Brochure

U
2
= r
2
ω
U
1
= r
1
ω
r
1
ω
r
2
64
64
4. Pump theory
4.2 Euler’s pump equation
Euler’s pump equation is the most important equation in connection with
pump design. The equation can be derived in many dierent ways. The met-
hod described here includes a control volume which limits the impeller, the
moment of momentum equation which describes flow forces and velocity
triangles at inlet and outlet.
A control volume is an imaginary limited volume which is used for setting
up equilibrium equations. The equilibrium equation can be set up for tor-
ques, energy and other flow quantities which are of interest. The moment
of momentum equation is one such equilibrium equation, linking mass flow
and velocities with impeller diameter. A control volume between 1 and 2, as
shown in figure 4.6, is often used for an impeller.
The balance which we are interested in is a torque balance. The torque (T)
from the drive shaft corresponds to the torque originating from the fluid’s
flow through the impeller with mass flow m=rQ:
(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
[ ]
⋅
By multiplying the torque by the angular velocity, an expression for the
shaft power (P
2
) is found. At the same time, radius multiplied by the
angular velocity equals the tangential velocity, r
2
w = U
2
. This results in:
(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
[ ]
⋅
According to the energy equation, the hydraulic power added to the fluid
can be written as the increase in pressure Δp
tot
across the impeller multi-
plied by the flow Q:
(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 4.6: Control volume for an impeller.
1
2
2
1