Thank you for the thoughtful comments about firebacks. I agree with your explanation about radiation shields and I think our differences can be attributed to slightly different assumptions and some over simplification.
I assume that most iron firebacks are not tight at the bottom or sides (most have legs) so that dilution air will flow in under and around the fireback, up between the iron fireback and the masonry fireback, becoming heated and wasted up the chimney.
As the object in view is to bring radiant heat into the room, it is clear that that material is best for the construction of a fireplace, which reflects the most, or which absorbs the least of it; for that heat which is absorbed cannot be reflected. - Now, as bodies which absorb radiant heat are necessarily heated in consequence of that absorption, to discover which of the various materials that can be employed for constructing fireplaces are best adapted for that purpose, we have only to find out by an experiment, very easy to be made, what bodies acquire least heat when exposed to the direct rays of a clear fire; for those which are least heated evidently absorb the least, and consequently reflect the most radiant heat. And hence it appears that iron, and, in general, metals of all kinds, which are well known to grow very hot when exposed to the rays projected by burning fuel, are to be reckoned among the very worst materials that it is possible to employ in the construction of fireplaces. The best materials I have hitherto been able to discover are fire-stone, and common bricks and mortar. Both these materials are, fortunately, very cheap; and as to their comparative merits, I hardly know to which of them the preference ought to be given. When bricks are used, they should be covered with a thin coating of plaster, which, when it is become perfectly dry, should be whitewashed.
If the fireback could be sealed around the edges to prevent this air flow, then I agree that it may result in the iron fireback becoming hotter and limiting the heat transferred through the firebox. So would an insulated masonry fireback.
The objective is to increase the temperature of the surface of the fireback so it should be easy to try it with and without an iron fireback while also measuring flue gas flows and temperatures to see if any extra heated air is wasted up; the chimney.
Radiation shields work much better when the temperature differential is greater - as between the sun and a space capsule. When the temperature differential is only a few hundred degrees and you have the presence of air and the possibility of convection losses, insulation may work better.
We will be back in a test lab soon. I will do some testing to verify your suggestions. Do you have any suggestions for what we might test? I would test the surface temperature of the iron fireback compared with the masonry fireback, with the edges sealed and not sealed and note the flue gas flows and temperatures in each test.
I think you make a good argument for limiting the heat lost into the masonry fireback but is it at the expense of more heated air lost up the chimney? And, if your fireplace is inside the exterior walls of the house that heat "lost" into the masonry mass is not so much lost as it is stored.
Iron firebacks do, of course, protect the masonry fireback - not so important these days with the availability of firebrick, but 200 years ago it saved a lot of maintenance.
I made some comments below in the body of your comments. Since you took so much time to think this through and write to me, I don't want to miss anything. I really appreciate it.
I had always assummed that firebacks were merely decorative or served some practical purpose besides increasing efficiency. In fact, an internet search for firebacks shows manufacturers suggesting that they protect the masonry surface of the firebox from heat damage.
I always thought that was the main reason before firebrick were common - and maybe also for looks. Rumford said the rich people hired ironmongers to make decorative iron firebacks but that ordinary masonry was better and less expensive.
That seems reasonable since it is a lot easier to replace the fireback than the masonry surface.
In your technical discussion section you wrote that an efficiency gain is not likely because half of the heat radiated from the fireback goes back against the masonry and is lost. This is not strictly true. Once absorbed by the masonry, that heat raises the temperature of the masonry surface which results in increased radiation from the masonry back to the fireback.
My statement was too simplistic and I think still essentially true. The heat radiated from the back side of the fireback heats the masonry fireback, as you say, but that is still mostly lost heat even if the heat loss is less as the masonry fireback heats up. I think your point is countered by the air flowing between the firebacks which is heated and lost up the chimney.
I specifically recall from my heat transfer classes that arranging parallel plates of any material of any conductivity with gaps in between would inhibit transmission of heat, and calculations of radiant energy were offered to show this was true. I also visited Oshkosh many years ago for their annual fly-in of homebuilt aircraft and while I was there I visited a booth set up I think by some branch of NASA. The guy manning the booth showed me a small sample of perforated aluminum sheet arranged in layers. He explained that it was essentially insulation for use in space.
That's true. Radiation shields are used to cut the clearance to combustibles in half for wood stoves too. Your point is well taken if your objective is to minimize the heat lost into the masonry. I just think that gain is probably wiped out by the extra air being heated and lost up the chimney. Maybe not. We should test it. I do think and insulated firebox would do the same thing without the air loss, and, as I said above, the "loss" can really be just heat storage if the chimney is inside the house.
Think about that. You're out in space in a space capsule. The sun is a black body at very high temperature and it is heating the side of your capsule. The capsule gets hot and radiates half the heat back at the sun and half in towards you. The capsule is hot but not as hot as the sun. Now add another layer. The first layer radiates half its heat back at the sun and half to the second layer. The second layer radiates half its heat back towards the first layer and half in towards you. It's an oversimplification because the first layer will stay hotter with the second layer in place than without the second layer in place but it is clear you come out ahead. Also, as a second way of looking at it intuitively, the second layer gets hot, but not as hot as the first layer, which itself is not as hot as the sun. It gets better with more layers. You could use sheets of highly conductive copper or aluminum and there would still be a sort of "insulation" going on in that due purely to radiative effects, there would be a resistance to the flow of heat. The layer closest to the fire would remain the hottest and would "glow" the brightest, heat wise, which is all we really care about.
The space engineers are not worried about efficiency and they aren't dealing with air and convection. All they want to do is protect the capsule from heat which is all radiant heat from a very hot source. In a fireplace situation I think insulation probably works better to reduce the heat lost into the masonry.
So, a fireback, separated from the back wall by an air gap and without significant thermally conductive bridges, should get hotter than the masonry surface alone and will tend to reradiate more heat out of the fireplace than the masonry surface alone. It also may survive high heat better than the masonry surface or at least be easier to replace, while keeping the masonry at a lower temperature.
Maybe it would get hotter if it's not cooled by the air flowing up behind it. Temperature is easy to measure, however, and so is flue gas flow and temperature. These days, with firebrick, there is no need to protect the masonry. It will serve longer than the iron fireback which can expand, warp and crack, especially if it's very large.
If you consider the firebox to be a "black box" (an analytical crutch where you don't concern yourself with what is going on inside the box but only with what is crossing the boundries of the box) then if the walls of the firebox are maintained at a lower temperature while the burning wood generates the same heat, then you have to ask where that heat is going, since losses through the walls will be less. Unless the fireback is transfering the heat to the air, more heat must be leaving as radiation to warm your knees. I speculate that the air is fairly stagnant at the back of the firebox so at least some of the air not leaving through masonry will also not leave with the draft and will in fact leave as increased radiant heat.
That's a speculation worth testing. I'm not sure, but I do think, for the iron fireback to add anything to the efficiency, the surface temperature of the iron fireback would have to be higher than the surface temperature of the masonry fireback with the same fire without the iron fireback. You also assume "fairly stagnant at the back of the firebox" but I don't know - unless you seal the fireback around the edges. Without an iron fireback the plume of hot gasses flowing up the fireback is pretty vigorous. Will it all be out in front of the iron fireback or will some of that draft be between the iron and masonry firebacks serving only to cool both firebacks and waste heated air up the chimney.