Brochure
W
2
W
1
W
1
C
1
C
1U
C
2
C
2U
C
2m
C
1m
C
1m
U
2
U
1
U
1
β
1
α
1
β
2
α
2
63
63
4.1.2 Outlet
As with the inlet, the velocity triangle at the outlet is drawn as shown in
figure 4.2 position 2. For a radial impeller, outlet area is calculated as:
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.12)
111
2 brA
⋅ ⋅
=
π
1
,1
,
1
1
2
2 b
rr
A
shroud hub
+
⋅ ⋅ ⋅=
π
1
1
A
Q
C
impeller
m
=
ωπ ⋅ = ⋅ ⋅ ⋅ =
111
60
2 r
n
rU
1
1
tan
U
C
m
=
1
β
222
2 brA ⋅ ⋅ = π
2
,2,2
2
2
2 b
rr
A
shroud hub
⋅
+
⋅ ⋅ = π
2
2
A
Q
C
impeller
m
=
ωπ ⋅ = ⋅ ⋅ ⋅ =
222
60
2 r
n
rU
2
βsin
2
2
m
C
W =
=
2
βtan
2
2
m
C
U −
2U
C
)(
1122 UU
CrCrmT ⋅ − ⋅ ⋅ =
)(
)(
)(
)(
1122
1122
1122
1122
UU
UU
UU
UU
2
CUCUQ
CUCUm
CrCrm
CrCrm
TP
−
. . ..
=
−
. . .
=
−
. . . . .
=
−
. . . .
=
=
ρ
ωω
ω
ω
QpP
tothyd
⋅ ∆ =
g
p
H
tot
⋅
∆
=
ρ
gH
m
gHQP
hyd
⋅ ⋅ = ⋅ ⋅ ⋅ = ρ
g
CUCU
H
CUCUmgHm
PP
UU
UU
2hyd
)(
)(
1122
1122
⋅ − ⋅
=
⋅ − ⋅ ⋅ = ⋅ ⋅
=
Static head as consequence
of the centrifugal force
Static head as consequence
of the velocity change
through the impeller
Dynamic head
g
CC
g
WW
g
UU
H
⋅
−
+
⋅
−
+
⋅
−
=
222
2
1
2
2
2
2
2
1
2
1
2
2
[ ]
m
2
[ ]
m
2
[ ]
m
2
[ ]
m
2
[ ]
Nm
[ ]
m
[ ]
W
[ ]
W
[ ]
W
[ ]
m
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
⋅
and for a semi axial impeller it is:
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.12)
111
2 brA
⋅ ⋅
=
π
1
,1
,
1
1
2
2 b
rr
A
shroud hub
+
⋅ ⋅ ⋅=
π
1
1
A
Q
C
impeller
m
=
ωπ ⋅ = ⋅ ⋅ ⋅ =
111
60
2 r
n
rU
1
1
tan
U
C
m
=
1
β
222
2 brA ⋅ ⋅ = π
2
,2,2
2
2
2 b
rr
A
shroud hub
⋅
+
⋅ ⋅ = π
2
2
A
Q
C
impeller
m
=
ωπ ⋅ = ⋅ ⋅ ⋅ =
222
60
2 r
n
rU
2
βsin
2
2
m
C
W =
=
2
βtan
2
2
m
C
U −
2U
C
)(
1122 UU
CrCrmT ⋅ − ⋅ ⋅ =
)(
)(
)(
)(
1122
1122
1122
1122
UU
UU
UU
UU
2
CUCUQ
CUCUm
CrCrm
CrCrm
TP
−
. . ..
=
−
. . .
=
−
. . . . .
=
−
. . . .
=
=
ρ
ωω
ω
ω
QpP
tothyd
⋅ ∆ =
g
p
H
tot
⋅
∆
=
ρ
gH
m
gHQP
hyd
⋅ ⋅ = ⋅ ⋅ ⋅ = ρ
g
CUCU
H
CUCUmgHm
PP
UU
UU
2hyd
)(
)(
1122
1122
⋅ − ⋅
=
⋅ − ⋅ ⋅ = ⋅ ⋅
=
Static head as consequence
of the centrifugal force
Static head as consequence
of the velocity change
through the impeller
Dynamic head
g
CC
g
WW
g
UU
H
⋅
−
+
⋅
−
+
⋅
−
=
222
2
1
2
2
2
2
2
1
2
1
2
2
[ ]
m
2
[ ]
m
2
[ ]
m
2
[ ]
m
2
[ ]
Nm
[ ]
m
[ ]
W
[ ]
W
[ ]
W
[ ]
m
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
⋅
C
2m
is calculated in the same way as for the inlet:
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.12)
111
2 brA
⋅ ⋅
=
π
1
,1
,
1
1
2
2 b
rr
A
shroud hub
+
⋅ ⋅ ⋅=
π
1
1
A
Q
C
impeller
m
=
ωπ ⋅ = ⋅ ⋅ ⋅ =
111
60
2 r
n
rU
1
1
tan
U
C
m
=
1
β
222
2 brA ⋅ ⋅ = π
2
,2,2
2
2
2 b
rr
A
shroud hub
⋅
+
⋅ ⋅ = π
2
2
A
Q
C
impeller
m
=
ωπ ⋅ = ⋅ ⋅ ⋅ =
222
60
2 r
n
rU
2
βsin
2
2
m
C
W =
=
2
βtan
2
2
m
C
U −
2U
C
)(
1122 UU
CrCrmT ⋅ − ⋅ ⋅ =
)(
)(
)(
)(
1122
1122
1122
1122
UU
UU
UU
UU
2
CUCUQ
CUCUm
CrCrm
CrCrm
TP
−
. . ..
=
−
. . .
=
−
. . . . .
=
−
. . . .
=
=
ρ
ωω
ω
ω
QpP
tothyd
⋅ ∆ =
g
p
H
tot
⋅
∆
=
ρ
gH
m
gHQP
hyd
⋅ ⋅ = ⋅ ⋅ ⋅ = ρ
g
CUCU
H
CUCUmgHm
PP
UU
UU
2hyd
)(
)(
1122
1122
⋅ − ⋅
=
⋅ − ⋅ ⋅ = ⋅ ⋅
=
Static head as consequence
of the centrifugal force
Static head as consequence
of the velocity change
through the impeller
Dynamic head
g
CC
g
WW
g
UU
H
⋅
−
+
⋅
−
+
⋅
−
=
222
2
1
2
2
2
2
2
1
2
1
2
2
[ ]
m
2
[ ]
m
2
[ ]
m
2
[ ]
m
2
[ ]
Nm
[ ]
m
[ ]
W
[ ]
W
[ ]
W
[ ]
m
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
⋅
The tangential velocity U is calculated from the following:
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.12)
111
2 brA
⋅ ⋅
=
π
1
,1
,
1
1
2
2 b
rr
A
shroud hub
+
⋅ ⋅ ⋅=
π
1
1
A
Q
C
impeller
m
=
ωπ ⋅ = ⋅ ⋅ ⋅ =
111
60
2 r
n
rU
1
1
tan
U
C
m
=
1
β
222
2 brA ⋅ ⋅ = π
2
,2,2
2
2
2 b
rr
A
shroud hub
⋅
+
⋅ ⋅ = π
2
2
A
Q
C
impeller
m
=
ωπ ⋅ = ⋅ ⋅ ⋅ =
222
60
2 r
n
rU
2
βsin
2
2
m
C
W =
=
2
βtan
2
2
m
C
U −
2U
C
)(
1122 UU
CrCrmT ⋅ − ⋅ ⋅ =
)(
)(
)(
)(
1122
1122
1122
1122
UU
UU
UU
UU
2
CUCUQ
CUCUm
CrCrm
CrCrm
TP
−
. . ..
=
−
. . .
=
−
. . . . .
=
−
. . . .
=
=
ρ
ωω
ω
ω
QpP
tothyd
⋅ ∆ =
g
p
H
tot
⋅
∆
=
ρ
gH
m
gHQP
hyd
⋅ ⋅ = ⋅ ⋅ ⋅ = ρ
g
CUCU
H
CUCUmgHm
PP
UU
UU
2hyd
)(
)(
1122
1122
⋅ − ⋅
=
⋅ − ⋅ ⋅ = ⋅ ⋅
=
Static head as consequence
of the centrifugal force
Static head as consequence
of the velocity change
through the impeller
Dynamic head
g
CC
g
WW
g
UU
H
⋅
−
+
⋅
−
+
⋅
−
=
222
2
1
2
2
2
2
2
1
2
1
2
2
[ ]
m
2
[ ]
m
2
[ ]
m
2
[ ]
m
2
[ ]
Nm
[ ]
m
[ ]
W
[ ]
W
[ ]
W
[ ]
m
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
⋅
In the beginning of the design phase, β
2
is assumed to have the same value
as the blade angle. The relative velocity can then be calculated from:
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.12)
111
2 brA
⋅ ⋅
=
π
1
,1
,
1
1
2
2 b
rr
A
shroud hub
+
⋅ ⋅ ⋅=
π
1
1
A
Q
C
impeller
m
=
ωπ ⋅ = ⋅ ⋅ ⋅ =
111
60
2 r
n
rU
1
1
tan
U
C
m
=
1
β
222
2 brA ⋅ ⋅ = π
2
,2,2
2
2
2 b
rr
A
shroud hub
⋅
+
⋅ ⋅ = π
2
2
A
Q
C
impeller
m
=
ωπ ⋅ = ⋅ ⋅ ⋅ =
222
60
2 r
n
rU
2
βsin
2
2
m
C
W =
=
2
βtan
2
2
m
C
U −
2U
C
)(
1122 UU
CrCrmT ⋅ − ⋅ ⋅ =
)(
)(
)(
)(
1122
1122
1122
1122
UU
UU
UU
UU
2
CUCUQ
CUCUm
CrCrm
CrCrm
TP
−
. . ..
=
−
. . .
=
−
. . . . .
=
−
. . . .
=
=
ρ
ωω
ω
ω
QpP
tothyd
⋅ ∆ =
g
p
H
tot
⋅
∆
=
ρ
gH
m
gHQP
hyd
⋅ ⋅ = ⋅ ⋅ ⋅ = ρ
g
CUCU
H
CUCUmgHm
PP
UU
UU
2hyd
)(
)(
1122
1122
⋅ − ⋅
=
⋅ − ⋅ ⋅ = ⋅ ⋅
=
Static head as consequence
of the centrifugal force
Static head as consequence
of the velocity change
through the impeller
Dynamic head
g
CC
g
WW
g
UU
H
⋅
−
+
⋅
−
+
⋅
−
=
222
2
1
2
2
2
2
2
1
2
1
2
2
[ ]
m
2
[ ]
m
2
[ ]
m
2
[ ]
m
2
[ ]
Nm
[ ]
m
[ ]
W
[ ]
W
[ ]
W
[ ]
m
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
⋅
and C
2U
as:
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.12)
111
2 brA
⋅ ⋅
=
π
1
,1
,
1
1
2
2 b
rr
A
shroud hub
+
⋅ ⋅ ⋅=
π
1
1
A
Q
C
impeller
m
=
ωπ ⋅ = ⋅ ⋅ ⋅ =
111
60
2 r
n
rU
1
1
tan
U
C
m
=
1
β
222
2 brA ⋅ ⋅ = π
2
,2,2
2
2
2 b
rr
A
shroud hub
⋅
+
⋅ ⋅ = π
2
2
A
Q
C
impeller
m
=
ωπ ⋅ = ⋅ ⋅ ⋅ =
222
60
2 r
n
rU
2
βsin
2
2
m
C
W =
=
2
βtan
2
2
m
C
U −
2U
C
)(
1122 UU
CrCrmT ⋅ − ⋅ ⋅ =
)(
)(
)(
)(
1122
1122
1122
1122
UU
UU
UU
UU
2
CUCUQ
CUCUm
CrCrm
CrCrm
TP
−
. . ..
=
−
. . .
=
−
. . . . .
=
−
. . . .
=
=
ρ
ωω
ω
ω
QpP
tothyd
⋅ ∆ =
g
p
H
tot
⋅
∆
=
ρ
gH
m
gHQP
hyd
⋅ ⋅ = ⋅ ⋅ ⋅ = ρ
g
CUCU
H
CUCUmgHm
PP
UU
UU
2hyd
)(
)(
1122
1122
⋅ − ⋅
=
⋅ − ⋅ ⋅ = ⋅ ⋅
=
Static head as consequence
of the centrifugal force
Static head as consequence
of the velocity change
through the impeller
Dynamic head
g
CC
g
WW
g
UU
H
⋅
−
+
⋅
−
+
⋅
−
=
222
2
1
2
2
2
2
2
1
2
1
2
2
[ ]
m
2
[ ]
m
2
[ ]
m
2
[ ]
m
2
[ ]
Nm
[ ]
m
[ ]
W
[ ]
W
[ ]
W
[ ]
m
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
m
s
[ ]
⋅
Hereby the velocity triangle at the outlet has been determined and can now
be drawn, see figure 4.5.
Figure 4.5: Velocity triangle at outlet.