64
2 Basic Components
Tangential velocity components are negative for the cascade in Fig. 2.12, with
lift components L a and L u being negative as well. We reason in the relative frame.
As centrifugal force and Coriolis force lie in the radial direction with axial machines, these forces do not intervene with momentum relations in the axial and
tangential directions. The derived relations thus automatically apply to stator cascades, as we always apply the momentum laws in an algebraically consistent way.
As a consequence, the derived relations also apply to cascades with accelerating
flow (this will be verified later on). The pump rotor cascade in Fig. 2.12 is the most
complex one to analyse ( w u , L a and L u being negative). That is the reason why we
take it as an example.
The mass conservation law is:
1 a
2 a
a
w s w s w s.
=
=
The work equation (relative system, no losses: Eq. (2.8) with τ = 0) is
from which
2
2
2
1
1
2
p
p
( w w ).
2
r
− =
−
With
and
2
1
2
1 1
a
2
2
1
1
w
p p p ,q
w
w cos
w cos ,
2
r
D
b
b
=
−
=
=
=
it follows that:
(2.11)
The term Dp/q 1 is often denoted by C p . A criterion for maximum cascade load with
decelerating flow is: C p = 0.5 ( w 2 /w 1 ≈ 0.7). This results in a relation between β 1
and β 2 . For instance, corresponding values are β 1 = − 60° and β 2 = − 45°, β 1 = − 55°
and β 2 = − 35°.
In order to apply the momentum conservation law, we consider the dashed lined
contour in Fig. 2.12. It forms a control volume with two periodic streamsurfaces and
two planes parallel to the cascade front. On the front and back surfaces, pressures
are p 1 and p 2 . Pressure forces on the periodic streamsurfaces counterbalance. The
force by the blade on the flow is −L.
The momentum balance in the tangential direction is
or
(2.12)
or
01r
02
1
r
2
2
1
2
2
p
p
p
0
w
p
w ,
2
2
r
r
+
=
+
−
=
2 1
p
2
1
2
cos
p
C
1
.
q
cos
b
D
b
=
= −
u
a
2u
1u
u
a
1u
2u
L
s w ( w
w ),
L
s w ( w
w ).
r
r
− =
−
=
−
2 Basic Components
Tangential velocity components are negative for the cascade in Fig. 2.12, with
lift components L a and L u being negative as well. We reason in the relative frame.
As centrifugal force and Coriolis force lie in the radial direction with axial machines, these forces do not intervene with momentum relations in the axial and
tangential directions. The derived relations thus automatically apply to stator cascades, as we always apply the momentum laws in an algebraically consistent way.
As a consequence, the derived relations also apply to cascades with accelerating
flow (this will be verified later on). The pump rotor cascade in Fig. 2.12 is the most
complex one to analyse ( w u , L a and L u being negative). That is the reason why we
take it as an example.
The mass conservation law is:
1 a
2 a
a
w s w s w s.
=
=
The work equation (relative system, no losses: Eq. (2.8) with τ = 0) is
from which
2
2
2
1
1
2
p
p
( w w ).
2
r
− =
−
With
and
2
1
2
1 1
a
2
2
1
1
w
p p p ,q
w
w cos
w cos ,
2
r
D
b
b
=
−
=
=
=
it follows that:
(2.11)
The term Dp/q 1 is often denoted by C p . A criterion for maximum cascade load with
decelerating flow is: C p = 0.5 ( w 2 /w 1 ≈ 0.7). This results in a relation between β 1
and β 2 . For instance, corresponding values are β 1 = − 60° and β 2 = − 45°, β 1 = − 55°
and β 2 = − 35°.
In order to apply the momentum conservation law, we consider the dashed lined
contour in Fig. 2.12. It forms a control volume with two periodic streamsurfaces and
two planes parallel to the cascade front. On the front and back surfaces, pressures
are p 1 and p 2 . Pressure forces on the periodic streamsurfaces counterbalance. The
force by the blade on the flow is −L.
The momentum balance in the tangential direction is
or
(2.12)
or
01r
02
1
r
2
2
1
2
2
p
p
p
0
w
p
w ,
2
2
r
r
+
=
+
−
=
2 1
p
2
1
2
cos
p
C
1
.
q
cos
b
D
b
=
= −
u
a
2u
1u
u
a
1u
2u
L
s w ( w
w ),
L
s w ( w
w ).
r
r
− =
−
=
−