Problem 76 - College Physics

Question

\(\bullet\) \(\bullet\) At the site of a wind farm in North Dakota, the average wind speed is \(9.3 \mathrm{m} / \mathrm{s},\) and the average density of air is 1.2 \(\mathrm{kg} / \mathrm{m}^{3} .\) (a) Calculate how much kinetic energy the wind contains, per cubic meter, at this location. (b) No wind turbine can capture all of the energy contained in the wind, the main reason being that capturing all the energy would require stop- ping the wind completely, meaning that air would stop flowing through the turbine. Suppose a particular turbine has blades with a radius of 41 \(\mathrm{m}\) and is able to capture 35\(\%\) of the avail- able wind energy. What would be the power output of this tur- bine, under average wind conditions?

Step-by-Step Solution

Answer

(a) 51.894 J, (b) 895 kW.

Step 1: Calculating Kinetic Energy per Cubic Meter

To find the kinetic energy of the wind per cubic meter, we use the formula for kinetic energy, which is \( KE = \frac{1}{2}mv^2 \). Here, the mass per cubic meter, \( m \), is the air density, \( \rho = 1.2 \, \text{kg/m}^3 \), and the velocity \( v = 9.3 \, \text{m/s} \). Plug these values into the formula:\[ KE = \frac{1}{2} \times 1.2 \, \text{kg/m}^3 \times (9.3 \, \text{m/s})^2 \].First, calculate the squared velocity: \( (9.3 \, \text{m/s})^2 = 86.49 \, \text{m}^2/\text{s}^2 \).Then, substitute the values into the equation:\[ KE = \frac{1}{2} \times 1.2 \times 86.49 = 51.894 \, \text{J/m}^3 \].Thus, the kinetic energy per cubic meter is 51.894 J.

Step 2: Calculate Turbine Swept Area

The turbine swept area is given by the formula for the area of a circle: \( A = \pi r^2 \). Here, the radius \( r = 41 \, \text{m} \). Substituting into the formula gives:\[ A = \pi \times (41)^2 \ = 5281.02 \, \text{m}^2 \].This is the area through which the wind passes as it crosses the turbine blades.

Step 3: Calculate Wind Power Through the Swept Area

Wind power through the swept area is calculated as:\[ P_{wind} = \frac{1}{2} \cdot \rho \cdot A \cdot v^3 \].Substituting the known values: \( \rho = 1.2 \, \text{kg/m}^3 \), \( A = 5281.02 \, \text{m}^2 \), and \( v = 9.3 \, \text{m/s} \), we calculate:First, compute \( v^3 \):\( (9.3)^3 = 804.357 \, \text{m}^3/\text{s}^3 \).Then, substitute into the power formula:\[ P_{wind} = \frac{1}{2} \times 1.2 \times 5281.02 \times 804.357 \approx 2556071.76 \, W \].So, the total energy available in the wind is approximately 2556071.76 watts.

Step 4: Calculate Power Output of the Turbine

The turbine captures 35\% of the available wind energy. The actual power output \( P_{turbine} \) is given by:\[ P_{turbine} = 0.35 \times P_{wind} \].Substituting the value from Step 3:\[ P_{turbine} = 0.35 \times 2556071.76 \approx 894625.11 \, \text{W} \].Therefore, the power output of the turbine under average wind conditions is approximately 894625.11 watts or about 895 kW.

Key Concepts

Wind Turbines

Wind turbines are fascinating structures that convert the invisible power of wind into useful electricity. They work by using large blades to capture the moving air, much like how a windmill harnesses wind to pump water. When wind flows over these blades, it causes them to spin, which then turns a generator to produce electricity.

These structures can range from small wind turbines, suitable for individual homes, to massive offshore turbines that stand hundreds of feet tall and power entire communities. Turbines are usually placed in areas where wind is strong and consistent, such as on hilltops, open plains, or offshore, to maximize their efficiency.

Power Output Calculation

Calculating the power output of a wind turbine involves several steps, but it's crucial to understand the potential energy it can harness from the wind. First, it's important to calculate the turbine's swept area, which for a turbine with blades creates a circular area where wind can be captured.

Once this area is known, the wind power potential through this area can be calculated using the formula: \[ P_{wind} = \frac{1}{2} \cdot \rho \cdot A \cdot v^3 \] where \( \rho \) is the air density, \( A \) is the swept area, and \( v \) is the wind speed. The most challenging part of such a calculation is understanding how these different factors interact to influence the power a wind turbine could potentially generate.

Air Density

Air density is a critical factor in calculating the wind energy that a turbine can capture. It refers to how much mass is contained in a given volume of air and is usually measured in kilograms per cubic meter (kg/m³). Higher air densities mean more mass, and thus more energy potential, is available per cubic meter of airflow.

Air density can be affected by several factors, including temperature, altitude, and atmospheric pressure. Colder air is denser than warmer air, which means locations with cooler climates or higher altitudes can potentially provide more energy-efficient wind conditions for turbines.

Wind Speed

Wind speed is a key factor in determining the kinetic energy that a wind turbine can capture. The energy available from the wind increases cubically with wind speed, which means even a small increase in speed can result in a large increase in potential energy.

Turbines are usually designed to operate optimally at certain wind speeds, and that's why wind farms are located in places where these optimum speeds are often met. It's also important to note that too much wind speed can be damaging, which is why turbines have mechanisms to slow down or stop operating in too-powerful winds to avoid damage.

Energy Capture Efficiency

Energy capture efficiency is a measure of how much of the wind's physical energy a turbine can convert into usable electrical energy. No turbine can capture 100% of the wind energy due to inevitable losses and the necessity of maintaining airflow through the turbine.

Generally, the efficiency of a turbine is expressed as a percentage. In the case studied, the turbine was able to capture 35% of the wind energy that flows through it. Increasing this efficiency is a significant area of research, as enhancing a turbine's design can help convert more wind power into electricity, making wind energy a more viable alternative to traditional, non-renewable sources.