44
1 Working Principles
Pressure Balance A first pressure drop occurs due to the generation of the velocity
in the suction pipe v 0 . The accompanying pressure drop is
2
0
v / 2
r
. The acceleration in the guide vanes generates an additional pressure drop
2
2
1
0
v / 2 v / 2
r
r
−
. The
pressure rise by the fan is
2
2
2
1
u / 2 u / 2
r
r
−
. The fan has to raise the pressure to the
atmospheric pressure.
The pressure balance is:
2
2
2
2
2
0
0
2
1
1
p
v
v
u
u
v
E
2
2
2
2
2
D =
−
=
+
−
.
So:
(1.35)
The work consists of three terms in kinetic energy with sum v 2
2
2
/ . The first term
is the useful term. That is the work required to compensate the pressure drop due to
the velocity generation within the suction pipe. The second term is the kinetic energy rise within the stator vane ring. This increase causes a second pressure drop to
be compensated by the fan. The second term is useless. The third term is the kinetic
energy increase within the rotor. This is also useless, as the fan ideally only should
generate a pressure rise. All kinetic energy generated within the rotor causes a larger
kinetic energy dissipated into the atmosphere downstream of the rotor. The useless
terms in (1.35) may be eliminated or reduced by adapting the rotor inlet and outlet,
as sketched below. The blade is leaned backward at the inlet, making the stator vane
ring unnecessary. The blade is leaned backward at the outlet in order to decrease
the outlet velocity in the absolute frame. Constant meridional velocity and constant
relative velocity are assumed (no diffusion).
The pressure balance is now
2
2
2
0
2
1
p
v
u u
E
2
2
D
−
=
=
.
So:
(1.36)
The first term is the useful term once more in (1.36). The second term is the kinetic
energy increase within the rotor. This term is not useful, but cannot be reduced to
zero, due to the set conditions concerning constant meridional velocity and relative velocity. Expression (1.36) is obviously much more favourable than expression
2
2
2
2
2
0
1
0
2
1
p
k
v
v v
v v
W
E
E
2
2
2
D
D
D
−
−
=
+
=
+
+
2 2u
1 1u
2 2u
2 2
1
W u v
u v
u v
u (u u ),
D =
−
=
=
−
2
2
2
2
2
2
2
2
2
2
1
2
1
2
1
1
2
2
1
k
p
v
v
(u u )
u u
w w
u u
E
, E
.
2
2
2
2
2
2
D
D
−
−
−
−
=
−
=
=
+
=
2
2
0
2
1
p
k
v
( u u )
W
E
E
.
2
2
D
D
D
−
=
+
=
+
1 Working Principles
Pressure Balance A first pressure drop occurs due to the generation of the velocity
in the suction pipe v 0 . The accompanying pressure drop is
2
0
v / 2
r
. The acceleration in the guide vanes generates an additional pressure drop
2
2
1
0
v / 2 v / 2
r
r
−
. The
pressure rise by the fan is
2
2
2
1
u / 2 u / 2
r
r
−
. The fan has to raise the pressure to the
atmospheric pressure.
The pressure balance is:
2
2
2
2
2
0
0
2
1
1
p
v
v
u
u
v
E
2
2
2
2
2
D =
−
=
+
−
.
So:
(1.35)
The work consists of three terms in kinetic energy with sum v 2
2
2
/ . The first term
is the useful term. That is the work required to compensate the pressure drop due to
the velocity generation within the suction pipe. The second term is the kinetic energy rise within the stator vane ring. This increase causes a second pressure drop to
be compensated by the fan. The second term is useless. The third term is the kinetic
energy increase within the rotor. This is also useless, as the fan ideally only should
generate a pressure rise. All kinetic energy generated within the rotor causes a larger
kinetic energy dissipated into the atmosphere downstream of the rotor. The useless
terms in (1.35) may be eliminated or reduced by adapting the rotor inlet and outlet,
as sketched below. The blade is leaned backward at the inlet, making the stator vane
ring unnecessary. The blade is leaned backward at the outlet in order to decrease
the outlet velocity in the absolute frame. Constant meridional velocity and constant
relative velocity are assumed (no diffusion).
The pressure balance is now
2
2
2
0
2
1
p
v
u u
E
2
2
D
−
=
=
.
So:
(1.36)
The first term is the useful term once more in (1.36). The second term is the kinetic
energy increase within the rotor. This term is not useful, but cannot be reduced to
zero, due to the set conditions concerning constant meridional velocity and relative velocity. Expression (1.36) is obviously much more favourable than expression
2
2
2
2
2
0
1
0
2
1
p
k
v
v v
v v
W
E
E
2
2
2
D
D
D
−
−
=
+
=
+
+
2 2u
1 1u
2 2u
2 2
1
W u v
u v
u v
u (u u ),
D =
−
=
=
−
2
2
2
2
2
2
2
2
2
2
1
2
1
2
1
1
2
2
1
k
p
v
v
(u u )
u u
w w
u u
E
, E
.
2
2
2
2
2
2
D
D
−
−
−
−
=
−
=
=
+
=
2
2
0
2
1
p
k
v
( u u )
W
E
E
.
2
2
D
D
D
−
=
+
=
+