483
13.3 Radial Flow Variation
From derivatives in the radial direction follows
With the definition of entropy follows
Combination of the three differential equations results in
(13.14)
A stage is normally designed for constant work over the radius. This implies constant total enthalpy over the radius. Further, it is reasonable to assume constant
losses over the radius, which then implies constant entropy over the radius. This approximation is adequate in the core part of the flow, but losses obviously are greater
within the end wall boundary layers. With these simplifications, (13.14) becomes
or
(13.15)
Equation (13.14) is denominated the non-isentropic simple radial equilibrium equation (NISRE). Equation (13.15), where the term dh 0 /dr might be kept, is denominated the isentropic simple radial equilibrium equation (ISRE). The term simple
expresses the assumed absence of a radial velocity component.
13.3.3 Free Vortex Blades
Equation (13.15) is met for constant values of u
rv and a
v . The corresponding blade
shape is denominated free vortex blade. The term means that the angular momentum
within the flow is constant. No twisting is then exerted on circumferential streamsurfaces. The variation of the velocity triangles over the radius is then determined
when velocity triangles on the average radius are chosen.
As a first example of kinematic parameters on the average radius we assume
0
.
a
u
a
u
dh
dv
dv
dh v
v
dr
dr
dr
dr
=
+
+
1
.
ds dh
dp
T dr dr
dr
=
− r
2
0
.
a
u
u
a
u
dh
dv
v
dv
ds
T
v
v
dr
dr
dr
r
dr
−
=
+ +
2
0,
+ +
=
a
u
u
a
u
dv
v
dv
v
v
dr
r
dr
2
( ) 0.
1/2
a
u
u
dv
v d rv
dr
r dr
+
=
0.4,
0.5,
0.5.
y
f
=
=
=
R