[MUSIC] The Betz law, also called the Betz limit, was derived almost one century ago by the German physicist Albert Betz. This theoretically meet, indicates the maximum power that can be extracted from the wind or any open flow independent of the design of the turbine. We will derive this flow using the Bernoulli equation and two fundamental principles the mass and the momentum conservation. Let consider a turbine and a steady and uniform wind. If we assume this turbine extract fraction of the extreme kinetic energy. We expect that the downstream wind will be, just after the turbine, smaller than the upstream wind. However, due to the mass conservation, the cross section area of the downstream winds should be larger than the upstream. Hence, we should consider a control surface who bounds the volume of the fluid passing through the turbine. This surface corresponds to a given set of streamlines, and there is no flow crossing the lateral surface. There is a net mass flux only through the upstream and the downstream disc. Outside this control surface, the freed parcels may be deflected, but without energy losses. While inside, the parcels interact with the turbine and transfer both momentum and kinetic energy to the turbine. If now we try to estimate the variation of the kinetic power between the upstream and the downstream surface. This kinetic power is equal to the flow height Q, times the kinetic energy density, one half times rho times V squared. Due to the energy extracted by the turbine, the downstream kinetic power is smaller than the upstream one but it cannot vanish due to some mask observation. Hence, the turbine cannot takes track when 100% of abstain kinetic energy. Then the question is, what is maximum kinetic power that can extracted from the flow by the turbine? We should now consider four different cross-sections of this controlled volume. The upstream cross-section is A0 with the unparted velocity and pressure V0 and P0. The area A just before the turbine with the pressure P1, the area A just after the turbine with the pressure P2. A is equal here to the turbine area and due to the mass conservation, the velocity V1 and V2 are equal and noted here, V. Then we consider finally the downstream cross section, where the flow recover the unturbulent pressure p 0. As discussed before this downstream velocity v 3 is smaller than the upstream v 0. While the area A3 is larger than the upstream area A0. We cannot apply the Bernoulli formula along the streamline crossing the turbine area A because a significant energy loss occurs. Hence we will apply the Bernoulli formula before the turbine, from the pressure P0 to P1. Then we will consider a second streamline after the turbine, from the pressure P2 to P0. To estimate the pressure drop induced by the turbine, we take into account the two binary formulas obtained before and after the turbine. We subtract them and found that the pressure drop, P1- P2 is equal to, The drop of kinetic energy density from the upstream to the downstream area. This pressure drop induced by the turbine is directly proportional to the actual thrust t exerted by the flow on the turbine. If now we consider the momentum conservation this axial thrust should be equal to the devist in the momentum flux from the upstream to the downstream area. This momentum flux deficit is obtained by the integrals of the momentum density crossing the upstream and the downstream areas. This calculation leads to rho times the flow rate times V0 minus V3. And the momentum conservation gives then, and with some little algebra, we get a very interesting formula. The flow velocity at the water maybe, taken as the average of the up stream, and the down stream velocities, Now, we will use this relation, to compute the kinetic power, extracted by the turbine, from the up stream flow. The difference between the up stream and the down stream kinetic energy flux, gives here the maximum amount of power, available for the turbine. To calculate the turbine efficiency, Betz suggests to divide the power extracted from the flow, by a reference power. Which is the kinetic power of the unperturbed upstream flow contained in a cylinder, having the same cross section on area as the turbine. The power coefficient cp corresponds to the ratio of the power generated by turbine over this reference power. This dimensionless ratio depends only on the upstream and the downstream velocities V0 and V3 respectively. We can introduce the variable A, the ratio of the down stream to the upstream velocity and we then write the power coefficient as a simple function of A. This function is equal to 0 when A is equal to unity. In other words, there is no energy extraction if the downstream speed is not reduced. According to the plot, the power coefficient reaches a maximum value around 0.6 when A is equal to one third. This theoretical value corresponds to the standard Betz limit. Within an open flow, no turbine can capture more than 16/27 of the kinetic energy of the upstream flow. The viscous of turbulent dissipation is neglected. The flow is non compressible. The thrust exerted by the flow on the turbine is assumed to be uniform on the whole turbine area. And the turbine rotation is not taken into an account. And there is no swimming motion in the turbine wake. Any of this non ideal effects will reduce the energy available in the incoming wind, lowering the overall efficiency. 20 year after the Betz calculation Glauert derived another explanation for the power coefficient taking into account the angular component of the free velocity in the turbine wake. He introduced one of the most important parameter for open flow turbine, the Tip Speed Ratio. It is the ratio between the tangential speed of the tip of a blade and the actual velocity of the upstream flow. Due to the Glauert model, the efficiency of a rotating turbine is below the base limit and depends on the tip speed ratio according to the following curve. The power coefficient is equal to 0. If there is no rotation of the blade and it approaches asymptotically the Betz limit when the tip speed ratio goes to infinity. For real turbines the power coefficient will reach the maximum value for a finite speed ratio. If the power coefficient of ancient windmills could hardly exceed 0.3, a modern three blade turbine could reach an optimal power coefficient around 0.5. This value is indeed quite close to the theoretical Betz limit of 0.6. Such maximal values will be reached for a tip speed ratio of around six to seven and this also means that as the speed changes the rotor speed must change to optimize the power extraction from the flow. Thank you.
