Introduction

Buildings consume approximately 40% of global energy and account for approximately 28% of carbon emissions1. In modern buildings, transmission and distribution systems are indispensable for various functions such as heating, ventilation, cooling, and water supply and drainage2,3. Building transmission and distribution systems account for more than 1/3 of the energy consumption of buildings4,5,6. The resistance generated by local components such as tees and elbows directly increases the energy consumption of water pumps and fans and is one of the main sources of energy consumption in building transmission and distribution systems7,8. Therefore, optimizing the flow shape of local components such as tees to reduce resistance is an effective way to reduce carbon emissions and is a critical issue that urgently needs to be addressed in buildings and the environment9,10,11.

The shape optimization of local components is an effective method for reducing the energy consumption of pumps. In recent years, many scholars have studied the optimization of local components. Gao et al.12 conducted full-scale experiments and studied a low-resistance tee with protrusions in rectangular air ducts, which reduced the resistance by 36%. Gao et al.13 combined the structural shape of a river flow channel and optimized the outer arc shape of a duct tee to achieve resistance reduction, reducing the local resistance of the tee by 21%. Liu et al.14 inserted an arched guide vane into the elbow of a rectangular air duct, reducing the local resistance by 24%. Yin et al.15 studied the use of guide vanes in circular water pipe confluence tees to reduce resistance, with the optimal result reducing local resistance by 44%. Currently, most direct optimizations of local component flow shapes have focused on rectangular air ducts, simplifying rectangular components into two-dimensional planes. Since circular local components do not have planar structures, they can be optimized in only three dimensions. Existing resistance reduction methods for circular components often involve adding guide vanes, which is essentially a three-dimensional problem. However, the guide vanes themselves have a planar structure, and when determining their shape and position, they can be simplified to the cross-section of the local component. In practical engineering applications, guide vanes consume materials and are prone to wear, limiting their adoption. Therefore, directly optimizing the flow shape of circular local components to reduce resistance is crucial.

Traditional optimization methods for building transmission and distribution systems predominantly use a priori trial-and-error approaches. Within a certain range, these methods repeatedly test to select the relatively best optimization scheme, making it difficult to obtain a global optimal solution16,17,18,19. Machine learning can effectively predict the global optimal solution, and after training is completed, the time cost of each calculation is much lower than the trial-and-error method. Previous researchers have conducted a lot of research on it. Renganathan et al.20 proposed an aerodynamic shape design method based on machine learning that could be used for the preliminary shape design of aircraft. Thummar et al.21 proposed a vortex flowmeter design method based on machine learning, which can predict the optimal bluff body shape for a given flow field characteristic, and validated the prediction accuracy through numerical simulation. Hart et al.22 summarized the integration of machine learning prediction of optimal solutions with alloy material research. Wang et al.23 combined machine learning and genetic algorithms to optimize the geometric structure of a finned heat pipe heat exchanger, resulting in an optimal scheme that reduced the heat source temperature by 9%. Gu et al.24 optimized the shape of a self-powered piezoelectric transmissive strain sensor on the basis of auxetic structures via machine learning to determine geometric design variables. Wang et al.25 proposed a salt cavern shape optimization method based on artificial neural networks and increased the capacity of a field cavern example by 31%. Wang et al.26 optimized the shape of a double-D coil in an inductive power transfer system via machine learning and experimentally verified the optimization accuracy, with the optimal shape achieving 97% peak efficiency. However, owing to the complexity of flow and spatial constraints, a posteriori shape optimization methods have not been fully applied in the field of resistance reduction optimization for local components. Therefore, predicting the optimal shape of local components via machine learning and then validating the resistance reduction effect via full-scale experiments and numerical simulations is important for the development of an a posteriori resistance reduction method for building transmission and distribution systems.

Energy dissipation has been widely studied by scholars in the field of resistance reduction. Corino et al.27 proposed a dilute polymer resistance-reducing solution and analysed the resistance reduction mechanism through energy dissipation. Prasad et al.28 explained the resistance-reducing effect of polymer additives by reducing energy dissipation. An et al.29 proposed a hydraulic retarder with bionic surface shape vanes that reduced rotor resistance and explained the resistance reduction mechanism through energy dissipation. Eyink et al.30 combined the principle of energy dissipation with vortex dynamics to explain the causes of resistance and described the necessary and sufficient conditions for reducing the turbulent resistance. Li et al.31 studied the effect of rib pitch angle on resistance reduction rates under different submersion heights on the basis of large eddy simulations and energy dissipation. Chen et al.32 inspired by butterfly wings, proposed a new bionic surface and analysed resistance reduction effects through vorticity and energy dissipation. Jiang et al.33 studied the resistance reduction effect of a superhydrophobic coating through simulation and experiment and analysed changes in energy dissipation. The magnitude of energy dissipation is related to the velocity gradient. Therefore, optimizing the resistance reduction effect of tees through energy dissipation analysis, combined with velocity vector and vorticity analysis to verify the reduction in the velocity gradient, is an effective method for intuitively explaining the resistance reduction mechanism.

This study designs a new type of low-resistance circular tee structure on the basis of an improved random forest model. The research employs a combination of numerical simulations and full-scale experiments. The resistance of the tee is first calculated through numerical simulations, and the accuracy of these simulations is verified by experiments. After the optimal shape is obtained via the improved random forest model, a physical model is 3D-printed for experimental validation to verify the effectiveness of the resistance reduction optimization. Finally, the resistance reduction mechanism is analysed via the energy dissipation principle, and the optimization results are nondimensionalized relative to the pipe diameter, elucidating the resistance reduction under different working conditions. This study investigates resistance reduction for a circular diverter tee, limiting the generalizability to similar configurations. This study provides new references for the a posteriori resistance reduction optimization of circular components in building transmission and distribution systems.

Methods

Optimized random forest model for tee design

This paper optimizes the flow shape of a circular water pipe confluence tee, which is a multifactor regression problem. The random forest (RF) model, as a powerful and flexible ensemble learning method, is suitable for various classification and regression tasks34,35,36. A diagram of a traditional RF is shown in Fig. 1a.

However, the traditional RF model has two disadvantages. First, since RF integrates the results of each decision tree without selection, if some decision tree models in training have average accuracy, this will affect the results of the RF model. Second, because the bootstrap sampling technique randomly samples with replacement from the original training set, this inevitably leads to excessive overlap of some training subsets, causing high correlation among the corresponding decision tree models, which reduces the generalizability of the RF model.

To address these shortcomings, this paper makes relevant improvements to the traditional RF model. The specific process is as follows: first, a larger number of independent decision tree models are trained. Then, the accuracy of the decision tree models is verified via a test set, and a certain percentage of decision tree models with low accuracy are eliminated. Next, the correlation between decision tree models is measured via the Spearman rank correlation coefficient—the larger the Spearman rank correlation coefficient is, the stronger the correlation between the two decision tree models. After a certain percentage of highly correlated decision tree models are eliminated, the remaining decision tree models are used to construct the improved RF model. The specific percentage values can be adjusted on the basis of the specific problem, ensuring that the number of decision tree models after elimination is roughly consistent with the original number. To address the resistance reduction problem of local components, this study recommends first training twice the number of decision tree models as the traditional RF does, eliminating 30% of the decision tree models with low accuracy, and for the remaining 1.4 times the number of decision tree models, 30% of the highly correlated decision tree models are eliminated. The remaining 0.98 times the number of decision tree models is used to construct the improved RF model. The principle diagram of the improved RF model is shown in Fig. 1b, and the flowchart of the improved RF model is shown in Fig. 1c.

Fig. 1
figure 1

(a) Principle diagram of the traditional random forest; (b) principle diagram of the improved random forest; (c) flowchart of the improved random forest.

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Figure 2 shows that the flow field of a tee can be considered a combination of a main straight pipe flow field and a branch elbow flow field. The shape of the main straight pipe flow field and the branch elbow flow field can be determined by the shape and position of their respective mid-sections. After the geometric models of the main straight pipe flow field and the branch elbow flow field are determined, they are combined using SCDM software, and the overlapping areas are deleted to obtain the geometric model of the tee. As shown in Fig. 2, the mid-section of the main straight pipe flow field can be considered an ellipse. The major axis of this ellipse is defined as parameter a, the minor axis as parameter b, and the upwards offset along the axis of the ellipse as parameter c (if moving downwards, it is counted as negative). Similarly, the mid-section of the branch elbow flow field can also be considered an ellipse. The major axis of this ellipse is defined as parameter d, the minor axis as parameter e, and the upwards offset along the axis of the ellipse as parameter f. By determining the values of these six shape parameters a, b, c, d, e, and f, the shape of the tee can be determined.

Fig. 2
figure 2

Schematic diagram of the tee shape parameters.

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Before training the improved RF model, an appropriate sample dataset needs to be obtained. The shape parameters are the features of the sample dataset, and the local resistance coefficient of each set of features corresponding to the tee is used as the label of the sample dataset. Before the sample dataset is determined, unreasonable values for each feature are initially excluded through computational fluid dynamics (CFD) numerical simulation. The tee area ratio is a term that measures the size of the tee branch line to the size of the tee main line, and is defined in this study as the pipe diameter of the tee branch line divided by the pipe diameter of the tee main line. For a DN80 circular diverter tee with an area ratio of 1:1, the initial value ranges for each feature are as follows: a[70 mm, 90 mm], b[70 mm, 90 mm], c[− 10 mm, 0 mm], d[70 mm, 90 mm], e[70 mm, 90 mm], and f[− 10 mm, 10 mm], and each feature is an integer. Specifically, preliminary numerical simulations were conducted in a wider range of parameters and found that it may lead to a sharp increase in resistance or unreasonable structure. A relatively broad and reasonable parameter range was screened out through numerical simulation. A value is randomly selected from each feature to obtain a set of feature values. A value is randomly selected from each feature to obtain a set of eigenvalues. The shape of the tee is determined based on the eigenvalues, and the resistance is calculated via CFD to obtain the label. This process is repeated to obtain 100 sets of sample data sets with improved RF.

This study optimizes the resistance reduction of tees on the basis of an improved RF model, predicts the values of shape parameters when the local resistance coefficient of the tee is minimized, and then validates the resistance reduction effect through full-scale experiments. 70% of the sample dataset is used as the training set, and 30% is used as the test set. For the resistance reduction problem of local components, the number of decision trees in the traditional RF can typically be set to 2003. In this study, 400 decision tree models are initially set up in the improved RF and trained with the corresponding training subsets. The accuracy of the decision tree models is calculated via the test set, and the 30% of decision tree models with the lowest accuracy are eliminated, leaving 280 decision tree models. The correlations between decision tree models are calculated via the Spearman rank correlation coefficient, and the 30% of decision tree models with the highest correlations are eliminated. The remaining 196 decision tree models are integrated to construct the improved RF. The prediction results of the improved RF are finally validated using the test set, with an accuracy of over 95%, confirming that the results of the improved RF are accurate and reliable. The determination of the optimal parameter combination is a process of exhaustive search within a given range. Using the improved RF model, under the constraint that shape parameters are limited to integers, the local resistance coefficient of all possible parameter combinations within the given range is predicted. The combination that minimizes the local resistance coefficient is then identified as the optimal solution. According to the prediction results of the improved RF, the optimal shape of a DN80 circular diverter tee with an area ratio of 1:1 is obtained when a = 73 mm, b = 88 mm, c = -9 mm, d = 74 mm, e = 74 mm, and f = -6 mm. Subsequent studies validated the resistance reduction effect of the optimized tee through full-scale experiments and numerical simulations.

The trial-and-error method does have the potential to yield effective results with minimal effort. However, it heavily relies on the designer’s experience, involves a high degree of randomness, and is prone to getting stuck in local optima. Compared to the trial-and-error method, the key advantage of the machine learning approach proposed in this study is its ability to obtain a global optimal solution within a given range. Determining the global optimal solution requires millions of computations. In the trial-and-error method, each manual calculation of different shape parameter combinations corresponds to a local resistance coefficient, involving geometric modelling, mesh generation, numerical simulation, and resistance computation, with each calculation taking approximately half an hour. For the machine learning method proposed in this study, after training the improved random forest model with the data set, although the time consumption of each calculation is affected by hardware factors such as the number of cores of the processor and the size of the memory, it can basically be done in microseconds or milliseconds. Therefore, the machine learning approach can significantly reduce both time and labor costs in determining the global optimal solution.

Full-size experiments

Compared with numerical simulations, full-scale experiments require more time and incur greater costs, but full-scale experiments are the most accurate method37,38. This paper used full-scale experiments to study the resistance characteristics of circular diverter tees in water pipes. The experimental system in this study consisted of butterfly valves, a vertical water tank, filters, rubber flexible joints, a pipeline circulation pump, a pressure tank, an electromagnetic flowmeter, pressure transmitters, a paperless recorder, elbows, straight pipes, and tees, as shown in Fig. 3. The nominal diameter of all straight pipes and tees is DN80. The experimental pipeline material was galvanized steel pipe. The local components were connected to the pipeline system through flanges, facilitating the replacement of the measured tee.

The full-scale experimental system was driven by a pipeline circulation pump with variable-frequency flow adjustment and was supplied with water from a vertical water tank. During each experiment, the water level in the tank was kept constant. To prevent impurities from entering the pipeline and minimize pipeline vibration, the experimental system was equipped with a filter and rubber flexible joints. To prevent the water temperature from rising and affecting the accuracy of the experiment, the water pump was powered off and cooled for 30 min after every two hours of operation.

The key to measuring the local resistance coefficient requires the determination of the dynamic pressure and total pressure difference. The key to measuring the dynamic pressure lies in measuring the flow velocity, which was performed via an electromagnetic flowmeter in this study. Based on the known dynamic pressure, the key to measuring the total pressure difference is the static pressure difference, which was measured via two pressure transmitters at the test points in this study. Table 1 shows the basic parameters of each device and measuring instrument.

Table 1 Experimental instrument parameters.
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Figure 3 shows the arrangement positions of the two electromagnetic flowmeters and five pressure transmitters. According to the principle of mass conservation, the flow rate in the pipeline at test point 3 can be calculated by the difference between the two electromagnetic flowmeters. Therefore, the flow rates at all test points, and thus the dynamic pressure at all test points, can be determined through the two electromagnetic flowmeters. Test points 4 and 5 are used to measure the static pressure drop of the 40D straight pipe under different flow conditions, which corresponds to the frictional resistance ΔPf of the 40D straight pipe section under different Reynolds numbers. Test points 1 and 2 measure the static pressure difference and frictional resistance of two 40D straight pipes in the main line of the tee. Let the static pressure difference between test points 1 and 2 be ΔPs,1−2 and the dynamic pressure difference be ΔPv,1−2. Let the frictional resistance of the two 40D straight pipes between test points 1 and 2 be ΔPf,1−2. The resistance calculation formula of the tee main line during the experiment is as follows:

$${\xi _{m}} = \frac{{{{\Delta }}{P_{s,1 - 2}} - {{\Delta }}{P_{f,1 - 2}} + {{\Delta }}{P_{v,1 - 2}}}}{{{P_v}}}$$
(1)

Similarly, test points 1 and 3 measure the static pressure difference of the tee branch line and the resistance along the two sections of the 40D straight pipe. The static pressure difference between test points 1 and 3 is ΔPs,1−3, and the dynamic pressure difference is ΔPv,1−3. The resistance along the two sections of the 40D straight pipe between test points 1 and 3 is ΔPf,1−3. The resistance calculation formula of the tee branch line during the experiment is as follows:

$${\xi _{b}} = \frac{{{{\Delta }}{P_{s,1 - 3}} - {{\Delta }}{P_{f,1 - 3}} + {{\Delta }}{P_{v,1 - 3}}}}{{{P_v}}}$$
(2)

The reduction rate of resistance is represented by η, and its calculation formula is as follows:

$$\begin{matrix}{\eta = \frac{{\left| {{\xi _{tra}} - {\xi _{opt}}} \right|}}{{\left| {{\xi _{tra}}} \right|}} = \frac{{\left| {\user2{\Delta }\xi } \right|}}{{\left| {{\xi _{tra}}} \right|}}} \\ \end{matrix}$$
(3)

where ξtra is the local resistance coefficient of the traditional tee; ξopt is the local resistance coefficient of the optimized tee; and Δξ is the reduction in the local resistance coefficient.

For each tested tee, the resistance was measured 10 times for each working condition, and the average value of the 10 results was taken as the experimental result. The experimental error is expressed by the standard deviation sx:

$${s_x} = \sqrt {\frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{({x_i} - \bar x)}^2}} }$$
(4)

where n is the number of measurements, xi is the result of each measurement, and \(\bar{x}\) is the average result.

Fig. 3
figure 3

Schematic diagram of the full-scale test bench system (① butterfly valve; ② water tank; ③ Y-type filter; ④ rubber flexible joint; ⑤ pipeline circulation pump; ⑥ pressure regulating tank; ⑦ electromagnetic flowmeter; ⑧ pressure transmitter).

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Numerical simulation

Compared with full-scale experiments, numerical simulations can be performed without consuming significant resources or time and can efficiently calculate the local resistance of various tees39,40. This study adopts a method that combines full-scale experiments and numerical simulations to optimize the shape of tees. Full-scale experiments were used to validate the accuracy of the numerical simulations. This combination significantly improves the efficiency of resistance reduction research.

This study uses Fluent software for CFD numerical simulation. The boundary condition at the inlet of the diverter tee was set to the velocity pressure inlet, and the boundary conditions at the two outlets were set to velocity outlets. The wall boundary condition was set to a no-slip boundary, and the wall roughness height was set to 0.15 mm. The convection term was discretized via the second-order upwind scheme, and the SIMPLE algorithm was used to determine the coupling between the pressure and velocity. The convergence criteria were defined as the change in the average velocity and average pressure of the section between two iterations being less than 10− 3 and the normalized residuals being less than 10− 6.

Fig. 4
figure 4

Numerical model of the local resistance coefficient of the diverter tee.

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The numerical model shown in Fig. 4 is used to calculate the local resistance coefficient of the diverter tee. The minimum distance for fully developed turbulence in the pipeline is 10D. Three computational sections are taken 40D upstream and downstream of the tee and labelled Sect. 1, Sect. 4, and Sect. 6. The area-weighted average is used to obtain the average total pressure at the three sections, denoted as P1, P4, and P6. The flow field of the tee has two directions: the main line and the branch line. The total pressure difference between the two sections in the main line direction is denoted as ΔP1 − 4, and the total pressure difference between the two sections in the branch line direction is denoted as ΔP1 − 6. Similarly, the frictional resistance losses in the main and branch lines are determined by removing the tee and using an equivalent straight pipe, denoted as ΔPf,1−4 and ΔPf,1−6. The calculation formula for resistance during numerical simulation is as follows:

$${\xi _m} = \frac{{\Delta {P_{1 - 4}} - \Delta {P_{f,1 - 4}}}}{{{P_v}}} = \frac{{({P_1} - {P_4}) - (P_1^{\prime} - P_2^{\prime}) - (P_3^{\prime} - P_4^{\prime})}}{{\rho ({v^2}/2)}}$$
(5)
$${\xi _b} = \frac{{\Delta {P_{1 - 6}} - \Delta {P_{f,1 - 6}}}}{{{P_v}}} = \frac{{({P_1} - {P_6}) - (P_1^{\prime} - P_2^{\prime}) - (P_5^{\prime} - P_6^{\prime})}}{{\rho ({v^2}/2)}}$$
(6)

where ξm is the local resistance of the tee main line; ξb is the local resistance of the tee branch line; Pv is the dynamic pressure, Pa; ρ is the density of the fluid, kg/m3; and v is the average velocity at the tee outlet, m/s.

To simulate the internal flow conditions of the tee more accurately, it is necessary to verify the results of different turbulence models and combine them with full-scale experiments to determine the most suitable turbulence model41,42,43. This paper uses various turbulence models for numerical simulation of a traditional diverter tee with a flow rate ratio of 1:1, calculates the local resistance coefficient ξb of the branch line under different inlet Reynolds numbers, and compares the results with those of full-scale experiments. This study calculates five common turbulence models applicable to local pipeline components, including the k–ε RNG model, k–ε realizable model, k–ω SST model, k–ε standard model, and RSM. As shown in Fig. 5, the results of the k-ε realizable model match the experimental results most closely. For example, at an inlet Reynolds number Re = 2.0 × 105, the errors between the full-scale experimental average and the five turbulence models are 3.2%, 0.8%, 4.0%, 3.0%, and 2.1%, respectively, with the k-ε realizable model having the smallest error. Therefore, this study determines that the k-ε realizable model is the most suitable turbulence model on the basis of experimental verification.

Fig. 5
figure 5

Comparison of the calculated results of different turbulence models with the experimental results.

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This study altered the flow shape of the tee, increasing its geometric complexity; therefore, an unstructured mesh was chosen. Polyhedral meshes are better suited to models with complex geometries and have more contact faces between cells, offering higher computational and convergence efficiency. Additionally, polyhedral meshes can cover the tee model with fewer mesh cells, reducing the use of computational resources. Therefore, this study uses polyhedral meshes and performs local encryption on the tee area and boundary layer. A schematic diagram of mesh division is shown in Fig. 6.

Fig. 6
figure 6

Schematic diagram of grid division.

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To ensure calculation accuracy while using the minimum number of mesh cells to improve research efficiency, this study conducted a mesh independence verification44,45. Six different mesh densities were used for the calculations, with mesh quantities of 307.2k, 566.1k, 1001.5k, 1362.7k, 2186.2k, and 3018.3k. When the flow rate ratio was 1:1 and the inlet Reynolds number was Re = 2.4 × 105, a centreline one metre long was taken in the main line of the tee. The velocity distribution along this centreline was calculated for the six different mesh densities, and the ξb of the traditional tee’s branch line was calculated under different mesh densities. Mesh independence was verified on the basis of these two indicators, as shown in Fig. 7a, b. When the fifth mesh density of 2186.2k was used, the results stabilized, and the calculation difference from the higher-density sixth mesh was less than 5%. Therefore, this study selects the mesh configuration with 2186.2k cells as the final mesh scheme.

Fig. 7
figure 7

(a) Centreline velocity values corresponding to different grid numbers; (b) tee branch line resistance corresponding to different grid numbers.

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Results

Resistance reduction effect of tee main line and branch line

The flow field of the tee has two directions, the branch line and the main line, with the local resistance of the diverter tee mainly concentrated in the branch line. Existing studies have achieved a high resistance reduction rate for the branch line of the diverter tee, but for different pipeline systems, the most adverse circuit may also pass through the branch line of the diverter tee. Therefore, ensuring a high resistance reduction effect for the main line while reducing resistance in the branch line is important. According to the “Optimized random forest model for tee design” section, for a DN80 circular diverter tee with an area ratio of 1:1, the optimal low-resistance tee shape parameters predicted by the improved RF are a = 73 mm, b = 88 mm, c=− 9 mm, d = 74 mm, e = 74 mm, and f=− 6 mm. This paper studies the resistance reduction effects of a low-resistance tee’s branch lines and main lines under different working conditions through CFD numerical simulations. Figure 8 shows the comparison of the local resistance coefficients between the low-resistance tee and the traditional tee branch line under 5 flow rate ratios and 4 typical Reynolds numbers. Q2 and Q3 represent the outlet flow rates of the main line and the branch line, respectively, whereas Q1 represents the inlet flow rate. The low-resistance tee achieves a reduction in the local resistance coefficient of the branch line under different flow rate ratios and Reynolds numbers, with a resistance reduction rate ranging from 28 to 66%. The resistance reduction rate of the tee branch line is positively correlated with the proportion of the branch flow rate. When Q2/Q1 is equal to 0.5, which is a typical flow rate ratio of 1:1 for the two inlet flow rates, the resistance reduction rate of the low-resistance tee branch line reaches 50%.

Fig. 8
figure 8

Comparison of branch line resistance under different flow ratios and Reynolds numbers values.

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The resistance reduction performance of the low-resistance tee’s main line is shown in Fig. 9. Under different working conditions, the low-resistance tee also achieves a reduction in the local resistance coefficient of the main line, with resistance reduction rates ranging from 16 to 93%. For the common working condition where Q2/Q1 equals 0.5, the resistance reduction rate of the low-resistance tee’s main line reaches 30%.

Fig. 9
figure 9

Comparison of the main line resistance under different flow ratios and Reynolds numbers values.

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Resistance reduction effect of the normalized shape at different area ratios

The local resistance of the diverter tees with different area ratios differs significantly, and in practical engineering applications, tees in water pipes are not limited to a single area ratio. Optimizing the resistance reduction for each area ratio separately would be highly inefficient and impractical. This study normalizes the shape parameters relative to the diameters of the two outlet pipes of the tee, D2 and D3, resulting in dimensionless numbers that enable similar shapes for tees with different area ratios. The normalized results for the six shape parameters are a/D2 = 0.913, b/D2 = 1.100, c/D2 = − 0.113, d/D3 = 0.925, e/D3 = 0.925, and f/D3 = − 0.075.

To verify the rationality and effectiveness of normalizing shape parameters, this study conducted CFD numerical simulations on four common area ratios of diverter tees. The total energy loss coefficient K was used to measure the total resistance of the tee main line and branch line46. The calculation formula of the total energy loss coefficient K is as follows:

$$K = \frac{{{Q_2}}}{{{Q_1}}}{\xi _m} + \frac{{{Q_3}}}{{{Q_1}}}{\xi _b}$$
(7)

When the flow rate ratio was 1:1, numerical simulations were conducted for traditional tees and low-resistance tees with normalized shapes under different area ratios, and their respective total energy loss coefficients were calculated, as shown in Fig. 10. Under different area ratios, the low-resistance tees with normalized shapes obtained in this study significantly reduced the total energy loss coefficient, with a resistance reduction rate ranging from 48 to 68%.

Fig. 10
figure 10

Comparison of the total energy loss coefficients of a traditional tee and optimized tee under different area ratios.

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Experimental verification and analysis

To verify the resistance reduction effect of the new tee predicted by the improved RF model in practical applications, this study conducted full-scale experiments on DN80 traditional and optimized diverter tees with an area ratio of 1:1, using 3D printing technology to create physical tees. As shown in Fig. 11, this study measured the branch resistance under different Reynolds numbers for both traditional and optimized tees and compared the results with those of numerical simulations. The results of the full-scale experiments and numerical simulations were consistent, with the local resistance coefficient of the optimized tee being lower than that of the traditional tee. The new tee developed in this study achieved reduced flow resistance, directly reducing the operating power consumption of the pump and meeting the demand for reducing carbon emissions in the building and environmental sectors.

Fig. 11
figure 11

Comparison and experimental verification analysis of traditional tee and optimized tee.

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Discussion

Energy dissipation analysis

Mechanical energy dissipation is often used to characterize the flow resistance of local components, as the formation of local resistance involves the conversion of mechanical energy into internal energy29,47. For circular water pipe diverter tees, lower energy dissipation means lower local resistance. This study proposes a new shape of low-resistance tee shape based on an improved RF, where the reduction in energy dissipation can serve as the mechanism for resistance reduction optimization. A comparison of the energy dissipation fields for the optimized tee and traditional tee can intuitively verify the effectiveness of resistance reduction. For incompressible fluids, the calculation formula for the energy dissipation function ϕ is as follows:

$$\phi = \mu \left\{ {2\left[ {{{\left( {\frac{{\partial {u_x}}}{{\partial x}}} \right)}^2} + {{\left( {\frac{{\partial {u_y}}}{{\partial y}}} \right)}^2} + {{\left( {\frac{{\partial {u_z}}}{{\partial z}}} \right)}^2}} \right] + {{\left( {\frac{{\partial {u_x}}}{{\partial y}} + \frac{{\partial {u_y}}}{{\partial x}}} \right)}^2} + {{\left( {\frac{{\partial {u_y}}}{{\partial z}} + \frac{{\partial {u_z}}}{{\partial y}}} \right)}^2} + {{\left( {\frac{{\partial {u_z}}}{{\partial x}} + \frac{{\partial {u_x}}}{{\partial z}}} \right)}^2}} \right\}$$
(8)

where \({u_x}\), \({u_y}\), and \({u_z}\) are the partial velocities in the x, y, and z directions, respectively, m/s; µ is the kinetic viscosity coefficient, \({\text{Ns}}/{{\text{m}}^2}\).

This study uses energy dissipation as an indicator to visualize the energy dissipation fields of traditional and optimized diverter tees under typical working conditions with a flow rate ratio of 1:1 and an inlet Re = 2.4 × 105. The energy dissipation fields were determined at the centre plane of the tee and at the branch outlet, as shown in Fig. 12. The traditional tee has higher energy dissipation values and larger dissipation areas in the flow field of the branch. The optimized tee obviously improved because the optimized tee improves the flow of fluid at the divergence and reduces the velocity gradient in the local area, thereby reducing energy dissipation and lowering the resistance of the tee.

Fig. 12
figure 12

Energy dissipation field of the tee centre section and the branch outlet; (a) energy dissipation field of the traditional tee; (b) energy dissipation field of the optimized tee.

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Velocity vector and vorticity analysis

Figure 13 shows the velocity vector cloud charts for five cross-sections within five times the diameter of the branch line of the diverter tee. The distances from the branch outlet to these sections are 0.2D, 0.4D, 0.8D, 2D, and 5D. (1) Both traditional and optimized tees exhibit vortices distributed symmetrically along the vertical axis after divergence. The vortices above rotate in the opposite direction to those below. Traditional tees have two large vortices at each section, reflecting strong collision and swirling movements of the fluid. In contrast, the optimized tees have four smaller vortices at each section, reducing the size of the vortices. (2) The velocity distribution in the flow field after convergence in traditional methods is uneven. For example, at Sect. 0.2D, 0.4D, and 0.8D downstream, the maximum velocity reaches four times the minimum velocity, indicating a significant velocity gradient in the flow field. The optimized tees have a more uniform velocity distribution across the sections, with no obvious high-speed or low-speed zones. The optimized tees reduce the velocity gradient in the flow field, which corroborates the results shown in Fig. 12. Thus, the optimized tees effectively reduce the turbulence and disorder of the fluid, improving the stability of the flow field in the diverter tee of the water pipe.

Fig. 13
figure 13

Velocity vector distribution within 5 times the pipe diameter of the tee branch line; (a) velocity vector distribution of the traditional tee; (b) velocity vector distribution of the optimized tee.

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3D graphics can better visualize complex flow behaviors in three-dimensional space. The Q-criterion is an effective method to evaluate whether there is vortex in the flow field. Figure 14 shows the comparison of vortex in the three-dimensional flow field between the traditional tee and the optimized tee under different Q-criterion iso-surfaces. It can be found that under different iso-surfaces, the optimized tee significantly reduces the vortex downstream of the main and branch.

Fig. 14
figure 14

Comparison of vortices in three-dimensional flow field corresponding to different Q-criterion iso-surfaces.

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Limitations and prospects

This study proposes an effective method for reducing resistance in local components; however, there are still some limitations. The results of this study are limited to circular diverging tees and do not consider rectangular tees or converging tees. Additionally, the study focuses on the Reynolds number range relevant to building plumbing systems, without accounting for extremely high or low Reynolds number conditions. Given the limited dataset, there is still a possibility that the performance of the tees designed using the proposed model may be inferior to those designed through repeated trial-and-error. Furthermore, the resistance reduction effect of the proposed method has only been compared to T- type tees and has not been evaluated against Y- type tees. We will further explore these aspects in future research.