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6.10 Exercises
6.10.5. Design the last stage of the LP part of the Arabelle turbine described in
the previous exercise.
a. Determine the number of stages necessary in the expansion from 11 bar, 275 °C
to 46 mbar. Take as hub diameter of the first IP-stage 2.50 m (Fig 6.26) and
assume a degree of reaction of 10 %. Consider also 10 % degree of reaction for
the hub part of the last stage and take again as hub diameter 2.50 m (Fig. 6.27).
Take into account that the hub diameter increases somewhat in the successive
stages of the IP part and then decreases in the successive stages of the LP part,
Compare with the number of stages in Figs. 6.26 and 6.27 (4 + 5).
A: The isentropic enthalpy drop of the IP part + LP part is 875 kJ/kg. For R = 10 %,
the isentropic enthalpy drop in the first and last stages is about 77 kJ/kg. With
this value, the number of stages is 11.4. The number of stages may certainly be
reduced to 10, taking into account the larger diameter in the middle stages. In
reality, there are 9 stages. These look somewhat more loaded than optimal.
b. Determine the variation of the velocity as a function of the radius in the space
between the stator and the rotor for the last stage of the LP part. Consider constant
total enthalpy over the radius downstream (and upstream) of the stator vanes.
Take as stator outlet angle 70º, constant over the radius (this is lower than the
geometric maximum of about 75º in order to enlarge the mass flow). Estimate
the stator loss coefficient with the formula of Soderberg, ignoring secondary loss.
Take as simplification that streamsurfaces are cylindrical (radial velocity component equal to zero), such that there is equilibrium between the radial pressure
gradient and the centrifugal force (so-called simple radial equilibrium). Derive
that the tangential velocity varies according to v u
1 ~ r
a
−
, with
2
s
1
a ( sin ) ,
= f
a
where s
f is the velocity coefficient of the nozzle vanes. Neglect the difference
between infinitesimal and isentropic efficiency such that
2
s .
∞ =
h
f
A: Cylindrical stream surfaces: force equilibrium:
2
1u
v
1 dp
.
dr
r
=
r
Constant enthalpy downstream of nozzle vanes (
):
v r
1
0
=
Fig. 6.37 Stator vane profile; subcritical flow; initial shape
6.10 Exercises
6.10.5. Design the last stage of the LP part of the Arabelle turbine described in
the previous exercise.
a. Determine the number of stages necessary in the expansion from 11 bar, 275 °C
to 46 mbar. Take as hub diameter of the first IP-stage 2.50 m (Fig 6.26) and
assume a degree of reaction of 10 %. Consider also 10 % degree of reaction for
the hub part of the last stage and take again as hub diameter 2.50 m (Fig. 6.27).
Take into account that the hub diameter increases somewhat in the successive
stages of the IP part and then decreases in the successive stages of the LP part,
Compare with the number of stages in Figs. 6.26 and 6.27 (4 + 5).
A: The isentropic enthalpy drop of the IP part + LP part is 875 kJ/kg. For R = 10 %,
the isentropic enthalpy drop in the first and last stages is about 77 kJ/kg. With
this value, the number of stages is 11.4. The number of stages may certainly be
reduced to 10, taking into account the larger diameter in the middle stages. In
reality, there are 9 stages. These look somewhat more loaded than optimal.
b. Determine the variation of the velocity as a function of the radius in the space
between the stator and the rotor for the last stage of the LP part. Consider constant
total enthalpy over the radius downstream (and upstream) of the stator vanes.
Take as stator outlet angle 70º, constant over the radius (this is lower than the
geometric maximum of about 75º in order to enlarge the mass flow). Estimate
the stator loss coefficient with the formula of Soderberg, ignoring secondary loss.
Take as simplification that streamsurfaces are cylindrical (radial velocity component equal to zero), such that there is equilibrium between the radial pressure
gradient and the centrifugal force (so-called simple radial equilibrium). Derive
that the tangential velocity varies according to v u
1 ~ r
a
−
, with
2
s
1
a ( sin ) ,
= f
a
where s
f is the velocity coefficient of the nozzle vanes. Neglect the difference
between infinitesimal and isentropic efficiency such that
2
s .
∞ =
h
f
A: Cylindrical stream surfaces: force equilibrium:
2
1u
v
1 dp
.
dr
r
=
r
Constant enthalpy downstream of nozzle vanes (
):
v r
1
0
=
Fig. 6.37 Stator vane profile; subcritical flow; initial shape