You don't seem to say how thick the slab is. How deep is the cavity?

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You need to estimate a lot of things. How hot is the heater? How large of an area of the slab is getting heated? What temp will the bottom surface reach? If the heater was close coupled to the slab this would be easy.

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And you may lose a considerable amount of heat by the slab conducting heat to the edges and away from your targeted heating space. You might want to look at that possibility. 400mm is a lot of concrete and conduction heat flow is not always directed upwards. It is possible that heat will tend to be conducted left, past the wall to the left and heat the lower slab there, especially if that lower slab is continuous and colder than the regions inside that wall.

I only have rough thermal images ...based on that it's around 30-41C ...locally. Of course the heat I'm assuming conducts radially in the slab somewhat (elliptically given line source) and decays with distance. Top surface inside is ambient. May have to instrument this with a heat flux sensor in the winter to provide better data.

It's a fairly large slab roughly 15 ft wide x 20 ft long. You get both conduction and radiation as it warms up the space significantly.

And yes do get loss in the winter as you get at that wall boundary thermal bridging. It's a continuous slab sans thermal break.

Cavity dimensions are hard to get as slab geometry is complex... basically it's a ceiling of a parking structure with drop ceiling drywall suspended and batts of insulation.

Is this a trick question? The heat generated is on your drawing, about 2 kW. The heater heats the air under the slab, which will rise and spread out very rapidly. So what is the actual question?

The cavity is air and slab is irradiated...so not quite direct. I could be wrong here...the slab has a lot of mass that continuously radiates the space above

Trick question is, You ask how much heat will be generated.
Then you tell us the answer. 275W/m x 7 m = 1925W
You will generate that much heat.

You need to ask a more specific question, such as how hot the room above the slab will get after 4 hours, or something like that.

So, obviously you will generate heat with your 1925W heater. The question is then probably something like, Is that a practical way to heat the space above the slab?

I didnt run any numbers, but I think not. 400mm of concrete may take two days or so to reach some kind of equilibrium temperature itself. That temperature depends on how much heat is conducted by the concrete to the outside the wall and how much heat is convected by air from both heater and concrete away from entering the space above. There isn't likely much heat left that will have a chance to get into the space above. It may also take several days to feel any difference, if ever. You have several tonnes of concrete to heat and basically the power of my wife's hair drier. Its like bright sunshine on 4 or 5 Square meters of surface. I'm guessing that after 3 days, you might notice some difference in the room, if that was a closed space and provided that little heat escaped to the slab on the left, or was convected away by air before it ever reached the bottom of the slab. Some radiation going on there, but that isn't as efficient as direct contact. Probably better to put the heater strip above the concrete, if indeed your goal is to warm that area.

Fair point I did not ask the correct question. I suppose what I meant to say is that the heater at the source is 2 kW (approx) and the amount coming through is much less obviously. But you're right there is a long transient to start feeling it above....and the floor temperature on the other side is generally 23-25 C so it very much warms up the room(s) above.
Without the heat source here, you're looking at a median temperature in the winter of 3.5C beneath the slab sampled during 3 month period (average ~ 3.8C). The last image here showing the slab and approx location of the heaters.

I was not thing that 3 day transient would be "very practical", but if that works for you, it must be so. I usually want to feel some heat within the hour or two, but if its for general storage space or something, that won't matter much.

OK, so you still have not really asked a specific question. You already know the top of slab temperature. At least in one or two spots. Do you need an isotherm map for the whole slab or something? Where do you want to go with this?

I suppose was looking for guidance on the heat flux, temperature distribution on the slab at steady state as it's fairly non-uniform.

Its relatively easy to solve an approximate 2D plan view isobar map (temperature contour plot), if you can define the temperatures at points along the the slab edges and of course know the location of the heater axis. For example, If its 23 degrees on the slab at all points immediately above the heater axis and the temperatures on the west and north edges of the slab are-5, then you could solve for temperatures of all other unknown points. The more temperatures you know along the edges, the better the solution will be.

The solution looks like this.
http://eng-web1.eng.famu.fsu.edu/~dommelen/pdes/st...

The problem can be solved using Excel.

The air temperature of the room would be useful

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I'd prefer to think that air temp would be dependent on slab slab temperature, as it doesn't seem to be the primary driver of what's heating up what there. There is a lot more heat in the concrete than in the air, just due to mass. Id tend to think it would pretty much be whatever the average concrete surface temperature is. So more likely the reverse, no? That seems to be who it works in my old house. Its got 500mm thick stone and concrete walls and an uninsulated floor slab on ground. The air temperature seems to parallel the average temp of the walls and floor. The avg interior wall temps lag the outside average air temp by 3 days, but extreme delta temps from interior to exterior are not usually more than 10°C. Do you think that to be too inaccurate?

Air temperature of the room is controlled via thermostat. It's maintained at 21 C. The void under the slab corresponds to a space that is obviously heated by the radiator that sits approx 8 inches below the slab, but above the drop ceiling as shown above with insulation. The drop ceiling corresponds to the ceiling of a parking garage that for all purposes is at ambient temperature (noted median in winter months 3.5C).

The space above the slab is a living space. It's a three storey "California style town home" and so with the thermostat on the main floor, along with the effect from the heated slab, the space is always 'warm' and rarely triggers operation of the furnace which is mounted in a mechanical room above the third floor.

Since there's only about a 2 degC delta between the top of the floor(23C) and the room air temperature (21C), you're getting around 15W/m^2 of heating.

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What did you use to get 15W/m^2?

(7.5W/m^2-K)*(2deC delta)

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