fluid_mechanics

12.4 The Centrifugal Pump 657
pumps, the blade angle b 2 falls in the range of 15°35°, with a normal range of 20° 6 b 2 6 25°,
and with 15° 6 b 1 6 50° 1Ref. 102. Blades with b 2 6 90° are called backward curved, whereas
blades with b 2 7 90° are called forward curved. Pumps are not usually designed with forward
curved vanes since such pumps tend to suffer unstable flow conditions.
E XAMPLE 12.2
Centrifugal Pump Performance Based on Inlet/Outlet
Velocities
GIVEN Water is pumped at the rate of 1400 gpm through a FIND Determine (a) the tangential velocity component, at
centrifugal pump operating at a speed of 1750 rpm. The impeller the exit, (b) the ideal head rise, and (c) the power, W # V u2 ,
h i ,
shaft, transferred
to the fluid. Discuss the difference between ideal and actual
has a uniform blade height, b, of 2 in. with r 1 1.9 in. and
r 2 7.0 in., and the exit blade angle b 2 is 23° 1see Fig. 12.82. Assume
ideal flow conditions and that the tangential velocity com-
head rise. Is the power, W # shaft, ideal or actual? Explain.
ponent, V u1 , of the water entering the blade is zero 1a 1 90°2.
SOLUTION
(a) At the exit the velocity diagram is as shown in Fig. 12.8c,
1107 ft s2 2
32.2 fts 1107 ft s215.11 fts2 cot 23°
where V 2 is the absolute velocity of the fluid, W 2 is the relative
2 32.2 fts 2
velocity, and U 2 is the tip velocity of the impeller with
316 ft
(Ans)
11750 rpm2
U 2 r 2 v 1712 ft212p radrev2 (c) From Eq. 12.11, with V u1 0, the power transferred to the
160 smin2
fluid is given by the equation
107 fts
W # shaft rQU 2 V u2
Since the flowrate is given, it follows that
11.94 slugs ft 3 211400 gpm21107 fts2195.0 fts2
Q 2pr 2 b 2 V r2
311slug # fts 2 2lb417.48 galft 3 2160 smin2
or
161,500 ft # lbs211 hp550 ft # lbs2 112 hp (Ans)
From Fig. 12.8c we see that
so that
(b)
V r2
Q
2pr 2 b 2
1400 gpm
17.48 galft 3 2160 smin212p21712 ft21212 ft2
5.11 fts
V u2 U 2 V r2 cot b 2
1107 5.11 cot 23°2 fts
95.0 fts
From Eq. 12.15 the ideal head rise is given by
h i U 2V u2
g
316 ft
cot b 2 U 2 V u2
V r2
Alternatively, from Eq. 12.16, the ideal head rise is
h i U 2 2
1107 ft s2195.0 fts2
32.2 fts 2
g U 2V r2 cot b 2
g
(Ans)
(Ans)
Note that the ideal head rise and the power transferred to the
fluid are related through the relationship
W # shaft rgQh i
COMMENT It should be emphasized that results given in
the previous equation involve the ideal head rise. The actual
head-rise performance characteristics of a pump are usually
determined by experimental measurements obtained in a testing
laboratory. The actual head rise is always less than the ideal head
rise for a specific flowrate because of the loss of available energy
associated with actual flows. Also, it is important to note that
even if actual values of U 2 and V r2 are used in Eq. 12.16, the ideal
head rise is calculated. The only idealization used in this example
problem is that the exit flow angle is identical to the blade angle
at the exit. If the actual exit flow angle was made available in
this example, it could have been used in Eq. 12.16 to calculate
the ideal head rise.
The pump power, W # shaft, is the actual power required to
achieve a blade speed of 107 fts, a flowrate of 1400 gpm, and the
tangential velocity, V 2 , associated with this example. If pump
losses could somehow be reduced to zero (every pump designer’s
dream), the actual and ideal head rise would have been identical
at 316 ft. As is, the ideal head rise is 316 ft and the actual head rise
something less.