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
62
62
4. Pump theory
4.1.1 Inlet
Usually it is assumed that the flow at the impeller inlet is non-rotational.
This means that α
1
=90°. The triangle is drawn as shown in figure 4.2 position
1, and C
1m
is calculated from the flow and the ring area in the inlet.
The ring area can be calculated in dierent ways depending on impeller type
(radial impeller or semi-axial impeller), see figure 4.3. For a radial impeller
this 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
[ ]
⋅
where
r
1
= The radial position of the impeller’s inlet edge [m]
b
1
= The blade’s height at the inlet [m]
and for a semi-axial impeller this 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
[ ]
⋅
The entire flow must pass through this ring area. C
1m
is then 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
[ ]
⋅
The tangential velocity U
1
equals the product of radius and angular frequency:
(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
[ ]
⋅
where
ω = Angular frequency [s
-1
]
n = Rotational speed [min
-1
]
When the velocity triangle has been drawn, see figure 4.4, based on α
1
, C
1m
and U
1
, the relative flow angle β
1
can be calculated. Without inlet rotation
(C
1
= C
1m
) this becomes:
(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
[ ]
⋅
Figure 4.3: Radial impeller at the top,
semi-axial impeller at the bottom.
Blade
Blade
Figure 4.4: Velocity triangle at inlet.
b
2
b
1
r
1
r
2
b
1
b
2
r
1, hub
r
2, hub
r
1, shroud
r
2, shroud