MP Board · Class 10 · Science · Sources of EnergyA wind energy farm consists of 10 wind turbines, each of blade sweep area $100\text{ m}^2$. If the wind is blowing at a constant velocity of $10\text{ m/s}$ and the density of air is $1.2\text{ kg/m}^3$, calculate: The kinetic energy of air passing through one turbine per second. The total electrical power generated by the wind farm, assuming the turbines convert $25\%$ of the wind's kinetic energy into electrical energy.

Step-by-Step Numerical Solution:

Given Data:

• Number of turbines ($N$) = $10$
• Area of blade sweep for one turbine ($A$) = $100\text{ m}^2$
• Velocity of wind ($v$) = $10\text{ m/s}$
• Density of air ($\rho$) = $1.2\text{ kg/m}^3$
• Efficiency of conversion ($\eta$) = $25% = 0.25$

Part 1: Kinetic Energy of air passing through one turbine per second

\nThe mass of air ($m$) passing through the blade sweep area per second is given by the formula: $$m = \frac{\text{Volume}}{\text{Time}} \times \text{Density} = A \times v \times \rho$$ \nSubstituting the given values: $$m = 100\text{ m}^2 \times 10\text{ m/s} \times 1.2\text{ kg/m}^3$$ $$m = 1200\text{ kg/s}$$ (This means 1200 kg of air passes through the turbine every second.) \nNow, the kinetic energy ($KE$) of this mass of air per second (which represents power of wind, $P_{\text{wind}}$) is calculated using the formula: $$P_{\text{wind}} = \frac{1}{2} m v^2$$ \nSubstitute the values of $m$ and $v$: $$P_{\text{wind}} = \frac{1}{2} \times 1200\text{ kg/s} \times (10\text{ m/s})^2$$ $$P_{\text{wind}} = 600 \times 100$$ $$P_{\text{wind}} = 60,000\text{ Watts} = 60\text{ kW}$$<br> Answer 1: The kinetic energy of air passing through one turbine per second is $60,000\text{ Joules}$ (or power is $60\text{ kW}$).

Part 2: Total electrical power generated by the wind farm

\nFirst, let's find the electrical power generated by one turbine: $$\text{Power of one turbine} = \eta \times P_{\text{wind}}$$ $$\text{Power of one turbine} = 0.25 \times 60,000\text{ W}$$ $$\text{Power of one turbine} = 15,000\text{ W} = 15\text{ kW}$$ \nSince the wind farm consists of 10 identical turbines, the total electrical power generated by the farm ($P_{\text{total}}$) is: $$P_{\text{total}} = 10 \times \text{Power of one turbine}$$ $$P_{\text{total}} = 10 \times 15\text{ kW} = 150\text{ kW}$$<br> Answer 2: The total electrical power generated by the wind farm is $150\text{ kW}$ (or $150,000\text{ Watts}$).