532
14 Radial Compressors
A: The mass flow rate through the blade channel may be approximated by
where ρ is an average value of the density and w is an average value of the relative
velocity at the position of the infinitesimal streamtube and β is the local angle of
the mean streamline with respect to the radial direction. The moment of momentum
transferred to the mass flow in the infinitesimal streamtube is
with v u averaged over the blade passage.
This is equal to the moment of the pressure force on the blades (neglecting a possible moment by friction forces):
where the subscripts p and s refer to pressure and suction sides. The Bernoulli equation on a streamline in a rotor is
Therefore, on a constant radius, neglecting losses, and assuming that the integration
constant of the Bernoulli equation is the same on all streamlines (which is satisfied
for constant enthalpy and free vortex flow upstream of the rotor inlet; see Chap. 3,
Sect. 3.3.2):
where an average value of the density has to used. From the last equation follows an
approximation of the pressure difference over a blade:
p
r
b
=
b
2 r
m
wcos ,
Z
M m
d
dr
(rv )dr
b
u
=
,
,
p
s
M (p
p )dr r
=
−
r
=
+
+
2
2
1
1
1
irr
2
2
d u
d w
dp dq .
r
−
=
−
2
2
1
1
s
p
p
s
2
( w w )
( p
p ),
p
s
s
p
p
p
w( w w ).
r
−
=
−