Question

Consider the flow of air through a wind turbine whose blades sweep an area of diameter \(D\) (in \(\mathrm{m}\) ). The average air velocity through the swept area is \(V\) (in \(\mathrm{m} / \mathrm{s}\) ). On the bases of the units of the quantities involved, show that the mass flow rate of air (in \(\mathrm{kg} / \mathrm{s}\) ) through the swept area is proportional to air density, the wind velocity, and the square of the diameter of the swept area.

Short answer

Answer: The mass flow rate of air through a wind turbine is proportional to the air density, wind velocity, and the square of the diameter of the swept area.

Step-by-step solution

Identify the given variables and their units

The variables given are: 1. Diameter of the swept area: \(D\) (in meters, m) 2. Wind velocity: \(V\) (in meters per second, m/s) 3. Air density: \(\rho\) (in kilograms per cubic meter, kg/m³) We need to find the proportionality of the mass flow rate of air \(Q\) (in kg/s) with respect to these variables.

Calculate the area of the wind turbine

In order to find the mass flow rate, we need to first calculate the area of the wind turbine where the air is flowing through. The area can be calculated as: \(A=\frac{\pi}{4}D^{2}\) Where \(A\) is the area of the swept area by the wind turbine and \(D\) is the diameter of the swept area.

Calculate the mass flow rate

The mass flow rate can be defined as the product of the air density, wind velocity, and the area of the wind turbine. Mathematically, it can be represented as: \(Q=\rho VA\) Now, we substitute the area formula from Step 2: \(Q = \rho V\left(\frac{\pi}{4}D^{2}\right)\)

Analyze the units for mass flow rate

Let's analyze the units of each term in the equation for mass flow rate: - Air density (\(\rho\)): kg/m³ - Wind velocity (\(V\)): m/s - Swept area (\(A\)): \(D^2\) has units of m² Now, multiply the units together: kg/m³ * m/s * m² = kg/s Since the mass flow rate (\(Q\)) has the same unit as the product of the air density, wind velocity, and the square of the diameter of the swept area (kg/s), we can conclude that: \(Q \propto \rho VD^{2}\) Thus, we have shown that the mass flow rate of air through the swept area is proportional to the air density, wind velocity, and the square of the diameter of the swept area.

Key concepts

Understanding Air Density

Air density, symbolized as \( \rho \) (rho), plays a crucial role in calculating the mass flow rate of air through a wind turbine. It's a measurement of how compact or concentrated the mass of air is within a certain volume and is typically expressed in kilograms per cubic meter (kg/m³). The higher the air density, the more mass of air is available to move through the turbine's swept area per unit of time.

Factors affecting air density include temperature, altitude, and humidity. Cooler air is denser than warmer air, higher altitudes have less dense air due to reduced atmospheric pressure, and higher humidity levels can decrease air density since water vapor is lighter than dry air. Understanding these variations is key because they directly impact the mass flow rate of air, affecting the potential energy a wind turbine can harness.

The Significance of Wind Velocity

Wind velocity, denoted as \( V \) and measured in meters per second (m/s), represents the speed at which air moves through the swept area of the wind turbine. This factor is fundamental in determining the mass flow rate (\( Q \) in kg/s) because it affects how much air passes through the turbine at any given moment.

The equation \( Q = \rho VA \) clearly indicates that as wind speed increases, assuming air density and the area remain constant, the mass of air passing through the turbine per second will also increase. Therefore, monitoring wind velocity is not just important for predicting the performance of a wind turbine but also crucial for optimizing its design and placement to maximize energy production.

Swept Area Diameter Impact

The swept area diameter, indicated by the variable \( D \) and measured in meters (m), describes the width of the circular area covered by the spinning blades of the wind turbine. As the formula for calculating the swept area (\( A = \frac{\pi}{4}D^{2} \) reveals, the area is directly proportional to the square of the diameter. This means that if the diameter doubles, the swept area quadruples, significantly increasing the potential for capturing wind energy.

A larger diameter enables the turbine to intersect more of the air's kinetic energy, which can be converted into electricity. However, it's worth noting that while larger swept areas can capture more wind energy, they also require stronger materials and more engineering considerations to withstand the increased forces involved.