# American Institute of Mathematical Sciences

December  2015, 8(6): 1373-1384. doi: 10.3934/dcdss.2015.8.1373

## A geometric inversion algorithm for parameters calculation in Francis turbine

 1 School of Water Conservancy and Electric Power, Hebei University of Engineering, Handan 056021, China, China, China 2 College of Mechanical Engineering, DongHua University, Shanghai 200051, China

Received  May 2015 Revised  September 2015 Published  December 2015

In terms of the structure and working~principle~of Francis turbine, a geometric inversion algorithm for parameters calculation in Francis turbine is proposed in this paper. Firstly through defining unit parameters the linear characteristics of turbine are derived in a certain opening, then the geometric parameters can be reversely calculated. The HL160-LJ--25 model turbine is used to verify the linear relation between the characteristic flow and the characteristic efficiency and reversely perform parameters calculation, and then the relation curves are established between the geometric parameters of turbine and the opening of guide blade, which can make us accurately acquire the energy characteristics of the prototype turbine. It is useful for us to acquire the proper parameters of turbine for purposes of reducing pressure fluctuation of turbine and improving its operating efficiency.
Citation: Liying Wang, Weiguo Zhao, Dan Zhang, Linming Zhao. A geometric inversion algorithm for parameters calculation in Francis turbine. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1373-1384. doi: 10.3934/dcdss.2015.8.1373
##### References:
 [1] R. K. Fisher and R. K. Donelson, Characteristics of francis turbines operating in cavitating regimes, in ASME Winter Annual Meeting, Massachusetts, 1983, 201-208. Google Scholar [2] R. Gerich and J. Raabe, Measurement of the unsteady and cavitating flow in a model francis turbine of high specific speed, Journal of Fluids Engineering, 97 (1975), 402-405. doi: 10.1115/1.3448042.  Google Scholar [3] J. G. I. Hellstrom, B. D. Marjavaara and T. S. Lundstrom, Parallel CFD simulations of an original and redesigned hydraulic turbine draft tube, Advances in Engineering Software, 38 (2007), 338-344. doi: 10.1016/j.advengsoft.2006.08.013.  Google Scholar [4] X. D. Lai, Analysis and estimation of hydraulic stability of Francis hydro turbine, Journal of Hydrodynamics, 16 (2004), 194-200. Google Scholar [5] J. Paik, F. Sotiropoulos and M. J. Sale, Numerical simulation of swirling flow in complex hydroturbine draft tube using unsteady statistical turbulence models, Journal of Hydraulic Engineering, 131 (2005), 441-456. doi: 10.1061/(ASCE)0733-9429(2005)131:6(441).  Google Scholar [6] D. Qian, W. Li, W. X. Huai and Y. L. Wu, The effect of runner cone design on pressure oscillation characteristics in a Francis hydraulic turbine, Journal of Power and Energy, 226 (2012), 137-150. doi: 10.1177/0957650911422865.  Google Scholar [7] R. A. Saeed, A. N. Galybin and V. Popov, Modeling of flow-induced stresses in a Francis turbine runner, Advances in Engineering Software, 41 (2010), 1245-1255. Google Scholar [8] R. Susan-resiga, G. D. Ciocan, I. Anton and F. Avellan, Analysis of the swirling flow downstream a Francis turbine runner, Journal of Hydraulic Engineering, 128 (2005), 177-189. doi: 10.1115/1.2137341.  Google Scholar [9] G. Wu, C. X. Wei, K. W. Zhang and L. R. Song, The relations between the operating condition and pressure fluctuation in draft tube of Francis turbine, Journal of Huazhong University of Science and Technology, 26 (1998), 88-91. Google Scholar [10] W. Zhao, L. Wang and L. Zhao, Pressure fluctuation identification of draft tube based on singular value decomposition and cascade correlation neural network, Journal of Vibroengineering, 16 (2014), 126-133. Google Scholar [11] W. Zhao and L. Wang, SVM multi-class classification based on binary tree for fault diagnosis of hydropower units, Information: An International Interdisciplinary, 15 (2012), 4615-4620. Google Scholar [12] L. M. Zhao, X. D. Wang and X. H. Wang, Calculation method on geometric parameters of runner outlet for model turbine and its application, Water Resources and Power, 29 (2011), 109-111. Google Scholar [13] L. M. Zhao and X. H. Wang, A new method to get the characteristics of francis turbines under small opening, Journal of Basic Science and Engineering, 18 (2010), 35-39. Google Scholar [14] Y. P. Zhu, X. Y. Shi and L. J. Zhou, Study on complete characteristic curve s based on internal characteristics, Journal of China Agricultural University, 11 (2006), 88-91. Google Scholar

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##### References:
 [1] R. K. Fisher and R. K. Donelson, Characteristics of francis turbines operating in cavitating regimes, in ASME Winter Annual Meeting, Massachusetts, 1983, 201-208. Google Scholar [2] R. Gerich and J. Raabe, Measurement of the unsteady and cavitating flow in a model francis turbine of high specific speed, Journal of Fluids Engineering, 97 (1975), 402-405. doi: 10.1115/1.3448042.  Google Scholar [3] J. G. I. Hellstrom, B. D. Marjavaara and T. S. Lundstrom, Parallel CFD simulations of an original and redesigned hydraulic turbine draft tube, Advances in Engineering Software, 38 (2007), 338-344. doi: 10.1016/j.advengsoft.2006.08.013.  Google Scholar [4] X. D. Lai, Analysis and estimation of hydraulic stability of Francis hydro turbine, Journal of Hydrodynamics, 16 (2004), 194-200. Google Scholar [5] J. Paik, F. Sotiropoulos and M. J. Sale, Numerical simulation of swirling flow in complex hydroturbine draft tube using unsteady statistical turbulence models, Journal of Hydraulic Engineering, 131 (2005), 441-456. doi: 10.1061/(ASCE)0733-9429(2005)131:6(441).  Google Scholar [6] D. Qian, W. Li, W. X. Huai and Y. L. Wu, The effect of runner cone design on pressure oscillation characteristics in a Francis hydraulic turbine, Journal of Power and Energy, 226 (2012), 137-150. doi: 10.1177/0957650911422865.  Google Scholar [7] R. A. Saeed, A. N. Galybin and V. Popov, Modeling of flow-induced stresses in a Francis turbine runner, Advances in Engineering Software, 41 (2010), 1245-1255. Google Scholar [8] R. Susan-resiga, G. D. Ciocan, I. Anton and F. Avellan, Analysis of the swirling flow downstream a Francis turbine runner, Journal of Hydraulic Engineering, 128 (2005), 177-189. doi: 10.1115/1.2137341.  Google Scholar [9] G. Wu, C. X. Wei, K. W. Zhang and L. R. Song, The relations between the operating condition and pressure fluctuation in draft tube of Francis turbine, Journal of Huazhong University of Science and Technology, 26 (1998), 88-91. Google Scholar [10] W. Zhao, L. Wang and L. Zhao, Pressure fluctuation identification of draft tube based on singular value decomposition and cascade correlation neural network, Journal of Vibroengineering, 16 (2014), 126-133. Google Scholar [11] W. Zhao and L. Wang, SVM multi-class classification based on binary tree for fault diagnosis of hydropower units, Information: An International Interdisciplinary, 15 (2012), 4615-4620. Google Scholar [12] L. M. Zhao, X. D. Wang and X. H. Wang, Calculation method on geometric parameters of runner outlet for model turbine and its application, Water Resources and Power, 29 (2011), 109-111. Google Scholar [13] L. M. Zhao and X. H. Wang, A new method to get the characteristics of francis turbines under small opening, Journal of Basic Science and Engineering, 18 (2010), 35-39. Google Scholar [14] Y. P. Zhu, X. Y. Shi and L. J. Zhou, Study on complete characteristic curve s based on internal characteristics, Journal of China Agricultural University, 11 (2006), 88-91. Google Scholar
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