117
Due to end effects at the inlet and outlet of the rotor, Pfleiderer’s reasoning leads
to some underestimation of the velocity difference near the outlet. With (Eq. 3.22),
criterion (Eq. 3.29) becomes
(3.30)
We assume a lower limit value compared to Eq. (3.29) to take the underestimation
into account. Equation (3.30) for a radial rotor is similar to the expression of the
Zweifel coefficient (Eq. 2.29 in Chap. 2) for an axial cascade. The factor
may be considered as the moment solidity of the meridional section, so that
Eq. (3.30) may be written as
(3.31)
We compare with the Zweifel tangential force coefficient:
with the axial solidity σ a
a
c s
= / .
From now on, we call the factor C M by Eq. (3.31) the Pfleiderer moment coefficient since it is based on the average pressure difference reasoning of Pfleiderer.
The criterion determines the necessary solidity for avoiding boundary layer separation in a centrifugal rotor. We will assume that optimal solidity corresponds with
a moment coefficient of about unity. For completeness, we should mention that
Pfleiderer only had the objective to derive a formula for slip effect and not a separation criterion. The extension to a separation criterion is, however, quite obvious, as
is clear from the reasoning above. Also, it should be mentioned that the separation
criterion (Eq. 3.31) is not generally used in the turbomachinery literature. Mostly,
C
v
w
r b
Z M
W
Z
r b
M
v
u
W
w
C
M
r
st
st
r
M
=
=
<
≈
2
2
2
2 2
2
2
2 2
2
2
2
2
2
1 4
π
π
∆
Ω
∆
,lim
. .
σ
π
M
st
Z M
r b r
= 2 2 2 2
,
C
v
u
W
w
C
M
r
M
M
=
<
≈
2
2
2
2
1 4
∆
σ
,lim
. .
C
v
w
w
C
Fu
a
u
a
Fu
=
<
≈
2
1 4
2
2
2
∆
σ
,lim
. ,
Fig. 3.13 Sketch of velocity
distribution on pressure and
suction sides on the blade of
a centrifugal rotor
3.3 Radial Fan Analysis for Lossless Two-Dimensional Flow …
Due to end effects at the inlet and outlet of the rotor, Pfleiderer’s reasoning leads
to some underestimation of the velocity difference near the outlet. With (Eq. 3.22),
criterion (Eq. 3.29) becomes
(3.30)
We assume a lower limit value compared to Eq. (3.29) to take the underestimation
into account. Equation (3.30) for a radial rotor is similar to the expression of the
Zweifel coefficient (Eq. 2.29 in Chap. 2) for an axial cascade. The factor
may be considered as the moment solidity of the meridional section, so that
Eq. (3.30) may be written as
(3.31)
We compare with the Zweifel tangential force coefficient:
with the axial solidity σ a
a
c s
= / .
From now on, we call the factor C M by Eq. (3.31) the Pfleiderer moment coefficient since it is based on the average pressure difference reasoning of Pfleiderer.
The criterion determines the necessary solidity for avoiding boundary layer separation in a centrifugal rotor. We will assume that optimal solidity corresponds with
a moment coefficient of about unity. For completeness, we should mention that
Pfleiderer only had the objective to derive a formula for slip effect and not a separation criterion. The extension to a separation criterion is, however, quite obvious, as
is clear from the reasoning above. Also, it should be mentioned that the separation
criterion (Eq. 3.31) is not generally used in the turbomachinery literature. Mostly,
C
v
w
r b
Z M
W
Z
r b
M
v
u
W
w
C
M
r
st
st
r
M
=
=
<
≈
2
2
2
2 2
2
2
2 2
2
2
2
2
2
1 4
π
π
∆
Ω
∆
,lim
. .
σ
π
M
st
Z M
r b r
= 2 2 2 2
,
C
v
u
W
w
C
M
r
M
M
=
<
≈
2
2
2
2
1 4
∆
σ
,lim
. .
C
v
w
w
C
Fu
a
u
a
Fu
=
<
≈
2
1 4
2
2
2
∆
σ
,lim
. ,
Fig. 3.13 Sketch of velocity
distribution on pressure and
suction sides on the blade of
a centrifugal rotor
3.3 Radial Fan Analysis for Lossless Two-Dimensional Flow …