# How do I correctly model a building to determine effectiveness of a heat source?

## The problem domain

You have a shelter that you use in a cold climate, and you want to provide it with some heat to take the edge off the chill.

You already have a structure in a location which gets cold winters, let's say in the -15C to -30C range. When it gets down to -30C you are used to turning on a heater, whose fire usually increases the temperature by approximately 10 degrees (to -20C).

You want to construct a similar structure - exactly like the first, or as close as possible - except this one will be in an even colder area near one of the planetary poles and getting winters that might reach temperatures as low as -80C. Similar to before, you plan to turn on the heater if you wake up to the lowest expected temperature of -80C.

I want to figure out how much the heater will warm the structure in the colder area. It very well may just heat the structure by 10 degrees as before up to -70C, but I do not know for sure.

## The question

This question does not seem to be as straightforward as I originally thought it would be, and there have been other comments and posts probing some specifics here and not getting me to a satisfactory answer, so the question is this...

What are all the non-negligible forces at play here which should be considered to decide if the secondary structure will be heated by the same amount (10 degrees) as the first structure, and in what way does each of these affect the result?

In case it matters: The building will likely be either a cold-weather tent or a human-made cave made from snow which is designed with the floor higher than the doorway to trap the warmer air inside. If the specific choice matters, feel free to either say so in comment or assume one or the other for your answer.

I think the primary concern might be the heat capacity (ie: the number of heat units needed to raise the temperature of a body by one degree) of air. See also my question Physics.SE: If two objects of different temperatures have the same heat source applied, do they heat up by the same amount? for the Q&A which led me to this (hopefully correct) assumption. One of the answers even seemed to suggest that the heat capacity change may be significant, assuming I read the complicated answer correctly (maybe not).

I tried to Google up whether the heat capacity of air changes with temperature. I found an interesting table labeled "Specific Heat Capacities of Air", though that does not go down far enough, and a calculator which does. I get 1.005 kJ/(kgK) for 0C and 1.009 kJ/(kgK) for -80C. In my naivety, I would guess that means the heat source might work 0.4% better at the colder temperature (making this tree I'm barking up a negligible one), but I would not be surprised if I am way off base.

• consider the materials used and the rate of heat transfer through them, surface area, quality of walls - do they leak air in / out etc – Solar Mike Jul 12 '18 at 19:43
• – Solar Mike Jul 12 '18 at 20:41

If we look at this as realistic experiment, the distribution of warmer air throughout the shelter is mainly due to convection of warm air from the walls and object near to the flow of warm air to colder surfaces and circulating back.

Convection in a space potentially partitioned by walls and openings and crowded by articles and furniture of deferent material and shape will be very complex. strata of layers of warm air with different temperature will create an union like geometry starting from warmer at center zones to colder at outer zones.

However one major component is the heat source. If it is going to be the same in both cases, meaning same surface temperature of heat element such as electrical heater or furnace or similar, because of the larger difference in heat source and ambient temperature, convection will happen in narrower, relatively warmer, and faster rising plums and the gradient of layers temperatures will be steeper. One would need to imagine a time laps image of the shelter's occupants movements, and usage of space density, then design the distribution of warm air efficiently to cover that usage density pattern closely.

Like others have mentioned, this can be complex depending on your conditions. Providing additional information helps us determine not only complicated things, but also determine if a simple method is satisfactory or not. I will provide some simple intuition on the topic.

Thermal conduction is a linear equation. If all of your heat transfer in this shelter was via conduction, your 10C temperature increase in both shelters would be correct.

Thermal radiation is an exponential equation that result in exponentially more heat being lost from a higher temperature source. In your case it will probably make no difference at a small 10C differential. (considering the 37C human in the shelter, it will mater a lot)

Thermal convection is more complicated and will result in exponentially more heat being lost from a higher temperature source. This will depend on the size and shape of the shelter, design of the heater (forced or natural convection), as well as the location of the human and how much heat they are giving off. Since it is not possible to calculate without significantly more information; your assumption of 10C increase in both locations is fine.

I'm not sure what your goal is in knowing this information, but if you plan to heat such a shelter I recommend just using the shelter to eliminate the outside wind (forced convection) and create a protected space. A temperature increase from -80C to -70C is kindof a waste of energy. It would be better to use this heat energy to heat the inside layer of an occupants sleeping bag or insulated clothing. This reduces the area the heat can be dissipated over, and the clothing sounds like it will have much better insulation properties than your structure. It is important to stay at least a few degrees below body temperature or sweating will create a new problem.