Research on Modal Behavior of Large Francis Turbine Runner

The water-added-mass can affect the modal behavior of the Francis turbine runner largely. The sealing rings in the band-chamber and the crown-chamber outside of the runner are utilized to prevent leakage of water around the Francis turbine. The sealing rings and the chambers filled with water can also affect the modal behavior of the turbine. When the turbine runner operates in the power plant at various operating conditions, the seal clearances of the band-chamber and the crown-chamber filled with water may change the modal behavior of the large Francis turbine runner. Firstly, the 3D geometrical structure of the studied large-scale Francis turbine runner was constructed, and then the CAD model was meshed to establish the finite-element model for numerical analysis. Next, the modal shapes and the natural-frequencies of runners in the air were calculated, which is the traditional method used for the structural modal behavior analysis of the turbine runners. Finally, this paper investigates the modal behavior of a large-scale Francis runner prototype with the combined effects of sealing rings and water added-mass by direct water-structure coupled analysis via the finite element method (FEM) to obtain more realistic results. The results reveal that the effects of water-added-mass and sealing rings cause a significant decrease in the natural-frequency of the large Francis turbine runner, which may lead to an increase in the amplitude of vibrations and even cracks. Technical suggestions are proposed to improve the monitoring and maintenance in the power stations to maintain the safe and efficient operation of the large Francis turbine units.


Introduction
Energy crises and environmental problems pose significant threats to global sustainability.To achieve carbon peaking and carbon neutrality goals, the existing energy system is dominated by fossils and must be gradually updated to a renewable energy system with water-power, solar and wind power, etc.
Hydropower is a major clean renewable energy source, and plenty of hydropower stations have been built in different developed and developing countries over the past few decades.Due to the lack of understanding of the hydraulic turbine operation mechanism, there are many reports of turbine damage in hydropower stations around the world [1][2][3][4], and many researchers conducted fatigue analyses of the hydraulic turbine runners numerically and experimentally [2][3][5][6].This in turn confirms the importance of evaluating the structural modal behavior of turbine runners during the design phase rather than during operation.
Hydraulic turbine runners operate underwater, so the water-added-mass effect on the turbine runners needs to be studied carefully in the design stage to avoid vibration resonance and material fatigue caused by hydraulic excitation under operation.Some researchers have investigated the structural behavior of the Francis runners and Francis type of pump-turbine runners with simulations and measurements [7][8][9][10].However, the research on the combined impacts of sealing rings and wateradded-mass on the modal behavior of large prototype Francis runners needs to be further strengthened.
A large prototype Francis runner is selected in this study for modal behavior study.After building the 3D CAD model for the runner with surrounding water and sealing rings, a finite-element (FE) model is built to obtain the mode shapes and their natural-frequencies of the runner in air and also in water.A detailed analysis of the combined impacts of water-added-mass and sealing on the structural modal characteristics of the large Francis runner is presented by comparing the results of the calculations.

Method of Numerical Modal Analysis
Structural modal behavior analysis is one of the fundamental techniques for understanding the essential behavior of turbines.It helps to identify the critical natural-frequencies and vibration modes and to avoid structural resonance and damage.
The inherent modal behavior of the turbine runner is related to the boundary conditions of the runner in addition to its own structural properties.The runner is underwater when the unit is in operation, so it needs to consider the influence of the water around the turbine runner and the seal clearances of the crown-chamber and the band-chamber filled with water.
The water added-mass effect represents the extra inertia experienced by the runner structure due to the surrounding water.Therefore, the analysis of the model behavior of a large-scale Francis turbine runner using the finite element method (FEM) needs to take into account the complex coupling behavior between the water and the runner structure.
For the water-structure-coupling analysis, the structural dynamic response equilibrium equation based on the FEM can be expressed as where: [Ks] are the runner's mass matrix, damping matrix, and stiffness matrix,  , , are the runner's displacement, velocity, and acceleration vectors,  () is the external excitation load vector on the runner,  is the equivalent stiffness coupling matrices of water,  is the fluid pressure vector.
Since the water in the Francis turbine is considered to be slightly compressible, non-viscous and of uniform density, the hydrodynamic response equilibrium equation based on the FEM can be expressed as where: [Kf] are the water's equivalent mass, damping, and stiffness matrix,  is the equivalent mass coupling matrices of water.
Considering the above dynamic equilibrium equations of the runner together with the momentum and continuity equations of the water, the joint fluid-structure coupling equations can be adopted [9].

Turbine Runner Model for Numerical Modal Analysis
The prototype Francis turbine runner adopted in this study is a middle-head Francis runner with 13 blades.As shown in Fig. 1, after the Francis turbine runner of a hydropower station is installed, a crown-chamber is formed between the crown of the turbine runner and the turbine head-cover, and a band-chamber is formed between the band of the turbine runner and the turbine bottom-ring.When the turbine unit is in operation, the runner flow passage, the band-chamber and the crown-chamber chamber will be filled with water.The hydraulic turbine strives to maximize the utilization of water flow energy and minimize water leakage from the crown and band-chambers.Therefore, sealing rings are installed in the band-chamber and crown-chamber.The rated power of the studied Francis turbine is 180MW, and Table 1 lists other parameters of the prototype Francis turbine unit.The FE meshes of the Francis runner and water have been created, including fluid domains inside the turbine runner and in the chambers (Fig. 2).The surface between the turbine shaft and the turbine runner is fixed to perform the numerical modal analysis.The Young's modulus of the runner is 206GPa, the runner's density is 7700kg/m 3 , and the Poisson's ratio is 0.3.

Results and Discussion
The first 3 modes with repeated frequencies of the Francis runner are calculated.The nodal diameter (ND) is used to describe the vibration pattern of the mode shapes.The mode vibration shapes of the turbine runner in air and water present the same vibration patterns as 1ND, 2ND and 3ND.The runner's natural-frequencies in air and water for different vibration modes are calculated (Table 2).The frequency-reduction-ratio Δ is calculated by Eq.2 to accurately assess the combined effect of water-added-mass and sealing rings on the modal behavior of the runner.Δ=100* (fa -fw)/ fa (4) The frequency results of various vibration modes shown in Table 2 indicate that the water-addedmass and sealing rings have a significant impact on the natural-frequencies of the Francis runner.The natural-frequency of 1ND mode drops from 27.5Hz in air down to 13.7Hz in water.The naturalfrequency of 2ND mode in air is 32.5Hz in air, but only 15.9Hz in water.For the 3ND mode, the corresponding natural-frequency decreases from 54.6Hz in the air to 26.3Hz in the water.
As shown in Table 2, the frequency-reduction-ratio is 50% for 1ND mode, 51% for 2ND, and 52% for 3ND.It reveals that the effect of water-added-mass and sealing rings on the modal behavior of the Francis runner have to be seriously considered in the design stage to have a guaranteed design and avoid the potential structural resonances during operation in hydropower stations.

Conclusions
In this study, a coupled water-structure analysis using the finite element method is carried out to investigate the structural modal behavior of a large-scale Francis runner considering the combined effect of water-added-mass and the sealing ring.The obtained results are more accurate than the calculation only in the air which provided too high frequency values and misled the turbine design.
The natural-frequencies of 1ND, 2ND and 3ND are decreased dramatically with the presence of the surrounding water and sealing rings on the Francis runner.The frequency-reduction-ratios of the mode shapes of 1ND, 2ND and 3ND are in the range of 50% to 52%.Therefore, during the design phase, the effect of water-added-mass and sealing rings on the modal behavior of the Francis runner must be taken into account and the runner's natural-frequencies should be accurately calculated.This will help to avoid potential resonance risks and fatigue damage on the turbine runners.
During operation, it needs to check all types of possible excitation on the turbine runners with the online and offline monitoring systems and keep regular maintenance of the turbine unit to maintain the safe operation of hydro-generation units and hydropower stations.The same method proposed in this study can be applied to other machines submerged in water or other dense fluids.By adopting the finite element model of the Francis runner generated in this study, the flow-induced vibration behavior of the turbine can be investigated in the next stage.

Figure 1 .
Figure 1.Francis runner CAD model prototype with water.

Figure 2 .
Figure 2. The FE meshes of the prototype Francis turbine runner.

Figure 3 .
Figure 3. Model shapes of the prototype Francis runner.

Table 1 .
Main parameters for the prototype Francis turbine.

Table 2 .
Natural-frequency comparison of the prototype Francis turbine runner.