A conventional horizontal-axis machine is sufficient.

8.4.19

Determine the power that may be delivered by a wind generator, assuming that the wind kinetic energy is completely converted into mechanical kinetic energy, and explain why this is impossible.

8.4.19 - Determine the power that may be delivered by a wind generator

The power delivered by a wind generator

Recognizing assumptions made

The power delivered by a wind generator

The power of the electrical energy delivered by a wind generator depends on the kinetic energy of its turbines, which depend on the kinetic energy of the wind that spins its turbines. The kinetic energy of the wind, in turn, comes from solar energy: the sun heats different parts of the earth to variable temperatures, and this temperature difference causes hot air to rise and cold air to fall in altitude. This results in a pressure difference that causes a general movememt of air, or wind.

A mathematical formula for the power delivered by a wind generator can be constructed by considering the energy transformation from the kinetic energy of wind to the kinetic energy of the turbines. Looking at the diagram below with several variables defined as:

Wind is moving horizontally towards a wind turbine (source: Rozenblat, Lazar. "How Wind Energy Works." Wind Energy. 2009. 29 Mar. 2009 <http://windpower.generatorguide.net/wind-energy.html>.)

ρ = density of air
D = Diameter of circular blade shape
v = horizontal wind speed
m = mass of air

The area in which the wind and turbines intersect
A = π(D/2)2

The volume of air that passes through the intersecting area in one second
V = (A)(v)(1s)
= πv(D/2)2

Since V=(m)/(ρ),

(m)/(ρ) = πv(D/2)2 m = ρπv(D/2)2

We have found out that air mass of ρπv(D/2)2 kg crosses the intersecting area every second. Therefore the total kinetic energy of the wind that crosses the intersecting area every second:
K = (1/2)(m)(v2 )
= (1/2)(ρπv(D/2)2)(v2)
= (1/2)(ρπ(D/2)2(v3)

Wind with total kinetic energy of (1/2)(ρπ(D/2)2(v3) J crosses the intersecting area every second. Assuming the kinetic energy of the wind is converted completely into kinetic energy of the turbines and that all kinetic energy of the turbines is converted into electrical energy through the generator, this is also the power of the wind generator.

Power developed by wind generator
= (1/2)(ρπ(D/2)2)(v3)

Note that since the intersecting area (A) = π(D/2)2, the power can be written in the simpler form:

Power developed by wind generator
= (1/2)(ρπ(D/2)2)(v3)
= (1/2)(ρA)(v3)

Unit analysis gives the units of power to be:
(kg/m3)(m2)(m/s)3
= kgm2/s3

This is our definition of Watts (W), the unit of power.

Recognizing assumptions made

The mathematical expression for power developed by a wind generator constructed above makes several assumptions.

One underlying assumption in the above calculations is that wind kinetic energy is completely transformed into turbine kinetic energy. This is impossible because friction between moving parts, for example between the turbine and the axis of rotation, causes some of the wind kinetic energy to be transformed into degraeded heat energy (energy that is no longer useful for doing work). Also, the very process by which the energy is transferred from the wind to the turbine- the collision of air molecules with the turbine surface- causes some of the wind's initial kinetic energy to be converted into degraded energy.

Another assumption that is made is that the kinetic energy of the turbine is completely transformed into electrical energy through the generator. However, energy is lost again as degraded energy because the generator is not 100% efficient, due to the presence of friction between the moving parts. This further reduces the percentage of the initial kinetic energy of the wind that is transformed into electrical energy by the wind generator.

As a result of the above assumptions, the mathematical model for the power of a wind generator is overstated. The actual power developed by a wind power will be the mathematical model developed above, P=(1/2)(ρAv3), reduced by the rate at which friction converts the kinetic energy (wind) input into degraded energy. Minimizing the friction between parts, for example by applying lubricants or smoothing out surfaces, will allow the actual power to be closer to the mathematical model above.

A sankey diagram for a highly efficient wind generator (source: Katopodi, Koroneos. "Energy analysis of the wind power hydrogen and electricity production." 2 Mar. 2006. EWEC. 3 Apr. 2009 <http://ewec2006proceedings.info/allfiles2/0376_Ewec2006fullpaper.pdf>.)

A sankey diagram is one that shows the transformations of energy as it is processed, in this case through a wind generator. Betz law, simply put, is a theory by the German physicist Albert Betz (in 1919) that states the maximum amount of energy that can be developed from a wind generator. According to the law,nont wind generator can only be more than 59.3% efficient. The diagram above shows that several more energy losses occur as the wind's kinetic energy is ultimately transformed into electrical energy.

Wind power8.4.19Determine the power that may be delivered by a wind generator, assuming that the wind kinetic energy is completely converted into mechanical kinetic energy, and explain why this is impossible.The power delivered by a wind generatorThe power of the electrical energy delivered by a wind generator depends on the kinetic energy of its turbines, which depend on the kinetic energy of the wind that spins its turbines. The kinetic energy of the wind, in turn, comes from solar energy: the sun heats different parts of the earth to variable temperatures, and this temperature difference causes hot air to rise and cold air to fall in altitude. This results in a pressure difference that causes a general movememt of air, or wind.

A mathematical formula for the power delivered by a wind generator can be constructed by considering the energy transformation from the kinetic energy of wind to the kinetic energy of the turbines. Looking at the diagram below with several variables defined as:

ρ = density of air

D = Diameter of circular blade shape

v = horizontal wind speed

m = mass of air

The area in which the wind and turbines intersect

A = π(D/2)2

The volume of air that passes through the intersecting area in one second

V = (A)(v)(1s)

= πv(D/2)2

Since V=(m)/(ρ),

(m)/(ρ) = πv(D/2)2

m = ρπv(D/2)2

We have found out that air mass of ρπv(D/2)2 kg crosses the intersecting area every second. Therefore the total kinetic energy of the wind that crosses the intersecting area every second:

K = (1/2)(m)(v2 )

= (1/2)(ρπv(D/2)2)(v2)

= (1/2)(ρπ(D/2)2(v3)

Wind with total kinetic energy of (1/2)(ρπ(D/2)2(v3) J crosses the intersecting area every second. Assuming the kinetic energy of the wind is converted completely into kinetic energy of the turbines and that all kinetic energy of the turbines is converted into electrical energy through the generator, this is also the power of the wind generator.

Power developed by wind generator

= (1/2)(ρπ(D/2)2)(v3)

Note that since the intersecting area (A) = π(D/2)2, the power can be written in the simpler form:

Power developed by wind generator

= (1/2)(ρπ(D/2)2)(v3)

= (1/2)(ρA)(v3)

Unit analysis gives the units of power to be:

(kg/m3)(m2)(m/s)3

= kgm2/s3

This is our definition of Watts (W), the unit of power.

Recognizing assumptions madeThe mathematical expression for power developed by a wind generator constructed above makes several assumptions.

One underlying assumption in the above calculations is that wind kinetic energy is completely transformed into turbine kinetic energy. This is impossible because friction between moving parts, for example between the turbine and the axis of rotation, causes some of the wind kinetic energy to be transformed into degraeded heat energy (energy that is no longer useful for doing work). Also, the very process by which the energy is transferred from the wind to the turbine- the collision of air molecules with the turbine surface- causes some of the wind's initial kinetic energy to be converted into degraded energy.

Another assumption that is made is that the kinetic energy of the turbine is completely transformed into electrical energy through the generator. However, energy is lost again as degraded energy because the generator is not 100% efficient, due to the presence of friction between the moving parts. This further reduces the percentage of the initial kinetic energy of the wind that is transformed into electrical energy by the wind generator.

As a result of the above assumptions, the mathematical model for the power of a wind generator is overstated. The actual power developed by a wind power will be the mathematical model developed above, P=(1/2)(ρAv3), reduced by the rate at which friction converts the kinetic energy (wind) input into degraded energy. Minimizing the friction between parts, for example by applying lubricants or smoothing out surfaces, will allow the actual power to be closer to the mathematical model above.

A sankey diagram is one that shows the transformations of energy as it is processed, in this case through a wind generator. Betz law, simply put, is a theory by the German physicist Albert Betz (in 1919) that states the maximum amount of energy that can be developed from a wind generator. According to the law,nont wind generator can only be more than 59.3% efficient. The diagram above shows that several more energy losses occur as the wind's kinetic energy is ultimately transformed into electrical energy.

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