- Shalmali

# Wind Turbine - momentum theory

**Betz is the limit!!!**

*Table of Contents:*

### General working of a wind turbine

In general, this is how a wind turbine works. The wind hits the blades, the blades rotate with the rotor. The rotor shaft rotates that is then connected to the gearbox in the nacelle. The gearbox increases the speed of the shaft. Finally, this shaft goes into the generator, where the mechanical energy is converted into electrical energy.

There are a few basic theories that make wind turbines understand easier. The Momentum theory is one of the most important theories to study to learn about a wind turbine as a mechanical engineer. This article will focus on the same. This article is an extension of the previous article that we strongly recommend reading__ here __before proceeding.

### Momentum theory

Momentum theory is also called actuator disk theory. In this theory, the rotor is considered a solid, infinitely thin disc. The disc is assumed to be frictionless, without any rotational wake component, i.e. an ideal rotor. It is assumed that there is constant velocity induced along the axis of rotation on the disc. The disk creates a flow around the rotor, and the streamlines move away as shown in the figure.

### Conservation of Momentum

The wind flows in the direction of the rotor. As the wind approaches the disc, it sees an obstacle in front of it, which means the area for the flow is reduced. Hence by __continuity__ equation, the velocity of the wind decreases. This wind speed lost is gained by the rotor, and wind energy is converted to mechanical energy. By __Bernoulli's principle__ as the velocity decreases the pressure increases.

But later suddenly there is no more rotor disk present as it is assumed to be thin. This means the area increases or is equal to before, hence pressure reduces drastically and then becomes constant as before the presence of the disk.

The thrust force (T) acting on the rotor is

where,

**∆p** is the pressure jump on the rotor.

**A** is the area of the disk facing the wind. (**A =** **π R^2**)

Applying Bernoulli's Principle at two places to get two equations as below, before and after the rotor. To find the pressure jump, (assuming that density is constant all over). Later the thrust can also be found as follows,

where p is the pressure on the disk.

### Conservation of mass

Applying conservation of mass to this system,

where,

The thrust has a negative sign because it is in the opposite direction. Now, we would want to keep the thrust force as small as possible so that the force acting on the turbine is as low as possible. In order to do that we might want to,

Keep the density as low as possible

Rotor disk as small as possible

The incoming wind speed is to be as low as possible

The velocity of the rotor is lower OR the velocity of the wind after crossing the rotor disk is the same as the wind speed.

### Power extracted

The power extracted by a wind turbine can be defined by the equation,

Now maximum power will be generated when

The density of the air is higher

The area of the rotor disk is higher

The velocity of the incoming wind is higher

The velocity of the rotor is higher OR the velocity of the wind after crossing the rotor disk is zero.

It is very important to know our aim here because if you compare the requirements to keep the thrust low and the lower high, they are exactly the opposite. We have to keep in mind that the first focus here is to increase the power produced. The thrust force can be managed by engineering. But also we cannot extract all the available wind. If the velocity of the wind after the disk is zero, then there will be no flow of the wind. So there has to be a middle ground/limit for a good balance between the Thrust and Power, which is taken care of by the axial induction factor.

### Axial induction factor

The axial induction factor, a is defined as,

The axial induction factor tells us by how much factor the incoming wind is not used. If we combine this equation with the one we saw previously we get,

The power and thrust equations later become,

The maximum power that can be extracted by a wind turbine from the wind is given by, considering all the wind is extracted,

### Power coefficient

It is a non-dimensionalized factor that is a ratio of the available power to the power actually extracted.

Similarly, the thrust coefficient can be written as,

### Betz Limit

Maximum power will be extracted when the value of the power coefficient is higher. The maximum possible limit to extract power can be found when the power coefficient is differentiated with respect to the axial induction factor.

In this way, for a = 1/3, the value for the power coefficient is 16/27. This is the theoretical maximum for an ideal wind turbine which is known as the ** BETZ limit**.

The Betz limit was found by a German physicist, Albert Betz. He concluded that only **59.3%** of the wind energy can be used to spin the rotor ideally. In reality, modern wind turbines cannot reach the Betz limit and have an efficiency of around 35-45%.

### Summary

Applying the momentum theory to the rotor we can find the power that can be extracted from the wind.

All the wind energy cannot be converted into the spinning of the rotor, but there is a limit called Betz Limit.

The wind that could not be extracted by the rotor can be defined by a factor called an axial induction factor.

*The next topic in continuation is the Blade element momentum theory. Let me know how you liked this article, and if you want to know something more related to this topic. Use the comments section to the fullest. Have a great day!*