Question

At a certain location, wind is blowing steadily at \(10 \mathrm{m} / \mathrm{s} .\) Determine the mechanical energy of air per unit mass and the power generation potential of a wind turbine with 60 -m-diameter blades at that location. Take the air density to be \(1.25 \mathrm{kg} / \mathrm{m}^{3}\)

Short answer

Answer: The power generation potential of a wind turbine with 60-meter-diameter blades in this location is approximately 52.2 MW.

Step-by-step solution

Calculate the kinetic energy per unit mass of the wind

Kinetic energy per unit mass can be calculated using the formula: $$ \frac{KE}{m} = \frac{1}{2}v^2 $$ Where \(KE\) is the kinetic energy, \(m\) is the mass, and \(v\) is the velocity of the wind. In this case, the wind velocity is given as \(v = 10 \mathrm{m} / \mathrm{s}\). Now we can calculate the kinetic energy per unit mass: $$ \frac{KE}{m} = \frac{1}{2}(10 \mathrm{m} / \mathrm{s})^2 = \frac{1}{2}(100 \mathrm{m}^2 / \mathrm{s}^2) = 50 \mathrm{J} / \mathrm{kg} $$

Calculate the power generation potential of the wind turbine

We are given the diameter of the wind turbine blades as 60 meters. In order to determine its power generation potential, we will first calculate the area swept by the blades, which is given by this formula: $$ A = \pi r^2 $$ Where A is the area and r is the radius of the circle. In this case, radius \(r\) is half the diameter, \(r = 30 \mathrm{m}\). Now we can calculate the area: $$ A = \pi (30 \mathrm{m})^2 = 900 \pi \mathrm{m}^2 $$ Now we can use the Betz limit to determine the maximum power that can be extracted from the wind, which is approximately 59.3%. Therefore, the potential power generation of a wind turbine can be calculated using this formula: $$ P = 0.593 \times \frac{1}{2} \rho A v^3 $$ Where P is the power, \(\rho\) is the air density, A is the area swept, and \(v\) is the wind velocity. In this case, the air density is given as \(\rho = 1.25 \mathrm{kg} / \mathrm{m}^3\). Therefore, we can plug all the numbers into the formula: $$ P = 0.593 \times \frac{1}{2} \cdot 1.25 \mathrm{kg} / \mathrm{m}^3 \cdot 900 \pi \mathrm{m}^2 \cdot (10 \mathrm{m} / \mathrm{s})^3 $$ $$ P = 0.593 \times \frac{1}{2} \cdot 1.25 \mathrm{kg} / \mathrm{m}^3 \cdot 900 \pi \mathrm{m}^2 \cdot 1000 \mathrm{m}^3 / \mathrm{s}^3 $$ $$ P = 0.593 \times 5625000 \pi \mathrm{W} $$ $$ P \approx 16637500 \pi \mathrm{W} $$ $$ P \approx 52213546 \mathrm{W} $$ So, the power generation potential of a wind turbine with 60-meter-diameter blades in this location is approximately 52.2 MW.

Key concepts

Kinetic Energy of Wind

Wind turbines harness the kinetic energy in wind and convert it into electrical energy. The kinetic energy of the wind is essentially the energy of motion held by the flowing air. For any given mass of air, this kinetic energy is quantified by the equation \[ \frac{KE}{m} = \frac{1}{2} v^2 \]where \(v\) is the wind velocity, and the expression \(\frac{KE}{m}\) represents the kinetic energy per unit mass of air.An increased wind velocity will result in a substantial increase in kinetic energy since the energy is proportional to the square of the velocity. Understanding how this energy translates into usable power is fundamental in designing efficient wind turbines. The example given in the exercise, with a wind speed of \(10 \mathrm{m/s}\), illustrates a practical scenario where this kinetic energy can be harnessed by a wind turbine to generate electricity.

Betz Limit

The Betz limit is a theoretical maximum that dictates how much kinetic energy in wind can be converted into mechanical power by a wind turbine. Named after the German physicist Albert Betz, who derived it in 1919, this limit is approximately 59.3% of the total kinetic energy available in the wind.The Betz limit recognizes that not all the wind's energy can be captured—some energy must remain in the wind to allow it to move away from the turbine, making room for more wind to come through. This concept is essential when calculating the power generation potential of a wind turbine, as it defines an upper boundary on efficiency. However, real-world turbines often operate at efficiencies lower than the Betz limit due to practical design considerations, losses in the turbine mechanism, and non-ideal air flow conditions.

Air Density

Air density plays a crucial role in determining the power generation of wind turbines. It is a measure of how much mass of air is contained within a specific volume and is influenced by factors such as temperature, altitude, and humidity. The formula for power generated by a wind turbine includes air density (\(\rho\)) because the amount of kinetic energy in the wind depends on how much mass is moving. The higher the air density, the more kinetic energy the wind will have and, consequently, the more power a turbine can generate from the same wind speed. In the given exercise, the air density is used as \(1.25 \mathrm{kg/m}^3\) to calculate the potential power output of the turbine. It is essential to use accurate air density values based on the local environmental conditions for precise calculations.

Swept Area Calculation

The swept area is the region through which the turbine blades move and intercept the wind, effectively acting as the 'collection area' for the wind's kinetic energy. The larger the swept area, the more wind the turbine can capture, leading to a greater potential power output.To calculate this area for a circular path of the rotor blades, the formula \[ A = \pi r^2 \]is applied, where \(A\) is the swept area, and \(r\) is the radius of the circle (half the diameter of the turbine rotor). For the exercise scenario with a turbine blade diameter of 60 meters, the swept area calculation is crucial for determining the amount of wind energy that can be harnessed and underlines the significance of blade size in turbine design. By selecting an appropriate diameter for turbine blades, energy production can be maximized for the prevailing wind conditions at a specific location.