Environmental Engineering Reference
In-Depth Information
Now, the
Bernoulli equation
may be used between the free-stream and the upwind side
of the turbine and again between the downwind side of the turbine and the far wake, so that
Equation (5-10) becomes
T
= 0.5r
A U
2
-
V
1
( )
(5-11)
Combining Equations (5-9) and (5-11) we obtain
V
= 0.5(
U
+
V
1
)
(5-12)
Thus, the wind velocity at the disk is the average of the free-stream and far-wake velocities,
so the total velocity change from free-stream to far-wake is twice the change from free-
stream to the disk. Let
U
-
V
=
aU
(5-13a)
U
-
V
1
= 2
aU
Then
(5-13b)
The term
a
is known as the
axial induction factor
(or the
retardation factor
) and is a
measure of the influence of the turbine on the wind. Because the minimum far-wake velocity
is zero, the maximum value of the axial induction factor is 0.5. The thrust is not of immedi-
ate importance, but the power is. From the
first law of thermodynamics
, assuming isothermal
flow and ambient pressure in the far wake, power is equal to
(
P
= 0.5r
A U
2
-
V
1
)
V
= 0.5r
AV
(
U
+
V
1
) (
U
-
V
1
)
(5-14)
Combining Equations (5-13) and (5-14), the power coefficient for the actuator disk, accord-
ing to the Rankine-Froude theory, is
P
0.5r
U
3
A
= 4
a
(1 -
a
)
2
(5-15a)
C
P
=
which has a maximum when
a
=
1/3
. Thus
)
C
P
, max
= 16/27 = 0.593
(
(5-15b)
When examining Equation (5-15a), note that the denominator is the kinetic energy of the
free-stream wind contained in a
streamtube
with an area equal to the disk area. However,
Equation (5-15b) does not represent maximum
efficiency
(power output/power input), since
the mass flow rate through the disk is not r
AU
but r
AV
. Instead,
P
0.5r
U
2
VA
= 4
a
(1 -
a
)
(5-16)
h
d
=
where
h
d
= actuator disk efficiency.
The maximum efficiency is 100 percent at
a
=
0.5
, which yields a power coefficient of 0.5.
The disk efficiency is 88.8 percent at the maximum power coefficient of 0.593.
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