Heat It Up

The heating ducts to our rooms not only keep us warm in the dead of winter, but also teach us important physics! A typical double room in Bryan (my room) has a volume of roughly 25 m² x 2.5 m = 62.5 m³, and length, y₂, of 4 m. The area of the heating duct on the wall is approximated to be 10 (cm) x 25 (cm) = 250 cm². A question that concerns a Physics student who has just learned the equation of continuity is that if the air is moving from the duct to the room at about 1.0 m/s, how long does it take to replenish the air in the room?

To make the question easier, we assume that the density of air is constant. Therefore, we can apply the equation of continuity. As shown in the graph, the air flows through the duct and then into the room. Suppose the replenishing time, t, equals x seconds, then applying the equation of continuity, we obtain that

Therefore, it takes about 42 min to replenish the air in the room.