReS and (b) ReW = ReS, ReN ≠ ReE.

The general trends of in Figure 4(b) increased with increasing, especially when , thereby indicating that has a negative impact on mixing. When values were equal to 7.5 and 10.0, the trends of were almost parallel to the axis, exhibiting the slight effect of the Reynolds number ratio of the outlets on the mixing at the junctions.

CONCLUSION

The research on mixing at double-Tee junction configurations could be extended to general cases. The cross junction configuration was a special case of double-Tee junctions when .

  • The mixing at double-Tee junctions was an incomplete mixing when the L/D was less than 5.0. When , the mixing behavior was much closer to complete mixing.

  • The mixing phenomenon at double-Tee junctions was conspicuously affected by the L/D and the Reynolds number ratio of the inlets and outlets. The L/D was the most important factor in mixing. In addition, had more impact on mixing than that of .

  • A conceptual model that described the mixing behavior of double-Tee junctions was developed. A dimensionless parameter was used to define the departure degree from complete mixing. The results showed that the value of was less than 0.5 for most situations at double-Tee junctions, thereby indicating that mixing at double-Tee junctions was closer to complete mixing than bulk mixing given the additional mixing in the connecting segment of double-Tee junctions.

ACKNOWLEDGEMENTS

The authors wish to thank three anonymous reviewers for their constructive comments. The present research is funded by the National Natural Science Foundation of China (Nos 51208457 and 51478417), the Major Science and Technology Program for Water Pollution Control and Treatment in China (2012ZX07408-002 and 2012ZX07403-004), and the National High Technology Research and Development Program of China (863 Program: 2012AA062608).

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