In order to survive brutal Colgate winter, heaters are essential. My roommate and I crank up the heater these days, because the temperature drops very low at night. I wanted to figure out how long it took the heater to replenish air in my room if the flow rates were equal. The density of the air remained constant. I assumed that hot air traveled at a speed of 4 m/s through the heater, which had an area of 0.15 m2. I assumed my room to be about 400 m3.
The equation of continuity: A1v1 = A2v2
v2 = l2t → A2v2 = A2 l2t = V2t
The volume flow rate in the room equals the volume of the room divided by the replenishing time.
A1v1 = A2v2 = V2/t
t = V2 / A1v1
t = 400 m3 / (0.15 m2)(4 m/s)
t = 666 seconds = 11.1 minutes
So the heater replenishes the air in the room every 11.1 minutes. This is why my roommate and I are able to sleep toasty warm each night.