542
15 Axial and Radial Turbines for Gases
value at the hub to a lower value at the casing. This rather strong variation cannot be
matched with the weak variation on the maximum efficiency line in the Smith chart:
Fig. 15.4. In using this figure, one has to take into account that high values of ψ and
ϕ occur at the hub, so where the degree of reaction is the lowest, and low values
at the casing, so where the degree of reaction is the highest. Examples are ψ = 2.4
and ϕ = 1 at R = 0.25 and ψ = 1 and ϕ = 0.3 at R = 0.75. These sets do not go together
with free vortex blades. But the radial equilibrium equation can be satisfied in other
ways, e.g. for constant stator outlet angle.
With
/
constant
1
1u
1a
tg
v v
a =
=
, (15.6) becomes for the stator outlet:
The equation is satisfied for
sin
2 1
1a
v
C r
a
−
=
.
The axial velocity v 1a is then higher at the hub and lower at the casing.
With a very high radius ratio, as with the final stage of the LP part of a steam
turbine, the stator outlet velocity variation is in between the results for free vortex
flow and constant stator angle, but close to this last one (Exercise 6.10.5 in Chap. 6
shows that the efficiency is close to optimum on every radius for α 1 = ct.). With gas
turbine stages with high radius ratio, a similar distribution is chosen.
15.1.3 Blade Profiles
The general flow in turbines is accelerating. This facilitates the blade profile design
compared to that of a compressor. Compressor blade profiles are strongly determined by the limitation of the loading due to the strong adverse pressure gradient
in the rear part of the suction side. Turbines allow, due to the generally accelerating
flow, much higher loading (work coefficient around 1.80 compared to 0.4 for compressors). This results in much higher flow turnings with turbines.
Subsonic and Supercritical Cascades
Subsonic and supercritical rotor cascades occur within the low pressure turbine (LP)
part of a high bypass ratio turbofan engine, as direct fan drive results in low running
speed in the LP turbine. Within the high pressure (HP) part, blade speed is higher and
rather determined by strength limitations, resulting in transonic cascades as a rule.
Subsonic or supercritical rotor cascades mostly also occur in the high-pressure and
the intermediate-pressure parts of steam turbines. Steam turbine blades are manufactured of cheaper materials (steel alloys), which results in lower blade speeds.
Figure 15.5, left, sketches the optimum Mach number distribution on a subsonic
rotor cascade. The right-hand part of the figure sketches the corresponding momentum loss thickness distribution [1].
(
)
(
) or
2
2
2
2
2
1a
1a
1
1
1a
1
1
1a
2
v
v
d
d
tg
v 1 tg
sin
v .
r
dr
r
dr
a
a
a
= −
+
= −
15 Axial and Radial Turbines for Gases
value at the hub to a lower value at the casing. This rather strong variation cannot be
matched with the weak variation on the maximum efficiency line in the Smith chart:
Fig. 15.4. In using this figure, one has to take into account that high values of ψ and
ϕ occur at the hub, so where the degree of reaction is the lowest, and low values
at the casing, so where the degree of reaction is the highest. Examples are ψ = 2.4
and ϕ = 1 at R = 0.25 and ψ = 1 and ϕ = 0.3 at R = 0.75. These sets do not go together
with free vortex blades. But the radial equilibrium equation can be satisfied in other
ways, e.g. for constant stator outlet angle.
With
/
constant
1
1u
1a
tg
v v
a =
=
, (15.6) becomes for the stator outlet:
The equation is satisfied for
sin
2 1
1a
v
C r
a
−
=
.
The axial velocity v 1a is then higher at the hub and lower at the casing.
With a very high radius ratio, as with the final stage of the LP part of a steam
turbine, the stator outlet velocity variation is in between the results for free vortex
flow and constant stator angle, but close to this last one (Exercise 6.10.5 in Chap. 6
shows that the efficiency is close to optimum on every radius for α 1 = ct.). With gas
turbine stages with high radius ratio, a similar distribution is chosen.
15.1.3 Blade Profiles
The general flow in turbines is accelerating. This facilitates the blade profile design
compared to that of a compressor. Compressor blade profiles are strongly determined by the limitation of the loading due to the strong adverse pressure gradient
in the rear part of the suction side. Turbines allow, due to the generally accelerating
flow, much higher loading (work coefficient around 1.80 compared to 0.4 for compressors). This results in much higher flow turnings with turbines.
Subsonic and Supercritical Cascades
Subsonic and supercritical rotor cascades occur within the low pressure turbine (LP)
part of a high bypass ratio turbofan engine, as direct fan drive results in low running
speed in the LP turbine. Within the high pressure (HP) part, blade speed is higher and
rather determined by strength limitations, resulting in transonic cascades as a rule.
Subsonic or supercritical rotor cascades mostly also occur in the high-pressure and
the intermediate-pressure parts of steam turbines. Steam turbine blades are manufactured of cheaper materials (steel alloys), which results in lower blade speeds.
Figure 15.5, left, sketches the optimum Mach number distribution on a subsonic
rotor cascade. The right-hand part of the figure sketches the corresponding momentum loss thickness distribution [1].
(
)
(
) or
2
2
2
2
2
1a
1a
1
1
1a
1
1
1a
2
v
v
d
d
tg
v 1 tg
sin
v .
r
dr
r
dr
a
a
a
= −
+
= −