Q12PE - College Physics (Urone)

Question

The main uptake air duct of a forced air gas heater is 0.300 m in diameter. What is the average speed of air in the duct if it carries a volume equal to that of the house’s interior every 15 min? The inside volume of the house is equivalent to a rectangular solid 13.0 m wide by 20.0 m long by 2.75 m high.

Step-by-Step Solution

Answer

The average air speed in duct is \({\bf{10}}{\bf{.33}}\;{\bf{m/s}}\).

Step 1: Determination of area of air duct

Given Data:

The diameter of air duct is \(d = 0.3\;{\rm{m}}\)

The time to fill the room by gas is \(t = 15\;\min   = 900\,{\rm{s}}\)

The size of house is \(V = 13\;{\rm{m}} \times 20\;{\rm{m}} \times 2.75\;{\rm{m}}\)

The average speed of air in duct is found by dividing the volume of house by time and area of air duct.

The area of the air duct is given as:

\(A = \frac{\pi }{4}{d^2}\)

Substitute all the values in the above equation.

\(\begin{aligned}A &= \frac{\pi }{4}{\left( {0.3\;{\rm{m}}} \right)^2}\\A &= 0.071\;{{\rm{m}}^2}\end{aligned}\)

Step 2: Determination of average air speed in air duct

The average air speed in duct is given as:

\(v = \frac{V}{{A \cdot t}}\)

Substitute all the values in the above equation.

\(\begin{aligned}v &= \frac{{\left( {13\;{\rm{m}} \times 20\;{\rm{m}} \times 2.75\;{\rm{m}}} \right)}}{{\left( {0.071\;{{\rm{m}}^2}} \right)\left( {900\;{\rm{s}}} \right)}}\\v &= 10.33\;{\rm{m}}/{\rm{s}}\end{aligned}\)

Therefore, the average air speed in duct is \(10.33\;{\rm{m}}/{\rm{s}}\).