# Finding the best height for windmill towers

• Imolopa
In summary, the conversation discusses the use of the natural logarithm relation to approximate windspeeds and the Betz limit for extracting power from the wind. It also explores the ideal height for wind turbines and how it is affected by factors such as roughness length and blade radius. Overall, it is found that larger wind turbines can be more efficient but may not always be practical due to cost and other limitations.
Imolopa
To approximate windspeeds we normally use the natural logarithm relation:
u_air = V ln [ (z-d) / z_0 ]Where:
V - Characteristic speed, constant
z - height from the ground
z_0 - roughness length, constant
d - zero plane displacement, constant

The Betz limit tells us that in theory that the max possible fraction of power that can be extracted from the wind is:

C_P = 16/27 = 0.59

Since this is equivalent to a 1/3 reduction of the kinetic energy of the wind we get that:

v_windturbine = (2/3) v_air = v_airthroughthewindturbine

Now the extracted power would be given by:

P = 0.5 * C_p * rho * A * [v_windturbine]3Where:

rho - density of air at given conditions (assume constant in this ideal case)

A = pi * R2 - area covered by the windmill blades of radius R

So both of the top expressions into the expression for power returns:

P = 0.5 * C_p * rho * A * [(2/3) V ln [ (z-d) / z_0 ] ]3So assuming that:
R = 3 m
rho = 1.2 kg/m3
V = 5 m/s
d = 0.0001 m
z_0 = 30 m

Also we assume this is an ideal case and that the above holds for the site in question.

Now if divide P/z, we find that the peak values for P/z are achieved between z = 500 and 600 m of tower/hub height. The same results for a windmill that is double that size (R = 6m). However I'm quite sure I've seen claims of small windturbines decreasing in performance gain after around hundred metres of hub/tower height.

This last claim could be due to material costs and disadvantages of building stuff that is that tall, also the tower for the windmill would have to be wider for increasing heights (which in itself increases the material need considerably since the shape is cylindrical). Also the windmills normally don't operate when the windspeeds go above a certain limit due to safety/limitations, which I guess should be between 30 and 40 m/s.

However if we disregard this last paragraph, should I get results that would indicate peak values for P/z at around 100 m or between 100 and 200 m, iow. are my above results in the ballpark/realistic at all?

Last edited:
Multiple units are wrong.

With your model, the ideal height would be hundreds of meters (z0*e3 neglecting d), but building such a large tower is impractical.

The height above ground where surface friction has a negligible effect on wind speed is called the "gradient height" and the wind speed above this height is assumed to be a constant called the "gradient wind speed".[11][15][16] For example, typical values for the predicted gradient height are 457 m for large cities, 366 m for suburbs, 274 m for open terrain, and 213 m for open sea.[17]

274 m for open terrain, and 213 m for open sea.
The largest wind turbines are approaching those values as tip height (but not with the hub).
230 m onshore, 220m offshore, and they are working on R=100m blades, which would probably be installed on wind turbines with a height exceeding 250 meters.

mfb said:
Multiple units are wrong.

With your model, the ideal height would be hundreds of meters (z0*e3 neglecting d), but building such a large tower is impractical.
Thanks corrected the units.
mfb said:
The largest wind turbines are approaching those values as tip height (but not with the hub).
230 m onshore, 220m offshore, and they are working on R=100m blades, which would probably be installed on wind turbines with a height exceeding 250 meters.

Interesting, so the critical factor here is the roughness length z_0. As we can see the results are highly sensitive to its value:
z0= 1 => z0e3 = 20 m
z0= 2.5 => z0e3 = 50 m
z0 = 5 => z0e3 = 100 m
z0 = 10 => z0e3 = 200 m
z0 = 20 => z0e3 = 402 m

And it seems like they don't even bother going above gradient height so that the entire "windmill-area" is above it since it's not even done on the tallest windmill in the world.
This might imply that the final gain does not win over the costs for building taller towers, even with much larger blades than the ones initially used in the problem here.

And yes it makes perfect sense to increase bladeradius R, since it increases the power P quite significantly since P is a function of R2

A larger blade radius is more challenging in terms of blade weight (square/cube law), and you need larger distances between larger windmills. You need taller cranes and so on. It is not immediately obvious that larger windmills win, but overall they do with recent advances in their design.

## What is the purpose of finding the best height for windmill towers?

The purpose of finding the best height for windmill towers is to optimize the efficiency and effectiveness of wind turbines. This can lead to higher energy production and lower costs for renewable energy.

## What factors are considered when determining the best height for windmill towers?

Factors that are considered when determining the best height for windmill towers include wind speed and direction, terrain and topography, and local climate conditions. Additionally, the size and type of wind turbine being used will also play a role in determining the optimal tower height.

## How does the height of a windmill tower affect its energy production?

The height of a windmill tower can significantly impact its energy production. As wind speed increases with height, taller towers can capture more energy from stronger winds. Additionally, taller towers can reach above any surface-level obstructions, such as buildings or trees, which can disrupt wind flow and decrease energy production.

## Are there any disadvantages to building taller windmill towers?

While taller windmill towers can increase energy production, they also come with some disadvantages. Taller towers require more materials and are more expensive to build and maintain. They may also face height restrictions and zoning regulations in certain areas. Additionally, taller towers may be more difficult to transport and install.

## How can scientists determine the best height for windmill towers?

Scientists use a combination of computer modeling, wind tunnel testing, and field testing to determine the best height for windmill towers. They analyze data on wind speeds and terrain to create simulations and models to predict the most efficient tower height for a specific location. Field testing involves installing and monitoring wind turbines at different heights to gather real-world data and make adjustments to the models.

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