4.3.2. Effect of Operating Condition on the Erosion Rate

Figure 13 shows the influence of the guide vane opening and flow velocity on the erosion rate at the stay vanes, guide vanes, and runner. The erosion rate on the walls is calculated as:

$$Er = \frac{1}{A} \int Er\_f dA\tag{32}$$

where *Erf* is the facet value of the erosion rate and A is the cell area. The erosion increases when increasing guide vane opening since the velocity and the amount of particles impacting the walls is multiplied due to the rise in water flow. A dramatic increase in the erosion rate was observed for operating conditions with guide vane openings over 90%. The sudden increase in erosion rate observed past a certain operating point may be related to the increase in the intensity of turbulent vortices near the outlet of the blade. Previous works [34,35] have found a direct relation among turbulent flow, vortex formation, and accelerated erosion. Vortices and recirculation accelerate particles in the flow and change

the impingement angle to critical values. Figure 14 shows the turbulence intensity of the flow surrounding the blade for the different operating conditions.

**Figure 13.** Erosion rate as a function of (**a**) guide vane opening and (**b**) inlet velocity.

**Figure 14.** Flow turbulence intensity for different operating conditions.

#### **5. Conclusions**

A CFD study replicating the operating conditions of the Francis turbines of San Francisco hydropower plant in Pastaza, Ecuador, was carried out. Flow conditions and erosion patterns were studied for different performance points, obtaining a detailed prediction of wear damage in different turbine components. From the results of the numerical analysis, the following can be concluded

Erosion damage increases significantly for higher flow rates, when the opening of the guide vane exceeds an 85% aperture considering the closed position as a reference.

Operating the turbines at the previously mentioned conditions would result in unnecessary and accelerated erosion wear since the best performance point was obtained at a lower flow rate.

The operation of Francis turbines in sediment-laden rivers should be carried out with particular consideration of the effect that guide vane opening has on the formation of turbulent flow and vorticity. This situation can lead to accelerated erosion rates since vortices and recirculation can accelerate particles in the flow and change the impingement angle to critical values.

CFD is a powerful tool that can be used to prevent such occurrences and analyze the operating conditions in hydropower plants that best harness the available power without sacrificing mechanical integrity.

This study was conducted with the aim of contributing to the creation of a clear and cost-effective strategy to prevent and reduce erosion in existing hydropower plants and proposing an effective erosion-based operating procedure for Francis turbines in the Andean region.

**Author Contributions:** Conceptualization, E.C.; methodology, E.C. and E.V.; software, C.C.; validation, C.C.; formal analysis, C.C. and D.J.; investigation, C.C. and D.J.; resources, E.C., C.M. and E.V.; data curation, E.V.; writing—original draft preparation, C.C. and D.J.; writing—review and editing, C.C., E.C., E.V. and X.L.; visualization, C.C. and D.J.; supervision, E.C. and I.Z.; project administration, E.C. and I.Z.; funding acquisition, E.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Escuela Politécnica Nacional (PIS 19-06).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** The authors gratefully acknowledge the financial support provided by Escuela Politécnica Nacional through the project PIS 19-06.

**Conflicts of Interest:** The authors declare no conflict of interest.
