Domain partitioned of characteristic curve in a Francis turbine under multiple boundary conditions

A complete characteristic curve was fundamental to the simulation of the transition process in a Francis turbine. However, only a high-efficiency region is provided in the efficiency hill diagram and combined characteristic curve in the turbine. It is necessary to expand the characteristic curve of the Francis turbine, thereby improving the simulation precision of the hydraulic turbine model, or the dynamic test. The main expanding was the external and the internal characteristics at present. The external characteristics with over-reliance on experience were only used in the mathematical way to increase the experimental data of the turbine efficiency hill diagram, particularly combing the characteristic curve and runaway speed curve. The internal characteristics were adopted in a theoretical way to process according to the internal laws of the turbine, but suffered from the difficult calculation and inaccurate results at present. In this research, a novel domain partitioned expanding method was proposed for the characteristic curve in a Francis turbine under multiple boundary conditions. The specific procedure was that: feature points were first taken from the boundary conditions of zero speed, zero discharge, runaway speed curve, and zero GVO, as well as the intersection point of unit torque, in order to divide the characteristic curve region, while constraining the extension range of each partition. Then, the structure parameters of a turbine were identified to calculate the boundary conditions of characteristic curves, according to the simplified mathematical model of internal and external characteristic data. Various fitting methods were also proposed for the characteristic curves of each region, according to the features of different regions. Finally, the fitting data on both sides of the boundary were connected smoothly, in order to form a complete turbine characteristic curve for the simulation of the transition process. The method in this study (MIS) was applied in an example of a hydraulic turbine, compared with the typical external characteristic method (TECM) and typical internal characteristic method (TICM). The performances of MIS and TICM were compared using the velocity triangle. The results showed that the MIS was more consistent with the inner structure of the runner. The fitting accuracy of experimental data was improved significantly via effectively integrating TECM and TICM, particularly considering the boundary conditions. The hydraulic characteristics and evolution of the turbine were qualitatively represented in the low-speed area, where the unit discharge was relatively stable. The unit discharge of each guide vane opening (GVO) gradually approached, with the increase of the unit speed and then decreased sharply after the runaway operation. The sign of unit discharge was changed, after passing the zero-discharge boundary. The MIS in the transition was reduced with the relative error of the maximum volute pressure from 2.03% to 1.69%, and the relative error of volute pressure in the small GVO region from 3.48% to 1.47%. It infers that the time-domain response of the dynamic process was closer to the measured data. The absolute average error of MIS was 0.010 MPa, when simulating the volute pressure oscillation in the range of small GVO region. It was reduced by 0.013MPa, compared with the absolute average error of 0.023MPa of TECM, while the relative error was reduced from 3.48% to 1.47%. The simulation of transition showed the time-domain response of the dynamic process was closer to the measured data. Hence, the expanding characteristic curve contributed to the calculation accuracy of transition in the modeling of hydropower units. The finding can provide strong support to the design of similar vane-type agricultural machinery, the expanding of characteristic curves, and the promotion of digital hydropower construction.