The amount of energy available in the wind at any instant is proportional to ___ of the wind speed.

Question

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Option D: Cube power

Explanation & Concepts:
The power available in wind is defined by the kinetic energy of the air mass flowing through a cross-sectional area per unit time. Mathematically, this power is proportional to the cube of the wind velocity, meaning a small increase in wind speed results in a significant increase in available power.

Working Principle: Wind power is derived from the kinetic energy of moving air: $E = \frac{1}{2} m v^2$. Since mass flow rate is $\dot{m} = \rho A v$ (where $\rho$ is air density, $A$ is rotor swept area, and $v$ is wind velocity), substituting $\dot{m}$ into the power equation yields $P = \frac{1}{2} \rho A v^3$. Thus, the power is fundamentally proportional to $v^3$.

Key Formulas: $P = \frac{1}{2} \rho A v^3$ — The power equation where $\rho$ is air density, $A$ is swept area, and $v$ is velocity

Key Points: Power is directly proportional to the cube of the wind speed ($P \propto v^3$).; Doubling the wind speed increases the available power by a factor of eight.; Air density ($\rho$) also directly impacts total power output.; The swept area ($A$) of the turbine blades is directly proportional to power.

Advantages: Renewable and inexhaustible source of energy.; Low operational and maintenance costs post-installation.

Disadvantages: Wind is intermittent and unpredictable.; Low energy density requires large geographic areas for wind farms.

Applications: Utility-scale electricity generation.; Off-grid power supply for remote areas and water pumping.

Additional Info: The Betz Limit states that no turbine can capture more than 59.3% of the kinetic energy in wind.. Option C (Square power) is incorrect as it relates to area scaling, not velocity scaling.

Answer: D is correct — The power available in a wind stream is directly proportional to the cube of the wind velocity.

Exam Tip: Remember that in fluid dynamics, power is often a cubic function of velocity; this applies to both wind turbines and hydro-turbines.

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Option A: Square root power of two

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Option B: Square root power of three

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Option C: Square power

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