Wind Resources 2
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| Part Number | 2 | |
| Number of Parts | 5 | |
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| License | CC Attribution - ShareAlike 4.0 International: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this | |
| Identifiers | 10.5446/64799 (DOI) | |
| Publisher | 014nnvj65 (ROR) | |
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00:00
Lecture/ConferenceComputer animation
Transcript: English(auto-generated)
00:00
Hello and welcome. In the second video we will have a look again to our wind behavior, but this time we are looking how we can use the wind. Is it possible to extract all the power from the wind or is there a natural limitation in this process? And therefore we are considering the Betz impulse theory.
00:25
So we have now the same cylinder of air that is moving around and now we are looking what happens when this cylinder approaches a wind turbine. And then when the wind turbine is reached, what we are doing,
00:42
we want to slow down the wind. And when we slow down the wind, we take power out of the wind and definitely the wind speed behind the turbine is then at lower value than before. And as a consequence it looks much more like we see here. So once more we assume that the wind
01:07
V1 approaching the wind turbine is quite large, then probably the wind speed at the wind term itself is V. And when we have slowed down the wind, definitely behind the
01:24
turbine, the wind velocity V2 must be slower. So we have V2 that is lower than V1. But now we see in the picture that this cylinder is getting wider. And why is that?
01:42
And that is due to the fact that finally the air mass remains the same before the wind turbine and behind the wind turbine. So we use the power of the wind, but we are not consuming the wind. The wind turbine is not eating or drinking wind, so therefore the amount behind must be the same.
02:03
And when the velocity here is fast, then the area A1 needs to be different from the area A2. That means when the speed behind is slower, then the area just must be bigger. And that means that the A2 is bigger than the A1. And the first one who ever had a look to
02:29
this question how much wind we can extract from the wind by wind turbine that was this Betz. And according to him the impulse theory is called the Betz impulse theory.
02:45
There we would like to have a closer look. First what we could express the power that we take out of the wind is the force the wind is giving to our rotor blades. And that we can
03:01
do by the impulse rule that we say the force F is equivalent the mass flow times the difference of the wind speeds V1 minus V2. And that would be then our equation five when we also consider the numeration of equations from the last video. So and when we also know that the power
03:29
is equivalent to our force times the velocity and we use our equations three and five then we can express that the power P is the mass flow times the velocity V1
03:50
minus V2 times the wind speed at the wind turbine itself. And we can express that
04:00
with the density times the rotor area times the velocity to the two V1 minus V2. We also can have a look to the power of the wind turbine that it extracts from the wind
04:21
by the different energy contents in the wind before and behind the wind turbine. So we also could express that the power that we are extracting is just the kinetic power of the wind that we have before the turbine minus the kinetic power that we have behind the wind
04:47
turbine that would be equation eight. And when we now set eight is equivalent to the expression of equation seven then we get that the air mass times the V1 minus the V2 times
05:08
the V that is equivalent then one half the air mass flow times V1 to the square minus V2 to
05:21
the square. And there we also could eliminate the mass flow and then when we solve the equation towards the wind velocity in the wind turbine area itself then we get here that is one
05:41
half times V1 to the square minus V2 to the square divided by V1 minus V2. And there we see binomial formula and that means that is V1 minus V2 times V1 plus V2 divided by V1 minus V2
06:07
and there we can eliminate those two and then we have the result that the velocity at place of the wind turbine is just one half times V1 plus V2. That is not really surprising
06:27
that the wind speed directly at the turbine is just the geometric mean of the wind speed before and behind the wind turbine. Nevertheless we use that now to express the power that we have
06:44
that we are able to extract and when we put here the equation nine in seven then we get here that is the density times the rotor area times one fourth times V1 plus V2 to the square
07:07
times V1 minus V2 and that is equation 10. And now we are interested what can we do or what is the maximum power that we can extract and when we have a function where we are interested
07:24
in the maximum value we need to have the derivation the derivative of this equation and setting that to equal to zero. The question is to what we are deviating the equation and for sure
07:42
density area are constants the V1 is the air velocity there we are almost not able to influence that. So what we are able to influence with our technology with the wind turbine that is the velocity behind the wind turbine. So in order to look for the maximum power we write dp and make
08:07
the dairy weight towards the velocity V2 and when we do so we get here by the product rule we get here that is one half rho times a times V1 plus V2 times V1 minus V2
08:32
and then it is minus one fourth rho times a times V1 plus V2 to the square.
08:44
And that when we are looking for a maximum we are setting equal zero. So we could make that a bit easier by formulating this into one half V1 plus V2 V1 minus V2 minus
09:13
one fourth rho times a V1 plus V2 and V1 plus V2. And then we can extract here one fourth
09:28
rho times a times this V1 plus V2 and then we have a second expression and there we have limited 2 V1 minus 2 V2 minus V1 minus V2 and that we can express then as one fourth rho times
09:56
a V1 plus V2 and then we have left here the V1 minus 3 times V2 and that still needs to become
10:09
a zero. And now we have two brackets expression and that means one of those two needs to be zero to become zero in total then we have the first solution that would be that the V2 is equal to
10:27
minus the V1. That is from a mathematical point of view a good solution but from physics are more or less impossible that would mean the wind that approaches the wind turbine is reflected
10:42
with the same value in the different in the opposite direction. At least up to now nobody has invented a wind turbine that works like that. So we have the second solution when the second bracket expression is becoming zero and that means we would have then the V2 is
11:15
that means when the wind is approaching with V1 we slow it down by two-third and behind the wind
11:23
turbine it is one-third of the wind velocity we already have had. And when we now put this 11 into our equation 10 we get the expression for the maximum
11:42
power we are able to extract and that is one-fourth row times a times 16 divided by 9 V1 to the square times two-third times V1 and that we can express in a different way
12:02
we use the expression one-half row times a times V1 to the three and that is exactly the power of the wind and then we have remaining here a factor of 16 divided by 27 and that is
12:21
exactly the maximum efficiency we or the maximum part that we can extract from the wind and that is approximately 60 percent it is more 59 dot something else percent so that is the
12:40
maximum in reality we are a bit below this maximum value because in reality the picture looks a bit like that here because that means that our wind turbine the rotor is influencing our air cylinder by its presence and that means all the cylinder is getting in a rotation itself
13:04
and when it's getting in a rotation that means there's also energy needed to put the air cylinder in rotation and that we cannot convert into electrical power and then we also observe at the rotor blades that we have such turbulences around the rotor blades that we
13:23
also find then here in those parts behind the wind turbine all that means that the efficiency in reality is even a bit lower than this theoretical 60 percent but with modern wind turbines we are approaching quite close to this theoretical limit and in the next video
13:47
we will then discuss why we need statistics to describe wind conditions and the energy yields of wind turbines thank you very much
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