Renewable Energy
Assignment 2 Fluid Mechanics of Wind Tu
ines
Due Sunday 11th September (This assignment should take a typical student around 10 hours to complete)
Notes:
• Submit this assignment as a professionally presented report. In your report you must present all of your
analysis clearly. Present, describe and discuss the equations and results as you would in an engineering
design report. Consider using LaTeX as this will make presentation of equations easier.
• Submit your report as a PDF document generated directly from your Word or LaTeX file, not scanned.
• Write a computer program in the language of your choice – eg. Matlab, Python etc – to do this
assignment. (You may not use a spreadsheet to do this assignment.) Include your complete computer
program in an appendix to your report as part of the same document. If you do not include your program
you will receive a mark of zero for this assignment.
• Submit your report electronically through Turnitin on the Canvas site.
• The terminology below is the same as that used in the Wind Tu
ine Analysis section of the lecture notes.
You are part of a design team doing a preliminary study for a wind farm. One of the features of the proposed
site is that the wind changes direction frequently. This may cause problems if horizontal axis tu
ines are
used. Instead you are considering using vertical axis gyromill-type tu
ines*. Each tu
ine will have blades of
length H = 10m and width (chord length) K = 0.6m, and will be mounted at a distance r = 5m from the central
axis.
* These are also known as Da
ieus H-type tu
ines.
The tu
ine will use aerofoil blades with a NACA 0015 profile. The lift and drag characteristics may be
approximated as follows. The drag coefficient is approximately sinusoidal and approximated by
??= 0.61− 0.58 cos(2?).
The lift coefficient data is shown below. You should approximate it as piece-wise linear with 8 linear segments.
The coordinates of the 8 end points of the linear segments are shown in Table 1.
The coordinates of the end-points for each linear segment are:
Point Angle of attack, ? (°) Lift coefficient CL
A XXXXXXXXXX
B XXXXXXXXXX
C XXXXXXXXXX
D XXXXXXXXXX
E XXXXXXXXXX
F 54 0.9
G XXXXXXXXXX
H XXXXXXXXXX
Part A
Consider a single blade. Assume air density = 1.2 kg/m3.
1) Plot CL and CD against the angle of attack α for angles from -180 to 180. Use intervals of 2 degrees.
2) For a wind speed V = 12m/s, blade speed U = 9m/s (ie a tip-speed ratio U/V = XXXXXXXXXXand set-up angle = 0,
plot the following parameters against the blade position angle at 2 degree intervals from = -180 to =
180:
a)
) CL and CD
c) the lift and drag forces FL and FD
d) the lift and drag forces in the direction of the blade motion FCF and FCB
e) the torque on the blade T
Hence determine the mean torque
(kN.m), mean delivered power PD (kW) and rotational speed (rad/s)
for these operating conditions.
[0.62kN.m, 1.11kW, 1.8rad/s]
3) For a wind speed V = 12m/s, blade speed U = 18m/s (ie a tip-speed ratio U/V = 1.5) and set-up angle = 0
plot the following parameters against the blade position angle at 2 degree intervals from = -180 to =
180:
a)
) CL and CD
c) the lift and drag forces FL and FD
d) the lift and drag forces in the direction of the blade motion FCF and FCB
e) the torque on the blade T
Hence determine the mean torque (kN.m), mean delivered power PD (kW) and rotational speed (rad/s)
for these operating conditions.
[1.14kN.m, 4.10kW, 3.6rad/s]
4) Plot and PD against U/V for U/V at intervals of 1.2 in the range 0 – 7.2 for a wind speed of 12m/s.
(Hint: Write a function and then call it varying U/V as an input argument.)
U/V (kN.m) PD (kW)
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
What do the negative torque and power for U/V = 7.2 mean in terms of the operation of the tu
ine?
Part B
Consider a tu
ine using three blades with the same aerofoil as in Part A.
1) Plot efficiency against U/V for a set up angle =0. (As before, plot for U/V at intervals of 1.2 in the range 0
– 7.2).
U/V efficiency(%)
0 0.00
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
6 0.14
XXXXXXXXXX
2) Play with your model. Try different values of the set up angle and observe the effect it has on efficiency and
power. Determine the optimum setup angle with this aerofoil profile. (I recommend a simple trial and e
or
approach.) Explain this finding. [0]
3) Try different numbers of tu
ine blades. What happens? In a real tu
ine would you expect this trend to
continue indefinitely? Explain your answer.
4) For the three blade tu
ine, plot and PD against at a wind speed V = 12m/s. On your vs plot add
the generator load line for a generator with the following torque vs speed characteristic: Tgen = 0.62 kN.m.*
Hence determine the operating speed , mean torque , and delivered power PD of the tu
ine for this wind
speed. ( Hint – the operating point is the where the tu
ine torque and generator torque are balanced. Just
find this point by eye.) [8.8rad/s, 5.5kN.m, 48kW] (Answers approximate)
5) Plot the operating speed , torque , and delivered power PD of the tu
ine against wind speed V for wind
speeds V =0, 2, 4, … , 24, 26 m/s. (Hint: Set up your program so that you can vary V and read these values off
the plots.) (Answers approximate).
V (m/s)
omega
(
s) T (kN.m)
PD
(kW)
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
XXXXXXXXXX
*NB: Real tu
ine controllers give a more complex generator torque curve. In particular, this enables a cut-in wind
speed which we have not included here. If you’re interested, I’ve put up an article on the Canvas site.
Part C
An eolergometric survey has been undertaken at the site and the wind speed probability distribution (in all
directions) has been fitted to a Weibull distribution. The shape and scale parameters for the site were found to be
k = 3.3 and A = 14.8m/s respectively.
The wind farm is to have 40 tu
ines of the type analyzed in parts A and B. Each tu
ine has three blades with a
set-up angle of = 0. Assume total electrical losses of 15% in the generators and power electronics. Neglect
mechanical inefficiencies such as wake interference, blade fouling etc.
1) Estimate the mean velocity and mean cubic velocity of the wind at the site. Which is more useful in choosing a
wind farm site? Why? Also determine the mean available power density of the wind at the site.
[13.28m/s, 14.62m/s, 1112W/m2]
2) Plot the wind speed probability density F x 100 (%/ms -1) against wind speed V for V at 2m/s intervals in the
ange 0 – 50m/s. On the same plot show the tu
ine power P D/100 that you derived in Part B and also XXXXXXXXXXF x
P D (kW/ms-1). Assume that the tu
ine shuts down at wind speeds above 27m/s.
3) Determine the mean power delivered by each tu
ine to its generator and the capacity factor and specific
ated capacity of the tu
ines. Hence, estimate the annual average electrical energy delivered by the wind
farm (in GWh).
[79kW, 20.9%, 3780W/m2, 23.5GWh] (Answers approximate)
4) Determine the mean efficiency of the tu
ines. [71%] (Answers approximate)
5) The capacity factor calculated in part 3 could be improved. If the tu
ine power is “capped” at its level for V =
22m/s (that is its power output remains constant for wind speeds in the range V = 22 – 27m/s) a smaller
generator could be used. Determine the effect of this change on the capacity factor and specific rated
capacity of the tu
ine and the annual average electrical energy delivered by the wind farm. Comment on this.
Show a plot similar to the one you generated for part 2 for this case. What is a simple way in which the tu
ine
power could be capped involving only the tu
ine blades?
[31.4%, 2480W/m2, 23.2GWh] (Answers approximate)
Try capping at different velocities and look at how the capacity factor and annual energy output change.
6) [OPTIONAL] Just for fun … play with your wind farm model ☺. For example:
a. How does the size of the generator affect the power and torque curves of the whole machine? And
how does that affect things like the capacity factor of