Question

(I)A \(12\;{\rm{cm}}\) radius air duct is used to replenish the air of a room \(8.2\;{\rm{m}} \times 5.0\;{\rm{m}} \times 3.5\;{\rm{m}}\) every \(12\;{\rm{min}}\). How fast does the air flow in the duct?

Short answer

The velocity of the air flow in the duct is \(4.4\;{\rm{m/s}}\).

Step-by-step solution

Concept of volumetric flow rate

Volumetric flow rate is the rate of fluid flow, where the volume of fluid flowing in a unit time is measured. It can be used in various flow measuring devices such as nozzles, siphons, orifices, etc.

Given data

The radius of air duct is \(r = 12\;{\rm{cm}}\).

The capacity of room is \(l \times b \times h = 8.2\;{\rm{m}} \times 5.0\;{\rm{m}} \times 3.5\;{\rm{m}}\).

The time to fill air in the room is \(t = 12{\rm{ min}}\).

Calculation of velocity

Air flow rate inside the duct must be equal to the rate at which air goes out.

\(\begin{array}{c}Q = \frac{V}{t}\\Av = \frac{V}{t}\\{\rm{\pi }}{r^2}v = \frac{{lbh}}{t}\\v = \frac{{lbh}}{{{\rm{\pi }}{r^2}t}}\end{array}\)

Here,\(V\)is the volume of the room,\(v\)is the velocity of the air,\(A\)is area of the air duct and\(Q\)is the air flow rate.

Substitute the values in the above equation to find the velocity of air.

\(\begin{array}{c}v = \left( {\frac{{8.2\;{\rm{m}} \times 5.0\;{\rm{m}} \times 3.5\;{\rm{m}}}}{{\pi {{\left( {12\;{\rm{cm}} \times \frac{{1\;{\rm{m}}}}{{100\;{\rm{cm}}}}} \right)}^2}12{\rm{ min}}\left( {\frac{{60\;{\rm{s}}}}{{1\;{\rm{min}}}}} \right)}}} \right)\\ = 4.4\;{\rm{m/s}}\end{array}\)

Hence, the velocity of air is \(4.4\;{\rm{m/s}}\).