# Flow and Fast Fourier Transform Analyses for Tip Clearance Effect in an Operating Kaplan Turbine

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Analysis

#### 2.1. Geometrical Model and Meshing

#### 2.2. Governing Equations

^{−}

^{5}was controlled by convection criteria. The unsteady simulation was carried out to investigate the dynamic behavior of the model Kaplan turbine. The time step of the 3° rotation of the runner blade was selected. In general, 4–5 cycles would be enough to get the stable unsteady flow, but because of the limitation of computer resources, two rotation cycles were made for case 1, and five cycles for case 2. Therefore, the time step was 7.29 × 10

^{−3}s. The total time for case 1 was 0.70 s, and 1.75 s for case 2. The transient stator–rotor was accounted to couple the rotation and stationary interface for the unsteady analysis. Furthermore, the FFT analysis of the Kaplan turbine investigated the stability of the operating conditions.

#### 2.3. Calculation of Hydraulic Performance

_{s}and obtained as the rate of the mechanical energy removal from the flowing fluid stream; it is defined as:

_{ni}is the calculated individual velocity along the face normal vector, and S

_{i}is the area of a face cell. As the neglecting friction and torque generated by pressure changes in turbomachinery, the shaft power L

_{s}of the hydraulic machine is defined as:

_{shaft}is the torque of the machine shaft. Lastly, the efficiency, ${\eta}_{t}$ of the turbine is expressed as:

## 3. Results and Discussion

#### 3.1. Validation of Numerical Results

#### 3.2. Performance Characteristics

^{3}/s; however, for case 2, the efficiency was only 84.06%, and power was 5.59 MW at a flow rate of 74.31 m

^{3}/s. It was expected to get the desired output at the rated condition of 6 MW, but the computed power was slightly less than 6 MW. It is also seen that the output difference was only 0.69%. Figure 12 shows the guide vane angle efficiency and versus flow rate performance characteristics of the CFD results for cases 1 and 2. It is seen from Figure 12 that the average deviation of the flow rate was only 1.46%. On top of this, the average efficiency was only 1.54%. In the case of efficiency at the rated condition, the difference was only 4.59% as shown in Figure 11 and Figure 12.

#### 3.3. Effect of Tip Clearance

#### 3.4. Pressure Pulsation Analysis

^{6}as shown in Figure 16a. Figure 16b shows the pressure pulsation results near the Kaplan turbine runner at unsteady intervals. Having an increased tip gap of unsteady flow, the oscillations on the horizontal axis are more than 10 times greater than the vertical axis, which is 10 times more than the normal Kaplan turbine pressure pulsations. For measuring the frequency of the generator, it was found that the pressure fluctuations generated by the generator were not significantly different from the pressure fluctuations of the vertical axis to the horizontal axis in the direction of the axial runner as shown in Figure 17a. As shown in Figure 17b, the pressure fluctuation occurs in the same cycle as the pressure fluctuation period of 6–7 Hz occurring in the runner vane. It was considered that the vibration caused by the unbalance flow of the runner vane of the tip gap of Kaplan turbine was transmitted to the generator.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

CFD | Computational Fluid dynamics |

FFT | Fast Fourier Transformation |

GCI | Grid convergence index |

g | Acceleration due to gravity, m/s^{2} |

H | Head, m |

L_{s} | Shaft power, kW |

p | Pressure, Pa |

Q | Flow rate, m^{3}/s |

r | Mesh ratio |

S_{i} | Area of a face cell |

T_{shaft} | Torque, N·m |

t | Time, s |

${u}_{i}$ | Velocity vector, m/s |

v | Velocity of fluid, m/s |

V_{ni} | Velocity along the face normal vector |

x | Component of position vector, m |

z | Elevation of water level, m |

Greek Symbols | |

ρ | Density, kg/m^{3} |

${\eta}_{t}$ | Turbine efficiency, % |

ω | Angular velocity, rad/s |

μ | Viscosity, Pa·s |

Subscript | |

i, j | Tensor indices |

1, 2 | Inlet, outlet |

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**Figure 9.**Comparison between computed and experimental output results as a function of runner vane opening angle.

**Figure 10.**Comparison between computed and experimental flow rates results as a function of guide vane opening angle.

**Figure 13.**Tip clearance (

**a**) velocity and (

**b**) pressure profiles between the runner blade and the shroud (case 2).

Description | Dimension |
---|---|

Runner outlet diameter | 1648.25 mm |

Head | 9.2 m |

Flow rate | 75.3 m^{3}/s |

Max. Power | 6000 kW |

Rotational speed | 171.4 rpm |

Runner blade | 4 |

Guide vane | 16 |

Description | Elements | Nodes | Y+ |
---|---|---|---|

Casing | 3,495,838 | 694,056 | ~478 |

Guide vane | 2,744,459 | 520,788 | ~276 |

Runner | 7,054,423 | 1,313,645 | ~597 |

Draft tube | 3,212,250 | 638,744 | ~121 |

Total | 1,6506,970 | 3,167,233 |

No. | Nodes | Grid Ratio, r | Efficiency (%) | Error, ε_{a} | GCI |
---|---|---|---|---|---|

1 | 1698866 | 1.31 | 88.602 | 0.11738 | 0.2047 |

2 | 2225771 | 1.14 | 88.706 | 0.00676 | 0.0269 |

3 | 2551448 | 1.08 | 88.712 | 0.03382 | 0.2359 |

4 | 2770562 | 1.14 | 88.742 | 0.00789 | 0.0321 |

5 | 3167233 | 1.07 | 88.749 | 0.00225 | 0.0171 |

6 | 2935178 | 0.97 | 88.747 | 0.02028 | 0.5708 |

7 | 3002617 | 0.86 | 88.729 | 0.03156 | 0.1619 |

No. | Guide Vane Angle (°) | Runner Vane Angle (°) | Elements | Nodes |
---|---|---|---|---|

1 | 23.5 | −2.25 | 6,292,574 | 1,213,316 |

2 | 34.5 | 3 | 11,690,834 | 2,142,607 |

3 | 46.87 | 7.95 | 8,418,222 | 1,576,519 |

4 | 55 | 13.07 | 6,119,611 | 1,185,878 |

5 | 61.82 | 18.3 | 6,118,300 | 1,185,676 |

6 | 67 | 23 | 16,506,970 | 3,167,233 |

7 | 68.5 | 23 | 11,685,688 | 2,141,639 |

8 | 69.1 | 25 | 12,626,332 | 2,305,743 |

9 | 72 | 25 | 12,628,024 | 2,306,005 |

No. | Guide Vane Angle (°) | Runner Vane Angle (°) | Elements | Nodes |
---|---|---|---|---|

1 | 23.5 | −2.25 | 11,109,862 | 2,007,133 |

2 | 34.5 | 3 | 42,082,030 | 7,466,191 |

3 | 46.87 | 7.95 | 8,783,048 | 1,643,035 |

4 | 55 | 13.07 | 6,647,494 | 1,281,519 |

5 | 61.82 | 18.3 | 14,069,184 | 2,603,090 |

6 | 67 | 23 | 27,291,793 | 4,937,129 |

7 | 68.5 | 23 | 12,210,473 | 2,236,618 |

8 | 69.1 | 25 | 18,238,111 | 3,309,172 |

9 | 72 | 25 | 18,239,803 | 3,309,434 |

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## Share and Cite

**MDPI and ACS Style**

Kim, H.-H.; Rakibuzzaman, M.; Kim, K.; Suh, S.-H.
Flow and Fast Fourier Transform Analyses for Tip Clearance Effect in an Operating Kaplan Turbine. *Energies* **2019**, *12*, 264.
https://doi.org/10.3390/en12020264

**AMA Style**

Kim H-H, Rakibuzzaman M, Kim K, Suh S-H.
Flow and Fast Fourier Transform Analyses for Tip Clearance Effect in an Operating Kaplan Turbine. *Energies*. 2019; 12(2):264.
https://doi.org/10.3390/en12020264

**Chicago/Turabian Style**

Kim, Hyoung-Ho, Md Rakibuzzaman, Kyungwuk Kim, and Sang-Ho Suh.
2019. "Flow and Fast Fourier Transform Analyses for Tip Clearance Effect in an Operating Kaplan Turbine" *Energies* 12, no. 2: 264.
https://doi.org/10.3390/en12020264