521
14.4 Rotor Shape
Thus
With M u0 = 1.3 it follows that M 2 ≈ 1.15. This means that, with pressure ratio equal to
4, the rotor outlet is slightly supersonic. Some deceleration within a vaneless space
may reduce the Mach number to just below 1 ( M≈ 0.9). This allows leading the flow
into a vaned diffuser, just avoiding choking.
The conclusion is that a pressure ratio equal to 4 may be attained with a singlestage radial compressor by applying radial end or slightly backward swept blades at
a high peripheral speed. The machine then has a limited operating range. In practice,
radial compressors with pressure ratio up to 4 and somewhat above this value are
applied. A larger pressure ratio is attainable, in principle, since the peripheral speed
can exceed 450 m/s, but at a pressure ratio above 4, supersonic inlet flow to the
diffuser becomes unavoidable. Shock waves occur then at the diffuser inlet (vaned
diffuser), impairing the efficiency. Moreover, the operating range becomes narrow.
Single-stage radial compressors with a pressure ratio 6 or even 8 do exist, but can
only be used for a limited number of applications.
14.4 Rotor Shape
14.4.1 Number of Blades
Determination of the minimum required number of blades with radial rotors was
already discussed in Chap. 3 (Sect. 3.3.4 and 3.3.5). We repeat some elements of
the reasoning, but adapted to centrifugal compressors. In Chap. 3, we analysed fans,
which are machines of low solidity and therefore we used an estimate of the velocity difference between the suction and pressure sides of a blade based on Pfleiderer’s
reasoning for rotors with small overlap of the blades. In compressors, the solidity is
high and the reasoning according to Stodola, considering the rotor as composed of
blade channels, is more appropriate.
The equation for the velocity difference over a blade at a given radius (Chap. 3,
Eq. 3 14) is
(14.6)
with Dq
p
= 2 Z and Z the number of blades. This equation was derived in Chap. 3
from flow analysis in the rotating frame and considering the circulation of the relative velocity on two different contours. The first term between the square brackets
originates from the Coriolis force, while the second term comes from the streamline
curvature, i.e. the lift. For radial end blades and blades with backward sweep at the
2
2
2
2
u0
2
2
2
2
2
2
2
1
1
2
00
2
u0
2
2
M
v
u
M
.
c
c
(
1 )(
)u
1 (
1 )(
)M
g
s
g
s
=
≈
=
+ −
−
+ −
−
s
p
u
d
w w
cos
2 r
( r w ) ,
dr
Dq
b W
−
=
+
14.4 Rotor Shape
Thus
With M u0 = 1.3 it follows that M 2 ≈ 1.15. This means that, with pressure ratio equal to
4, the rotor outlet is slightly supersonic. Some deceleration within a vaneless space
may reduce the Mach number to just below 1 ( M≈ 0.9). This allows leading the flow
into a vaned diffuser, just avoiding choking.
The conclusion is that a pressure ratio equal to 4 may be attained with a singlestage radial compressor by applying radial end or slightly backward swept blades at
a high peripheral speed. The machine then has a limited operating range. In practice,
radial compressors with pressure ratio up to 4 and somewhat above this value are
applied. A larger pressure ratio is attainable, in principle, since the peripheral speed
can exceed 450 m/s, but at a pressure ratio above 4, supersonic inlet flow to the
diffuser becomes unavoidable. Shock waves occur then at the diffuser inlet (vaned
diffuser), impairing the efficiency. Moreover, the operating range becomes narrow.
Single-stage radial compressors with a pressure ratio 6 or even 8 do exist, but can
only be used for a limited number of applications.
14.4 Rotor Shape
14.4.1 Number of Blades
Determination of the minimum required number of blades with radial rotors was
already discussed in Chap. 3 (Sect. 3.3.4 and 3.3.5). We repeat some elements of
the reasoning, but adapted to centrifugal compressors. In Chap. 3, we analysed fans,
which are machines of low solidity and therefore we used an estimate of the velocity difference between the suction and pressure sides of a blade based on Pfleiderer’s
reasoning for rotors with small overlap of the blades. In compressors, the solidity is
high and the reasoning according to Stodola, considering the rotor as composed of
blade channels, is more appropriate.
The equation for the velocity difference over a blade at a given radius (Chap. 3,
Eq. 3 14) is
(14.6)
with Dq
p
= 2 Z and Z the number of blades. This equation was derived in Chap. 3
from flow analysis in the rotating frame and considering the circulation of the relative velocity on two different contours. The first term between the square brackets
originates from the Coriolis force, while the second term comes from the streamline
curvature, i.e. the lift. For radial end blades and blades with backward sweep at the
2
2
2
2
u0
2
2
2
2
2
2
2
1
1
2
00
2
u0
2
2
M
v
u
M
.
c
c
(
1 )(
)u
1 (
1 )(
)M
g
s
g
s
=
≈
=
+ −
−
+ −
−
s
p
u
d
w w
cos
2 r
( r w ) ,
dr
Dq
b W
−
=
+