Question

Large wind turbines with blade span diameters of over \(100 \mathrm{m}\) 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 -hour period, determine the amount of electric energy and the revenue generated per day for a unit price of \(\$ 0.09 / \mathrm{kWh}\) for electricity.

Short answer

Answer: The wind turbine produces 24.576 MWh of electric energy in a 24-hour period and generates a revenue of 2211.84 USD.

Step-by-step solution

Calculate the area swept by the wind turbine's blades

First, we need to find the area of the circle formed by the sweeping motion of the blades. The area can be calculated using the formula \(A=\pi \cdot R^{2}\), where \(A\) is the area and \(R\) is the radius of the circle. The radius is half of the blade span diameter, which is given as \(100 \mathrm{m}\), so \(R = 50 \mathrm{m}\). \(A=\pi \cdot (50 \mathrm{m})^{2}\) \(A=7850 \mathrm{m}^{2}\) So, the area swept by the wind turbine's blades is \(7850 \mathrm{m}^{2}\).

Determine the power available in the wind

The available wind power can be calculated using the formula \(P_{wind}=\frac{1}{2} \cdot \rho \cdot A \cdot v^{3}\), where \(P_{wind}\) is the available wind power, \(\rho\) is the air density, \(A\) is the swept area, and \(v\) is the wind speed. We are given the air density \(\rho = 1.25 \mathrm{kg} / \mathrm{m}^{3}\) and wind speed \(v = 8 \mathrm{m} / \mathrm{s}\). \(P_{wind}=\frac{1}{2} \cdot 1.25 \mathrm{kg} / \mathrm{m}^{3} \cdot 7850 \mathrm{m}^{2} \cdot (8 \mathrm{m} / \mathrm{s})^{3}\) \(P_{wind}=3200000 \mathrm{W}\) So, the power available in the wind is \(3200000 \mathrm{W}\) (or \(3.2 \mathrm{MW}\)).

Calculate the electric power generated by the wind turbine

Now we can find the electric power generated by the wind turbine by factoring in the overall efficiency, which is given as 32 percent. \(P_{generated} = P_{wind} \cdot \text{efficiency}\) \(P_{generated} = 3200000 \mathrm{W} \cdot 0.32\) \(P_{generated} = 1024000 \mathrm{W}\) The electric power generated by the wind turbine is \(1024000 \mathrm{W}\) (or \(1.024 \mathrm{MW}\)).

Determine the total energy produced during a 24-hour period

To find the total energy produced in a 24-hour period, we need to multiply the power generated by the wind turbine by the number of hours in a day. \(E = P_{generated} \cdot \text{time}\) \(E = 1024000 \mathrm{W} \cdot 24 \mathrm{h}\) \(E = 24576000 \mathrm{Wh}\) The total energy produced during a 24-hour period is \(24576000 \mathrm{Wh}\) (or \(24.576 \mathrm{MWh}\)).

Calculate the revenue generated

Finally, we need to calculate the revenue generated per day at a unit price of \(0.09 \mathrm{USD} / \mathrm{kWh}\) for electricity. \(R = E \cdot \text{unit price}\) \(R = 24576 \mathrm{kWh} \cdot \)(0.09)"USD / kWh"$ \(R = 2211.84\mathrm{USD}\) The revenue generated per day is \(2211.84\mathrm{USD}\).

Key concepts

Wind Energy

Wind energy harnesses the power of the wind to generate electricity, making it a key player in sustainable development initiatives. At the heart of wind energy technology are wind turbines, which convert the kinetic energy of wind into electrical energy. This transformation is achieved through the turning of turbine blades connected to a generator. The blade span, or diameter, of a turbine heavily influences its capacity to capture wind. Larger turbines with spans exceeding 100 meters, like the one in the exercise, are designed to maximize energy capture from the wind.

Wind speed is another critical factor. The kinetic energy in wind is proportional to the cube of the wind speed, meaning small increases in wind speed translate into greatly increased energy potential. Therefore, sites with steady winds, such as the one described in the exercise with a constant speed of 8 m/s, are ideal for efficient wind turbine operation. Understanding this relationship is crucial as it guides placement and design of wind farms to optimize for the most productive wind conditions.

Renewable Energy Efficiency

Renewable energy efficiency is a measure of how well energy sources like wind, solar, and hydro are converted into usable power. The overall efficiency of a wind turbine, highlighted in the exercise as 32%, is a reflection of how much of the wind's kinetic energy can be transformed into electrical energy.

This efficiency takes into account various losses that occur during energy conversion, including aerodynamic losses from the blades, electrical losses from the generator and gearbox, and other system losses. Students should be aware that renewable energy systems often have a range of efficiencies, influenced by design, technology, and environmental factors. Consequently, while renewable energy is a vital component in the quest for cleaner power, maximizing the efficiency of these systems is essential for sustainable growth.

Electric Power Generation

Electric power generation refers to the process of producing electrical energy from various energy sources, among which wind energy has become increasingly prominent. The calculation process laid out in the exercise includes determining the area swept by the turbine's blades to calculate the available wind power, and then applying the turbine's efficiency to find the actual generated electric power.

Understanding Power and Energy

Power, measured in watts (W), is the rate at which energy is generated or consumed. Energy, measured in watt-hours (Wh), is the total amount of power produced or used over a period of time. The step-by-step solution in the exercise provides a clear methodology for computing both the instantaneous electric power output of a wind turbine and the total electric energy generated over a day, demonstrating how the reliable wind speeds at the site translate to predictable, renewable electric power generation.

Air Density Effects on Wind Power

Air density plays a significant role in the calculation of wind power, as it directly impacts the amount of kinetic energy available in the wind. Given a particular wind speed, denser air contains more mass and hence more energy. In the exercise, air density is factored in at 1.25 kg/m³, which is a typical value at sea level under standard atmospheric conditions.

The fluctuation of air density with altitude, temperature, and humidity can influence wind power output. For instance, turbines at higher altitudes or in warmer climates face less dense air, potentially reducing power generation efficiency. Students must be aware of local atmospheric conditions when calculating wind power, as the precision of these calculations affects energy predictions and potential revenue, as shown in the exercise's financial assessment of energy generated over a 24-hour period.