Basic .
. ....
..........................by .Claus Nybroe
...........................................Windmission
.........

CONTENTS:
1. INPUT DATA*
2. VELOCITIES IN THE ROTOR PLANE
3. TIP SPEED RATIO
4. MATCHING FORMULAS
5. SELECTING BLADE CHORD AND PROFILE
6. ANGLES

With this rather simple method we over the years have made very efficient blades (Cp-max measured = 0.46). Do not be afraid of the mathematics. There are only 6 formulas and a couple of curves. All Calculations can be performed by hand or by the means of a pocket calculator, a spreadsheet or another small computer programme.

The paper is made as an example based on known generator data from Ian Cummings, Putnam CT.

In Ian's case we start with known generator data. Alternatively you can also start with known wind, rotor or profile data. The formulas are all there.

1. INPUT DATA
The metric (m) system is used.
1000 meter = 0.625 US miles
Power is measured in Watts (W)
1 HP = 736 W

Ian Cummings has provided us with these approximate data:

PM-generator.
220 W at 700 rpm (revolutions per minute)

2. VELOCITIES IN THE ROTOR PLANE
To get a first grip of things please have a look at
the velocities in the rotor plane

3. TIP SPEED RATIO
We start by selecting a value for the Tip Speed Ratio (TSR) which is defined as

(Formula 1) :
TIP SPEED RATIO (TSR) =
(tip speed of blade)/(wind speed).

The tip speed ratio is a very important factor in the different formulas of blade design.

Generally can be said, that slow running multi bladed wind turbine rotors operate with tip speed ratios like 1-4, while fast runners use 5-7 as tip speed ratios.

Ian Cummings wants to cut a two bladed rotor. This rotor type usually runs very fast, so let's choose a tip speed ratio of 7.

4. MATCHING FORMULAS
The task is now to fit the known generator capacity and revolutions to the wind speed and to the swept rotor area. Two formulas are needed:

(Formula 2) :
Power (W) = 0.6 x Cp x N x A x V3

(Formula 3):
Revolutions (rpm) = V x TSR x 60 / (6.28 x R)

Cp = Rotor efficiency
N = Efficiency of driven machinery
A = Swept rotor area (m2)
V = Wind speed (m/s)
TSR = Tip Speed Ratio
R = Radius of rotor

Rotor efficiency can go as high as Cp = 0.48, but Cp = 0.4 is often used in this type of calculations.

This concept works without transmission. If a transmission with an efficiency of 0.95 was to be included this means that
N = 0.95 x 0.7

In Ian Cummings case the following values fit into formula (2) and (3):
Tip speed ratio "TSR" = 7"
Wind speed "V" = 8.6 m/s
Rotor efficiency "Cp" = 0.4
Generator efficiency "N" = 0.7
Swept rotor area "A" = 2.11 M2
Radius of rotor = 0.82 m
Revolutions = 701 rpm
Power output = 226 W

It took about 20 minutes to perform these calculations and make them match on the pocket calculator. A simple spreadsheet can also be useful.

5. SELECTING BLADE CHORD AND PROFILE

The width of the blade is also called the blade chord. A good formula for computing this is:

(Formula 4):
Blade Chord (m) = 5.6 x R2 / (i x Cl x r x TSR xTSR)

R = Radius at tip
r = radius at point of computation
i = number of blades
Cl = Lift coefficient
TSR = Tip Speed Ratio

As can be seen from formula (4) we need to know the lift coefficient "Cl " in order to compute the blade cord. This means that we have to select a profile. A lot of good profile data can be found in model airplane (gliders) literature.

We have chosen the NACA 2412 profile
The side facing the wind is flat, which makes the profile easy to construct. It is an effective profile with a good thickness, which makes the blade strong.

In order to determine the lift coefficient we must have a look at the profile curves.

By checking the NACA 2412 profile curves Cl is determined to be 0,85. Ian Cummings formula now looks like this:

"Chord" = 5.6 x 0.82 x 0.82/(2 x 0.85 x 0.82 x 7 x 7))...(m)
..Tip

"Chord " = 55 mm
..Tip

Now, calculate blade chord at 2/3 x R. On a paper choose a center line at at distance 1/3 from the leading edge. Connect the the two blade chords, and you can measure all the cords of the blade. (Illustration)

The closer you come to the hub you might choose thicker profile to increase strength. Close to the hub you should also consider an extra increase in chord in order to make the blade start easier.

6. ANGLES

Have a look at the angles of the blade where the angles for Ian Cummings are calculated.

Close to the hub you should consider an extra increase in the angle of attack, in order to make the blade start easier.