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Turbo Turbines

Stage: 5 Challenge Level: Challenge Level:1

 

A landowner wants to calculate whether it would be worthwhile for her to install a triple-blade wind turbine. The turbine would face the wind, which is parallel to the ground and of speed $V$. However, a wind turbine changes the speed locally around it, so that $V_{local} = V \left(\frac{3}{4} + \frac{x^2}{4L^2}\right)$, where $x$ is the distance along the blade from the centre, and the blades are of length $L$.

The force in the direction of the blade rotation is $F = k V_{local}$ at each position $x$ along each blade ($k$ is a coefficient determined by the shape of the blade).

The central pivot is resisted by a torque $T$. It is directly connected to a large gear, which drives a smaller gear (gearing ratio 1:50). The frictional torque that the small gear exerts on the larger one is $4T$.

Draw a diagram to accurately represent the turbine and the wind.

What is the minimum wind speed $V_{crit}$ in terms of $T$, $k$, and $L$, that will produce power?

How could you decrease this minimum wind speed, assuming the mechanical torque is fixed?

Power is generated by the small gear. It has a torque $A\omega_g$, where $\omega_g$ is its angular velocity, and $A$ is a constant. The angular velocity of the blades can be approximated by $\omega = B k V$, when $V > V_{crit}$.

You may know the formula

Power = Force $\times$ Velocity

There is a rotational analogy for many such formulae. Can you find an equation for the power produced by this turbine?