Say I have a rectangular room (so its volume is easy to find) and an air conditioner that cools this room. I know the initial state of the room without the air conditioner and cool this room eventually to the same temperature as the air conditioner. I turn off the air conditioner. How much time does it take for the room to heat up to its initial state?

My idea: Maybe we can use Newton's Law of Cooling (or Heating) here somehow? But I think the volume has something to do with the solution to this problem.

You would normaly assume that the rate of change of temprature with the

air conditioner turned off is:

\(\displaystyle

\frac{dT_{room}}{dt}=k(T_{ambient}-T_{room})

\)

and (oversimplifying) that when the airconditioner is on that:

\(\displaystyle

\frac{dT_{room}}{dt}=k(T_{ambient}-T_{room})-K

\)

The essential assumption is that the air conditioner removes heat at a constant rate,

while heat flows into the room at a rate proportional to the temprature difference.

The problem is that the thermal capacity and heat loss constant for a room

are difficult things to calculate. There are tools to do so but I doubt you

will want to look for them.

RonL