541
15.1 Axial Turbines
the pressure drop over the stator is larger than over the rotor. There is only little
leakage over a stator. At the hub side, vanes are shrouded and there is a sealing
(typically a labyrinth sealing) between the vane shroud and the hub. Leakage loss
encompasses two parts. The first is overflow from the pressure to the suction side
of a blade or vane. This component depends on the pressure difference between the
pressure and the suction sides, which does not specifically depend on the degree
of reaction. This component does not exist when a shroud is applied. It is often reduced with an unshrouded blade by a tip treatment (a partial form of shroud). The
second component originates from the through-flow, so from the pressure drop in
the axial direction. The axial component is strongly reduced, but does not disappear
completely, with shrouded blades. This component decreases considerably with
unshrouded blades by lowering the degree of reaction. Note that also with shrouded
blades, leakage losses decrease somewhat with a degree of reaction lower than
50 %, because the radius at which leakage occurs is lower with a stator than with a
rotor [8]. With shrouded blades, the optimal degree of reaction on the mean radius
is close to 50 %, but somewhat smaller, typically 0.40–0.45. For a stage with long
unshrouded rotor blades, a degree of reaction much lower than 50 % on the mean
radius is optimal.
Combining all considerations, the conclusion is that there is no universal optimum of the kinematic parameters at the mean radius of a turbine stage in a gas turbine. Turbine optimisation is delicate, with a big role of three-dimensional effects.
The topic is far above the possibilities of the present book. The above analysis only
allows demonstration of tendencies, but not derivation of precise results. It should
also be noticed that stages in a real turbine are not always repeating stages. At the
outlet of a first stage, negative post-swirl may be given, enlarging the stage work
(an example is Fig. 11.5; even more post-swirl may be chosen). We keep in mind
that the work coefficient is always very large. Referring to Figs. 15.2 and 15.4,
this means that there is typically strong negative post-swirl at the outlet of a stage
(Fig. 15.1). At the outlet of a turbine part, sometimes outlet guide vanes are required
for turning the flow into the meridional direction. For a recent design example of the
LP part of an aero-engine, we refer to [2].
15.1.2 Radial Variation of Flow Parameters
The equation for simple radial equilibrium (Eq. 13.15) reads:
(15.6)
The equation is satisfied for u
rv
cst
=
together with a
v
cst
=
. For axial outlet flow
is
0
2u
v = . Then follows
constant.
2a
v =
For constant work distribution over the radius,
1u
W uv
∆ =
, the equation is also satisfied for 1u
rv
cst
=
together with 1a
v
cst
=
. The blade form is then called a free vortex blade. The disadvantage with a great
radius change is a rather strong variation of the stator outlet angle a 1 from a higher
(
)
( ) ( )
or
2
2
2
2
u
u
1
1
1
a
u
u
a
2
2
2
v
v
d
d
d
v + v
0
rv
v
0.
r dr
r dr
dr
+
=
+
=
15.1 Axial Turbines
the pressure drop over the stator is larger than over the rotor. There is only little
leakage over a stator. At the hub side, vanes are shrouded and there is a sealing
(typically a labyrinth sealing) between the vane shroud and the hub. Leakage loss
encompasses two parts. The first is overflow from the pressure to the suction side
of a blade or vane. This component depends on the pressure difference between the
pressure and the suction sides, which does not specifically depend on the degree
of reaction. This component does not exist when a shroud is applied. It is often reduced with an unshrouded blade by a tip treatment (a partial form of shroud). The
second component originates from the through-flow, so from the pressure drop in
the axial direction. The axial component is strongly reduced, but does not disappear
completely, with shrouded blades. This component decreases considerably with
unshrouded blades by lowering the degree of reaction. Note that also with shrouded
blades, leakage losses decrease somewhat with a degree of reaction lower than
50 %, because the radius at which leakage occurs is lower with a stator than with a
rotor [8]. With shrouded blades, the optimal degree of reaction on the mean radius
is close to 50 %, but somewhat smaller, typically 0.40–0.45. For a stage with long
unshrouded rotor blades, a degree of reaction much lower than 50 % on the mean
radius is optimal.
Combining all considerations, the conclusion is that there is no universal optimum of the kinematic parameters at the mean radius of a turbine stage in a gas turbine. Turbine optimisation is delicate, with a big role of three-dimensional effects.
The topic is far above the possibilities of the present book. The above analysis only
allows demonstration of tendencies, but not derivation of precise results. It should
also be noticed that stages in a real turbine are not always repeating stages. At the
outlet of a first stage, negative post-swirl may be given, enlarging the stage work
(an example is Fig. 11.5; even more post-swirl may be chosen). We keep in mind
that the work coefficient is always very large. Referring to Figs. 15.2 and 15.4,
this means that there is typically strong negative post-swirl at the outlet of a stage
(Fig. 15.1). At the outlet of a turbine part, sometimes outlet guide vanes are required
for turning the flow into the meridional direction. For a recent design example of the
LP part of an aero-engine, we refer to [2].
15.1.2 Radial Variation of Flow Parameters
The equation for simple radial equilibrium (Eq. 13.15) reads:
(15.6)
The equation is satisfied for u
rv
cst
=
together with a
v
cst
=
. For axial outlet flow
is
0
2u
v = . Then follows
constant.
2a
v =
For constant work distribution over the radius,
1u
W uv
∆ =
, the equation is also satisfied for 1u
rv
cst
=
together with 1a
v
cst
=
. The blade form is then called a free vortex blade. The disadvantage with a great
radius change is a rather strong variation of the stator outlet angle a 1 from a higher
(
)
( ) ( )
or
2
2
2
2
u
u
1
1
1
a
u
u
a
2
2
2
v
v
d
d
d
v + v
0
rv
v
0.
r dr
r dr
dr
+
=
+
=