I'm trying to model a problem where a pump is being used to remove heat from a room at a rate of $R$. This pump uses an automated system that only activates when the outside temperature reaches some point $T_{outside}$. The pump then continues to operate so that the temperature inside the room $T_{inside}$ reaches a balance between the heat being conducted in (through walls) and the heat being pumped out.
I want to find the "steady/balance/equilibrium" temperature $T_{steady}$ which the pump can maintain for the room if a certain surface area of the room, say $A_{surface}$, is exposed to the outside.
The wall materials would have certain thermal conductivities, thickness, etc., and these would all have their own values.
How would I model this problem? What type of problem is this? What particular equations would I use? Is this modelled using partial differential equations? Reference to a specific textbook chapter or online document that deals with this specific type of problem would be greatly appreciated, so that I may study this. I've done my own research, but there's such a large amount of seemingly relevant material that it's difficult to be sure of exactly what I should be using, since I'm not familiar with the mathematics or physics.
In this case, would $T_{outside}$ and $T_{steady}$ be boundary conditions in the hypothetical model? It seems to me that this is reasonable?
I would greatly appreciate it if the modellers here could please take the time to provide information and clarity about this.
You want to balance the heat flows into and out of the room. The heat flow out is the capacity of the heat pump, which may depend on the temperature differential it is working against. The heat flow in is conduction through the walls. You can assess the R-value of each wall based on its construction. The heat flow through that wall is then the temperature difference across the wall times the wall area divided by the R value. If you have windows in the sun, you also need to compute the heat that comes in as light and is absorbed in the room.