# Temperature in a Room

• Sep 11th 2007, 07:12 PM
ThePerfectHacker
Temperature in a Room
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.
• Sep 11th 2007, 10:29 PM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
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
• Sep 12th 2007, 05:42 AM
topsquark
Quote:

Originally Posted by ThePerfectHacker
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.

And a spherical room would be much simpler to work with. :p

-Dan