# Betz Limit

## The Betz Limit for Wind Turbine Power

Among the various renewable energy sources, the wind provides an environmentally friendly but intermittent source of renewable energy. Wind turbine generators (WTG’s) of different sizes and designs are successfully used to convert the kinetic energy of the wind into both mechanical and electrical energy. We may think that the bigger and more efficient these wind turbine generators are, the more energy they will produce. But there is a maximum limit to the efficiency of any wind turbine generator in converting the energy of the wind into usable energy and this is known as the *Betz Limit*.

The **Betz Limit**, or *Betz Law*, calculated by German physicist *Albert Betz* nearly a century ago, states that no wind turbine generator can convert more than about 60% of the kinetic energy of the wind into mechanical (or electrical) energy simply be turning the blades of a rotor.

Since wind moves irratically around from place to place carrying with it kinetic energy, wind direction and wind velocity are therefore subject to both rapid and frequent changes. So any wind turbine must be designed to respond to both conditions as it rotates about a central axis.

Generally, a wind turbine generator (WTG) generates power (P) proportional to the volumetric flow of the wind through it. This results in a pressure difference bewteen the wind entering the rotor blades swept area and leaving it as energy is extracted as it does so.

Thus the overall efficiency of a wind turbine is measured by how effectively the turbine can convert the energy within the wind into mechanical or electrical power.

## A Turbines Coefficient of Power

The *Coefficient of Power* (C_{P}) allows us to calculate the total amount of power a wind turbine is producing from the total energy available in the wind at a particular wind speed. For example:

### Coefficient of Power, C_{P}

C_{P} = | Energy produced by the wind turbine |

Total Energy available in the wind |

Clearly there are basic physical processes that limit a wind turbines maximum rotor power with the efficiency of a wind turbine at a particular site or location. Thus the energy extracted will depend on various factors, such as:

- The velocity of the oncoming wind, which is constantly changing.
- The rotational speed of the wind turbine blades.
- The type and design of wind turbine.
- The length and swept diameter of the blades.

So the conversion power of a wind turbine is related to design and is defined as the ratio of the net power output to the wind energy input. But the actual power available to rotate the turbine blades is only a fraction of the total theoretical power available in the wind as friction, drag and the swirling action of the air around the wind turbine rotor blades all contribute in reducing efficiency. Therefore, not all the wind’s energy is converted into usable mechanical (or electrical) energy due to these additional losses.

We know that wind turbines convert the winds energy into mechanical or electrical energy as a direct result of slowing down the oncoming wind as it transfers part of its linear kinetic energy to the rotor blades.

But a turbines rotor blades captures only part of the available oncoming wind energy passing through the swept area of the rotor blades. For a wind turbine to be 100% efficient, it would have to stop 100% of the available wind which is clearly not possible, as the rotor blades would have to rotate as one solid disk.

Clearly, if the turbines rotor was constructed as one solid disk design it could not pass any wind energy at all, so it would not be rotated by the wind and therefore no kinetic energy would be extracted or converted.

If we go to the other extreme where a wind turbine had just one single thin rotor blade. Possibly 90% or more of the oncoming wind energy would pass directly through the rotational (swept) area of the single turbine blade missing it completely. Therefore, very little of the winds linear kinetic energy would be converted using a turbine generator with just one single rotor blade.

So there has to be some middle ground in which the number of rotor blades, the volume of air passing through the swept area, as well as the limitations and vortex effect behind the rotor blades produces the ideal or theoretical maximum efficiency from a available wind.

## Wind Power Calculations

It is commonly known that the theoretical wind power per given swept area by a wind turbine is generally given by the equation:

^{3}× C

_{P}

Where: P is the power in watts, ρ (rho) is the air density in Kg/m^{3}, A is the circular area (πr^{2} or πd^{2}/4) in m^{2} swept by the rotor blades, V is the oncoming wind velocity in m/s, and C_{P} is the power coefficient (efficiency) which is the percentage of power in the wind that is converted into usable energy.

Thus, the wind power output is directly proportional to the cubic power of the wind velocity and to the square of the diameter of the wind turbine. Which is interesting to note, because, when the wind velocity is doubled, the power output is increased by a factor of eight.

Equally as mentioned above, it is not possible to extract all the energy from the oncoming air which passes through the swept area of the rotor blades. As to do so would mean that the air would stop and pile up behind the rotor blades preventing any additional air flow through the blades, irrespective of the specific wind turbine used.

So German physicist Albert Betz carried out some experiments and concluded that no wind turbine could convert more than 59.3% of the kinetic energy of the oncoming wind into usable energy simply by turning rotor blades. So how did he come up with this 59.3% maximum value to produce what is now taken as being the Betz limit.

## The Betz Limit

If we assume that the direction of wind velocity through the rotor blades is centric, uniform and of a constant velocity over the entire rotor area, the turbines efficiency would be the ratio between the output power, P_{OUT}, and the wind energy available flowing into the blades, P_{IN}. Based on these two values, the efficiency of the turbine is thereby given as: η = P_{OUT}/P_{IN}.

Thus for a standard wind turbine to be 100% efficient, η must be equal to “1” (η = 1). However, due to the effect of design, friction and windage, the maximum efficiency that could be extracted is actually given as a percentage of the Betz limit. Consider the folllowing wind turbine image showing how the air flows through it.

### Wind Turbine Air Flow

The airflow through the rotor blades consists of two parts. The oncoming wind before passing through the rotor blades, and the outgoing unused wind expelled or retarded by the rotor blades. If the oncoming wind speed (before the rotor) is defined as V_{IN} and outgoing wind speed (after the rotor) is defined as V_{OUT}, then the reduction in airflow would be given as: V_{OUT} - V_{IN}.

Clearly then, the maximum extraction of kinetic energy from the wind is achieved when the largest wind slowdown is produced but still maintaining sufficient air speed (velocity) past the turbines blades. From his observations and calculations, Betz found that no more than about 59.3% of the kinetic energy can be extracted from the incoming (V_{IN}) wind using an optimal three bladed design.

Then we can modify the above equation for the “*theoretical*” power output from a wind turbine as being:

^{3}× 0.593

Where: C_{P} has been replaced with the *Betz Limit* of 59.3% (0.593) that corresponds to the maximum of the power coeffcient. However, this assumes that the power extraction of the turbine generator is without any loss at 100% efficiency. In reality, turbines with 100% efficiency is not possible even when rotating at full speed due to the mechanical nature of their design and operation.

Modern wind turbines operate at between 60 and 80% efficiency depending on type and maunfacturer. So if we assume our brand new wind turbine generator is declared as being 80% efficient by the manufacturer, then it will convert 80% of the Betz Limit into electricity. Therefore, the C_{P} of our wind turbine would be: 0.8 x 0.593 = 0.474 (or 47.4%). In other words our wind turbine would convert only 47.4% of the available wind energy into usable electricity and NOT 80%.

## Example No1.

The specifications given by a manufacturer of its 3.5kW horizontal-axis wind turbine generator, states that it will operate at its peak efficiency when rotated by a wind velocity of 12m/s. At this wind speed it reaches 69% efficiency. What will be its overall coefficient of power taking into account the Betz limit.

_{P}= 0.69 x 0.593 = 0.41 or 41% at 12m/s wind speed

## Tutorial Summary

Wind turbines are designed to extract the kinetic energy contained in movement of air through its blades. The efficiency of wind power extraction is determined by the Power Coefficient (Cp) which is the ratio of power extracted by the turbine to the total power available in the wind.

100% extraction of the winds kinetic energy is not possible as the flow of air would be reduced to a complete stop and no wind speed would remain available to sustain the flow of air through the turbines blades. German physicist *Albert Betz* calculated that any wind turbine could not convert more than 59.3% of the kinetic energy available in the wind into mechanical energy simply by turning a rotor blade. This theoretical maximum coefficient of power is known as the Betz Limit.

There are many factors that prevent wind turbines from reaching the theoretical Betz limit such as blade losses, mechanical losses, friction, airfoil drag loss, and many more. Reducing these losses will help increase the overall efficiency of a practical wind turbine.

To learn more about how the Betz limit relates to the power produced by wind turbine generators, or to obtain more wind energy information about the various wind turbine generating systems available. Or if you just want to explore the advantages and disadvantages of wind energy and wind power and how you could use it to power your home, then Click Here to get your Dummies copy of one of the top “Wind Turbine Guides” today direct from Amazon.