9 Hydraulic Turbines
330
choice is reasonable, although not optimal, may be demonstrated by considering the
losses and deriving some guidelines for optimisation.
9.4.2 Optimisation of the Velocity Triangles
Losses first occur in the distributor components. These are the volute and the adjustable guide vanes. Flow is accelerating in these parts, causing only small losses,
namely of the 5 % order of the kinetic energy at the stator outlet (v 1
2
2
/ ). We represent the distributor losses by ξ s v
1 2 1
2
with ξ s ≈ 0 0
. .
5 The spaces in between blades
in the rotor constitute channels. Loss is mainly caused by friction and may be represented by ξ r w
1 2 2
2
, where again the order of magnitude of the loss coefficient is a
few percent: ξ r
5
≈ 0 0
. . The loss representation is acceptable, unless the rotor would
have diffusion ( w 2 < w 1 ). In that case, diffusion loss should be added. Incorporating
diffusion is possible, but it is contrary to turbine functioning. So we do not expect it.
Kinetic energy at the rotor outlet ( v 2
2
2
/ ) is not completely lost, as the draught tube
also functions as a diffuser. Recovery of this kinetic energy is limited to about 75 %,
however. The outlet loss is thus very high and dominantly determines the optimisation. Diffuser loss is represented by:
As diffuser loss is dominant, measures have to be taken to minimise the kinetic energy at the rotor outlet. This first implies that the outlet velocity should be oriented
axially, as drawn in Fig. 9.10. Moreover, the value of v 2 must be kept low. This
implies a low flow coefficient v
u
1m
1
/ . As a consequence, the stator inlet angle α 1
must be rather large. This cannot be exaggerated, as a greater α 1 means a larger
turning in the distribution elements and thus larger losses. It also implies greater
through-flow areas, which increases friction surfaces in both the rotor and the stator.
An exact value for the stator angle α 1 cannot be determined with a simple reasoning. The whole optimisation should be studied for that. In real Francis turbines, α 1
varies between about 60 and 75°. 70° is applied in Fig. 9.10.
Work may thus be represented by
(9.3)
Kinematic parameters should follow from minimising the sum of the losses. With
the relation (9.3), it is impossible to perform an exact optimisation, if fixed loss
coefficients are assumed. But some guidelines may be derived from it.
Given the dominance of outlet losses, we may assume an axial outlet flow
(α 2 0
= ). With regard to the reduction of the outlet loss, it is clearly advantageous
to choose the ratio of the meridional components v v
2m 1m
/
below 1. v 2 is reduced by
q
v
irr d
d
,
. .
=
≈
ξ
ξ
1 2 2
2
0
with d
25
∆W gH
q
gH
v
w
v
irr
s
r
d
=
−
=
−
−
−
∑
ξ
ξ
ξ
1
2
2
2
2
2
2
2
2
.
330
choice is reasonable, although not optimal, may be demonstrated by considering the
losses and deriving some guidelines for optimisation.
9.4.2 Optimisation of the Velocity Triangles
Losses first occur in the distributor components. These are the volute and the adjustable guide vanes. Flow is accelerating in these parts, causing only small losses,
namely of the 5 % order of the kinetic energy at the stator outlet (v 1
2
2
/ ). We represent the distributor losses by ξ s v
1 2 1
2
with ξ s ≈ 0 0
. .
5 The spaces in between blades
in the rotor constitute channels. Loss is mainly caused by friction and may be represented by ξ r w
1 2 2
2
, where again the order of magnitude of the loss coefficient is a
few percent: ξ r
5
≈ 0 0
. . The loss representation is acceptable, unless the rotor would
have diffusion ( w 2 < w 1 ). In that case, diffusion loss should be added. Incorporating
diffusion is possible, but it is contrary to turbine functioning. So we do not expect it.
Kinetic energy at the rotor outlet ( v 2
2
2
/ ) is not completely lost, as the draught tube
also functions as a diffuser. Recovery of this kinetic energy is limited to about 75 %,
however. The outlet loss is thus very high and dominantly determines the optimisation. Diffuser loss is represented by:
As diffuser loss is dominant, measures have to be taken to minimise the kinetic energy at the rotor outlet. This first implies that the outlet velocity should be oriented
axially, as drawn in Fig. 9.10. Moreover, the value of v 2 must be kept low. This
implies a low flow coefficient v
u
1m
1
/ . As a consequence, the stator inlet angle α 1
must be rather large. This cannot be exaggerated, as a greater α 1 means a larger
turning in the distribution elements and thus larger losses. It also implies greater
through-flow areas, which increases friction surfaces in both the rotor and the stator.
An exact value for the stator angle α 1 cannot be determined with a simple reasoning. The whole optimisation should be studied for that. In real Francis turbines, α 1
varies between about 60 and 75°. 70° is applied in Fig. 9.10.
Work may thus be represented by
(9.3)
Kinematic parameters should follow from minimising the sum of the losses. With
the relation (9.3), it is impossible to perform an exact optimisation, if fixed loss
coefficients are assumed. But some guidelines may be derived from it.
Given the dominance of outlet losses, we may assume an axial outlet flow
(α 2 0
= ). With regard to the reduction of the outlet loss, it is clearly advantageous
to choose the ratio of the meridional components v v
2m 1m
/
below 1. v 2 is reduced by
q
v
irr d
d
,
. .
=
≈
ξ
ξ
1 2 2
2
0
with d
25
∆W gH
q
gH
v
w
v
irr
s
r
d
=
−
=
−
−
−
∑
ξ
ξ
ξ
1
2
2
2
2
2
2
2
2
.