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Wind Energy Calculations

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And power is dE by dt which is what we did and so once you do that dE by dt you will get rho A dl by t v square and that is rho A v cube right. So, if you plot power versus v the velocity you will get a function which is a cubic function, something like that right. So, it is strongly dependent on, p is proportional to v cube this is basically what we get. So, that is a very important thing to remember. This curve is very I mean important curve to keep in mind when you think of a windmill. So, that is the point, so in fact, a lot of work that goes on concerning windmills. At some point I mean fundamentally acknowledges this curve and then you know accounts for this in the design. So, that is the point that we have to keep in mind right. (Refer Slide Time: 18:53) So, the same parameter that we just saw I am plotting here. So, for example, if you just look go back here we can take for some values to try out. So, for example, rho for air is about 1.225 kilograms per meter cube. So, that is a value you can use. To calculate A, the area we need to make some assumption on the radius of the wind turbine blade right. So, some assumption we have to make, modern wind turbine blades that you see including the photograph that I am showing you, that I showed you and what you see dotted you know around in various locations in various places are and you know 50 60 70 meter kind of range in terms of radii. So, we can assume that is, we will assume that the length of the blade is 60 meters, so, and then we can have a velocity in of course, in some unit we can select and I have just plotted kilometres per hour because that is often the that is a value that we can more easily you know understand or visualize. So, what we are looking at is power equals half rho is 1.225 into the area will be pi which is 3.14, r square, 60 square, so pi r square into so velocity we will take some velocity in kilometres per hour. So, kilometres into 1000 will give it to us in meters by 3600, will give it to us in meters per second, this cubed. So, this is the; if you do this calculation for every velocity v in kilometres per hour you will get power in watts. So, this is in watts and this v here is kilometres per hour right. So, that is half the factor half that is out here we have put in a value for row 1.225 kilograms per meter cube, Awe have calculated as pi r square, where r we have assumed is 60 meters and v we have assumed is some velocity in kilometres per hour. So, to convert that into meters per second we have multiplied by 1000 and divided by 3600 and the whole thing is cubed. So, this is what is you know the equation that we need to use for power. So, for various values of v, we can now get various values of P right. So, that is the plot that I am I am showing you here the power W, I mean power in watts. So, this is the P that I just spoke about and wind speed in kilometres per hour which is the v that we just discussed. So, power versus wind speed. So, you can see again as I said you know this is proportional P is proportional to v cube and that is the function that you are seeing on your screen you see this plot which is you know steadily climbing up. So, for example, if you look at it at say roughly around 50, 50 kilometres per hour, the power that is available in the wind is about 20 megawatts. So, this is 10 power 7, 20 megawatts is what is available in the wind at 50 kilometres an hour right. So, this curve I have gone to a very wide range of velocities from 0 to about you know nearly 100 kilometres per hour is what we have drawn. But that is something we will discuss in just a moment. So, please remember this is not the power that has been captured by the windmill, this is not yet at that point. We are simply discussing what is the power that is available in the wind. So, this is what is available in the wind. So, the wind is blowing at 50 kilometres an hour then the power that is available in that wind is about 20 megawatts right. So, as faced by a windmill of 60, I mean 60-meter radius right. So, this is as a power, as a function of wind speed, so this is power as a function of wind speed in a cross-section corresponding to a circle of radius 60 meters. So, that is also something that we have to remember because 50 kilometres an hour if you have a huge amount of wind going at kilometres an hour, correspondingly power, is more right. So, this is you have to limit the size of it and that size is limited by this circle of radius 60 meters. (Refer Slide Time: 24:12) So, now we will, I need to discuss that curve a little bit more we will come back to that curve in just a moment. Before that, I just wanted to discuss a few things about performance characteristics. Some terms are used in the context of windmills. So, we will you know briefly look at those terms, and on the basis, we will again revisit that curve that we just saw. So, the first thing is there is this thing called a tip speed ratio, tip speed ratio. It is the ratio of the rotational speed of the blade to the wind speed and usually, it is a maximum of about 10 for blades that are considered lift type blades. So, we will see an in a little bit later on what we mean by lift type blades. So, there is a certain type of blade which is called a lift type blade and if that kind of a blade is used then you will get a top speed of, the tip speed ratio of 10 which is and we will see that in greater detail, but fundamentally it conveys the idea of how quickly the windmill is rotating. So, it is a rotational speed of the blade to the wind speed, for the same wind speed if the blade is rotating faster and faster and faster you have a higher tip speed right. And based on the design of the windmill some windmills will have the tip speed that is low and some will have it high, so in some cases, it is only 1, it is a factor of you know it is just the same as the wind speed some cases it is much higher. And there are some consequences of this based on what you are trying to do with the windmill so that we will see in a little bit. The second thing is there is something called a cut-in speed. This is the minimum speed that should be there in the wind before which the blades will turn. So, if the wind speed is too low, it does not have enough power there is not enough energy in the wind to start pushing those blades. So, the blades look, appear stationary. So, even though there is energy in the wind, even though there is power in the wind it is insufficient to get the blade to get past whatever frictional is aspects are there, whatever you know loads are there on the blade which prevent it from rotating because it is going to be attached to a gear that gear is going to be attached to a generator. So, there is a lot of you know in the back there is some load that is there mechanical load which it has to overcome before it starts rotating. So, you need a certain amount of wind speed before typical windmill starts rotating till then it will not respond to the wind it will look like there is no wind, but there is wind it is just not enough to get the windmill to move and that is usually somewhere between 10 to 16 kilometres an hour. So, based on various aspects associated with the windmill it will take you at least 10 to 16 kilometres per hour wind speed before the windmill begins to rotate. So, that is quality cut-in speed. Then there is something called the rated speed. The rated speed is the wind speed at which the windmill generates its rated power. So, rated power it will have when they say that you know this is windmill is a 1-megawatt windmill or you know half a megawatt windmill whatever it is some rating they give for that windmill right. So, you once they give you that rating for the twin mill. it usually it will require at least a certain level of wind speed before it starts delivering at that rated capacity right. So, sometimes the rated capacity may be a nice number, but you will not be getting that rated capacity most of the time because wind speeds may be less than whatever is that optimum speed that is required for the treated capacity. So, you do not always get that energy from the windmill, but there is that rating is there so you have to be aware that there is such a thing as the rated speed, and only when you reach that speed you will start getting that power. Generally what happens is the profile of the power generated by the windmill is such that till you reach that speed once you cross the cutting speed as the wind speed keeps increasing the power generated by the windmill will keep climbing up. Then it will reach this rated speed, once it reaches the rated speed at which it's delivering its rated power which is usually the best power, the best condition for it, it is sort of level of it will even though you may have a range of wind velocity is higher than that for various other aspects associated with the design of that windmill it will sort of level off at that value and it will only deliver that amount of power. And then and, therefore, it is sort of levels off, pass the point. And this is usually around 40 kilometres per hour speed is where you will get this levelling off and after that, it stays level, it does not do much. Then there is a cutout speed and this is usually at wind speeds above for 70 kilometres per hour. So, this is mostly a safety issue. So, the wind windmill or the wind turbine is made of certain types of materials, they can handle certain types of you know a certain level of forces on them, torque on them or all these things there will be some rated capacity to for these turbines for those long blades that are present there etcetera. And if you if the wind speed is too high, this let us say there is a storm and you can easily have you know storms which come which have 100 kilometres per hour wind speeds right. So, those are significantly higher than this 70 that I am talking about. So, 40, at 40 it is already delivering its rated power. So, they will allow it leave way up to about 70 kilometres an hour beyond 70 and there will be days in the year when there is a storm or cyclone or whatever you want to call it where the wind speed will be higher than this and at that point, it is not safe for the windmill to operate. So, they usually have a mechanism by which they stop the windmill, the windmill is stopped to prevent damage. That is you know the very practical issue, some of the things that we are talking of here are practical issues. A speed cut is a practical issue. In a hypothetical perfect you know you know friction-free situation this should just start running even the slowest wind it should be running, and it should get you you know once you even will go past the rated speed it should continue to give you higher and higher power. And even cut out speed in principle you should never have a cut or speed higher the wind you should be you know very happy and generating you know at the cube of the velocity is what we are looking at. So, you should be able to generate a significantly high amount of power. So, these restrictions that I am showing you here are all practical restrictions. So, there is a cutout speed; and this cutout speed is about 70 kilometres per hour after which the windmill is stopped. So, for this there are various mechanisms inside there is usually sometimes they have a break in the system which will physically you know act like a brake which will stop the rotation of the windmill. They also have a way in which they can rotate the windmill so that you know normally the working very effective when it is facing the wind. So, they can turn the windmill away from the wind and make it now face perpendicular to the wind, so it is seeing less of you know action from the wind so that also enables you to slow it down they can add spoilers to it to slow down the process and various other things can be done. So, there is a standard set of mechanisms they have by which they will shut down the windmill if the wind conditions are very severe. And in many cases they can even automate it so that it automatically senses that the wind speed has crossed some limit it will get into the shutdown mode and then when once the wind speed drops below this you know condition this high-speed condition, once it drops below that threshold value of 70 kilometres an hour and stays stable there for some time if the wind will again start operating. So, these are some practical aspects that we have to keep in mind. So, just to keep in mind, these are 3 numbers here about 10 kilometres an hour about 40 kilometres an hour for rated speed and about 70 kilometres per hour for the cutout speed right. So, the same plot that we just saw here where I had power versus wind speed where we went up two 100 kilometres an hour. I am going to show you another plot which is the same thing in you know the magnified way where we are sort of magnifying the lower end of this graph here so that we can know to look at this cut-in speed, rated speed and cut out speed and so that is basically what we have here. (Refer Slide Time: 32:03) So, this is what we will look at here. We will come to that theoretical limit in a moment. So, you can see here the same thing this is power as a function of wind speed for the cross-sectional area, for a cross-sectional area corresponding to the radius of 60 meters. So, now, if you see those values I have wind speed here I have power here and power you can see here since we are taken a smaller section of that diagram that we previously saw, we have you know some values in megawatts, 2 megawatts here, 4 megawatts, 6 megawatts, 10 megawatts and then it continues up to about 20 megawatts. So, that is basically what we have here. So, at 10 kilometres per hour which this is your cut-in speed, so up to this the windmill does not even move you just sort of stays stationary. So, that is our cutting speed and then about 40 kilometres an hour it is where it reaches it is at a rated capacity right and then finally, at about 70 kilometres an hour we have the cutout speed. So, in practice what we see if I just mark this up here say that is the value there. So, what we will see is this is the actual curve that we will be using as the windmill operates. So, what you see here is that till 10 nothing is happening to the windmill, till 10 kilometres per hour nothing is happening in the windmill no power is being generated it is just stationary. Once you cross about 10 kilometres an hour the power it starts generating power than that power, this is the power available in the wind, this is not yet the power that the windmill is generating. But this is the kind of power that is there available in the windmill. There is some factor that is that needs to be put into this which we will just see in a moment. This is the power available in the wind and then so, it will reach some rated power and at that point, at 40 kilometres an hour this is the power available in the wind, and that is the power that the windmill is experiencing and then at 70 kilometres an hour, we shut it down. So, that is what is happening here, shut it down and so, even if the speed is higher than that the windmill stays shut down. So, this is the power in the wind corresponding to those speeds that we just spoke about. It is not yet the power that the windmill is capturing. (Refer Slide Time: 35:21) So, there is some theoretical limit. So, this has come under a law that is referred to as the Betz law, it was proposed in 1920 that is what you see here. But historically if you go and see the records at that point there were actually 2 3 people who came up with this law, at the roughly the same time there is one person I think who is credited to have even indicated this around 1915 and then there were 2 people who indicated this in 1920 and somehow it got associated with Betz and so, it is called the Betz law. And where analysis has been done on the idea that there is a wind that is blowing it has some energy what is the maximum energy from that energy that is available in the wind that we can capture. So, if energy, it comes with an energy e, if it has an energy e what fraction of that e can be captured; can we capture 100 per cent of it or can we can only capture 50 per cent of it; can we only capture 30 per cent of it, what is the theoretical limit of this. Is there a theoretical limit and if so what is the theoretical limit. So, a very nice analysis was done. So, just to understand why there are limits to this process we can just consider the idea that you know this wind is coming with some energy and for the windmill to capture the energy the wind has to interact with the windmill right. So, you have a windmill. So, you have wind incident on this windmill. So, now, we are trying to capture this energy. So, what are the options? So, this is the wind is arriving at some energy. Let us assume that the windmill somehow you know completely captures the energy of that of the wind that is coming towards it. So, let us say some we just assume hypothetically we will consider this situation that wind arrives with some energy e which is basically what, its kinetic energy. Its kinetic energy, it is arriving it's arriving with the kinetic energy half mv square and it arrives with that energy arrives incident on the windmill. And let us say we have managed to capture 100 per cent of it. What is the implication of it? The implication of it is that the wind immediately after the windmill. So, I will just say it has come with a velocity v1. So, half mv1 square right, so that is the energy it has come with. So, if you say v2, is the velocity of the wind immediately after the windmill if all the energy of the wind has been captured then v2 equals 0 right. So, if all the energy that is available in the wind has been captured successfully then the wind comes to a complete halt because all there it has lost all its energy, everything has been captured by the windmill and therefore, v2 is equal to 0. Now, in principle that is possible, but the issue is that if you have brought the wind to a complete halt then there is no further wind flowing through the windmill right. So, it is like hitting out, it is like hitting a wall, you brought the wind and the wind of the first a first little bit of wind that crossed it reached 0 velocities. So, that wind is parked there it is not going anywhere and it is just building up there. Any other wind that is going to come cannot go further past the windmill because this wind is just parked there, it is not moving anywhere right. So, that is the implication of saying that you have captured all the energy present in the wind. You have stopped the wind. So, the moment you stop the wind there is no further energy to capture. So, if you ever manage to capture all the energy in the wind you can do so, only momentarily. The moment you do it all the energy you will no longer have a wind that is flowing because you stopped the wind and therefore, there is no further interaction of the wind with the windmill and therefore, there is no further energy that is captured, right. So, when you go to this condition that you have captured the entire energy the windmill stops. The wind is fully stopped by the windmill stops because the wind stops. Another option is you have wind arriving with same thing half mv1 square and let us say it is completely unaffected by the windmill right. So, completely unaffected by the windmill which means it comes out of the side out of the windmill also with velocity v1, v2 is equal to v1. The velocity with which it comes past the windmill is the same as a velocity with which it arrived with the windmill. What is the implication of that? The energy that it had before it reached the windmill is completely preserved as it comes past the windmill; that means, no energy was delivered to the windmill. So, both when the wind is unaffected by the windmill. So, no energy transferred to the windmill. So, both when the velocity is completely stopped and when there is you know where the velocity is completely protected in both these instances there is no energy transfer to the windmill. So, these then, therefore, set the limits. So, therefore, if you think that you know you can capture 100 per cent of the energy you cannot, we have just seen that you cannot capture 100 per cent of them. So, we also see that if we capture nothing, of course, you have captured nothing. So, there is 0 on this site and there is 0 on that side, based on what is the velocity of the wind that is there, what is the velocity that is possessed by the wind that has just cross the windmill right. So, as you can see there must be some optimum value of velocity in the middle where a fair bit of energy has transfer has been transferred to the windmill and still the wind is moving and therefore, more breeze can keep coming here right. So, that is the idea. So, it turns out that if you do the calculation and there is some, I mean you write all the equations associated with this flow of wind and put in the parameters are appropriate to capture all these ideas together when you do that you will arrive at a factor of 16 by 27 or 0.59 almost 60 per cent, that is the best that you can capture from the wind. So, we had the half rho A v cube right as the power that is available in the wind if you go back here, half rho A v cube that is the power that is available in the wind. So, if you take that half rho A v cube that is the power that is available named wind. We can capture only 0.59 into this thing theoretical limit. This is a theoretical limit of how much energy or how much power in this case that you can capture from the power that is available in the wind that is coming to the wind turbine. And in reality actually what you capture as you can expect is less than this theoretical limit, so often we are capturing only 10 to 30 per cent of the energy that is available at the wind. So, only 10 to 30 per cent of the energy that is available in the wind is being captured which is less than the theoretical limit which is what you will expect. Some nice designs will enable you to capture you know, this itself at the at 30 per cent, this is 50 per cent of the what is possible in the theoretical limit, 50 per cent of what is the theoretical limit, you can get to 60 or 70 per cent of what is possible of the theoretical limit. So, instead of 0.6, you will have 0.6 to 0.7. So, correspondingly you will have you know 40 per cent of the energy that is available in the wind. So, something like this can be done, this is the limit and so, this is what we have to keep in mind. So, if we go back here, this is the power that is available in the wind that has arrived. So, as we have seen the theoretical limit is about 60 per cent of this. So, it is distinctly less than this and what is being captured is only about 10 per cent or so, of this. So, only about 10 per cent of the graph that you see here is being captured at any given point in time. So, if you take, for example, at 40 kilometres an hour this is almost about 10 megawatts. So, actually what you are going to capture is only about around 1 megawatt right. So, in reality, this is what you will capture. So, if you plot those points correspondingly the actual curve that you will see is this one right. So, we will get about 1 megawatt from this plant, after taking into account the theoretical limit that is there and all the practical inefficiencies that are there. So, the curve on top is what is there in the wind, what is available in the wind, and what you see below is the power captured by the windmill is this value that the lower curve that you see. So, this is what I would say as you know performance characteristic of that windmill which sort of nicely captures the main aspects of how you know power available in the wind varies as a function of velocity and the fact that there are some limits on cut in speed-rated speed and cut out speed and within that framework is what the windmill operates. And so, this plot here shows you what is the framework within which the windmill is operating. So, I will just add a couple more comments here as you know sort of wind up this discussion on the windmill and what energy is associated with it. (Refer Slide Time: 45:51) We spoke about this you know the tip speed ratio here, tip speed ratio here and it is a ratio of the rotational speed of the blade to the wind speed and I told you that it has some significance. So in fact, what you will see is that if you look at windmill designs in olden days there was a certain type of design which was referred to as the drag type design and usually that would have many more blades. So, it would have many more blades and the phenomenon that was occurring there was the breeze or the wind would physically push the blade out of the way. So, that is how the original windmills operated, the wind would flow in one direction and it pushed the blade away, physically push the blade away. Generally, those kinds of windmills have much lower rotational speed, but they have much greater torque and it was, therefore, very convenient for the use that it was being put to then which was primarily is you know, therefor you know grinding grains or you know pumping water, where torque was a major requirement. The speed was not such a major requirement, but the torque was required because it had to push against the grains or lift water from deep well etcetera and so that you now torque was required to enable that to happen. So, it is better suited for mechanical work of that nature where you are you know grinding something, putting all water from deep well etcetera. The modern type of windmills is of the lift type. So, they use something that is referred to as the aerofoil design which is similar to the design of the wings of aircraft etcetera and there it is got to do, how the blade moves have got to do with the velocity of the wind velocity and pressure of the wind on one side of the blade relative to the velocity and pressure on the other side of the blade. So, that is how that is designed and it creates what is called a lift. And when it does that you get much higher rotational speeds. And as a result of the power generation part when you want to do power generation the rate of rotation of that turbine is a very critical component which decides how effective your power generation process is and therefore, the tip speed has to be high, it cannot be slow, it has to rotate rapidly, if it keeps rotating rapidly you are in a better position to generate power from it. So, the modern wind windmills are sort of optimized towards the power generation aspect of it. The old windmills by just by chance they were designed in such a way based on whatever knowledge was available then that they generated a lot of torque which was very useful for the activity that they were being used for which was primarily for grinding or pumping water. So, there is some difference there in the design and so that is worth you know keeping in mind or you know even examining in greater detail, but that is something that we should be aware of. (Refer Slide Time: 49:00) So, to conclude we have we have noted through this class that there is power available in the wind and it is proportional to the third power of wind velocity. So, it is not linearly proportional to the velocity of the wind, so if you double the wind velocity, the power that is available in the wind is going up by a factor of 8, right. So, what is the velocity goes up by a factor of 2, the power and the wind go up by a factor of 8? So, that is a huge difference, a factor of you know I mean cube, v cube is, the power is proportional to v cube. So, it is a very important result to keep in mind concerning wind energy capture and usage. We also noted that there are practical aspects that limit the range of wind velocities that can be effectively tapped. So, just because we have P is proportional to v cube, at least we cannot just assume that you know it would be great to you know capture the energy that is available in a storm, but there are practical aspects that prevent us from capturing the energy that is available in a storm because that is very strong winds and it may not even be steady winds. So, you will have a gust of winds, sudden stop, sudden gust of wind etcetera. So, it is not a very convenient way to capture the power and that may be based on that limits, the material limits of that windmill structure because it should not you know break up. So, therefore, some practical aspects limit the range of wind velocities that can effectively be tapped for you know energy generation. And also we noted that there is a theoretical limit where we took into account the possibilities that neither the wind does not interact at all with the windmill and the other extreme where it interacts to the point with the wind has come to a complete halt. So, we saw that on both I mean both extremes you know power being captured by the windmill is 0, and therefore in between, you have a range of velocities where naturally you expect that the power will keep increasing, it will reach some kind of an optimal value and then it will start decreasing some high value and then start decreasing, a maximum will be reached and then it will start decreasing. Some optimal value of the velocity at which the maximum will be reached at which; this is the velocity that at the exit point after the windmill. So, it turns out that the best that we will get is about 60 per cent, 59 points some per cent and that is a limit, the theoretical limit to what can be accomplished. Practically, we get less than that, maybe half of that maybe two-thirds of that and so on. So, there is a theoretical limit and where which we need to be aware of, which limits the extent to which energy can be energy that is available in the wind can be captured.