130
3 Fans
is the incidence loss at the entrance of the volute. The mixing loss is minimal for
minimum relative velocity at outlet of the rotor ( w 2 ). This implies minimum relative
inlet velocity ( w 1 ) and strongest possible deceleration in the rotor. The incidence
loss at entrance in the volute is mostly minimal for the outlet velocity of the rotor in
the absolute frame ( v 2 ) as near to the tangential direction as possible, which again,
implies minimum w 2 .
The deceleration in the eye of the rotor may be expressed by a velocity factor
ζ =
v
v
r
b
1
0
with v
Q
d b
r
b
rotor
1
1 1 1
= π
τ
and v
Q
d
0
0
2
4
= π
.
Thus:
The diameter of the eye d 0 is somewhat smaller than the inlet diameter of the rotor
d 1 and Q rotor is somewhat larger than Q. Say d 0 = 0.9 d 1 ; η V
r otor
Q Q
=
≈
/
.
0 9 (volumetric efficiency); τ 1 = 0.9 (obstruction factor), such that
The strongest possible deceleration between the inlet of the eye and the inlet of the
rotor is about 0.60, so that 1 1
/
0.25 to 0.40
b d =
. Then:
v
Q
k d b
k
Q
d
r
b
1
1 1
1
2
4
=
=
π
ζ
π
, with k
V
=
≈
η τ 1
2
0 9
( . ) and u
d
1
1
2
=
Ω .
Thus:
with
4
a
(
)Q
k
z
p
=
and b =
Ω
2
.
With the minimum of w 1 corresponds an optimum value of d 1 :
(3.35)
The corresponding velocity components are:
(3.36)
The remarkable result is that the expressions of the velocity components and the
diameter d 1 are all proportional to the same factor ( / )
/
ζ π
k
1 3
. This means that it
is advantageous to set the velocity factor to the lowest possible value. The second
2
0
1 1 1
.
4
rotor
Q
d
Q
d b
z
t
=
1
1
1
1
1
1
or
.
4
4
d
b
b
d
z
z
≈
≈
2
2
2
2
2 2
2
2
1
1
1
1
1
( )
( )
( )
( ),
b
b
r
w
v
u
a d
b d
−
=
+ =
+
( )
.
/
/
/
d
k
Q
o
1
1 6
1 3
1 3
2 2
=
ζ
π
Ω
1/3
1/3
1/6
1/3 2/3
1/3
1/3 2/3
1
1
2
and
2
.
b
r
u
Q
v
Q
k
k
z
z
p
p
−
=
Ω
=
Ω
3 Fans
is the incidence loss at the entrance of the volute. The mixing loss is minimal for
minimum relative velocity at outlet of the rotor ( w 2 ). This implies minimum relative
inlet velocity ( w 1 ) and strongest possible deceleration in the rotor. The incidence
loss at entrance in the volute is mostly minimal for the outlet velocity of the rotor in
the absolute frame ( v 2 ) as near to the tangential direction as possible, which again,
implies minimum w 2 .
The deceleration in the eye of the rotor may be expressed by a velocity factor
ζ =
v
v
r
b
1
0
with v
Q
d b
r
b
rotor
1
1 1 1
= π
τ
and v
Q
d
0
0
2
4
= π
.
Thus:
The diameter of the eye d 0 is somewhat smaller than the inlet diameter of the rotor
d 1 and Q rotor is somewhat larger than Q. Say d 0 = 0.9 d 1 ; η V
r otor
Q Q
=
≈
/
.
0 9 (volumetric efficiency); τ 1 = 0.9 (obstruction factor), such that
The strongest possible deceleration between the inlet of the eye and the inlet of the
rotor is about 0.60, so that 1 1
/
0.25 to 0.40
b d =
. Then:
v
Q
k d b
k
Q
d
r
b
1
1 1
1
2
4
=
=
π
ζ
π
, with k
V
=
≈
η τ 1
2
0 9
( . ) and u
d
1
1
2
=
Ω .
Thus:
with
4
a
(
)Q
k
z
p
=
and b =
Ω
2
.
With the minimum of w 1 corresponds an optimum value of d 1 :
(3.35)
The corresponding velocity components are:
(3.36)
The remarkable result is that the expressions of the velocity components and the
diameter d 1 are all proportional to the same factor ( / )
/
ζ π
k
1 3
. This means that it
is advantageous to set the velocity factor to the lowest possible value. The second
2
0
1 1 1
.
4
rotor
Q
d
Q
d b
z
t
=
1
1
1
1
1
1
or
.
4
4
d
b
b
d
z
z
≈
≈
2
2
2
2
2 2
2
2
1
1
1
1
1
( )
( )
( )
( ),
b
b
r
w
v
u
a d
b d
−
=
+ =
+
( )
.
/
/
/
d
k
Q
o
1
1 6
1 3
1 3
2 2
=
ζ
π
Ω
1/3
1/3
1/6
1/3 2/3
1/3
1/3 2/3
1
1
2
and
2
.
b
r
u
Q
v
Q
k
k
z
z
p
p
−
=
Ω
=
Ω