For flap type turbines the sail width can be calculated in terms of TPR (TIP POWER RATIO) s defined in here. We first assumed that the sail of the flap-turbine is sum of infinite number of Pelton turbines on the sail with infinitesimal thickness. Then we defined the TIP POWER RATIO s as the ratio of the power produced at the tip strip to the center strip. Since these strips are parallel to rotation axis as shown in the following figure and their thickness are infinitesimal, it is safe to assume that the sail velocity along a strip is not changing. However each strip has different velocity depending on the location on the sail. While strips away and outward from the center strip moving faster than the center strip, inner ones move slower.
Power produced on any Pelton strip can be found from equation given below. Where Vw is the wind speed and U is the sail strip which is assumed constant along the strip. Here ρ is the density of the air in that location, and ΔQ is the volume rate of air acting on the infinitisimal strip.
ΔP = 2*ρ*ΔQ(Vw - U) * U (1)The power produced at center strip where the sail velocity is 0.5*Vw, the power can be expressed as
ΔPc = 2*ρ*ΔQ*(Vw-0.5*Vw)*Vw*0.5 = 0.5*ρ*ΔQ*Vw2 (2)
At the tip strip where sail is moving at the velocity Ut the power generated would be acording the pelton power formula as
ΔPt = 2*ρ*ΔQ*(Vw-Ut)*Ut (3)
We also know from definition of Power Tip Ratio s that
ΔPt = ΔPc * s (4)
From equation (2), (3) and (4) we can find following equation.
(Vw - Ut) * Ut = 0.25 * s * Vw2
or, it can be expressed as
Ut2 - Vw * Ut + 0.25 * Vw2 * s = 0 (5)
When this equation is solved for Ut, we find following two values.
Uto = 0.5 * Vw * ( 1 + Sqrt(1 - s)) (6) Uti = 0.5 * Vw * ( 1 - Sqrt(1 - s)) (7)
Where Uto and Uti correspond to sail velocity at the outer and inner tip of the sail respectively. We know that for rotating bodies
U = ω * R (8)
We also know that at the center of the sail velocity is 0.5*Vw and R is equal to Rc, substituting these into equation (8) we get
Vw = 2 * ω *Rc (9)
By using equation (8) we can express Uto and Uti as
Uto = ω * Ro (10) Uti = ω * Ri (11)
By substituting equation (9), (10) and (11) to (6) and (7) we get
Ro = Rc * ( 1 + Sqrt(1 - s)) (12) Ri = Rc * ( 1 - Sqrt(1 - s)) (13)
By taking the difference of Ro and Ri we can find the width ot the sail as
sail_width = W = 2 * Rc * Sqrt(1 - s) and called as Seyhan's Sail Width Equation for the Flap-Turbine (14)
The Plot 1 given below shows the ratio of sail width to Rc ( distance from rotation center line to the sail center line) as function of s. Important observation are shown under the plot.
In the following flash animation by changing the value as inside green box, you can observe how the width of the sail changes.
We should emphasize that these calculation should not be used for any other drag type wind turbines. By opening its flaps on up moving sail, the flap-turbine becomes almost like a Pelton turbine. However any other drag type turbines which does not have opening flaps, creates so much resistance during up movement, the resembles to Pelton is eliminated. These type turbines, especially Savonius are so inefficient that one should never consider building one.