17.13 WIND POWER
713
Pt- P2
P3 =pt
-
3
-
Figure 17.20 A schematic diagram illustrating a venturi flowmeasuring device.
velocity may be determined by knowing the pressure change and the flow rate,
determined from rh = pAv. The throat contraction is small, so the pressure change is
also small. If the flow is compressible, the velocity change may be found as follows. If
a density change occurs and must be accounted for, the equation pvk = C can be used
to determine the functional relationship in the following integral.
The conservation of energy applied between state 1 and state 2 yields for isentropic flow
h1 + (k.e.)1 =h.,.+ (k.e.)2
v2
2
v21 + !.2 vdp
2
1
__1=-
where dh
= T ds + v dp = v dp for isentropic flow. For incompressible flow, v = C.
v2 = [vj + 2v 1(P2 - p 1)] 112 m/s
(17.52a}
v2 = [vj + 2gcP.(P2- Pt)P12 ft/sec
(17.52b}
The velocity, specific volume, and area being known, the mass flow rate may be
calculated.
There are other flow-measuring devices, such as orifices, but we will not discuss
these or the empirical coefficients that must be included in any study of flow-measuring devices. Most fluid dynamics texts cover this area in greater detail.
17.13 WIND POWER
The use of wind power as a power source appears attractive for several reasons. First,
wind power is a renewable resource, for which the required technology has already
been developed. Second, no air, water, or thermal pollution is associated with it.
Third, weather modification due to wind utilization is negligible. The drawbacks to
wind-power utilization are relatively low efficiencies, high capital costs, noise, aesthetic problems, and reliability of equipment.
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CHAPTER 17 / FLUID FLOW IN NOZZLES AND TURBOMACHINERY
Wind power is not a reliable energy source, the output being a function of the
wind velocity. Since wind-velocity fluctuations do not normally coincide with powerrequirement variations, most wind-power systems must include energy storage systems.
.
There are several types of wind-driven machines, operating on several different
principles. Although each type has its advantages, the one type that stands out as the
most promising is the horizontal-axis, two-bladed propeller wind turbine, illustrated
in Figure 17.21.
A wind turbine should have the following characteristics: the ability to maintain
optimum alignment with the wind; a low starting torque; the ability to endure high
winds; and, if used to drive a generator, a high rotational speed.
The propeller windmill is almost always either a two- or three-bladed design; the
two-bladed design is more widely used because it is strong, simple, and less expensive.
The horizontal-axis windmill can be positioned so that the blades lie upwind or
downwind of the tower. The downwind design is usually preferred for larger machines, where a tail vane is not practical. On smaller models, a tail vane keeps the
blades pointing into the wind. Large windmills are usually steered by a pilot wind
vane, coupled to the drive gear, which operates to keep the windmill in constant
alignment with the wind. The pilot wind vane is more sensitive to wind shifts than is
Rotor axis
Figure 11.21 A two-bladed horizontal
windmill.
715
17.13 WIND POWER
Pt
Figure 17.22
Windmill propeller in a flow field.
the large windmill. At the cutout wind speed, the blades are turned edgewise to the
wind to protect the machine against damage due to high winds.
Let us develop the equations governing the windmill. Figure 17.22 illustrates a
windmill propeller located in a moving fluid.
When the fluid between planes 1 and 4 is isolated, the only force acting is that
exerted by the fluid on the propeller. The force acting on the windmill is equal to the
pressure drop across the blades.
(17.53)
The force is also equal to
F= m(v1 - v4)
If we let v be the mean velocity across the blades and p be the air density,
F= pvA(vi- v4)
(17.54)
Combining equations (17.53) and (17.54) yields
P2- P3
= pv(vi- v4)
(17.55)
Applying the first law to the flow between planes 1 and 2 yields
U1
+ P1V1 + !v~ =
U2
+ P2V2 + !v~
(17.56)
The temperature is constant; hence u1 = u2 and
Pt
+ lP1V~ = P2 + fP2V~
(17.57a)
Similarly, for planes 3 and 4,
P3
+ lP3V~ = P4 + lP4vi
(17.57b)
For a windmill operating in an unconfined fluid, pressures p 1 and p 4 are equal. Since
this is so, equations ( 17.56) and ( 17 .57) may be combined, on the assumption that the
density is constant and by virtue of the conservation of mass v3 = v2 •
P2 - P3 = !p(v~-
vi)
(17.58)
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CHAPTER 17 / FLUID FLOW IN NOZZLES AND TURBOMACHINERY
Combining equations (17.55) and (17.58) yields
vi +v4
v=--=--...::.
(17.59)
2
The mean velocity across the propeller blades is equal to the average of the upstream
and downstream velocities, measured at some distance from the windmill. Thus, the
velocity drop through the propeller is the same ahead as behind.
The windmill efficiency is the ratio of the power output to the total power
available in the airstream of area A and velocity v. This was derived in Chapter 3.
Power available =
Power output =
T
rizy2
(v2v2)Avp
1
4
2
q= (v·I + v4)(v I
2 -
v 42)
(17.60)
2v~
The maximum efficiency is found by differentiating ttwith respect to {v4 /v 1) and
setting the result equal to zero. Let
Then
'1 = (v I
+ yv l )(v 2l -
y 2v 2I )
2vf
(1 + y)(1 - y )
= ----'---.;.......;...
2
2
(17.61)
dtt =0=3y2 +2y-1
dy
The only physically possible solution is y = t. This results in a value of v4 jv1 = t,
which when substituted in equation (17.61) yields a maximum efficiency of59.3%.
This is the maximum percentage ofenergy that can be used from the available energy,
the inlet wind's kinetic energy.
17.14 ENERGY TRANSFER IN A TURBOMACHINE
The word turbomachine is derived from two Latin words; turbo, a spinning object,
and mach ina, a solid device. If these are combined, the result is a "machine that spins
or rotates." The rotating portion, or rotor, interacts with the fluid that encompasses it.
A turbomachine is, therefore, a device that imparts energy to (does work on) or
receives energy from (has work done by) a fluid. See Figure 17.23 for a simplified
sketch noting the rotor; fixed portion, or stator; and fluid flow direction. Thermodynamically, the effect of the energy transfer is manifested as a change in the stagnation
enthalpy.