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- Wind energy
- Wind-Turbine and Wind-Farm Flows: A Review
- Blade element momentum theory
- Natural snowfall reveals large-scale flow structures in the wake of a 2.5-MW wind turbine

## Wind energy

Blade element momentum theory is a theory that combines both blade element theory and momentum theory. It is used to calculate the local forces on a propeller or wind-turbine blade.

Blade element theory is combined with momentum theory to alleviate some of the difficulties in calculating the induced velocities at the rotor. This article emphasizes application of BEM to ground-based wind turbines, but the principles apply as well to propellers.

Whereas the streamtube area is reduced by a propeller, it is expanded by a wind turbine. For either application, a highly simplified but useful approximation is the Rankine—Froude "momentum" or "actuator disk" model , This article explains the application of the "Betz limit" to the efficiency of a ground-based wind turbine. A development came in the form of Froude's blade element momentum theory , later refined by Glauert In blade element momentum theory, angular momentum is included in the model, meaning that the wake the air after interaction with the rotor has angular momentum.

That is, the air begins to rotate about the z-axis immediately upon interaction with the rotor see diagram below. Angular momentum must be taken into account since the rotor, which is the device that extracts the energy from the wind, is rotating as a result of the interaction with the wind. The "Betz limit," not yet taking advantage of Betz' contribution to account for rotational flow with emphasis on propellers, applies the Rankine—Froude " actuator disk " theory to obtain the maximum efficiency of a stationary wind turbine.

The following analysis is restricted to axial motion of the air:. In our streamtube we have fluid flowing from left to right, and an actuator disk that represents the rotor. We will assume that the rotor is infinitesimally thin. The fluid interacts with the rotor, thus transferring energy from the fluid to the rotor.

The fluid then continues to flow downstream. Before interaction with the rotor, the total energy in the fluid is constant. Furthermore, after interacting with the rotor, the total energy in the fluid is constant. Bernoulli's equation describes the different forms of energy that are present in fluid flow where the net energy is constant i.

The energy consists of static pressure, gravitational potential energy, and kinetic energy. Mathematically, we have the following expression:. For the purposes of this analysis, we will assume that gravitational potential energy is unchanging during fluid flow from left to right such that we have the following:. Now let us return to our initial diagram. Consider pre-actuator flow. Thus, if mass flow rate is constant, increases in area must result in decreases in fluid velocity along a streamline.

This means the kinetic energy of the fluid is decreasing. If the flow is expanding but not transferring energy, then Bernoulli applies. Thus the reduction in kinetic energy is countered by an increase in static pressure energy. This can be described mathematically using Bernoulli's equation:.

We are treating the rotor as an actuator disk that is infinitely thin. Thus we will assume no change in fluid velocity across the actuator disk. Since energy has been extracted from the fluid, the pressure must have decreased.

Assuming no further energy transfer, we can apply Bernoulli for downstream:. If the rotor is the only thing absorbing energy from the fluid, the rate of change in axial momentum of the fluid is the force that is acting on the rotor. The rate of change of axial momentum can be expressed as the difference between the initial and final axial velocities of the fluid, multiplied by the mass flow rate:.

This force is acting at the rotor. The power taken from the fluid is the force acting on the fluid multiplied by the velocity of the fluid at the point of power extraction:. Suppose we are interested in finding the maximum power that can be extracted from the fluid. The power in the fluid is given by the following expression:. The power extracted from the fluid by a rotor in the scenario described above is some fraction of this power expression. We can see that the power extracted is dependent on the axial induction factor.

If we have maximised our power extraction, we can set the above to zero. In other words, the rotor cannot extract more than 59 per cent of the power in the fluid. Compared to the Rankine—Froude model, Blade element momentum theory accounts for the angular momentum of the rotor. Consider the left hand side of the figure below. We have a streamtube, in which there is the fluid and the rotor.

We will assume that there is no interaction between the contents of the streamtube and everything outside of it. That is, we are dealing with an isolated system. In physics, isolated systems must obey conservation laws. An example of such is the conservation of angular momentum. Thus, the angular momentum within the streamtube must be conserved.

Consequently, if the rotor acquires angular momentum through its interaction with the fluid, something else must acquire equal and opposite angular momentum. As already mentioned, the system consists of just the fluid and the rotor, the fluid must acquire angular momentum in the wake.

Consider the following setup. We will break the rotor area up into annular rings of infinitesimally small thickness. We are doing this so that we can assume that axial induction factors and tangential induction factors are constant throughout the annular ring. An assumption of this approach is that annular rings are independent of one another i. The velocity is the velocity of the fluid along a streamline. The streamline may not necessarily run parallel to a particular co-ordinate axis, such as the z-axis.

Thus the velocity may consist of components in the axes that make up the co-ordinate system. NOTE: We will in fact, be working in cylindrical co-ordinates for all aspects e.

Now consider the setup shown above. As before, we can break the setup into two components: upstream and downstream. Written in cylindrical polar co-ordinates, we have the following expression:. This is exactly the same as the upstream equation from the Betz model.

As can be seen from the figure above, the flow expands as it approaches the rotor, a consequence of the increase in static pressure and the conservation of mass. However, for the purpose of this analysis, that effect will be neglected. The radial component of the velocity will be zero; this must be true if we are to use the annular ring approach; to assume otherwise would suggest interference between annular rings at some point downstream.

Angular momentum must be conserved in an isolated system. Thus the rotation of the wake must not die away. Thus Bernoulli simplifies in the downstream section:. In other words, the Bernoulli equations up and downstream of the rotor are the same as the Bernoulli expressions in the Betz model.

Therefore, we can use results such as power extraction and wake speed that were derived in the Betz model i. This allows us to calculate maximum power extraction for a system that includes a rotating wake.

This can be shown to give the same value as that of the Betz model i. This method involves recognising that the torque generated in the rotor is given by the following expression:.

Consider fluid flow around an airfoil. The flow of the fluid around the airfoil gives rise to lift and drag forces. By definition, lift is the force that acts on the airfoil normal to the apparent fluid flow speed seen by the airfoil.

Drag is the forces that acts tangential to the apparent fluid flow speed seen by the airfoil. What do we mean by an apparent speed? Consider the diagram below:. That is, the apparent fluid velocity is given as below:. Consider the annular ring, which is partially occupied by blade elements. Remember that these forces calculated are normal and tangential to the apparent speed.

Thus we need to consider the diagram below:. Recall that for an isolated system the net angular momentum of the system is conserved. If the rotor acquired angular momentum, so must the fluid in the wake. Thus the torque in the air is given by. By the conservation of angular momentum, this balances the torque in the blades of the rotor; thus,. From momentum theory, the rate of change of linear momentum in the air is as follows:.

Let us make reference to the following equation which can be seen from analysis of the above figure:. Thus, with these three equations, it is possible to get the following result through some algebraic manipulation: [2]. This allows us to understand what is going on with the rotor and the fluid. Equations of this sort are then solved by iterative techniques.

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## Wind-Turbine and Wind-Farm Flows: A Review

Wind energy, together with other renewable energy sources, are expected to grow substantially in the coming decades and play a key role in mitigating climate change and achieving energy sustainability. One of the main challenges in optimizing the design, operation, control, and grid integration of wind farms is the prediction of their performance, owing to the complex multiscale two-way interactions between wind farms and the turbulent atmospheric boundary layer ABL. From a fluid mechanical perspective, these interactions are complicated by the high Reynolds number of the ABL flow, its inherent unsteadiness due to the diurnal cycle and synoptic-forcing variability, the ubiquitous nature of thermal effects, and the heterogeneity of the terrain. Particularly important is the effect of ABL turbulence on wind-turbine wake flows and their superposition, as they are responsible for considerable turbine power losses and fatigue loads in wind farms. These flow interactions affect, in turn, the structure of the ABL and the turbulent fluxes of momentum and scalars. This review summarizes recent experimental, computational, and theoretical research efforts that have contributed to improving our understanding and ability to predict the interactions of ABL flow with wind turbines and wind farms.

Request PDF | Wind Turbine and Propeller Aerodynamics—Analysis and This technique requires the flow field to be discretized into several zones, and the.

## Blade element momentum theory

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Blade Element Propeller Theory. A relatively simple method of predicting the performance of a propeller as well as fans or windmills is the use of Blade Element Theory. In this method the propeller is divided into a number of independent sections along the length. At each section a force balance is applied involving 2D section lift and drag with the thrust and torque produced by the section. At the same time a balance of axial and angular momentum is applied.

### Natural snowfall reveals large-scale flow structures in the wake of a 2.5-MW wind turbine

Blade element momentum theory is a theory that combines both blade element theory and momentum theory. It is used to calculate the local forces on a propeller or wind-turbine blade. Blade element theory is combined with momentum theory to alleviate some of the difficulties in calculating the induced velocities at the rotor. This article emphasizes application of BEM to ground-based wind turbines, but the principles apply as well to propellers. Whereas the streamtube area is reduced by a propeller, it is expanded by a wind turbine.

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