Environmental Engineering Reference
In-Depth Information
3.5 Combining the Equations for the Streamtube
and the Blade Element
Equations
3.2
and
3.5
for the wake must be solved with (
3.10
) and (
3.11
) for the
blade element. One way of doing this is through the following relationship for /:
U
1
Xr 1
þ
a
0
1
a
k
r
1
þ
a
0
tan /
¼
Þ
¼
ð
3
:
12
Þ
ð
ð
Þ
which follows from the velocity triangle Fig.
3.2
. k
r
is the local speed ratio from
Eq.
3.7b
which obviously varies from zero at the axis of rotation to k at the blade
tip. If values are assumed for a and a
0
, / can be found from (
3.12
) and then C
l
and
C
d
determined and so on. This is an iterative process because (
3.2
) and (
3.10
) can
be combined to give a
n
, the new estimate for a for the nth iteration, as
Þ
f
a
¼
U
T
C
a
r
=
4U
0
a
n
1
a
n
ð
ð
3
:
13
Þ
where the quantities on the right hand side of (
3.13
) are those from the (n - 1)
iteration. In Eq.
3.13
, r is defined as
r
¼
Nc
=
2p
ðÞ
3
:
14
Þ
which is the local solidity mentioned in
Sect. 3.2
. f
a
in (
3.13
) plays the same role
for the blade element as does C
T
for the whole turbine in the analysis of
Chap. 2
.
Thus, for example, it is easy to modify (
3.13
) for the high thrust region in the same
way that Eqs.
2.20
and
2.21
can modify (
2.19
).
Combining (
3.5
) and (
3.11
) gives
a
0
¼
aC
a
0
=
C
a
k
r
ð
Þ
ð
3
:
15
Þ
This allows the iteration to be solely based on a until the momentum and
angular momentum fluxes in the flow over the blades balance the thrust and torque,
respectively, on a blade element.
3.6 Matlab Programs for Blade Element Analysis
This section describes and lists three simple Matlab programs that implement the
equations derived in the last two chapters. Use of the programs will be postponed
until aerofoil properties are more extensively discussed in
Chap. 4
, including the
circulation, and their consequences. The programs are available from the online
materials site (
http://extras.springer.com
).
All lengths are normalised by the tip radius and all velocities by the wind speed,
unless specifically stated otherwise in the definition of the appropriate variable in
the comments at the start of each program.
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