Journal of Hydrodynamics

, Volume 30, Issue 2, pp 203–217 | Cite as

Master equation and runaway speed of the Francis turbine

  • Zh. Zhang


The master equation of the Francis turbine is derived based on the combination of the angular momentum (Euler) and the energy laws. It relates the geometrical design of the impeller and the regulation settings (guide vane angle and rotational speed) to the discharge and the power output. The master equation, thus, enables the complete characteristics of a given Francis turbine to be easily computed. While applying the energy law, both the shock loss at the impeller inlet and the swirling loss at the impeller exit are taken into account. These are main losses which occur at both the partial load and the overloads and, thus, dominantly influence the characteristics of the Francis turbine. They also totally govern the discharge of the water through the impeller when the impeller is found in the standstill. The computations have been performed for the discharge, the hydraulic torque and the hydraulic efficiency. They were also compared with the available measurements on a model turbine. Excellent agreement has been achieved. The computations also enable the runaway speed of the Francis turbine and the related discharge to be determined as a function of the setting angle of the guide vanes.

Key words

Francis turbine master equation runaway speed hydraulic efficiency shock loss swirling loss streamline similarity method 


  1. [1]
    Drtina P., Sallaberger M. Hydraulic turbines–basic prin-ciples and state-of-the-art computational fluid dynamics applications [J]. Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science, 1999, 213(1): 85–102.CrossRefGoogle Scholar
  2. [2]
    Zeng W., Yang J., Cheng Y. Construction of pump-turbine characteristics at any specific speed by domain-partitioned transformation [J]. Journal of Fluids Engineering, 2015, 137(3): 031103.CrossRefGoogle Scholar
  3. [3]
    Zeng W., Yang J. D., Cheng Y. G. et al. Formulae for the intersecting curves of pump-turbine characteristic curves with coordinate planes in three-dimensional parameter space [J]. Proceedings of the Institution of Mechanical Engineers Part A Journal of Power and Energy, 2015, 229(3): 324–336.CrossRefGoogle Scholar
  4. [4]
    Zhang Zh., Titzschkau M. Self-validated calculation of characteristics of a Francis turbine and the mechanism of the S-shape operational instability [J]. IOP Conference Series: Earth and Environmental Science, 2012, 15(3): 032036CrossRefGoogle Scholar
  5. [5]
    Zhang Zh. Streamline similarity method for flow distribu-tions and shock losses at the impeller inlet of the centrifugal pump [J]. Journal of Hydrodynamics, 2018, 30(1): 140–152CrossRefGoogle Scholar
  6. [6]
    Hasmatuchi V., Roth S., Botero F. et al. High-speed flow visualization in a pump-turbine under off-design operating conditions [C]. 25th IAHR Symposium on Hydraulic Machinery and Systems, Timisoara, Romania, 2010.Google Scholar
  7. [7]
    Staubli T., Senn F., Sallaberger S. Instability of pump-turbines during start-up in turbine mode [C]. Hydro 2008, Ljubljana, Slovenia, 2008.Google Scholar
  8. [8]
    Martinm C. Instability of pump-turbines with S-shaped characteristics [C]. 20th IAHR Symposium on Hydraulic Machinery and Systems, Charlotte, USA, 2000.Google Scholar
  9. [9]
    Zuo Z., Fan H., Liu S. et al. S-shaped characteristics on the performance curves of pump-turbines in turbine mode–A review [J]. Renewable and Sustainable Energy Reviews, 2016, 60: 836–851.CrossRefGoogle Scholar
  10. [10]
    Sun H., Xiao R., Liu W. et al. Analysis of S characteristics and pressure pulsations in a pump-turbine with misaligned guide vanes [J]. Journal of Fluids Engineering, 2013, 135(5): 511011.CrossRefGoogle Scholar
  11. [11]
    Cavazzini G., Covi A., Pavesi G. et al. Analysis of the unstable behavior of a pump-turbine in turbine mode: Fluid-dynamical and spectral characterization of the S-shape characteristic [J]. Journal of Fluids Engineering, 2016, 138(2): 021105.CrossRefGoogle Scholar
  12. [12]
    Merino J. M., Lopez A. ABB Varspeed generator boosts efficiency and operating flexibility of hydropower plant [J]. Fuel and Energy Abstracts, 1996, 37(4): 272.Google Scholar
  13. [13]
    Terens L., Schäfer R. Variable speed in hydropower generation utilizing static frequency converters [C]. Pro-ceedings of the International Conference on Hydropower, Nashville, Tennessee, 1993, 1860–1869.Google Scholar
  14. [14]
    Käling B., Schütte T. Adjustable speed for hydropower applications [C]. ASME Joint International Power Gene-ration Conference, Phoenix, Arizona, 1994.Google Scholar
  15. [15]
    Susan-Resiga R., Ciocan G., Anton I. et al. Analysis of the swirling flow downstream a Francis turbine runner [J]. Journal of Fluids Engineering, 2006, 128(1): 177–189.CrossRefGoogle Scholar
  16. [16]
    Nielsen T. K., Olimstad G. Dynamic behaviour of rever-sible pump-turbines in turbine mode of operation [C]. The 13th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Honolulu, USA, 2010.Google Scholar
  17. [17]
    ESHA. Guide on how to develop a small hydropower plant [R]. European Small Hydropower Association (ESHA), 2004.Google Scholar

Copyright information

© China Ship Scientific Research Center 2018

Authors and Affiliations

  1. 1.Free ResearcherZurichSwitzerland

Personalised recommendations