486
13 Axial Compressors
ΔW is kept constant over the radius. The variation of v um with free vortex blades is
inversely proportional to the radius. This variation may be chosen however. In order
to establish the influence of this variation, we assume
n
um
v ~ r . With the exponent
n = − 1, free vortex blades are recovered. Hereafter, we calculate the variation of the
flow parameters for n = 0. Velocities are written in dimensionless form by dividing them by the blade speed on the mean radius. The Eq. (13.15) stays valid with
dimensionless quantities. The variation of tangential velocity components follows
from
or
(13.16)
Analogously it applies that
(13.17)
The parameters R m and ψ m are chosen on the mean radius, assuming that the axial
velocity on the mean radius is constant, with R
w
u
m
m u
m
= −
(
/ ) . On the mean radius we further choose ψ = 0.4, ϕ = 0.5 and R = 0.7. We omit the overbars indicating
dimensionless values.
For n = 0, (13.16) becomes
Equation (13.15) becomes
The constant c follows from the value of v a1 on the mean radius: c = 0.37.
The results are shown in Table 13.4.
The flow variation has become more favourable. The blade twist is moderate: 7°
on the stator and 26° on the rotor. Maximal flow turnings are acceptable: 40° on the
stator hub with Mach number 0.8 and 37° on the rotor hub with Mach number 0.88.
The Mach number at the rotor blade tip is certainly acceptable: 1.18. It may even
be somewhat higher. We further note that values of the velocity reductions at the
stator and rotor hub are 0.62 and 0.64 with a free vortex blade. Actually, such strong
reductions are not realisable. Vortex forcing with n = 0 improves these reductions.
The reduction values are 0.63 for the stator and 0.70 for the rotor. Only at the stator
hub, the velocity reduction stays too strong.
,
u1
um
m
u1
2
m
m
m
v
v
u
W
v
1/ 2
u
u
u
u
D
=
=
−
1
(1
)
1/2
/ .
n
u
m
m
v
R r
r
y
= −
−
y
= −
+
n 1
u2
m
m
2
v
( 1 R )r
/ r .
1
1
0.3 0.2 /
or
0.3 0.2.
=
−
=
−
u
u
v
r
rv
r
2
2
1
1
2
0.6 0.4 0.3 or
0.18ln
0.12 / .
= −
−
= −
−
a
a
d v
v
c
r
r
dr
r
r
Analogously we obtain
ln
=
−
+
2
a2
v
0 .13 0 .18 r 0 .12 / r.
13 Axial Compressors
ΔW is kept constant over the radius. The variation of v um with free vortex blades is
inversely proportional to the radius. This variation may be chosen however. In order
to establish the influence of this variation, we assume
n
um
v ~ r . With the exponent
n = − 1, free vortex blades are recovered. Hereafter, we calculate the variation of the
flow parameters for n = 0. Velocities are written in dimensionless form by dividing them by the blade speed on the mean radius. The Eq. (13.15) stays valid with
dimensionless quantities. The variation of tangential velocity components follows
from
or
(13.16)
Analogously it applies that
(13.17)
The parameters R m and ψ m are chosen on the mean radius, assuming that the axial
velocity on the mean radius is constant, with R
w
u
m
m u
m
= −
(
/ ) . On the mean radius we further choose ψ = 0.4, ϕ = 0.5 and R = 0.7. We omit the overbars indicating
dimensionless values.
For n = 0, (13.16) becomes
Equation (13.15) becomes
The constant c follows from the value of v a1 on the mean radius: c = 0.37.
The results are shown in Table 13.4.
The flow variation has become more favourable. The blade twist is moderate: 7°
on the stator and 26° on the rotor. Maximal flow turnings are acceptable: 40° on the
stator hub with Mach number 0.8 and 37° on the rotor hub with Mach number 0.88.
The Mach number at the rotor blade tip is certainly acceptable: 1.18. It may even
be somewhat higher. We further note that values of the velocity reductions at the
stator and rotor hub are 0.62 and 0.64 with a free vortex blade. Actually, such strong
reductions are not realisable. Vortex forcing with n = 0 improves these reductions.
The reduction values are 0.63 for the stator and 0.70 for the rotor. Only at the stator
hub, the velocity reduction stays too strong.
,
u1
um
m
u1
2
m
m
m
v
v
u
W
v
1/ 2
u
u
u
u
D
=
=
−
1
(1
)
1/2
/ .
n
u
m
m
v
R r
r
y
= −
−
y
= −
+
n 1
u2
m
m
2
v
( 1 R )r
/ r .
1
1
0.3 0.2 /
or
0.3 0.2.
=
−
=
−
u
u
v
r
rv
r
2
2
1
1
2
0.6 0.4 0.3 or
0.18ln
0.12 / .
= −
−
= −
−
a
a
d v
v
c
r
r
dr
r
r
Analogously we obtain
ln
=
−
+
2
a2
v
0 .13 0 .18 r 0 .12 / r.