Brochure

U
A
W
B
U
B
C
B
W
A
C
A
C
m,B
C
U,B
C
m,A
C
U,A
W
2,B
U
C
2,B
W
2
C
2
C
2m,B
C
2U,B
C
2m
C
2U
U
W C
C
m
C
U
β
2
β
2
H
Q
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
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
68
68
In the following, the eect of reducing the outlet width b
2
on the velocity
triangles is discussed. From e.g. (4.6) and (4.8), the velocity C
2m
can be seen
to be inversely proportional to b
2
. The size of C
2m
therefore increases when b
2
decreases. U
2
in equation (4.9) is seen to be independent of b
2
and remains
constant. The blade angle β
2
does not change when changing b
2
.
The velocity triangle can be plotted in the new situation, as shown in figure
4.10. The figure shows that the velocities C
2U
and C
2
will decrease and that
W
2
will increase. The head will then decrease according to equation (4.21).
The power which is proportional to the flow multiplied by the head will
decrease correspondingly. The head at zero flow, see formula (4.20), is
proportional to U
2
2
and is therefore not changed in this case. Figure 4.11
shows a sketch of the pump curves before and after the change.
Similaranalysiscanbemadewhenthebladeformischanged,seesection
4.3, and by scaling of both speed and geometry, see section 4.5.
4.5 Anity rules
By means of the so-called anity rules, the consequences of certain changes
in the pump geometry and speed can be predicted with much precision.
The rules are all derived under the condition that the velocity triangles are
geometrically similar before and after the change. In the formulas below,
derived in section 4.5.1, index
A
refers to the original geometry and index
B
to
the scaled geometry.
Figure 4.10: Velocity triangle at changed
outlet width b
2
.
Figure 4.11: Change of head curve as
consequence of changed b
2
.
4. Pump theory