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The turbulent flow through a small horizontal Francis turbine is solved by means of Ansys-CFX at different operating points, with the determination of the hydrodynamic performance and the best efficiency point. The flow structures at different regimes reveal a large flow eddy in the runner and a swirl in the draft tube. The use of the mixture model for the cavity/liquid two-phase flow allowed studying the influence of cavitation on the hydrodynamic performance and revealed cavitation pockets near the trailing edge of the runner and a cavitation vortex rope in the draft tube. By maintaining a constant dimensionless head and a distributor vane opening while gradually increasing the cavitation number, the output power and efficiency reached a critical point and then had begun to stabilize. The cavitation number corresponding to the safety margin of cavitation is also predicted for this hydraulic turbine.

The inward-flow reaction water turbines known as Francis turbines are the result of many years of gradual development, which has resulted in very large units of hydraulic efficiency in excess of 80% and capable of transforming up to 95% [

The present day state-of-the-art CFD technique is considered as an alternative tool to provide insight into the flow characteristics of hydropower components, Helmut et al. [

This paper contributes to predicting the hydraulic performance and cavitating flows in the full passages of a small model of a Francis turbine at different operating conditions, by means of Ansys-CFX solver and considering

The model of a Francis turbine is taken from a hydraulic laboratory workbench, which includes a spiral case and a distributor with six adjustable vanes to make the right-angle turn and to control and feed the flow to the runner which has ten blades. The draft tube (transparent wall) decelerates the water flow leaving the runner and converts the excess of kinetic energy into a static pressure rise. The obtained CAD model is shown in Figure

Geometrical parameters of Francis turbine model.

Parameter | Value |
---|---|

Inlet diameter of spiral case | 38 |

Outlet diameter of spiral case | 149 |

Height of distributor vanes | 8 |

Inlet diameter of runner | 83 |

Outlet diameter of runner | 38 |

Outlet diameter of draft tube | 80 |

Number of blades of guide vane | 6 |

Number of blades of runner | 10 |

CAD model of Francis turbine.

The computational model is separated into four domains meshed separately and connected by interfaces. The distributor vanes and the runner used Turbo-Grid. Local refinements to boundary layers in the runner were applied to ensure values of

CFD solution is greatly affected by the size of grid elements, and this is why a grid dependency study was done for five different grid sizes varied from 1.2 to 3.8 million. Accordingly, the hydraulic efficiency revealed a stability for a total grid size equal to 3.2 million. The number of nodes in each part is as follows: in the distributor vanes it is equal to 870116, in the runner it is equal to 1689290, and in the spiral case, meshed by tetrahedral and hexahedral elements, the nodes number is equal to 250354, and for the draft tube it is equal to 390804 nodes. Figure

Grids of the computational domains.

Spiral case

Distributor vane

Runner

Draft tube

The simulation was carried out with complete flow passages consisting of spiral casing, distributor vanes, runner blades, and draft tube. The boundary conditions needed in the present simulations are as follows: mass flow set at the casing inlet and a static pressure at the outlet of draft tube. These are widely accepted boundary conditions for the simulation of hydraulic machines [

The Ansys-CFX solver [

The governing equations are integrated over each control volume. The volume’s integral is evaluated by considering the flow properties as constant on a control volume and set equal to the central value (mesh node). The surface integrals are evaluated at the integration points which are located at the center of each surface segment of the control volume. The solution field or solution gradients are approximated at the integration points by the nodal values, performed by using finite-elements. The advection scheme in Ansys-CFX is presented in the form of

In the mixture model, it is assumed that there exist the dynamic balance and the diffusion balance in both the liquid phase and the cavity phase of the cavitating flow and the velocities; temperatures and densities of both phases are identical to each other at every position in the whole two-phase flow field. The present used mixture model of cavity/liquid two-phase flow [

Continuity equation of mixture is

If ^{3} and the initial nuclei radius

The present study was performed for the entire flow rate range of this small Francis turbine at a constant head (

With the cavitation model, the suction head and the vapor pressure that depends on temperature are considered with respect to the operating conditions. The solution obtained without cavitation was used as initial result to carry out the computation of cavitating flows.

Firstly, the results of the global hydrodynamic performance parameters characterizing this Francis turbine are presented for different rotational speeds of the runner (1100, 1500, 1900, and 2360 rpm) and for four distributor vanes openings (32%, 64%, 72%, and 100%). Secondly, the visualization of internal flows through the turbine components with and without cavitation model is presented. Finally, the hydraulic efficiency and power were calculated based on the data obtained from the simulations without and under cavitation for the purpose of evaluating the performance drop. Figure

Produced power and comparison between prediction and test.

The hydraulic efficiency is calculated using the relation defined as follows, where

Produced power.

Hydraulic efficiency.

Figure

Ratio of efficiency versus coefficient of discharge.

Ratio of unit speed versus ratio of unit discharge.

The predicted performance curves are compared with the work of Aggidis and Židonis [

Figure

Streamlines at midplane of spiral casing and distributor at operating point (BEP).

The static pressure distribution (Figure

Static pressure contours at operating point (BEP) over the vanes and runner blades.

Figure

Velocity contours in distributor vanes and runner at three spans, for optimal point (BEP).

In the interblade channels, the decrease in flow speed near hub and shroud makes the fluid more susceptible to pressure gradient, with a fluid migration from pressure side to suction side leading to radial movements and a passage vortex. As observed from Figure

Streamlines in the runner at midspan: (a) optimal BEP, (b) low flow rate, and (c) large flow rate.

The incorporated elbow type draft tube with a single outflow channel decelerates the flow leaving the runner and converts the excess of outlet kinetic energy into a static pressure. Figure

Static pressure contour in draft tube.

The swirl component of the flow velocity appearing downstream of runner at low flow rates due to the radial component of flow velocity affects greatly the flow condition in the draft tube, so an unsymmetrical and nonuniform flow is obtained with an appearance of a vortex core. This decelerated flow with a swirl results in a vortex breakdown which is considered as the main reason for a severe pressure fluctuation. Francis turbines with a fixed pitch runner have a high level of residual swirl at the inlet of draft tube due to the mismatch between the swirl generated by the distributor vanes and due to angular momentum extracted by the runner [

Streamlines in draft tube at midplane for different operating conditions.

The performance of the draft tube is quantified by the pressure recovery coefficient

Static pressure recovery in the draft tube.

Cavitation plays an important role in reaction water turbines such as Kaplan and Francis turbines, but the main difference between them is the design of the runner which has a clear influence on the cavitation phenomenon and its location. The other two important parameters influencing its inception and development are the machine setting level and the operation at off-design conditions, when a liquid reaches a state at which vapor cavities are formed and grow due to dynamic pressure reduction to the liquid pressure vapor. In a flowing liquid, these cavities are subjected to pressure increase that reverses their growth, collapsing implosively and disappearing. The violent process of cavity collapse takes place in a very short time of about several nanoseconds, resulting in an emission of large amplitude shock waves as demonstrated by Avellan and Farhat [^{-6 }m, isothermal temperature equal to 198 K, and nuclei volume fraction: 0.5 × 10^{−5}.

At small flow rate, the formation of cavitation pockets is near the trailing edge of the runner blade. Figure

Water vapor volume fraction at the small flow rate.

If the flow from the runner has a strong swirl, the cavitated vortex rope is observed in the draft tube at a large flow rate (215 l/min). It is seen (Figure

Cavitated vortex rope at large flow rate in the outlet of runner.

Prediction of produced power and efficiency.

Power (W) |
Power (W) |
Efficiency % |
Efficiency % | |
---|---|---|---|---|

Optimal flow rate | 189.963 | 189.970 | 79.28 | 79.28 |

Large flow rate | 434.114 | 434.128 | 73 | 72.3 |

Small flow rate | 98.315 | 96.553 | 72 | 70 |

The setting level of a machine determines the pressure field in relation to the vapor pressure threshold. The bubble cavitation can appear even at the best efficiency operating point because it has a strong dependence on this level. To analyze the influence of cavitation number and the setting level on the hydraulic efficiency and power, simulations for three different operating points (Table

Characteristics of the operating conditions.

Low flow rate | Optimal BEP | Large flow rate | |
---|---|---|---|

Efficiency (%) | 78.87 | 79% | 78.8 |

Produced power (W) | 112.1 | 189.96 | 347 |

Rotational speed (rpm) | 1500 | 1900 | 2360 |

Unit discharge (m^{3}/s) |
0.12 | 0.14 | 0.17 |

Unit speed (rpm) | 37.95 | 48.06 | 59.7 |

Unit power (W) | 413.46 | 710.4 | 1305 |

For each operating point, eight simulations were performed at different values of cavitation number defined as follows:

Cavitation number and boundaries conditions.

Simulations for operating points | Cavitation number | Inlet head (m) | Static pressure at outlet (bar) |
---|---|---|---|

1 | 0.08 | 12 | 0.89 |

2 | 0.11 | — | 0.85 |

3 | 0.15 | — | 0.8 |

4 | 0.20 | — | 0.74 |

5 | 0.25 | — | 0.68 |

6 | 0.33 | — | 0.59 |

7 | 0.40 | — | 0.51 |

8 | 0.48 | — | 0.41 |

Figure

Cavitation effects on hydraulic efficiency and unit power: (a) at optimal point (BEP), (b) at large flow rate, and (c) at small flow rate.

The role of each component of this horizontal small Francis turbine and its effect on the hydrodynamic performance were investigated by simulating the single phase and the cavitating turbulent flows considering SST turbulence model and mixture cavitation model over the entire flow passages. The predicted performance depicts that the nominal point corresponds to a maximum of hydraulic efficiency of 79.28% and an important drop with increased discharge. The details of flow structures show that the most of losses are located in the runner where there are large vortices affecting the stability of operation. The velocity at inlet of draft tube has a substantial circumferential component that initiates a precession motion of a vortex of helical shape. According to the results without cavitation, it is clear that the off-design operating points are more affected, and from the simulations performed at different values of cavitation number the following conclusions are drawn:

The hydraulic efficiency is more affected by cavitation number compared to unit power.

The loss of hydraulic efficiency is shown to increase with the suction head.

The value of cavitation number for the optimal point (BEP) is smaller than that at off-design.

The cavitation number for the safety margin for a plant is equal to 0.48.

Discharge coefficient

Energy coefficient

Flow rate

External diameter of runner

Angular velocity

Vector of angular velocity

Head, total enthalpy

Static enthalpy

Rotational speed

Efficiency

Recovery coefficient

Specific speed

Static pressure

Power

Density

Thoma number

Correction coefficients

Torque

Radius of the bubble

Gas constant

Temperature

Kinetic energy of turbulence

Moore molecule weight

Coordinates

Volume fraction

Mass transfer rate

Stress

Kinematic viscosity of liquid phase

Surface tension

Gravitational acceleration

Velocity.

Optimal point

Inlet

Outlet

Liquid

Cavity phase

Specific

Nondissolved gas

Vapor.

The authors declare no competing interests.