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Wind Speed, vT

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Persian windmill or anemometer

Aerodynamics of the Persian Windmill:
Persian Windmill Power Calculator

The Oldest Windmill Design
The Persian windmill from about 1000 b.c. is the oldest known windmill design. Basically it works almost the same way as the Drag Machine in this chapter, but the sails rotate around a vertical shaft instead of moving in a straight line with the wind.*)
In the image, the wind is blowing towards you, the reader. The reason why the machine will work is that a wall blocks the right side of the entrance for the wind.
In a sense, it is not so important whether the shaft of the turbine is vertical or horizontal. The turbine can work either way. The advantage of the vertical axis is that we avoid fatigue loads on the sails due to gravity. An essential difference from modern propeller-based wind turbines is that the air flow is tangential to the plane of rotation, rather than axial (flowing along the axis of rotation).
The basic inconvenience of having a tangential air flow is that the sails or blades will be moving in the wrong direction (against the wind) during one half of the rotation of the turbine. Although this particular design avoids moving the sails directly against the wind, it unfortunate that the sails are subject to fatigue loads from the changing wind direction in relation to each sail for every rotation of the turbine.

*) The Persian windmill has been reinvented many times, and several inventors have patented many, rather useless variations on the theme with parachutes, yawing wind shields etc. Drag based windmills are very inefficient, and really only of interest for educational or hobby purposes. (With all due respect, think before you act - there are thousands of worthless wind turbine patents around. Contrary to popular belief patenting guarantees neither efficiency, usefulness nor workability).

How to Use the Calculator
This calculator works exactly the same way as the Drag Power Calculator. If you know how that works, you can skip this section.
You can easily change the values in the input boxes to the left up or down by one unit by clicking the arrows, or you may enter a number. If you enter a number, press the tab key or click outside the box to tell the programme to calculate the result. If you are a non-English speaker, remember to use decimal points, and not commas when entering decimal fractions in the boxes. The calculator will not accept unreasonable quantities.
You can read the input and output data for each experiment in the Test Results to the right. Each variable is explained in the section How does the Calculator Work? at the bottom of this page.
You can sort the test results by any column by clicking on the variable name at the top of the column. You can delete a row by clicking on the row number (in the first column). You can plot any two variables against one another by selecting the variables from the popup menus at the top of the list and then clicking the "Plot" button. Remember to sort your data by the variable on the horizontal (x-) axis before you plot. Finally, you can delete all of the test results by clicking the "Delete" button.

OwlSimplified Aerodynamic Analysis
Let us make the simplified assumption, that the sail which is exposed to the wind on average is perpendicular to the wind. If we disregard the fact that the drag force brakes the sails on the way back, then the analysis is exactly the same as on the page Drag Machine: Optimal Load and Optimal Operating Speed:
The maximum efficiency cP is 16%, when the machine operates at its optimal operating speed ratio (lambda), λ = 1/3 of the wind speed. You should note, however, that when we say that the machine can capture 16% of the energy in the wind, we consider only the sail area which is directly facing the wind, not the sail area behind the wall.
The operating speed ratio, λ, should be measured as the speed of the middle of the sail, vP, divided by the wind speed, vT.

OwlHow does the Calculator Work?
We need the same variables which we used for our drag machine:

vT = wind velocity in m/s
vP = (ground) velocity of the centre of each sail in m/s
λ = vP / vT = the operating speed ratio (lambda)
A = area of each sail in m2
= the density of air, 1.225 kg/m3 (at 15° C, and standard atmospheric pressure at sea level)
F = the force applied to put a load on the windmill measured in N (Newton)
FD= the aerodynamic drag force on the sail measured in N
cD1 = the drag coefficient, N/m2 (Newton per square metre frontal area) when the sail is running with a tailwind (useful drag).
cD2 = the drag coefficient, N/m2 (Newton per square metre frontal area) when the sail is running with a headwind (undesirable drag).
P = the mechanical power produced by the machine
cP = the power coefficient, i.e. the share of the power of the wind, which the machine is able to capture

Now, let us try a more realistic assumption than in the simplified analysis, i.e. that we subtract the drag on the sails behind the wall from the output of the machine.

In order to find the propulsion speed of the Persian windmill, we solve this equation for vP:

0 = cD1 0.5 A (vT - vP)2 - cD2 0.5 ρ A vP2 - F

The second half of the expression to the right of the equal sign (shown in red) is the undesirable drag from the rotor sails. The formula is simply the definition of the drag force explained previously. We subtract the propulsion speed from the wind speed (vT - vP) in order to get the speed of the sail relative to the wind in the first half of the expression. The drag coefficients cD1 and cD2 are equal, and about 1.1, although some sources *) give values of 1.18.

The equation for the Anemometer, which we can solve for vP, is exactly the same, only the values of the drag coefficients are different.

0 = cD1 0.5 A (vT - vP)2 - cD2 0.5 ρ A (vT + vP)2 - F

The second half of the expression to the right (shown in dark red) is the undesirable drag force from the anemometer cups moving upwind. The drag coefficients cD1 and cD2 are 1.33 and 0.33 respectively.

*) = The figure is from Gash (1996). Frank M. White (1999) has 1.18-1.2, see the bibliography.


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© Copyright 2002 Danish Wind Industry Association
Updated 8 December 2002

Test Results