A 12-cm-radius air duct is used to replenish the air of a room 8.2 m x 4.5 m x 3.5 m every 12 min. How fast does the air flow in the duct?

Verified step by step guidance

1. Calculate the volume of the room using the formula for the volume of a rectangular prism: \( V = l \cdot w \cdot h \), where \( l = 8.2 \ \text{m} \), \( w = 4.5 \ \text{m} \), and \( h = 3.5 \ \text{m} \).
2. Determine the air flow rate required to replenish the room's air every 12 minutes. Convert 12 minutes to seconds (\( t = 12 \cdot 60 \ \text{s} \)) and use the formula \( Q = \frac{V}{t} \), where \( Q \) is the volumetric flow rate.
3. The air duct is cylindrical, so calculate the cross-sectional area of the duct using the formula for the area of a circle: \( A = \pi r^2 \), where \( r = 12 \ \text{cm} = 0.12 \ \text{m} \).
4. Relate the volumetric flow rate \( Q \) to the air velocity \( v \) in the duct using the formula \( Q = A \cdot v \). Rearrange this equation to solve for \( v \): \( v = \frac{Q}{A} \).
5. Substitute the values for \( Q \) and \( A \) into the equation for \( v \) to find the air velocity in the duct. Ensure all units are consistent (e.g., meters, seconds).

Key Concepts

Airflow Rate

Airflow rate is the volume of air that moves through a duct per unit of time, typically measured in cubic meters per second (m³/s). To find the airflow rate, one can divide the total volume of air exchanged by the time taken for that exchange. In this case, the volume of the room must be calculated first to determine how much air is replenished every 12 minutes.

Cross-sectional Area of the Duct

The cross-sectional area of a duct is the area of the circular opening through which air flows. It can be calculated using the formula A = πr², where r is the radius of the duct. For a 12-cm radius duct, this area is crucial for determining the velocity of the air flowing through the duct, as it relates the volume flow rate to the speed of the air.

Velocity of Airflow

The velocity of airflow is the speed at which air moves through the duct, typically expressed in meters per second (m/s). It can be calculated by dividing the airflow rate by the cross-sectional area of the duct. Understanding this concept is essential for determining how quickly air is replenished in the room, which is a key part of the problem.