Brochure

β
2
< 90
o
β
2
β
1
β
2
β
1
β
2
β
1

2
>90
o
β
2
= 90
o
β
2
< 90
o
β
2
β
1
β
2
β
1
β
2
β
1

2
>90
o
β
2
= 90
o
β
2
< 90
o
β
2
β
1
β
2
β
1
β
2
β
1

2
>90
o
β
2
= 90
o
H
Q
H for b
2
> 90°
Forward-swept blades
H for b
2
= 90°
H for b
2
< 90°
Backward-swept blades
66
66
When designing a pump, it is often assumed that there is no inlet rotation
meaning that C
1U
equeals zero.
4.3 Blade shape and pump curve
If it is assumed that there is no inlet rotation (C
1U
=0), a combination of Eul-
er’s pump equation (4.17) and equation (4.6), (4.8) and (4.11) show that the
head varies linearly with the flow, and that the slope depends on the outlet
angle β
2
:
n
n
n
(4.21)
(4.22)
(4.23)
vmF ⋅ =
⋅
2
vAvmI ⋅⋅=⋅=∆ ρ
⋅
FI = ∆
(4.21)
(4.22)
(4.23)
Q
g bD
U
g
U
H
⋅
⋅ ⋅⋅
⋅
−=
22
2
2
2
2
) tan(
π
β
Scaling of
rotational speed
n
PP
QQ
HH

















⋅ =






⋅ =






⋅ =
3
2
Geometric
scaling
bD
bD
PP
bD
bD
Q
D
D
HH



















⋅
⋅
⋅=







⋅
⋅
⋅=






⋅=
4
4
2
2
2
u,Am,A
C
A
B
AB
CU
= =
(4.24)
(4.25)
(4.26)
(4.27)
(4.28)
2,A
DnU ⋅
=
2,A AA
2
2,A2,AA2,A2,A2,A2,A
222
nb
2,B B
nb
D
2,B
D
DnbDC
C
bDQ
B
Q
CbDCAQ
2m,A
2m,B
2m2m
⋅ ⋅








=
⋅⋅⋅⋅
= ⋅
⋅⋅
=
⋅⋅⋅ = ⋅ =
π
2,BB2,B2,B
DnbD ⋅⋅⋅⋅ π
π
2,B2,B
bD ⋅⋅ π
π
2 2
2,A2,A2,A22,A22,A
22,A
⋅















=
⋅⋅⋅
=
⋅
=
⋅⋅
=
⋅
=
nDDnDnCU gCU H
H
g
CU
H
U,A
22,B
⋅ CU
U,B
AA
B
A
2,B
nD
B
A
2,B2,B
⋅⋅⋅ DnDn
B B
U,A
22,B
⋅⋅ gCU
U,B
U,A
3
4
2,A 2,A A22,A22,A
22
⋅














= ⋅ =
⋅
⋅ =
⋅ ⋅ ⋅
=
⋅⋅⋅=
n
B
n
b
2,B
b
D
2,B
D
HQCUQ
Q
CU
QP
CUQP
U,A
22,B
⋅ CU
U,B
A A
HQ
B B
U,AA
P
B
A A
B
U
ρ
22,B
⋅ ⋅ ⋅ CUQ
U,BB
ρ
ρ
4
3
2
1
H
Q
nn
q
⋅=
(4.29)
(4.20)
g
U
H =
2 2U
[ ]
⋅ C
m
[ ]
N
[ ]
N
[ ]
N
[ ]
m
B
B
P
A
n
B
A
n
B
A
B
A
A
B A
A
Q
B
AB
AA
BB
AA
BB
A
u,B
d
d
d
m,B
CCU
B
A A
2,B
DnU ⋅
B B
Figure 4.7 and 4.8 illustrate the connection between the theoretical pump
curve and the blade shape indicated at β
2
.
Real pump curves are, however, curved due to dierent losses, slip, inlet
rotation, etc., This is further discussed in chapter 5.
Figure 4.7: Blade shapes depending on outlet angle
Figure 4.8: Theoretical pump curves calcu-
lated based on formula (4.21).
4. Pump theory
n
n
n
(4.21)
(4.22)
(4.23)
vmF ⋅ =
⋅
2
vAvmI ⋅⋅=⋅=∆ ρ
⋅
FI = ∆
(4.21)
(4.22)
(4.23)
Q
g bD
U
g
U
H
⋅
⋅ ⋅⋅
⋅
−=
22
2
2
2
2
) tan(
π
β
Scaling of
rotational speed
n
PP
QQ
HH

















⋅ =






⋅ =






⋅ =
3
2
Geometric
scaling
bD
bD
PP
bD
bD
Q
D
D
HH



















⋅
⋅
⋅=







⋅
⋅
⋅=






⋅=
4
4
2
2
2
u,Am,A
C
A
B
AB
CU
= =
(4.24)
(4.25)
(4.26)
(4.27)
(4.28)
2,A
DnU ⋅
=
2,A AA
2
2,A2,AA2,A2,A2,A2,A
222
nb
2,B B
nb
D
2,B
D
DnbDC
C
bDQ
B
Q
CbDCAQ
2m,A
2m,B
2m2m
⋅ ⋅








=
⋅⋅⋅⋅
= ⋅
⋅⋅
=
⋅⋅⋅ = ⋅ =
π
2,BB2,B2,B
DnbD ⋅⋅⋅⋅ π
π
2,B2,B
bD ⋅⋅ π
π
2 2
2,A2,A2,A22,A22,A
22,A
⋅















=
⋅⋅⋅
=
⋅
=
⋅⋅
=
⋅
=
nDDnDnCU gCU H
H
g
CU
H
U,A
22,B
⋅ CU
U,B
AA
B
A
2,B
nD
B
A
2,B2,B
⋅⋅⋅ DnDn
B B
U,A
22,B
⋅⋅ gCU
U,B
U,A
3
4
2,A 2,A A22,A22,A
22
⋅














= ⋅ =
⋅
⋅ =
⋅ ⋅ ⋅
=
⋅⋅⋅=
n
B
n
b
2,B
b
D
2,B
D
HQCUQ
Q
CUQP
CUQP
U,A
22,B
⋅ CU
U,B
A A
HQ
B B
U,AA
P
B
A A
B
U
ρ
22,B
⋅ ⋅ ⋅ CUQ
U,BB
ρ
ρ
4
3
2
1
H
Q
nn
q
⋅=
(4.29)
(4.20)
g
U
H =
2 2U
[ ]
⋅ C
m
[ ]
N
[ ]
N
[ ]
N
[ ]
m
B
B
P
A
n
B
A
n
B
A
B
A
A
B A
A
Q
B
AB
AA
BB
AA
BB
A
u,B
d
d
d
m,B
CCU
B
A A
2,B
DnU ⋅
B B