116
3 Fans
The curvature factor, f R , expresses the effect of the Coriolis force and the lift force
(= curvature) together relative to the effect of the Coriolis force alone. This factor is
difficult to estimate at the outlet of a rotor with blades with small overlap.
With (Eq. 3.26) it follows
Flow reversal at the pressure side is prevented for
(3.27)
Prevention of boundary layer separation at the suction side may be expressed by a
local diffusion factor criterion as
(3.28)
This criterion also gives result (Eq. 3.27) for D loc = 0.5. This D loc value constitutes
the critical value for separation with axial cascades. Deceleration within the boundary layer at the suction side is less favourable with a centrifugal rotor, due to a
strong turbulence segregation effect by the Coriolis force. Similar to the centrifugal
force (Chap. 2, Sect. 2.3.2), the Coriolis force drives high-energy cores within the
turbulent motion from the suction side to the pressure side. This migration effect
lowers the turbulence level in the suction side boundary layer (it becomes more
laminar), which weakens its capacity to withstand an adverse pressure gradient. In
practice, this rather causes separation at the suction side than flow reversal at the
pressure side. Therefore, we further apply criterion (Eq. 3.28) for separation at the
suction side, but with w 2 in the right-hand side multiplied by the reduction factor
0.8 ( D loc = 0.45).
We note the criterion to prevent separation at the suction side as
(3.29)
As already said, it is difficult to estimate the velocity difference near the trailing
edge with a formula like (Eq. 3.26). We might set the curvature factor to unity,
as in the Stodola slip reasoning, but this is certainly a quite crude assumption for
fans. For pumps and compressors, the Stodola reasoning is more appropriate, which
makes that (Eq. 3.26), together with (Eq. 3.29) may be used for estimating the minimum number of blades (see Chap. 14: radial compressors). For fans, a way out is
using the estimate (Eq. 3.22) from Pfleiderer’s reasoning based on the average pressure difference. Usually, the velocity difference over a blade in a centrifugal rotor
is nearly constant as a function of the radius and the velocity distribution on the
pressure and the suction sides is as sketched in Fig. 3.13.
2
2
2
2
2
2
2
2
cos
and
cos
.
s
R
p
R
w w
u
f
w
w
u
f
Z
Z
p
p
b
b
=
+
=
−
2
cos
2
2
2
π
β
Z
u
f
w
R < .
0.5 or
.
s
2
loc
loc
s
2
2
s
loc
w w
D
D
w w
w
w
1 D
−
<
≈
−
< −
C
w w
w
C
s
p
=
−
<
≈
(
)
. .
lim
2
2
1 6
3 Fans
The curvature factor, f R , expresses the effect of the Coriolis force and the lift force
(= curvature) together relative to the effect of the Coriolis force alone. This factor is
difficult to estimate at the outlet of a rotor with blades with small overlap.
With (Eq. 3.26) it follows
Flow reversal at the pressure side is prevented for
(3.27)
Prevention of boundary layer separation at the suction side may be expressed by a
local diffusion factor criterion as
(3.28)
This criterion also gives result (Eq. 3.27) for D loc = 0.5. This D loc value constitutes
the critical value for separation with axial cascades. Deceleration within the boundary layer at the suction side is less favourable with a centrifugal rotor, due to a
strong turbulence segregation effect by the Coriolis force. Similar to the centrifugal
force (Chap. 2, Sect. 2.3.2), the Coriolis force drives high-energy cores within the
turbulent motion from the suction side to the pressure side. This migration effect
lowers the turbulence level in the suction side boundary layer (it becomes more
laminar), which weakens its capacity to withstand an adverse pressure gradient. In
practice, this rather causes separation at the suction side than flow reversal at the
pressure side. Therefore, we further apply criterion (Eq. 3.28) for separation at the
suction side, but with w 2 in the right-hand side multiplied by the reduction factor
0.8 ( D loc = 0.45).
We note the criterion to prevent separation at the suction side as
(3.29)
As already said, it is difficult to estimate the velocity difference near the trailing
edge with a formula like (Eq. 3.26). We might set the curvature factor to unity,
as in the Stodola slip reasoning, but this is certainly a quite crude assumption for
fans. For pumps and compressors, the Stodola reasoning is more appropriate, which
makes that (Eq. 3.26), together with (Eq. 3.29) may be used for estimating the minimum number of blades (see Chap. 14: radial compressors). For fans, a way out is
using the estimate (Eq. 3.22) from Pfleiderer’s reasoning based on the average pressure difference. Usually, the velocity difference over a blade in a centrifugal rotor
is nearly constant as a function of the radius and the velocity distribution on the
pressure and the suction sides is as sketched in Fig. 3.13.
2
2
2
2
2
2
2
2
cos
and
cos
.
s
R
p
R
w w
u
f
w
w
u
f
Z
Z
p
p
b
b
=
+
=
−
2
cos
2
2
2
π
β
Z
u
f
w
R < .
0.5 or
.
s
2
loc
loc
s
2
2
s
loc
w w
D
D
w w
w
w
1 D
−
<
≈
−
< −
C
w w
w
C
s
p
=
−
<
≈
(
)
. .
lim
2
2
1 6