9.4 Francis and Kaplan Turbines: Reaction Turbines
333
velocity component causes some deviation of the real degree of reaction. A higher
degree of reaction implies less freedom to decrease v 2 with respect to v m
1 . Constant
mass flow rate then requires an increase of the outlet area. This is more difficult
with a diameter ratio near to 1 (see Fig. 9.12). With degrees of reaction 0.55 and
0.65, the work by lift has been put exactly to zero: u w
u w
u
u
1 1
2 2
0
−
= ; it is difficult
to avoid negative values. With the degree of reaction of 0.75, u 2 was determined by
interpolation between the values of the other cases. The part of the lift work comes
out at 15 %.
Realisation of a Francis turbine with a degree of reaction under R = 0.55 is possible. Rotor turning must be high in that case in order to obtain a sufficient outlet
diameter. This impairs the efficiency. Therefore, turbines with a low degree of reaction are not used in practice. The flow coefficient v u
1m
1
/ increases somewhat with
an increasing degree of reaction. The value varies from 0.23 to 0.27 with Francis
turbines (this is approximately constant). This ratio changes more strongly with
Kaplan turbines, namely v u
1m
1
/ ≈ 0.30–0.40 with a degree of reaction varying from
0.75 to 0.90. A rudimentary argumentation cannot demonstrate that this is optimal.
Approximate stator angle values are given in Table 9.2.
9.4.5 Specific Speed and Meridional Shape of Francis Turbines
With Ω
Ω
s
Q
gH
= (
)/ (
) ,
/
3 4
we express Ω and Q as functions of geometric parameters. With b 1 being the stator vane height at the stator outlet and d 1 the rotor
inlet diameter:
and
where we applied 1m 1
v / u 0.25.
≈
After substitution we obtain
Ω =
=
×
−
2
2 0 48
1
2
1
1
1
u
d
d
R
gH
.
,
Q
d b v
d b
R
gH
m
=
=
−
π
π
1 1 1
1 1 0 25
0 48
1
2
.
.
,
Ω s
R
b
d
≈ −
1
1
3 4
1
1
(
)
.
/
Table 9.2 Stator angles as a function of degree of reaction with Francis and Kaplan turbines
R = 0.55
R = 0.65
R = 0.75
R = 0.85
α 1 = 75°
α 1 = 70°
α 1 = 60°
α 1 = 45°
333
velocity component causes some deviation of the real degree of reaction. A higher
degree of reaction implies less freedom to decrease v 2 with respect to v m
1 . Constant
mass flow rate then requires an increase of the outlet area. This is more difficult
with a diameter ratio near to 1 (see Fig. 9.12). With degrees of reaction 0.55 and
0.65, the work by lift has been put exactly to zero: u w
u w
u
u
1 1
2 2
0
−
= ; it is difficult
to avoid negative values. With the degree of reaction of 0.75, u 2 was determined by
interpolation between the values of the other cases. The part of the lift work comes
out at 15 %.
Realisation of a Francis turbine with a degree of reaction under R = 0.55 is possible. Rotor turning must be high in that case in order to obtain a sufficient outlet
diameter. This impairs the efficiency. Therefore, turbines with a low degree of reaction are not used in practice. The flow coefficient v u
1m
1
/ increases somewhat with
an increasing degree of reaction. The value varies from 0.23 to 0.27 with Francis
turbines (this is approximately constant). This ratio changes more strongly with
Kaplan turbines, namely v u
1m
1
/ ≈ 0.30–0.40 with a degree of reaction varying from
0.75 to 0.90. A rudimentary argumentation cannot demonstrate that this is optimal.
Approximate stator angle values are given in Table 9.2.
9.4.5 Specific Speed and Meridional Shape of Francis Turbines
With Ω
Ω
s
Q
gH
= (
)/ (
) ,
/
3 4
we express Ω and Q as functions of geometric parameters. With b 1 being the stator vane height at the stator outlet and d 1 the rotor
inlet diameter:
and
where we applied 1m 1
v / u 0.25.
≈
After substitution we obtain
Ω =
=
×
−
2
2 0 48
1
2
1
1
1
u
d
d
R
gH
.
,
Q
d b v
d b
R
gH
m
=
=
−
π
π
1 1 1
1 1 0 25
0 48
1
2
.
.
,
Ω s
R
b
d
≈ −
1
1
3 4
1
1
(
)
.
/
Table 9.2 Stator angles as a function of degree of reaction with Francis and Kaplan turbines
R = 0.55
R = 0.65
R = 0.75
R = 0.85
α 1 = 75°
α 1 = 70°
α 1 = 60°
α 1 = 45°