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The bulb tubular unit has advantages over other units in the low head, large discharge, and tidal energy. The development and utilization of low head and ultralow head hydraulic resources can increase the runner’s energy parameters, make up for the defects of large size and low speed of the unit, and reduce the construction investment. The bulb tubular turbine’s operation stability and structural strength are compared and analyzed by the orthogonal test method to improve the perfomance of bulb tubular turbine. The results show that the optimal design scheme can be obtained when the number of guide vane is 16, the distance between the guide vane and the blade is 1995 mm, the hub ratio is 0.31, the relative twist angle of the blade is 27°, and the cascade density is 0.72. After optimization, the bulb tubular turbine’s efficiency and output are increased by 5.72% and 2.86% at a low flow rate, respectively. The vortex at the tailpipe is reduced, and the blade work effect and the cavitation performance are well. The maximum pressure pulsation amplitude in the flow passage is reduced by 73.15%, the maximum blade deformation is reduced by 39.59%, and the maximum blade static stress is reduced by 13.16%. Finally, the reliability of numerical simulation results is verified by the model test.

Tubular turbines have been widely used and developed rapidly due to their excellent technical and economic characteristics and applicability since they came out in the 1930s. Tubular turbine technology, including bulb tubular generator, has become increasingly mature. The development, design, operation technology, and experience of the tubular hydropower stations are becoming more abundant [

The development of CFD has brought great convenience to the design and optimization analysis of hydraulic machinery [

Most of the above studies carried out structural optimization by changing a certain design parameter (guide vane or runner), while few studies on structural optimization by changing multiple design parameters were carried out. This paper uses the orthogonal test method to optimize the bulb tubular turbine to improve the bulb tubular turbine’s overall optimization level, and find out the main geometric parameters that affect the performance of the bulb tubular turbine device. Based on the _{16} (4^{5}) orthogonal table, 16 sets of schemes are obtained. The in-depth study on the number of guide vanes, the distance between the guide vanes and the blades, the ratio of the hub part of the runner, the relative twist angle of the blades, and the density of the cascade density on the bulb flow was carried out. The five evaluation indexes’ influence rules, such as the efficiency, output, pressure pulsation, blade deformation, and blade static stress of the hydraulic turbine, were determined by a comprehensive frequency analysis method. The performance of the unit before and after optimization was compared and analyzed.

The orthogonal test method is an efficient test design method based on the basic principles of probability theory and mathematical statistics [

In orthogonal experimental design, the test index, test level, and the test factors are the parameters that need to be carefully considered. The test index can judge the test result, the test factor is the crucial variable in the experiment process, and the test level represents the different states of the test factors [

The orthogonal test table is the core of the orthogonal test method’s essential analysis tool [_{16} (4^{5}) with five factors and four levels is selected to optimize the turbine’s design [

The purpose of the test is to improve the hydraulic performance and operating stability of the bulb tubular turbine unit, and ensure the flow components’ structural strength. Five evaluation indexes are selected: efficiency, output, pressure pulsation, blade deformation, and blade static stress.

According to the structural design requirements of the bulb tubular turbine, the number of guide vanes (factor A), the distance between the guide vanes and blades (factor B), the hub ratio (factor C), the relative twist angle of the blades (factor D), and cascade density (factor E) have a significant impact on the performance of the unit. Therefore, they are selected for this test to optimize the orthogonal test.

The conical guide vane in the bulb tubular turbine meets the requirement of the required amount of circulation in front of the runner and must meet the requirement of closing and sealing. The number of original guide vanes is 14, and the number of guide vanes of large-sized and medium-sized tubular turbines is usually 12, 16, and 20 [

The distance between the guide vane and the blade is defined as L1, and its value is generally (0.65 ∼ 0.8) D1 [

Schematic diagram of the distance between guide vane and blade.

The hub ratio of the runner is an important design parameter in the bulb tubular turbine. It is related to the arrangement of the blade operating mechanism and has a great impact on the hydraulic performance of the turbine and the structural rigidity of the runner. This article mainly studies the influence of the change of the runner’s hub ratio on the performance of the hydraulic turbine. The calculation method of the runner hub ratio is shown in formula (

Schematic diagram of the hub ratio of the runner.

The relative twist angle of the blade affects the overflow when the unit is running. For the tubular turbine runner, the blade’s optimal relative twist angle is 18° ∼ 28° [

Schematic diagram of blade relative torsion angle.

The cascade density is an essential parameter in the design of the runner. Its value not only affects the runner’s flow but also affects the runner’s cavitation performance and hydraulic stability. The density of the hub cascade is generally 1.3 to 1.4 times that of the rim cascade. When the number of blades is 3, the rim cascade’s density ranges from 0.75 to 0.85 [

Schematic diagram of the density of the cascade at the blade rim.

_{16} (4^{5}) orthogonal table is shown in Table

Orthogonal test plan.

Test plan | Experimental factors | ||||
---|---|---|---|---|---|

Number of guide vanes _{0} ( | Distance between guide vane and blade | Hub ratio | Relative twist angle of blade △_{0} ( | Cascade solidity | |

1 | 14 | 1710 | 0.29 | 20 | 0.72 |

2 | 14 | 1852.5 | 0.31 | 23.5 | 0.76 |

3 | 14 | 1995 | 0.33 | 27 | 0.80 |

4 | 14 | 2137.5 | 0.35 | 30.5 | 0.84 |

5 | 16 | 1710 | 0.31 | 27 | 0.84 |

6 | 16 | 1852.5 | 0.29 | 30.5 | 0.80 |

7 | 16 | 1995 | 0.35 | 20 | 0.76 |

8 | 16 | 2137.5 | 0.33 | 23.5 | 0.72 |

9 | 17 | 1710 | 0.33 | 30.5 | 0.76 |

10 | 17 | 1852.5 | 0.35 | 27 | 0.72 |

11 | 17 | 1995 | 0.29 | 23.5 | 0.84 |

12 | 17 | 2137.5 | 0.31 | 20 | 0.80 |

13 | 20 | 1710 | 0.35 | 23.5 | 0.80 |

14 | 20 | 1852.5 | 0.33 | 20 | 0.84 |

15 | 20 | 1995 | 0.31 | 30.5 | 0.72 |

16 | 20 | 2137.5 | 0.29 | 27 | 0.76 |

This article takes a three-blade bulb tubular turbine as the research object. The main design parameters of the turbine model are shown in Table

Basic parameters of bulb tubular turbine.

Number | Basic parameters | Numerical value | Units |
---|---|---|---|

1 | Runner diameter | 2.85 | m |

2 | Nominal output | 3000 | kW |

4 | Rated head | 5.2 | m |

5 | Minimum head | 3.5 | m |

6 | Rated speed | 150 | r/min |

7 | Rated flow | 64.83 | m^{3}/s |

UG NX is used to model the bulb tubular unit, which is mainly divided into four parts: the inlet flow channel section, the guide vane section, the runner section, and the draft tube section, as shown in Figure

Model diagram of bulb tubular.

ICEM is used to divide the grid of fluid computation domain. Mesh division is the basic way to realize regional discretization in computational fluid dynamics. The quality and quantity of the mesh division will affect the calculation time and calculation accuracy. The unit’s complex structure is divided by an unstructured grid with good adaptability, and the mesh of critical parts is locally refined [

Grid number and unit efficiency.

Case | Cell numbers | Efficiency/% |
---|---|---|

1 | 4.4 million | 88.9 |

2 | 4.8 million | 89.7 |

3 | 5.2 million | 90.2 |

4 | 5.6 million | 90.8 |

5 | 6.0 million | 90.9 |

The results of Table

Mesh division: (a) guide vane section; (b) runner section.

The solid computing domain includes the hub and blades of the runner chamber. Using the mesh module in ANSYS Workbench, the solid computing domain is discreted. The tetrahedral mesh is selected, with a grid size of 30 mm. It is found that when the number of grid cells is changing, the maximum displacement value changes very little, but the maximum stress value differs greatly. This is because the blade root is prone to stress concentration and the gradient of stress change is large [

Grid division of solid computing domain.

Meshing scheme | Units | Number of nodes | Maximum stress value/MPa | Maximum displacement value/mm |
---|---|---|---|---|

30 mm | 8568 | 16244 | 14.11 | 1.54 |

20 mm | 23007 | 39932 | 15.78 | 1.56 |

10 mm | 188415 | 291972 | 19.24 | 1.62 |

Refined 1 time | 223936 | 341825 | 20.06 | 1.62 |

Refined 2 times | 315840 | 471858 | 20.24 | 1.62 |

Refined 3 times | 430935 | 626276 | 20.25 | 1.62 |

The settings of the flow field’s boundary conditions are as follows: the numerical simulation calculation is carried out on the ANSYS CFX code. The RNG k-

In the steady calculation, the dynamic and static interface is set to the frozen rotor type. In the unsteady calculation, the dynamic and static interface is set to the transient frozen rotor type. The interface grid is set as a GGI connection to ensure the transmission of energy on both sides of the interface. In the unsteady calculation, the rotation period

Layout of monitoring points.

The boundary conditions of the structural field are set as follows: (1) apply fixed constraints to the cylindrical surface of the hub shaft; (2) apply gravity and rotation constraints; (3) load water pressure on the fluid-solid coupling surface [

The boundary conditions of structural field: (a) boundary conditions; (b) water pressure loading.

In order to verify the reliability of the numerical simulation in this paper, a model test of the optimized hydraulic turbine was carried out on a multifunctional hydraulic test rig. The test rig is designed and constructed in accordance with China’s water conservancy industry standards. The comprehensive accuracy of the multifunctional hydraulic test rig is ±0.35% (level A) accuracy. Figures

Schematic diagram of the three-dimensional model of the multifunctional test stand.

Photo of the multifunctional test rig.

During the experiment, the model turbine’s assembly and testing are strictly in accordance with the relevant regulations of the Turbine Model Acceptance Test Regulations (DL 446-91). After ensuring that the model turbine is running for a while and the water flow is gradually stabilized, the external characteristic parameters are measured and improve the accuracy of the measurement.

The model test mainly measures and calculates the efficiency of the unit. The size of the whole flow passage of the model turbine is geometrically similar to that of the prototype turbine, and the diameter of the runner of the model turbine is 0.38 m. In the test, six working conditions with increasing flow rates are selected, and the specific parameters are shown in Table

Comparison of numerical simulation calculation and test results under different working conditions.

Working condition | _{11}/ | _{11}/m^{3}/s | Guide vane opening/° | Blade angle/° | Effectiveness | |
---|---|---|---|---|---|---|

Calculated | Test value | |||||

1 | 179.0 | 1.38 | 48.0 | 15.0 | 80.53 | 78.12 |

2 | 169.8 | 1.65 | 52.0 | 20.3 | 84.87 | 82.50 |

3 | 170.6 | 2.01 | 56.1 | 25.3 | 90.79 | 88.36 |

4 | 190.0 | 2.84 | 68.0 | 33.5 | 89.92 | 87.25 |

5 | 180.0 | 3.20 | 71.5 | 38.6 | 85.56 | 83.01 |

6 | 187.6 | 3.52 | 73.7 | 39.8 | 81.71 | 79.37 |

It can be seen from Table

Under the turbine’s rated operating conditions, 16 sets of schemes are subjected to steady, unsteady, and unidirectional fluid-solid coupling calculations, and the calculation results are analyzed. The operating efficiency and output of the units under each scheme are obtained through the steady flow calculation. The time-domain information of pressure pulsation is obtained through unsteady calculation, and the frequency domain characteristics are obtained through Fourier transform. The maximum static stress and maximum deformation of the runner blade under various design schemes are obtained through the unidirectional fluid-solid coupling calculation. Table _{p} [

Calculation results of various design schemes.

Plan | Judgment index | ||||
---|---|---|---|---|---|

Effectiveness | Output | Pressure pulsation coefficient _{P} | Maximum deformation of blade/mm | Maximum static stress of blade/MPa | |

1 | 71.34 | 1727.41 | 4.4510 | 1.62 | 20.25 |

2 | 80.53 | 2576.65 | 1.1472 | 1.44 | 24.20 |

3 | 91.02 | 3127.65 | 0.7879 | 0.99 | 27.07 |

4 | 84.57 | 3087.91 | 1.1761 | 1.04 | 26.79 |

5 | 85.92 | 2982.89 | 0.6178 | 3.05 | 50.25 |

6 | 87.78 | 3297.23 | 1.1185 | 0.91 | 26.12 |

7 | 75.19 | 1940.22 | 7.9660 | 1.67 | 20.41 |

8 | 80.59 | 2768.03 | 2.1571 | 1.17 | 22.09 |

9 | 86.98 | 3185.20 | 2.6395 | 1.02 | 25.26 |

10 | 83.16 | 2934.21 | 2.4201 | 0.74 | 19.47 |

11 | 79.90 | 2266.58 | 1.3565 | 0.96 | 27.21 |

12 | 75.19 | 1926.01 | 1.4233 | 2.95 | 27.38 |

13 | 72.49 | 1977.54 | 3.9701 | 1.22 | 20.76 |

14 | 70.04 | 1172.33 | 2.4499 | 3.01 | 27.38 |

15 | 88.83 | 3079.09 | 1.0581 | 0.87 | 22.16 |

16 | 85.78 | 2948.61 | 0.7246 | 0.97 | 21.25 |

After intuitive analysis of the calculation results of 16 schemes, the better schemes were obtained as (a), (b), (c), (d), and (e), as shown in Table

Intuitive analysis of the better solution.

Evaluation indexes | Efficiency | Output | Pressure pulsation coefficient | Deformation of blade | Static stress of blade |
---|---|---|---|---|---|

Plan | 3 | 6 | 5 | 10 | 10 |

Factor level | A_{1}B_{3}C_{3}D_{3}E_{3} | A_{2}B_{2}C_{1}D_{4}E_{3} | A_{2}B_{1}C_{2}D_{3}E_{4} | A_{3}B_{2}C_{4}D_{3}E_{1} | A_{3}B_{2}C_{4}D_{3}E_{1} |

Better plan | (a) | (b) | (c) | (d) | (e) |

By calculating the range value, the weight of each factor level’s influence on each index can be obtained. The larger the range value is, the greater the influence weight of the selected level on the test index under this factor is. The influence of indicators is shown in Figure

The relationship between various factor levels and indicators: (a) efficiency index; (b) output index; (c) pressure coefficient index; (d) blade deformation index; (e) blade static stress index.

Based on the data of range analysis and the relationship between each factor level and the index, an in-depth analysis of each factor level’s impact on the index under rated conditions is performed. An optimal plan under the range analysis is determined.

The water guide mechanism is an essential part of the hydraulic turbine. The number of guide vanes Z0 involves its own processing volume and economic investment and affects the uniformity of the water flow into the runner. It can be seen from Figure

It can be seen from Figure _{1} between the guide vane and the blade gradually increases from 0.60_{1} ⟶ 0.65_{1} ⟶ 0.70_{1} ⟶ 0.75_{1}, the operating efficiency of the unit increases first and then decreases, and the output gradually increases. When _{1} is large, the hydraulic performance of the unit is well; the change of L1 has no obvious effect on the pressure pulsation coefficient, but when _{1} = 0.75_{1}, the pressure pulsation coefficient is the smallest; with the increase of _{1}, the maximum deformation of the blade decreases first and then increases. When _{1} = 0.70_{1}, the blade deformation is the minimum. When _{1} > 0.60_{1}, the blade’s maximum static stress does not change with the change of _{1}. The larger the distance between the guide vane and the blade is, the more stable and uniform the water flow into the runner is, and the hydraulic performance is improved to some extent. However, when the distance is too large, the flow circulation in front of the runner will be weakened, weakening the blade’s work effectively. Therefore, it is crucial to choose the appropriate distance between the guide vane and blade for the unit’s operation.

It can be seen from Figure

It can be seen from Figure _{0} has the most obvious effect on the unit’s operating efficiency and output. As the blade relative torsion angle increases, the efficiency and output gradually increase, and the rate increases when △_{0} is small. The change of the blade relative torsion angle also significantly influences the pressure pulsation coefficient. When △_{0} increases from 20° to 30.5°, the pressure pulsation coefficient decreases significantly and then increases slowly. Besides, the blade deformation is affected by the blade relative twist. When △_{0} = 30.5°, the blade deformation is the smallest, but when △_{0} = 27°, the stress value is the smallest. It can be concluded that increasing the relative twist angle of the blade can increase the runner’s flow capacity, improve the flow pattern of the unit, and reduce the deformation of the blade. However, when the blade’s relative twist angle is too large, the blade is difficult to process, and the processing technology is difficult to guarantee.

It can be seen from Figure

When the cascade’s density is small, the flow capacity of the runner is large, the friction area is small, and the unit’s efficiency is improved. When the cascade density is large, the stress area of the blade increases, and the deformation and stress value of the blade increase, which requires the increase of blade strength and stiffness.

Through the range analysis, the better schemes obtained are (f), (g), (h), (i), (j), as shown in Table

The better solution of range analysis.

Evaluation indexes | Efficiency | Output | Pressure pulsation coefficient | Blade deformation | Blade static pressure value |
---|---|---|---|---|---|

Factor level | A_{2}B_{3}C_{2}D_{4}E_{2} | A_{2}B_{4}C_{2}D_{4}E_{2} | A_{1}B_{4}C_{2}D_{3}E_{4} | _{1}B_{3}C_{1}D_{4}E_{1} | A_{4}B_{3}C_{4}D_{2}E_{1} |

Better plan | (f) | (g) | (h) | (i) | (j) |

From the magnitude of the range

The five evaluation indexes of this orthogonal experiment are of equal importance. According to the better plan obtained by intuitive analysis and the better plan obtained by range analysis, a comprehensive frequency analysis of each factor is performed, as shown in Table

Frequency table of each factor level.

Factor level | _{1} | _{2} | _{3} | _{4} | _{1} | _{2} | _{3} | _{4} | _{1} | _{2} | C_{3} | _{4} | _{1} | _{2} | _{3} | _{4} | _{1} | _{2} | _{3} | _{4} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Frequency | 0 |

It can be seen from the table that level 2 of factor A, level 3 of factor B, level 2 of factor C, level 3 of factor D, and level 1 of factor E occur most frequently. Therefore, the optimal solution was determined to be A2B3C2D3E1. The specific parameters were as follows: the number of guide vanes was 16, the distance between guide vanes and blades was 1995 mm, the hub ratio was 0.31, the relative twist angle of the blades was 27°, and the cascade density was 0.72.

The following focuses on comparing the calculation results of the steady flow characteristics, unsteady flow characteristics, and unidirectional fluid-solid coupling of the bulb tubular turbine with the flow rate of 0.38Qr under low flow conditions before and after optimization.

After optimization, the unit’s small flow conditions’ efficiency increases from 86.01% to 90.93%, an increase of 5.72%, and the output increases from 1234.10 kW to 1269.42 kW, an increase of 2.86%. It can be concluded that the optimal scheme obtained through orthogonal test optimization can effectively improve the operating performance of the bulb tubular turbine unit.

Figure

Streamline pressure diagram of

Figure

Blade surface pressure and streamline diagram: (a) blade front before optimization; (b) blade back before optimization; (c) optimized blade front; (d) optimized blade back.

The thickness of the blade gradually decreases from the hub to the rim. The three cross-sections are evenly taken in the blade’s circumferential direction to analyze and optimize the change of the blade’s surface pressure from the hub to the rim. The cross-sections’ distribution positions from the hub to the rim are span = 0.25, span = 0.50, and span = 0.75. The specific cross-sectional position is shown in Figure

Location of blade cross-section.

Figure

Static pressure distribution on the blade surface before and after optimization: (a) span = 0.25; (b) span = 0.5; (c) span = 0.75.

From Figure

Table

Maximum amplitude of pressure pulsation at each monitoring point before and after optimization.

Monitoring points | A1–A4 | B1–B4 | C1–C4 | D1–D4 | |
---|---|---|---|---|---|

Frequency/Hz | 3 fn | 3 fn | 3 fn | 0.167 fn | |

Amplitude/% | 0.591 | 5.121 | 1.741 | 0.656 | |

Frequency/Hz | 3 fn | 3 fn | 0.167 fn | 0.167 fn | |

Amplitude/% | 0.308 | 1.375 | 0.512 | 0.484 |

Before optimization, the main frequency at the runner outlet is 3 fn, and the pressure pulsation amplitude is large. After optimization, the pressure pulsation amplitude at the monitoring point at this location is significantly reduced. The main frequency becomes 0.167 fn, indicating that the flow pattern is better when the optimization scheme is selected and the vortex is reduced. At this time, the runner outlet is more affected by the low-frequency pulsation of the draft tube.

After the unit is optimized, the maximum amplitude of pressure pulsation is significantly reduced, especially at the runner entrance, which is reduced by 73.15%, and the stability is improved.

Figure

Optimization of pressure pulsation at the inlet and outlet monitoring points of the runner: (a) time-domain graph before optimization; (b) frequency domain graph before optimization; (c) time-domain graph after optimization; (d) optimized frequency domain graph.

The structural stress and deformation of the runner blades before and after the orthogonal test optimization are compared and analyzed through one-way fluid-solid coupling calculation. Figures

Distribution of blade deformation before and after optimization: (a) before optimization; (b) optimized.

Static stress cloud of the blade before and after optimization: (a) before optimization; (b) optimized.

It can be seen from Figure

From Figure

After the turbine unit’s optimization, the maximum deformation value of the blade is reduced from 1.0754 mm to 0.64971 mm, a decrease of 39.59%; the maximum static stress value of the blade is reduced from 26.130 MPa to 22.691 MPa, a decrease of 13.16%. It can be seen that the optimal scheme obtained by the orthogonal test improves the structural strength of the runner part of the bulb tubular turbine.

In this article, the orthogonal test method is used to optimize the multiobjective bulb turbine. The five indexes of unit efficiency, output, pressure pulsation, blade deformation, and blade static stress are used as judgment criteria, revealing the impact of these five factors (number of guide vanes, the distance between the guide vanes and the blades, hub ratio, blade relative torsion angle, and cascade density) on the hydraulic performance, operating stability, and structural strength of the bulb tubular turbine. The performance of the turbine before and after the optimization is compared and analyzed. The conclusion is as follows:

The optimal scheme of bulb tubular turbine is as follows: the number of guide vane is 16, the distance between guide vane and the blade is 1995 mm, the hub ratio is 0.31, the relative twist angle of the blade is 27°, and the cascade density is 0.72.

Range analysis shows that blade relative torsion angle is an important geometric parameter in the hydraulic design of bulb tubular turbine. The number of guide vanes mainly affects the unit output; the distance between guide vane and blade mainly affects efficiency; hub ratio mainly affects pressure pulsation and blade deformation; cascade density mainly affects blade static stress. The relative twist angle change of the blade has the most significant influence on the four test evaluation indexes except the blade’s static stress.

After optimization, the hydraulic performance, stability, and structural strength of the unit are all improved, verifying the feasibility of the integrated frequency analysis method in the multiobjective orthogonal test of bulb tubular turbine. After optimization, the bulb tubular turbine’s efficiency and output are increased by 5.72% and 2.86% at a low flow rate, respectively. The vortex at the tailpipe is reduced, and the cavitation performance is well. The maximum pressure pulsation amplitude in the flow passage is reduced by 73.15%, the maximum blade deformation is reduced by 39.59%, and the maximum blade static stress is reduced by 13.16%.

The numerical simulation results are consistent with the model test data, and the error is within ±3%, which verifies the reliability of the numerical simulation calculation, indicating that the numerical calculation model and method used in this article can more accurately predict the performance of the bulb tubular turbine. In future work, the turbulence model’s optimization to further reduce the error is worthy of the effort. Moreover, the analysis and comparison of different design optimization methods should be the focus of the research.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

The project was supported by the Central University Fundamental Research Funds (2015B02814) and the IWHR Research and Development Support Program (HM0145B182017).