Designing a Windmill
Question
Windmills tend to have 3 blades. Why is this?
The speed of air through a windmill is the average of the speed of air into the turbine and the speed of air exiting the turbine, $v_{mid}=\frac{v_{in}+v_{out}}{2}$. By considering the mass of air through the windmill, and the change in kinetic energy, find an estimate for the most efficient ratio $r=v_{out}/v_{in}$.
Hints
The mass through the turbine can be found by considering the conservation of mass. For a constant cross sectional area (i.e. the cross-section of the turbine blades), the mass flow rate is a factor of the speed $v_{mid}$. That is, if the speed of the air is halved, then the mass flow rate through the turbine is halfed.
Answer
There are two competing factors when designing a windmill:
- Mass flow rate needs to be maximised, to increase the amount of energy input to the windmill.
- Energy extraction efficiency needs to be maximised, meaning the energy of the output airflow - and therefore the output velocity - need to be minimised.
These two competing factors mean that the number of turbine blades is a compromise between extracting energy, and allowing mass to flow.
The factor by which the mass flow rate is reduced is given by $$v_{mid} = \frac{v_{in}}{2} (1 + r)$$
The energy extracted per unit mass by the turbine is given by $$\frac{E}{\dot m} =\frac{1}{2} (v_{in}^2-v_{out}^2) \propto v_{in}^2(1-r^2)$$
The total efficiency is given by the product of these two terms. Differentiate and set to zero to find the maximum efficiency is achieved at $r=\frac{1}{3}$.
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