# Temperature in a Room

#### ThePerfectHacker

MHF Hall of Fame
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.

#### CaptainBlack

MHF Hall of Fame
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

#### topsquark

Forum Staff
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. -Dan

#### Monoxdifly

And a spherical room would be much simpler to work with. -Dan
How is that so? Why? Or was it a sarcasm?

#### topsquark

Forum Staff
How is that so? Why? Or was it a sarcasm?
I frankly don't recall. On the other hand since heat radiates outward from a point a spherical room would be simpler to model.

Odds are I was picking on TPH. There's an old joke about a Mathematician, and Engineer, and a Physicist at the horse races. They were tasked to find out which horse would win and why. I don't recall most of it but the Physicist started off with "Let's assume a spherical horse..." (Spheres are nice to work with but often don't show up in Nature.)

-Dan

#### Monoxdifly

There's an old joke about a Mathematician, and Engineer, and a Physicist at the horse races. They were tasked to find out which horse would win and why. I don't recall most of it but the Physicist started off with "Let's assume a spherical horse..." (Spheres are nice to work with but often don't show up in Nature.)
If I recall correctly, it was the mathematician who said that.

#### topsquark

Forum Staff
If I recall correctly, it was the mathematician who said that.
Really? That sounds so much like a Physicst.

-Dan