Environmental Engineering Reference
In-Depth Information
Use Equation 6.3:
$
$
p
t
$
$
(
m
T
T
vv
)
R
uvv
(
)
(6.13)
0
2
0
2
Also, the thrust loading on the disk due to the pressure difference across the disk is
TAp p
(
)
(6.14)
Bernoulli's theorem relates the velocity and pressures in streamline flow (kinetic energy and
pressure are constants for horizontal flow). If the velocity increases, then the pressure decreases;
the two are related through conservation of energy and momentum. The wind speed and pressure
upstream and downstream of the disk are related by:
Upstream
Disk
Downstream
2
2
+
2
2
0.5
R
vp
0.5
R
up
0.5
R
up
0.5
R
vp
0
0
2
0
From the two equations, take the pressure difference (
p
p
) and substitute into Equation 6.14:
2
2
T
0.5R
A v
v
(6.15)
0
2
The thrusts are equal, so set Equation 6.13 equal to Equation 6.15:
2
2
R
uvv
(
)
0.5
R
Avv
0.5
R
Avvvv
(
)(
2
)
(6.16)
0
2
0
2
0
2
0
From Equation 6.14 the wind speed at the disk is the average of the wind speeds before and after
the disk (wake).
u
0.5 (
v
0
v
2
)
(6.17)
The axial interference factor is defined by what ratio the wind speed is reduced by the disk.
vu
v
u
v
0
A
1
or
uv
(
A
(6.18)
0
0
0
Substitute into Equation 6.17 and the wake wind speed is
vv
v
0
2
vv
(1
2AA
) or
(6.19)
2
0
2
0
If the disk or rotor absorbs all the energy,
v
2
0 and
]
0.5. That is physical nonsense, as all the
mass would pile up at the rotor. The power is equal to the change in kinetic energy from upstream
to downstream:
KE
KE
$
KE
t
m
t
m
t
us
ds
2
2
2
2
P
0.5
v
0.5
v
0.5
R
Au v
v
t
0
2
0
2
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