109
The balances thus become
(3.8)
(3.9)
The momentum balance (Eq. 3.8) is the Bernoulli equation on a streamline in the
relative frame, as derived in Chap. 1. A new relation, compared to the one-dimensional analysis in Chap. 1, is the force balance in the normal direction (Eq. 3.9). The
Bernoulli equation may be integrated along streamlines if ρ depends only on p. This
particularly applies to ρ = constant. The result is (  u = Ω r):
(3.10)
From the velocity triangle follows: w u v
uv u
2
2
2
2
= + −
. The constant is the same
on all streamlines if the incoming flow at rotor inlet is a free vortex flow ( rv u =
constant) with a uniform total pressure p v
ρ
+


 


 
2
2
. This condition is mostly met,
especially when inlet guide vanes are absent. When the integration constant in
Eq. (3.10) is the same for all streamlines, this relation implies
(3.11)
An analogous result for a compressible fluid is obtained for an isentropic flow
(no entropy creation), the entropy definition being T s
h
p
∇ = ∇ − ∇
1
ρ
. For constant
entropy on the streamline, Eq. (3.8) may be integrated to
Under analogous conditions as for an incompressible fluid, constant total enthalpy
2
0
(
1/2 )
h h
v
= +
and free vortex flow ( rv u = constant) at the inlet, the integration
constant is the same on all streamlines, which implies constant entropy within the
entire flow area (homentropic flow). So Eq. (3.11) applies. Combination of Eq. (3.9)
and Eq. (3.11) results in
(3.12)
2
1
,
w
p
r
w
r
x
x
x
W
r
∂
∂
∂
= −
+
∂
∂
∂
0
1
2
2
2
= −
∂
∂
+
∂
∂
−
+
ρ
p
y
r
r
y
w
w
R
Ω
Ω
.
2
2
constant.
2
2
w
p u
r
+ −
=
2
1
0.
w
p
r
w
r
y
y
y
W
r
∂
∂
∂
+
−
=
∂
∂
∂
1/2
1/2
constant.
2
2
w + h
u
−
=
2
2
0 or
2
.
w
w
w
w
w
w
y
R
y
R
∂
∂
− Ω +
=
= Ω −
∂
∂
3.3 Radial Fan Analysis for Lossless Two-Dimensional Flow …