205
6.4 The Single Impulse Stage or Laval Stage
(6.9)
This efficiency cannot be expressed elementarily. However, if the stage is supposed
to supply kinetic energy
2
1 2
2 v to the next stage, it seems obvious to suppose that the
stage receives a same kinetic energy:
2
2
1
1
0
2
2
2
v
v .
=
Especially, this is satisfied when
the stage inlet and the stage outlet velocities are equal in magnitude and direction:
v v
0
2
= . Such a stage is called a repeating stage. It is a model for a stage of a multistage machine. For a repeating stage, (Eq. 6.9) reduces to
(6.10)
With impulse turbines, the outlet kinetic energy is completely lost or almost completely lost in some applications. With partial admission, rotor blades not exposed
to steam produce a windage flow. The fluid between the blades is driven to the
periphery by centrifugal effect. This generates a circulating flow (analogous with
the peripheral pump and the side-channel pump: see Chap. 8). The windage flow
consumes wheel power (= windage loss). It also perturbs the outlet flow from
steam-exposed blade channels and partially dissipates the outlet kinetic energy. An
impulse stage with partial admission is sometimes used as a first stage in a multistage turbine. There is then typically a diameter reduction after this stage, intended
to diminish the through-flow area in order to obtain full admission on the second
and further stages. In such a case, the outlet kinetic energy in the first stage is completely lost (such a first stage will be discussed in Sect. 6.8.2). With a single-stage
impulse turbine as illustrated in Fig. 6.1, outlet kinetic energy is largely a loss, as the
post-connected diffuser can only recover a small amount of it. In other applications,
outlet kinetic energy is fully recovered, as in full admission stages of multistage
turbines. Both efficiency definitions (Eqs. 6.8 and 6.9) thus are relevant. Efficiency
according to (Eq. 6.8) is called total-to-static efficiency as the reference enthalpy
drop goes from a total to a static state: h
h s
00
2
− . Efficiency according to Eq. (6.9) is
analogously called total-to-total efficiency. With a repeating stage the total-to-total
efficiency equals a static-to-static efficiency (expression 6.10), but the latter name
is not generally used.
6.4.4 Blade Profile Shape
Figure 6.8 represents the velocity triangles with a given nozzle angle a 1 and a given
blade speed u. Without rotor losses, the path by which w 1 in the rotor can be turned
to w 2 would be the dashed line (circle). The theoretically maximum turning, and
thus maximum rotor work, is reached for a purely tangential outlet. This is practically unachievable, as there would be no axial velocity. Taking losses into account,
i
00
02s
W .
h
h
=
−
D
h
i
0
2s
0
1s
W
W .
h h
h h
=
=
−
−
D
D
h
6.4 The Single Impulse Stage or Laval Stage
(6.9)
This efficiency cannot be expressed elementarily. However, if the stage is supposed
to supply kinetic energy
2
1 2
2 v to the next stage, it seems obvious to suppose that the
stage receives a same kinetic energy:
2
2
1
1
0
2
2
2
v
v .
=
Especially, this is satisfied when
the stage inlet and the stage outlet velocities are equal in magnitude and direction:
v v
0
2
= . Such a stage is called a repeating stage. It is a model for a stage of a multistage machine. For a repeating stage, (Eq. 6.9) reduces to
(6.10)
With impulse turbines, the outlet kinetic energy is completely lost or almost completely lost in some applications. With partial admission, rotor blades not exposed
to steam produce a windage flow. The fluid between the blades is driven to the
periphery by centrifugal effect. This generates a circulating flow (analogous with
the peripheral pump and the side-channel pump: see Chap. 8). The windage flow
consumes wheel power (= windage loss). It also perturbs the outlet flow from
steam-exposed blade channels and partially dissipates the outlet kinetic energy. An
impulse stage with partial admission is sometimes used as a first stage in a multistage turbine. There is then typically a diameter reduction after this stage, intended
to diminish the through-flow area in order to obtain full admission on the second
and further stages. In such a case, the outlet kinetic energy in the first stage is completely lost (such a first stage will be discussed in Sect. 6.8.2). With a single-stage
impulse turbine as illustrated in Fig. 6.1, outlet kinetic energy is largely a loss, as the
post-connected diffuser can only recover a small amount of it. In other applications,
outlet kinetic energy is fully recovered, as in full admission stages of multistage
turbines. Both efficiency definitions (Eqs. 6.8 and 6.9) thus are relevant. Efficiency
according to (Eq. 6.8) is called total-to-static efficiency as the reference enthalpy
drop goes from a total to a static state: h
h s
00
2
− . Efficiency according to Eq. (6.9) is
analogously called total-to-total efficiency. With a repeating stage the total-to-total
efficiency equals a static-to-static efficiency (expression 6.10), but the latter name
is not generally used.
6.4.4 Blade Profile Shape
Figure 6.8 represents the velocity triangles with a given nozzle angle a 1 and a given
blade speed u. Without rotor losses, the path by which w 1 in the rotor can be turned
to w 2 would be the dashed line (circle). The theoretically maximum turning, and
thus maximum rotor work, is reached for a purely tangential outlet. This is practically unachievable, as there would be no axial velocity. Taking losses into account,
i
00
02s
W .
h
h
=
−
D
h
i
0
2s
0
1s
W
W .
h h
h h
=
=
−
−
D
D
h