Brochure

65
65
The head is defined 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 the expression for hydraulic power can therefore be transcribed to:
(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
[ ]
⋅
If the flow is assumed to be loss free, then the hydraulic and mechanical
power can be equated:
(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
[ ]
⋅
This is the equation known as Euler’s equation, and it expresses the impel-
ler’s head at tangential and absolute velocities in inlet and outlet.
If the cosine relations are applied to the velocity triangles, Euler’s pump
equation can be written as the sum of the three contributions:
• Staticheadasconsequenceofthecentrifugalforce
• Staticheadasconsequenceofthevelocitychangethroughtheimpeller
• Dynamichead
(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
[ ]
⋅
If there is no flow through the impeller and it is assumed that there is no
inlet rotation, then the head is only determined by the tangential velocity
based on (4.17) where C
2U
= U
2
:
(4.19)
g
U
H =
2
2
0
[ ]
m