fluid_mechanics

12.4 The Centrifugal Pump 655
Centrifugal pump
impellers involve an
increase in blade
velocity along the
flow path.
As discussed in Section 12.3, the moment-of-momentum equation indicates that the shaft
torque, required to rotate the pump impeller is given by equation Eq. 12.2 applied to a pump
with m # T shaft
1 m # ,
2 m # . That is,
or
(12.9)
T shaft rQ1r 2 V u2 r 1 V u1 2
(12.10)
where V u1 and V u2 are the tangential components of the absolute velocities, V 1 and V 2 1see Figs.
12.8b,c2.
For a rotating shaft, the power transferred, W # shaft, is given by
and therefore from Eq. 12.10
Since U 1 r 1 v and U 2 r 2 v we obtain
T shaft m # 1r 2 V u2 r 1 V u1 2
W # shaft T shaft v
W # shaft rQv1r 2 V u2 r 1 V u1 2
W # shaft rQ1U 2 V u2 U 1 V u1 2
(12.11)
Equation 12.11 shows how the power supplied to the shaft of the pump is transferred to the flowing
fluid. It also follows that the shaft power per unit mass of flowing fluid is
w shaft W# shaft
rQ
For incompressible pump flow, we get from Eq. 5.82
U 2V u2 U 1 V u1
w shaft a p out
r V 2
out
2 gz outb a p in
r V in 2
2 gz inb loss
(12.12)
W 2
β 2
V 2
V θ 2
α 2
Vr2
(c)
U 2
W 2
β 2
V 2
U 2
α 2
W 1
V 1
r 2
1
W 1
V 1
r 1
β 1
β
α 1
U 1
(a)
+
ω
V θ 1
α 1
Vr1
U 1
(b)
F I G U R E 12.8
centrifugal pump impeller.
Velocity diagrams at the inlet and exit of a