fluid_mechanics

12.9 Compressible Flow Turbomachines 685
F l u i d s i n t h e N e w s
Cavitation damage in hydraulic turbines The occurrence of
cavitation in hydraulic pumps seem to be an obvious possibility
since low suction pressures are expected. Cavitation damage can
also occur in hydraulic turbines even though they do not seem
obviously prone to this kind of problem. Local acceleration of
liquid over blade surfaces can be sufficient to result in local pressures
low enough to cause fluid vaporization or cavitation.
Further along the flow path, the fluid can decelerate rapidly
enough with accompanying increase in local pressure to make
cavitation bubbles collapse with enough intensity to cause blade
surface damage in the form of material erosion. Over time, this
erosion can be severe enough to require blade repair or replacement
which is very expensive. (See Problem 12.80.)
E XAMPLE 12.9
Use of Specific Speed to Select Turbine Type
GIVEN A hydraulic turbine is to operate at an angular velocity
of 6 revs, a flowrate of 10 ft 3 s, and a head of 20 ft.
SOLUTION
The most efficient type of turbine to use can be obtained by calculating
the specific speed, , and using the information of Fig.
12.32. To use the dimensional form of the specific speed indicated
in Fig. 12.32 we must convert the given data into the appropriate
units. For the rotor speed we get
To estimate the shaft power, we assume all of the available head
is converted into power and multiply this amount by an assumed
efficiency 194%2.
W # shaft 21.3 hp
Thus for this turbine,
N¿ sd v2W# shaft
1h a 2 5 4
According to the information of Fig. 12.32,
A mixed-flow Francis turbine would
probably give the highest efficiency and
an assumed efficiency of 0.94 is appropriate.
(Ans)
COMMENT What would happen if we wished to use a Pelton
wheel for this application? Note that with only a 20-ft head, the
maximum jet velocity, V 1 , obtainable 1neglecting viscous effects2
would be
As shown by Eq. 12.52, for maximum efficiency of a Pelton
wheel the jet velocity is ideally two times the blade velocity.
Thus, V 1 2vR, or the wheel diameter, D 2R, is
N¿ sd
FIND What type of turbine should be selected? Explain.
v 6 revs 60 smin 360 rpm
W # 20 ft10.942
shaft gQzh 162.4 lbft 3 2110 ft 3 s2c
550 ft # lbs # hp d
1360 rpm2 221.3 hp
120 ft2 5 4
39.3
V 1 12 gz 22 32.2 fts 2 20 ft 35.9 fts
To obtain a flowrate of Q 10 ft 3 s at a velocity of
V 1 35.9 fts, the jet diameter, d 1 , must be given by
or
D V 1
v 35.9 fts
0.952 ft
16 revs 2p radrev2
Q p 4 d 2 1 V 1
d 1 c 4Q 12
d c 4110 ft3 12
s2
pV 1 p135.9 fts2 d 0.596 ft
A Pelton wheel with a diameter of D 0.952 ft supplied with
water through a nozzle of diameter d 1 0.596 ft is not a practical
design. Typically d 1 66 D 1see Fig. 12.222. By using
multiple jets it would be possible to reduce the jet diameter.
However, even with 8 jets, the jet diameter would be 0.211 ft,
which is still too large 1relative to the wheel diameter2 to be
practical. Hence, the above calculations reinforce the results
presented in Fig. 12.32—a Pelton wheel would not be practical
for this application. If the flowrate were considerably smaller,
the specific speed could be reduced to the range where a Pelton
wheel would be the type to use 1rather than a mixed-flow reaction
turbine2.
12.9 Compressible Flow Turbomachines
Compressible flow turbomachines are in many ways similar to the incompressible flow pumps and
turbines described in previous portions of this chapter. The main difference is that the density of
the fluid 1a gas or vapor2 changes significantly from the inlet to the outlet of the compressible flow
machines. This added feature has interesting consequences, benefits, and complications.