Problem 11.98
Francis and Kaplan turbines are often provided with draft tubes, which lead the exit flow into the
tailwater region, as in Fig. P11.98. Explain at least two advantages to using a draft tube.
Solution 11.98
Draft tubes have two big ad-vantages:
Problem 11.99
Turbines can also cavitate when the pressure at point 1 in Fig. P11.98 drops too low. With NPSH
defined by Eq. (11.20), the empirical criterion given by Wislicenus [4] for cavitation is
1/2
ss 3/4
(rpm)(gal / min)
N 11,000
[NPSH(ft)]
=
Use this criterion to compute how high z1 − z2, the impeller eye in Fig. P11.98, can be placed for
a Francis turbine, with a head of 300 ft, Nsp = 40, and pa = 14 lbf/in2 absolute before cavitation
occurs in 60F water.
Solution 11.99
For water at 60F, take
g = 62.4 lbf/ft3 and pv 37 psfa = 0.25 psia. Then
Problem 11.100
The manufacturer of the wind turbine in the chapter-opener photo claims that it develops exactly
100 kW at a wind speed of 15 m/s. Compare this with an estimate from the correlations in
Fig. 11.32.
Solution 11.100
It is near sea level, so take ρ = 1.2 kg/m3. There is a curve for a 3-blade HAWT in Fig. 11.32,
with a maximum CP ≈ 0.43 at ωr/V ≈ 5.0. The predicted maximum power is thus
Problem 11.101
A Darrieus VAWT in operation in Lumsden, Saskatchewan, that is 32 ft high and 20 ft in
diameter sweeps out an area of 432 ft2. Estimate (a) the maximum power and (b) the rotor speed
if it is operating in 16 mi/h winds.
Solution 11.101
For air in Saskatchewan (?), take
0.0023 slug/ft3. Convert 16 mi/h = 23.5 ft/s. From Fig. 11.34 for
Problem 11.102
An American 6-ft diameter multiblade HAWT is used to pump water to a height of 10 ft through
3-in-diameter cast-iron pipe. If the winds are 12 mi/h, estimate the rate of water flow in gal/min.
Solution 11.102
For air in “America” (?), take
0.0023 slug/ft3. Convert 12 mi/h = 17.6 ft/s. For water, take
= 1.94 slug/ft3 and
= 2.09E−5 slug/(fts). From Fig. 11.34 for the American multi-blade
Problem 11.103
Only a mile from the wind turbine in the chapter-opener photo is a 100-ft-high, 23-ft-diameter
HAWT, in Fig. P11.103. It is rated at 10 kW and provides half of the electricity for the Salty
Brine State Beach bathhouse. From the data in Fig. 11.32, at a wind velocity of 20 mi/h,
estimate (a) the maximum power developed, and (b) the rotation speed, in r/min?
Figure 11.103:
Solution 11.103
Assume sea-level, ρ ≈ 0.00237 slug/ft3. Convert 20 mi/h to 29.3 ft/s. The curve for a 3-blade
HAWT in Fig. 11.32 shows a maximum CP ≈ 0.43 at ωr/V ≈ 5.0. (a) The predicted maximum
power is
Problem 11.104
The controversial Cape Wind project proposes 130 large wind turbines in Nantucket Sound,
intended to provide 75 percent of the electric power needs of Cape Cod and the Islands. The
turbine diameter is 328 ft. For an average wind velocity of 14 mi/h, what are the best rotation
rate and total power output estimates for (a) a HAWT; and (b) a VAWT?
Solution 11.104
Convert 14 mi/h = 20.53 ft/s. Use Fig. 11.32 of the text to make your estimates. (a) HAWT
geometry. Best efficiency is at (
r/V) 5.8, where CP 0.45:
Problem 11.105
In 2007, a wind-powered-vehicle contest, held in North Holland [64], was won with a design by
students at the University of Stuttgart. A schematic of the winning three-wheeler is shown in
Fig. P11.105. It is powered by a shrouded wind turbine, not a propeller, and, unlike a sailboat,
can move directly into the wind. (a) How does it work? (b) What if the wind is off to the side?
(c) Cite some design questions you might have.
Solution 11.105
(a) The wind turns the turbine. The turbine turns a shaft that is geared to the wheels and drives
the vehicle.
Problem 11.106
Analyze the wind-powered-vehicle of Fig. P11.105 with the following data: turbine diameter
D = 6 ft, power coefficient (Fig. 11.32) = 0.3, vehicle CDA = 4.5 ft2, and turbine rotation
240 r/min. The vehicle moves directly into a head wind, W = 25 mi/h. The wind backward
thrust on the turbine is approximately T CT(
)Vrel2 Aturbine, where Vrel is the air velocity
relative to the turbine and CT 0.7. Eighty per cent of the turbine power is delivered by gears to
the wheels, to propel the vehicle. Estimate the sea-level vehicle velocity V, in mi/h.
Solution 11.106
At sea level take
= 0.00237 slug/ft3. Recall the definition of power coefficient from
Eq. (11.46). The air velocity relative to the moving turbine is (W+V). Hence
Problem 11.107
Figure 11.32 showed the typical power performance of a wind turbine. The wind also causes a
thrust force that must be resisted by the structure. The thrust coefficient CT of a wind turbine
may be defined as follows:
2 2 2
( / 2) ( / 2)[( / 4) ]
T
Thrust force T
C
A V D V
==
Values of CT for a typical horizontal-axis wind turbine are shown in Fig. P11.107. The abscissa
is the same as in Fig. 11.32. Consider the turbine of Prob. P11.103. If the wind is 20 mi/h and
the rotation rate 115 r/min, estimate the bending moment about the tower base.
Problem 11.103
Only a mile from the wind turbine in the chapter-opener photo is a 100-ft-high, 23-ft-diameter
HAWT, in Fig. P11.103. It is rated at 10 kW and provides half of the electricity for the Salty
Brine State Beach bathhouse. From the data in Fig. 11.32, at a wind velocity of 20 mi/h,
estimate (a) the maximum power developed, and (b) the rotation speed, in r/min?
Figure 11.103:
Solution 11.107
Assume sea-level density of 0.00237 slug/ft3 – all of Rhode Island is pretty much sea-level.
Convert 115 r/min x 2/60 =
= 12.04 rad/s. Convert wind speed = 20 mi/h = 29.3 ft/s. Now
find where we are on the graph:
Problem 11.108
To avoid the bulky tower and impeller and generator in the HAWT of the chapter-opener photo,
we could instead build a number of Darrieus turbines of height 4 m and diameter 3 m. (a) How
many of these would we need to match the HAWT’s 100 kW output for 15 m/s wind speed and
maximum power? (b) How fast would they rotate? Assume the area swept out by a Darrieus
turbine is two-thirds the height times the diameter.
Solution 11.108
Again take ρ = 1.2 kg/m3. From Fig. 11.32, for a Darrieus turbine, CP,max = 0.42 at ωr/V = 4.2.
The total power generated by one turbine is thus