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# Turbo Turbines

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### Maximum Flow

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Challenge Level

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?

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

Given the graph of a supply network and the maximum capacity for flow in each section find the maximum flow across the network.