What the exponent is doing
A radiator gives up heat by convection from its surfaces and by radiation from the face. Neither one is proportional to the temperature difference. Natural convection depends on the buoyancy of the air the surface has already warmed, which itself grows with the difference, and radiation follows a fourth-power law on absolute temperature. Add the two together over the range radiators actually work in, and the result is described well by a single power law: output varies as the temperature difference raised to an exponent a bit above one.
So the conversion between a published rating and your conditions is one line of arithmetic. Divide your temperature difference by the rated one, raise that ratio to the exponent, and multiply the rating by the result. With the defaults on this page — 5,000 BTU/h at a 100 degree difference, exponent 1.30, mean water 140 against a 68 degree room — the ratio is 0.72, raised to 1.30 it is 0.652, and the output is 3,262 BTU/h. A straight proportional guess would have said 3,600, which is 10 percent high. At lower temperatures the gap between the two widens.
Why the units cancel and why that still catches people
The ratio of two temperature differences has no units, so long as both are measured in the same size of degree. Fifty Celsius degrees over a hundred Celsius degrees is the same 0.5 as ninety Fahrenheit degrees over a hundred and eighty. The exponent then applies identically. This is genuinely convenient and it is also where the common error lives: European radiators are usually rated at a 50 K difference and American cast iron ratings are often quoted around a 100 degree Fahrenheit difference, and comparing a K-based rating against a Fahrenheit-based difference without converting is out by 1.8. The answer does not look wrong, it just looks like a radiator that is far too big or far too small. Pick a unit on this page and keep every temperature in it.
Reading a rating backwards
The same relation inverts. If you know the load a radiator has to cover, the difference it needs is the rated difference times the ratio of loads raised to one over the exponent. For a 4,500 BTU/h room on a single 5,000 BTU/h radiator with exponent 1.30, that is a ratio of 0.9 to the power 0.769, which is 0.922, so the difference needed is 92.2 degrees and the mean water temperature is 160.2 with a 68 degree room. Add half the drop and the supply is 170.2. That is the number worth having before anyone specifies a boiler, because it is the water temperature the emitters demand rather than the one the boiler prefers.
Where the number stops being trustworthy
Test ratings are produced in a chamber with defined air temperature, a specified connection arrangement and clear space around the emitter. Real installations differ. A radiator with a shelf over it, a decorative cover, a deep window recess or a curtain across the face loses a meaningful share of its convective output, and the loss is not in any published table. Top-and-bottom-opposite-end connections, bottom-opposite-end and single-pipe arrangements do not all perform the same either. None of that is in the arithmetic on this page.
For fin-tube baseboard the same physics applies but the published data is usually a table of outputs per foot rather than a rating plus an exponent, so the baseboard heat length calculator interpolates that table directly. For where the load comes from, the heat loss calculator. For what a reset control does to the water temperature across a season — and therefore to everything on this page — the outdoor reset curve calculator.
Questions people ask
What exponent should I use if the data sheet does not give one?
Then you have a gap in the data rather than a number to guess, and the honest move is to ask the manufacturer or find a rating at a second temperature difference and solve for the exponent from the two points. Published exponents differ by emitter type, and the difference between a low value and a high one is not trivial: at half the rated temperature difference, an exponent of 1.2 gives 43.5 percent of the rating and 1.4 gives 37.9 percent, which is a sixth of the output between them. If you must proceed with an assumption, note it as an assumption and see how much your answer moves when you change it.
Is a rating at delta T 50 the same as a rating at 50 degrees F?
No, and treating them as the same is the most expensive mistake on this page. Delta T 50 means fifty Celsius degrees, which is ninety Fahrenheit degrees. If your rating is at 50 K and you compute your operating difference in Fahrenheit, every ratio is 1.8 times too large and the radiator looks far more capable than it is. Select the units at the top of this page to match your rating and enter every temperature in the same scale.
Do I use supply temperature or mean water temperature?
Mean, which is supply minus half the drop across the emitter. The rating was measured that way. There is a subtlety worth knowing: some standards use a logarithmic mean rather than a simple average, and the two diverge as the drop grows. At a 20 degree drop the difference is small. At a 40 degree drop on low-temperature water it is not, and the arithmetic mean flatters the radiator. If your system runs a large drop and low water temperatures, check which mean the rating standard used.
Can I run old cast iron radiators on a heat pump?
That is exactly the question this page is for, and the answer is a number rather than a yes or no. Work out the output of each radiator at the mean water temperature the heat pump will actually supply, total it up, and compare it to the room loads. Cast iron often does better here than baseboard does, because the sections were frequently generous when installed and because a large water volume and a large surface help at low temperature. It still usually falls short somewhere, and the shortfall is per room rather than per house. Whether the heat pump can supply the temperature you assumed, and at what efficiency, is a separate question for its performance data.
Does adding a second radiator double the output?
It doubles the emitter surface, which is not the same thing. Two radiators in parallel each see roughly the supply temperature, so the total is close to double. Two in series see different water — the second gets what the first returned, at a lower mean temperature and therefore a lower output — so the total is less than double, and how much less depends on the drop. The room also warms up, which reduces the temperature difference and pulls output back for both. The count on this page assumes parallel connection at the same mean water temperature and says so, and it is an upper bound on what a series arrangement gives.