# Experimental and Numerical Analysis of the Clearance Effects between Blades and Hub in a Water Wheel Used for Power Generation

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## Abstract

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## 1. Introduction

## 2. Mathematical Model and Entropy Production Theory

#### 2.1. Governing Equation

_{i}represents speed, t is for time, x

_{i}and x

_{j}are coordinates, and μ and μ

_{t}represent molecular viscosity and turbulence viscosity, respectively.

#### 2.2. The VOF Method

_{q}represents the volume fraction of water in each grid. If α

_{q}= 1, the grid is full of water. If α

_{q}= 0, then there is no liquid in the grid. When between 0 and 1 is the free surface between water and air. When the volume fraction of water α

_{q}is determined, the volume fraction of gas α

_{p}= 1 − α

_{q}. According to the volume fraction determined by Equation (5), the density and dynamic viscosity coefficient in Equation (5) are modeled as follows:

#### 2.3. Standard k-ε Turbulence Model

_{k}= 1.0, C

_{1ɛ}= 1.44, C

_{2ɛ}= 1.92.

#### 2.4. Entropy Production Theory

_{v}means the mass fraction, ρ

_{m}Y is formulated from radiation, chemical reaction, and so forth. Moreover, s represents the specific entropy production, q stands for heat flux, and ϕ

_{l}and ϕ

_{v}represent chemical potentials.

_{total}:

## 3. Simulation Setup and Experimental Validation

#### 3.1. Computation Domain and Boundary Conditions

- (1)
- Inlet boundary: the water inlet boundary is set as the velocity inlet. The single variable method is adopted in that the inlet of water flow was set at 4 m/s to simulate the ultra-low head site condition. The direction of the inlet velocity is perpendicular to the inlet, and the turbulence intensity is set to a medium value of 5%. Considering that the water and air interface should be separated, the free surface level is set to the same height as the axis of rotation.
- (2)
- Outlet boundary: the outlet of water flow is set as the pressure outlet, which value is the same as atmospheric pressure.
- (3)
- Wall boundary: a no-slip wall is imposed on all other walls except the top of the area, which is set for symmetry [10].
- (4)
- During the calculation process, the influence of gravity is considered. The unsteady calculation method is used to simulate the gas–liquid two-phase, in which the sliding grid is used in the rotation field to ensure that the wheel rotates at 4 rpm.

#### 3.2. Numerical Method

#### 3.3. Grid Verification and Validation of Results

## 4. Performance and Flow Characteristics of Water Wheels

#### 4.1. Water Wheels Performance and Power Fluctuation

_{P}) was evaluated as [10]:

_{p}of water wheels under different clearance effects in a cycle. Observing that the water wheel with a clearance of 86 mm showed the optimal performance, with a calculated power coefficient of up to 0.528. This value is successively followed by the water wheels with clearance of 120 mm, 20 mm, and 50 mm, with power coefficients, which are 0.496, 0.445, and 0.441, respectively. The results of the torque curves show merely a slight difference between the water wheels with 20 mm and 50 mm clearances in the ability to capture water energy. The average power coefficient of water wheels, with clearance of 86 mm and 120 mm, proves the result of the torque fluctuation for a rotational cycle that the water wheel with the 86 mm clearance has the best performance. The torque fluctuation of the 86 mm configuration is more stable than the 120 mm one, which means that the efficiency trend is quite constant, and excessive clearance may increase leakage losses.

#### 4.2. Analysis of Flow Fields

_{p}is defined as:

_{p}= (Δd

_{i}− Δd

_{ave})/Δd

_{ave}

_{i}is defined as the torque at each time step. Δd

_{ave}is a time-average depth between the upstream and downstream.

_{P}) of water wheels, the results are divided into two groups of data for comparison, within the water area is α = 1, the air area is α = 0, and α = 0.5 is the air-water interface. Figure 7a shows a comparison between water wheels with 20 mm clearance and 50 mm clearance, and water wheels with 86 mm clearance and 120 mm clearance are demonstrated in Figure 7b.

#### 4.3. Energy Dissipation Analysis and Coupling Mechanism of Vorticity-Pressure

_{p}of the water wheel with 20 mm clearance fluctuates strongly, as shown in Figure 6, and is consistent with the complicated air–liquid mixing region shown in Figure 7a. Moreover, it can be observed that the outlet exhibits a large number of turbulent viscosity owning to an unsteady flow field under the operation of the 120 mm clearance water wheel. With the clearance enlarged to 50 mm, the turbulent viscosity distributed in the vicinity of the inlet between the left blades decreases; besides, the turbulent viscosity under the hub enhances to the largest during the operation of the water wheel with 50 mm clearance. Further, when the clearance increases to 86 mm, the turbulent viscosity distributed between the left blades abates to the minimum. The 86 mm clearance water wheel has the smallest turbulent viscosity over all the configurations, which presents that the 86 mm clearance water wheel has the steadiest operation flow field and the least energy losses. When the clearance increases to 120 mm, the excessive clearance will let a larger amount of air enters the water, causing unnecessary kinetic energy dissipation.

## 5. Characteristics and Distribution of Entropy Production

#### 5.1. Distributions of EPR of Water Wheels

#### 5.2. Distributions of EPWS of Water Wheels

## 6. Conclusions

- The numerical simulation and experimental results represent a relatively good agreement, with a maximum error of 3.5%.
- The torque of the water wheel has periodic transformation within a cycle of rotation, the maximum torque is reached at approximately 70.5° per period. Under the same working conditions, the performance of water wheels can be elevated by 8.7% by setting appropriate clearance.
- It was found that the average difference in water level is the highest in the water wheel with the 86 mm clearance configuration, and its fluctuation amplitude is the gentlest. The water wheels with 20 mm clearance and 120 mm clearance have a slight difference in the average difference in water level but the efficiency of maximum clearance (120 mm) is greater than that of minimum clearance (20 mm). The water wheel with optimal clearance can reduce air intruding into the water and attenuate the complexity of the water flow, potentially enhancing the performance of water wheels.
- It proved that the setting of water wheel clearance can attenuate outcomes of the blocking effect of the river channel and reduce the potential energy loss and kinetic energy dissipation while maintaining high efficiency. By comparing the most efficient two configurations of the vortex evolution and pressure distribution, it is illustrated that the optimal clearance can target eliminating vortical flow and improving flow adaptability, avoiding unnecessary energy losses, and decreasing the uneven pressure.
- As a new method, the irreversible energy loss characteristics can be intuitively diagnosed by using the entropy production method in water wheels under different clearance effects. The coupling mechanism of vorticity–pressure which will induce irreversible energy loss of the water wheel under different clearance effects is investigated. Hence, this research can provide a reference for the optimization of clearance between the hub and blade of water wheel performance.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

U_{i}, V | velocity [m/s] |

μ | dynamic viscosity [N·s/m^{2}] |

ρ | density [kg/m^{3}] |

F | Force [N] |

T | torque [N·m] |

g | Gravitational acceleration [m/s^{2}] |

α | volume fraction |

r | radius [m] |

q | heat flux [W/m^{2}] |

s | specific entropy [J/kg·K] |

P | power [W] |

ω | angular velocity [rev/s] |

A | submerged water area [m^{2}] |

Δd | water depths [m] |

θ | circumferential angle [deg] |

C_{P} | power coefficient |

ɛ | turbulent dissipation rate [m^{2}/s^{3}] |

${S}_{\overline{D}}$ | entropy production by direct dissipation [W/K] |

${S}_{D\prime}$ | entropy production by turbulent dissipation [W/K] |

${S}_{W}$ | entropy production by wall shear stress [W/K] |

Abbreviations | |

HPZ | high-pressure zone |

EPDD | entropy production rate by direct dissipation [W/(m^{3}·K)] |

EPTD | entropy production rate by turbulent dissipation [W/(m^{3}·K)] |

EPWS | entropy production rate by wall shear stress [W/(m^{2}·K)] |

Subscripts | |

t | turbulent |

- | Time-averaged value |

i, j | direction of Cartesian coordinates |

eff | Effective value |

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**Figure 6.**Difference in water level, Δd in a cycle (

**a**); Difference in non-dimensional parameter Δd

_{p}in a cycle (

**b**).

**Figure 7.**Water volume fractions at the wheel’s outer circumference. (

**a**) Between the clearance of 20 mm and 50 mm; (

**b**) Between the clearance of 86 mm and 120 mm.

**Figure 8.**Profiles of water volume fraction and turbulent viscosity of water wheels under different clearance effects at circumferential angle θ = 288°.

**Figure 9.**Contours of velocity vectors, water volume fraction, and vorticity in a cycle of the 86 mm clearance water wheel. (

**a**) The circumferential angle θ = 0°; (

**b**) The circumferential angle θ = 72°; (

**c**) The circumferential angle θ = 144°; (

**d**) The circumferential angle θ = 216°; (

**e**) The circumferential angle θ = 288°; (

**f**) The circumferential angle θ = 360°.

**Figure 10.**Contours of velocity vectors, water volume fraction, and vorticity in a cycle of the 120 mm clearance water wheel. (

**a**) The circumferential angle θ = 0°; (

**b**) The circumferential angle θ = 72°; (

**c**) The circumferential angle θ = 144°; (

**d**) The circumferential angle θ = 216°; (

**e**) The circumferential angle θ = 288°; (

**f**) The circumferential angle θ = 360°.

**Figure 11.**Pressure distribution on water wheels from bird’s eye views of the pressure side and suction side under different clearance configurations.

**Figure 15.**Distribution of EPWS in the hub of water wheels under different clearance configurations.

**Figure 16.**Distribution of EPWS in the blades of water wheels under different clearance configurations.

Elements (×10^{6}) | Averaged Torque (×10^{5} N·m) | |
---|---|---|

Mesh 1 | 1.26 | 2.24 |

Mesh 2 | 1.46 | 2.20 |

Mesh 3 | 2.43 | 2.44 |

Mesh 4 | 3.01 | 2.36 |

Material | Number of Blades | Flow Velocity (m/s) | Rotational Speed (rpm) | Output in Experiments (KW) | Output in Numerical Result (KW) | Error (%) |
---|---|---|---|---|---|---|

Q235 steel | 5 | 1.3 | 5 | 0.34 | 0.35 | 2.9 |

6 | 0.55 | 0.57 | 3.5 | |||

7 | 0.67 | 0.69 | 4.3 |

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**MDPI and ACS Style**

Feng, W.; Zheng, Y.; Yu, A.; Tang, Q. Experimental and Numerical Analysis of the Clearance Effects between Blades and Hub in a Water Wheel Used for Power Generation. *Water* **2022**, *14*, 3640.
https://doi.org/10.3390/w14223640

**AMA Style**

Feng W, Zheng Y, Yu A, Tang Q. Experimental and Numerical Analysis of the Clearance Effects between Blades and Hub in a Water Wheel Used for Power Generation. *Water*. 2022; 14(22):3640.
https://doi.org/10.3390/w14223640

**Chicago/Turabian Style**

Feng, Wenjin, Yuan Zheng, An Yu, and Qinghong Tang. 2022. "Experimental and Numerical Analysis of the Clearance Effects between Blades and Hub in a Water Wheel Used for Power Generation" *Water* 14, no. 22: 3640.
https://doi.org/10.3390/w14223640