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For economic purposes, I am interested in studying the equilibrium of heat in a typical rectangular room from various sources, such as a classic radiator or vents.

If I have a radiator sitting next to a wall in just a single box room, which on the other size is exposed to frigid temperatures, how would I calculate the equilibrium temperature? I've heard of Newton's laws of cooling, I don't know if those are still relevant anymore or if they work for box-like structures.

I imagine for extending this to other rooms that this somewhat resembles constructing a matrix for a heat plate, or at least, a matrix for differential equations instead.

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    $\begingroup$ Hi PhiEarl, welcome to Engineering.SE. Your question is too broad to be answered here, you can start with Cengel's Heat Transfer: A Practical Approach on the various factors considered in heat transfer calculations and different modes, and may be come back with a specific problem we can assist you with. $\endgroup$
    – Algo
    Jan 5 at 9:08
  • $\begingroup$ I think you find find the question is specific if you might consider that it is only a box room with typical air at standard pressure at either 32F or 0C, and nothing more. If your site advertises that this is for students but then can't answer a basic introductory question, that is very misleading. I also find it suspicious that you would recommend a book that is over $100. This is a ".com" site, and while I am not necessarily making assumptions, the conflict of interest seems obvious here in claiming to be for information yet deceptively choosing to capitalize on it. $\endgroup$
    – PhiEarl
    Jan 5 at 10:53
  • $\begingroup$ It is also important to note that this is a question based on industry experience as well. If you are in fact qualified to give an assessment, then you should be able to address this question rather easily by drawing upon your vast experience in referencing what are merely typical calculations for room efficiency in one of the easiest possible scenarios of your profession. $\endgroup$
    – PhiEarl
    Jan 5 at 11:00
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  • Thermal equilibrium will be reached when heat in = heat lost.
  • Find the U-value for each surface and calculate the heat loss rate in W/K (watts/kelvin) for each surface.
  • Add them up and you've got the heat loss rate for the whole room.
  • From this you can calculate the ΔT between inside and outside at equilibrium.
  • For a give outside temperature you can now calculate the room's equilibrium temperature.
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  • $\begingroup$ It's not a bad summary in the right context, the only thing missing now is the context. I am guessing you are referencing Newton's laws of cooling in some way maybe, the differential equation associated with it. Why don't you need 3 equations for 3 dimensions? Or a PDE for both spatial and temporal dimensions? $\endgroup$
    – PhiEarl
    Jan 5 at 22:36
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Room heat loss/gain are almost always calculated via empirical methods. In the US, ASHRAE guides provide the standard methodologies, although there are other sources. I'm sure you could find an ASHRAE manual either via ebay or a library. HVAC design contractors use purpose-built software. Other than that, this is a classical heat transfer problem and would be covered by almost any introduction to heat transfer college textbook. You could make it as complicated or as simple as you like based on how hard you want to work the problem.

Do you really need to have different heat transfer rates for the wall next to the radiator and the wall away from it? If so, you will need to probably integrate the heat transfer equations, and combine them with material heat capacities and conduction properties. These types of analyses typically include total heat loads on the room including air infiltration. I suggest that the accuracy of the equations and insulation values are not as accurate as use of multiple equations and matrices would warrant. These are empirical values, not to be used for precision work.

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