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I'm currently in the process of designing a numerical model for a vertical axis wind turbine, more specifically, a straight blade giromill. I'm currently having trouble because depending on the variables I choose, I can produce more power than available from the wind.

My Calculations are based off the following diagrams:

http://imageshack.us/photo/my-images/851/selection002y.png/

http://imageshack.us/photo/my-images/31/selection003r.png/

I can post the code (MATLAB), but I'm sure no one wants to sift through that, but here's my design methodology:

User Defined Variables:

Airfoil (NACA00XX)

Wind Speed, [itex]U[/itex]

Tip Speed Ratio, [itex]\lambda[/itex]

Chord, [itex]c[/itex]

Radius, [itex]R[/itex]

Number of Blades, [itex]N[/itex]

Change in Azimuthal Position, [itex]d\theta[/itex]

Swept Area, [itex]A[/itex]

From these variables, I have a loop that iterates [itex]\theta[/itex], the azimuthal position, and calculated the following variables each time:

Chord Velocity

[itex]V_c=U(\lambda+\cos(\theta)[/itex]

Normal Velocity

[itex]V_n=U\sin(\theta)[/itex]

Angle of Attack

[itex]\alpha=\arctan\left(\frac{V_n}{V_c}\right)[/itex]

Relative Wind Speed

[itex]W=\sqrt{V_c^2+V_n^2}[/itex]

Coefficient of Lift and Drag

Calculated using XFoil

Tangential Force Coefficient

[itex]C_t=C_l\sin(\alpha)-C_d\cos(\alpha)[/itex]

Normal Force Coefficient

[itex]C_n=C_l\cos(\alpha)+C_d\sin(\alpha)[/itex]

Tangential Force

[itex]F_t=\frac{C_t \rho c h W^2}{2}[/itex]

Normal Force

[itex]F_n=\frac{C_n \rho c h W^2}{2}[/itex]

As I said, the above variables are calculated for every [itex]\theta_i[/itex]. Once the loop is finished, the following variables are calculated:

Average Tangential Force

[itex]\bar{F}_t=\frac{1}{2\pi}\int_{i=0}^{2\pi} F_t(\theta) \mathrm{d}\theta[/itex]

Numerical Approximation

[itex]\bar{F}_t=\frac{1}{n}\sum_{i=1}^n F_t[/itex]

Total Torque

[itex]T=N\bar{F}_tR[/itex]

Total Power

[itex]P=T\omega[/itex]

I have checked the numbers individually, and my [itex]\alpha[/itex]'s range from [itex]0-13^{\circ}[/itex], [itex]C_l[/itex] and [itex]C_d[/itex] range from [itex]-1.8-1.8[/itex], [itex]C_t[/itex] from [itex]0-0.34[/itex], and [itex]C_n[/itex] from [itex]0-1.22[/itex].

For some reason, if I choose parameters such as:

NACA0015

[itex]U=4.5\,m/s[/itex]

[itex]\lambda=5[/itex]

[itex]c=0.5\,m[/itex]

[itex]R=1.0\,m[/itex]

[itex]h=10\,m[/itex]

[itex]N=3[/itex]

I get a power output of:

[itex]P=10\,kW[/itex]

However, I don't believe I should be getting more than:

[itex]P_{max}=\frac{\rho AU^3}{2}[/itex]

Can someone please help me? I'm pulling my hair out here. If the code would actually help, let me know and I can try and post it.

Thanks So Much.