# Analytical Implementation and Prediction of Hydraulic Characteristics for a Francis Turbine Runner Operated at BEP

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. AMOC Methodology Implementation

#### 2.1. Basic Assumptions

- The liquid is inviscid, single-phase and incompressible.
- The flow field in the runner is only subject to kinematics and described based on monistic theory, namely the velocity at a certain point is determined only by the streamwise distance from the origin of a streamline.
- The fluid flows smoothly in the blade channel, i.e., the relative velocity component is aligned with the local tangential direction of the relative spatial streamline.

#### 2.2. Implementation Procedures

- A model of the runner and the main geometry of the blade, such as wooden figures.
- Meridional channel geometry, such as the crown, band, and blade leading-edge and trailing-edge curves.
- The operational parameters at the BEP, such as available unit speed, head, and discharge.

#### 2.2.1. Coordinate Transformation

#### 2.2.2. Generatrix of Flow Section and Meridional Velocity

#### 2.2.3. Streamline Cluster Generation

_{m}′, the relative flow angle β

_{i}and the blockage factor 1/k

_{i}at each node on the SSL.

_{Ui}denotes the Eulerian energy at node i.

_{1}and S

_{2,}respectively, denote the cross sections upstream and downstream of the runner.

_{Ub}

_{1}and E

_{Ub}

_{2}separately denote Eulerian energies at the blade inlet and outlet. H

_{0}is the rated head and η

_{r}

_{0}is the runner’s hydraulic efficiency at the BEP as computed by the AMOC.

_{UJ}denotes the Eulerian energy on the Jth SSL. L

_{J}denotes the length of the subsection. ξ

_{J}denotes the weighted factor of each segment.

#### 2.3. Transformation of Velocity Component

## 3. Case Study and Results Discussion

#### 3.1. Basic Parameters of Runner Model

#### 3.2. Validation and Result Analyses

#### 3.2.1. Time Consumption and Turbine Hydraulic Efficiency

#### 3.2.2. Velocity Distribution in Blade Channel

#### 3.2.3. Eulerian Energy in Blade Channel

_{0}are used to normalize the coordinate and Eulerian energy, respectively, of each point on the corresponding SSL. Figure 8a displays the resultant distribution of the dimensionless Eulerian energy on five spatial streamlines. The contour of this energy on the S2 surface is illustrated in Figure 8b. Obviously, Eulerian energy shows both a spanwise ascent and a streamwise descent trend. On the other hand, the fluid potential energy is converted to mechanical energy by pushing the runner. This process is reflected in the figures as the Eulerian energy declines from the blade inlet towards the outlet. Hence, the slope of the curves in Figure 8a can be regarded as a criterion to define the local ability of energy conversion. This ability is proportional to the absolute value of the specific slope. By extension, a larger absolute value corresponds to the steep decline of the curve, triggering more energy conversion. Therefore, the corresponding location on the blade works more efficiently. As for the five curves in the figure, the gradient varies with the streamwise position. In a similar manner, it can be inferred that the whole blade surface converts energy unevenly, as seen in the color map in Figure 8b. This non-uniform energy transition affects the runner performance from a micro perspective and, in turn, provides a decisive step in the runner optimization, design and manufacturing process. Additionally, the right figure schematically highlights two special locations of maximal and minimal Eulerian energy, showing the same behavior as the absolute velocity in Figure 6b, and the energy value is observed to be close to zero on the outlet curve. This happens to coincide with the absence of residual momentum downstream of the runner in the optimal case. Similar distribution patterns can be found between Figure 7b and Figure 8b. Due to the high dimensional interpolation required to obtain Figure 8b, the shape of the S2 surface is slightly changed. The general contour distribution is still in accord with the Fluent results.

#### 3.2.4. Pressure Difference over the Blade

^{+}and p

^{−}separately denote the static pressure on the pressure surface and the suction surface of the blade.

_{p}distribution seems to be chaotic due to the shape of the spatially twisted blade, while the streamwise distribution basically declines. During the first half chord, namely 0 < m’ < 0.5, most of the load was applied to the region adjacent to both the crown and the band, whereas the medium region was subjected to less load. On the contrary, the liquid load mainly worked on the downstream region near the band for 0.5 < m’ < 1. This coincides with the higher Eulerian energy near the lower part at the blade outlet, as shown in Figure 7b and Figure 8b. It is notable that on the edge of the blade outlet (m′ = 1) the pressure difference is slightly away from zero. This is probably attributed to the enlarged turbulence by the blade-channel vortex and to the outlet residual circulation, which ensured a preferable wake flow and further, a higher runner efficiency. In such a case, the flow pattern leaving the blade fails to meet the Kutta–Joukowski condition. In addition, when C

_{p}reaches the peak value, the corresponding velocity difference over the blade also becomes the largest. The flow is likely to separate from the blade surface, and this separation can be arithmetically predicted by the definition of stagnation enthalpy in rotational machinery. This scenario will be quantitatively analyzed in the future work combining CFD technology. In summary, the C

_{p}distribution mainly depends on the blade geometry and flow pattern in the blade channel. By calculating this value, the specific load distribution can be obtained, and the local energy conversion ability on the blade can be clearly evaluated.

## 4. Conclusions and Outlook

## 5. Patents

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Coordinate Transformation

_{i}, y

_{i}, z

_{i}) denotes the coordinates of the ith node in the Cartesian system. (m

_{i}, θ

_{i}) denotes the coordinates of the ith node in the curved system. (x

_{0}, y

_{0}) are the coordinates of the SSL origin. Δz

_{i}= z

_{i}− z

_{i}

_{−1}is the axial interval of the adjacent nodes. Δr

_{i}= r

_{i}− r

_{i}

_{−1}is the radial interval of the adjacent nodes.

#### Appendix A.2. Generatrix of Flow Section and Meridional Velocity

_{m}are subsequently computed:

_{0}denotes the discharge at the BEP.

#### Appendix A.3. Streamline Cluster Generation

#### Appendix A.4. Relative Flow Angle

_{i}denotes the relative flow angle at the ith node and τ

_{i}denotes the interpolating coefficient.

**Figure A5.**Velocity triangle and flow angle at different locations on SSL: (

**a**) velocity triangle; (

**b**) origin of streamline; (

**c**) endpoint of streamline; (

**d**) medium positions.

#### Appendix A.5. Meridional Velocity at SSL Nodes and Effects of Blade Thickness

_{m}on SL. Therefore, based on the (m, V

_{m}) pairs on the SL, V

_{m}at each node on the SSL is subsequently interpolated.

_{i}denotes the radius of the ith node. N denotes the blade number. e

_{i}denotes the corresponding thickness derived from diameter of the incircles in the wooden blade figure. V

_{m}′ denotes the revised meridional velocity at node i. Then the blockage factor, accounting for the thickness effect, is defined as:

#### Appendix A.6. Velocity Triangle

_{0}is the rated rotation speed.

_{i}and the corresponding absolute flow angle α

_{i}can be calculated. Then, V

_{i}is projected in the circumferential direction as:

_{Ui}denotes the tangential component of V

_{i}at node i.

## Appendix B

#### Transformation of Velocity Component

_{X}, V

_{Y}and V

_{Z}. Specific decomposition is shown in Figure A6, and the positive direction of each component coincides with the coordinate axis.

_{Z}. Recalling the vector operation:

_{r}is the projection of W onto vector ${F}_{i+1}{F}_{i+1}\prime $. Then the following two equations are deduced:

_{Z}defines its vertically downward direction. The angle between the Y-axis and vector ${F}_{i+1}\prime {F}_{i+1}$ is set as ${\gamma}_{4}$ as shown in Figure A6b. Then

**j**is the positive unit vector of the Y-axis. The above equation is later elaborated by the coordinate operation and ${\gamma}_{4}$ is solely determined in [0, π]. That is:

_{X}and V

_{Y}can be obtained by decomposing V

_{r}and V

_{U}along the X-axis and Y-axis, respectively:

_{r}and V

_{U}are presented in Figure A6b. In particular, at the origin of SSL, namely i = 1, the angle between W

_{1}and ${F}_{1}{F}_{1}\prime $ is approximated by the angle between ${F}_{1}{F}_{2}$ and ${F}_{1}{F}_{1}\prime $. Using the above procedures in this section, one can successfully convert the velocity components from the velocity triangle to the Cartesian coordinate system.

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**Figure 6.**Distribution of absolute velocity on the SSL by AMOC: (

**a**) Velocity vector and streamline; (

**b**) Velocity scalar.

**Figure 7.**Distribution of results obtained by the Navier–Stokes solver on S2 surface: (

**a**) Absolute velocity; (

**b**) Dimensionless Eulerian energy.

**Figure 9.**Distribution of C

_{p}along spatial streamlines: (

**a**) AMOC implementation; (

**b**) ANSYS Fluent Navier–Stokes solver.

Geometric Dimensions | D1 mm | D2 mm | N | Blade shape |

372.9 | 366.0 | 15 | X | |

BEP Parameters | n_{0} r/min | Q_{0} m^{3}/s | H_{0} m | ρ kg/m^{3} |

1122 | 0.492 | 30 | 997 |

SSL | V m/s | W m/s | U m/s | β ° | α ° | E_{U}m ^{2}/s^{2} | L mm | $\overline{{\mathit{E}}_{\mathit{U}}}\phantom{\rule{0ex}{0ex}}{\mathbf{m}}^{2}/{\mathbf{s}}^{2}$ | |
---|---|---|---|---|---|---|---|---|---|

Blade inlet | SSL-0 | 16.229 | 4.195 | 17.085 | 71.236 | 14.169 | 268.829 | 7.905 | 305.507 |

SSL-0.0625 | 16.361 | 4.179 | 17.100 | 72.828 | 14.124 | 271.325 | |||

7.960 | |||||||||

SSL-0.125 | 16.445 | 4.186 | 17.133 | 73.538 | 14.131 | 273.225 | |||

15.770 | |||||||||

SSL-0.25 | 16.614 | 4.230 | 17.274 | 74.005 | 14.166 | 278.274 | |||

15.577 | |||||||||

SSL-0.375 | 16.852 | 4.290 | 17.534 | 73.849 | 14.153 | 286.505 | |||

15.288 | |||||||||

SSL-0.5 | 17.219 | 4.343 | 17.934 | 73.603 | 14.004 | 299.634 | |||

15.122 | |||||||||

SSL-0.625 | 17.602 | 4.436 | 18.446 | 72.174 | 13.881 | 315.193 | |||

14.758 | |||||||||

SSL-0.75 | 17.932 | 4.594 | 19.067 | 68.891 | 13.826 | 332.012 | |||

14.080 | |||||||||

SSL-0.875 | 18.273 | 4.776 | 19.853 | 64.022 | 13.591 | 352.623 | |||

13.066 | |||||||||

SSL-1 | 18.511 | 5.081 | 20.726 | 57.639 | 13.406 | 373.192 | |||

Blade outlet | SSL-0 | 5.385 | 8.943 | 5.921 | 35.663 | 104.470 | −7.966 | 23.288 | 26.846 |

SSL-0.0625 | 5.533 | 10.272 | 6.761 | 29.733 | 112.968 | −14.597 | |||

21.686 | |||||||||

SSL-0.125 | 5.160 | 10.012 | 7.741 | 30.516 | 99.864 | −6.843 | |||

35.446 | |||||||||

SSL-0.25 | 5.392 | 9.451 | 9.767 | 32.539 | 70.507 | 17.574 | |||

26.720 | |||||||||

SSL-0.375 | 5.853 | 10.144 | 11.594 | 30.310 | 61.004 | 32.895 | |||

20.423 | |||||||||

SSL-0.5 | 6.031 | 11.290 | 13.249 | 26.973 | 58.104 | 42.221 | |||

17.256 | |||||||||

SSL-0.625 | 6.450 | 12.061 | 14.771 | 25.331 | 53.139 | 57.150 | |||

14.977 | |||||||||

SSL-0.75 | 6.983 | 12.593 | 16.175 | 24.245 | 47.777 | 75.907 | |||

13.097 | |||||||||

SSL-0.875 | 6.720 | 14.220 | 17.475 | 21.495 | 50.842 | 74.151 | |||

12.220 | |||||||||

SSL-1 | 7.104 | 14.881 | 18.681 | 20.737 | 47.879 | 89.001 | |||

η_{r}_{0} % | 94.69 |

Case and Variable | n_{11} r/min | Q_{11} m^{3}/s | η % | Cavitation Coefficient |
---|---|---|---|---|

BEP | 73.7 | 670 | 94.63 | / |

Limitation case | / | 992 | 86.60 | 0.08 |

Methods | Efficiency Value % | ||
---|---|---|---|

AMOC | 94.69 | 94.69 | |

Test | 94.63 | 94.63 | |

CFD | 94.57 | 94.57 | |

Relative error % | 0.06 | 0.06 | 0.13 |

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## Share and Cite

**MDPI and ACS Style**

Chen, Y.; Zhou, J.; Karney, B.; Guo, Q.; Zhang, J.
Analytical Implementation and Prediction of Hydraulic Characteristics for a Francis Turbine Runner Operated at BEP. *Sustainability* **2022**, *14*, 1965.
https://doi.org/10.3390/su14041965

**AMA Style**

Chen Y, Zhou J, Karney B, Guo Q, Zhang J.
Analytical Implementation and Prediction of Hydraulic Characteristics for a Francis Turbine Runner Operated at BEP. *Sustainability*. 2022; 14(4):1965.
https://doi.org/10.3390/su14041965

**Chicago/Turabian Style**

Chen, Yu, Jianxu Zhou, Bryan Karney, Qiang Guo, and Jian Zhang.
2022. "Analytical Implementation and Prediction of Hydraulic Characteristics for a Francis Turbine Runner Operated at BEP" *Sustainability* 14, no. 4: 1965.
https://doi.org/10.3390/su14041965