Question Bank • QB_FP_016 NAT • 2 Marks

Section 3: Farm Power

Sources of Power

Subtopic: Sources of power on the farm – human, animal, mechanical, electrical, wind, solar and biomass

Question Statement:

A farm wind turbine has a rotor blade diameter of $D = 8.0\text{ m}$. It operates in wind blowing at a steady velocity of $v = 7.5\text{ m/s}$. The ambient air density is $\rho = 1.20\text{ kg/m}^3$. According to the Betz limit (maximum theoretical aerodynamic power coefficient $C_{p,\max} = 16/27$), the maximum mechanical power that can be extracted from the wind by this turbine in $\text{kW}$ is ________ (round off to 2 decimal places). Take $\pi = 3.1416$.

Answer Key:

7.54

Step-by-Step Solution & Explanation

1. Rotor swept area ($A$): $A = \frac{\pi}{4} D^2 = \frac{3.1416}{4} (8.0)^2 = 50.2656\text{ m}^2$ 2. Total wind power flux ($P_{wind}$): $P_{wind} = \frac{1}{2} \rho A v^3 = 0.5 \times 1.20 \times 50.2656 \times (7.5)^3$ $(7.5)^3 = 421.875$ $P_{wind} = 0.60 \times 50.2656 \times 421.875 = 12,723.6\text{ W} = 12.7236\text{ kW}$ 3. Maximum extractable power by Betz limit: $P_{\max} = \frac{16}{27} \times P_{wind} = \frac{16}{27} \times 12.7236 \approx 7.5399\text{ kW} \approx 7.54\text{ kW}$
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