Brochure
7070
4.5.1 Derivation of the anity rules
The anity method is very precise when adjusting the speed up and down
and when using geometrical scaling in all directions (3D scaling). The ani-
ty rules can also be used when wanting to change outlet width and outlet
diameter (2D scaling).
When the velocity triangles are similar, then the relation between the
corresponding sides in the velocity triangles is the same before and after
a change of all components, see figure 4.13. The velocities hereby relate to
each other as:
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
The tangential velocity is expressed by the speed n and the impeller’s outer
diameter D
2
. The expression above for the relation between components
before and after the change of the impeller diameter can be inserted:
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
Figure 4.13: Velocity triangle
at scaled pump.
4. Pump theory
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