BELTZ LIMIT - WIND TURBINE ENERGY ABSORPTION

Any moving object has energy. This type of energy is called kinetic energy. For example, a car, a bicycle, or a ball, when moving, all have kinetic energy. If they stop, that energy is gone. The same is true for moving air, that is, the wind.

The amount of energy of a moving object depends on two factors, its mass and its speed. Using the proper units for measuring mass and speed, the relationship to determine the energy of a moving object is as follows:

Energy ½ (mass)(speed)2.

This equation implies that, for example, if mass is doubled (that is, if you have two objects the mass of one of which is two times the mass of the other, boTh having the same speed), the energy doubles.

But, if the speed is doubled (if you have two objects of the same mass, but one has a speed two times that of the other), then the energy is four times more. It is very important to understand the relationship given by an equation similar to that in.

This energy can be converted to work. In other words, it can be used to do work (see the text on energy and power in this chapter for the technical meaning of work).

As was mentioned earlier, for wind or any other moving fl uid it is more practical to fi nd the power rather than the energy. In this sense, in equation we can substitute for the mass of the air that fl ows in 1 sec, and that gives the power in the wind.

Note that
Mass (Density)(Volume) (2.2)

The power in a tunnel of wind is proportional to the air density, the cross-sectional area of the tunnel of wind, and the cubic power of the wind speed.

The cross-sectional area in this equation refers to the size of a turbine.

The following conclusions can be derived from equation:
1. For the same turbine and at the same time, if the wind speed doubles, the power in the wind increases by a factor of 8.

2. For the same turbine and the same wind speed, if the weather is cold (higher density), more power exists in the wind available for the turbine.

3. In the same weather conditions and for the same wind speed, a turbine that is two times larger in cross sectional area than a smaller one has twice as much wind power available to it.

From item 2 above it can also be concluded that a turbine at a given wind speed can produce more power in the winter than in the summer because the weather is colder. The same conclusion can be extended from day to night if the temperature diff erence is signifi cant. This diff erence due to temperature change is, nevertheless, not very much.

Power absorption by a turbine
In the previous section the available power in a stream of wind was defined in terms of the wind speed, turbine size, and the air density. This power, however, is the power that exists in the wind when it is blowing.

A turbine cannot necessarily capture all of this power; it can only absorb a portion of it. This depends on the type of turbine, the efficiency, and other conditions in the operation of a turbine.

In order to show the fraction of power in the wind that a particular wind turbine can harness from wind (i.e., wind harnessing), a coeffi cient is used in equation (2.4). Thus, we can say that

Power of a wind turbine (A coeffi cient)(Power in the wind) This coeffi cient must, obviously, be smaller than 1. It is called power coeffi cient.

Wind turbine power ½ (Power coeffi cient)(Density)(Cross–sectional area)(Speed)3 .The power coeffi cient depends on how good a turbine is in design and how well it can grasp the wind energy.

Thus, its value can be small or large. Nevertheless, there is a maximum value that no turbine in its best performance can exceed. It can be theoretically determined and is called the Betz limit. The value for Betz limit is =16/27 = 0.59.

Betz limit = 16/27 = 0.59 The power coeffi cients for certain turbines can reach values near the Betz limit. A value of 0.50 for a good design is acceptable. For others, this number can be smaller, say 0.25 or even 0.20. Any claim for coeffi cients greater than the Betz limit are groundless and can be rejected.