Welcome to this session on power, thrust and optimum rotors. We will look at 1D momentum theory for wind turbines and my name is Henrik Bredmose. A wind turbine is a device, which is made to extract energy from the wind. It does so by taking other kinetic energy. And therefore, the wind loses velocity as it passes through the turbine. After this lecture, you will be able to explain how the flow is braked by the turbine and explain also the theory behind the optimum power production of an idealized rotor. Rotors come in many shapes. This is an old one from the 80's. This one here is a modern one. This one here is a specialized one for experiments with floating wind turbines. The role of these rotors is to stop the wind not fully, but enough to take out some kinetic energy, which can then go in to the power. When we look at the 1D momentum theory to analyze how much power can you actually extract from the wind. Because if we think of it, then of course, we could try to block all of the wind, but that would also mean that no more wind would through to us and we would have zero energy. The other side is to extract no energy at all, then we wouldn't break it wind but no energy. So there is some optimum in between and that's the optimum we want to find. Let's look at the flow here through the rotor. I already said that we stop some of the velocity. That means that the flow occupies a bigger area as it passes through, because no wind is stocked or stored here on the way. So velocity starts with a V_0 velocity and then it kind of goes down to a terminal velocity. We know from the Bernoulli's principle that falling velocity leads to bigger pressures. So the pressure has to increase until it gets to the point of the rotor. Over the rotor, there's a drop in pressure here. And then say, the pressure recovers to atmospheric pressure again, and that also means that the velocity keeps falling until we are back at the atmospheric pressure. This is something that we can analyse with the Bernoulli equation. The Bernoulli equation states that you can follow a streamline and along that streamline the sum of pressure and one-half rho velocity squared, is constant. So if we go from the start, and then to the start of the rotor, we would have that this initial term here equals p plus one-half rho u squared at the front of the rotor. Now from the back of the rotor and to the end, we can apply the Bernoulli principle again. We would have the same pressure but now when the pressure drop included, the same velocity term that's here. And then in the far end, we would say, p0 plus one-half rho u1 squared. Now by subtracting these two equations, we can actually derive an expression for delta p. And that's the one written here, which kind of expresses, delta p in terms of the inflow velocity and the terminal velocity. On this next step, we'll be to look at the momentum conservation. And we have drawn our control volume here, which is the red box. And we will look at the balance between inflow of momentum, rho A_0 V_0 squared. Outflow of momentum, rho A1 u1 squared. And forces, the forces are delta p times the area. Now on this delta p, we have an expression for that from the previous slide. And we're also able to utilize mass conservation, because rho A_0 V_0 equals rho A_1 u_1, which also equals u times A times rho. And we can decide that out as a common factor. And that leads to this important result which says that the velocity through the rotor plane is actually the average of the inflow speed and the terminal speed u1. We could define this velocity in terms of an induction factor, the actual induction factor a. So we say that that velocity in the rotor plane is 1 minus a times V_0. In turn, we could define the thrust. So the thrust would be one-half rho A V_0 squared C_T, where C_T is this function of a. And finally, for the power, we know that the power is velocity times force. So that would be thrust times u, or that can be expressed in terms of a. We get one half rho A V_0 cubed C_p, where C_p is this function here of a. So now if we're after getting the most power out of the wind, wouldn't we like to find the optimum value of a? We can do that by differentiation. So dC_p da is written here. It has a serial value for a equals one-half and 1. 1 means no energy at all. It's a minimum, so not relevant. Our interesting optimum is the one-third, which is the one that gives us the maximum power. At that value C_p is 16 divided by 27. This is called the Betz limit. So it tells us, let say, the best performance we can have from an idea wind turbine is to extract 59% of the available energy in the wind. So in summary, in this session, we have learned that a rotor is a device that breaks the wind and thereby extracts kinetic energy. We have seen how the basic fluid mechanics laws can be used to describe the flow. Bernoulli's equation, mass conservation, momentum conservation, and we have found that an optimal rotor can extract 59% of the incoming kinetic energy.