Rayleigh Distribution - Wind Speed

The purpose of this page is to illustrate the use of the Rayleigh distribution to estimate the energy recovered by a medium sized wind turbine.  The parameters are illustrative and it is important to obtain site specific parameters for a project evaluation.

Basic Equation

The equation for energy recovery from the wind is as follows:

Coefficient of Performance

Not all the energy can be recovered from a wind stream.  The theoretical maximum value for the coefficient of performance is 0.593.  An "ideal" wind turbine with this maximum value is known as a Rayleigh-Betz machine.  In practice the value of the maximum values of coefficient is in the range 0.25 to 0.45.  In general, the larger the machine the higher the value.  Also the use of variable pitch rotors can optimise the coefficient of performance for a range of wind speeds.  The curve used in the example is shown below.

The maximum value of the coefficient has been set close to the modal wind speed for Rayleigh averages in the range 5 - 7 m/sec.  The rotor design should be optimised for the site.

Conversion Efficiency

This the fraction of the energy available at the turbine hub which is converted into electricity.  For simplicity, the example value has been set at 0.7.  In practice, the value is composed of two elements, mechanical and electrical.  The mechanical element is quite high with losses coming from items like bearings, gearboxes (if any) etc. whilst the electrical part would come from the specification of the generator.

Area

This is the area swept by the rotor blades, the example calculation shows the power generated per square metre.  The range of sizes for wind turbines is shown in the table below:

Description	Diameter (m)	Area (m2)	Power (kw)
Small	< 2.5	5	<1
Medium	5 - 15	20 - 200	10 - 50
Large	> 50	2000	> 1000

Density of Air

The density of air at sea level is 1.23 kg/m³.  However, this density declines with altitude as shown in the graph below:

Thus a turbine located at an altitude of 1,000m would produce approximately 80% of the power an equivalent installation would produce at sea level.

Wind Speed - Effect of Height

Wind speed varies with height.  At ground level (zero metres) the speed is low and turbulent and at some higher altitude (say, 100m) it is faster and smoother.  This is due to friction as wind passes across the earth's surface.  Whilst the nature of surface varies, it is common practice to use an empirical relationship between height and speed:

The standard height for meteorological observation wind speed data is10m.  This type of data is the most readily available.  As the power generated is proportional to the velocity cubed, there is an advantage to be gained by locating the turbine on some form of tower, typically in the range 30 to 80 metres high.  If site specific data is not available at the proposed height, an initial estimate can be gained from scaling up data collected at 10m to the height of the proposed tower.  Using the above equation, the effect of height is shown in the graph below:

The sample calculation assumes that the turbine is mounded on a 30 metre tower, thus if the Rayleigh wind speed is 5 m/sec at 10m, this can be expected to increase by a factor of 1.17, allowing a value of 5.85 to be used for the estimate, giving a 60% increase in output.

Wind Speed - Operating Range

At very low speeds (say less than 2 m/sec) the turbine will not rotate at all whilst at high speeds (say greater than 25 m/s) it is necessary to limit or stop the turbine to prevent damage from over-speeding.

In the sample calculation, it has been assumed that the turbine has an operating range of 2 - 25 m/sec and does not generate any power outside this range.

Form of Calculation

The estimation of power output with varying Rayleigh wind speed has been performed using a table based on intervals of 1 m/sec.  This method makes it relatively simple to handle variations in coefficient of performance and operating range.

Page Updated 31-Jan-2005

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