Generalized Kirchhoff and Riabouchinsky models with semepermeable obstacles and their application for estimating the efficiency of hydraulic turbines in open flow

Valentin M. Silantyev, Northeastern University, Boston MA

Classical and generalized Kirchhoff and Riabouchinsky models Open flow hydraulic turbines

Helical turbine invented by Prof. A.M.Gorlov(Northeastern University, MIME Department)

The conceptual view of the floating tidal power plant for Uldolmok Strait (South Korea)

A power plant being constructed in South Korea

Generalized Riabouchinsky model with a partially penetrable energy absorbing laminaClassical Riabouchinsky model with an impervious lamina

Generalized Kirchhoff model with a partially penetrable energy absorbing lamina

Classical Kirchhoff model with an impervious laminaIntroduction and basic definitions

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References:

[1] Silantyev V.M., Explicitly solvable Kirchhoff and Riabouchinsky models with partially penetrable obstacles and their application for estimating the efficiency of free flow turbines, Vychislitel’nye tekhnologii (to appear) [2] Gorban’A.N., Gorlov A.M., Silantyev V. Limits of the turbine efficiency for free fluid flow, ASME Journal of Energy Resources Technology, Dec. 2001.[3] Gorlov A.M., The Helical turbine: a new idea for low-head hydropower, Hydro Review, 14(1995), No. 5, pp. 44-50.[4] Gorban’A.N., Braverman M.E. and Silantyev V., Modified Kirchhoff flow with a partially penetrable obstacle and its application to the efficiency of free flow turbines, Math. Comput. Modelling, 35 (2002), no.13, pp. 1371–1375.[5] Gorban’ A.N. and Silantyev V., Riabouchinsky flow with partially penetrable obstacle, Math. Comput. Modelling 35 (2002), no.13, 1365 – 1370[6] Milne–Thomson L.M., Theoretical Hydrodynamics, 4th ed., Macmillan, New York 1960, 632pp. [7] Friedman A. Variational principles and free-boundary problems, 2nd ed. Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1988

Inclination angle, a

Efficiency, E Flow trough the lamina, s

0.00000 0.00000 0.00000 0.07854 0.01761 0.02294 0.15708 0.03646 0.04785 0.23562 0.06922 0.09168 0.31416 0.07771 0.10405 0.39270 0.09998 0.13559 0.47124 0.12320 0.16961 0.54978 0.14717 0.20623 0.62832 0.17164 0.24562 0.70686 0.19625 0.28793 0.78540 0.22050 0.33333 0.86394 0.24371 0.38199 0.94248 0.26494 0.43409 1.02102 0.28292 0.48983 1.09956 0.29582 0.54940 1.17810 0.30113 0.61302 1.25664 0.29521 0.68091 1.33518 0.27274 0.75331 1.41372 0.22569 0.83044 1.49226 0.14158 0.91259 1.57080 0.00000 1.00000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Efficiency E versus inclination angle α

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Table 1

The tables and graphs for the generalized Kirchhoff model

The table for the generalized Riabouchinsky model

Table 2Cavitation number, σ Inclination

angle, a 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

0.07854 0.01761 0.01787 0.01814 0.01841 0.01867 0.01894 0.01921 0.01949 0.01976 0.02003 0.02031

0.15708 0.03646 0.03700 0.03755 0.03810 0.03866 0.03922 0.03978 0.04034 0.04090 0.04147 0.04204

0.23562 0.06922 0.05735 0.05821 0.05906 0.05992 0.06079 0.06165 0.06252 0.06340 0.06427 0.06516

0.31416 0.07771 0.07887 0.08005 0.08123 0.08241 0.08360 0.08479 0.08599 0.08719 0.08839 0.08961

0.39270 0.09998 0.10148 0.10299 0.10451 0.10603 0.10756 0.10909 0.11063 0.11218 0.11373 0.11529

0.47124 0.12320 0.12504 0.12690 0.12877 0.13065 0.13253 0.13442 0.13632 0.13822 0.14014 0.14206

0.54978 0.14717 0.14938 0.15160 0.15383 0.15607 0.15832 0.16058 0.16285 0.16512 0.16741 0.16970

0.62832 0.17164 0.17421 0.17681 0.17941 0.18202 0.18465 0.18728 0.18993 0.19258 0.19525 0.19793

0.70686 0.19625 0.19919 0.20216 0.20513 0.20812 0.21112 0.21414 0.21716 0.22020 0.22325 0.22632

0.78540 0.22050 0.22381 0.22714 0.23048 0.23384 0.23722 0.24061 0.24401 0.24743 0.25086 0.25430

0.86394 0.24371 0.24737 0.25105 0.25475 0.25846 0.26220 0.26595 0.26971 0.27350 0.27730 0.28111

0.94248 0.26494 0.26892 0.27293 0.27695 0.28100 0.28506 0.28915 0.29325 0.29738 0.30152 0.30568

1.02102 0.28292 0.28717 0.29145 0.29575 0.30008 0.30443 0.30881 0.31321 0.31763 0.32207 0.32654

1.09956 0.29582 0.30028 0.30476 0.30927 0.31381 0.31838 0.32298 0.32761 0.33227 0.33695 0.34167

1.17810 0.30113 0.30567 0.31025 0.31486 0.31951 0.32420 0.32893 0.33370 0.33851 0.34335 0.34823

1.25664 0.29521 0.29967 0.30418 0.30875 0.31337 0.31804 0.32277 0.32756 0.33239 0.33729 0.34224

1.33518 0.27274 0.27688 0.28110 0.28541 0.28981 0.29429 0.29886 0.30352 0.30827 0.31310 0.31803

1.41372 0.22569 0.22917 0.23280 0.23660 0.24057 0.24470 0.24900 0.25346 0.25809 0.26288 0.26783

1.49226 0.14158 0.14392 0.14671 0.14995 0.15363 0.15773 0.16224 0.16714 0.17241 0.17803 0.18398

1.57080 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

Efficiency E versus flow through the lamina s

Figure 3Figure 2

Figure 4 Figure 5

Acknowledgements

The author is very grateful to Prof. A.M.Gorlov (MIME Dept., Northeastern University, Boston MA USA), whose oustanding achievements in the open flow turbine technology initiated this study and Prof. A.N.Gorban' (Institute of Computational Modeling, Krasnoyarsk, Russia) and Prof. A.S. Demidov (Moscow State University, Moscow, Russia) for helpful discussion.