Large wind turbines with a power capacity of 8 MW and blade span diameters of over 160 m are av

step 1 Our question is question number 70 of chapter number two. A large wind turbine is installed at a

Large wind turbines with a power capacity of 8 MW and blade...

Large wind turbines with a power capacity of $8 \mathrm{MW}$ and blade span diameters of over $160 \mathrm{~m}$ are available for electric power generation. Consider a wind turbine with a blade span diameter of $100 \mathrm{~m}$ installed at a site subjected to steady winds at $8 \mathrm{~m} / \mathrm{s}$. Taking the overall efficiency of the wind turbine to be 32 percent and the air density to be $1.25 \mathrm{~kg} / \mathrm{m}^{3},$ determine the electric power generated by this wind turbine. Also, assuming steady winds of $8 \mathrm{~m} / \mathrm{s}$ during a 24 -h period, determine the amount of electric energy and the revenue generated per day for a unit price of $\$ 0.09 / \mathrm{kWh}$ for electricity.

Key Concepts

Turbine Efficiency

Turbine efficiency, often referred to as the power conversion efficiency, is the ratio of the electrical power output to the wind power available. It accounts for real-world losses due to aerodynamic, mechanical, and electrical inefficiencies, and is critical for determining the actual energy production of the turbine.

Wind Power Equation

This is the mathematical relationship defined by P = ½ ? A v³, where P is the power available from the wind, ? is the air density, A is the swept area, and v is the wind speed. This equation is used to estimate the theoretical maximum power that a wind turbine can extract from the wind before considering efficiency losses.

Energy Production and Revenue Calculation

This involves extending the power output over a specific time period to calculate total energy produced, typically converting the power (in watts) to energy (in kilowatt-hours). Once the total energy is determined, multiplying by the unit price of electricity provides the revenue generated from the energy production.

Kinetic Energy in Wind

This concept is about the inherent energy contained in moving air. The kinetic energy per unit mass of wind is proportional to the square of its speed, and the power available in the wind that passes through an area is dependent on the cube of the wind speed. Understanding this is foundational to calculating how much energy can be harnessed by a wind turbine.

Wind Turbine Swept Area

The swept area of a wind turbine, which is the area covered by the rotating blades (calculated as ?r²), determines the volume of air that interacts with the turbine. This area directly influences the amount of wind energy that can be intercepted and converted into mechanical energy.

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