Flow (liquid or gas) across a rotating surface's face

[QuarkDecay]

<< Mentor Note -- Two threads on the same question merged into one thread >>

How does the maximum Power equation change if there's an angle to the way the wind falls into the wind turbine's blades?

Example, when it falls vertically to the blades, it's
P_max= 8/27Sρu₁³

But if there's for example an angle of θ=150^ο then what? My book says you need to do S=Scosθ, but cos(150^o)= -0.89. Which is negative and doesn't make sense for the Power to be negative?

I thought maybe I should do S=πR²=π(Rcosθ)² instead. Is that wrong or what am I supposed to do?

 

[russ_watters]

Example, when it falls vertically to the blades, it's
P_max= 8/27Sρu₁³

But if there's for example an angle of θ=150^ο then what? My book says you need to do S=Scosθ, but cos(150^o)= -0.89. Which is negative and doesn't make sense for the Power to be negative?

150 degrees would be behind the turbine.

 

[QuarkDecay]

Suppose we have a surface that scans a cycle, with an S=πR², when its axis faces the gas' flow vertically.
Now if the gas' flow gets an angle of θ=150^ο, what will the S be? My book says it's Scosθ, but with an angle of 150^o it gives me a negative number, which doesn't make a lot of sense considering the nature of the problem (needs to calculate the power output of a wind turbine)
Should I do S=πR²= π(Rcosθ)² instead?

Also why should it be cosθ and not sinθ? Can someone explain the math behind this?

 

[vanhees71]

Just draw the disk and the flow vector with the definition of the surface normal vector such to give an angle of $150^{\circ}$ and see for yourself that the book is right with it's definition. The flow through the surface element $\mathrm{d}^2 \vec{f}$ is given given by $\vec{j} \cdot \mathrm{d}^2 \vec{f}$ it gives the amount of fluid flowing per unit time through the little surface element. This gets negative in your case, and it has a very simple physical meaning, but you have to consider the (arbitrary) choice between the two possibilities to orient the surface-normal vectors along the surface!

Hint: This should also become clear, if you simply flip the direction of the surface normal then the angle is $30^{\circ}$ and your current gets positive!

 

[QuarkDecay]

Just draw the disk and the flow vector with the definition of the surface normal vector such to give an angle of $150^{\circ}$ and see for yourself that the book is right with it's definition. The flow through the surface element $\mathrm{d}^2 \vec{f}$ is given given by $\vec{j} \cdot \mathrm{d}^2 \vec{f}$ it gives the amount of fluid flowing per unit time through the little surface element. This gets negative in your case, and it has a very simple physical meaning, but you have to consider the (arbitrary) choice between the two possibilities to orient the surface-normal vectors along the surface!

Hint: This should also become clear, if you simply flip the direction of the surface normal then the angle is $30^{\circ}$ and your current gets positive!

So Scos(150^o)=πR²cos(150^o) is the right choice?

 

[vanhees71]

Yes, but draw a picture and understand it (including the sign)! Physics is not about using some formula to get a result but it's about understanding why you use this formula and what it means for the phenomenon described!