Abstract
To make the Kaplan turbine technology comparable to both the Pelton and the Francis turbine, the master equation for the Kaplan turbine has been established by analyses similar to that-for the Francis turbine. The analysis begins with the descriptions of free vortex flows at the runner inlet and the swirl flow at the impeller exit. By considering the Euler equation for specific work and by further evaluating the most significant shock and swirling losses, the first and the second energy equations in the form of hydraulic efficiency were formulated. The master equation is then established by combining both energy equations. In addition, three design equations and a new design parameter are presented. The master equation relates the turbine hydromechanics to the geometrical design of both the runner and the guide-vane parameters. It enables the complete hydraulic characteristics of a given Kaplan turbine to be analytically and simply computed. A computation example demonstrates the functionality and applicability of the method. With the reconstructed master equation, the runaway speed of the Kaplan turbine and its dependence on the guide-vane setting can be easily and precisely computed. For bulb turbines with guide vanes directly ahead of the turbine runner in the same tube, all computations are also applicable using another equivalent control parameter.
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References
Zhang Zh. Pelton turbines [M]. Cham, Switzerland: Springer-Verlag, 2016.
Zhang Zh. Master equation and runaway speed of the Francis turbine [J]. Journal of Hydrodynamics, 2018, 30(2): 203–217.
Gieseck J., Mosonyi E. Wasserkraftanlagen [M]. 6 Auflage, Berlin, Germany: Springer-Verlag, 2014.
Bohl W. Strömungsmaschinen 2 [M]. 8 Auflage, Würzburg, Germany: Vogel Fachbuch, 2012.
Quantz L., Meerwarth K. Wasserkraftmaschinen [M]. 11 Auflage, Berlin, Germany: Springer-Verlag, 1963.
Jehle C. Bau von Wasserkraftanlagen [M]. 6 Auflage, Berlin, Germany: Springer-Verlag, 2016.
Rai A., Kumar A. Analyzing hydro abrasive erosion in Kaplan turbine: A case study from India [J]. Journal of Hydrodynamics, 2016, 28(5): 863–872.
Tran C. T., Long X. P., Ji B. et al. Prediction of the precessing vortex core in the Francis-99 draft tube under off-design conditions by using Liutex/Rortex method [J]. Journal of Hydrodynamics, 2020, 32(4): 623–628.
Yang J., Zhou L. J., Wang Z. W. Numerical investigation of the cavitation dynamic parameters in a Francis turbine draft tube with columnar vortex rope [J]. Journal of Hydrodynamics, 2019, 31(5): 931–939.
Mulu B. G., Cervantes M. J., Devals C. et al. Simulation-based investigation of unsteady flow in near-hub region of a Kaplan Turbine with experimental comparison [J]. Engineering Applications of Computational Fluid Mechanics, March 2015, 9(1): 139–156.
Muntean S., Balint D., Susan-Resiga R. et al. 3D flow analysis in the spiral case and distributor of a Kaplan turbine [C]. 22nd IAHR Symposium on Hydraulic Machinery and Systems, Stockholm, Sweden, 2004.
Muntean S., Balint D., Susan-Resiga R. et al. Analytical representation of the swirling flow upstream the Kaplan turbine runner for variable guide vane opening [C]. 23rd IAHR Symposium on Hydraulic Machinery and Systems, Yokohama, Japan, 2006.
Zhang Zh. Hydraulic transients and computations [M]. Cham, Switzerland, Springer-Verlag, 2020.
ESHA. Guide on how to develop a small hydropower plant [R]. European Small Hydropower Association (ESHA), 2004.
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The author kindly and appreciatively asks the interested readers or researchers to make validations of the method presented in this paper with their experimental data.
Biography: Zh. Zhang (1957-), Male, Ph. D.
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Zhang, Z. Master equation, design equations and runaway speed of the Kaplan turbine. J Hydrodyn 33, 282–300 (2021). https://doi.org/10.1007/s42241-021-0020-1
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DOI: https://doi.org/10.1007/s42241-021-0020-1