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How many blades should a HAWT have?

Recently a participant asked the question '3 blades? 5 blades? 20 blades?' in the Wind Professionals Group I am member
off in Linkedin. There are many variables that effect number of blade selection, such as wind speed, TSR ( Tip Speed Ratio ),
weight, drag, cost and so on. Commertial three bladed turbines operate very efficiently in wide range of wind speed and TSR due
to their pitch adjustment mechanisms. Due to so many variables it is hard to answer this question easly. However if we keep
wind speed and TSR constant and not to wory about cost, this question becomes reletively easy to answer.

We know from physics that the power is the work done in the unit time. Unit time being a second, the more work a
turbine does in a given second the more power we can extract from it. Lets us ask following question to our
selves. What is the lowest RPM (Revolution Per Minute) a single bladed turbine can make without lousing its
efficiency drastically? Please note the emphesize on word lowest. It is 60 RPM or one revolution per second.
Why? Because it should sweep whole rotor area in one second. If turbine rotates slower than this, it will miss
some air particles and it's output power will be low. However if it rotates faster than one revolution per
second it will produce more power which is better.
But faster rotating blades create more stresses on overall structure. Therefore one bladed turbine
should make at least one revolution per second to cover all the rotor surface to produce optimum power.
However if rotates faster, it will produce more power.

What will be RPM of two bladed turbine to produce optimum power?
With same thinking, it would be 30 RPM, that is half revolution per second for the two bladed
turbine. This is because while one blade sweeping one half of the circle area, the other blade is sweeping
the other half. By using same analogy a three bladed turbine will make 20 RPM (one third of revolution per
second) and a four bladed turbine will make 15 RPM (one fourth revolution per second). I will STRESS one more
time that, if turbines rotates faster then these, up to a certain point, they will be more efficient. For
example if a single bladed turbine making 2 revolution per second will be more efficient than same turbine
making one revolution per second. This is because wherever the blade is during its revolution, wind is
escaping in the apposite end, by turning faster it will catch some of them. But there is an upper limit
how fast a blade should turn, if it turns much faster, it would act like a solid surface and stresses
will be enormous in the system. Following video shows what happens when a three bladed wind turbine rotation speed
exceeds its design limit for a given wind speed.

Now we know that the more blade the turbine has the slower it can rotate to cover the rotor sweep area.
Since the rotor rotation also depent on wind speed and TSR, now we know that there are some relationship
between wind speed, TSR and the blade numbers. But we still don't know how many blades we should have
for a given wind speed and TSR. The wind speed and TSR are the two constrains to answer to this question.
The wind speed is the biggest and the most important constrain nature put for the wind turbine and we do
not have any control on it. You can build worlds most efficient and powerful turbine and put it
in a location where there is no wind, you will get nothing from it. The second constrain is the TSR and
PLEASE DO NOT FORGET THAT TSR IS NOT APPLICABLE TO DRAG TYPE VAWT.
Unlike to the wind speed which we have no control, TSR constrain is set by designer of HAWT.
The TSR number for a given wind speed is always taken as high as possible so that we extract more power
due to high rotation. Please note that for 10m/s wind speed TSR could be 7 or 8 ( I do not know actual number)
but say for 30m/s wind speed, TSR could be 1 or less. This is because high wind put a lot of pressure on
system. If we kept TSR at 8, blade tip speed would be 240m/s. I hope you watched the wind turbine destruction
video given above. That destruction happened because TSR vas high for that wind speed. I assume that this
happened because pitch control mechanism failed to adjust TSR to lower values.
By changing pitch angle in one direction the blade TSR can be reduced and turning it in the apposite direction
TSR can be increased.
Now by using the wind speed constrain put by nature and TSR constrain set by the designer and in light of RPM
discussion we explained before we can find maximum rotor diameter for given number of blades.

For the argument sake lets build a HAWT which will work with optimum performance for the wind speed
in between 10m/s to 20m/s with TSR being 6 ( this is our choice, we as well chose another number such
as 5 or any reasonable number ). Since this turbine will work between 10m/s and 20m/s and our turbine does
not have pitch control mechanism, we should design the turbine for average wind speed of 15m/s.

The tip speed of a given blade can be found seperatly
from two formula given below.
TSR = V_tip_speed/V_wind_speed = 6
V_tip_speed = 6 * V_wind_speed = 6* 15 = 90 (1)
V_tip speed = Angular_velocity * R (2)
and
Angular_velocity = 2*PI*n/60 (3)
where n is being revolution per minute (RPM) and R is the radius of the blade.
By using formula (1), (2) and (3) we can find that
2*R*PI*n/60 = 90 ==> n*R = 860 (approximetly) (4)

From our previous discussion we know that for a single bladed turbine lowest RPM is 60.

If we use formula (4) with n = 60 we will find that R = 14.33.
This is optimum diameter for one bladed turbine for 15m/s wind velocity and TSR = 6. Note that
circumference of the circle with radius 14.33m is 90m which is equal to turbine blade tip speed. This
means that turbine blade comes its starting point in a second. If we take smaller diameter for one
bladed turbine, it will rotate faster and efficiency will increase but area will decrease. Since area
decreased by square of radius, power output will drop accordingly. If the blade get longer than 14.33
the efficiency of the turbine will drop due to blade's inability to sweep all the rotor surface.
However the extra length added to the blade will create an annulus segment around the optimum circle
which will increase power output. However in some point efficiency loss by sweeping less area and
increased annulus will be balanced out and further increasing the length will not be feasible to
produce single bladed turbine. When this point is reached, the blade number should be increased.
There will be always a window where N bladed turbine and N+1 bladed turbine can perform equally well.

Lets find optimum diameter for 1,2,3 and 4 bladed HAWT.
By using the formula (4) and minimum RPM for a given blade number we can conclude that
For one bladed turbine optimum radius is,
60*R = 860 ==> R = 14.33m.
For two bladed turbine optimum radius is,
30*R = 860 ==> R = 28.66m.
For three bladed turbine optimum radius is,
20*R = 860 ==> R = 43m.
And for four bladed turbine it is,
15*R = 860 ==> R = 57.33m.

To demonstrate this concept following flash animation for 1,2,3 and 4 bladed HAWT has been made. Please note
that from the time you press "Rotate" button, to the time the rotation stops, represent 1 second time interval.
Note that during this time interval 1 bladed turbine makes 1 revolution, 2 blade turbine makes half
revolution, 3 bladed turbine makes one third and 4 bladed turbine makes quarter rotation. However in
each case whole rotor area swept by the blades. It is also important to note that tips of every blade
travel a distance of 90m. This is because V_tip_speed is 90m/s and time interval is 1 second. The
circumference of each swept area is 90,180,270 and 360 respectively for 1,2,3 and 4 bladed turbines
for this design.

Please note that for a 4 bladed turbine it has been said that, optimum diameter should be 57.33m, but we may
see 3 bladed turbine rotates in this range. This is because, while due to less area the 3 bladed
turbine sweep it might louse some power, however by increased annulus around optimum radius it gains
additional power. Since annulus increase by square of the radius gain will be much higher than lost.
If the wind speed, TSR, density of the air and efficiency kept constant the only one variable determines the output of
the turbine is the area that extract power from the wind. Note that I did not say the rotor sweep area.
Assume that we have a 3 bladed turbine operating its optimum radius of 43m. The power output of this turbine
will be C*43**2 where C is a constant including in it air density, PI and cube of the wind velocity which we assumed all fixed values.

Therefore power output of three bladed turbine in its optimum radius will be,
P = 43*43*C = 1849*C Hp.
Now assume that we used same 3 bladed turbine in the optimum diameter
of 4 bladed turbine which is 57.33.
What will be the power output? Is it
P = 57.33*57.33*C = 3286.73*C Hp ?
We think that it will be,
P = 3*3286.73*C/4 = 2465*C Hp.

Why? This is because blades are not sweeping whole rotor area. Please look at following animation
and figure out what will be the output of one and two bladed turbines if their rotor diameter is
taken to be 57.33m.

Hint: For 1 blade power output will be 821.7*C Hp and for two blade turbine it will be 1643.4*C Hp.

What will happen if we keep TSR at 6 and change wind speed to 30m/s? Since wind speed doubled and TSR kept constant, the tip speed of blades will be 180m/s rather than 90m/s. This will cause rotation speed (angular velocity) to double too. Following flash animation demonstrates what happens when rotation speed is increased.

Note that even this doubled speed one bladed turbine is not efficient when its rotor diameter is taken
as 57,33m it sweep half of the rotor area but misses air particles in the other half. The two bladed
turbines perform its optimum level since all the rotor area swept. For three bladed turbine blades
sweeps 1.5 times of its rotor area per second. You can observe blade colors (shown on yellow blade)
as Dark where the blade passed from there twice and Light where the blade passes once in one second.
For four bladed turbines two blades pass from any given point per second. Blades sweep whole rotor area
twice. Here there are only darker regions since blades pass every location twice.

It should be emphasized that the discussions given above is not representation of real world application.
This is very crude approximation for how many blades we might need for given design consideration. The
turbines we see far away lumbering slowly are a technological marvels and very sophisticated machines
with their variable pitch and computer control mechanisms. They work very efficiently in wide range of
wind speeds. However approximation given above gives you some logical explanation what is the appropriate
blade number for a given turbine.