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

1. 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 g Open flow hydraulic turbines eneralized Kirchhoff and Riabouchinsky models
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
Classical Riabouchinsky model with an Generalized Riabouchinsky model with a partially penetrable energy absorbing lamina
impervious lamina
Generalized Kirchhoff model with a partially
penetrable energy absorbing lamina
Introduction and basic definitions Classical Kirchhoff model with an impervious lamina
g
g ¢
Wake
Flow
domain
g
Cavity
g ¢
Flow
domain
Wg
Wg
Virtual
obstacle
a) Kirchhoff model b) Riabouchinsky model
y
C = ¥ g
A
O
A¢
1
-1
x O
g ¢
a) z-plane b) Potential w-plane
p
-p
A¢
c) Hodograph V - plane d) t-plane
A
A¢
C = ¥
u
v
2
2
x
h
O = ¥ C
A
-1 1
C = ¥
A¢ O A
Wg
y
A
A¢
1
-1
O
C = ¥
g
s
x O
g ¢
A
a
A¢
C = ¥
u
v
a) z-plane b) Potential w-plane
-1 1
C = ¥
A¢ O A
h
p -a
2
- p +a
2
x
A¢
Wg
O = ¥ C
A
c) Hodograph V - plane d) t-plane
A
O
A¢
1
-1
a) z-plane b) Potential w-plane
O
c) Hodograph V - plane d) t-plane
e) T-plane f) a-plane
C = ¥
g
g ¢
x
y M
M¢
A
A¢
O
C = ¥
M
M¢
u
v
2 p
2 p
-
x
h
O = ¥ C
A¢
A
M
M¢
g lnV M¢ A¢ A M
0 t 0 - t
C = ¥
-1 1
O = ¥
A M C M¢ A¢
-1 1
C
A M M¢ A¢
O = ¥
0
1
t
1
t 0
-
Wg
A
O
A¢
1
-1
a) z-plane b) Potential w-plane
O
c) Hodograph V - plane d) t-plane
e) T-plane f) a-plane
C = ¥
g
g ¢
x
y
M
M¢
A
A¢
O
C = ¥
M
M¢
u
v
x
h
O = ¥ C
A¢
A
M
M¢
g lnV
p -a
2
a
p
- +
2
s
a
M¢ A¢ A M
0 t 0 - t
C = ¥
-1 1
O = ¥
A M C M¢ A¢
-1 1
C
A M M¢ A¢
O = ¥
0
1
t 0
1
t
-
Wg
Figure 1
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
The tables and graphs for the
generalized Kirchhoff model
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
Efficiency E versus inclination angle
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
α
Efficiency E versus flow
through the lamina s
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Table 1
The table for the generalized
Riabouchinsky model
Table 2
Inclination Cavitation number, σ
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
Figure 2 Figure 3
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.