93
2.5 Exercises
The energy extraction according to Euler’s formula is
1u
2u
W u( v
v ) u( 2bu ).
D =
−
=
(2.36)
Equality of (Eq. 2.35) and (Eq. 2.36) results in
2
2 2
b
a( 1 a ) b
.
l
l
=
− −
(2.37)
The momentum balance in the tangential direction locally on the blade segment is
u
1a
2u
1u
1a 2u
1a
L
v s( w
w )
v sv
v s2bu.
r
r
r
− =
−
=
= −
(2.38)
The velocity triangle at the position of the blade segment is sketched in the figure:
From this:
(2.39)
Combination of Eq. (2.38) and Eq. (2.39) results in
or
(2.40)
With λ = 3 and sC L = 1/12, (Eq. 2.37) and (Eq. 2.40) result in: a = 0.2700, b = 0.0214.
The power exchanged per surface unit (Eq. 2.34) is
2.5.9. Verify the tangential force coefficient (Zweifel) of the steam turbine cascades shown in Fig. 1.11 of Chap. 1.
A: R = 0: stator C Fu = 0.50 (low load), rotor C Fu = 0.90 (moderate load); R = 0.50,
stator and rotor: C Fu = 0.50 (low load).
2.5.10. A cylindrical tube should undergo a diameter increase with a factor 2.
For this increase, there is only a length of 1 diameter available. Design an optimum
2
1a
1
u
m
L
m
2
m
v
L
L cos
C
w c w
b
r
=
=
,
1
1a
L 1a m
2
v s2bu
C cv w
=
2
2
L
2
C ( 1 a )
2b
( 1 b ) .
2
s
l
−
=
+ +
3
3
2
2
0
0
0
v
v
v ( 1 a )2bu
( 1 a )4b
0.5624.
2
2
r
r
l
r
−
=
−
=
×