Heat rises, but how much? Say I have a a room of a certain dimension and uniform temperature. How can I calculate what the temperature difference will he between the floor and the ceiling once it has reached a stable point?
I think this is going to vary a lot by specifics. Even assuming no air flow and exterior walls in contact with a uniform heat sink, i think it’s going to vary depending on the temperature outside those walls and how different that is from the internal temperature.
Also, at true equalibrium, the interior is going to match that heat sink uniformly, so you need to assume some heat source, and some method of dispersion, i would think.
If by “stable point” you mean thermal equilibrium, there won’t be any temperature difference between the floor and ceiling.
Under these conditions, there is no hot air to rise. That’s not the way it works.
The concept of hot air rising means a “packet” of air that has been heated, and will rise under an assumption of adiabatic conditions - no heat transfer between that packet of air and the surrounding air.
Eventually mixing and heat transfer occurs, but the adiabatic assumption is valid for modeling many processes in the atmosphere. You will come across it a lot in meteorology.
Trouble with the question is the premise. Heat doesn’t rise.
Hot air may rise when surrounded by colder air. But that already says that the system isn’t in equilibrium.
Ok, assume you have a pocket of hot air starting out. Does that mean that the hot air rises to the ceiling, and then eventually mixes back with the cold air at the floor again. Unless lost trough the ceiling/walls before that?
It’s a “packet” of air, not pocket. Yes, basically, it will rise - but then eventually mix with the colder air.
You could end up with a dynamic equilibrium with a heat gradient if you are constantly inputting heat in one place.
I think what I am really after is the difference between heating a house from the top or from the bottom. I realize that there are many factors at play here, but I was looking for some simple formulas or calculators I could use.
“Hot air” relative to what?
If you mean there’s a pocket of air that is at a higher temperature than the surrounding air, then it will rise due to buoyancy: the hot air is less dense than the surrounding air. It will eventually transfer its heat to the surrounding air, and then all the air will (essentially) be at the same temperature.
You are not going to get a simple calculator. It gets desperately horrid really quickly. Design engineers will use CFD programs to model heat transfer in buildings, but you are quickly into the realm of very difficult to model problems.
Transporting heat around inside a house is going to be done by shifting air. Just letting the air move about on its own volition due to convection is usually not going to be the best use. It is slow, and prone to not do what you hope. Forced air movement will win out. Air behaves as an incompressible fluid in these regimes. Every thought you might have about where air is going has to be balanced by thoughts of where air is coming back. And why.
If the air was still you could calculate using the basic conductivity of air. But this is very weak in comparison to convective transport, and as soon as the air starts to move, all bets are off about what it is going to do. And it won’t stand still. In a single room you can reason. But a whole house, forget it.
Force the air to do your bidding and you have some chance.
This isn’t much different than your basic coffee cup. The lower stuff rises if it’s hotter, then reaches the top where it cools, and falls to the floor. If you assume an insulated wall of common temperature, then if the air in contact with the wall or ceiling is colder, it will be warmed and rise. If warmer, it will cool and fall. Etc, etc, etc.
The system reaches a steady state - no more convection - when the whole is at a similar temperature. You can construct a steady-state gradient, but that would require a gradient in the container walls, floor, and ceiling itself - which kind of defeats the purpose of your thought experiment.
If you use the magical room where the walls and ceiling are 100% insulated and neither add nor subtract heat (the though experiment I assume you are looking for), then the air will slowly come to a steady state where heat percolates by conduction downward from the hotter higher up air to the lower air, and over time by conduction, just like a solid object, without outside interference it will achieve maximum entropy - no heat differential. Just note that heat conduction in air (in gases) is sufficiently slow it will take a while, which is why convection tends to dominate in the real world.
But if there is indeed a differential temperature, then it is not yet at steady state. No different than if you heat one end of a metal rod - eventually, slowly, the heat will work its way along the whole rod.
Of course, not taking into account adiabatic effects but in a room-sized enclosure, that should be irrelevant.
First you have to address the variables. Is the home sealed well? Windows are snug, no leakage and closed or open? Where are the cooling vents located, and is a fan running, either on a table or floor, or from the ceiling? Is a kitchen adjacent to the room? Where is the heating/cooling unit located?
Or say the hell w/ all that and put up a low cost ceiling fan or two. Run them one way in winter, and the other way in summer. That will even out the temps unless you live in Alaska or somewheres.
Heh. No.
Thermodynamics gives me the willies. It gets very complicated very quickly, and I was quite happy when I gave up on one problem and the mechanical engineer I asked to help me responded “well, at least you had the right formulas.”
Please excuse my brain fart, the usual terminology is a “parcel” of air.
I think the field in question is “fluid dynamics.”
Our daughter took a fluid dynamics when she was getting her undergraduate in chemical engineering. She said it was “pure hell.” 
Or, as my lab partner called it, fluid goddamnics.
Thanks!
That was indeed the thought experiment I was looking for. I have a new build home that is very well insulated, but is open between the floors. I may have to run some experiments for my house specifically, but sometimes I stay on the second floor and only heat up that floor and vice versa.
So in the first case it should take longer before the temperature evens out.
Thermofluid mechanics for incompressible and isentropic flow is actually pretty straightforward; there is a lot of math but, at least for the kind of problems you are presented in the classroom setting, it is reducible to a closed form solution. I won’t say that it is the easiest course I took in my undergraduate curriculum and it does require a level of comfort with calculus and differential equations but I found it pretty straightfoward. On the other hand, when you get to compressible flows in virtually any regime other than purely laminar and steady state flows, it becomes very complicated and full of approximation methods and models.
It should be understood that it isn’t ‘heat’ (which is actually the property of a system) that rises, but that a parcel of air that is warmer than the surrounding air will expand, becoming less dense and rising due to relative buoyancy in the denser ambient air as @Crafter_Man explained. As that parcel of air rises, it will diffuse into the ambient air (provided it isn’t restrained by a boundary like a balloon) and will ultimately come to equilibrium, although in the interim the rising warm air will displace colder air above it, causing it to move down. Although there is a slight gradient in air density due to gravity between the floor and ceiling, it is inconsequential and easily overwhelmed by any displacement of the air or heat exchange with the boundaries of a room. The reason air in a house is often persistently colder at the floor is that walls and attic spaces are well insulated while floors typically are not, so there is a constant transfer of thermal energy from the warm mass of air to the floor and thus to the outside world.
It would be relatively easy to create a ‘simple’ diffusion model of air in a box assuming appropriate heat transfer coefficient to through the walls, ceiling, and floor since the air isn’t moving fast enough to have any significant entropic effects or complex flows; however, you’d have to have some kind of measurements to come up with those coefficients.
Stranger
I would dispute that. Most floors are part of the house and have no exposure to the outside world temperature, unless you have a crawlspace or mobile home (trailer). More likely, the walls are the cold spots; the air along the outside of the walls cools and falls to the floor, thus the floor will cool to the wall temperature. This sets up a convection, where the warm air rises in the center of the room, descends along the outside-exposed walls, thus cooling the floor. The air cools as it descends (or conversely, transfer the ambient heat to the walls where it leaks outward) so theoretically the walls are coldest closest toward the floor where the air is already cooler as it reaches there.
Well designed forced air heating (and hot water radiator heat) tend to put the heater - air duct or radiator) under the windows, which historically have the least insulation value and steal heat more. Also, my newer furnace will run the fan 24-7 to help prevent stagnant air and the accompanying temperature differential the OP is wondering about - otherwise, the areas with the most outside wall exposure (and/or most windows) will cool off faster than the rest of the house.
I anticipate the energy-micromanaged house of the future will have computer controlled dampers for each room’s air feed, and sensors to tell when someone is in the room. No need to heat unoccupied rooms as high as occupied. The enemy of this is of course modern “open concept” houses. The house I grew up in had separate French doors to the dining room, living room, a vestibule to prevent heat loss, etc.
If there is no heat loss or gain through the walls, floors or ceilings then you can reach an uniform temperature. But if you have heat loss through the ceiling then the air at the top of the room will begin to cool and migrate to a lower level. If someone walks through the room then that migration will be interfered with. If the circulation fan comes on with the heater or AC unit then that will also affect the migration.
I would say there are too many things to come up with any formula that will be correct much less simple. There is the insulation of each wall, ceiling, floors, windows and doors. The there is the temperature difference between inside and outside. Now add heat sources. Equipment, lights, people.
The only way that I could think to determine the difference put a thermometer on the floor and one near the ceiling.